Exponential mixing and essential spectral gaps for Anosov subgroups

Michael Chow Department of Mathematics, Yale University, New Haven, CT 06511, USA [email protected] and Pratyush Sarkar Department of Mathematics, UC San Diego, La Jolla, CA 92093, USA [email protected]

(Date: 24 décembre 2024)

Abstract.

Let $\Gamma$ be a Zariski dense $\Theta$ -Anosov subgroup of a connected semisimple real algebraic group for some nonempty subset of simple roots $\Theta$ . In the Anosov setting, there is a natural compact metric space $\mathcal{X}$ equipped with a family of translation flows $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ , parameterized by vectors $\mathsf{v}$ in the interior of the $\Theta$ -limit cone $\mathcal{L}_{\Theta}$ of $\Gamma$ , and they are Hölder conjugate to Hölder reparametrizations of the Gromov geodesic flow. We prove that for all $\mathsf{v}$ outside an exceptional cone $\mathscr{E}\subset\operatorname{int}\mathcal{L}_{\Theta}$ , which is a smooth image of the walls of the Weyl chamber, the translation flow is exponentially mixing with respect to the Bowen–Margulis–Sullivan measure associated to $\mathsf{v}$ . Moreover, the exponential rate is uniform for a compact set of such $\mathsf{v}$ . We also obtain an essential spectral gap formulated in terms of the Selberg zeta function and a prime orbit theorem with a power saving error term. Our proof relies on Lie theoretic techniques to prove the crucial local non-integrability condition (LNIC) for the translation flows and thereby implement Dolgopyat’s method in a uniform fashion. The exceptional cone $\mathscr{E}$ arises from the failure of LNIC for those vectors.

1. Introduction

The main result of this paper is exponential mixing of translation flows associated to Anosov subgroups. Anosov subgroups, first introduced by Labourie [36] and later generalized by Guichard–Wienhard [25], includes many interesting geometric examples such as the images of Hitchin representations [36] into $\operatorname{PSL}_{n}(\mathbb{R})$ , strongly convex cocompact projective representations into $\operatorname{PGL}_{n}(\mathbb{R})$ [19], maximal representations into $\mathrm{PSp}_{2n}(\mathbb{R})$ and $\mathrm{PO}(2,n)$ [10, 11], and Barbot representations into $\operatorname{PSL}_{n}(\mathbb{R})$ [2]. In recent times, there has been a lot activity in understanding the dynamics related to Anosov subgroups (see for instance [60, 9, 39, 40, 23, 12, 18, 61, 31], etc.). Mixing properties are of particular interest because existing techniques allow one to readily derive various counting and equidistribution results. What is more, exponential mixing allows one to effectivize these results, i.e., obtain precise error terms.

Let us first provide some basic setup for the main theorem. Let $G$ be a connected semisimple real algebraic group with $\mathfrak{g}:=\operatorname{Lie}(G)$ , $A$ be a maximal real split torus of $G$ and fix a closed positive Weyl chamber $\mathfrak{a}^{+}\subset\mathfrak{a}:=\operatorname{Lie}(A)$ . Let $\Pi\subset\mathfrak{a}^{*}$ denote the set of all simple roots for $(\mathfrak{g},\mathfrak{a}^{+})$ and let $\mathsf{i}:\mathfrak{a}\to\mathfrak{a}$ denote the opposition involution preserving $\mathfrak{a}^{+}$ and acting on $\Pi$ by precomposition. Fix a nonempty subset $\Theta\subset\Pi$ . Let $P_{\Theta}$ denote the standard parabolic subgroup of $G$ corresponding to $\Theta$ and $P_{\Theta}=S_{\Theta}A_{\Theta}N_{\Theta}$ be its Langlands decomposition where $S_{\Theta}$ centralizes $A_{\Theta}:=\exp(\mathfrak{a}_{\Theta})$ and $\mathfrak{a}_{\Theta}:=\bigcap_{\alpha\in\Pi-\Theta}\ker\alpha\subset\mathfrak{a}$ . The $\Theta$ -Furstenberg boundary is defined as $\mathcal{F}_{\Theta}:=G/P_{\Theta}$ and we denote the unique open $G$ -orbit in $\mathcal{F}_{\Theta}\times\mathcal{F}_{\mathsf{i}\Theta}$ by $\mathcal{F}_{\Theta}^{(2)}$ .

Let $\Gamma<G$ be a torsion-free Zariski dense $\Theta$ -Anosov subgroup, that is, $\Gamma$ is a torsion-free Gromov hyperbolic group with Gromov boundary $\partial\Gamma$ and continuous $\Gamma$ -equivariant maps $\zeta_{\Theta}:\partial\Gamma\to\mathcal{F}_{\Theta}$ and $\zeta_{\mathsf{i}\Theta}:\partial\Gamma\to\mathcal{F}_{\mathsf{i}\Theta}$ such that $(\zeta_{\Theta}(x),\zeta_{\mathsf{i}\Theta}(y))\in\mathcal{F}_{\Theta}^{(2)}$ for all distinct $x,y\in\partial\Gamma$ . The images $\Lambda_{\Theta}:=\zeta_{\Theta}(\partial\Gamma)$ and $\Lambda_{\mathsf{i}\Theta}:=\zeta_{\mathsf{i}\Theta}(\partial\Gamma)$ are the unique $\Gamma$ -minimal subsets of $\mathcal{F}_{\Theta}$ and $\mathcal{F}_{\mathsf{i}\Theta}$ . Let $\Lambda_{\Theta}^{(2)}=(\Lambda_{\Theta}\times\Lambda_{\mathsf{i}\Theta})\cap\mathcal{F}_{\Theta}^{(2)}$ . Unless $\Theta=\Pi$ (in which case $S_{\Pi}$ is compact), $\Gamma$ does not necessarily act on $G/S_{\Theta}\cong\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ properly discontinuously (see [24, §1.5] for an extensive discussion). However, the $\Gamma$ -action restricted to the $\Gamma$ -invariant subset $\Lambda_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ is always properly discontinuous.

Given $\mathsf{v}\in\mathfrak{a}_{\Theta}$ , the translation action of $\mathbb{R}\mathsf{v}$ on the $\mathfrak{a}_{\Theta}$ -coordinate of $\Lambda_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ induces the one-parameter diagonal flow on $\Omega:=\Gamma\backslash\bigl{(}\Lambda_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}\bigr{)}$ which we denote by $\{a_{t\mathsf{v}}\}_{t\in\mathbb{R}}$ . We further assume that $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ where $\mathcal{L}_{\Theta}$ denotes $\Theta$ -limit cone of $\Gamma$ (see Definition 2.6). In this case, the dynamical system $(\Omega,m^{\mathrm{BMS}}_{\mathsf{v}},\{a_{t\mathsf{v}}\}_{t\in\mathbb{R}})$ is known to be locally mixing where $m^{\mathrm{BMS}}_{\mathsf{v}}$ is the Bowen–Margulis–Sullivan measure associated to $\mathsf{v}$ . That is, there exists $\kappa_{\mathsf{v}}>0$ such that for all $\phi_{1},\phi_{2}\in C_{\mathrm{c}}(\Omega)$ , we have

\displaystyle\lim_{t\to+\infty}t^{\frac{\#\Theta-1}{2}}\int_{\Omega}\phi_{1}(a_{t\mathsf{v}}x)\phi_{2}(x)\,dm^{\mathrm{BMS}}_{\mathsf{v}}(x)=\kappa_{\mathsf{v}}m^{\mathrm{BMS}}_{\mathsf{v}}(\phi_{1})m^{\mathrm{BMS}}_{\mathsf{v}}(\phi_{2}).

When $\Theta=\Pi$ , Thirion [69] first proved the above for Schottky subgroups of $\operatorname{SL}_{n}(\mathbb{R})$ and this was later generalized by Sambarino [60] to fundamental groups of compact negatively curved manifolds. Generalizing their work, the authors [18] proved local mixing on $\Gamma\backslash G$ (which is a principal $S_{\Pi}$ -bundle over $\Gamma\backslash\bigl{(}\mathcal{F}_{\Pi}^{(2)}\times\mathfrak{a}\bigr{)}$ ) for general $\Pi$ -Anosov subgroups. The arguments in [18] easily generalize for arbitrary $\Theta$ without the $S_{\Pi}$ -valued holonomy (cf. [61, Appendix B] for a general theorem and a proof sketch amending the one in [60] using strategies similar to [18]). It is an interesting (and still open) question as to what the rate of mixing above should be in general. It is known to the authors that Dolgopyat’s method cannot be carried out as the non-concentration property or its generalization (cf. [64, 17]) fails miserably whenever $G$ is of higher rank; and hence, the rate is expected to be subexponential when $\#\Theta>1$ . See the results below for the case $\#\Theta=1$ .

On the other hand, the situation is completely different for the dynamical system $(\mathcal{X},m_{\mathcal{X}}^{\mathsf{v}},\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}})$ . Here, $\mathcal{X}$ is a compact metric space over which $\pi:\Omega\to\mathcal{X}$ is a trivial $\mathbb{R}^{\#\Theta-1}$ -bundle, $m_{\mathcal{X}}^{\mathsf{v}}$ is a probability measure on $\mathcal{X}$ such that we have the product structure $m^{\mathrm{BMS}}_{\mathsf{v}}=m_{\mathcal{X}}^{\mathsf{v}}\otimes\mathrm{Leb}_{\mathbb{R}^{\#\Theta-1}}$ , and $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ is the translation flow on $\mathcal{X}$ induced by $\{a_{t\mathsf{v}}\}_{t\in\mathbb{R}}$ via the bundle projection (see Subsection 2.4 for details). So far in the literature on Anosov subgroups, translation flows have been studied more often as an auxiliary tool rather than a dynamical system of intrinsic interest. However, translation flows are natural dynamical systems which mimic (and in fact generalize) the geodesic flow for convex cocompact hyperbolic manifolds and they are Hölder conjugate to Hölder reparametrizations of the Gromov geodesic flow associated to $\Gamma$ . In this vein, we prove that translation flows are exponentially mixing for generic vectors $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ . Indeed, we obtain a detailed characterization of the exponential mixing property for the whole family of translation flows.

The set of generic $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ for which we can prove exponential mixing can be described in terms of a fundamental object associated to $\Gamma$ called its $\Theta$ -growth indicator $\psi_{\Theta}:\mathcal{L}_{\Theta}\to[0,+\infty)$ (see Definition 2.7) which is the higher rank generalization of critical exponent in rank one. We say $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ is $\psi_{\Theta}$ -regular if $\nabla\psi_{\Theta}(\mathsf{v})\notin\partial\mathfrak{a}_{\Theta}^{+}$ and we define the exceptional cone

\displaystyle\mathscr{E}=\{\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}:\mathsf{v}\text{ is not }\psi_{\Theta}\text{-regular}\}.

It follows from the properties of $\psi_{\Theta}$ (see Theorem 2.11 (2)(4) and Proposition B.1) that $\mathscr{E}$ is the image of $\partial\mathfrak{a}_{\Theta}^{+}$ under a diffeomorphism on $\mathfrak{a}_{\Theta}$ and bounds an open strictly convex cone.

We are now ready to state the main theorem regarding exponential mixing of the translation flows associated to $\psi_{\Theta}$ -regular $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ . We remark that the translation flows are homothety equivariant, i.e., $a^{c\mathsf{v}}_{t}=a^{\mathsf{v}}_{ct}$ for all $t\in\mathbb{R}$ and $c>0$ . For $\alpha\in(0,1]$ , we denote by $C^{0,\alpha}(\mathcal{X})$ the space of $\alpha$ -Hölder continuous functions on $\mathcal{X}$ endowed with the $\alpha$ -Hölder norm $\|\cdot\|_{C^{0,\alpha}}$ .

Theorem 1.1.

Let $\alpha\in(0,1]$ . There exist

—

a continuous piecewise-smooth function $\eta_{\Theta,\alpha}:\mathcal{L}_{\Theta}-\{0\}\to[0,+\infty)$ which is positive except possibly on $(\partial\mathcal{L}_{\Theta}\cup\mathscr{E})-\{0\}$ ,
—

a smooth positive function $C_{\bullet}:\operatorname{int}\mathcal{L}_{\Theta}\to\mathbb{R}$ (independent of $\alpha$ ),

which are both homothety-invariant, such that for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ , and $\phi_{1},\phi_{2}\in C^{0,\alpha}(\mathcal{X})$ , and $t>0$ , we have

\displaystyle\left|\int_{\mathcal{X}}\phi_{1}(a^{\mathsf{v}}_{t}x)\phi_{2}(x)\,dm_{\mathcal{X}}^{\mathsf{v}}(x)-m_{\mathcal{X}}^{\mathsf{v}}(\phi_{1})m_{\mathcal{X}}^{\mathsf{v}}(\phi_{2})\right|\leq C_{\mathsf{v}}e^{-\|\mathsf{v}\|\eta_{\Theta,\alpha}(\mathsf{v})t}\|\phi_{1}\|_{C^{0,\alpha}}\|\phi_{2}\|_{C^{0,\alpha}}.

Remark 1.2.

An imprecise version of Theorem 1.1 had been claimed in a talk by the second named author at Institut des Hautes Études Scientifiques in June 2023 as it was well-known to the authors and much of this paper had been written at the time. However, a sticking point delayed the authors in the distribution of the manuscript. See Remark 4.9.

We obtain the following simple corollary when $\#\Theta=1$ because in this case $\dim(\mathfrak{a}_{\Theta})=1$ and so the family of translation flows collapses to a trivial one-dimensional family $\{\{a_{ct}\}_{t\in\mathbb{R}}\}_{c>0}$ coinciding with rescalings of the one-parameter diagonal flow. This theorem already generalizes the case that $\Gamma$ is a projective Anosov subgroup of $\operatorname{PSL}_{n}(\mathbb{R})$ which now follows from the combination of the works of Delarue–Montclair–Sanders [20] and Stoyanov [67] (see the discussion below) and strengthens the local mixing result above. It also generalizes the case that $G$ is of rank one which gives the geodesic flow for convex cocompact locally symmetric spaces and is contained in the work of Stoyanov [67] (see [17] of the authors for the frame flow).

Theorem 1.3.

Suppose $\#\Theta=1$ . Let $\alpha\in(0,1]$ . Then, there exist $\eta_{\alpha}>0$ and $C>0$ (independent of $\alpha$ ) such that for all $\phi_{1},\phi_{2}\in C^{0,\alpha}(\mathcal{X})$ and $t>0$ , we have

\displaystyle\left|\int_{\mathcal{X}}\phi_{1}(a_{t}x)\phi_{2}(x)\,dm_{\mathcal{X}}(x)-m_{\mathcal{X}}(\phi_{1})m_{\mathcal{X}}(\phi_{2})\right|\leq Ce^{-\eta_{\alpha}t}\|\phi_{1}\|_{C^{0,\alpha}}\|\phi_{2}\|_{C^{0,\alpha}}.

We also obtain the following corollary regarding uniform exponential mixing.

Theorem 1.4.

Let $\alpha\in(0,1]$ . Suppose $\mathcal{K}\subset\operatorname{int}\mathcal{L}_{\Theta}-\mathscr{E}$ is a compact subset. Then, there exist $\eta_{\alpha,\mathcal{K}}>0$ and $C_{\mathcal{K}}>0$ (independent of $\alpha$ ) such that for all $\mathsf{v}\in\mathcal{K}$ , and $\phi_{1},\phi_{2}\in C^{0,\alpha}(\mathcal{X})$ , and $t>0$ , we have

\displaystyle\left|\int_{\mathcal{X}}\phi_{1}(a^{\mathsf{v}}_{t}x)\phi_{2}(x)\,dm_{\mathcal{X}}^{\mathsf{v}}(x)-m_{\mathcal{X}}^{\mathsf{v}}(\phi_{1})m_{\mathcal{X}}^{\mathsf{v}}(\phi_{2})\right|\leq C_{\mathcal{K}}e^{-\eta_{\alpha,\mathcal{K}}t}\|\phi_{1}\|_{C^{0,\alpha}}\|\phi_{2}\|_{C^{0,\alpha}}.

Using the spectral bounds of transfer operators in Theorem 4.7, we actually prove the more detailed theorem below from which the above theorems follow. To describe the behavior of the correlation function especially near the boundary of the limit cone $\partial\mathcal{L}_{\Theta}$ , we introduce the map $\kappa_{\Theta}$ . Equip $\mathfrak{a}$ with the inner product and norm induced by the Killing form on $\mathfrak{g}$ . Define $\kappa_{\Theta}:\operatorname{int}\mathcal{L}_{\Theta}\to\mathbb{R}$ by

\displaystyle\kappa_{\Theta}(\mathsf{v}):=\frac{\psi_{\Theta}(\mathsf{v})}{\|\nabla\psi_{\Theta}(\mathsf{v})\|}E_{\Theta}(\|\nabla\psi_{\Theta}(\mathsf{v})\|)\qquad\text{for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$}

where $E_{\Theta}:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ is the bounded elementary function (in the formal sense) with vanishing limit at $+\infty$ provided by Theorem 4.1 which can be computed explicitly from the proofs in the paper. Note that $\kappa_{\Theta}$ is homogeneous of degree $1$ because so is $\psi_{\Theta}$ . The complex numbers $\{\lambda_{k}(\mathsf{v})\}_{k=1}^{k_{\mathsf{v}}}$ which appear are the Pollicott–Ruelle resonances for the translation flow associated to $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ . The concept of Pollicott–Ruelle resonances originated in [47, 48, 56, 55].

Theorem 1.5.

Let $\alpha\in(0,1]$ . There exist

—

a continuous piecewise-smooth function $\eta_{\Theta,\alpha}:\mathcal{L}_{\Theta}-\{0\}\to[0,+\infty)$ which is positive except possibly on $\mathscr{E}$ ,
—

a smooth positive function $C_{\bullet}:\operatorname{int}\mathcal{L}_{\Theta}\to\mathbb{R}$ (independent of $\alpha$ ),

such that for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ , there exist $k_{\mathsf{v}}\in\mathbb{N}$ and

—

a finite set $\{\lambda_{k}(\mathsf{v})\}_{k=1}^{k_{\mathsf{v}}}\subset(-1,0)\times i(-1,1)$ which come in conjugate pairs,
—

a finite set of finite-rank positive bilinear forms $\{\mathcal{B}_{\mathsf{v},k}\}_{k=1}^{k_{\mathsf{v}}}$ ,

which are all homothety-invariant, such that for all $\phi_{1},\phi_{2}\in C^{0,\alpha}(\mathcal{X})$ and $t>0$ , we have

\left|\int_{\mathcal{X}}\phi_{1}(a^{\mathsf{v}}_{t}x)\phi_{2}(x)\,dm_{\mathcal{X}}^{\mathsf{v}}(x)-\left(m_{\mathcal{X}}^{\mathsf{v}}(\phi_{1})m_{\mathcal{X}}^{\mathsf{v}}(\phi_{2})+\sum_{k=1}^{k_{\mathsf{v}}}e^{\lambda_{k}(\mathsf{v})t}\mathcal{B}_{\mathsf{v},k}(\phi_{1},\phi_{2})\right)\right|\\ \leq C_{\mathsf{v}}e^{-\kappa_{\Theta}(\mathsf{v})\eta_{\Theta,\alpha}(\mathsf{v})t}\|\phi_{1}\|_{C^{0,\alpha}}\|\phi_{2}\|_{C^{0,\alpha}}.

1.1. Applications

Again using the powerful Theorem 4.7, we also obtain the following applications.

For all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ and $T>0$ , define

	$\displaystyle\mathcal{P}_{\mathsf{v}}:={}$	$\displaystyle\{\gamma:\text{$\gamma$ is a primitive closed $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$-orbit in $\mathcal{X}$}\},$
	$\displaystyle\mathcal{P}_{\mathsf{v}}(T):={}$	$\displaystyle\{\gamma\in\mathcal{P}_{\mathsf{v}}:\ell_{\mathsf{v}}(\gamma)\leq T\},$

where $\ell_{\mathsf{v}}(\gamma)$ denotes the primitive period of $\gamma$ . We may drop the subscript for the former and write $\mathcal{P}$ since the set of closed orbits coincide for all translation flows. Moreover, there is a canonical bijective correspondence

\displaystyle\mathcal{P}\leftrightarrow[\Gamma_{\mathrm{prim}}],

where the latter is the set of conjugacy classes of $\Gamma$ corresponding to the subset of primitive loxodromic elements $\Gamma_{\mathrm{prim}}\subset\Gamma$ . Note that in our setting, all nontorsion elements of $\Gamma$ are loxodromic (see Theorem 2.11). Under the above bijective correspondence, we also have

\displaystyle\mathcal{P}_{\mathsf{v}}(T)\leftrightarrow\{[\gamma]\in[\Gamma_{\mathrm{prim}}]:\psi_{\mathsf{v}}(\lambda(\gamma))\leq T\}

where we have used the notation $\psi_{\mathsf{v}}:=\langle\delta_{\mathsf{v}}^{-1}\nabla\psi_{\Theta}(\mathsf{v}),\cdot\rangle$ and $\delta_{\mathsf{v}}:=\psi_{\Theta}(\mathsf{v})$ , which is the topological entropy of the translation flow $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ , from Subsection 2.4.

For all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ , recall the Selberg zeta function $Z_{\mathsf{v}}:\{\xi\in\mathbb{C}:\Re(\xi)>\delta_{\mathsf{v}}\}\to\mathbb{C}$ [65] defined by

\displaystyle Z_{\mathsf{v}}(\xi)=\prod_{k=0}^{\infty}\prod_{\gamma\in\mathcal{P}}\bigl{(}1-e^{-(\xi+k)\ell_{\mathsf{v}}(\gamma)}\bigr{)}\qquad\text{for all $\Re(\xi)>\delta_{\mathsf{v}}$}

which converges absolutely using the noneffective prime orbit theorem in [16, Eq. (1.8)] (cf. [26, Chapter 2, Definition 4.1] and the subsequent remark). By the work of Pollicott [48], it then extends meromorphically to the domain $\{\xi\in\mathbb{C}:\Re(\xi)>\delta_{\mathsf{v}}-\eta_{\mathsf{v}}\}$ for some $\eta_{\mathsf{v}}>0$ with a simple zero at $\delta_{\mathsf{v}}$ . He also gives a simple explicit formula for $\eta_{\mathsf{v}}$ in terms of the expansion/contraction rate from the Anosov property and the topological entropy from which we see that $\eta_{\mathsf{v}}\asymp\kappa_{\Theta}(\mathsf{v})$ on some conical neighborhood of $\partial\mathcal{L}_{\Theta}$ .

Theorem 1.6.

There exist a homothety-invariant continuous piecewise-smooth function $\eta_{\Theta}:\mathcal{L}_{\Theta}-\{0\}\to[0,+\infty)$ which is positive except possibly on $\mathscr{E}$ such that for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ , the Selberg zeta function $Z_{\mathsf{v}}$ has only finitely many zeros on $\{\xi\in\mathbb{C}:\Re(\xi)>\delta_{\mathsf{v}}-\kappa_{\Theta}(\mathsf{v})\eta_{\Theta}(\mathsf{v})\}$ .

We say that the dynamical system $(\mathcal{X},m_{\mathcal{X}}^{\mathsf{v}},\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}})$ has an essential spectral gap of $\kappa_{\Theta}(\mathsf{v})\eta_{\Theta}(\mathsf{v})>0$ . We emphasize that $\kappa_{\Theta}$ is an explicit function which dictates the behavior of the decay of the essential spectral gap near the boundary of the limit cone $\partial\mathcal{L}_{\Theta}$ whereas we have no information on the zeros of the Selberg zeta function $Z_{\mathsf{v}}$ .

Remark 1.7.

The above could also be formulated equivalently in terms of the Ruelle zeta function $\zeta_{\mathsf{v}}:\{\xi\in\mathbb{C}:\Re(\xi)>\delta_{\mathsf{v}}\}\to\mathbb{C}$ defined by $\zeta_{\mathsf{v}}(\xi):=\frac{Z_{\mathsf{v}}(\xi+1)}{Z_{\mathsf{v}}(\xi)}=\prod_{\gamma\in\mathcal{P}}\bigl{(}1-e^{-\xi\ell_{\mathsf{v}}(\gamma)}\bigr{)}^{-1}$ for all $\Re(\xi)>\delta_{\mathsf{v}}$ and then extended meromorphically as above.

Recall the offset logarithmic integral function $\operatorname{Li}:(2,+\infty)\to\mathbb{R}$ defined by $\operatorname{Li}(x)=\int_{2}^{x}\frac{1}{\log(t)}\,dt$ for all $x>2$ . We have the following prime orbit theorem with a power saving error term.

Theorem 1.8.

There exist

—

a continuous piecewise-smooth function $\eta_{\Theta}:\mathcal{L}_{\Theta}-\{0\}\to[0,+\infty)$ which is positive except possibly on $(\partial\mathcal{L}_{\Theta}\cup\mathscr{E})-\{0\}$ ,
—

a smooth positive function $C_{\bullet}:\operatorname{int}\mathcal{L}_{\Theta}\to\mathbb{R}$ ,

which are both homothety-invariant, such that for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ and $T>0$ , we have

\displaystyle\bigl{|}\#\mathcal{P}_{\mathsf{v}}(T)-\operatorname{Li}\bigl{(}e^{\delta_{\mathsf{v}}T}\bigr{)}\bigr{|}\leq C_{\mathsf{v}}e^{(\delta_{\mathsf{v}}-\|\mathsf{v}\|\eta_{\Theta}(\mathsf{v}))T}.

Both the theorems above generalize those in [51, 20] for the case that $\Gamma$ is a projective Anosov subgroup of $\operatorname{PSL}_{n}(\mathbb{R})$ (see the discussion below). The above theorem also effectivizes the prime orbit theorem of Sambarino [59, Theorem 7.8] and its generalization by Chow–Fromm [16, Eq. (1.8)] (cf. [16, Corollary 1.4]). In fact, another approach to proving it is by effectivizing the work of [16] with Theorem 1.1 as input.

1.2. Related works

We discuss the related works of Pollicott–Sharp [51] and Delarue–Montclair–Sanders [20]. Both of these works are in the projective Anosov setting.

In [51], the results are for the case that $\Gamma$ is a Hitchin surface subgroup of $\operatorname{PSL}_{n}(\mathbb{R})$ , i.e., the image of a Hitchin representation of a surface group into $\operatorname{PSL}_{n}(\mathbb{R})$ , viewed as a projective Anosov subgroup (in which case $\#\Theta=1$ ). The authors use a very different kind of coding from that of ours to introduce transfer operators. They then use it to study the Selberg zeta function and obtain both an essential spectral gap and a prime orbit theorem with a power saving error term. This was generalized in [20] to the general projective Anosov setting.

In [20], the results are for the case that $\Gamma$ is a projective Anosov subgroup of $\operatorname{PSL}_{n}(\mathbb{R})$ (in which case $\#\Theta=1$ ). The authors obtain a nice result which shows the existence of a $\Gamma$ -invariant and flow-invariant open subset $\tilde{\mathcal{M}}\subset\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ containing $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ where the $\Gamma$ -action is properly discontinuous and the translation flow is a contact Axiom A flow on $\mathcal{M}:=\Gamma\backslash\tilde{\mathcal{M}}$ with $\mathcal{X}$ as its basic set [20, Theorem A]. With this construction in hand, exponential mixing follows at once via Stoyanov’s work [67]. They also obtain related theorems already mentioned above. The main common difficulty to both [20] and our paper is that the actions $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ which naturally appear are not necessarily properly discontinuous (see Appendix A). This poses an obstruction to using the smooth structure on $G$ to prove a local non-integrability condition (LNIC) and also to invoking [67]. Delarue–Montclair–Sanders overcome this by constructing $\tilde{\mathcal{M}}$ . We take a more direct approach and use weaker properties (which is often advantageous) and do not invoke [67] (see Subsection 1.3 for more details).

Thus, a key point in our paper compared to the aforementioned works is that we handle the vastly more general setting of arbitrary $\Theta$ -Anosov subgroups which turns out to have major differences compared to the setting of projective Anosov subgroups. In the general setting, there is a nontrivial family of translation flows parametrized by vectors in $\operatorname{int}\mathcal{L}_{\Theta}$ and we discover that we must delete an exceptional set $\mathscr{E}$ for the expected results, i.e., exponential mixing, essential spectral gap, and prime orbit theorem with a power saving error term hold for the generic vectors in $\operatorname{int}\mathcal{L}_{\Theta}-\mathscr{E}$ . We also obtain the precise nature of the dependence on the vectors in $\operatorname{int}\mathcal{L}_{\Theta}$ .

1.3. On the proof of Theorem 1.1

For general $\Theta$ -Anosov subgroups, we have a family of translation flows $\{\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}\}_{\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}}$ on $\mathcal{X}$ and we begin by proving that they are metric Anosov. We note that when $\Theta\neq\Pi$ , the space $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ does not naturally inherit a $\Gamma$ -invariant metric from the Riemannian metric on $G$ and so the metric on $\mathcal{X}$ needs careful treatment. Moreover, we also show that the strong (un)stable foliations in $\mathcal{X}$ of a translation flow are projections of the horospherical foliations of $G$ (Theorem 3.2). Our proofs of these properties, following [18], use the fact that one can always construct a projective Anosov representation of $\Gamma$ using the Plücker representation and the results of Bridgeman–Canary–Labourie–Sambarino [9] on the metric Anosov property in this case. It is likely that the arguments in [9] can be extended to general Anosov subgroups to give more direct proofs of these properties but we do not do so for expedition reasons. An immediate consequence of the metric Anosov property is that there exist Markov sections for the translation flows which are compatible with the corresponding strong (un)stable foliations. The constructed framework is an important tool which do not appear in previous works and for which reason our approach is fruitful in the general $\Theta$ -Anosov setting.

Thanks to the Markov sections, we may freely introduce symbolic dynamics and thermodynamic formalism, and use transfer operators. More specifically, we define transfer operators $\mathcal{L}_{\xi\tau^{\mathsf{v}}}:C\bigl{(}U,\mathbb{C}\bigr{)}\to C\bigl{(}U,\mathbb{C}\bigr{)}$ defined by

\displaystyle\mathcal{L}_{\xi\tau^{\mathsf{v}}}(H)(u)=\sum_{u^{\prime}\in\sigma^{-1}(u)}e^{\xi\tau^{\mathsf{v}}(u^{\prime})}H(u^{\prime})

where $\tau^{\mathsf{v}}:U\to\mathbb{R}$ is the first return time map associated to the translation flow $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ on $\mathcal{X}$ . We wish to then execute Dolgopyat’s method. Stoyanov [67] has done related work for Axiom A flows but we do not prove this or other strong properties and rely on weaker properties instead. In any case, we need to prove the crucial local non-integrability condition (LNIC) (Proposition 5.11). We wish to use Lie theoretic arguments for this. However, as alluded to in Subsection 1.2, we need to carefully deal with some topological issues due to the fractal nature of the limit set $\Lambda_{\Theta}$ and the $\Gamma$ -action on $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ not being properly discontinuous in many cases. Following [64], the relationship between the strong (un)stable foliations and horospherical foliations allows us to lift and extend the rectangles of the Markov section to open sets and use auxiliary smooth extensions of the Poincaré map and the first return time map. We then generalize the techniques in [64, 18] to prove first a smooth version and then a reverse Lipschitz version of LNIC on the original Markov section which is of fractal nature. In the process, it becomes clear that the aforementioned derivation holds for all $\mathsf{v}$ but those in a certain exceptional set $\mathscr{E}\subset\operatorname{int}\mathcal{L}_{\Theta}$ . Fortunately, Dolgopyat’s method is robust enough that the above ingredients suffice to be able to run the classical arguments (for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}-\mathscr{E}$ ).

Throughout the paper, we also take a more unified viewpoint and carefully investigate the dependence of the family of translation flows on the parameter $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ . This requires additional important properties. The first one is the existence of a compatible family of Markov sections for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ . This allows us to identify the corresponding rectangles with each other and therefore use the same domain $U$ to define the transfer operators and only vary the first return time maps $\tau^{\mathsf{v}}$ which is smooth in $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ . We also show some bounds for $\tau^{\mathsf{v}}$ which is used to carry through the dependence on $\mathsf{v}$ throughout the proofs.

1.4. Organization of the paper

Section 2 covers background on semisimple real algebraic groups, Anosov subgroups, and translation flows. In Section 3, we prove that the translation flow is metric Anosov for a suitable metric and describe the foliations in terms of the $\Theta$ -horospherical subgroups to establish Markov sections compatible with the Lie structure. Section 4 contains the formulation of a key theorem for a Lipschitz version of Dolgopyat’s method from which exponential mixing for Lipschitz continuous functions can be derived. In Section 5 we prove the crucial LNIC. Finally, in Section 6, we prove the aforementioned key theorem. We also include some appendices after that which contain proofs of some basic facts.

Acknowledgements

We thank Hee Oh for her encouragements. We thank Rafael Potrie for useful correspondence regarding metric Anosov flows. We also thank Institut des Hautes Études Scientifiques and the Fields Institute for their hospitality and facilitating conversations. Sarkar acknowledges support by an AMS-Simons Travel Grant.

2. Anosov subgroups and their translation flows

2.1. Lie theoretic preliminaries

Let $G$ be a connected semisimple real algebraic group with identity element $e$ and Lie algebra $\mathfrak{g}$ . Let $B:\mathfrak{g}\times\mathfrak{g}\to\mathbb{R}$ denote the Killing form and $\theta:\mathfrak{g}\to\mathfrak{g}$ be a Cartan involution, i.e., the symmetric bilinear form $B_{\theta}:\mathfrak{g}\times\mathfrak{g}\to\mathbb{R}$ defined by $B_{\theta}(x,y)=-B(x,\theta(y))$ for all $x,y\in\mathfrak{g}$ is positive definite. We write $\langle*_{1},*_{2}\rangle=B_{\theta}(*_{1},*_{2})$ for the inner product and $\|\cdot\|$ for the induced norm. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the associated eigenspace decomposition corresponding to the eigenvalues $+1$ and $-1$ of $\theta$ respectively. Then $K=\exp(\mathfrak{k})<G$ is a maximal compact subgroup. Let $\mathfrak{a}\subset\mathfrak{p}$ be a maximal abelian subalgebra and let $A=\exp\mathfrak{a}$ . Identifying $\mathfrak{a}\cong\mathfrak{a}^{*}$ via the Killing form, let $\Phi\subset\mathfrak{a}^{*}$ be the restricted root system of $(\mathfrak{g},\mathfrak{a})$ and $\Phi^{\pm}\subset\Phi$ be a choice of sets of positive and negative roots. Let $\mathfrak{a}^{+}\subset\mathfrak{a}$ be the corresponding closed positive Weyl chamber. We have the associated restricted root space decomposition

\displaystyle\mathfrak{g}=\mathfrak{a}\oplus\mathfrak{m}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha},

where $\mathfrak{m}$ is the centralizer of $\mathfrak{a}$ in $\mathfrak{k}$ .

Let $\Pi\subset\Phi^{+}$ denote the set of all simple roots. We denote by $\langle\Theta\rangle$ the root subsystem generated by any subset $\Theta\subset\Pi$ (it is empty if so is $\Theta$ ). Fix a nonempty subset $\Theta\subset\Pi$ . Define Lie subalgebras $\mathfrak{a}_{\Theta}$ and $\mathfrak{n}_{\Theta}^{\pm}$ of $\mathfrak{g}$ by

\displaystyle\mathfrak{a}_{\Theta}=\bigcap_{\alpha\in\Pi-\Theta}\ker\alpha,

\displaystyle\mathfrak{n}^{\pm}_{\Theta}=\bigoplus_{\alpha\in\Phi^{+}-\langle\Pi-\Theta\rangle}\mathfrak{g}_{\mp\alpha}

where for the former, we take the empty intersection to be $\mathfrak{a}$ . Let $A_{\Theta}=\exp(\mathfrak{a}_{\Theta})$ , $\mathfrak{a}^{+}_{\Theta}=\mathfrak{a}_{\Theta}\cap\mathfrak{a}^{+}$ and $A^{+}_{\Theta}=\exp(\mathfrak{a}^{+}_{\Theta})\subset A_{\Theta}$ . We denote by $\operatorname{int}\mathfrak{a}^{+}_{\Theta}$ the interior of $\mathfrak{a}^{+}_{\Theta}$ in the topology of $\mathfrak{a}_{\Theta}$ . Let $\mathcal{W}=N_{K}(A)/Z_{K}(A)$ be the Weyl group which is a finite group. The adjoint action $\operatorname{Ad}$ induces an action $\mathcal{W}\curvearrowright\mathfrak{a}$ which we also denote by $\operatorname{Ad}$ . Let $p_{\Theta}:\mathfrak{a}\to\mathfrak{a}_{\Theta}$ denote the orthogonal projection map which is in fact the unique projection map invariant under precomposition by all elements of the Weyl group $\mathcal{W}$ which act trivially on $\mathfrak{a}_{\Theta}$ . Let $w_{0}\in\mathcal{W}$ be the longest element with respect to the generating set consisting of root reflections. Then, $\operatorname{Ad}_{w_{0}}(\mathfrak{a}^{+})=-\mathfrak{a}^{+}$ and the map $\mathsf{i}=-\operatorname{Ad}_{w_{0}}:\mathfrak{a}\to\mathfrak{a}$ is called the opposition involution of $\mathfrak{a}$ . The opposition involution preserves $\mathfrak{a}^{+}$ and acts on $\Pi$ by precomposition.

For a fixed $v\in\operatorname{int}\mathfrak{a}_{\Theta}^{+}$ , the expanding/contracting $\Theta$ -horospherical subgroups are

\displaystyle N^{\pm}_{\Theta}

\displaystyle=\exp(\mathfrak{n}^{\pm}_{\Theta})=\left\{h\in G:\lim_{t\to\pm\infty}\exp(tv)h\exp(-tv)=e\right\}.

The definition is independent of the choice of $v\in\operatorname{int}\mathfrak{a}_{\Theta}^{+}$ and $N_{\Theta}^{\pm}=w_{0}N_{\mathsf{i}\Theta}^{\mp}w_{0}^{-1}$ .

Let $P_{\Theta}$ denote the standard parabolic subgroup associated to $\Theta$ , i.e., $P_{\Theta}$ is the normalizer of $N^{-}_{\Theta}$ in $G$ . The Lie algebra of $P_{\Theta}$ is given by $\mathfrak{p}_{\Theta}=\mathfrak{a}\oplus\mathfrak{m}\oplus\bigoplus_{\alpha\in\Phi^{+}\cup\langle\Pi-\Theta\rangle}\mathfrak{g}_{\alpha}$ . We denote the $\Theta$ -Furstenberg boundary of $G$ by

\displaystyle\mathcal{F}_{\Theta}=G/P_{\Theta}.

Note that we also have an action $\mathcal{W}\curvearrowright\mathcal{F}_{\Theta}$ induced by the left translation action. For all $g\in G$ , we denote

\displaystyle g^{+}

\displaystyle:=g[P_{\Theta}]\in\mathcal{F}_{\Theta},

\displaystyle g^{-}

\displaystyle:=gw_{0}[P_{\mathsf{i}\Theta}]\in\mathcal{F}_{\mathsf{i}\Theta}.

There is a unique open $G$ -orbit of $\mathcal{F}_{\Theta}\times\mathcal{F}_{\mathsf{i}\Theta}$ given by

\displaystyle\mathcal{F}_{\Theta}^{(2)}:=G\cdot(e^{+},e^{-}).

The Levi subgroup $L_{\Theta}$ is the centralizer of $A_{\Theta}$ in $G$ . It satisfies $P_{\Theta}=L_{\Theta}N_{\Theta}^{-}$ , $w_{0}P_{\mathsf{i}\Theta}w_{0}^{-1}=L_{\Theta}N_{\Theta}^{+}$ , $L_{\Theta}=P_{\Theta}\cap w_{0}P_{\mathsf{i}\Theta}w_{0}^{-1}$ and $L_{\Theta}=A_{\Theta}S_{\Theta}$ where $S_{\Theta}$ is an almost direct product of a connected semisimple real algebraic group and a compact center. In all of the above and below notations, when $\Theta=\Pi$ , we omit the subscript/superscript $\Pi$ .

Definition 2.1 (Iwasawa cocycle).

The Iwasawa cocycle is the map $\sigma:G\times\mathcal{F}\rightarrow\mathfrak{a}$ which assigns to a pair $(g,\xi)\in G\times\mathcal{F}$ the unique element $\sigma(g,\xi)\in\mathfrak{a}$ such that $gk\in K\exp(\sigma(g,\xi))N^{-}$ where $k\in K$ such that $\xi=kP$ .

Definition 2.2 ( $\Theta$ -Busemann function).

The $\Theta$ -Busemann function is the map $\beta^{\Theta}:\mathcal{F}_{\Theta}\times G\times G\to\mathfrak{a}_{\Theta}$ defined by

\displaystyle\beta^{\Theta}_{\xi}(g,h):=p_{\Theta}(\sigma(g^{-1},\tilde{\xi})-\sigma(h^{-1},\tilde{\xi}))

for all $\xi\in\mathcal{F}_{\Theta}$ and $g,h\in G$ , where $\tilde{\xi}\in\mathcal{F}$ is any lift of $\xi$ under the natural projection $\mathcal{F}\to\mathcal{F}_{\Theta}$ . This is well-defined, i.e., independent of the choice of $\tilde{\xi}$ [52, Lemma 6.1], $G$ -equivariant, and satisfies the identity

\displaystyle\beta^{\Theta}_{\xi}(g_{1},g_{2})=\beta^{\Theta}_{\xi}(g_{1},h)+\beta^{\Theta}_{\xi}(h,g_{2})

for all $\xi\in\mathcal{F}_{\Theta}$ and $g_{1},g_{2},h\in G$ .

Define a left $G$ -action on $\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ by

\displaystyle g\cdot(\xi,\eta,v)=\bigl{(}g\xi,g\eta,v+\beta^{\Theta}_{g\xi}(e,g)\bigr{)}=(g\xi,g\eta,v+p_{\Theta}(\sigma(g,\xi)))

for all $g\in G$ and $(\xi,\eta,v)\in\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ . Then the stabilizer of $(e^{+},e^{-},0)$ is $S_{\Theta}$ and we have the following diffeomorphism.

Definition 2.3 ( $\Theta$ -Hopf parameterization).

The $\Theta$ -Hopf parameterization is the left $G$ -equivariant diffeomorphism $G/S_{\Theta}\to\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ defined by

\displaystyle gS_{\Theta}\mapsto\bigl{(}g^{+},g^{-},\beta^{\Theta}_{g^{+}}(e,g)\bigr{)}.

We note that the right $A_{\Theta}$ -action on $G/S_{\Theta}$ corresponds to the $\mathfrak{a}_{\Theta}$ -translation on the $\mathfrak{a}_{\Theta}$ -coordinate of $\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ .

Definition 2.4 ( $\Theta$ -Gromov product).

For $(\xi,\eta)=(g^{+},g^{-})\in\mathcal{F}_{\Theta}^{(2)}$ , the $\Theta$ -Gromov product is defined as

\displaystyle[\xi,\eta]_{\Theta}:=\beta^{\Theta}_{g^{+}}(e,g)+\mathsf{i}\beta^{\mathsf{i}\Theta}_{g^{-}}(e,g)

which is independent of choice of $g$ .

2.2. Zariski dense discrete subgroups

Let $\Gamma<G$ be a Zariski dense discrete subgroup.

Definition 2.5 ( $\Theta$ -limit set).

The $\Theta$ -limit set $\Lambda_{\Theta}$ is the set of limit points of $\Gamma e^{+}$ in $\mathcal{F}_{\Theta}$ . It is the unique minimal nonempty closed $\Gamma$ -invariant subset of $\mathcal{F}_{\Theta}$ [52, Corollary 5.2, Lemma 6.3], [40, Lemma 2.13].

The Jordan projection of an element $g\in G$ is the unique element $\lambda(g)\in\mathfrak{a}^{+}$ such that the hyperbolic component of the Jordan decomposition of $g$ is conjugate to $\exp(\lambda(g))$ .

Definition 2.6 ( $\Theta$ -limit cone).

The $\Theta$ -limit cone $\mathcal{L}_{\Theta}\subset\mathfrak{a}_{\Theta}^{+}$ of $\Gamma$ is the unique minimal closed cone in $\mathfrak{a}_{\Theta}^{+}$ containing $p_{\Theta}(\lambda(\Gamma))$ . It is a convex cone with nonempty interior [3].

The Cartan projection of an element $g\in G$ is the unique element $\mu(g)\in\mathfrak{a}^{+}$ such that $g\in Ka_{\mu(g)}K$ for all $g\in G$ .

Definition 2.7 ( $\Theta$ -growth indicator).

The growth indicator $\psi_{\Theta}:\mathfrak{a}^{+}_{\Theta}\to\mathbb{R}\cup\{-\infty\}$ of $\Gamma$ is defined by

\displaystyle\psi_{\Theta}(v)=\|v\|\inf_{\begin{subarray}{c}\text{open cones }\mathcal{C}\subset\mathfrak{a}_{\Theta}^{+}\\ v\in\mathcal{C}\end{subarray}}\tau_{\mathcal{C}}^{\Theta}\qquad\text{ for all }v\in\mathfrak{a}_{\Theta}

with the convention that $0\cdot(-\infty)=0$ , where $\|\cdot\|$ is any Euclidean norm on $\mathfrak{a}_{\Theta}$ and $\tau_{\mathcal{C}}^{\Theta}$ is the abscissa of convergence of the Poincaré series

\displaystyle\sum_{\gamma\in\Gamma,\,p_{\Theta}(\mu(\gamma))\in\mathcal{C}}e^{-t\|p_{\Theta}(\mu(\gamma))\|}.

This definition does not depend on the choice of norm $\|\cdot\|$ (though we have fixed the one induced by the Killing form). Quint [53] showed that $\psi_{\Theta}$ is homogeneous of degree $1$ , concave, upper semicontinuous, and satisfies $\psi_{\Theta}|_{\operatorname{ext}(\mathcal{L}_{\Theta})}=-\infty$ , $\psi_{\Theta}|_{\mathcal{L}_{\Theta}}\geq 0$ , and $\psi_{\Theta}|_{\operatorname{int}(\mathcal{L}_{\Theta})}>0$ . Denote by $\mathsf{u}_{\Theta}\in\operatorname{int}\mathcal{L}_{\Theta}$ any maximal growth direction of $\psi_{\Theta}$ .

The following generalization of Patterson–Sullivan measures [45, 68] is also due to Quint [52] (see also [1]).

Definition 2.8 ( $(\Gamma,\psi)$ -conformal measure).

Let $\psi\in\mathfrak{a}_{\Theta}^{*}$ . A Borel probability measure $\nu$ on $\mathcal{F}_{\Theta}$ is called a $(\Gamma,\psi)$ -conformal measure if

\displaystyle\frac{d\gamma_{*}\nu}{d\nu}(\xi)=e^{\psi(\beta^{\Theta}_{\xi}(e,\gamma))}\qquad\text{ for all }\gamma\in\Gamma,\,\xi\in\mathcal{F}_{\Theta}.

We say that $\psi\in\mathfrak{a}_{\Theta}^{*}$ is tangent to $\psi_{\Theta}$ at $v\in\mathcal{L}_{\Theta}$ if $\psi\geq\psi_{\Theta}$ and $\psi(v)=\psi_{\Theta}(v)$ . We also denote the dual limit cone

\displaystyle\mathcal{L}_{\Theta}^{*}:=\{\psi\in\mathfrak{a}_{\Theta}^{*}:\psi|_{\mathcal{L}_{\Theta}}\geq 0\}.

(1)

Theorem 2.9 (53, Theorem 8.4).

If $\psi\in\mathfrak{a}_{\Theta}^{*}$ is tangent to $\psi_{\Theta}$ at $v\in\operatorname{int}\mathfrak{a}_{\Theta}^{+}\cap\mathcal{L}_{\Theta}$ , then there exists a $(\Gamma,\psi)$ -conformal measure.

2.3. Anosov subgroups

The notation introduced in the remainder of this section will be fixed throughout the paper. The following definition of Anosov subgroup is due to Guichard–Wienhard [25, Corollary 4.16] (see also [36, 24, 27] for other equivalent characterizations).

Definition 2.10 (Anosov subgroup).

We call a Zariski dense discrete subgroup $\Gamma<G$ a ( $\Theta$ -)Anosov subgroup if it is a finitely generated Gromov hyperbolic group with Gromov boundary $\partial\Gamma$ and it admits continuous $\Gamma$ -equivariant boundary maps $\zeta_{\Theta}:\partial\Gamma\to\mathcal{F}_{\Theta}$ and $\zeta_{\mathsf{i}\Theta}:\partial\Gamma\to\mathcal{F}_{\mathsf{i}\Theta}$ such that $(\zeta_{\Theta}(x),\zeta_{\mathsf{i}\Theta}(y))\in\mathcal{F}_{\Theta}^{(2)}$ for all $(x,y)\in\partial\Gamma^{(2)}:=\{(x,y)\in\partial\Gamma^{2}:x\neq y\}$ .

Let $\Gamma$ be a Zariski dense $\Theta$ -Anosov subgroup with boundary maps $\zeta_{\Theta}$ and $\zeta_{\mathsf{i}\Theta}$ . We record some basic properties in the following theorem. The first 3 properties are [9, Theorem 6.1], [46, Proposition 4.6], [25, Lemma 3.1], respectively, and the last three properties follow from [61, Theorem A] and [58, Lemma 4.8] (cf. [58, Corollary 4.9] and [60, Theorem 4.20]).

Theorem 2.11.

We have the following properties:

((1))

the boundary maps $\zeta_{\Theta}$ and $\zeta_{\mathsf{i}\Theta}$ are Hölder homeomorphisms onto the limit sets $\Lambda_{\Theta}$ and $\Lambda_{\mathsf{i}\Theta}$ , respectively;
((2))

the limit cone $\mathcal{L}_{\Theta}$ is contained in $\operatorname{int}\mathfrak{a}_{\Theta}^{+}\cup\{0\}$ ;
((3))

every nontorsion element $\gamma\in\Gamma$ is loxodromic;
((4))

the growth indicator $\psi_{\Theta}$ is analytic and strictly concave except along radial directions on $\operatorname{int}\mathcal{L}_{\Theta}$ and vertically tangent on $\partial\mathcal{L}_{\Theta}$ ;
((5))

if $\psi\in\mathfrak{a}_{\Theta}^{*}$ is tangent to $\psi_{\Theta}$ , then $\psi|_{\mathcal{L}_{\Theta}-\{0\}}>0$ ;

((6))

if $\psi\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ , then there exists a unique

\displaystyle\delta_{\psi}:=\limsup_{t\to+\infty}\frac{1}{t}\log\#\{[\gamma]:\psi(\lambda(\gamma))\leq t\}\in(0,+\infty),

(2)

where $[\gamma]$ denotes the conjugacy class of $\gamma\in\Gamma$ , such that $\delta_{\psi}\psi$ is tangent to $\psi_{\Theta}$ .

For $\psi\in\mathfrak{a}_{\Theta}^{*}$ , let $\pi_{\psi}:\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}\to\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ be the projection map defined by

\displaystyle\pi_{\psi}(x,y,v)=(x,y,\psi(v))\qquad\text{ for all }(x,y,v)\in\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}.

(3)

The $\Gamma$ -action on $G/S_{\theta}\cong\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ descends to an action on $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ via the projection $\pi_{\psi}$ given explicitly by

\displaystyle\gamma\cdot(x,y,s)

\displaystyle=\bigl{(}\gamma x,\gamma y,s+\psi(\beta^{\Theta}_{\gamma x}(e,\gamma))\bigr{)}\qquad\text{ for all }\gamma\in\Gamma,\,(x,y,s)\in\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}.

(4)

The left $\Gamma$ -action on $G/S_{\theta}\cong\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ is not necessarily properly discontinuous. For example, when $G=\operatorname{SL}_{3}(\mathbb{R})$ and $\#\Theta=1$ , the $\Gamma$ -action is not properly discontinuous (see [4, 35, 19] for a general criterion) but on the other hand, when $G=\operatorname{SL}_{n}(\mathbb{R})$ for some $n\geq 4$ , there are examples where the $\Gamma$ -action is properly discontinuous (see [24, Corollaries 1.10 and 1.11]). The $\Gamma$ -action on $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ is also not necessarily properly discontinuous (see Theorem A.2 for a self-contained construction of numerous examples). However, restricting the $\Gamma$ -action to $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ where $\Lambda_{\Theta}^{(2)}:=(\Lambda_{\Theta}\times\Lambda_{\mathsf{i}\Theta})\cap\mathcal{F}_{\Theta}^{(2)}$ , we have the following theorem.

Theorem 2.12.

If $\psi\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ , then the $\Gamma$ -action on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ induced by $\pi_{\psi}$ is properly discontinuous and cocompact.

In light of the above theorem, for $\psi\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ , let

\displaystyle\mathcal{X}_{\psi}:=\Gamma\backslash\bigl{(}\Lambda_{\Theta}^{(2)}\times\mathbb{R}\bigr{)}

where the $\Gamma$ -action is the one induced by $\pi_{\psi}$ . The space $\mathcal{X}_{\psi}$ is equipped with a Bowen–Margulis–Sullivan (BMS) measure $m_{\mathcal{X}}^{\psi}$ . Let $\delta_{\psi}>0$ be as in Theorem 2.11 (6). By [33, Theorem 1.11], there exists a unique $(\Gamma,\delta_{\psi}\psi)$ -conformal measure $\nu_{\psi}$ (resp. $(\Gamma,\delta_{\psi}\psi\circ\mathsf{i})$ -conformal measure $\nu_{\psi\circ\mathsf{i}}$ ) on $\mathcal{F}_{\Theta}$ (resp. $\mathcal{F}_{\mathsf{i}\Theta}$ ) and moreover, $\nu_{\psi}$ (resp. $\nu_{\psi\circ\mathsf{i}}$ ) is supported on $\Lambda_{\Theta}$ (resp. $\Lambda_{\mathsf{i}\Theta}$ ). We define a locally finite Borel measure $\tilde{m}^{\psi}_{\mathcal{X}}$ on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ by

d\tilde{m}^{\psi}_{\mathcal{X}}(x,y,s)=e^{\delta_{\psi}\psi([x,y]_{\Theta})}\,d\nu_{\psi}(x)\,d\nu_{\psi\circ\mathsf{i}}(y)\,ds\qquad\text{ for all }(x,y,s)\in\Lambda_{\Theta}^{(2)}\times\mathbb{R}

(5)

where $ds$ denotes the Lebesgue measure on $\mathbb{R}$ . Observe that $\tilde{m}^{\psi}_{\mathcal{X}}$ is left $\Gamma$ -invariant, so it descends to a probability measure $m_{\mathcal{X}}^{\psi}$ on $\mathcal{X}_{\psi}$ (after renormalization). By [61, Proposition 3.3.2], $m_{\mathcal{X}}^{\psi}$ is the measure of maximal entropy for the translation flows on $\mathcal{X}_{\psi}$ .

A consequence of Theorem 2.12 is that the restriction of the $\Gamma$ -action on $\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ to $\Lambda_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ is properly discontinuous. Moreover, $\Omega:=\Gamma\backslash(\Lambda_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta})$ is a trivial $\ker\psi$ -vector bundle over $\mathcal{X}_{\psi}$ by a standard argument.

2.4. Translation flows

Given a vector $\mathsf{v}\in\mathfrak{a}_{\Theta}$ , we have a flow $\{a^{\psi}_{t\mathsf{v}}\}_{t\in\mathbb{R}}$ on $\Omega$ which is given by translation by $t\mathsf{v}$ in the $\mathfrak{a}_{\Theta}$ -coordinate. The flow $\{a^{\psi}_{t\mathsf{v}}\}$ descends via $\pi_{\psi}$ to what we call a translation flow $\{a^{\psi,\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ on $\mathcal{X}_{\psi}$ given explicitly by

\displaystyle a^{\psi,\mathsf{v}}_{t}\cdot\Gamma(x,y,s):=\Gamma(x,y,s+\psi({\mathsf{v}})t)\qquad\text{ for all }t\in\mathbb{R},\,(x,y,s)\in\Lambda_{\Theta}^{(2)}\times\mathbb{R}.

By [61], the topological entropy of the flow $\bigl{\{}a^{\psi,\mathsf{v}}_{t}\bigr{\}}_{t\in\mathbb{R}}$ is $\delta_{\psi}\psi(\mathsf{v})$ (see Eq. 2). Note that only the value of $\psi(\mathsf{v})$ is required to determine the translation flow on $\mathcal{X}_{\psi}$ and furthermore, rescaling $\psi$ simply rescales the $\mathbb{R}$ -coordinate of $\mathcal{X}_{\psi}$ . We introduce the following notation to fix a family of translation flows without the redundacies from scaling $\psi$ or $\mathsf{v}$ . For $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ , set

\displaystyle\psi_{\mathsf{v}}=\frac{\langle\nabla\psi_{\Theta}(\mathsf{v}),\cdot\rangle}{\psi_{\Theta}(\mathsf{v})};

\displaystyle\mathcal{X}_{\mathsf{v}}:=\mathcal{X}_{\psi_{\mathsf{v}}};

\displaystyle a^{\mathsf{v}}_{t}:=a^{\psi_{\mathsf{v}},\mathsf{v}}_{t};

\displaystyle m_{\mathcal{X}}^{\mathsf{v}}:=m_{\mathcal{X}}^{\psi_{\mathsf{v}}}.

(6)

Note that $\psi_{\Theta}(\mathsf{v})\psi_{\mathsf{v}}=\langle\nabla\psi_{\Theta}(\mathsf{v}),\cdot\rangle$ is the unique linear form tangent to $\psi_{\Theta}$ at $\mathsf{v}$ . Then, the aforementioned redundancies are removed if $\mathsf{v}$ is chosen so that $\psi_{\mathsf{v}}(\mathsf{v})=1$ and the topological entropy is then $\delta_{\mathsf{v}}:=\delta_{\psi_{\mathsf{v}}}=\psi_{\Theta}(\mathsf{v})$ by Theorem 2.11 (6).

We recall the relation between the translation flow and the (unique up to Hölder reparametrization (see Definition 3.9)) Gromov geodesic flow $\bigl{(}\mathcal{G},\bigl{\{}a^{\mathcal{G}}_{t}\bigr{\}}_{t\in\mathbb{R}}\bigr{)}$ associated to $\Gamma$ .

Theorem 2.13.

For all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ , there exists a Hölder homemorphism $\mathcal{X}_{\mathsf{v}}\to\mathcal{G}$ conjugating $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ to a Hölder reparametrization of $\{a^{\mathcal{G}}_{t}\}_{t\in\mathbb{R}}$ .

Henceforth, fix $\mathsf{v}_{0}\in\operatorname{int}\mathcal{L}_{\Theta}$ , say $\mathsf{v}_{0}:=\mathsf{u}_{\Theta}$ . The above theorem in particular shows that for any $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ the dynamical system $(\mathcal{X}_{\mathsf{v}},\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}})$ is conjugate to a reparametrization of $(\mathcal{X}_{\mathsf{v}_{0}},\{a^{\mathsf{v}_{0}}_{t}\}_{t\in\mathbb{R}})$ . In fact, we will see that the conjugating homeomorphism and reparametrizations are bi-Lipschitz (see Theorem 3.13). In light of this fact, we take a unified viewpoint where we have a single compact metric space

\displaystyle\mathcal{X}:=\mathcal{X}_{\mathsf{v}_{0}}

equipped with a family of pairs of BMS measures and translation flows

\displaystyle\{(m_{\mathcal{X}}^{\mathsf{v}},\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}})\}_{\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}}.

To distinguish the family for $\mathsf{v}=\sf{v_{0}}$ , we set

\displaystyle\psi:=\psi_{\mathsf{v}_{0}},

\displaystyle m_{\mathcal{X}}:=m_{\mathcal{X}}^{\mathsf{v}_{0}}

\displaystyle\{a_{t}:=a^{\mathsf{v}_{0}}_{t}\}_{t\in\mathbb{R}}.

The translation flows are then homothety equivariant, i.e., we have

\displaystyle a^{c\mathsf{v}}_{t}=a^{\mathsf{v}}_{ct}\qquad\text{for all $t\in\mathbb{R}$ and $c>0$}.

(7)

When $\Theta=\Pi$ , Theorems 2.12 and 2.13 follow from the authors’ prequel work [18, Theorem 4.15]. Such theorems first appeared in the work of Sambarino [59, Theorem 3.2] for fundamental groups of compact negatively curved manifolds and in [9, Proposition 4.2] for projective Anosov subgroups whose generalization was outline in [13, Theorem A.2]. The proofs can be generalized for arbitrary $\Theta$ (cf. [61] and see [33, Theorem 9.1] for a complete alternative proof).

3. Anosov property of the translation flow

The purpose of this section is to prove that the translation flow $\{a_{t}\}_{t\in\mathbb{R}}$ on $\mathcal{X}$ is metric Anosov (see Definition 3.7). Before stating the precise theorem, we first define the metric on $\mathcal{X}$ we will be using. Fix a bi-invariant Riemannian metric on the compact subgroup $K$ . Since $\mathcal{F}_{\Theta}$ and $\mathcal{F}_{\mathsf{i}\Theta}$ are quotients of $K$ , the metric on $K$ along with Euclidean metric on $\mathbb{R}$ induces a product metric $d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}}\times d_{\mathbb{R}}$ on $\mathcal{F}_{\Theta}\times\mathcal{F}_{\mathsf{i}\Theta}\times\mathbb{R}$ which we restrict to $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ . We call any metric on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ which is locally bi-Lipschitz equivalent to the product metric on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ a locally product-like metric. Construction of a $\Gamma$ -invariant locally product-like metric on $\mathcal{X}$ can be done as in [14, Lemma 4.11] (cf. [9, Lemma 5.2]). We equip $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ (resp. $\mathcal{X}$ ) with any $\Gamma$ -invariant locally product-like metric $\tilde{d}$ (resp. $d$ ).

Remark 3.1.

When $\Theta=\Pi$ , we have the diffeomorphism $G/M\cong\mathcal{F}^{(2)}\times\mathfrak{a}$ where $M$ is a subgroup of $K$ . Then using a $\Gamma$ -equivariant continuous section $\Lambda^{(2)}\times\mathbb{R}\to\Lambda^{(2)}\times\mathfrak{a}$ , any left $G$ -invariant and right $K$ -invariant Riemannian metric on $G$ descends to a metric on $G/M$ pulls back to a locally product-like metric on $\Lambda^{(2)}\times\mathbb{R}$ .

This section is devoted to proving the following theorem.

Theorem 3.2.

The translation flow $\{a_{t}\}_{t\in\mathbb{R}}$ on $(\mathcal{X},d)$ is a metric Anosov flow with respect to the pair of foliations $(W^{\mathrm{su}},W^{\mathrm{ss}})$ induced by the $\Theta$ -horospherical foliations (see Subsection 3.2).

We obtain the following as a consequence of Theorems 3.2 and 3.8.

Corollary 3.3.

The translation flow $\{a_{t}\}_{t\in\mathbb{R}}$ on $(\mathcal{X},d)$ has a Markov section with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ (see Definition 3.6).

3.1. Metric Anosov flows, Markov sections, and reparametrizations

In this subsection, we briefly recall the definition of metric Anosov flows and the fact that they admit Markov sections due to Pollicott [49]. The reader can also refer to [9, §3.2] and [18, §4.1] for details omitted in this subsection. We also discuss reparametrizations and give a Lipschitz criterion for when a reparametrization of a metric Anosov flow is also metric Anosov.

For this subsection, let $\{\phi_{t}\}_{t\in\mathbb{R}}$ be an arbitrary continuous flow on an arbitrary metric space $(\mathcal{Y},d)$ . Given a foliation $W$ of $\mathcal{Y}$ , a point $x\in\mathcal{Y}$ and $\epsilon>0$ , let $W(x)$ denote the leaf through $x$ and $W_{\epsilon}(x)=\{y\in W(x):d(x,y)<\epsilon\}$ . Given a foliation $W$ transverse to $\{\phi_{t}\}_{t\in\mathbb{R}}$ , we define the corresponding central foliation $W^{\mathrm{c}}$ such that for all $x,y\in\mathcal{Y}$ , we have $y\in W^{\mathrm{c}}(x)$ if and only if $\phi_{t}(y)\in W(x)$ for some $t\in\mathbb{R}$ .

Definition 3.4 (Local product structure).

A pair of foliations $(W,W^{\prime})$ of $\mathcal{Y}$ is said to have local product structure if there exists $\epsilon>0$ such that for all $x\in\mathcal{Y}$ , there exists a homeomorphism $[\cdot,\cdot]:W_{\epsilon}(x)\times W^{\prime}_{\epsilon}(x)\to O_{x}$ where $O_{x}\subset\mathcal{Y}$ is a neighborhood of $x$ such that $[\cdot,\cdot]^{-1}$ is a chart for both $W$ and $W^{\prime}$ .

Let $(W^{\mathrm{su}},W^{\mathrm{ss}})$ be a pair of foliations transverse to $\{\phi_{t}\}_{t\in\mathbb{R}}$ . Denote $W^{\mathrm{wu}}=(W^{\mathrm{su}})^{\mathrm{c}}$ and $W^{\mathrm{ws}}=(W^{\mathrm{ss}})^{\mathrm{c}}$ and suppose $(W^{\mathrm{wu}},W^{\mathrm{ss}})$ and $(W^{\mathrm{ws}},W^{\mathrm{su}})$ have local product structures given by $[\cdot,\cdot]$ defined on $W^{\mathrm{ws}}_{\epsilon_{0}}\times W^{\mathrm{su}}_{\epsilon_{0}}$ for some $\epsilon_{0}\in(0,1)$ sufficiently small so that the image sets are always contained in a unit ball. For subsets $U\subset W_{\epsilon_{0}}^{\mathrm{su}}(x)$ and $S\subset W_{\epsilon_{0}}^{\mathrm{ss}}(x)$ such that $x\in U=\overline{\operatorname{int}(U)}$ and $x\in S=\overline{\operatorname{int}(S)}$ , we call

\displaystyle R=[U,S]=\{[u,s]\in\mathcal{Y}:u\in U,s\in S\}\subset\mathcal{Y}

a rectangle of size $\hat{\delta}$ if $\operatorname{diam}(R)\leq\hat{\delta}$ for some $\hat{\delta}>0$ , and $x$ the center of $R$ . For any rectangle $R=[U,S]$ , we can extend the map $[\cdot,\cdot]$ to $[\cdot,\cdot]:R\to R$ defined by $[v_{1},v_{2}]=[u_{1},s_{2}]$ for all $v_{1}=[u_{1},s_{1}]\in[U,S]$ and $v_{2}=[u_{2},s_{2}]\in[U,S]$ .

Definition 3.5 (Complete set of rectangles).

A set $\mathcal{R}=\{R_{1},R_{2},\dotsc,R_{N}\}=\{[U_{1},S_{1}],[U_{2},S_{2}],\dotsc,[U_{N},S_{N}]\}$ for some $N\in\mathbb{N}$ consisting of rectangles of size $\hat{\delta}$ with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ in $\mathcal{Y}$ is called a complete set of rectangles of size $\hat{\delta}$ if:

((1))

$R_{j}\cap R_{k}=\varnothing$ for all $1\leq j,k\leq N$ with $j\neq k$ ;
((2))

$\mathcal{Y}=\bigcup_{j=1}^{N}\bigcup_{t\in[0,\hat{\delta}]}\phi_{t}(R_{j})$ .

Let $\mathcal{R}=\{R_{1},R_{2},\dotsc,R_{N}\}=\{[U_{1},S_{1}],[U_{2},S_{2}],\dotsc,[U_{N},S_{N}]\}$ be a complete set of rectangles of size $\hat{\delta}\in(0,\epsilon_{0})$ in $\mathcal{Y}$ . We introduce some notation related to $\mathcal{R}$ . Let

\displaystyle R

\displaystyle=\bigsqcup_{j=1}^{N}R_{j},

\displaystyle U

\displaystyle=\bigsqcup_{j=1}^{N}U_{j}.

Define the first return time map $\tau:R\to\mathbb{R}$ by

\displaystyle\tau(u)=\inf\{t\in\mathbb{R}_{>0}:\phi_{t}(u)\in R\}\qquad\text{for all $u\in R$}.

Define

\displaystyle\overline{\tau}

\displaystyle=\sup_{u\in R}\tau(u),

\displaystyle\underline{\tau}=\inf_{u\in R}\tau(u).

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Define the Poincaré first return map $\mathcal{P}:R\to R$ by

\displaystyle\mathcal{P}(u)=\phi_{\tau(u)}(u)\qquad\text{for all $u\in R$}.

Let $\sigma=(\operatorname{proj}_{U}\circ\mathcal{P})|_{U}:U\to U$ be its projection where $\operatorname{proj}_{U}:R\to U$ is the projection defined by $\operatorname{proj}_{U}([u,s])=u$ for all $[u,s]\in R$ . We define the cores

	$\displaystyle\hat{R}$	$\displaystyle=\{u\in R:\mathcal{P}^{k}(u)\in\operatorname{int}(R)\text{ for all }k\in\mathbb{Z}\},$
	$\displaystyle\hat{U}$	$\displaystyle=\{u\in U:\sigma^{k}(u)\in\operatorname{int}(U)\text{ for all }k\in\mathbb{Z}_{\geq 0}\}.$

Definition 3.6 (Markov section).

We say the complete set of rectangles $\mathcal{R}$ is a Markov section (with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ ) if is satisfies the Markov property: $[\operatorname{int}(U_{k}),\mathcal{P}(u)]\subset\mathcal{P}([\operatorname{int}(U_{j}),u])$ and $\mathcal{P}([u,\operatorname{int}(S_{j})])\subset[\mathcal{P}(u),\operatorname{int}(S_{k})]$ for all $u\in R$ such that $u\in\operatorname{int}(R_{j})\cap\mathcal{P}^{-1}(\operatorname{int}(R_{k}))\neq\varnothing$ , for all $1\leq j,k\leq N$ .

Observe that if $\mathcal{R}$ is a Markov section, then $\tau$ is constant on $[u,S_{j}]$ for all $u\in U_{j}$ and $1\leq j\leq N$ .

Definition 3.7 (Metric Anosov flow).

The flow $\{\phi_{t}\}_{t\in\mathbb{R}}$ is said to be metric Anosov (with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ ) if there exist $\epsilon>0$ , $\eta>0$ , and $C>0$ such that

((1))

$(W^{\mathrm{wu}},W^{\mathrm{ss}})$ and $(W^{\mathrm{ws}},W^{\mathrm{su}})$ have local product structures;

((2))

it satisfies the Anosov property: for all $x\in\mathcal{Y}$ , $y\in W_{\epsilon}^{\mathrm{su}}(x)$ , and $z\in W_{\epsilon}^{\mathrm{ss}}(x)$ , we have

\displaystyle d(\phi_{-t}(x),\phi_{-t}(y))

\displaystyle\leq Ce^{-\eta t}d(x,y),

\displaystyle d(\phi_{t}(x),\phi_{t}(z))

\displaystyle\leq Ce^{-\eta t}d(x,z)

for all $t\geq 0$ .

The existence of Markov sections for metric Anosov flows is due to Pollicott [49] and generalizes results of Bowen and Ratner [5, 54].

Theorem 3.8.

If $\{\phi_{t}\}_{t\in\mathbb{R}}$ is a metric Anosov flow with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ , then there exists a Markov section with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ (see Definition 3.6).

We now recall the definition of reparametrization of a flow and give a Lipschitz criterion to verify if a given reparametrization of a metric Anosov flow is also metric Anosov with respect to a given pair of foliations.

Definition 3.9 (Reparametrization).

A flow $\{\hat{\phi}_{t}\}_{t\in\mathbb{R}}$ on $\mathcal{Y}$ is called a reparametrization of $\{\phi_{t}\}_{t\in\mathbb{R}}$ if it is of the form $\hat{\phi}_{t}(x)=\phi_{\kappa(x,t)}(x)$ for all $x\in\mathcal{Y}$ and $t\in\mathbb{R}$ , where $\kappa:\mathcal{Y}\times\mathbb{R}\to\mathbb{R}$ is a continuous map satisfying

((1))

positivity: $\kappa(x,t)>0$ for all $x\in\mathcal{Y}$ and $t>0$ ;
((2))

the cocycle condition: $\kappa(x,s+t)=\kappa(\hat{\phi}_{s}(x),t)+\kappa(x,s)$ for all $x\in\mathcal{Y}$ and $s,t\in\mathbb{R}$ .

In that case, $\phi$ is itself a reparametrization of $\hat{\phi}$ for some continuous map $\kappa^{*}:\mathcal{Y}\times\mathbb{R}\to\mathbb{R}$ which we call the inverse of $\kappa$ and satisfies

((1))

$(\kappa^{*})^{*}=\kappa$ ,
((2))

$\kappa(x,\kappa^{*}(x,t))=\kappa^{*}(x,\kappa(x,t))=t$ for all $(x,t)\in\mathcal{Y}\times\mathbb{R}$ .

We say that a reparametrization is Lipschitz (resp. Hölder) if $\kappa(\cdot,t)$ is Lipschitz (resp. Hölder) continuous for all $t\in\mathbb{R}$ .

Proposition 3.10.

Let $\{\phi_{t}\}_{t\in\mathbb{R}}$ be a metric Anosov flow with respect to $(W^{\mathrm{su}},W^{\mathrm{ss}})$ and with constants $(\eta,C)$ . Suppose that $\bigl{\{}\hat{\phi}_{t}:=\phi_{\kappa(\cdot,t)}\bigr{\}}_{t\in\mathbb{R}}$ is a reparametrization of $\{\phi_{t}\}_{t\in\mathbb{R}}$ and $(\hat{W}^{\mathrm{su}},\hat{W}^{\mathrm{ss}})$ is a pair of foliations transverse to $\hat{\phi}$ such that $\hat{W}^{\mathrm{su}}\subset W^{\mathrm{wu}}$ and $\hat{W}^{\mathrm{ss}}\subset W^{\mathrm{ws}}$ leafwise. Suppose there exists $\epsilon>0$ , $c\geq 1$ , and $c^{\prime}>0$ such that

((1))

for all $x\in\mathcal{Y}$ , $y\in\hat{W}_{\epsilon}^{\mathrm{su}}(x)$ , and $z\in\hat{W}_{\epsilon}^{\mathrm{ss}}(x)$ , if $y^{\prime}\in\phi_{\mathbb{R}}(y)\cap W^{\mathrm{su}}(x)$ and $z^{\prime}\in\phi_{\mathbb{R}}(z)\cap W^{\mathrm{ss}}(x)$ , then

\displaystyle c^{-1}d(x,y^{\prime})\leq d(x,y)\leq cd(x,y^{\prime}),

\displaystyle c^{-1}d(x,z^{\prime})\leq d(x,z)\leq cd(x,z^{\prime});

((2))

$\kappa(x,t)\geq c^{\prime}t$ for all $t>0$ and $x\in\mathcal{Y}$ .

Then, $\hat{\phi}$ is also metric Anosov with respect to $(\hat{W}^{\mathrm{su}},\hat{W}^{\mathrm{ss}})$ and with constants $(c^{\prime}\eta,c^{2}C)$ .

Proof.

Let $\{\phi_{t}\}_{t\in\mathbb{R}}$ , $\{\hat{\phi}_{t}\}_{t\in\mathbb{R}}$ , $\epsilon$ , $c$ , and $c^{\prime}$ be as in the proposition. It suffices to check the Anosov property for $\hat{\phi}$ . We check it for the stable foliation and the unstable foliation is done similarly. We may assume $\epsilon>0$ is small enough so that by the Anosov property of $\phi$ , there exist $\eta>0$ and $C>0$ such that for all $x\in\mathcal{Y}$ , $z^{\prime}\in W_{c^{-1}\epsilon}^{\mathrm{ss}}(x)$ , and $t>0$ we have

\displaystyle d(\phi_{t}(x),\phi_{t}(z^{\prime}))\leq Ce^{-\eta t}d(x,z^{\prime}).

Then, for all $x\in\mathcal{Y}$ , $z\in\hat{W}_{\epsilon}^{\mathrm{ss}}(x)$ , $t>0$ , and $z^{\prime}\in\phi_{\mathbb{R}}(z)\cap W^{\mathrm{ss}}(x)$ we have

	$\displaystyle d(\hat{\phi}_{t}(x),\hat{\phi}_{t}(z))$	$\displaystyle=d(\phi_{\kappa(x,t)}(x),\phi_{\kappa(z,t)}(z))$
		$\displaystyle\leq cd(\phi_{\kappa(x,t)}(x),\phi_{\kappa(x,t)}(z^{\prime}))\qquad\text{by \lx@cref{creftype~refnum}{itm:FoliationProperty}}$
		$\displaystyle\leq cCe^{-\eta\kappa(x,t)}d(x,z^{\prime})\qquad\text{by the Anosov property}$
		$\displaystyle\leq c^{2}Ce^{-c^{\prime}\eta t}d(x,z)\qquad\text{by \lx@cref{creftypeplural~refnum}{itm:FoliationProperty} and~\lx@cref{refnum}{itm:KappaProperty}}.$

∎

3.2. Stable and unstable foliations for the translation flow

In this subsection, we introduce a pair of natural foliations on $\mathcal{X}$ transverse to the translation flow. The right $N_{\Theta}^{\pm}$ -orbits in $G$ give the $\Theta$ -horospherical foliations. Recall that $P_{\Theta}$ is the normalizer of $N_{\Theta}^{-}$ in $G$ . In particular, $S_{\Theta}\subset L_{\Theta}=P_{\Theta}\cap w_{0}P_{\mathsf{i}\Theta}w_{0}^{-1}$ normalizes both $N_{\Theta}^{-}$ and $N_{\Theta}^{+}=w_{0}N_{\mathsf{i}\Theta}^{-}w_{0}^{-1}$ . Then the $\Theta$ -horospherical foliations descend to foliations of $G/S_{\Theta}\cong\mathcal{F}_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ which we denote by $H^{\mathcal{F}_{\Theta},\pm}$ , and its restriction to $\Lambda_{\Theta}^{(2)}\times\mathfrak{a}_{\Theta}$ by $H^{\pm}$ . More explicitly,

	$\displaystyle H^{\mathcal{F}_{\Theta},+}(gS_{\Theta}):=$	$\displaystyle\,\bigl{\{}\bigl{(}(gh)^{+},(gh)^{-},\beta^{\Theta}_{(gh)^{+}}(e,gh)\bigr{)}\,:\,h\in N_{\Theta}^{+}\bigr{\}}$
	$\displaystyle=$	$\displaystyle\,\bigl{\{}\bigl{(}(gh)^{+},g^{-},\beta^{\Theta}_{g^{+}}(e,g)+[(gh)^{+},g^{-}]_{\Theta}-[g^{+},g^{-}]_{\Theta}\bigr{)}:h\in N_{\Theta}^{+}\bigr{\}},$
	$\displaystyle H^{\mathcal{F}_{\Theta},-}(gS_{\Theta}):=$	$\displaystyle\,\bigl{\{}\bigl{(}(gn)^{+},(gn)^{-},\beta^{\Theta}_{(gn)^{+}}(e,gn)\bigr{)}\,:\,n\in N_{\Theta}^{-}\bigr{\}}$
	$\displaystyle=$	$\displaystyle\,\bigl{\{}\bigl{(}g^{+},(gn)^{-},\beta^{\Theta}_{g^{+}}(e,g)\bigr{)}\,:\,n\in N_{\Theta}^{-}\bigr{\}},$

(see [32, Lemma 7.4] for the equalities) for all $gS_{\Theta}\in G/S_{\Theta}$ . We denote the image foliations induced by $H^{\mathcal{F}_{\Theta},\pm}$ and $H^{\pm}$ under the projection $\pi_{\psi}$ (see Eq. 3) by $W^{\mathcal{F}_{\Theta},\mathrm{su}/\mathrm{ss}}$ and $W^{\mathrm{su}/\mathrm{ss}}$ and call them the strong unstable and strong stable foliations, respectively. Finally, we use the same notation $W^{\mathrm{su}/\mathrm{ss}}$ for the image foliation under the vector bundle projection $\Lambda_{\Theta}^{(2)}\times\mathbb{R}\to\mathcal{X}$ . The following lemma ensures that the above procedure produces well-defined foliations on $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ , $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ , and $\mathcal{X}$ .

Lemma 3.11.

Let $g_{1},g_{2}\in G$ such that

\displaystyle\bigl{(}g_{1}^{+},g_{1}^{-},\psi\bigl{(}\beta^{\Theta}_{g_{1}^{+}}(e,g_{1})\bigr{)}\bigr{)}=\bigl{(}g_{2}^{+},g_{2}^{-},\psi\bigl{(}\beta^{\Theta}_{g_{2}^{+}}(e,g_{2})\bigr{)}\bigr{)}.

Then, for all $h_{1},h_{2}\in N_{\Theta}^{+}$ with $(g_{1}h_{1})^{+}=(g_{2}h_{2})^{+}$ , we have

\displaystyle\psi\bigl{(}\beta^{\Theta}_{(g_{1}h_{1})^{+}}(e,g_{1}h_{1})\bigr{)}=\psi\bigl{(}\beta^{\Theta}_{(g_{2}h_{2})^{+}}(e,g_{2}h_{2})\bigr{)}.

Proof.

Let $g_{1},g_{2}\in G$ as in the lemma and suppose $h_{1},h_{2}\in N_{\Theta}^{+}$ with $(g_{1}h_{1})^{+}=(g_{2}h_{2})^{+}$ . Since $(g_{1}^{+},g_{1}^{-})=(g_{2}^{+},g_{2}^{-})$ , there exists $ar\in A_{\Theta}S_{\Theta}=L_{\Theta}$ such that $g_{1}=g_{2}ar$ . Then, by an elementary computation, we have

\displaystyle\beta^{\Theta}_{g_{1}^{+}}(e,g_{1})=\beta^{\Theta}_{g_{2}^{+}}(e,g_{2})+a.

Since $\psi\bigl{(}\beta^{\Theta}_{g_{1}^{+}}(e,g_{1})\bigr{)}=\psi\bigl{(}\beta^{\Theta}_{g_{2}^{+}}(e,g_{2})\bigr{)}$ by hypothesis, we have

\displaystyle\psi(a)=0.

Since $((g_{1}h_{1})^{+},(g_{1}h_{1})^{-})=((g_{1}h_{1})^{+},g_{1}^{-})=((g_{2}h_{2})^{+},g_{2}^{-})=((g_{2}h_{2})^{+},(g_{2}h_{2})^{-})$ , we also have $g_{1}h_{1}=g_{2}h_{2}a^{\prime}r^{\prime}$ for some $a^{\prime}r^{\prime}\in A_{\Theta}S_{\Theta}$ . Combining with $g_{1}=g_{2}ar$ , we obtain $arh_{1}=h_{2}a^{\prime}r^{\prime}=a^{\prime}r^{\prime}((a^{\prime}r^{\prime})^{-1}h_{2}(a^{\prime}r^{\prime}))$ . Since $L_{\Theta}$ normalizes $N_{\Theta}^{+}$ and $L_{\Theta}N_{\Theta}^{+}$ is a direct product, it follows that

\displaystyle a^{\prime}=a.

Similar to before, we have

\displaystyle\beta^{\Theta}_{(g_{1}h_{1})^{+}}(e,g_{1}h_{1})=\beta^{\Theta}_{(g_{2}h_{2})^{+}}(e,g_{2}h_{2})+a^{\prime}=\beta^{\Theta}_{(g_{2}h_{2})^{+}}(e,g_{2}h_{2})+a

and using the hypothesis $\psi\bigl{(}\beta^{\Theta}_{(g_{1}h_{1})^{+}}(e,g_{1})\bigr{)}=\psi\bigl{(}\beta^{\Theta}_{(g_{2}h_{2})^{+}}(e,g_{2})\bigr{)}$ , we conclude that $\psi\bigl{(}\beta^{\Theta}_{(g_{1}h_{1})^{+}}(e,g_{1}h_{1})\bigr{)}=\psi\bigl{(}\beta^{\Theta}_{(g_{2}h_{2})^{+}}(e,g_{2}h_{2})\bigr{)}$ . ∎

Hence, we obtain (weak/strong) (stable/unstable) foliations on $\mathcal{X}$ as follows: for all $z=\Gamma(x,y,s)\in\mathcal{X}$ , we have

\displaystyle\begin{aligned} W^{\mathrm{su}}(z)&=\bigl{\{}\Gamma\bigl{(}(gh)^{+},g^{-},s+\psi\bigl{(}[(gh)^{+},g^{-}]_{\Theta}-[g^{+},g^{-}]_{\Theta}\bigr{)}\bigr{)}:h\in N_{\Theta}^{+},(gh)^{+}\in\Lambda_{\Theta}\bigr{\}},\\ W^{\mathrm{ss}}(z)&=\bigl{\{}\Gamma\bigl{(}g^{+},(gn)^{-},s\bigr{)}:n\in N_{\Theta}^{-},(gn)^{-}\in\Lambda_{\mathsf{i}\Theta}\bigr{\}},\\ W^{\mathrm{wu}}(z)&=\bigcup_{t\in\mathbb{R}}a_{t}W^{\mathrm{ss}},\\ W^{\mathrm{ws}}(z)&=\bigcup_{t\in\mathbb{R}}a_{t}W^{\mathrm{ss}}\end{aligned}

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for any choice of $g\in G$ such that $z=\bigl{(}g^{+},g^{-},\psi\bigl{(}\beta^{\Theta}_{g^{+}}(e,g)\bigr{)}\bigr{)}$ . It is easy to see that $W^{\mathrm{ss}}$ and $W^{\mathrm{su}}$ are transverse to the translation flow, and $(W^{\mathrm{ss}},W^{\mathrm{wu}})$ and $(W^{\mathrm{su}},W^{\mathrm{ws}})$ have local product structures.

3.3. Projective Anosov representations

In this subsection, we recall from [9] facts about projective Anosov representations that we will need for the proof of Theorem 3.2. Let $\mathfrak{p}_{\Theta}$ denote the Lie algebra of $P_{\Theta}$ . Then the adjoint representation $\operatorname{Ad}:G\to\operatorname{Aut}(\mathfrak{g})$ induces the representation $\rho_{0}:G\to\operatorname{PSL}\bigl{(}\bigwedge^{\dim\mathfrak{p}_{\Theta}}\mathfrak{g}\bigr{)}$ . Define the vector space $V:=\rho_{0}(G)\bigl{(}\bigwedge^{\dim\mathfrak{p}_{\Theta}}\mathfrak{p}_{\Theta}\bigr{)}$ (one can see that $V$ is a vector space using the Iwasawa decomposition $G=KAN$ and the fact that $\langle\cdot,\cdot\rangle$ is $\operatorname{Ad}_{K}$ -invariant). Then, we have the induced irreducible representation $\rho:G\to\operatorname{PSL}(V)$ , called the Plücker representation, and it induces diffeomorphisms $\mathcal{F}_{\Theta}\to\mathbb{P}(V)$ and $\mathcal{F}_{\mathsf{i}\Theta}\to\mathbb{P}(V^{*})$ [38, Theorem 7.25]. By [25, Proposition 4.3]), the restriction $\rho|_{\Gamma}:\Gamma\to\operatorname{PSL}(V)$ is a projective Anosov representation in the sense of [9, Definition 2.2] and in particular, the previous diffeomorphisms restrict to $\Gamma$ -equivariant bi-Lipschitz maps $\zeta_{\rho}:\Lambda_{\Theta}\to\mathbb{P}(V)$ and $\zeta_{\rho}^{*}:\Lambda_{\mathsf{i}\Theta}\to\mathbb{P}(V^{*})$ .

The $\mathbb{R}$ -bundle

\displaystyle\tilde{\mathcal{G}}_{\rho}:=\bigl{\{}(x,y,(v,\Psi)):(x,y)\in\Lambda_{\Theta}^{(2)},(v,\Psi)\in\zeta_{\rho}(x)\times\zeta_{\rho}^{*}(y),\Psi(v)=1\bigr{\}}/{\sim}

over $\Lambda_{\Theta}^{(2)}$ , where $(v,\Psi)\sim(-v,-\Psi)$ is equipped with a $\Gamma$ -action and a flow $\{\tilde{a}_{\rho,t}\}_{t\in\mathbb{R}}$ that commute with each other which are given as follows: for all $(x,y,(v,\Psi))\in\tilde{\mathcal{G}}_{\rho}$ , $\gamma\in\Gamma$ , and $t\in\mathbb{R}$ , let

	$\displaystyle\gamma(x,y,(v,\Psi))$	$\displaystyle:=(\gamma x,\gamma y,(\rho(\gamma)v,\Psi\circ\rho(\gamma)^{-1}));$
	$\displaystyle\tilde{a}_{\rho,t}(x,y,(v,\Psi))$	$\displaystyle:=(x,y,(e^{t}v,e^{-t}\Psi)).$

Let $\mathcal{G}_{\rho}:=\Gamma\backslash\tilde{\mathcal{G}}_{\rho}$ . Then $\{\tilde{a}_{\rho,t}\}_{t\in\mathbb{R}}$ descends to a flow $\{a_{\rho,t}\}_{t\in\mathbb{R}}$ on $\mathcal{G}_{\rho}$ called the $\rho$ -geodesic flow.

Any Euclidean metric on $V$ induces a metric on $\mathbb{P}(V)\times\mathbb{P}(V^{*})\times(V\times V^{*})$ and this further induces a metric on $\tilde{\mathcal{G}}_{\rho}$ in a natural way. Any metric on $\tilde{\mathcal{G}}_{\rho}$ obtained in this fashion is called a linear metric.

Theorem 3.12 (9, Proposition 5.7).

There exists a $\Gamma$ -invariant metric $\tilde{d}_{\rho}$ on $\tilde{\mathcal{G}}_{\rho}$ , which is bi-Lipschitz equivalent to any linear metric, such that it descends to a metric $d_{\rho}$ on $\mathcal{G}_{\rho}$ for which the flow $\{a_{\rho,t}\}_{t\in\mathbb{R}}$ is metric Anosov with respect to the pair of foliations $\bigl{(}W^{\mathrm{su}}_{\rho},W^{\mathrm{ss}}_{\rho}\bigr{)}$ which are defined as follows: for all $z=\Gamma(x,y,(v,\Psi))\in\mathcal{G}_{\rho}$ ,

	$\displaystyle W^{\mathrm{su}}_{\rho}(z)$	$\displaystyle:=\bigl{\{}\Gamma(x^{\prime},y,(v^{\prime},\Psi)):(x^{\prime},y)\in\Lambda_{\Theta}^{(2)},(v^{\prime},\Psi)\in\zeta_{\rho}(x^{\prime})\times\zeta_{\rho}^{*}(y),\Psi(v^{\prime})=1\bigr{\}};$
	$\displaystyle W^{\mathrm{ss}}_{\rho}(z)$	$\displaystyle:=\bigl{\{}\Gamma(x,y^{\prime},(v,\Psi^{\prime})):(x,y^{\prime})\in\Lambda_{\Theta}^{(2)},(v,\Psi^{\prime})\in\zeta_{\rho}(x)\times\zeta_{\rho}^{*}(y^{\prime}),\Psi^{\prime}(v)=1\bigr{\}}.$

3.4. Proof of Theorem 3.2

Theorem 3.13.

There exists a bi-Lipschitz homeomorphism $\mathcal{H}:(\mathcal{G}_{\rho},d_{\rho})\to(\mathcal{X},d)$ which conjugates the translation flow $\{a_{t}\}_{t\in\mathbb{R}}$ to a Lipschitz reparametrization $\{a_{\rho,\kappa(\cdot,t)}\}_{t\in\mathbb{R}}$ of the $\rho$ -geodesic flow $\{a_{\rho,t}\}_{t\in\mathbb{R}}$ .

Proof.

We proceed in the following two steps.

Step 1: Construction of $\mathcal{H}$ . We only give an overview (omitting details) of the construction of $\mathcal{H}$ as it is similar to the full construction in [18, Theorem 4.15] which deals with the case $\Theta=\Pi$ . Consider the trivial $\mathbb{R}$ -bundle

\displaystyle\tilde{\mathcal{I}}:=\tilde{\mathcal{G}}_{\rho}\times\mathbb{R}_{>0}

equipped with a left $\Gamma$ -action defined by

\displaystyle\gamma\cdot(z,r):=\bigl{(}\gamma z,re^{\psi(\beta^{\Theta}_{x}(\gamma^{-1},e))}\bigr{)}\qquad\text{ for all }\gamma\in\Gamma\text{ and }(z,r)\in\tilde{\mathcal{I}}

and a flow $\{\tilde{\phi}^{\mathcal{I}}_{t}\}_{t\in\mathbb{R}}$ defined by

\displaystyle\tilde{\phi}^{\mathcal{I}}_{t}(z,r):=(\tilde{a}_{\rho,t}z,r)\qquad\text{ for all }\gamma\in\Gamma,(z,r)\in\tilde{\mathcal{I}},\text{ and }t\in\mathbb{R}

which commutes with the $\Gamma$ -action.

Let $\mathcal{U}=\{\gamma\mathcal{U}_{j}\}_{\gamma\in\Gamma,\,1\leq j\leq j_{0}}$ for some $j_{0}\in\mathbb{N}$ be a locally finite open cover of $\tilde{\mathcal{G}}_{\rho}$ such that $\gamma\mathcal{U}_{j}\cap\gamma^{\prime}\mathcal{U}_{j}=\varnothing$ for all distinct $\gamma,\gamma^{\prime}\in\Gamma$ and $1\leq j\leq j_{0}$ . Using a partition of unity $\{\varphi_{i}\}_{i\in I}$ , for some index set $I$ , subordinate to $\mathcal{U}$ such that $\varphi_{i}:\Lambda_{\Theta}^{(2)}\times\mathbb{R}\to[0,+\infty)$ is smooth along the $\rho$ -geodesic flow for all $i\in I$ , we can construct a $\Gamma$ -equivariant section $u_{0}:\tilde{\mathcal{G}}_{\rho}\to\tilde{\mathcal{I}}$ of the form $u_{0}(z):=(z,\hat{u}_{0}(z))$ . Explicitly, we can take

\displaystyle\log\hat{u}_{0}(z)=\sum_{i\in I,\,\operatorname{supp}\varphi_{i}\subset\gamma\mathcal{U}_{j}}\psi\bigl{(}\beta^{\Theta}_{x}(e,\gamma)\bigr{)}\cdot\varphi_{i}(z)\qquad\text{ for all }z=(x,y,(v,\Psi))\in\tilde{\mathcal{G}}_{\rho}.

Let $\tilde{\mathcal{L}}:=\tilde{\mathcal{G}}_{\rho}\times\mathbb{R}_{>0}$ denote the image of $\tilde{\mathcal{I}}$ under the exponential map on the $\mathbb{R}$ -coordinate. Equip $\tilde{\mathcal{L}}$ with the $\Gamma$ -action and flow $\{\tilde{\phi}^{\mathcal{L}}_{t}\}_{t\in\mathbb{R}}$ it inherits from $\tilde{\mathcal{I}}$ and the unique $\Gamma$ -invariant bundle norm $\|\cdot\|^{\tilde{\mathcal{L}}}_{0}$ satisfying $\|u_{0}(z)\|_{0}^{\tilde{\mathcal{L}}}=1$ for all $z\in\tilde{\mathcal{G}}_{\rho}$ . Then, $\mathcal{L}:=\Gamma\backslash\tilde{\mathcal{I}}$ is an $\mathbb{R}_{>0}$ -bundle over $\mathcal{G}_{\rho}$ , $\{\tilde{\phi}^{\mathcal{L}}_{t}\}_{t\in\mathbb{R}}$ descends to a flow $\{\phi^{\mathcal{L}}_{t}\}_{t\in\mathbb{R}}$ on $\mathcal{L}$ , and $\|\cdot\|_{0}^{\tilde{\mathcal{L}}}$ descends to a norm $\|\cdot\|_{0}^{\mathcal{L}}$ on $\mathcal{L}$ . Using the Morse property [21, Theorem 4.13] of Kapovich–Leeb–Porti [27, Proposition 5.16] and an analogue of Sullivan’s Shadow Lemma [32, Lemma 3.1], one can show that $\{\phi^{\mathcal{L}}_{t}\}_{t\in\mathbb{R}}$ is discretely contracting with respect to $\|\cdot\|_{0}^{\mathcal{L}}$ , i.e., there exists $T>0$ and $\eta>0$ such that $\|\phi_{T}^{\mathcal{L}}(\ell)\|_{0}^{\mathcal{L}}\leq e^{-\eta}\|\ell\|_{0}^{\mathcal{L}}$ for all $\ell\in\mathcal{L}$ . By [9, Lemma 4.3], the norm $\|\cdot\|^{\tilde{\mathcal{L}}}=\int_{0}^{T}e^{s/T}\|\phi_{s}^{\mathcal{L}}(\cdot)\|_{0}^{\tilde{\mathcal{L}}}\,ds$ on $\tilde{\mathcal{L}}$ descends to a norm $\|\cdot\|^{\mathcal{L}}$ on $\mathcal{L}$ such that $\{\phi_{t}^{\mathcal{L}}\}_{t\in\mathbb{R}}$ is uniformly contracting with respect to $\|\cdot\|^{\mathcal{L}}$ . Note that $\|(z,\hat{u}(z))\|^{\tilde{\mathcal{L}}}=1$ where

\displaystyle\frac{1}{\hat{u}(z)}=\int_{0}^{T}\frac{e^{s/T}}{\hat{u}_{0}(\tilde{a}_{\rho,s}(z))}\,ds\qquad\text{ for all }z\in\tilde{\mathcal{G}}_{\rho}.

Let $\tilde{\mathcal{H}}:\tilde{\mathcal{G}}_{\rho}\to\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ be the $\Gamma$ -equivariant map defined by

\displaystyle\tilde{\mathcal{H}}(z)=(x,y,\log\hat{u}(z))\qquad\text{ for all }z=(x,y,(v,\Psi))\in\tilde{\mathcal{G}}_{\rho}.

Then, it can be shown as in [18, Theorem 4.15] that $\hat{u}$ is locally Lipschitz and $\tilde{\mathcal{H}}$ descends to a Lipschitz homeomorphism $\mathcal{H}:\mathcal{G}_{\rho}\to\mathcal{X}$ and the reparametrization is Lipschitz. (Here, $\mathcal{H}$ is Lipschitz instead of Hölder as in [18] since the induced maps $\mathcal{F}_{\Theta}\to\mathbb{P}(V)$ and $\mathcal{F}_{\mathsf{i}\Theta}\to\mathbb{P}(V^{*})$ are Lipschitz.)

Step 2: $\mathcal{H}$ is bi-Lipschitz. By the above, it suffices to show that $\tilde{\mathcal{H}}^{-1}$ is locally Lipschitz with respect to $\tilde{d}$ and $\tilde{d}_{\rho}$ . Recall that $\tilde{d}$ is a locally bi-Lipschitz equivalent to the product metric on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ and $\tilde{d}_{\rho}$ is locally bi-Lipschitz equivalent to any linear metric on $\tilde{\mathcal{G}}_{\rho}$ . Thus, it suffices to show that $\tilde{\mathcal{H}}^{-1}$ is locally Lipschitz with respect to the product metric on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ and any linear metric on $\tilde{\mathcal{G}}_{\rho}$ . Fix a Euclidean metric $d_{\mathrm{E}}$ on $V$ . We also use $d_{\mathrm{E}}$ to denote the induced metrics on $V^{*}$ and $V\times V^{*}$ . Fix a compact subset $\mathcal{K}\subset\tilde{\mathcal{G}}_{\rho}$ that is “radially symmetric” in the sense that if $(x_{i},y_{i},(v_{i},\Psi_{i}))\in\mathcal{K}$ for $i\in\{1,2\}$ , then $p(z_{1},z_{2}):=\Bigl{(}x_{1},y_{1},\Bigl{(}\tfrac{\|v_{2}\|}{\|v_{1}\|}v_{1},\tfrac{\|v_{1}\|}{\|v_{2}\|}\Psi_{1}\Bigr{)}\Bigr{)}\in\mathcal{K}$ as well. We need to show that there exists $c_{1}>0$ such that for any $z_{i}=(x_{i},y_{i},(v_{i},\Psi_{i}))\in\mathcal{K}$ for $i\in\{1,2\}$ , we have

\displaystyle d_{\mathrm{E}}((v_{1},\Psi_{1}),(v_{2},\Psi_{2}))\leq c_{1}((d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}})((x_{1},y_{1}),(x_{2},y_{2}))+|\log\hat{u}(z_{1})-\log\hat{u}(z_{2})|).

For convenience, let $t(z_{1},z_{2}):=\log(\|v_{2}\|/\|v_{1}\|)$ so that

\displaystyle p(z_{1},z_{2})=\bigl{(}x_{1},y_{1},\bigl{(}e^{t(z_{1},z_{2})}v_{1},e^{-t(z_{1},z_{2})}\Psi_{1}\bigr{)}\bigr{)}=\tilde{a}_{\rho,t(z_{1},z_{2})}(z_{1}).

Observe that since $\mathcal{K}$ is compact, there exists constants $c_{3}>c_{2}>0$ such that

\displaystyle d_{\mathrm{E}}\bigl{(}(v_{1},\Psi_{1}),\bigl{(}e^{t(z_{1},z_{2})}v_{1},e^{-t(z_{1},z_{2})}\Psi_{1}\bigr{)}\bigr{)}\leq c_{2}t(z_{1},z_{2})

and

	$\displaystyle d_{\mathrm{E}}\Bigl{(}\Bigl{(}\tfrac{\\|v_{2}\\|}{\\|v_{1}\\|}v_{1},\tfrac{\\|v_{1}\\|}{\\|v_{2}\\|}\Psi_{1}\Bigr{)},(v_{2},\Psi_{2})\Bigr{)}$	$\displaystyle\leq c_{2}d_{\mathcal{F}_{\Theta}}(x_{1},x_{2})+\tfrac{1}{\\|v_{2}\\|}d_{\mathrm{E}}\bigr{(}\\|v_{1}\\|\Psi_{1},\\|v_{2}\\|\Psi_{2}\bigl{)}$
		$\displaystyle\leq c_{3}(d_{\mathcal{F}_{\Theta}}(x_{1},x_{2})+d_{\mathcal{F}_{\mathsf{i}\Theta}}(y_{1},y_{2})),$

where the last inequality uses the observation that for $i\in\{1,2\}$ , the linear form $\|v_{i}\|\Psi_{i}\in\zeta_{\rho}^{*}(y_{i})$ which satisfies $\|v_{i}\|\Psi_{i}\bigl{(}\tfrac{v_{i}}{\|v_{i}\|}\bigr{)}=1$ is smoothly determined by $y_{i}$ and the unit vector $\tfrac{v_{i}}{\|v_{i}\|}$ (or equivalently, $x_{i}$ ). The triangle inequality gives

\displaystyle d_{\mathrm{E}}((v_{1},\Psi_{1}),(v_{2},\Psi_{2}))\leq c_{3}((d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}})((x_{1},y_{1}),(x_{2},y_{2}))+|t(z_{1},z_{2})|).

Hence, it suffices to show that there exists a constant $c_{4}>0$ such that for all $z_{1},z_{2}\in\mathcal{K}$ , we have

\displaystyle|t(z_{1},z_{2})|\leq c_{4}((d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}})((x_{1},y_{1}),(x_{2},y_{2}))+|\log\hat{u}(z_{1})-\log\hat{u}(z_{2})|).

Since $\{\phi_{t}^{\mathcal{L}}\}_{t\in\mathbb{R}}$ is uniformly contracting with respect to $\|\cdot\|^{\mathcal{L}}$ , there exists $\eta>0$ such that

\displaystyle\frac{1}{\hat{u}(\tilde{a}_{\rho,t}(z))}\leq\frac{e^{-\eta t}}{\hat{u}(z)}\qquad\text{ for all }t>0\text{ and }z\in\tilde{\mathcal{G}}_{\rho},

i.e., $\eta t\leq\log\hat{u}(\tilde{a}_{\rho,t}(z_{1}))-\log\hat{u}(z_{1})$ . Then, for all $z_{1},z_{2}\in\mathcal{K}$ , we have

		$\displaystyle\|\log\hat{u}(z_{1})-\log\hat{u}(z_{2})\|$
	$\displaystyle\geq$	$\displaystyle\|\log\hat{u}(z_{1})-\log\hat{u}(\tilde{a}_{\rho,t(z_{1},z_{2})}(z_{1}))\|-\|\log\hat{u}(p_{1}(z_{1},z_{2}))-\log\hat{u}(z_{2})\|$
	$\displaystyle\geq$	$\displaystyle\eta\|t(z_{1},z_{2})\|-\|\log\hat{u}(p_{1}(z_{1},z_{2}))-\log\hat{u}(z_{2})\|.$

Lastly, since $\hat{u}$ is locally Lipschitz with respect to any linear metric, there exists $c_{5}>0$ such that $|\log\hat{u}(p_{1}(z_{1},z_{2}))-\log\hat{u}(z_{2})|\leq c_{5}(d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}})((x_{1},y_{1}),(x_{2},y_{2}))$ for all $z_{1},z_{2}\in\mathcal{K}$ and this completes the proof. ∎

The following is an immediate consequence of Theorems 3.12 and 3.13.

Proposition 3.14.

The reparametrization $\{a_{\kappa^{*}(\mathcal{H}^{-1}(\cdot),t)}\}_{t\in\mathbb{R}}$ of $\{a_{t}\}_{t\in\mathbb{R}}$ is a metric Anosov flow with respect to $\bigl{(}\mathcal{H}(W^{\mathrm{su}}_{\rho}),\mathcal{H}(W^{\mathrm{ss}}_{\rho})\bigr{)}$ .

Proof of Theorem 3.2.

In view of Proposition 3.14, by considering $\{a_{t}\}_{t\in\mathbb{R}}$ as a reparametrization of $\{a_{\kappa^{*}(\mathcal{H}^{-1}(\cdot),t)}\}_{t\in\mathbb{R}}$ , it suffices to check Properties (1) and (2) in Proposition 3.10.

Proof of Property (1). Fix a compact fundamental domain $D\subset\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ for the $\Gamma$ -action. Fix $r>0$ sufficiently small so that the closed $r$ -balls in $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ injectively project into $\mathcal{X}$ and that the projection $D^{\prime}\subset\Lambda_{\Theta}^{(2)}$ of the closed $r$ -neighborhood of $D$ is compact. Since $D$ is compact and $\tilde{d}$ is locally bi-Lipschitz equivalent to the product metric, there exists $c_{1}>1$ such that for all $z\in D$ and $z^{\prime}\in\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ with $\tilde{d}(z,z^{\prime})<r$ , we have

c_{1}^{-1}(d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}}\times d_{\mathbb{R}})(z,z^{\prime})\leq d(\Gamma z,\Gamma z^{\prime})=\tilde{d}(z,z^{\prime})\leq c_{1}(d_{\mathcal{F}_{\Theta}}\times d_{\mathcal{F}_{\mathsf{i}\Theta}}\times d_{\mathbb{R}})(z,z^{\prime}).

(10)

Let $\epsilon\in(0,r)$ which will be specified later. Fix $z=(x,y,t)\in D$ , $z_{1}=(x^{\prime},y,t_{1})\in\tilde{W}_{\epsilon}^{\mathrm{su}}(z)$ , and $z_{2}=(x^{\prime},y,t_{2})\in\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ such that $\Gamma z_{2}\in a_{\mathbb{R}}(\Gamma z_{1})\cap\mathcal{H}(W^{\mathrm{su}}_{\rho}(\mathcal{H}^{-1}(\Gamma z))$ . By compactness of $D$ , we can uniformly choose $\epsilon$ sufficiently small so that $\tilde{d}(z,z_{2})<r$ . We want to show that there exists a uniform constant $c>1$ such that

c^{-1}d(\Gamma z,\Gamma z_{2})\leq d(\Gamma z,\Gamma z_{1})\leq cd(\Gamma z,\Gamma z_{2}).

(11)

By (10), for each $j\in\{1,2\}$ , we have

\displaystyle c_{1}^{-1}(d_{\mathcal{F}_{\Theta}}(x,x^{\prime})+|t-t_{i}|)\leq\tilde{d}(z,z_{i})\leq c_{1}(d_{\mathcal{F}_{\Theta}}(x,x^{\prime})+|t-t_{i}|).

By (9), we have $|t-t_{1}|=|\psi([x,y]_{\Theta}-[x^{\prime},y]_{\Theta})|\leq c_{2}d_{\mathcal{F}_{\Theta}}(x,x^{\prime})$ for a uniform constant $c_{2}>0$ by smoothness of the $\Theta$ -Gromov product restricted to $D^{\prime}$ . Then

\displaystyle d(\Gamma z,\Gamma z_{1})\leq c_{1}(d_{\mathcal{F}_{\Theta}}(x,x^{\prime})+|t-t_{1}|)\leq c_{1}(1+c_{2})d_{\mathcal{F}_{\Theta}}(x,x^{\prime})\leq c_{1}^{2}(1+c_{2})d(\Gamma z,\Gamma z_{2}).

This establishes the right hand inequality of (11).

For the left hand inequality of (11), we observe that compactness of $D$ and the Lipschitz properties of the maps $\hat{u}$ and $\mathcal{H}$ in the proof of Theorem 3.13 implies that there exists a uniform constant $c_{3}>0$ such that

\displaystyle|t-t_{2}|=|\log\hat{u}(\mathcal{H}^{-1})(z)-\log\hat{u}(\mathcal{H}^{-1})(z_{2})|\leq c_{3}\tilde{d}(z,z_{2}).

This establishes Property (1) for the strong unstable foliations. The argument for the strong stable foliations is similar.

Proof of Property (2). Property (2) is immediate from the fact that $\kappa(\cdot,t)=\int_{0}^{t}f(a_{\rho,s}(\cdot))\,ds$ for all $t\in\mathbb{R}$ for some positive continuous function $f:\mathcal{G}_{\rho}\to\mathbb{R}$ (see [18, Remark 4.11]). ∎

Remark 3.15.

Although we have only shown that the distinguished translation flow $\{a_{t}\}_{t\in\mathbb{R}}$ on $\mathcal{X}$ is metric Anosov, it is not difficult to see that the above work can be adapted to show that any translation flow $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ with $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ on $\mathcal{X}$ is a Lipschitz reparametrization of $\{a_{t}\}_{t\in\mathbb{R}}$ and also metric Anosov. Indeed, the proof of Theorems 3.13 and 3.14 is easily modified to show that for all $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ there exists a bi-Lipschitz homeomorphism $\mathcal{H}^{\mathsf{v}_{0},\mathsf{v}}:\mathcal{X}\to\mathcal{X}_{\mathsf{v}}$ conjugating $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ to a Lipschitz reparametrization of $\{a_{t}\}_{t\in\mathbb{R}}$ and that $\{a^{\mathsf{v}}_{t}\}_{t\in\mathbb{R}}$ is metric Anosov. In fact, such a homeomorphism $\mathcal{H}^{\mathsf{v}_{0},\mathsf{v}}$ can be explicitly described as follows. Let $\pi_{\ker\psi}:\mathfrak{a}_{\Theta}\to\ker\psi$ be the projection map determined by the decomposition $\mathfrak{a}_{\Theta}=\mathbb{R}\mathsf{v}_{0}\oplus\ker\psi$ , $\mathcal{U}=\{\gamma\mathcal{U}_{j}\}_{\gamma\in\Gamma,\,1\leq j\leq j_{0}}$ be a locally finite open cover of $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ such that $\gamma\mathcal{U}_{j}\cap\gamma^{\prime}\mathcal{U}_{j}=\varnothing$ for all distinct $\gamma,\gamma^{\prime}\in\Gamma$ and $1\leq j\leq j_{0}$ and $\{\varphi_{i}\}_{i\in I}$ for some index set $I$ be a partition of unity subordinate to $\mathcal{U}$ such that $\varphi_{i}:\Lambda_{\Theta}^{(2)}\times\mathbb{R}\to[0,+\infty)$ is smooth along the $\rho$ -geodesic flow for all $i\in I$ and $\{\varphi_{i}\}_{i\in I}$ be a partition of unity subordinate to the cover $\{\gamma U_{j}\}_{\gamma\in\Gamma,1\leq i\leq j_{0}}$ . Define the map $\tilde{\mathcal{H}}^{\mathsf{v}_{0},\mathsf{v}}:\Lambda_{\Theta}^{(2)}\times\mathbb{R}\to\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ by $\tilde{\mathcal{H}}^{\mathsf{v}_{0},\mathsf{v}}(x,y,r)=(x,y,r\psi_{\mathsf{v}}({\mathsf{v}}_{0})+\log\hat{u}(x,y,r))$ where $\hat{u}$ is given by

\displaystyle\frac{1}{\hat{u}(z)}=\int_{0}^{T}\frac{e^{s/T}}{\exp\left(\sum_{i\in I,\,\operatorname{supp}\varphi_{i}\subset\gamma U_{j}}(\psi_{\mathsf{v}}\circ\pi_{\ker\psi})(\beta^{\Theta}_{x}(e,\gamma))\cdot\varphi_{i}(x,y,r+t)\right)}\,ds

for all $z=(x,y,r)\in\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ , for some $T>0$ . Then, it descends to the desired map $\mathcal{H}^{\mathsf{v}_{0},\mathsf{v}}:\mathcal{X}\to\mathcal{X}_{\sf{v}}$ .

4. Transfer operators, Dolgopyat’s method, and the proof of Theorem 1.5 and applications

In this section, we cover the necessary background for transfer operators in order to state Theorem 4.7 which is the main technical theorem regarding spectral bounds. We also outline how to derive the main theorem stated in Theorem 1.5 from Theorem 4.7. A key point in this section is that all the theorems include the precise dependence on a parameter associated to $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ .

4.1. Reduction of Theorem 1.5 by rescaling

Due to the homothety equivariance property for the family of translation flows (see Eq. 7), it is convenient to fix a scaling for each direction in $\operatorname{int}\mathcal{L}_{\Theta}$ for the purpose of proving Theorem 1.5. It turns out that there is a particular scaling so that we have additional properties. Using the isomorphism $\mathfrak{a}_{\Theta}\cong\mathfrak{a}_{\Theta}^{*}$ induced by the inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak{a}_{\Theta}$ , we abuse notation and identify the dual limit cone $\mathcal{L}_{\Theta}^{*}\subset\mathfrak{a}_{\Theta}^{*}$ with a dual limit cone $\mathcal{L}_{\Theta}^{*}\subset\mathfrak{a}_{\Theta}$ (see Eq. 1). Note that $\mathcal{L}_{\Theta}\subset\operatorname{int}\mathfrak{a}_{\Theta}^{+}\cup\{0\}$ (see Theorem 2.11 (2)) and Proposition B.1 implies that $\partial\mathfrak{a}_{\Theta}^{+}\subset\operatorname{int}\mathcal{L}_{\Theta}^{*}\cup\{0\}$ . For all $\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\|\mathsf{w}\|=1$ , we define $\mathsf{v}(\mathsf{w})\in\operatorname{int}\mathcal{L}_{\Theta}$ to be the unique vector such that $\nabla\psi_{\mathsf{v}(\mathsf{w})}=\mathsf{w}$ . Then, Eq. 6 gives

\displaystyle\frac{\|\nabla\psi_{\Theta}(\mathsf{v}(\mathsf{w}))\|}{\psi_{\Theta}(\mathsf{v}(\mathsf{w}))}=1.

(12)

Therefore, Theorem 1.5 follows from the following exponential mixing theorem. Theorem 1.4 also follows either directly from the following or via Theorem 1.5. Note that one can repeat a standard convolution argument as in [34, Appendix] using Lipschitz continuous bump functions to handle general $\alpha$ -Hölder functions (see also [42, Corollary 5.2] and [62, Theorem 3.1.4]).

Theorem 4.1.

There exists a bounded elementary function $E_{\Theta}:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ with vanishing limit at $+\infty$ such that for all open neighborhoods $\mathcal{N}\supset\partial\mathfrak{a}_{\Theta}^{+}$ and $\mathsf{v}\in\operatorname{int}\mathcal{L}_{\Theta}$ with $\|\nabla\psi_{\mathsf{v}}\|=1$ and $\nabla\psi_{\mathsf{v}}\notin\mathcal{N}$ , there exist $C_{\mathsf{v}}>0$ , $k_{\mathsf{v}}\in\mathbb{N}$ , a finite set $\{\lambda_{k}(\mathsf{v})\}_{k=1}^{k_{\mathsf{v}}}\subset(-1,0)\times i(-1,1)$ which come in conjugate pairs, and a finite set of finite-rank positive bilinear forms $\{\mathcal{B}_{\mathsf{v},k}\}_{k=1}^{k_{\mathsf{v}}}$ , such that for all $\phi_{1},\phi_{2}\in L(\mathcal{X})$ and $t>0$ , we have

\left|\int_{\mathcal{X}}\phi_{1}(a^{\mathsf{v}}_{t}x)\phi_{2}(x)\,dm_{\mathcal{X}}^{\mathsf{v}}(x)-\left(m_{\mathcal{X}}^{\mathsf{v}}(\phi_{1})m_{\mathcal{X}}^{\mathsf{v}}(\phi_{2})+\sum_{k=1}^{k_{\mathsf{v}}}e^{\lambda_{k}(\mathsf{v})t}\mathcal{B}_{\mathsf{v},k}(\phi_{1},\phi_{2})\right)\right|\\ \leq Ce^{-E_{\Theta}(\|\nabla\psi_{\Theta}(\mathsf{v})\|)t}\|\phi_{1}\|_{\operatorname{Lip}}\|\phi_{2}\|_{\operatorname{Lip}}.

The rest of the paper is devoted to the proof of the above theorem. To this end, we fix an open neighborhood $\mathcal{N}\supset\partial\mathfrak{a}_{\Theta}^{+}$ henceforth and define the subset of unit vectors

\displaystyle\mathscr{w}=\{\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}:\|\mathsf{w}\|=1,\mathsf{w}\notin\mathcal{N}\}.

Note that we have a corresponding compact subset of unit vectors

\displaystyle\overline{\mathscr{w}}=\{\mathsf{w}\in\mathcal{L}_{\Theta}:\|\mathsf{w}\|=1,\mathsf{w}\notin\mathcal{N}\}.

4.2. Compatible family of Markov sections

Corollary 3.3 gives a Markov section for any fixed translation flow. Recalling that we have a family of translation flows $\bigl{\{}\bigl{\{}a^{\mathsf{v}(\mathsf{w})}_{t}\bigr{\}}_{t\in\mathbb{R}}\bigr{\}}_{\mathsf{w}\in\mathscr{w}}$ , we fix a corresponding family of Markov sections $\{\mathcal{R}^{\mathsf{w}}\}_{\mathsf{w}\in\mathscr{w}}$ on $\mathcal{X}$ . They have several other corresponding objects (see Subsection 3.1) all of which we denote using a superscript ‘ $\mathsf{w}$ ’ on the same notations. By the same observation as in [18, §4.2] regarding Markov sections of reparametrized flows, we deduce the remarkable property that the family of Markov sections can be chosen such that for different unit vectors $\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}$ , the Markov sections $\mathcal{R}^{\mathsf{w}}$ and $\mathcal{R}^{\mathsf{w}^{\prime}}$ consist of corresponding rectangles which project to each other along the translation flow foliation while fixing their centers. Consequently, for some fixed $N\in\mathbb{N}$ , we denote

\displaystyle\mathcal{R}^{\mathsf{w}}=\{R^{\mathsf{w}}_{1},R^{\mathsf{w}}_{2},\dotsc,R^{\mathsf{w}}_{N}\}=\{[U^{\mathsf{w}}_{1},S^{\mathsf{w}}_{1}],[U^{\mathsf{w}}_{2},S^{\mathsf{w}}_{2}],\dotsc,[U^{\mathsf{w}}_{N},S^{\mathsf{w}}_{N}]\}\qquad\text{for all $\mathsf{w}\in\mathscr{w}$}

and the family $\{\mathcal{R}^{\mathsf{w}}\}_{\mathsf{w}\in\mathscr{w}}$ comes equipped with a family

\displaystyle\{\Psi^{\mathsf{w},\mathsf{w}^{\prime}}:R^{\mathsf{w}}\to R^{\mathsf{w}^{\prime}}\}_{\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}}

of compatibility maps which are Lipschitz homeomorphisms and vary smoothly in $\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}$ . More precisely, there exists a family $\{t^{\mathsf{w},\mathsf{w}^{\prime}}:R^{\mathsf{w}}\to\mathbb{R}\}_{\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}}$ of Lipschitz continuous functions which vary smoothly in $\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}$ such that $t^{\mathsf{w},\mathsf{w}^{\prime}}(w_{j})=0$ at the centers $w_{j}\in R^{\mathsf{w}}_{j}$ for all $1\leq j\leq N$ and

\displaystyle\Phi^{\mathsf{w},\mathsf{w}^{\prime}}(u)=a^{\mathsf{v}(\mathsf{w})}_{t^{\mathsf{w},\mathsf{w}^{\prime}}(u)}u\in R^{\mathsf{w}^{\prime}}\qquad\text{for all $u\in R^{\mathsf{w}}$ and $\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}$}.

All the properties are clear save injectivity which we now justify. For all $\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}$ , the map $\Psi^{\mathsf{w},\mathsf{w}^{\prime}}$ moves the points in $R^{\mathsf{w}}$ along the translation flow foliation to which the strong stable and unstable foliations corresponding to both $\mathsf{w},\mathsf{w}^{\prime}\in\mathscr{w}$ are transverse and hence preserves the time ordering along the translation flow foliation. Now, if $\Psi^{\mathsf{w},\mathsf{w}^{\prime}}(u_{1})=\Psi^{\mathsf{w},\mathsf{w}^{\prime}}(u_{2})$ for some distinct $u_{1},u_{2}\in R^{\mathsf{w}}$ , then the time ordering would be violated since $u_{1}$ and $u_{2}$ must lie on the same translation flow orbit; implying injectivity.

We now use the compatibility maps $\Psi^{\mathsf{w}_{0},\mathsf{w}}$ to identify the Markov sections $\mathcal{R}^{\mathsf{w}}$ for all $\mathsf{w}\in\mathscr{w}$ with a fixed Markov section $\mathcal{R}:=\mathcal{R}^{\mathsf{w}_{0}}$ of some size $\hat{\delta}\in(0,\epsilon_{0})$ , where $\mathsf{w}_{0}$ corresponds to $\mathsf{v}_{0}$ , i.e., $\mathsf{v}(\mathsf{w}_{0})\in\mathbb{R}_{>0}\mathsf{v}_{0}$ . We need to address a subtle point. For any $\mathsf{w}\in\mathscr{w}$ outside of some open neighborhood of $\mathsf{w}_{0}\in\mathscr{w}$ , the corresponding Markov sections need not be of size $\hat{\delta}$ . However, apart from that, they are legitimate Markov sections and come from the local product structures on the image $\Psi^{\mathsf{w}_{0},\mathsf{w}}\bigl{(}\bigsqcup_{j\in\mathcal{A}}[W^{\mathrm{su}}_{\epsilon_{0}}(w_{j}),W^{\mathrm{ss}}_{\epsilon_{0}}(w_{j})]\bigr{)}$ , where $w_{j}\in R_{j}^{\mathsf{w}_{0}}$ are the centers for all $j\in\mathcal{A}$ . Thus, we can drop all superscripts henceforth, except for $\{\tau^{\mathsf{w}}\}_{\mathsf{w}\in\mathscr{w}}$ which we view as a family of first return time maps on the same set $U$ ; since they still vary in $\mathsf{w}$ , they need to be distinguished. A useful property is that they remain positive, again due to preservation of time ordering. We record this and other useful properties in the following lemma (recall Eq. 8).

Lemma 4.2.

The family $\{\tau^{\mathsf{w}}\}_{\mathsf{w}\in\mathscr{w}}$ is smooth in $\mathsf{w}\in\mathscr{w}$ and there exists $C_{\tau}>0$ such that

((1))

$0<\tau^{\mathsf{w}}(u)\leq\frac{C_{\tau}}{2\overline{\tau^{\mathsf{w}_{0}}}}\tau^{\mathsf{w}_{0}}(u)\leq\frac{C_{\tau}}{2}$ for all $u\in U$ and $\mathsf{w}\in\mathscr{w}$ ,
((2))

$\max_{(j,k)}\operatorname{Lip}\bigl{(}\tau^{\mathsf{w}}|_{\mathtt{C}[j,k]}\bigr{)}\leq\frac{C_{\tau}}{2}$ for all $\mathsf{w}\in\mathscr{w}$ .

Consequently, $\bigl{\{}\max_{(j,k)}\|\tau^{\mathsf{w}}|_{\mathtt{C}[j,k]}\|_{\operatorname{Lip}}\bigr{\}}_{\mathsf{w}\in\mathscr{w}}$ is bounded above by $C_{\tau}$ .

Proof.

Positivity of $\tau^{\mathsf{w}}$ for all $\mathsf{w}\in\mathscr{w}$ , as mentioned above, is clear and so we prove the other properties. We use notations from Subsection 5.1 and terminology from Subsection 4.3 to shorten the proof. Namely, we use the first return vector map $\mathsf{K}:U\to\mathfrak{a}_{\Theta}$ descended from the one in Subsection 5.1 which is Lipschitz continuous on cylinders of length $1$ . Then, we have the identity $\tau^{\mathsf{w}}=\psi_{\mathsf{v}(\mathsf{w})}\circ\mathsf{K}$ for all $\mathsf{w}\in\mathscr{w}$ . Thus, the lemma follows from this and the property that $\|\nabla\psi_{\mathsf{v}(\mathsf{w})}\|=1$ . ∎

Due to the above lemma, we conclude that the smooth family $\{\tau^{\mathsf{w}}\}_{\mathsf{w}\in\mathscr{w}}$ can be extended to a smooth family of uniformly essentially Lipschitz (i.e., with a uniform Lipschitz constant on cylinders of length $1$ ) nonnegative first return time maps $\{\tau^{\mathsf{w}}\}_{\mathsf{w}\in\overline{\mathscr{w}}}$ .

Using a compatible family of Markov sections as constructed in this subsection is necessary to execute Dolgopyat’s method in a uniform fashion for the family of translation flows. Although we cannot make sense of a translation flow on $\mathcal{X}$ or even transfer operators associated to any $\mathsf{v}\in\partial\mathcal{L}_{\Theta}$ , the above uniform bounds and the extended family of first return time maps is used to obtain other Lipschitz bounds and a uniform version of LNIC to derive the precise form of the exponential decay rate in Theorem 4.7 for the family of transfer operators in $\mathsf{w}\in\mathscr{w}$ .

4.3. Symbolic dynamics

Let $\mathcal{A}=\{1,2,\dotsc,N\}$ be the alphabet of $\mathcal{R}$ . Define the $N\times N$ transition matrix $T$ by

\displaystyle T_{j,k}=\begin{cases}1,&\operatorname{int}(R_{j})\cap\mathcal{P}^{-1}(\operatorname{int}(R_{k}))\neq\varnothing\\ 0,&\text{otherwise}\end{cases}\qquad\text{for all $1\leq j,k\leq N$}.

The transition matrix $T$ is topologically mixing as a consequence of [18, Theorem 8.1], i.e., there exists $N_{T}\in\mathbb{N}$ such that $T^{N_{T}}$ consists only of positive entries. This was assumed in [18, §4.3] as well which is a minor inaccuracy; see Appendix C for the correction. Define the spaces of bi-infinite and infinite admissible sequences

	$\displaystyle\Sigma$	$\displaystyle=\{(\dotsc,x_{-1},x_{0},x_{1},\dotsc)\in\mathcal{A}^{\mathbb{Z}}:T_{x_{j},x_{j+1}}=1\text{ for all }j\in\mathbb{Z}\};$
	$\displaystyle\Sigma^{+}$	$\displaystyle=\{(x_{0},x_{1},\dotsc)\in\mathcal{A}^{\mathbb{Z}_{\geq 0}}:T_{x_{j},x_{j+1}}=1\text{ for all }j\in\mathbb{Z}_{\geq 0}\}$

respectively. We will use the term admissible sequences for finite sequences as well in the natural way. For any fixed $\beta_{0}\in(0,1)$ , we endow $\Sigma$ with the metric $d$ defined by $d(x,y)=\beta_{0}^{\inf\{|j|\in\mathbb{Z}_{\geq 0}:x_{j}\neq y_{j}\}}$ for all $x,y\in\Sigma$ . We similarly endow $\Sigma^{+}$ with a metric which we also denote by $d$ .

Definition 4.3 (Cylinder).

For all admissible sequences $x=(x_{0},x_{1},\dotsc,x_{k})$ of length $\operatorname{len}(x)=k\in\mathbb{Z}_{\geq 0}$ , we define the cylinder of length $\operatorname{len}(\mathtt{C}[x])=k$ to be

\displaystyle\mathtt{C}[x]

\displaystyle=\{u\in U:\sigma^{j}(u)\in\operatorname{int}(U_{x_{j}})\text{ for all }0\leq j\leq k\}.

We denote cylinders simply by $\mathtt{C}$ (or other typewriter style letters) when we do not need to specify the admissible sequence.

By a slight abuse of notation, let $\sigma$ also denote the shift map on $\Sigma$ or $\Sigma^{+}$ . There exist natural continuous surjections $\eta:\Sigma\to R$ and $\eta^{+}:\Sigma^{+}\to U$ defined by $\eta(x)=\bigcap_{j=-\infty}^{\infty}\overline{\mathcal{P}^{-j}(\operatorname{int}(R_{x_{j}}))}$ for all $x\in\Sigma$ and $\eta^{+}(x)=\bigcap_{j=0}^{\infty}\overline{\sigma^{-j}(\operatorname{int}(U_{x_{j}}))}$ for all $x\in\Sigma^{+}$ . Define $\hat{\Sigma}=\eta^{-1}(\hat{R})$ and $\hat{\Sigma}^{+}=(\eta^{+})^{-1}(\hat{U})$ . Then the restrictions $\eta|_{\hat{\Sigma}}:\hat{\Sigma}\to\hat{R}$ and $\eta^{+}|_{\hat{\Sigma}^{+}}:\hat{\Sigma}^{+}\to\hat{U}$ are bijective and satisfy $\eta|_{\hat{\Sigma}}\circ\sigma|_{\hat{\Sigma}}=\mathcal{P}|_{\hat{R}}\circ\eta|_{\hat{\Sigma}}$ and $\eta^{+}|_{\hat{\Sigma}^{+}}\circ\sigma|_{\hat{\Sigma}^{+}}=\sigma|_{\hat{U}}\circ\eta^{+}|_{\hat{\Sigma}^{+}}$ .

We can take $\beta_{0}\in(0,1)$ sufficiently close to $1$ so that $\eta$ and $\eta^{+}$ are Lipschitz continuous [6, Lemma 2.2]. Let $L(\Sigma,\mathbb{R})$ denote the space of Lipschitz continuous functions $f:\Sigma\to\mathbb{R}$ . We use similar notations for Lipschitz function spaces with domain $\Sigma^{+}$ and other codomains. For all function spaces, we suppress the codomain when it is $\mathbb{R}$ .

Let $\mathsf{w}\in\overline{\mathscr{w}}$ . Since the horospherical foliations on $G$ are smooth, we conclude that $\tau^{\mathsf{w}}$ is Lipschitz continuous on cylinders of length $1$ . Then the maps $(\tau^{\mathsf{w}}\circ\eta)|_{\hat{\Sigma}}$ and $(\tau^{\mathsf{w}}\circ\eta^{+})|_{\hat{\Sigma}^{+}}$ are Lipschitz continuous and hence there exist unique Lipschitz extensions which, by abuse of notation, we also denote by $\tau^{\mathsf{w}}:\Sigma\to\mathbb{R}$ and $\tau^{\mathsf{w}}:\Sigma^{+}\to\mathbb{R}$ , respectively.

4.4. Thermodynamics

Definition 4.4 (Pressure).

For all $f\in L(\Sigma)$ , called the potential, the pressure is defined by

\displaystyle\operatorname{Pr}_{\sigma}(f)=\sup_{\nu\in\mathcal{M}^{1}_{\sigma}(\Sigma)}\left\{\int_{\Sigma}f\,d\nu+h_{\nu}(\sigma)\right\}

where $\mathcal{M}^{1}_{\sigma}(\Sigma)$ is the set of $\sigma$ -invariant Borel probability measures on $\Sigma$ and $h_{\nu}(\sigma)$ is the measure theoretic entropy of $\sigma$ with respect to $\nu$ .

For all $f\in L(\Sigma)$ , there exists a unique $\sigma$ -invariant Borel probability measure $\nu_{f}$ on $\Sigma$ which attains the supremum in Definition 4.4, called the $f$ -equilibrium state [7, Theorems 2.17 and 2.20]. It satisfies $\nu_{f}(\hat{\Sigma})=1$ [15, Corollary 3.2].

Let $\mathsf{w}\in\mathscr{w}$ . Associated to $\psi_{\mathsf{v}(\mathsf{w})}$ , we relabel $\delta_{\mathsf{w}}:=\delta_{\mathsf{v}(\mathsf{w})}=\delta_{\psi_{\mathsf{v}(\mathsf{w})}}$ (see Eq. 2) and recall from Subsection 2.4 that $\delta_{\mathsf{w}}=\psi_{\Theta}(\mathsf{v}(\mathsf{w}))=\|\nabla\psi_{\Theta}(\mathsf{v}(\mathsf{w}))\|$ where we have used Eq. 12. Thanks to Corollary 3.3, we can use [13, Theorem A.2] which states that $m_{\mathcal{X}}^{\mathsf{v}(\mathsf{w})}$ is a measure of maximal entropy for the translation flow which attains the maximal entropy of $\delta_{\mathsf{w}}$ . We will consider in particular the probability measure $\nu_{-\delta_{\mathsf{w}}\tau^{\mathsf{w}}}$ on $\Sigma$ with corresponding pressure $\operatorname{Pr}_{\sigma}(-\delta_{\mathsf{w}}\tau^{\mathsf{w}})=0$ [8, Proposition 3.1] (cf. [15, Theorem 4.4]), which we will denote simply by $\nu_{\Sigma}^{\mathsf{w}}$ . Define $\nu_{R}^{\mathsf{w}}=\eta_{*}(\nu_{\Sigma}^{\mathsf{w}})$ and $\nu_{U}^{\mathsf{w}}=(\operatorname{proj}_{U})_{*}(\nu_{R}^{\mathsf{w}})$ . Note that $\nu_{U}^{\mathsf{w}}(\tau^{\mathsf{w}})=\nu_{R}^{\mathsf{w}}(\tau^{\mathsf{w}})$ . We refer the reader to [18, §4.4] for the relation between $m_{\mathcal{X}}^{\mathsf{v}(\mathsf{w})}$ and $\nu_{R}^{\mathsf{w}}$ , which we do not require directly in this paper.

4.5. Transfer operators

Throughout the paper, we will use the notation $\xi=a+ib\in\mathbb{C}$ for the complex parameter for the transfer operators.

Definition 4.5 (Transfer operators).

For all $\xi\in\mathbb{C}$ and $\mathsf{w}\in\overline{\mathscr{w}}$ , the transfer operator $\mathcal{L}_{\xi\tau^{\mathsf{w}}}:C\bigl{(}U,\mathbb{C}\bigr{)}\to C\bigl{(}U,\mathbb{C}\bigr{)}$ is defined by

\displaystyle\mathcal{L}_{\xi\tau^{\mathsf{w}}}(H)(u)=\sum_{u^{\prime}\in\sigma^{-1}(u)}e^{\xi\tau^{\mathsf{w}}(u^{\prime})}H(u^{\prime})

for all $u\in U$ and $H\in C\bigl{(}U,\mathbb{C}\bigr{)}$ .

We recall the Ruelle–Perron–Frobenius (RPF) theorem along with the theory of Gibbs measures in this setting [7, 44]. For all $a\in\mathbb{R}$ and $\mathsf{w}\in\mathscr{w}$ , there exist a unique positive function $h_{a,\mathsf{w}}\in L(U)$ and a unique Borel probability measure $\nu_{a,\mathsf{w}}$ on $U$ such that $\nu_{a,\mathsf{w}}(h_{a,\mathsf{w}})=1$ and

\displaystyle\mathcal{L}_{-(\delta_{\mathsf{w}}+a)\tau^{\mathsf{w}}}(h_{a,\mathsf{w}})

\displaystyle=\lambda_{a,\mathsf{w}}h_{a,\mathsf{w}},

\displaystyle\mathcal{L}_{-(\delta_{\mathsf{w}}+a)\tau^{\mathsf{w}}}^{*}(\nu_{a,\mathsf{w}})

\displaystyle=\lambda_{a,\mathsf{w}}\nu_{a,\mathsf{w}}

where $\lambda_{a,\mathsf{w}}:=e^{\operatorname{Pr}_{\sigma}(-(\delta_{\mathsf{w}}+a)\tau^{\mathsf{w}})}$ is the maximal simple eigenvalue of $\mathcal{L}_{-(\delta_{\mathsf{w}}+a)\tau^{\mathsf{w}}}$ and the rest of the spectrum of $\mathcal{L}_{-(\delta_{\mathsf{w}}+a)\tau^{\mathsf{w}}}|_{L(U,\mathbb{C})}$ is contained in a disk of radius strictly less than $\lambda_{a,\mathsf{w}}$ . Moreover, $d\nu_{U}^{\mathsf{w}}=h_{0,\mathsf{w}}\,d\nu_{0,\mathsf{w}}$ and $\lambda_{0,\mathsf{w}}=1$ (see Subsection 4.4).

Let $\mathsf{w}\in\mathscr{w}$ . As usual, it is convenient to normalize the transfer operators. For all $a\in\mathbb{R}$ define the map

\displaystyle\tau^{\mathsf{w},(a)}=-(\delta_{\mathsf{w}}+a)\tau^{\mathsf{w}}+\log\circ h_{0,\mathsf{w}}-\log\circ h_{0,\mathsf{w}}\circ\sigma\in L(U).

For all $k\in\mathbb{N}$ , we define the Lipschitz continuous maps $\tau_{k}^{\mathsf{w}}:U\to\mathbb{R}$ and $\tau_{k}^{\mathsf{w},(a)}:U\to\mathbb{R}$ by

\displaystyle\tau_{k}^{\mathsf{w}}(u)

\displaystyle=\sum_{j=0}^{k-1}\tau^{\mathsf{w}}(\sigma^{j}(u)),

\displaystyle\tau_{k}^{\mathsf{w},(a)}(u)

\displaystyle=\sum_{j=0}^{k-1}\tau^{\mathsf{w},(a)}(\sigma^{j}(u))

and $\tau_{0}^{\mathsf{w}}(u)=\tau_{0}^{(a),\mathsf{w}}(u)=0$ for all $u\in U$ . For all $\xi\in\mathbb{C}$ , we define $\mathcal{L}_{\xi,\mathsf{w}}:C(U,\mathbb{C})\to C(U,\mathbb{C})$ and its $k$ ^th iterates for all $k\in\mathbb{Z}_{\geq 0}$ by

\displaystyle\mathcal{L}_{\xi,\mathsf{w}}^{k}(H)(u)=\sum_{u^{\prime}\in\sigma^{-k}(u)}e^{(\tau_{k}^{\mathsf{w},(a)}+ib\tau_{k}^{\mathsf{w}})(u^{\prime})}H(u^{\prime})

for all $u\in U$ and $H\in C(U,\mathbb{C})$ . With this normalization, for all $a\in\mathbb{R}$ , the maximal simple eigenvalue of $\mathcal{L}_{a,\mathsf{w}}$ is $\lambda_{a,\mathsf{w}}$ with eigenvector $\frac{h_{a,\mathsf{w}}}{h_{0,\mathsf{w}}}$ . Moreover, we have $\mathcal{L}_{0,\mathsf{w}}^{*}(\nu_{U}^{\mathsf{w}})=\nu_{U}^{\mathsf{w}}$ .

We fix some related constants. Let $\mathsf{w}\in\mathscr{w}$ . By perturbation theory of operators as in [28, Chapter 7] and [44, Proposition 4.6], we can fix $a_{0}^{\prime}>0$ such that the map $[-a_{0}^{\prime},a_{0}^{\prime}]\to\mathbb{R}$ defined by $a\mapsto\lambda_{a,\mathsf{w}}$ and the map $[-a_{0}^{\prime},a_{0}^{\prime}]\to C(U,\mathbb{R})$ defined by $a\mapsto h_{a,\mathsf{w}}$ are Lipschitz uniformly in $\mathsf{w}\in\mathscr{w}$ . To see uniformity in $\mathsf{w}\in\mathscr{w}$ , a standard calculation using the eigenvalue equation and Lemma 4.2 gives $\bigl{|}\left.\frac{d}{da^{\prime}}\right|_{a^{\prime}=a}\lambda_{a^{\prime},\mathsf{w}}\bigr{|}=|\nu_{a,\mathsf{w}}(\tau^{\mathsf{w}}h_{a,\mathsf{w}})|\leq C_{\tau}$ . Similar estimates in the construction of the family of eigenvectors $\{h_{a,\mathsf{w}}\}_{a\in[-a_{0}^{\prime},a_{0}^{\prime}]}$ (see [44, Theorem 2.2]) gives the latter. Fix $A_{\tau}>0$ such that it is greater that the aforementioned Lipschitz constants and so that $|\tau^{\mathsf{w},(a)}(u)-\tau^{\mathsf{w},(0)}(u)|\leq A_{\tau}|a|$ for all $u\in U$ and $|a|\leq a_{0}^{\prime}$ . Again by similar estimates, we also have $\|h_{a,\mathsf{w}}\|_{\operatorname{Lip}}\ll\delta_{\mathsf{w}}=\|\nabla\psi_{\Theta}(\mathsf{v}(\mathsf{w}))\|$ which tends to $+\infty$ as $\mathsf{w}$ tends to the boundary of $\operatorname{int}\mathcal{L}_{\Theta}^{*}$ . Fix

\displaystyle T_{\mathsf{w}}>{}

\displaystyle\max\bigg{(}\max_{\mathsf{w}\in\overline{\mathscr{w}}}\max_{(j,k)}\|\tau^{\mathsf{w}}|_{\mathtt{C}[j,k]}\|_{\operatorname{Lip}},\sup_{|a|\leq a_{0}^{\prime}}\max_{(j,k)}\bigl{\|}\tau^{\mathsf{w},(a)}|_{\mathtt{C}[j,k]}\bigr{\|}_{\operatorname{Lip}}\bigg{)}

which can be chosen as some elementary function evaluated at $\delta_{\mathsf{w}}=\|\nabla\psi_{\Theta}(\mathsf{v}(\mathsf{w}))\|$ , for all $\mathsf{w}\in\mathscr{w}$ (see also [46, Lemma 4.1] for taking the supremum over $[-a_{0}^{\prime},a_{0}^{\prime}]$ ).

4.6. Dolgopyat’s Method and the proof of Theorem 4.1 and applications

Recall that $L(U,\mathbb{C})$ denotes the space of complex Lipschitz continuous functions on $U$ . It is a Banach space with the Lipschitz seminorm and norm

\displaystyle\operatorname{Lip}(H)

\displaystyle=\sup_{u,u^{\prime}\in U,u\neq u^{\prime}}\frac{|H(u)-H(u^{\prime})|}{d(u,u^{\prime})},

\displaystyle\|H\|_{\operatorname{Lip}}

\displaystyle=\|H\|_{\infty}+\operatorname{Lip}(H)

for all $H\in L(U,\mathbb{C})$ , where $\|\cdot\|_{\infty}$ denotes the usual $L^{\infty}$ -norm. For all $b\in\mathbb{R}$ , we also generalize the above norm to

\displaystyle\|H\|_{1,b}=\|H\|_{\infty}+\frac{1}{\max(1,|b|)}\operatorname{Lip}(H)

for all $H\in L(U,\mathbb{C})$ , which will be required later.

Following Stoyanov [67, §5], we define the convenient new metric $D$ on $U$ by

\displaystyle D(u,u^{\prime})=\begin{cases}0,&u=u^{\prime}\\ \displaystyle\min_{\begin{subarray}{c}u,u^{\prime}\in\mathtt{C}\\ \text{cylinder }\mathtt{C}\end{subarray}}\operatorname{diam}(\overline{\mathtt{C}}),&u\neq u^{\prime}\text{ and }u,u^{\prime}\in U_{j}\text{ for some }j\in\mathcal{A}\\ 1,&\text{otherwise}.\end{cases}

We denote $L_{D}(U,\mathbb{R})$ to be the space of $D$ -Lipschitz continuous functions and $\operatorname{Lip}_{D}(h)$ to be the $D$ -Lipschitz constant for all $h\in L_{D}(U,\mathbb{R})$ . We define the cones

	$\displaystyle\mathcal{C}_{B}(U)$	$\displaystyle=\{h\in L_{D}(U,\mathbb{R}):h>0,\|h(u)-h(u^{\prime})\|\leq Bh(u)D(u,u^{\prime})\text{ for all }u,u^{\prime}\in U\}$
	$\displaystyle\tilde{\mathcal{C}}_{B}(U)$	$\displaystyle=\Big{\{}h\in L_{D}(U,\mathbb{R}):h>0,e^{-BD(u,u^{\prime})}\leq\frac{h(u)}{h(u^{\prime})}\leq e^{BD(u,u^{\prime})}\text{ for all }u,u^{\prime}\in U\Big{\}}.$

Remark 4.6.

If $h\in\mathcal{C}_{B}(U)$ , then using the convexity of $-\log$ , we can derive that $h$ is $\log$ - $D$ -Lipschitz, i.e., $|(\log\circ h)(u)-(\log\circ h)(u^{\prime})|\leq BD(u,u^{\prime})$ for all $u,u^{\prime}\in U$ . It follows that $\mathcal{C}_{B}(U)\subset\tilde{\mathcal{C}}_{B}(U)$ , however the reverse containment is not true.

We now state Theorem 4.7 regarding spectral bounds of transfer operators. Such bounds are the key for obtaining exponential mixing results.

Theorem 4.7.

There exist an elementary function $\eta:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ with vanishing limit at $+\infty$ , $a_{0}>0$ , and $b_{0}>0$ such that for all $\mathsf{w}\in\mathscr{w}$ , there exists $C_{\mathsf{w}}>0$ such that for all $\xi\in\mathbb{C}$ with $|a|<a_{0}$ and $|b|>b_{0}$ , $k\in\mathbb{N}$ , and $H\in L(U,\mathbb{C})$ , we have

\displaystyle\big{\|}\mathcal{L}_{\xi,\mathsf{w}}^{k}(H)\big{\|}_{2}\leq C_{\mathsf{w}}e^{-\eta(\delta_{\mathsf{w}})k}\|H\|_{1,b}.

By a standard inductive argument (see the proof after [64, Theorem 5.4]) which goes back to Dolgopyat [22], Theorem 4.7 is a consequence of the following theorem which records the mechanism of Dolgopyat’s method.

Theorem 4.8.

There exist $m\in\mathbb{N}$ , an elementary function $\eta:\mathbb{R}_{>0}\to(0,1)$ which tends to $1$ at $+\infty$ , $E>\max\left(1,\frac{1}{b_{0}}\right)$ , $a_{0}>0$ , $b_{0}>0$ , and a set of operators $\{\mathcal{N}_{a,J}^{\mathsf{w}}:L_{D}(U,\mathbb{R})\to L_{D}(U,\mathbb{R}):\mathsf{w}\in\mathscr{w},|a|<a_{0},J\in\mathcal{J}(b)\text{ for some }|b|>b_{0}\}$ , where $\mathcal{J}(b)$ is some finite set for all $|b|>b_{0}$ , such that

((1))

$\mathcal{N}_{a,J}^{\mathsf{w}}(\mathcal{C}_{E|b|}(U))\subset\mathcal{C}_{E|b|}(U)$ for all $\mathsf{w}\in\mathscr{w}$ , $|a|<a_{0}$ , $J\in\mathcal{J}(b)$ , and $|b|>b_{0}$ ;
((2))

$\left\|\mathcal{N}_{a,J}^{\mathsf{w}}(h)\right\|_{2}\leq\eta(\delta_{\mathsf{w}})\|h\|_{2}$ for all $h\in\mathcal{C}_{E|b|}(U)$ , $\mathsf{w}\in\mathscr{w}$ , $|a|<a_{0}$ , $J\in\mathcal{J}(b)$ , and $|b|>b_{0}$ ;
((3))
for all $|a|<a_{0}$ , $|b|>b_{0}$ , and $\mathsf{w}\in\mathscr{w}$ , if $H\in L(U,\mathbb{C})$ and $h\in\mathcal{C}_{E|b|}(U)$ satisfy
1. (1a)
  
  $|H(u)|\leq h(u)$ for all $u\in U$ ;
2. (1b)
  
  $|H(u)-H(u^{\prime})|\leq E|b|h(u)D(u,u^{\prime})$ for all $u,u^{\prime}\in U$ ;
then there exists $J\in\mathcal{J}(b)$ such that
1. (2a)
  
  $\bigl{|}\mathcal{L}_{\xi,\mathsf{w}}^{m}(H)(u)\bigr{|}\leq\mathcal{N}_{a,J}^{\mathsf{w}}(h)(u)$ for all $u\in U$ ;
2. (2b)
  
  $\bigl{|}\mathcal{L}_{\xi,\mathsf{w}}^{m}(H)(u)-\mathcal{L}_{\xi,\mathsf{w}}^{m}(H)(u^{\prime})\bigr{|}\leq E|b|\mathcal{N}_{a,J}^{\mathsf{w}}(h)(u)D(u,u^{\prime})$ for all $u,u^{\prime}\in U$ .

Theorem 4.1 for Lipschitz continuous functions is derived from the spectral bounds of transfer operators in Theorem 4.7 using arguments of Pollicott and Paley–Wiener. The derivation is similar to that of [64, §10] and so we omit it and describe the differences. One minor difference is that in our case the derivation would be a Lipschitz version of loc. cit. (cf. [41, §9]). Another difference is that we include the Pollicott–Ruelle resonances since we cannot control simple eigenvalue of the transfer operators $\mathcal{L}_{\xi,\mathsf{w}}$ for the complex parameter $\xi$ in a fixed neighborhood of $0\in\mathbb{C}$ uniformly in $\mathsf{w}\in\mathscr{w}$ . For this, we also refer the reader to [47]; more specifically, [47, Proposition 4], the proof of [47, Theorem 1], and the remark after [47, Corollary 1].

Let us give some more details for integrating out the stable direction in the derivation mentioned above. The constants $\eta>0$ and $C>0$ which come from the metric Anosov property vary smoothly in $\mathsf{w}$ . However, the decay rate $\eta$ could, a priori, get worse as $\mathsf{w}$ tends to $\partial\mathcal{L}_{\Theta}^{*}$ . The exact behavior comes from the lower bound for the derivative of the reparametrization $\kappa(x,t)$ in $t$ . But this comes exactly from the upper bound for $\sup_{v\in\mathcal{L}_{\Theta},\|v\|=1}\psi_{\mathsf{v}(\mathsf{w})}$ which, by construction, is bounded above by $1$ uniformly in $\mathsf{w}\in\mathscr{w}$ . Thus, one can choose a decay rate $\eta$ uniformly in $\mathsf{w}\in\mathscr{w}$ .

Remark 4.9.

We warn the reader of a subtle point. The derivation in [64, §10] crucially uses the Anosov property of the frame flow; the spectral bounds of transfer operators on their own are not enough. Analogously, although the proof of Theorem 4.7 does not require the metric Anosov property, the proof of Theorem 4.1 would be incomplete without it. Though the metric Anosov property of the translation flow is claimed in the literature on Anosov subgroups, it is far from a triviality. In our paper, the metric Anosov property was stated in Theorem 3.2 and a complete proof was provided in Subsection 3.4, thus, justifying the described derivation above of Theorem 4.1 for Lipschitz continuous functions.

Let us now describe the derivations of Theorems 1.6 and 1.8. The former is proved using the relationship between the zeros of the Selberg zeta function $Z_{\mathsf{v}(\mathsf{w})}$ and the transfer operators $\mathcal{L}_{\xi,\mathsf{w}}$ stated in [47, Proposition 4], and of course the spectral bounds in Theorem 4.7 (cf. proof of [47, Theorem 1]). The latter is proved using the number theoretic techniques in [50] where the weaker spectral bounds in [50, Eq. (2.1)] (cf. [50, Proposition 2]) is replaced again by the stronger spectral bounds in Theorem 4.7.

As a result of our discussion, it suffices to focus only on the proof of Theorem 4.8 for the rest of the paper.

5. Local non-integrability condition

This section is devoted to a crucial ingredient for Dolgopyat’s method stated in Proposition 5.11 called the local non-integrability condition (LNIC) as in [67], which is then upgraded to Proposition 5.12. Although Proposition 5.12 is stated in reverse Lipschitz form on $\operatorname{int}(U)$ , we obtain it via Proposition 5.11 which requires Lie theory and hence the smooth structure on $G$ . Thus, we first resolve technical issues related to the incompatibility of $\Lambda_{\Theta}$ with the smooth structure on $G$ .

5.1. Smooth extensions of coding maps

Fix $\mathsf{w}\in\mathscr{w}$ throughout this subsection. As alluded to above, due to the fractal nature of $\Lambda_{\Theta}$ , the Markov section does not inherit the smooth structure on $G$ . In order to circumvent this issue, we need to enlarge $U$ to an open set and smoothly extend the maps associated to the coding, $\tau^{\mathsf{w}}$ , $\mathcal{P}$ , and $\sigma$ . This type of construction was done in [64, §5.1] based on [57, Lemma 1.2]. However, even this procedure has many more technical issues in our setting. One problem is that $\mathcal{X}=\Gamma\backslash\bigl{(}\Lambda_{\Theta}^{(2)}\times\mathbb{R}\bigr{)}$ is not naturally situated in a larger smooth manifold with a natural extension of the translation flow. Note that in general the action $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ is not properly discontinuous (see Theorem A.2). To avoid this problem, we work on the cover $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ instead and use the inclusion

\displaystyle\Lambda_{\Theta}^{(2)}\times\mathbb{R}\hookrightarrow\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}

which is a Lipschitz embedding into a smooth manifold where the translation flow is still defined. Another problem is that it is difficult to gain control on the $\Gamma$ -orbit of the open neighborhood of $U=\bigsqcup_{j\in\mathcal{A}}U_{j}$ in $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ (e.g., diameter estimates, mutual disjointness, seperation distance) since we do not have a $\Gamma$ -invariant metric on $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ and $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ may not be properly discontinuous. To avoid these topological obstructions, we allow dependence of the open neighborhoods on the arbitrarily large length $m\in\mathbb{N}$ of sections of $\sigma^{m}$ which will appear in Proposition 5.11. We do not follow [57, Lemma 1.2] because, although we believe that it is true, it would take significantly more work to properly formulate and establish the eventually contracting property of the smooth extension of $\sigma^{-1}$ on some fixed open neighborhood of $U$ . See the recent related work of Delarue–Montclair–Sanders [20] for the case that $\Gamma$ is a projective Anosov subgroup of $\operatorname{PSL}_{n}(\mathbb{R})$ which gives a construction of a flow-invariant smooth manifold $\mathcal{M}\supset\mathcal{X}$ on which the translation flow is a contact Axiom A flow.

In light of the above discussion, let us first fix a connected fundamental domain $\mathsf{D}\subset\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ for $\mathcal{X}$ , unique isometric lifts $\mathsf{R}_{j}=[\mathsf{U}_{j},\mathsf{S}_{j}]\subset\mathsf{D}$ of $R_{j}=[U_{j},S_{j}]$ for all $j\in\mathcal{A}$ , and write $\mathsf{R}:=\bigsqcup_{j\in\mathcal{A}}\mathsf{R}_{j}$ and $\mathsf{U}:=\bigsqcup_{j\in\mathcal{A}}\mathsf{U}_{j}$ . Denote by $\tilde{u}\in\mathsf{R}$ the unique lift of $u\in R$ . Abusing notation, we denote the lift of cylinders by the same symbol. Since the translation flow is defined on $\Lambda_{\Theta}^{(2)}\times\mathbb{R}$ , the maps $\tau^{\mathsf{w}}$ , $\mathcal{P}$ , and $\sigma$ have natural lifts $\tau^{\mathsf{w}}:\Gamma\mathsf{R}\to\mathbb{R}$ , $\mathcal{P}:\Gamma\mathsf{R}\to\Gamma\mathsf{R}$ , and $\sigma:\Gamma\mathsf{U}\to\Gamma\mathsf{U}$ , abusing notation. For all $u\in\mathsf{R}$ we define $\mathsf{g}(u)\in\Gamma$ to be the unique element such that $\mathcal{P}(u)\in\mathsf{g}(u)\mathsf{R}$ . Similarly, for all admissible pairs $(j,k)$ , we write $\mathsf{g}^{(j,k)}\in\Gamma$ for the unique element such that

\displaystyle\sigma(\operatorname{int}(\mathsf{U}_{j}))\cap\mathsf{g}^{(j,k)}\operatorname{int}(\mathsf{U}_{k})\neq\varnothing.

Then, $\hat{\mathsf{g}}=\{\mathsf{g}^{(j,k)}:(j,k)\text{ is an admissible pair}\}\subset\Gamma$ is a generating subset. For all admissible sequences $\alpha=(\alpha_{0},\alpha_{1},\dotsc,\alpha_{k})$ for some $k\in\mathbb{Z}_{\geq 0}$ , we extend the notation and write

\displaystyle\mathsf{g}^{\alpha}:=\prod_{j=0}^{k-1}\mathsf{g}^{(\alpha_{j},\alpha_{j+1})}

(13)

in ascending order if $k>0$ , and $\mathsf{g}^{\alpha}:=e$ if $k=0$ , and $\mathsf{g}^{-\alpha}:=(\mathsf{g}^{\alpha})^{-1}$ .

Let $j\in\mathcal{A}$ , $w_{j}\in\mathsf{U}_{j}$ be the center, and $\gamma\in\Gamma$ . We have a metric on $W^{\mathcal{F}_{\Theta},\mathrm{ss}}(\gamma w_{j})$ (resp. $W^{\mathcal{F}_{\Theta},\mathrm{wu}}(\gamma w_{j})$ ) which is induced by $d_{\mathcal{F}_{\mathsf{i}\Theta}}$ (resp. $d_{\mathcal{F}_{\Theta}}\times d_{\mathbb{R}}$ ) using coordinate pullbacks and denoted by the same notation. Then, $\bigl{(}W_{\epsilon_{\gamma}}^{\mathcal{F}_{\Theta},\mathrm{wu}}(\gamma w_{j}),W_{\epsilon_{\gamma}}^{\mathcal{F}_{\Theta},\mathrm{ss}}(\gamma w_{j})\bigr{)}$ has a local product structure such that $\bigl{[}W_{\epsilon_{\gamma}}^{\mathcal{F}_{\Theta},\mathrm{su}}(\gamma w_{j}),W_{\epsilon_{\gamma}}^{\mathcal{F}_{\Theta},\mathrm{ss}}(\gamma w_{j})\bigr{]}$ contains $\gamma\mathsf{R}_{j}$ , for some $\epsilon_{\gamma}>0$ . Now, fix open neighborhoods ${}^{\gamma}\tilde{\mathsf{U}}_{j}\subset W_{\epsilon_{\gamma}}^{\mathcal{F}_{\Theta},\mathrm{su}}(\gamma w_{j})$ of $\gamma\mathsf{U}_{j}$ and define ${}^{\gamma}\tilde{\mathsf{R}}_{j}=[{}^{\gamma}\tilde{\mathsf{U}}_{j},\gamma\mathsf{S}_{j}]$ such that $\{{}^{\gamma}\tilde{\mathsf{R}}_{j}\}_{j\in\mathcal{A},\gamma\in\Gamma}$ consists of mutually disjoint rectangles. Define $\tilde{\mathsf{R}}_{j}:={}^{e}\tilde{\mathsf{R}}_{j}$ , $\tilde{\mathsf{R}}:=\bigsqcup_{j\in\mathcal{A}}\tilde{\mathsf{R}}_{j}$ , ${}^{\Gamma}\tilde{\mathsf{R}}_{j}:=\bigsqcup_{\gamma\in\Gamma}{}^{\gamma}\tilde{\mathsf{R}}_{j}$ , and ${}^{\Gamma}\tilde{\mathsf{R}}:=\bigsqcup_{j\in\mathcal{A},\gamma\in\Gamma}{}^{\gamma}\tilde{\mathsf{R}}_{j}$ , and similarly define $\tilde{\mathsf{U}}_{j}$ , $\tilde{\mathsf{U}}$ , ${}^{\Gamma}\tilde{\mathsf{U}}_{j}$ , and ${}^{\Gamma}\tilde{\mathsf{U}}$ . We similarly omit the superscript ‘ $e$ ’ and use the superscript ‘ $\Gamma$ ’ for other sets.

Let $m\in\mathbb{Z}_{\geq 0}$ . Recall that the translation flow is defined on $\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ and smooth. Thus, for all $j\in\mathcal{A}$ and $\gamma\in\Gamma$ , we obtain compactly contained open neighborhoods ${}^{\gamma}\tilde{\mathsf{U}}_{j}^{(m)}\subset{}^{\gamma}\tilde{\mathsf{U}}_{j}$ of $\gamma\mathsf{U}_{j}$ which are decreasing in $m$ and the natural smooth injective extensions $\sigma^{-\alpha}:{}^{\Gamma\mathsf{g}^{\alpha}}\tilde{\mathsf{U}}_{\alpha_{m}}^{(m)}\to{}^{\Gamma}\tilde{\mathsf{U}}_{\alpha_{0}}$ of $\bigl{(}\sigma|_{\mathtt{C}[\alpha]}\bigr{)}^{-1}$ with cylinders ${}^{\gamma}\tilde{\mathtt{C}}[\alpha]^{(m)}:=\sigma^{-\alpha}\bigl{(}{}^{\gamma\mathsf{g}^{\alpha}}\tilde{\mathsf{U}}_{\alpha_{m}}^{(m)}\bigr{)}\subset{}^{\gamma}\tilde{\mathsf{U}}_{\alpha_{0}}$ ; and we define ${}^{\gamma}\tilde{\mathsf{R}}_{j}^{(m)}:=[{}^{\gamma}\tilde{\mathsf{U}}_{j}^{(m)},\gamma\mathsf{S}_{j}]$ and $\sigma^{\alpha}=(\sigma^{-\alpha})^{-1}:{}^{\Gamma}\tilde{\mathtt{C}}[\alpha]^{(m)}\to{}^{\Gamma\mathsf{g}^{\alpha}}\tilde{\mathsf{U}}_{\alpha_{m}}^{(m)}$ , for all admissible sequences $\alpha=(\alpha_{0},\alpha_{1},\dotsc,\alpha_{m})$ .

There are also natural smooth extensions $\tau^{\mathsf{w}}_{(j,k)}:{}^{\Gamma}\tilde{\mathtt{C}}[j,k]^{(1)}\to\mathbb{R}$ for all admissible pairs $(j,k)$ . We define the smooth maps $\tau^{\mathsf{w}}_{\alpha}:{}^{\Gamma}\tilde{\mathtt{C}}[\alpha]^{(m)}\to\mathbb{R}$ by

\displaystyle\tau^{\mathsf{w}}_{\alpha}(u)

\displaystyle=\sum_{j=0}^{m-1}\tau^{\mathsf{w}}_{(\alpha_{j},\alpha_{j+1})}(\sigma^{(\alpha_{0},\alpha_{1},\dotsc,\alpha_{j})}(u))

(14)

for all admissible sequences $\alpha=(\alpha_{0},\alpha_{1},\dotsc,\alpha_{m})$ .

Following [18, §5] we can construct a smooth section

\displaystyle\mathsf{F}:{}^{\Gamma}\tilde{\mathsf{R}}\to G

in a similar fashion such that:

((1))

for all $j\in\mathcal{A}$ and $\gamma\in\Gamma$ , and $u,u^{\prime}\in{}^{\gamma}\tilde{\mathsf{U}}_{j}$ , there exists unique $n^{+}\in N^{+}_{\Theta}$ such that $\mathsf{F}(u^{\prime})=\mathsf{F}(u)n^{+}$ ;
((2))

for all $j\in\mathcal{A}$ and $\gamma\in\Gamma$ , $u\in{}^{\gamma}\tilde{\mathsf{U}}_{j}$ , and $s,s^{\prime}\in\gamma\mathsf{S}_{j}$ , there exists unique $n^{-}\in N^{-}_{\Theta}$ such that $\mathsf{F}([u,s^{\prime}])=\mathsf{F}([u,s])n^{-}$ .

Observe that $\mathsf{F}$ can indeed be constructed such that it does not depend on $\mathsf{w}$ due to the properties of the compatible Markov sections.

We introduce the first return vector map and holonomy. Although we only need the first return time map, since it does not require excess work, we provide the general definitions and the subsequent fundamental lemmas in anticipation that it will turn out useful elsewhere.

Definition 5.1 (First return vector map, Holonomy).

The first return $\Theta$ -vector map and $\Theta$ -holonomy are $\Gamma$ -invariant maps $\mathsf{K}:\Gamma\mathsf{R}\to\mathfrak{a}_{\Theta}$ and $\vartheta:\Gamma\mathsf{R}\to S_{\Theta}$ respectively that associate to each $u\in\gamma\mathsf{R}$ for some $\gamma\in\Gamma$ the unique elements $\mathsf{K}(u)\in\mathfrak{a}_{\Theta}$ and $\vartheta(u)\in S_{\Theta}$ which satisfy

\displaystyle\mathsf{F}(\mathcal{P}(u))=\mathsf{g}(u)\mathsf{F}(u)a_{\mathsf{K}(u)}\vartheta(u).

We often drop the prefix “ $\Theta$ -”.

Again, $\mathsf{K}$ and $\vartheta$ are independent of $\mathsf{w}$ . Similar to previous constructions, we have smooth extensions $\mathsf{K}_{(j,k)}:{}^{\Gamma}\tilde{\mathtt{C}}[j,k]^{(1)}\to\mathfrak{a}_{\Theta}$ and $\vartheta^{(j,k)}:{}^{\Gamma}\tilde{\mathtt{C}}[j,k]^{(1)}\to S_{\Theta}$ for all admissible pairs $(j,k)$ . Again, for all $m\in\mathbb{Z}_{\geq 0}$ and admissible sequences $\alpha=(\alpha_{0},\alpha_{1},\dotsc,\alpha_{m})$ , we define the smooth maps $\mathsf{K}_{\alpha}:{}^{\Gamma}\tilde{\mathtt{C}}[\alpha]^{(m)}\to\mathfrak{a}_{\Theta}$ and $\vartheta^{\alpha}:{}^{\Gamma}\tilde{\mathtt{C}}[\alpha]^{(m)}\to S_{\Theta}$ as in Eq. 14 and in the style of Eq. 13 for holonomy.

5.2. Local non-integrability condition

We are now ready to prove Propositions 5.11 and 5.12. The proof follows the techniques developed in [64, 17]. The version here is actually more simplified in some parts and we obtain a stronger LNIC because we are dealing with the “geodesic flow” (see Remark 5.10).

We start with a slight generalization of [64, Definition 6.1]. Recall the identifications from Subsection 4.2.

Definition 5.2 (Associated sequence in $G$ ).

Let $\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\|\mathsf{w}\|=1$ . Let $z_{1}\in\tilde{\mathsf{R}}_{1}$ be the center. Consider a sequence $(z_{1},z_{2},z_{3},z_{4},z_{1})\in(\tilde{\mathsf{R}}_{1})^{5}$ such that $z_{2}\in\mathsf{S}_{1}$ , $z_{4}\in\tilde{\mathsf{U}}_{1}$ , and $z_{3}=[z_{4},z_{2}]$ . We define an associated sequence in $G$ to be the unique sequence $(g_{1},g_{2},\dotsc,g_{5})\in G^{5}$ where

	$\displaystyle g_{1}$	$\displaystyle=\mathsf{F}(z_{1}),$
	$\displaystyle g_{2}$	$\displaystyle=\mathsf{F}(z_{2})\in g_{1}N^{-}_{\Theta}\text{ such that }\pi_{\psi_{\mathsf{v}(\mathsf{w})}}(g_{2}S_{\Theta})=z_{2},$
	$\displaystyle g_{3}$	$\displaystyle\in g_{2}N^{+}_{\Theta}\text{ such that }a^{\mathsf{v}(\mathsf{w})}_{t}\bigl{(}\pi_{\psi_{\mathsf{v}(\mathsf{w})}}(g_{3}S_{\Theta})\bigr{)}=z_{3}\text{ for some }t\in(-\underline{\tau},\underline{\tau}),$
	$\displaystyle g_{4}$	$\displaystyle\in g_{3}N^{-}_{\Theta}\text{ such that }a^{\mathsf{v}(\mathsf{w})}_{t}\bigl{(}\pi_{\psi_{\mathsf{v}(\mathsf{w})}}(g_{4}S_{\Theta})\bigr{)}=z_{4}\ \text{ for some }t\in(-\underline{\tau},\underline{\tau}),$
	$\displaystyle g_{5}$	$\displaystyle\in g_{4}N^{+}_{\Theta}\text{ such that }a^{\mathsf{v}(\mathsf{w})}_{t}\bigl{(}\pi_{\psi_{\mathsf{v}(\mathsf{w})}}(g_{5}S_{\Theta})\bigr{)}=z_{1}\text{ for some }t\in(-\underline{\tau},\underline{\tau}).$

Remark 5.3.

In the above definition, the associated sequence $(g_{1},g_{2},g_{3},g_{4},g_{5})$ corresponding to each $(z_{1},z_{2},z_{3},z_{4},z_{1})$ is independent of $\mathsf{w}$ .

We continue using the notation in the above definition. Define the subsets

	$\displaystyle(N^{+}_{\Theta})_{1}$	$\displaystyle=\{n^{+}\in N^{+}_{\Theta}:\mathsf{F}(z_{1})n^{+}\in\mathsf{F}(\tilde{\mathsf{U}}_{1})\}\subset N^{+}_{\Theta},$
	$\displaystyle(N^{-}_{\Theta})_{1}$	$\displaystyle=\{n^{-}\in N^{-}_{\Theta}:\mathsf{F}(z_{1})n^{-}\in\mathsf{F}(\mathsf{S}_{1})\}\subset N^{-}_{\Theta},$

where the first is open and the second is compact. Now, if the above sequence $(z_{1},z_{2},z_{3},z_{4},z_{1})$ corresponds to some $n^{+}\in(N^{+}_{\Theta})_{1}$ and some $n^{-}\in(N^{-}_{\Theta})_{1}$ such that $\mathsf{F}(z_{4})=\mathsf{F}(z_{1})n^{+}$ and $\mathsf{F}(z_{2})=\mathsf{F}(z_{1})n^{-}$ respectively, then we can define the map $\Xi:(N^{+}_{\Theta})_{1}\times(N^{-}_{\Theta})_{1}\to A_{\Theta}S_{\Theta}$ by

\displaystyle\Xi(n^{+},n^{-})=g_{5}^{-1}g_{1}\in A_{\Theta}S_{\Theta}.

To view it as a function of the first coordinate for a fixed $n^{-}\in(N^{-}_{\Theta})_{1}$ , we write $\Xi_{n^{-}}:(N^{+}_{\Theta})_{1}\to A_{\Theta}S_{\Theta}$ .

Let $z_{1}\in\tilde{\mathsf{R}}_{1}$ be the center. Let $j\in\mathbb{N}$ and $\alpha=(\alpha_{0},\alpha_{2},\dotsc,\alpha_{j-1},1)$ be an admissible sequence. Then, there exists an element which we denote by $n_{\alpha}\in(N^{-}_{\Theta})_{1}$ such that

\displaystyle\mathsf{F}(\mathcal{P}^{j}(\sigma^{-\alpha}(z_{1})))=\mathsf{F}(z_{1})n_{\alpha}.

This is well-defined because $\sigma^{-\alpha}(z_{1})\in\mathsf{g}^{-\alpha}\mathtt{C}[\alpha]\subset\mathsf{g}^{-\alpha}\mathsf{U}$ .

Let $\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\|\mathsf{w}\|=1$ . Let $\pi_{\mathfrak{a}_{\Theta}\oplus\mathfrak{s}_{\Theta}}:\mathfrak{g}\to\mathfrak{a}_{\Theta}\oplus\mathfrak{s}_{\Theta}$ and $\pi_{\mathfrak{a}_{\Theta}}:\mathfrak{g}\to\mathfrak{a}_{\Theta}$ be orthogonal projection maps. Let

\displaystyle\pi_{\mathsf{w}}:\mathfrak{g}\to\mathbb{R}\mathsf{w}

denote the orthogonal projection map, i.e., the projection with respect to the decomposition $\mathfrak{g}\cong\mathbb{R}\mathsf{w}\oplus\ker\psi_{\mathsf{v}(\mathsf{w})}\oplus\mathfrak{a}_{\Theta}^{\perp}$ . Similarly, let

\displaystyle\tilde{\pi}_{\mathsf{w}}:A_{\Theta}S_{\Theta}\to\exp(\mathbb{R}\mathsf{w})

be the Cartesian projection map of $A_{\Theta}S_{\Theta}\cong\exp(\mathbb{R}\mathsf{w})\times\exp(\ker\psi_{\mathsf{v}(\mathsf{w})})\times S_{\Theta}$ onto the $\exp(\mathbb{R}\mathsf{w})$ factor. Note that $(d\tilde{\pi}_{\mathsf{w}})_{e}=\pi_{\mathsf{w}}|_{\mathfrak{a}_{\Theta}\oplus\mathfrak{s}_{\Theta}}$ .

Also define $(N^{-}_{\Theta})_{1,\epsilon_{e}}=\left\{n^{-}\in N^{-}_{\Theta}:\mathsf{F}(z_{1})n^{-}\in\mathsf{F}\left(W_{\epsilon_{e}}^{\mathcal{F}_{\Theta},\mathrm{ss}}(z_{1})\right)\right\}$ where $\epsilon_{e}$ is as in Subsection 5.1 and $z_{1}\in\tilde{\mathsf{R}}_{1}$ is the center.

In order to derive the LNIC in Proposition 5.11, we need the following two lemmas regarding $\Xi$ . We omit the proofs since they are almost a verbatim repetition of that of [64, Lemmas 6.2 and 6.3].

Lemma 5.4.

Let $m\in\mathbb{N}$ , $\alpha=(\alpha_{0},\alpha_{1},\dotsc,\alpha_{m-1},1)$ be an admissible sequence, and $n^{-}=n_{\alpha}\in(N^{-}_{\Theta})_{1}$ . Let $u\in\tilde{\mathsf{U}}_{1}^{(m)}$ and $n^{+}\in(N^{+}_{\Theta})_{1}$ such that $\mathsf{F}(u)=\mathsf{F}(z_{1})n^{+}$ where $z_{1}\in\tilde{\mathsf{R}}_{1}$ is the center. Then, we have

\displaystyle\Xi(n^{+},n^{-})=a_{-\mathsf{K}_{\alpha}(\sigma^{-\alpha}(z_{1}))}\vartheta^{\alpha}(\sigma^{-\alpha}(z_{1}))^{-1}a_{\mathsf{K}_{\alpha}(\sigma^{-\alpha}(u))}\vartheta^{\alpha}(\sigma^{-\alpha}(u)).

In particular, for all $\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\|\mathsf{w}\|=1$ , we have

\displaystyle\log\tilde{\pi}_{\mathsf{w}}(\Xi(n^{+},n^{-}))=\bigl{(}\tau^{\mathsf{w}}_{\alpha}(\sigma^{-\alpha}(u))-\tau^{\mathsf{w}}_{\alpha}(\sigma^{-\alpha}(z_{1}))\bigr{)}\mathsf{w}.

Lemma 5.5.

For all $n^{-}\in(N^{-}_{\Theta})_{1}$ , we have

\displaystyle(d\Xi_{n^{-}})_{e}=\pi_{\mathfrak{a}_{\Theta}\oplus\mathfrak{s}_{\Theta}}\circ\operatorname{Ad}_{n^{-}}|_{\mathfrak{n}^{+}_{\Theta}}\circ(dh_{n^{-}})_{e},

and in particular, for all $\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\|\mathsf{w}\|=1$ , we have

\displaystyle d(\log\circ\tilde{\pi}_{\mathsf{w}}\circ\Xi_{n^{-}})_{e}=\pi_{\mathsf{w}}\circ\operatorname{Ad}_{n^{-}}|_{\mathfrak{n}^{+}_{\Theta}}\circ(dh_{n^{-}})_{e},

where $h_{n^{-}}:(N^{+}_{\Theta})_{1}\to N^{+}_{\Theta}$ is a diffeomorphism onto its image which is also smooth in $n^{-}\in(N^{-}_{\Theta})_{1,\epsilon_{e}}$ and satisfies $h_{n^{-}}(e)=e$ and $h_{e}=\operatorname{Id}_{(N^{+}_{\Theta})_{1}}$ .

For all $\alpha\in\Pi$ , let $H_{\alpha}\in\mathfrak{a}$ be the corresponding root direction, i.e., $\alpha=\langle\cdot,H_{\alpha}\rangle$ . The following lemma is more or less a standard Lie algebra fact. We give a proof for the sake of completeness and for comparison with the analogous statement in [17, Proposition 4.4]. Using loc. cit. more directions can be produced (including those in $\mathfrak{m}$ ) whereas our lemma below gives the stronger quantifier “for all $\alpha\in\Pi$ and $x_{\alpha}\in\mathfrak{g}_{\alpha}$ ” (cf. Remark 5.10).

Lemma 5.6.

Let $\alpha\in\Pi$ . For all nonzero $x_{\alpha}\in\mathfrak{g}_{\alpha}$ , there exists a nonzero $x_{-\alpha}\in\mathfrak{g}_{-\alpha}$ such that $[x_{-\alpha},x_{\alpha}]\in\mathbb{R}_{>0}H_{\alpha}$ .

Proof.

Let $\alpha\in\Pi$ and $x_{\alpha}\in\mathfrak{g}_{\alpha}$ be nonzero. We will show that $x_{-\alpha}:=\theta(x_{\alpha})$ satisfies the lemma. Indeed $x_{-\alpha}\in\mathfrak{g}_{-\alpha}$ because for all $H\in\mathfrak{a}$ , we have

\displaystyle[H,x_{-\alpha}]=\theta[\theta(H),\theta(x_{-\alpha})]=\theta[-H,x_{\alpha}]=-\alpha(H)\theta(x_{\alpha})=-\alpha(H)x_{-\alpha}.

Thus, $[x_{-\alpha},x_{\alpha}]\in[\mathfrak{g}_{-\alpha},\mathfrak{g}_{\alpha}]\subset\mathfrak{g}_{0}=\mathfrak{a}\oplus\mathfrak{m}$ . Moreover, we have

\displaystyle\theta[x_{-\alpha},x_{\alpha}]=[\theta(x_{-\alpha}),\theta(x_{\alpha})]=[x_{\alpha},x_{-\alpha}]=-[x_{-\alpha},x_{\alpha}]

which implies $[x_{-\alpha},x_{\alpha}]\in\mathfrak{p}$ . Hence, $[x_{-\alpha},x_{\alpha}]\in(\mathfrak{a}\oplus\mathfrak{m})\cap\mathfrak{p}=\mathfrak{a}$ . We use the symbol $\propto$ to mean proportionality with a positive constant. Recalling that $\langle\cdot,\cdot\rangle|_{\mathfrak{a}\times\mathfrak{a}}\propto B|_{\mathfrak{a}\times\mathfrak{a}}$ , for all $H\in\mathfrak{a}$ , we have

\displaystyle\langle H,[x_{-\alpha},x_{\alpha}]\rangle\propto B(H,[x_{-\alpha},x_{\alpha}])=B([H,x_{-\alpha}],x_{\alpha})=-\alpha(H)B(x_{-\alpha},x_{\alpha})

which implies $[x_{-\alpha},x_{\alpha}]\propto-B(x_{-\alpha},x_{\alpha})H_{\alpha}=B_{\theta}(x_{\alpha},x_{\alpha})H_{\alpha}\in\mathbb{R}H_{\alpha}-\{0\}$ since $B_{\theta}$ is positive definite. ∎

Repeating an analogous proof for arbitrary $\Theta$ , we have the following generalization of [23, Lemma 2.11] which is itself a generalization of [70, Proposition 3.12]. Alternatively, [29, Theorem 1.1] (which is a stronger measure theoretic statement but for subvarieties) also suffices for our purposes in the proof of Lemma 5.8.

Lemma 5.7.

For any open subset $\mathcal{O}\subset\mathcal{F}_{\Theta}$ with $\Lambda_{\Theta}\cap\mathcal{O}\neq\varnothing$ , the intersection $\Lambda_{\Theta}\cap\mathcal{O}$ is not contained in any smooth submanifold of $\mathcal{F}_{\Theta}$ of lower dimension.

We write the next lemma and its corollary in its most general form, though we apply it for $\mathsf{w}\in\overline{\mathscr{w}}$ . Let $\mathsf{w}\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\|\mathsf{w}\|=1$ . We define

	$\displaystyle\Theta_{\mathsf{w}}$	$\displaystyle:=\{\alpha\in\Theta:\mathsf{w}\notin\ker\alpha\}\subset\Theta,$
	$\displaystyle\mathfrak{S}^{\pm}_{\mathsf{w}}$	$\displaystyle:=\bigoplus_{\alpha\in\Theta_{\mathsf{w}}}\mathfrak{g}_{\mp\alpha}\subset\mathfrak{n}^{\pm}_{\Theta}.$

Since $\Theta_{\mathsf{w}}$ is always nonempty, if $\#\Theta=1$ , then we trivially have $\Theta_{\mathsf{w}}=\Theta$ . Morevoer, $\mathsf{w}\in\overline{\mathscr{w}}$ implies $\Theta_{\mathsf{w}}=\Theta$ and $\mathfrak{S}^{\pm}_{\mathsf{w}}=\mathfrak{n}^{\pm}_{\Theta}$ .

Lemma 5.8.

There exists $C_{\mathscr{w}}>0$ such that the following holds. Let $\mathsf{w}\in\overline{\mathscr{w}}$ . For all nonzero $n^{+}\in\mathfrak{S}^{+}_{\mathsf{w}}$ , there exists $n^{-}\in(N^{-}_{\Theta})_{1}$ such that

\displaystyle|\pi_{\mathsf{w}}(\operatorname{Ad}_{n^{-}}(n^{+}))|\geq C_{\mathscr{w}}.

(15)

Proof.

By compactness of $\overline{\mathscr{w}}$ and continuity of $\pi_{\mathsf{w}}(v)$ in $\mathsf{w}\in\overline{\mathscr{w}}$ for all $v\in\mathfrak{g}$ , it suffices to show that for all $\mathsf{w}\in\overline{\mathscr{w}}$ and nonzero $n^{+}\in\mathfrak{S}^{+}_{\mathsf{w}}$ , there exists $n^{-}\in(N^{-}_{\Theta})_{1}$ such that $\pi_{\mathsf{w}}(\operatorname{Ad}_{n^{-}}(n^{+}))\neq 0$ . Let $n^{+}\in\mathfrak{S}^{+}_{\mathsf{w}}$ be nonzero. Let $\mathsf{w}\in\overline{\mathscr{w}}$ and suppose for the sake of contradiction that $\pi_{\mathsf{w}}\bigl{(}\operatorname{Ad}_{(N^{-}_{\Theta})_{1}}(n^{+})\bigr{)}=0$ . Without loss of generality, we may assume that $\mathsf{F}(z_{1})=e$ at the center $z_{1}\in\tilde{\mathsf{R}}_{1}$ . Consider the smooth map

	$\displaystyle L:N^{-}_{\Theta}$	$\displaystyle\to\mathbb{R}$
	$\displaystyle n^{-}$	$\displaystyle\mapsto\pi_{\mathsf{w}}(\operatorname{Ad}_{n^{-}}(n^{+})).$

Then, we have $dL_{e}(\hat{n}^{-})=\pi_{\mathsf{w}}([\hat{n}^{-},n^{+}])$ for all $\hat{n}^{-}\in\mathfrak{n}^{-}_{\Theta}$ .

Note that $\pi_{\mathfrak{a}_{\Theta}}(H_{\alpha})\notin\ker\psi_{\mathsf{v}(\mathsf{w})}=\mathsf{w}^{\perp}$ for all $\alpha\in\Theta_{\mathsf{w}}$ . Let $x_{\alpha}$ for some $\alpha\in\Theta_{\mathsf{w}}$ be a nonzero component in the decomposition of $n^{+}$ according to $\mathfrak{S}^{+}_{\mathsf{w}}=\bigoplus_{\alpha\in\Theta_{\mathsf{w}}}\mathfrak{g}_{-\alpha}$ . By Lemma 5.6, there exists $x_{-\alpha}\in\mathfrak{g}_{-\alpha}\subset\mathfrak{n}^{-}_{\Theta}$ such that $[x_{-\alpha},x_{\alpha}]\in\mathbb{R}_{>0}H_{\alpha}$ . Using the definition of $\mathfrak{S}^{+}_{\mathsf{w}}$ , we have $\pi_{\mathsf{w}}([x_{-\alpha},x_{\alpha}])\neq 0$ . Then $dL_{e}(x_{-\alpha})=\pi_{\mathsf{w}}([x_{-\alpha},n^{+}])=\pi_{\mathsf{w}}([x_{-\alpha},x_{\alpha}])\neq 0$ so there is a neighborhood $O\subset N^{-}_{\Theta}$ containing $e$ such that $L^{-1}(0)\cap O$ is a smooth submanifold (in fact a subvariety) of $N^{-}_{\Theta}$ of strictly lower dimension. However, $(N^{-}_{\Theta})_{1}\cap O\subset L^{-1}(0)\cap O$ , but on the other hand $((N^{-}_{\Theta})_{1})^{-}$ contains $\Lambda_{\mathsf{i}\Theta}\cap\mathcal{O}$ for some open set $\mathcal{O}\subset\mathcal{F}_{\mathsf{i}\Theta}$ . It follows via the diffeomorphism $N^{-}_{\Theta}\to N^{-}_{\Theta}e^{-}$ that $\Lambda_{\mathsf{i}\Theta}\cap\mathcal{O}$ is contained in a smooth submanifold of $\mathcal{F}_{\mathsf{i}\Theta}$ of strictly lower dimension, contradicting Lemma 5.7. ∎

For any normed vector space $(V,\|\cdot\|)$ , let $\mathbb{S}(V)$ denote the unit sphere in $V$ centered at $0\in V$ . Since Eq. 15 is an open condition and $\mathbb{S}(\mathfrak{n}^{+}_{\Theta})$ is compact, a straightforward compactness argument gives the following corollary.

Corollary 5.9.

There exists $C_{\mathscr{w}}>0$ such that the following holds. Let $\mathsf{w}\in\overline{\mathscr{w}}$ . There exist nontrivial $n_{1}^{-},n_{2}^{-},\dotsc,n_{j_{0}}^{-}\in(N^{-}_{\Theta})_{1}$ for some $j_{0}\in\mathbb{N}$ and $\delta>0$ such that if $\eta_{1}^{-},\eta_{2}^{-},\dotsc,\eta_{j_{0}}^{-}\in(N^{-}_{\Theta})_{1}$ with $d_{N^{-}_{\Theta}}(\eta_{j}^{-},n_{j}^{-})\leq\delta$ for all $1\leq j\leq j_{0}$ , then for all $n^{+}\in\mathbb{S}(\mathfrak{S}^{+}_{\mathsf{w}})$ , there exists $1\leq j\leq j_{0}$ such that

\displaystyle\bigl{|}\pi_{\mathsf{w}}\bigl{(}\operatorname{Ad}_{\eta_{j}^{-}}(n^{+})\bigr{)}\bigr{|}\geq C_{\mathscr{w}}.

Remark 5.10.

It is important to note that although Corollary 5.9 is analogous to [64, Lemma 6.4], when projected to $\mathbb{R}\mathsf{w}$ for both results, the former has a stronger quantifier “for all $n^{+}\in\mathbb{S}(\mathfrak{S}^{+}_{\mathsf{w}})$ ”. As a result, Proposition 5.11 also has a stronger quantifier and is hence stronger than the $\mathbb{R}\mathsf{w}$ projection of [64, Proposition 6.5]. Note that the stronger proposition is possible only for the “geodesic flow” and is ultimately rooted in Lemma 5.6. Due to this stronger proposition, we do not require the non-concentration property as in [64]. At a conceptual level, this is similar to what happens in [67].

With Lemmas 5.5 and 5.9 in hand, the proof of Proposition 5.11 is nearly the same as that of [64, Proposition 6.5]. Note that in the proposition, the open neighborhood $\mathcal{U}_{m}$ can be chosen to be convex simply by shrinking it if necessary to an appropriate open ball. Also, its dependence on $m$ is not an issue since the proposition can be upgraded as discussed below.

Proposition 5.11 (LNIC).

There exist $\varepsilon\in(0,1)$ , $m_{0}\in\mathbb{N}$ , and $j_{0}\in\mathbb{N}$ such that for all $m\geq m_{0}$ , there exists a convex open neighborhood $\mathcal{U}_{m}\subset\tilde{\mathsf{U}}_{1}^{(m)}$ of the center $z_{1}\in\tilde{\mathsf{R}}_{1}$ such that there exist sections $v_{j}=\sigma^{-x_{j}}:\tilde{\mathsf{U}}_{1}^{(m)}\to{}^{\mathsf{g}^{-x_{j}}}\tilde{\mathsf{U}}_{x_{j,0}}$ for some mutually distinct admissible sequences $x_{j}=(x_{j,0},x_{j,1},\dotsc,x_{j,m-1},1)$ for all integers $0\leq j\leq j_{0}$ such that for all $\mathsf{w}\in\overline{\mathscr{w}}$ , $u\in\mathcal{U}_{m}$ , and $Z\in\mathbb{S}(\operatorname{T}_{u}(\mathcal{U}_{m}))$ , there exist $1\leq j\leq j_{0}$ such that

\displaystyle|(d\varphi_{j,u})_{u}(Z)|\geq\varepsilon

where we define $\varphi_{j}:\tilde{\mathsf{U}}_{1}^{(m)}\times\tilde{\mathsf{U}}_{1}^{(m)}\to\mathbb{R}$ by

\displaystyle\varphi_{j}(u,u^{\prime})

\displaystyle=\bigl{(}\tau^{\mathsf{w}}_{x_{j}}\circ v_{j}-\tau^{\mathsf{w}}_{x_{0}}\circ v_{0}\bigr{)}(u)-\bigl{(}\tau^{\mathsf{w}}_{x_{j}}\circ v_{j}-\tau^{\mathsf{w}}_{x_{0}}\circ v_{0}\bigr{)}(u^{\prime})

and we denote $\varphi_{j,u}=\varphi_{j}(u,\cdot)$ for all $u,u^{\prime}\in\tilde{\mathsf{U}}_{1}^{(m)}$ and $1\leq j\leq j_{0}$ .

We immediately derive the following corollary from Proposition 5.11 first on the open neighborhood $\operatorname{int}(\mathsf{U}_{1})\cap\mathcal{U}_{m}\subset\mathsf{U}_{1}$ of the center $z_{1}\in\mathsf{R}_{1}$ for a given $m\geq m_{0}$ since $\mathcal{U}_{m}$ is convex and then upgrade it to $\operatorname{int}(U)$ using the topological mixing property of the transition matrix $T$ (see Subsection 4.3). The mutual disjointness in Proposition 5.12 follows from mutual distinctness of the admissible sequences in Proposition 5.11.

Proposition 5.12.

There exist $\varepsilon\in(0,1)$ , an unbounded subset $\mathscr{m}\subset\mathbb{N}$ , and $j_{0}\in\mathbb{N}$ such that for all $m\in\mathscr{m}$ , there exists a set of Lipschitz sections $\{v_{j}:U\to U\}_{j=0}^{j_{0}}$ of $\sigma^{m}$ such that for all $\mathsf{w}\in\overline{\mathscr{w}}$ and $u,u^{\prime}\in\operatorname{int}(U)$ , there exist $1\leq j\leq j_{0}$ such that

\displaystyle|(\tau^{\mathsf{w}}_{m}\circ v_{j}-\tau^{\mathsf{w}}_{m}\circ v_{0})(u)-(\tau^{\mathsf{w}}_{m}\circ v_{j}-\tau^{\mathsf{w}}_{m}\circ v_{0})(u^{\prime})|\geq\varepsilon d(u,u^{\prime}).

Moreover, $v_{0}(U),v_{1}(U),\dotsc,v_{j_{0}}(U)$ are mutually disjoint.

Fix $\varepsilon$ , $\mathscr{m}$ , and $j_{0}$ to be the ones provided by Proposition 5.12 henceforth.

6. Dolgopyat operators and the proof of Theorem 4.8

In this section we carry out Dolgopyat’s method to prove Theorem 4.8. Many of the techniques used here go all the way back to Dolgopyat [22]. Since it is sensitive to the setting at hand, some of the arguments require careful treatment in each instance. Over the years, it has been made cleaner and more efficient in some settings such as by Stoyanov [67].

Thanks to Remark 5.10, we are able to follow Stoyanov’s strategy using cylinders and the new distance $D$ . One especially important advantage of Stoyanov’s version of Dolgopyat’s method is that we do not require the Federer/doubling property. This is not just a convenience—there is no appropriate metric to use on $\mathfrak{n}^{+}_{\Theta}$ which is also compatible with $U$ and a doubling property is not known yet for PS measures in higher rank to the best of the authors’ knowledge. The reader may also find it useful to consult [43, 62, 63] where many full proofs are provided in a similar framework.

We fix $\mathsf{w}\in\mathscr{w}$ for the rest of the section. Recall that this implies $\Theta_{\mathsf{w}}=\Theta$ and hence, Proposition 5.12 applies in this section.

6.1. Preliminary lemmas and constants

We obtain the following eventually contracting property, which also implies the same for the new distance function $D$ , from the metric Anosov property of the translation flow in Theorem 3.2, and a compactness argument for the lower bound. Another lemma follows from it as in [67, Proposition 3.3].

Lemma 6.1.

There exist $c_{0}\in(0,1)$ and $\kappa_{1}>\kappa_{2}>1$ such that for all cylinders $\mathtt{C}\subset U$ with $\operatorname{len}(\mathtt{C})=j\in\mathbb{N}$ , and $u,u^{\prime}\in\mathtt{C}$ , we have both

	$\displaystyle c_{0}\kappa_{2}^{j}d(u,u^{\prime})\leq d(\sigma^{j}(u),\sigma^{j}(u^{\prime}))\leq{c_{0}}^{-1}\kappa_{1}^{j}d(u,u^{\prime})$
	$\displaystyle c_{0}\kappa_{2}^{j}D(u,u^{\prime})\leq D(\sigma^{j}(u),\sigma^{j}(u^{\prime}))\leq{c_{0}}^{-1}\kappa_{1}^{j}D(u,u^{\prime}).$

Lemma 6.2.

There exist $p_{0}\in\mathbb{N}$ and $\rho\in(0,1)$ such that for all cylinders $\mathtt{C}\subset U$ with $\operatorname{len}(\mathtt{C})=l\in\mathbb{Z}_{\geq 0}$ and subcylinders $\mathtt{C}^{\prime},\mathtt{C}^{\prime\prime}\subset\mathtt{C}$ with $\operatorname{len}(\mathtt{C}^{\prime})=l+1$ and $\operatorname{len}(\mathtt{C}^{\prime\prime})=l+p_{0}$ , we have

\displaystyle\operatorname{diam}(\mathtt{C}^{\prime\prime})\leq\rho\operatorname{diam}(\mathtt{C})\leq\operatorname{diam}(\mathtt{C}^{\prime}).

We fix constants $c_{0}$ , $\kappa_{1}$ , $\kappa_{2}$ , $p_{0}$ , and $\rho$ provided by Lemmas 6.1 and 6.2 henceforth. We use these bounds extensively without comments. We also fix $p_{1}\in\mathbb{N}$ such that

\displaystyle\frac{1}{2}-2\rho^{p_{1}-1}\geq\frac{1}{16}.

(16)

The following is a Lasota–Yorke [37] type of lemma whose proof is similar to that of [67, Lemma 5.4] and [43, Lemma 3.9]. Its proof gives the explicit constant $A_{0}>2c_{0}^{-1}e^{\frac{T_{\mathsf{w}}}{c_{0}(\kappa_{2}-1)}}\max\bigl{(}1,\frac{T_{\mathsf{w}}}{\kappa_{2}-1}\bigr{)}>2$ which we fix henceforth.

Lemma 6.3.

There exists $A_{0}>0$ such that for all $|a|<a_{0}^{\prime}$ , $|b|>1$ , and $k\in\mathbb{N}$ , we have

((1))

if $h\in\mathcal{C}_{B}(U)$ (resp. $h\in\tilde{\mathcal{C}}_{B}(U)$ ) for some $B>0$ , then $\mathcal{L}_{a,\mathsf{w}}^{k}(h)\in\mathcal{C}_{B^{\prime}}(U)$ (resp. $\mathcal{L}_{a,\mathsf{w}}^{k}(h)\in\tilde{\mathcal{C}}_{B^{\prime}}(U)$ ) for $B^{\prime}=A_{0}\left(\frac{B}{\kappa_{2}^{k}}+1\right)$ ;

((2))

if $H\in C(U,\mathbb{R})$ and $h\in B(U,\mathbb{R})$ satisfy

\displaystyle\|H(u)-H(u^{\prime})\|_{2}\leq Bh(u)D(u,u^{\prime})

for all $u,u^{\prime}\in U_{j}$ , for all $j\in\mathcal{A}$ , for some $B>0$ , then for all $j\in\mathcal{A}$ , for all $u,u^{\prime}\in U_{j}$ , we have

\displaystyle\left\|\mathcal{L}_{\xi,\mathsf{w}}^{k}(H)(u)-\mathcal{L}_{\xi,\mathsf{w}}^{k}(H)(u^{\prime})\right\|_{2}\leq A_{0}\left(\frac{B}{\kappa_{2}^{k}}\mathcal{L}_{a,\mathsf{w}}^{k}(h)(u)+|b|\mathcal{L}_{a,\mathsf{w}}^{k}\|H\|(u)\right)D(u,u^{\prime})

for all $u,u^{\prime}\in U$ .

Now, we fix $b_{0}=1$ and some other significant positive constants

$\displaystyle E$	$\displaystyle>2A_{0}>4,$	(17)
$\displaystyle\epsilon_{1}$	$\displaystyle<\min\left(\hat{\delta},\frac{\pi c_{0}(\kappa_{2}-1)}{2T_{\mathsf{w}}}\right),$	(18)
$\displaystyle m$	$\displaystyle\in\mathscr{m}\text{ such that }\kappa_{2}^{m}>\max\left(8A_{0},\frac{4E\epsilon_{1}\rho^{p_{1}}}{c_{0}},\frac{4\cdot 128E}{c_{0}\varepsilon\rho}\right),$	(19)
$\displaystyle\mu$	$\displaystyle<\min\left(\frac{2E\epsilon_{1}c_{0}\rho^{p_{0}p_{1}+1}}{\kappa_{1}^{m}},\frac{1}{4},\frac{1}{16\cdot 16e^{2mT_{\mathsf{w}}}}\left(\frac{\varepsilon\rho\epsilon_{1}}{64}\right)^{2}\right).$	(20)

Having fixed $m\in\mathscr{m}$ , we also fix the corresponding set of Lipschitz sections $\{v_{j}:U\to U\}_{j=0}^{j_{0}}$ of $\sigma^{m}$ provided by Proposition 5.12.

6.2. Construction of Dolgopyat operators

Let $|b|>b_{0}$ . We define the set $\{\mathtt{C}_{1}(b),\mathtt{C}_{2}(b),\dotsc,\mathtt{C}_{c_{b}}(b)\}$ for some $c_{b}\in\mathbb{N}$ consisting of maximal cylinders $\mathtt{C}\subset U$ with $\operatorname{diam}(\mathtt{C})\leq\frac{\epsilon_{1}}{|b|}$ so that $U=\bigcup_{l=1}^{c_{b}}\overline{\mathtt{C}_{l}(b)}$ . We define the set $\{\mathtt{D}_{1}(b),\mathtt{D}_{2}(b),\dotsc,\mathtt{D}_{d_{b}}(b)\}$ for some $d_{b}\in\mathbb{N}$ consisting of subcylinders $\mathtt{D}\subset\mathtt{C}_{l}(b)$ with $\operatorname{len}(\mathtt{D})=\operatorname{len}(\mathtt{C}_{l}(b))+p_{0}p_{1}$ and $1\leq l\leq c_{b}$ . We define the index set $\Xi(b)=\{0,1,\dotsc,j_{0}\}\times\{1,2,\dotsc,d_{b}\}$ . We define $\mathtt{X}_{j,k}(b)=\overline{v_{j}(\mathtt{D}_{k}(b))}$ which satisfies $\mathtt{X}_{j,k}(b)\cap\mathtt{X}_{j^{\prime},k^{\prime}}(b)=\varnothing$ for all distinct $(j,k),(j^{\prime},k^{\prime})\in\Xi(b)$ . For all $J\subset\Xi(b)$ , we define the function

\displaystyle\beta_{J}=\chi_{U}-\mu\sum_{(j,k)\in J}\chi_{\mathtt{X}_{j,k}(b)}.

We record a number of basic facts derived from Lemmas 6.1 and 6.2.

—

For all $1\leq l\leq c_{b}$ and $(j,k)\in\Xi(b)$ , we have the diameter bounds:

	$\displaystyle\rho\frac{\epsilon_{1}}{\|b\|}\leq\operatorname{diam}(\mathtt{C}_{l}(b))\leq\frac{\epsilon_{1}}{\|b\|}$		(21)
	$\displaystyle\rho^{p_{0}p_{1}+1}\frac{\epsilon_{1}}{\|b\|}\leq\operatorname{diam}(\mathtt{D}_{k}(b))\leq\rho^{p_{1}}\frac{\epsilon_{1}}{\|b\|}$		(22)
	$\displaystyle\frac{\epsilon_{1}c_{0}\rho^{p_{0}p_{1}+1}}{\|b\|\kappa_{1}^{m}}\leq\operatorname{diam}(\mathtt{X}_{j,k}(b))\leq\frac{\epsilon_{1}\rho^{p_{1}}}{\|b\|c_{0}\kappa_{2}^{m}}.$		(23)

—

For all $J\subset\Xi(b)$ , we have $\beta_{J}\in L_{D}(U,\mathbb{R})$ with Lipschitz constant (cf. [67, Lemma 5.2]):

\displaystyle\operatorname{Lip}_{D}(\beta_{J})\leq\frac{\mu}{\min_{(j,k)\in J}\operatorname{diam}(\mathtt{X}_{j,k}(b))}\leq\frac{\mu|b|\kappa_{1}^{m}}{\epsilon_{1}c_{0}\rho^{p_{0}p_{1}+1}}.

(24)

—

For all $1\leq l\leq c_{b}$ , if $u,u^{\prime}\in\mathtt{C}_{l}(b)$ , then

$\displaystyle D(v_{j}(u),v_{j}(u^{\prime}))\leq\frac{\epsilon_{1}}{|b|c_{0}\kappa_{2}^{m}}.$ (25)

Definition 6.4.

For all $\xi\in\mathbb{C}$ with $|a|<a_{0}^{\prime}$ and $|b|>b_{0}$ , and $J\subset\Xi(b)$ , we define the Dolgopyat operators $\mathcal{N}_{a,J}^{\mathsf{w}}:L_{D}(U,\mathbb{R})\to L_{D}(U,\mathbb{R})$ by

\displaystyle\mathcal{N}_{a,J}^{\mathsf{w}}(h)=\mathcal{L}_{a}^{m}(\beta_{J}h)

for all $h\in L_{D}(U,\mathbb{R})$ .

Definition 6.5.

For all $|b|>b_{0}$ , a subset $J\subset\Xi(b)$ is said to be dense if for all $1\leq l\leq c_{b}$ , there exists $(j,k)\in J$ such that $\mathtt{D}_{k}(b)\subset\mathtt{C}_{l}(b)$ . Denote $\mathcal{J}(b)$ to be the set of all dense subsets of $\Xi(b)$ .

6.3. Proof of Theorem 4.8

Properties (1) and (3)(3)(2b) in Theorem 4.8 are derived from Lemma 6.3 using estimates from Eqs. 17, 19, and 20. We omit the proofs of since they are almost identical to those in [67, 43, 62, 63].

Similarly, we also omit the proof of Property (2) in Theorem 4.8 and refer the reader to [67, 62, 63]. However, we mention that the proof of Property (2) uses the Gibbs property of the measure $\nu_{U}^{\mathsf{w}}$ which is automatically satisfied since it comes from a equilibrium state. This replaces the role of the Federer/doubling property.

For the sake of completeness, we include the proof of Property (3)(3)(2a) in Theorem 4.8 where the crucial LNIC from Proposition 5.12 is used. Here we use $\Theta_{\mathsf{w}}=\Theta$ .

Lemma 6.6.

Let $|b|>b_{0}$ . Suppose $\mathtt{D}_{k}(b),\mathtt{D}_{k^{\prime}}(b)\subset\mathtt{C}_{l}(b)$ for some $1\leq k,k^{\prime}\leq d_{b}$ and $1\leq l\leq c_{b}$ such that $d(u_{0},u_{0}^{\prime})\geq\frac{1}{2}\operatorname{diam}(\mathtt{C}_{l}(b))$ for some $u_{0}\in\mathtt{D}_{k}(b)$ and $u_{0}^{\prime}\in\mathtt{D}_{k^{\prime}}(b)$ . Then, there exists $1\leq j\leq j_{0}$ such that

\displaystyle\frac{\varepsilon\rho\epsilon_{1}}{16}\leq|b|\cdot|(\tau^{\mathsf{w}}_{m}\circ v_{j}-\tau^{\mathsf{w}}_{m}\circ v_{0})(u)-(\tau^{\mathsf{w}}_{m}\circ v_{j}-\tau^{\mathsf{w}}_{m}\circ v_{0})(u^{\prime})|\leq\pi

for all $u\in\mathtt{D}_{k}(b)$ and $u^{\prime}\in\mathtt{D}_{k^{\prime}}(b)$ .

Proof.

Let $u_{0}$ , $u_{0}^{\prime}$ , $u$ , and $u^{\prime}$ be as in the lemma. The upper bound always holds on $\mathtt{C}_{l}(b)$ by Eq. 18. Recall that $\mathsf{w}\in\mathscr{w}$ and hence $\Theta_{\mathsf{w}}=\Theta$ in this section. Thus, the lemma follows from Proposition 5.12 and the estimate

\displaystyle d(u,u^{\prime})\geq d(u_{0},u_{0}^{\prime})-d(u,u_{0})-d(u^{\prime},u_{0}^{\prime})\geq\frac{\rho\epsilon_{1}}{2|b|}-2\frac{\rho^{p_{1}}\epsilon_{1}}{|b|}\geq\frac{\rho\epsilon_{1}}{16|b|}

using Eq. 16. ∎

Now, for all $\xi\in\mathbb{C}$ with $|a|<a_{0}^{\prime}$ and $|b|>b_{0}$ , for all integers $1\leq j\leq j_{0}$ , for all $H\in L(U,\mathbb{C})$ and for all $h\in\mathcal{C}_{E|b|}(U)$ , we define the functions $\chi_{j,0}^{[\xi,H,h]},\chi_{j,1}^{[\xi,H,h]}:U\to\mathbb{R}$ by

	$\displaystyle\chi_{j,0}^{[\xi,H,h]}(u)$	$\displaystyle=\frac{\bigl{\|}e^{(\tau_{m}^{\mathsf{w},(a)}+ib\tau^{\mathsf{w}}_{m})(v_{j}(u))}H(v_{j}(u))+e^{(\tau_{m}^{\mathsf{w},(a)}+ib\tau^{\mathsf{w}}_{m})(v_{0}(u))}H(v_{0}(u))\bigr{\|}}{e^{\tau_{m}^{\mathsf{w},(a)}(v_{j}(u))}h(v_{j}(u))+(1-\mu)e^{\tau_{m}^{\mathsf{w},(a)}(v_{0}(u))}h(v_{0}(u))},$
	$\displaystyle\chi_{j,1}^{[\xi,H,h]}(u)$	$\displaystyle=\frac{\bigl{\|}e^{(\tau_{m}^{\mathsf{w},(a)}+ib\tau^{\mathsf{w}}_{m})(v_{j}(u))}H(v_{j}(u))+e^{(\tau_{m}^{\mathsf{w},(a)}+ib\tau^{\mathsf{w}}_{m})(v_{0}(u))}H(v_{0}(u))\bigr{\|}}{(1-\mu)e^{\tau_{m}^{\mathsf{w},(a)}(v_{j}(u))}h(v_{j}(u))+e^{\tau_{m}^{\mathsf{w},(a)}(v_{0}(u))}h(v_{0}(u))}$

for all $u\in U$ . We need another lemma before proving Property (3)(3)(2a) in Theorem 4.8. See [67, 62, 63, 43] for a proof.

Lemma 6.7.

Let $|b|>b_{0}$ . Suppose $H\in L(U,\mathbb{C})$ and $h\in\mathcal{C}_{E|b|}(U)$ satisfy Properties (3)(3)(1a) and (3)(3)(1b) in Theorem 4.8. Then for all $(j,k)\in\Xi(b)$ , we have

\displaystyle\frac{1}{2}\leq\frac{h(v_{j}(u))}{h(v_{j}(u^{\prime}))}\leq 2\qquad\text{for all $u,u^{\prime}\in\mathtt{D}_{k}(b)$}

and also either of the alternatives

(1)

$|H(v_{j}(u))|\leq\frac{3}{4}h(v_{j}(u))$ for all $u\in\mathtt{D}_{k}(b)$ ,
(2)

$|H(v_{j}(u))|\geq\frac{1}{4}h(v_{j}(u))$ for all $u\in\mathtt{D}_{k}(b)$ .

For any $w_{1},w_{2}\in\mathbb{C}-\{0\}$ , let $\angle(w_{1},w_{2})\in[0,\pi]$ denote the angle between $w_{1}$ and $w_{2}$ viewed as vectors in $\mathbb{C}\cong\mathbb{R}^{2}$ .

Lemma 6.8.

Suppose $w_{1},w_{2}\in\mathbb{C}-\{0\}$ such that $\angle(w_{1},w_{2})\geq\alpha$ and $\frac{|w_{1}|}{|w_{2}|}\leq L$ for some $\alpha\in[0,\pi]$ and $L\geq 1$ . Then we have

\displaystyle|w_{1}+w_{2}|\leq\left(1-\frac{\alpha^{2}}{16L}\right)|w_{1}|+|w_{2}|.

Lemma 6.9.

Let $\xi\in\mathbb{C}$ with $|a|<a_{0}^{\prime}$ and $|b|>b_{0}$ . Suppose $H\in L(U,\mathbb{C})$ and $h\in\mathcal{C}_{E|b|}(U)$ satisfy Properties (3)(3)(1a) and (3)(3)(1b) in Theorem 4.8. For all integers $1\leq l\leq c_{b}$ , there exists $(j,k)\in\Xi(b)$ such that $\mathtt{D}_{k}(b)\subset\mathtt{C}_{l}(b)$ and such that $\chi_{j,0}^{[\xi,H,h]}(u)\leq 1$ or $\chi_{j,1}^{[\xi,H,h]}(u)\leq 1$ for all $u\in\mathtt{D}_{k}(b)$ .

Proof.

Let $\xi$ , $H$ , $h$ , and $l$ be as in the lemma. Suppose Alternative (1) in Lemma 6.7 holds for some $(j,k)\in\Xi(b)$ . Then it is a straightforward calculation to check that $\chi_{j,1}^{[\xi,H,h]}(u)\leq 1$ for all $u\in\mathtt{D}_{k}(b)$ , using Eq. 20. Otherwise, Alternative (2) in Lemma 6.7 holds for all $(j,k)\in\Xi(b)$ . Choose $1\leq k,k^{\prime}\leq d_{b}$ satisfying the hypotheses in Lemma 6.6 and $1\leq j\leq j_{0}$ satisfying the conclusion of Lemma 6.6. Let $u\in\mathtt{D}_{k}(b)$ and $u^{\prime}\in\mathtt{D}_{k^{\prime}}(b)$ . Note that $|H(v_{\ell}(u))|,|H(v_{\ell}(u^{\prime}))|>0$ for all $\ell\in\{0,j\}$ . We would like to apply Lemma 6.8 but first we need to establish bounds on relative angle and relative size. We start with the former. For all $\ell\in\{0,j\}$ , let $u_{\ell}\in\{u,u^{\prime}\}$ such that $|H(v_{\ell}(u_{\ell}))|=\min(|H(v_{\ell}(u))|,|H(v_{\ell}(u^{\prime}))|)$ . Then recalling the hypotheses for $H$ and $h$ , and Eqs. 25 and 19, for all $\ell\in\{0,j\}$ , we have

	$\displaystyle\frac{\|H(v_{\ell}(u))-H(v_{\ell}(u^{\prime}))\|}{\min\left(\|H(v_{\ell}(u))\|,\|H(v_{\ell}(u^{\prime}))\|\right)}$	$\displaystyle\leq\frac{E\|b\|h(v_{\ell}(u_{\ell}))D(v_{\ell}(u),v_{\ell}(u^{\prime}))}{\|H(v_{\ell}(u_{\ell}))\|}$
		$\displaystyle\leq 4E\|b\|\cdot\frac{\epsilon_{1}}{\|b\|c_{0}\kappa_{2}^{m}}\leq\frac{\varepsilon\rho\epsilon_{1}}{128}.$

Using some elementary geometry, the above shows that $\sin(\angle(H(v_{\ell}(u)),H(v_{\ell}(u^{\prime}))))\leq\frac{\varepsilon\rho\epsilon_{1}}{128}$ with $\angle(H(v_{\ell}(u)),H(v_{\ell}(u^{\prime})))\in[0,\frac{\pi}{2})$ , for all $\ell\in\{0,j\}$ and hence,

\angle(H(v_{\ell}(u)),H(v_{\ell}(u^{\prime})))\leq 2\sin(\angle(H(v_{\ell}(u)),H(v_{\ell}(u^{\prime}))))\leq\frac{\varepsilon\rho\epsilon_{1}}{64}

for all $\ell\in\{0,j\}$ . For notational convenience, we define $\varphi:U\to\mathbb{R}$ by

\varphi(w)=b(\tau^{\mathsf{w}}_{m}\circ v_{j}(w)-\tau^{\mathsf{w}}_{m}\circ v_{0}(w))\qquad\text{for all $w\in U$}.

By Lemma 6.6, we have $\frac{\varepsilon\rho\epsilon_{1}}{16}\leq|\varphi(u)-\varphi(u^{\prime})|\leq\pi$ . The second inequality is to ensure that we take the correct branch of angle in the following calculations. We will use these bounds to obtain a lower bound for $\angle(V_{j}(u),V_{0}(u))$ or $\angle(V_{j}(u^{\prime}),V_{0}(u^{\prime}))$ where we define

\displaystyle V_{\ell}(w)=e^{(\tau_{m}^{\mathsf{w},(a)}+ib\tau^{\mathsf{w}}_{m})(v_{\ell}(w))}H(v_{\ell}(w))\qquad\text{for all $w\in U$ and $\ell\in\{0,j\}$}.

Using the triangle inequality and previously computed bounds, we have

		$\displaystyle\angle\left(V_{j}(u),V_{0}(u)\right)=\angle\bigl{(}e^{i\varphi(u)}H(v_{j}(u)),H(v_{0}(u))\bigr{)}$
	$\displaystyle\geq{}$	$\displaystyle\angle\bigl{(}e^{i\varphi(u)}H(v_{j}(u)),e^{i\varphi(u^{\prime})}H(v_{j}(u))\bigr{)}-\angle\bigl{(}e^{i\varphi(u^{\prime})}H(v_{j}(u)),e^{i\varphi(u^{\prime})}H(v_{j}(u^{\prime}))\bigr{)}$
		$\displaystyle{}-\angle(H(v_{0}(u)),H(v_{0}(u^{\prime})))-\angle\bigl{(}e^{i\varphi(u^{\prime})}H(v_{j}(u^{\prime})),H(v_{0}(u^{\prime}))\bigr{)}$
	$\displaystyle={}$	$\displaystyle\|\varphi(u)-\varphi(u^{\prime})\|-\angle\left(H(v_{j}(u)),H(v_{j}(u^{\prime}))\right)-\angle(H(v_{0}(u)),H(v_{0}(u^{\prime})))$
		$\displaystyle{}-\angle(V_{j}(u^{\prime}),V_{0}(u^{\prime}))$
	$\displaystyle\geq{}$	$\displaystyle\frac{\varepsilon\rho\epsilon_{1}}{16}-\frac{\varepsilon\rho\epsilon_{1}}{64}-\frac{\varepsilon\rho\epsilon_{1}}{64}-\angle(V_{j}(u^{\prime}),V_{0}(u^{\prime}))$

and hence,

\displaystyle\angle\left(V_{j}(u),V_{0}(u)\right)+\angle\left(V_{j}(u^{\prime}),V_{0}(u^{\prime})\right)\geq\frac{\varepsilon\rho\epsilon_{1}}{32}

for all $u\in\mathtt{D}_{k}(b)$ and $u^{\prime}\in\mathtt{D}_{k^{\prime}}(b)$ . Without loss of generality, we may assume that $\angle(V_{j}(u),V_{0}(u))\geq\frac{\varepsilon\rho\epsilon_{1}}{64}$ for all $u\in\mathtt{D}_{k}(b)$ , which establishes the required bound on relative angle. For the bound on relative size, let $(\ell,\ell^{\prime})\in\{(0,j),(j,0)\}$ such that $h(v_{\ell}(u_{0}))\leq h(v_{\ell^{\prime}}(u_{0}))$ for some $u_{0}\in\mathtt{D}_{k}(b)$ . Then by Lemma 6.7, we have

\displaystyle\frac{|V_{\ell}(u)|}{|V_{\ell^{\prime}}(u)|}

\displaystyle=\frac{e^{\tau_{m}^{\mathsf{w},(a)}(v_{\ell}(u))}\big{|}H(v_{\ell}(u))\big{|}}{e^{\tau_{m}^{\mathsf{w},(a)}(v_{\ell^{\prime}}(u))}\big{|}H(v_{\ell^{\prime}}(u))\big{|}}\leq\frac{4e^{\tau_{m}^{\mathsf{w},(a)}(v_{\ell}(u))-\tau_{m}^{\mathsf{w},(a)}(v_{\ell^{\prime}}(u))}h(v_{\ell}(u))}{h(v_{\ell^{\prime}}(u))}\leq 16e^{2mT_{\mathsf{w}}}

for all $u\in\mathtt{D}_{k}(b)$ , which establishes the required bound on relative size. Now applying Lemmas 6.8 and 20 and $|H|\leq h$ on $|V_{j}(u)+V_{0}(u)|$ gives $\chi_{j,0}^{[\xi,H,h]}(u)\leq 1$ or $\chi_{j,1}^{[\xi,H,h]}(u)\leq 1$ for all $u\in\mathtt{D}_{k}(b)$ . ∎

Lemma 6.10.

There exists $a_{0}>0$ such that for all $\xi\in\mathbb{C}$ with $|a|<a_{0}$ and $|b|>b_{0}$ , if $H\in L(U,\mathbb{C})$ and $h\in\mathcal{C}_{E|b|}(U)$ satisfy Properties (3)(3)(1a) and (3)(3)(1b) in Theorem 4.8, then there exists $J\in\mathcal{J}(b)$ such that

\displaystyle\left|\mathcal{L}_{\xi,\mathsf{w}}^{m}(H)(u)\right|\leq\mathcal{N}_{a,J}^{\mathsf{w}}(h)(u)

for all $u\in U$ .

Appendix A Failure of proper discontinuity of $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$

In Theorem A.2, we exhibit numerous instances where the action $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ is not properly discontinuous. We remind the reader that the action is of course properly discontinuous whenever $G$ is of rank one, in which case $\mathfrak{a}_{\Theta}=\mathfrak{a}\cong\mathbb{R}$ , $S_{\Theta}=S$ is compact, and $\Gamma<G$ is convex cocompact.

We will use the following lemma. Recall the Weyl group $\mathcal{W}=N_{K}(A)/Z_{K}(A)$ . Note that for any loxodromic element $g\in h(\operatorname{int}A^{+})h^{-1}$ for some $h\in G$ , its set of fixed points is $h\mathcal{W}e^{+}\subset\mathcal{F}$ among which the attracting and repelling ones are $h^{+}=he^{+}\in\Lambda_{\Gamma}$ and $h^{-}=he^{-}=hw_{0}e^{+}\in\Lambda_{\Gamma}$ .

Lemma A.1.

Let $G$ be a semisimple real algebraic group. Let $g\in G$ be a loxodromic element so that $g\in h(\operatorname{int}A^{+})h^{-1}$ for some $h\in G$ . Let $w\in\mathcal{W}=N_{K}(A)/Z_{K}(A)$ be an element in the Weyl group. Then

\displaystyle\beta_{(hw^{-1})^{+}}(h,g^{k}h)=k\operatorname{Ad}_{w}(\lambda(g))\qquad\text{for all $k\in\mathbb{Z}_{\geq 0}$}.

Proof.

Let $G$ and $w$ be as in the lemma. By $G$ -equivariance, it suffices to show that $\beta_{(w^{-1})^{+}}(e,a_{v})=\operatorname{Ad}_{w}(v)$ for all $v\in\operatorname{int}\mathfrak{a}^{+}$ . Indeed, we calculate that

\displaystyle a_{-v}w^{-1}=w^{-1}\cdot wa_{-v}w^{-1}\in Ka_{-\operatorname{Ad}_{w}(v)}

and so by definition, $\beta_{(w^{-1})^{+}}(e,a_{v})=-\sigma(a_{-v},(w^{-1})^{+})=\operatorname{Ad}_{w}(v)$ . ∎

Theorem A.2.

There exist infinitely many triples $(G,\Gamma,\psi)$ where $G$ is a connected semisimple real algebraic group of rank at least $2$ , $\Gamma<G$ is a Zariski dense $\Theta$ -Anosov subgroup for some subset $\Theta\subset\Pi$ with $\#\Theta\geq 2$ , and $\psi\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ , such that the action $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ defined in Eq. 4 is not properly discontinuous.

Proof.

Let $G_{0}$ be any rank one connected simple real algebraic group (of which there are infinitely many), and $\Gamma_{0}<G_{0}$ be any convex cocompact subgroup. We simply write $\operatorname{Id}:\Gamma_{0}\to G_{0}$ for the representation obtained by restricting the identity map $\operatorname{Id}_{G_{0}}:G_{0}\to G_{0}$ . Take a different convex cocompact discrete faithful representation $\rho:\Gamma_{0}\to G_{0}$ and consider the induced representation

\displaystyle\operatorname{Id}\times\rho:\Gamma_{0}\to G_{0}\times G_{0}.

Take the rank two connected semisimple real algebraic group $G=G_{0}\times G_{0}$ . Denote by $\mathcal{W}_{0}=\{e,w_{00}\}$ and $\mathcal{F}_{0}$ the Weyl group and the Furstenberg boundary associated to $G_{0}$ , and by $\mathcal{W}=\mathcal{W}_{0}\times\mathcal{W}_{0}\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and $\mathcal{F}=\mathcal{F}_{0}\times\mathcal{F}_{0}$ the same for $G$ , respectively. Note that $w_{0}=(w_{00},w_{00})\in\mathcal{W}$ . Take the discrete subgroup $\Gamma=(\operatorname{Id}\times\rho)(\Gamma_{0})<G$ which is the self-joining of $\Gamma_{0}$ via $\rho$ . It is known that such discrete subgroups are $\Pi$ -Anosov either by checking that any orbit of $\Gamma\curvearrowright G/K$ gives a quasiisometric embedding of $\Gamma_{0}$ or by checking the antipodality condition (in the current setting, $\mathcal{F}^{(2)}\cong\mathcal{F}_{0}^{(2)}\times\mathcal{F}_{0}^{(2)}$ ). Hence, $\Gamma$ is also $\Theta$ -Anosov for any nonempty subset $\Theta\subset\Pi$ . We also assume that $\Gamma<G$ is Zariski dense (cf. [30, Lemma 2.2] for an equivalent criterion). Denote by $\Lambda_{\Gamma_{0}}\subset\mathcal{F}_{0}$ the limit set of $\Gamma_{0}$ and let $\zeta_{\rho}:\Lambda_{\Gamma_{0}}\to\mathcal{F}_{0}$ be the induced (continuous) boundary map.

Now, take any loxodromic element $\gamma=(\gamma_{0},\rho(\gamma_{0}))\in\Gamma$ for some $\gamma_{0}\in\Gamma_{0}$ , so that $\gamma=h(\operatorname{int}A^{+})h^{-1}$ for some $h\in G$ . Write $\xi^{+},\xi^{-}\in\Lambda_{\Gamma_{0}}$ for the attracting and repelling fixed points of $\gamma_{0}$ , respectively, which are distinct. Then, the set of fixed points of $\gamma$ is

\displaystyle h\mathcal{W}e^{+}=\{(\xi^{+},\zeta_{\rho}(\xi^{+})),(\xi^{-},\zeta_{\rho}(\xi^{+})),(\xi^{+},\zeta_{\rho}(\xi^{-})),(\xi^{-},\zeta_{\rho}(\xi^{-}))\}\subset\mathcal{F}

among which the attracting and repelling ones are $(\xi^{+},\zeta_{\rho}(\xi^{+})),(\xi^{-},\zeta_{\rho}(\xi^{-}))\in\Lambda_{\Gamma}$ . Take any Weyl group element $w\in\mathcal{W}-\{e,w_{0}\}$ , say $w=(e,w_{00})$ for the sake of concreteness, and also the fixed point $x=hw^{-1}e^{+}=hwe^{+}=(\xi^{+},\zeta_{\rho}(\xi^{-}))\in\mathcal{F}-\Lambda_{\Gamma}$ , where $x\notin\Lambda_{\Gamma}=(\operatorname{Id}\times\zeta_{\rho})(\Lambda_{\Gamma_{0}})$ since $\xi^{-}\neq\xi^{+}$ . By Lemma A.1, we have

\displaystyle\beta_{x}(e,\gamma^{k})=k\operatorname{Ad}_{w}(\lambda(\gamma))\in\operatorname{Ad}_{w}(\operatorname{int}\mathfrak{a}^{+})\qquad\text{for all $k\in\mathbb{N}$}.

Now, simply choose $\psi\in\operatorname{int}\mathcal{L}_{\Theta}^{*}$ with $\operatorname{Ad}_{w}(\lambda(\gamma))\in\ker\psi$ so that

\displaystyle\psi\bigl{(}\beta_{\gamma^{k}x}(e,\gamma^{k})\bigr{)}=\psi(\beta_{x}(e,\gamma^{k}))=0\qquad\text{for all $k\in\mathbb{N}$}.

(26)

Take any $y\in\mathcal{F}$ such that $(x,y)\in\mathcal{F}^{(2)}$ , say $y=hw^{-1}e^{-}=(\xi^{-},\zeta_{\rho}(\xi^{+}))$ . Now, by compactness of $\mathcal{F}$ , we could pass to subsequences of $\{\gamma^{k}\}_{k\in\mathbb{N}}$ to get $\{k_{j}\}_{j\in\mathbb{N}}\subset\mathbb{N}$ such that $\{\gamma^{k_{j}}(x,y)\}_{j\in\mathbb{N}}$ converges; however, for our choice of $(x,y)$ , the original sequence is already just a constant sequence. This imples the easier property that $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}$ is not properly discontinuous. Using Eq. 26 as well, we have $\gamma^{k}(x,y,0)=\bigl{(}\gamma^{k}x,\gamma^{k}y,\psi\bigl{(}\beta_{\gamma^{k}x}(e,\gamma^{k})\bigr{)}\bigr{)}=(x,y,0)$ for all $k\in\mathbb{N}$ , so we similarly conclude that $\{\gamma^{k}(x,y,0)\}_{j\in\mathbb{N}}$ is a constant sequence and hence converges. Thus, $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$ is not properly discontinuous. ∎

Appendix B Non-obtuseness of Weyl chambers

We give a quick proof of a fundamental proposition below regarding angles in Weyl chambers. Although a stronger statement holds regarding roots, the proposition below suffices for our purposes. We refer the reader to the concise book of Serre [66, Chapter V] for the basics of root systems used here. In fact the proof given here is along the lines of [66, Chapter V, §7, Proposition 3].

We prefer to work on $\mathfrak{a}^{*}$ and so we endow it with the inner product induced by $\mathfrak{a}$ . Let $(\mathfrak{a}^{*})^{+}$ be the closed positive Weyl chamber corresponding to $\Phi^{+}$ . We write $s_{\alpha}:=\operatorname{Id}_{\mathfrak{a}^{*}}-2\frac{\langle\cdot,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha\in\mathfrak{a}^{**}$ for the symmetry associated to the root $\alpha\in\Phi$ . It follows from the definition of root systems that for all $\alpha,\beta\in\Phi$ , the number $n(\beta,\alpha):=2\frac{\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in\mathbb{Z}$ is an integer such that $s_{\alpha}(\beta)=\beta-n(\beta,\alpha)\alpha\in\Phi$ is also a root.

Proposition B.1.

We have $\langle v_{1},v_{2}\rangle\geq 0$ for all $v_{1},v_{2}\in\mathfrak{a}^{+}$ . Equivalently, if $\dim(\mathfrak{a})\neq 1$ , then $\alpha\notin\operatorname{int}((\mathfrak{a}^{*})^{+})$ for all $\alpha\in\Pi$ .

Proof.

Suppose that $\dim(\mathfrak{a})\neq 1$ . Then, there exist two distinct simple roots $\alpha,\beta\in\Pi$ and for the sake of contradiction, suppose that $\langle\beta,\alpha\rangle>0$ . Then $n(\beta,\alpha)>0$ and $n(\alpha,\beta)>0$ . It is a standard fact of root systems that $n(\beta,\alpha)n(\alpha,\beta)\in\{0,1,2,3,4\}$ . Now, by our initial assumption, the product cannot vanish. Next, by the Cauchy–Schwarz inequality, if the product is $4$ , then $\alpha$ and $\beta$ are collinear in which case $\beta=2\alpha$ or $\alpha=2\beta$ by integrality of $n(\beta,\alpha)$ and $n(\alpha,\beta)$ , contradicting simplicity of $\alpha$ and $\beta$ . Thus, $n(\beta,\alpha)n(\alpha,\beta)\in\{1,2,3\}$ . In any case, integrality again implies that $n(\beta,\alpha)=1$ or $n(\alpha,\beta)=1$ . Therefore, either $s_{\alpha}(\beta)=\beta-\alpha\in\Phi$ or $s_{\beta}(\alpha)=\alpha-\beta\in\Phi$ , both of which contradict simplicity of $\alpha$ and $\beta$ once again. ∎

Appendix C A minor correction

We take the opportunity to make a minor correction in our prequel [18]. In [18, §4.3], the transition matrix $T$ is not topologically mixing a priori. However, it is topologically transitive (also called irreducible) as a consequence of the density result in [18, Proposition 6.3] which only uses ergodicity from [18, Proposition 3.7] due to Lee–Oh. One can then arrange the Markov section, by inserting extra rectangles if necessary as in the beginning of the proof of [18, Theorem 6.10], so that coprime primitive periods exist for the Poincaré first return map, implying that $T$ is aperiodic. Then, towards the end of the proof of [18, Theorem 6.10], “for any $n\geq N_{T}$ ” should be replaced with “for some coprime $n_{1},n_{2}\geq N_{T}$ ” and again we conclude $\theta\in\{0,1\}$ .

With the above correction, the topological mixing result in [18, Theorem 8.1] is valid. Only now, one derives as a consequence that $T$ is indeed topologically mixing.

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		$\displaystyle\|\log\hat{u}(z_{1})-\log\hat{u}(z_{2})\|$
	$\displaystyle\geq$	$\displaystyle\|\log\hat{u}(z_{1})-\log\hat{u}(\tilde{a}_{\rho,t(z_{1},z_{2})}(z_{1}))\|-\|\log\hat{u}(p_{1}(z_{1},z_{2}))-\log\hat{u}(z_{2})\|$
	$\displaystyle\geq$	$\displaystyle\eta\|t(z_{1},z_{2})\|-\|\log\hat{u}(p_{1}(z_{1},z_{2}))-\log\hat{u}(z_{2})\|.$

Exponential mixing and essential spectral gaps for Anosov subgroups

Abstract.

1. Introduction

Theorem 1.1.

Remark 1.2.

Theorem 1.3.

Theorem 1.4.

Theorem 1.5.

1.1. Applications

Theorem 1.6.

Remark 1.7.

Theorem 1.8.

1.2. Related works

1.3. On the proof of Theorem 1.1

1.4. Organization of the paper

Acknowledgements

2. Anosov subgroups and their translation flows

2.1. Lie theoretic preliminaries

Definition 2.1 (Iwasawa cocycle).

Definition 2.2 (Θ\Theta-Busemann function).

Definition 2.3 (Θ\Theta-Hopf parameterization).

Definition 2.4 (Θ\Theta-Gromov product).

2.2. Zariski dense discrete subgroups

Definition 2.5 (Θ\Theta-limit set).

Definition 2.6 (Θ\Theta-limit cone).

Definition 2.7 (Θ\Theta-growth indicator).

Definition 2.8 ((Γ,ψ)(\Gamma,\psi)-conformal measure).

Theorem 2.9 (53, Theorem 8.4).

2.3. Anosov subgroups

Definition 2.10 (Anosov subgroup).

Theorem 2.11.

Theorem 2.12.

2.4. Translation flows

Theorem 2.13.

3. Anosov property of the translation flow

Remark 3.1.

Theorem 3.2.

Corollary 3.3.

3.1. Metric Anosov flows, Markov sections, and reparametrizations

Definition 3.4 (Local product structure).

Definition 3.5 (Complete set of rectangles).

Definition 3.6 (Markov section).

Definition 3.7 (Metric Anosov flow).

Theorem 3.8.

Definition 3.9 (Reparametrization).

Proposition 3.10.

Proof.

3.2. Stable and unstable foliations for the translation flow

Lemma 3.11.

Proof.

3.3. Projective Anosov representations

Theorem 3.12 (9, Proposition 5.7).

3.4. Proof of Theorem 3.2

Theorem 3.13.

Proof.

Proposition 3.14.

Proof of Theorem 3.2.

Remark 3.15.

4. Transfer operators, Dolgopyat’s method, and the proof of Theorem 1.5 and applications

4.1. Reduction of Theorem 1.5 by rescaling

Theorem 4.1.

4.2. Compatible family of Markov sections

Lemma 4.2.

Proof.

4.3. Symbolic dynamics

Definition 4.3 (Cylinder).

4.4. Thermodynamics

Definition 4.4 (Pressure).

4.5. Transfer operators

Definition 4.5 (Transfer operators).

4.6. Dolgopyat’s Method and the proof of Theorem 4.1 and applications

Remark 4.6.

Theorem 4.7.

Theorem 4.8.

Remark 4.9.

5. Local non-integrability condition

5.1. Smooth extensions of coding maps

Definition 5.1 (First return vector map, Holonomy).

5.2. Local non-integrability condition

Definition 5.2 (Associated sequence in GG).

Definition 2.2 ( $\Theta$ -Busemann function).

Definition 2.3 ( $\Theta$ -Hopf parameterization).

Definition 2.4 ( $\Theta$ -Gromov product).

Definition 2.5 ( $\Theta$ -limit set).

Definition 2.6 ( $\Theta$ -limit cone).

Definition 2.7 ( $\Theta$ -growth indicator).

Definition 2.8 ( $(\Gamma,\psi)$ -conformal measure).

Definition 5.2 (Associated sequence in $G$ ).

Appendix A Failure of proper discontinuity of $\Gamma\curvearrowright\mathcal{F}_{\Theta}^{(2)}\times\mathbb{R}$