Pressure gaps, Geometric potentials, and nonpositively curved manifolds

We shall introduce notations and the setup to contextualize our results. Throughout the paper, we assume the following conditions on the Riemannian manifold $M$:

1. (C1)

   $M$ is a closed rank 1 manifold with nonpositive sectional curvature.

2. (C2)

   $\mathrm{Sing}$ is either $T^{1}T_{0}$ or the flat strip case $T^{1}T_{0}\times[-1,1]$, where $T_{0}$ is a codimension 1 totally geodesic flat torus.

3. (C3)

   $M$ is negatively curved outside $T_{0}$ or the flat strip $T_{0}\times[-1,1]$.

For these types of manifolds, it is convenient to study the behavior of geodesics near $T_{0}$ or $T_{0}\times[-1,1]$ using Fermi coordinates (see Section 2.3 for more details). For $p\in M$, let $x(p)$ be the signed distance from $p$ to $T_{0}$ or $T_{0}\times[-1,1]$, and $s(p)$ the closest point on $T_{0}$ or $T_{0}\times[-1,1]$ from $p$. The map $s:M\to T_{0}$ is a projection onto $T_{0}$, which induces a projection $ds:TM\to TT_{0}$.

For $v\in T_{p}M$, we consider two components from its Fermi coordinates: $x_{v}=x(p)$, and the signed angle $\phi_{v}$ between $v$ and the hypersurface $x=x(p)$. We define the radial curvature $K_{\perp}(v)$ at $v$ as the sectional curvature of the tangent plane spanned by $\{v,X\}$ where $X=\partial/\partial x$.

We are particularly interested in the following two types of manifolds:

• •

  $M$ is type 1 if $M$ has an order $m$ uniform curvature bound, i.e., $K_{\perp}$ vanishes uniformly to order $m-1$ over $T_{0}$ or the flat strip (see (4.2) for more details).

• •

  $M$ is type 2 if $\dim M=2$ and $M$ has an order $m$ nonuniform curvature bound, i.e., the Gaussian curvature vanishes up to order $m-1$ over $T_{0}$ or the flat strip (see (5.1) and (5.2)).

In Examples subsection 1, we discuss manifolds satisfying the above hypotheses in more detail. For example, surfaces of genus greater than one with analytic Riemannian metrics are type 2 manifolds. See Figure 1.1 and Figure 1.2 for some illustrations.

In this paper, in addition to conditions (C1)-(C3), we assume the following condition on the continuous potential $\varphi:T^{1}M\to\mathbb{R}$:

1. (C4)

   If $M$ is type 1, then $\varphi$ is constant on $\mathrm{Sing}$. If $M$ is type 2, then $\varphi$ is transversally constant on $\mathrm{Sing}$, meaning $\varphi$ depends only on the image of the projection $\mathrm{Sing}\to T^{1}T_{0}$.

Note that when $M$ is type 2, we do not assume that $\varphi$ is constant on $\mathrm{Sing}$.

Any potential function $\varphi:T^{1}M\to\mathbb{R}$ can be extended to $\varphi:TM\to\mathbb{R}$ via

With this extension, the integral of $\varphi$ along any curve is independent of parametrization.

We are now ready to state our first result on obtaining the pressure gap. We denote by $N_{R}(\mathrm{Sing})$ the radius $R$ neighborhood of $\mathrm{Sing}$ in $T^{1}M$.

In particular, any potential that is Hölder continuous with sufficiently large exponents at $\mathrm{Sing}$ satisfies (1.2). We also note that it is sufficient to have a lower bound on $\varphi(v)-\varphi(ds(v))$ near $\mathrm{Sing}$, since when $\varphi(v)$ is larger than $\varphi(ds(v))$, $\varphi$ accumulates more pressure outside the singular set. When $\varphi$ is locally constant, we achieve [BCFT18, Theorem B] in our setup:

We say that a potential $\varphi$ has a phase transition at $q_{0}$ if the pressure map $q\mapsto P(q\varphi)$ fails to be differentiable at $q_{0}$. It is well known that the uniqueness of equilibrium states implies the differentiability of the pressure map; see [Rue78].

Our second main result is that no phase transition appears if $\varphi$ decays rapidly near $\mathrm{Sing}$:

The proof of Theorem A relies on an abstract pressure criterion (Proposition 7.2). For readability, we defer the detailed exposition of this abstract result to Section 7. In essence, Proposition 7.2 demonstrates that if the geodesic flow $\mathcal{F}$ satisfies the following conditions: (1) a "strong" specification property, (2) the singular set can be shadowed by nearby vectors, and (3) the potential $\varphi$ decays rapidly enough, then $\varphi$ exhibits a pressure gap.

The second theme of this paper is dedicated to studying the behavior of the geometric potential $\varphi^{u}:T^{1}M\to\mathbb{R}$ near $T_{0}$. Recall that the geometric potential $\varphi^{u}$ is defined as

where $E^{u}(v)$ is the unstable subspace (see Section 2 for details).

For type 2 surfaces without flat strips, Gerber and Niţică [GN99, Theorem 3.1] and Gerber and Wilkinson [GW99, Lemma 3.3] provided Hölder continuity estimates for the geometric potential $\varphi^{u}$ at $\mathrm{Sing}$. The following result shows that, under a natural Ricci curvature constraint, similar Hölder continuity estimates can be extended to higher-dimensional cases.

In what follows, we denote by $a(v)\approx b(v)$ near $S$, if there exists a neighborhood $N$ of $S$ and $C>1$ such that $C^{-1}b(v)\leq a(v)\leq Cb(v)$ for $v\in N$. We say $M$ has order $m$ Ricci curvature bounds if the Ricci curvature $\mathrm{Ric}(v)$ vanishes uniformly to order $m-1$ over $T_{0}$ (see (6.3)).

The no-flat-strip condition is necessary for the Hölder continuity. See Remark 6.5 for more details.

In most cases, the Hölder continuity of geometric potentials is unknown for nonuniformly hyperbolic systems, especially in higher dimensions. As an immediate consequence of Theorem C, we have the following partial result for higher-dimensional manifolds:

We note that our method, inspired by [GW99], currently only establishes the Hölder continuity of the geometric potential $\varphi^{u}$ at $\mathrm{Sing}$. Achieving global Hölder continuity of $\varphi^{u}$ may require the Hölder continuity of the unstable Jacobian tensor $U^{u}$, which is still unclear in higher-dimensional cases.

On the other hand, as a consequence of Theorem C, we know that $\varphi^{u}$ is a borderline case of the pressure gap criterion given in Theorem A (i.e., $\varepsilon_{1}=\varepsilon_{2}=0$). Moreover, it is known that for manifolds (including surfaces) whose singular sets are unit tangent bundles of flat, totally geodesic codimension 1 tori, the geometric potential $\varphi^{u}$ exhibits a phase transition at $q=1$ (see [BBFS21, p. 530] and [BCFT18, Theorem C]).

In other words, Theorem C shows that the pressure gap criterion given in Theorem A is optimal in the sense that there are examples at the boundary of our criterion that do not have pressure gaps (see Figure 1.3).

In Figure 1.3, each point corresponds to potentials $\varphi$ satisfying $-\varphi(v)\approx|x_{v}|^{a}+|\phi_{v}|^{b}$ near $\mathrm{Sing}$. The shaded region represents potentials that have a pressure gap and no phase transitions by Theorem A. The geometric potential $\varphi^{u}$ lies at the vertex $(\frac{m}{2},\frac{m}{m+2})$ of the shaded region.

We conclude this subsection by posing an open question:

The prototype of type 1 manifolds is the surface of revolution with profile $f(x)=1+|x|^{r}$, which is the main example discussed in Lima, Matheus, and Melbourne [LMM]. For higher dimensions, an important example is the Heintze example (see Ballman, Brin, and Eberlein [BBE85] or [BCFT18, Section 10.2]). The simplest version of the Heintze example starts with a finite volume hyperbolic 3-manifold with one cusp, then removes the cusp and flattens the region near the cross-section. Recall that the cross-section of the cusp is a codimension 1 totally geodesic flat torus. The Heintze example is obtained by gluing two identical copies of the above 3-manifold along the cross-section (see Figure 1.1).

Type 2 surfaces (see Figure 1.2) were introduced in Gerber and Niţică [GN99] and Gerber and Wilkinson [GW99]. The archetype is a rank 1 nonpositively curved surface with an analytic metric. In such cases, it is well-known that $\mathrm{Sing}$ consists of unit tangent bundles of finitely many closed geodesics (see [BCFT18, Section 10.1] for a sketched proof).

We remark that for nonpositively curved surfaces, Coudène and Schapira [CS14, Theorem 3.2] (inspired by the unpublished work of Cao and Xavier [CX08]) showed that flat strips for nonpositively curved manifolds close up. However, in general, it is unknown if the singular geodesics or the (higher dimensional) zero curvature strips close up. Nevertheless, in all known examples, to the best of the authors’ knowledge, the singular sets do close up, leading to our hypothesis on the existence of $T_{0}$.

There is no singular set when the dynamical system is uniformly hyperbolic. Hence, the pressure gap persists for a broad class of potentials, allowing one to derive ergodic properties for associated equilibrium states. The origin of this fact traces back to the work of Bowen [Bow74] for maps and Franco [Fra77] for flows. For nonuniformly hyperbolic systems, Climenhaga and Thompson [CT16] proposed using the pressure gap as a condition to obtain ergodic properties of equilibrium states, particularly uniqueness, similar to uniform hyperbolic cases.

Burns et al. [BCFT18] applied the argument from [CT16] and derived a necessary condition for the pressure gap. Specifically, they showed that for closed rank 1 nonpositively curved manifolds, if $\varphi$ is locally constant near $\mathrm{Sing}$, then $\varphi$ has a pressure gap. This work was inspired by Knieper [Kni98], where the entropy gap $h_{\mathrm{top}}(\mathrm{Sing})<h_{\mathrm{top}}(\mathcal{F})$ was established as a consequence of the uniqueness of the measure of maximal entropy. Recall that the topological entropy $h_{\mathrm{top}}(\mathcal{F})$ is the pressure of the zero potential.

Gelfert and Schapira [GS14] compared different notions of pressure for closed rank 1 nonpositively curved manifolds, such as topological pressure, Gurevich pressure (or periodic orbit pressure), and their restrictions on singular and regular sets. They pointed out that under certain conditions, these different notions of pressure are identical. Similar discussions can be found in [BCFT18, Propositions 2.8 and 6.4].

In geometry, the Liouville measure is an equilibrium state for the geometric potential. Ergodic properties of equilibrium states have been extensively studied. Several recent contributions have employed the Climenhaga-Thompson strategy (see [CT21] for a survey on the strategy). For example, Chen, Kao, and Park [CKP20, CKP21] worked on no focal points settings; Climenhaga, Knieper, and War focused on no conjugate points manifolds [CKW21]; Call, Constantine, Erchenko, and Sawyer [CCE⁺] discussed flat surfaces with singularities.

There are many other relevant discussions on the ergodic properties of equilibrium states. For example, uniqueness is discussed in [GR19], the Kolmogorov property in [CT22], the Bernoulli property in [Pes77, BG89, OW98, LLS16, CT22, ALP], and the central limit theorem in [TW21, LMM].

For geometric potentials of rank 1 surfaces, Burns and Gelfert [BG14] pointed out the existence of a phase transition. This was also confirmed in [BCFT18] using a different approach. A recent work by Burns, Buzzi, Fisher, and Sawyer [BBFS21] further investigated the edge case of $\varphi=q\varphi^{u}$ for $q=1$. They showed that the Liouville measure is the only equilibrium state not supported on $\mathrm{Sing}$. However, the Hölder continuity of $\varphi^{u}$ is even less known. It is only assured by Gerber and Wilkinson [GW99, Theorem I] for type 2 surfaces. Much less is known about $\varphi^{u}$ in higher dimensional cases.

In Section 2, we recall some relevant background material from geometry and dynamics. In Section 3, we introduce the main shadowing technique and a key inequality to prove Theorem A. Sections 4 and 5 are devoted to technical estimates in type 1 and type 2 settings by analyzing the relevant Riccati equations. Section 6 focuses on the geometric potential and the proof of Theorem C. The proof of Theorem A is presented in Section 7 as a consequence of a more general pressure gap criterion, Proposition 7.2. The proof draws inspiration from [BCFT18, Theorem B]. However, our specific setup allows us to circumvent several technicalities and arrive at a more straightforward proof than that presented in [BCFT18]. In the appendix, we show that for warped product metrics, radial curvature bounds and Ricci curvature bounds are equivalent, and we provide a proof of Peres’ lemma for flows.

This subsection will survey relevant geometric features of nonpositively curved manifolds. A more comprehensive survey of these results can be found in [Bal95, Ebe01].

Let $M$ be a closed nonpositively curved manifold, and $\{f_{t}\}_{t\in\mathbb{R}}$ the geodesic flow on the unit tangent bundle $T^{1}M$. The tangent bundle $TT^{1}M$ contains three $df_{t}$-invariant continuous bundles $E^{s},E^{u}$, and $E^{c}$. The bundle $E^{c}$ is one-dimensional along the flow direction, and the other two bundles $E^{s/u}$, which are orthogonal to $E^{c}$ with respect to the Sasaki metric, can be described using Jacobi fields. If $M$ is negatively curved, these three bundles form a splitting of $TT^{1}M$.

A Jacobi field $J$ along a geodesic $\gamma$ is a vector field along $\gamma$ satisfying the Jacobi equation

where $R$ is a Riemannian curvature tensor. A Jacobi field $J$ is orthogonal if there exists $t_{0}\in\mathbb{R}$ such that $J(t_{0})$ and $J^{\prime}(t)$ are perpendicular to $\gamma^{\prime}(t_{0})$. It is well known that when this orthogonal property holds at some $t_{0}\in\mathbb{R}$, then it holds for all $t\in\mathbb{R}$. A Jacobi field $J$ is parallel if $J^{\prime}(t)=0$ for all $t\in\mathbb{R}$.

Denoting the space of orthogonal Jacobi fields along $\gamma$ by $\mathcal{J}^{\perp}(\gamma)$, we can identify $T_{v}T^{1}M$ with $\mathcal{J}^{\perp}(\gamma_{v})$ as follows. Consider a vector $v\in T_{p}M$. Using the Levi-Civita connection the tangent space $T_{v}TM$ at $v$ may be identified with the direct sum $H_{v}\oplus V_{v}$ of horizontal and vertical subspace, respectively, each isomorphic to $T_{p}M$ equipped with the norm induced from the Riemannian metric on $M$. The Sasaki metric on $TM$ is defined by declaring $H_{v}$ and $V_{v}$ to be orthogonal. Restricted to $T^{1}M$, the tangent space $T_{v}T^{1}M$ corresponds to $H_{v}\oplus v^{\perp}$ under this identification. Then any vector $\xi\in T_{v}T^{1}M$ for $v\in T^{1}M$ may be written as $(\xi_{h},\xi_{v})$ and can be identified with an orthogonal Jacobi field $J_{\xi}\in\mathcal{J}^{\perp}(\gamma_{v})$ along $\gamma_{v}$ with the initial conditions $(J_{\xi}(0),J^{\prime}_{\xi}(0))=(\xi_{h},\xi_{v})$. Moreover, the Sasaki norm of $df_{t}(\xi)$ satisfies

The stable subspace $E^{s}(v)$ at $v\in T^{1}M$ is then defined as

Similarly, the unstable subspace $E^{u}(v)$ consists of vectors $\xi\in T_{v}T^{1}M$ where $\|J_{\xi}(t)\|$ is bounded for $t\leq 0$. Notice that the subbundles $E^{u}(v)$ and $E^{s}(v)$ are integrable to the respective foliations $W^{u}(v)\subset T^{1}M$ and $W^{s}(v)\subset T^{1}M$. The footprints of $W^{u}(v)$ and $W^{s}(v)$ on $M$ are called the unstable and stable horospheres, which are denoted by $H^{u}(v)$ and $H^{s}(v)$, respectively.

The rank of a vector $v\in T^{1}M$ is the dimension of the space of parallel Jacobi fields along $\gamma_{v}$, which coincides with the number $1+\dim(E^{s}\cap E^{u})$. We say the manifold is rank 1 if it has at least one rank 1 vector. This paper will focus mainly on closed rank 1 nonpositively curved manifolds.

The singular set is a set of vectors on which the geodesic flow fails to display uniform hyperbolicity, and it is defined by

The singular set is closed and $\mathcal{F}$-invariant, and in the case where $M$ is a surface the singular set can be characterized as the set of vectors $v$ where the Gaussian curvature $K(\gamma_{v}(t))$ vanishes for all $t\in\mathbb{R}$. The complement of the singular set is the regular set

The geodesic flow restricted to the regular set is hyperbolic, but the degree of hyperbolicity, which can be measured by the function

where $\lambda^{u}(v)$ is the minimum eigenvalue of the shape operator on the unstable horosphere $H^{u}(v)$ at $v$. Using $\lambda$ we can define nested compact subsets $\{\mathrm{Reg}(\eta)\}_{\eta>0}$ of $\mathrm{Reg}$ where

These subsets may be viewed as uniformity blocks in the sense of Pesin’s theory, where the hyperbolicity is uniform. More details and properties of the function $\lambda$ can be found in [BCFT18].

The geometric potential $\varphi^{u}:T^{1}M\to\mathbb{R}$ is an important potential that measures the infinitesimal volume growth in the unstable direction:

In order to study the geometric potential, it is convenient to study Riccati equations. Interestingly, the shape operator of unstable (and unstable) horosphere is a solution of a Riccati equation. To see this, we start by introducing terminologies.

Let $H\subset M$ be a hypersurface orthogonal to $\gamma_{v}$ at $\pi v$ where $\pi:T^{1}M\to M$ is the canonical projection. An orthogonal Jacobi field $J\in\mathcal{J}^{\perp}(\gamma_{v})$ is called a $H$-Jacobi field along $\gamma_{v}$, if $J$ comes from varying $\gamma_{v}$ through unit speed geodesics orthogonal to $H$. We denote the set of $H$- Jacobi fields by $\mathcal{J}_{H}(\gamma_{v})$ . The shape operator on $H$ is the symmetric linear operator $U:T_{\pi v}H\to T_{\pi v}H$ defined by $U(v)=\nabla_{v}N$, where $N$ is the unit normal vector field toward the same side as $v$.

We are particularly interested in the unstable horosphere $H=H^{u}(v)$ at $v$. In this case, $J_{H^{u}}(\gamma_{v})$ coincides with the space of unstable Jacobi field $J^{u}(\gamma_{v})$. For $t\in\mathbb{R}$, let $U_{v}^{u}(t):T_{\pi v}H^{u}(f_{t}v)\to T_{\pi v}H^{u}(f_{t}v)$ be the shape operator of the unstable horosphere $H^{u}(f_{t}v)$. We know $U_{v}^{u}(t)$ is a a positive semidefinite symmetric linear operator on $(f_{t}v)^{\perp}$, and for any unstable Jacobi field $J(t)$ it satisfies $J^{\prime}(t)=U_{v}^{u}(t)J(t)$; see [BCFT18, Lemma 2.9].

For any vector $v\in T^{1}M$, let $\mathcal{K}(v):v^{\perp}\to v^{\perp}$ is the symmetric linear map defined via $\langle\mathcal{K}(v)X,Y\rangle:=\langle R(X,v)v,Y\rangle$ for $X,Y\in v^{\perp}$. Using the Jacobi equation, for an unstable Jacobi field $J(t)$ we know $J^{\prime\prime}(t)+K(f_{t}v)J(t)=0$ and $J^{\prime}(t)=U_{v}^{u}(t)J(t)$. Thus, we get the operator-valued Riccati equation:

see [BCFT18, (7.6)]. Using the above notation, the Ricci curvature ${\rm Ric}(v)$ at $v$ is defined as the trace of the map ${\mathcal{K}}(v)$.

We now briefly survey relevant results in thermodynamic formalism. The general notion of topological entropy and pressure described in the following can be defined for an arbitrary flow $\mathcal{F}=\{f_{t}\}_{t\in\mathbb{R}}$ in a compact metric space $(X,d)$.

For any $t>0$, we define a metric $d_{t}$ on $X$ via

and the corresponding $\delta$-ball around $x\in X$ in $d_{t}$-metric will be denoted by $B_{t}(x,\delta)$. We say a subset $E$ of $X$ is $(t,\delta)$-separated if $d_{t}(x,y)\geq\delta$ for distinct $x,y\in E$. Moreover, we will identify $(x,t)\in X\times[0,\mathbb{R}^{+})$ with the orbit segment of length $t$ starting at $x$.

Let $\varphi\colon X\to\mathbb{R}$ be a continuous function on $X$, which we often call a potential. We define ${\displaystyle\Phi(x,t):=\int_{0}^{t}\varphi(f_{\tau}x)\,d\tau}$ to be the integral of $\varphi$ along an orbit segment $(x,t)$. For any subset $\mathcal{C}\in X\times[0,\mathbb{R}^{+})$, we let $\mathcal{C}_{t}$ be the subset of $\mathcal{C}$ consisting of orbit segments of length $t$. We define

The topological pressure of $\varphi$ on $\mathcal{C}$ is then defined by

When $\mathcal{C}$ is the entire orbit space $X\times[0,\mathbb{R}^{+})$, then we denote it by $P(\varphi)$ and call it the topological pressure of $\varphi$. In the case where $\varphi\equiv 0$, the resulting pressure $P(0)$ is called the topological entropy of the flow $\mathcal{F}$ denoted by $h_{top}(\mathcal{F})$.

Denoting by $\mathcal{M}(\mathcal{F})$ the set of all $\mathcal{F}$-invariant measures on $X$, the pressure $P(\varphi)$ satisfies the variational principle

where $h_{\mu}(\mathcal{F})$ is the measure-theoretic entropy of $\mu$. Any invariant measure $\mu\in\mathcal{M}(\mathcal{F})$ attaining the supremum is called an equilibrium state for $\varphi$. Likewise, any invariant measure attaining the supremum when $\varphi\equiv 0$ is called a measure of maximal entropy.

Let $M$ be an $n$-dimensional closed rank 1 nonpositively curved manifold and $T_{0}$ a totally geodesic $(n-1)$-torus in $M$ with $K\equiv 0$ on any $x\in T_{0}$. We further suppose that the complement of $T_{0}$ is negatively curved and that curvature away from a small neighborhood of $T_{0}$ admits a uniform upper bound strictly smaller than 0. A more precise control of the curvature of the neighborhood will be specified later, depending on the setting under consideration.

In what follows, we fix a fundamental domain in $\widetilde{M}$ the universal covering of $M$ and (abusing the notation) continue denoting the lifts of $p\in M$ and $v\in T_{p}M$ by $p$ and $v$, respectively. Recall that the Fermi coordinate of $p$ is given by $(s,x)$ where $s$ is an $(n-1)$-dimensional coordinate on $\widetilde{T}_{0}$ and $x$ measures the signed distance on $\widetilde{M}$ to $\widetilde{T}_{0}$.

For $p\in\widetilde{M}$ near $\widetilde{T}_{0}$, by $x(p)$ we mean the $x$-coordinate of $p$. For any $v\in T_{p}\widetilde{M}$ with $p$ near $\widetilde{T}_{0}$, we define $x_{v}:=x(p)$ and denote by $\phi_{v}$ the signed angle between $v$ and the hypersurface $x=x(p)$; we adopt the convention that $\phi_{X}=\pi/2$ when $X=\partial/\partial x$. We also define

When there is no confusion, we may write $x_{v}$ and $\phi_{v}$ for $x_{v}(0)$ and $\phi_{v}(0)$, respectively.

For $\varepsilon>0$ small, the Riemannian metric near $\widetilde{T}_{0}$ can be written as

where $g_{x}$ is the Riemannian metric on $\widetilde{T}_{x}:=\widetilde{T}_{0}\times\{x\}$. In particular, $g_{0}$ is the Euclidean metric on $\widetilde{T}_{0}$.

Denoting by $X:=\partial/\partial x$ the vertical vector field, the second fundamental form on $\widetilde{T}_{x}$ is defined via

for any $v,w\in T_{(s,x)}\widetilde{T}_{x}$. The shape operator $U(s,x):T_{(s,x)}\widetilde{T}_{x}\to T_{(s,x)}\widetilde{T}_{x}$ is defined via

As II is bilinear and symmetric, the shape operator $U(s,x)$ is diagonalizable. Its eigenvalues

are called principal curvatures at $(s,x)$.

For any geodesic $\gamma(t)=(s(t),x(t))$ near $\widetilde{T}_{0}$, by the first variation formula, we have

where $\phi(t):=\phi(\gamma^{\prime}(t))$, which then gives $x^{\prime\prime}=\phi^{\prime}\cos\phi$.

We denote by

the component of $\gamma^{\prime}(t)$ that is orthogonal to $X$. Then $|\gamma^{\prime}_{\perp}|=\cos\phi$.

To distinguish vectors in $\mathrm{Sing}$ and generic vectors in $T^{1}M$, we will use different fonts to denote them; more precisely, we will write $\mathsf{v}\in\mathrm{Sing}$ and $v\in T^{1}M$. Given an orbit segment $(\mathsf{v},t)\in T^{1}M\times\mathbb{R}^{+}$ with $\mathsf{v}\in\mathrm{Sing}$, we now describe a method for constructing a new orbit segment that shadows $(\mathsf{v},t)$. Though simple, this construction will be crucial in proving Theorem A and Theorem D.

For any $\mathsf{v}\in\text{Sing}$, suppose $\pi(\mathsf{v})=(s_{0},0)$ for some $s_{0}$. For any $t>0$ and any $R>0$ such that the Fermi coordinates $(s,x)$ are well-defined for $|x|<R$, there exists $s_{1}\in\mathbb{R}$ such that the distance on $\widetilde{M}$ between $(s_{0},R)$ and $((s\circ\pi)(f_{s_{1}}\mathsf{v}),R)$ is equal to $t$; see Figure 3.1. From the triangle inequality we know that $|t-s_{1}|<2R$.

Denoting by $\gamma$ the geodesic connecting these two points, we define

Throughout the paper, we will often write $w_{\mathsf{v}}$, or simply $w$, to denote $\Pi_{t,R}(\mathsf{v})$ whenever the context is clear.

The next few lemmas establish a few properties on the map $\Pi_{t,R}$.

In this subsection, we provide a brief outline of what consists of the remaining sections. Recall from Lemma 3.1 that $x_{w}(\tau)$ is a convex function on $\tau\in[0,t]$ for any $w=\Pi_{t,R}(\mathsf{v})$, and hence there exists a well-defined number $\tilde{t}\in[0,t]$ such that

and that $\tilde{t}$ is the smallest among all such numbers. In the case where $x_{w}(\tau)$ is strictly convex, there is a unique $\tilde{t}$ which attains the minimum of $x_{w}(\tau)$. Note that $x^{\prime}_{w}(\tau)\leq 0$ and $\phi_{w}(\tau)\leq 0$ for $\tau\in[0,\tilde{t}\,]$.

The goal of the next few sections is to establish bounds on the distance $x_{w}(\tau)$ and the angle $\phi_{w}(\tau)$ under the assumption that the radial curvature $K_{\perp}$ vanishes to the order of $m-1$ at $T_{0}$ and that the curvature near $T_{0}$ is controlled; see Section 4 and 5 for the precise description of the setting. In particular, we will show that for suitable $R>0$, there exists $Q=Q(R)>1$ independent of $t$ such that the shadowing vector $w=\Pi_{t,R}(\mathsf{v})$ for any $\mathsf{v}\in\mathrm{Sing}$ and any $t>0$ satisfies

for any $\tau\in[0,\tilde{t}\,]$. From its derivation in Proposition 4.2 and 5.3, it will be clear that the analogous inequality holds for $\tau\in[\tilde{t},t]$ by simply applying the symmetric argument starting from $\tau=t$ instead of $\tau=0$: