A Mañé-Manning formula for expanding
measures for endomorphisms of $\mathbb{P}^{k}$

Let $f:\mathbb{P}^{k}\to\mathbb{P}^{k}$ be a holomorphic endomorphism of algebraic degree $d\geq 2$. The Julia set $J(f)$ of $f$ is the support of the unique measure of maximal entropy of $f$ [Lyu83, BD01, DS10]. Let $\mathcal{M}^{+}(f)$ (resp. $\mathcal{M}^{+}_{J}(f)$) be the set of ergodic invariant probability measures on $\mathbb{P}^{k}$ (resp. on $J(f)$) with strictly positive Lyapunov exponents. The set $\mathcal{M}_{J}^{+}(f)$ contains the set $\mathcal{M}_{e}^{+}(f)$ of all ergodic probability measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$ [deT08, Dup12], which are the natural generalization of the ergodic measures with strictly positive entropy in dimension $1$. Large classes of examples of measures in $\mathcal{M}^{+}_{e}(f)$ were constructed and studied in [Dup12, UZ13, SUZ14, BD23, BD22].

We introduce a volume dimension for measures $\nu\in\mathcal{M}^{+}(f)$; see Section 1.2 for an overview and Section 4 for precise definitions. The volume dimension is dynamical in nature and generalizes the notion of Hausdorff dimension in dimension 1 to higher dimensions to incorporate the failure of Koebe’s theorem and the non-conformality of holomorphic endomorphisms.

For $\nu\in\mathcal{M}^{+}(f)$, we denote by $\operatorname{VD}(\nu)$ the volume dimension, $h_{\nu}(f)$ the measure-theoretic entropy, and $L_{\nu}(f)$ the sum of the Lyapunov exponents of $\nu$. The main result of this paper relates these three quantities and generalizes the Mañé-Manning formula to any $k\geq 1$.

When $k=1$, Theorem 1.1 reduces to the Mañé-Manning formula (1.1), as in this case we have $2\operatorname{VD}(\nu)=\operatorname{HD}(\nu)$; see Proposition 4.20. The factor $2=2k/k$ is due to the fact that we weight open sets of covers by their volume instead of their diameter and we have $k$ Lyapunov exponents, counting multiplicities.

As an application of Theorem 1.1, we study a number of natural dimensions and quantities associated to an endomorphism $f$. In dimension 1, these quantities are already defined and well studied; see for example [DU91, DU91a, PU10, McM00]. We first define a dynamical dimension $\operatorname{DD}^{+}_{J}(f)$ of $f$ as

$\displaystyle\operatorname{DD}^{+}_{J}(f)$

$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup\left\{\operatorname{VD}(\nu)\colon\nu\in\mathcal{M}_{J}^{+}(f)\right\}.$

For $k=1$, recall that the pressure function is defined as

$$P(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup_{\nu}\left\{h_{\nu}(f)-t\chi_{\nu}(f)\right\},$$

where $t\in\mathbb{R}$ and the supremum is taken over the set of invariant probability measures on $J(f)$. In fact, the supremum can be taken over $\nu\in\mathcal{M}^{+}_{J}(f)=\mathcal{M}^{+}(f)$. This can be seen by combining Ruelle’s inequality [Rue78] with a theorem of Przytycki [Prz93] stating that all invariant measures supported on the Julia set of a rational map have non-negative Lyapunov exponent.

For any $k\geq 1$, we define in a similar way a pressure function $P_{J}^{+}(t)$ as

$$P^{+}_{J}(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup\left\{h_{\nu}(f)-tL_{\nu}(f)\colon\nu\in\mathcal{M}_{J}^{+}(f)\right\}.$$

By the above, we have $P^{+}_{J}(t)=P(t)$ when $k=1$. We remark that, for any $k\geq 2$, there may exist ergodic probability measures $\nu$ on $J(f)$ with $L_{\nu}(f)<0$; see Section 2.4 for examples and further comments. However, as in the case of $k=1$, the pressure function $P_{J}^{+}(t)$ is still non-increasing and convex for all $k\geq 1$; see Lemma 2.14. We define

$$p_{J}^{+}(f)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\inf\left\{t>0\colon P_{J}^{+}(t)\leq 0\right\}.$$

As a consequence of Theorem 1.1, we have the following result which generalizes a theorem due to Denker-Urbański [DU91, DU91a] in the case of rational maps to any dimension.

Finally, in the spirit of the celebrated Bowen-Ruelle formula for hyperbolic maps [Bow79, Rue82], we give an interpretation of $p_{J}^{+}(f)$, when $f$ is hyperbolic (i.e., uniformly expanding on $J(f)$; see Section 2.1) in terms of (volume-)conformal measures. Given $t\geq 0$, we say that a probability measure $\nu$ on $J(f)$ is $t$-volume-conformal on $J(f)$ if, for every Borel subset $A\subset J(f)$ on which $f$ is invertible, we have

$$\nu(f(A))=\int_{A}|\operatorname{Jac}f|^{t}d\nu$$

and define

$\displaystyle\delta_{J}(f)$

$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\inf\left\{t\geq 0\colon\mbox{ there exists a $t$-volume-conformal measure on }J(f)\right\}.$

For $k=1$, the definitions of $t$-volume-conformal measures and $\delta_{J}(f)$ reduce to those of conformal measures and conformal dimension for rational maps; see [DU91, DU91a, McM00, PU10]. In this case, owing to Bowen [Bow79], one sees that

$$\delta_{J}(f)=p_{J}^{+}(f)=\operatorname{HD}(J(f))$$

for every hyperbolic rational map $f$ on $\mathbb{P}^{1}(\mathbb{C})$, and that there exists a unique ergodic measure $\nu$ on $J(f)$ such that $\operatorname{HD}(\nu)=\operatorname{HD}(J(f))$. We have here the following result in any dimension, which further motivates the definition of the volume dimension as a natural generalization of the Hausdorff dimension for all $k\geq 1$. Observe that, if $f$ is hyperbolic, every invariant probability measure $\nu$ on $J(f)$ belongs to $\mathcal{M}^{+}(f)$.

Let us first recall the idea of the proof of the Mañé-Manning formula (1.1) in dimension 1. It essentially consists of two steps.

1. (1)

   The first step consists of defining a local dimension at a point $x$ by setting

   $$\delta_{x}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{r\to 0}\frac{\log\nu(B(x,r))}{\log r}$$

   (whenever the limit exists), where $B(x,r)$ denotes the balls of radius $r$ centred and $x$, and proving that the limit is well-defined and equal to the ratio $h_{\nu}(f)/\chi_{\nu}(f)$ for $\nu$-almost every $x$. In particular, $\nu$ is exact-dimensional.

2. (2)

   The second step is to prove that the Hausdorff dimension of $\nu$ must be equal to the common value of the local dimensions found in the first step [You82].

Let us describe how the one-dimensional setting plays a crucial role in Step (1). By [BK83] and [Mañ81], for $\nu$-almost every $x$ we have

$$h_{\nu}(f)=\lim_{\kappa\to 0}\lim_{n\to\infty}\frac{-\log\nu(B_{n}(x,\kappa))}{n},$$

where $B_{n}(x,\kappa)$ is the Bowen ball of radius $\kappa$ and depth $n$. This is defined as

$$B_{n}(x,\kappa)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left\{y\colon|f^{j}(y)-f^{j}(x)|<\kappa,0\leq j\leq n\right\}.$$

The crucial observation is that, for large $n$, the Bowen ball $B_{n}(x,\kappa)$ is comparable (up to precisely quantifiable errors) to the ball $B(x,\kappa\,e^{-n\chi_{\nu}(f)})$ of the same center and radius $\kappa\,e^{-n\chi_{\nu}(f)}$. Fixing a $\kappa_{0}$ for simplicity, and setting $n(r)\sim|\log r|/\chi_{\nu}(f)$, it then follows that

$$\lim_{r\to 0}\frac{\log\nu(B(x,r))}{\log r}=\lim_{r\to 0}\frac{-\log\nu(B_{n(r)}(x,\kappa_{0}))}{n(r)}\frac{n(r)}{-\log r}=\frac{h_{\nu}(f)}{\chi_{\nu}},$$

which in particular shows that $\delta_{x}$ is well-defined. The precise relation between geometric balls and Bowen balls is a consequence of Koebe’s theorem and related distortion estimates, which imply that images of balls by holomorphic maps (and in particular by their inverse branches) are still comparable to balls. As a consequence, in complex dimension $1$, there is a natural interplay between the Hausdorff dimension and the dynamics of a rational map. Observe in particular that one may define the Hausdorff dimension of $\nu$ by using covers consisting of Bowen balls, indexed over their depth $n$, and sending $n$ to infinity; see also [CPZ19].

All the above is in sharp contrast with the higher-dimensional situation, where, due to the lack of conformality of holomophic maps, preimages of balls can be arbitrarily distorted, and far from being balls. In the best possible scenario (e.g., for hyperbolic product maps), the preimages of balls are approximately ellipses whose axes reflect the contraction rate of the inverse branches in the different directions.

On the other hand, when $\nu\in\mathcal{M}^{+}(f)$, a result by Berteloot-Dupont-Molino (see [BDM08, BD19] and Theorem 2.1 below) states that the best possible scenario described above is actually true, in an infinitesimal sense, for preimages of balls along generic orbits of $\nu$. More precisely, there exists an increasing (as $\epsilon\to 0$) measurable exhaustion $\{Z^{\star}_{\nu}(\epsilon)\}_{\epsilon}$ of a full-measure subset $Z^{\star}_{\nu}$ of the space of orbits for $f$ such that the preimages of sufficiently small balls along orbits in $Z^{\star}_{\nu}(\epsilon)$ are approximately ellipses, and the contraction rate for their volume is essentially given (up to further controllable error terms) by $e^{-nL_{\nu}(f)+n{O(\epsilon)}}$. This is a consequence of very refined estimates on the convexity defect of such preimages. Such property was already exploited in [BB18] to give bounds on the Hausdorff dimension of the bifurcation locus of families of endomorphisms of $\mathbb{P}^{k}$ [BBD18, Bia19], and in particular to prove that this is maximal near isolated Lattés maps, i.e., maps for which all the Lyapunov exponents are equal and minimal, i.e., equal to $(\log d)/2$ [BD99, BD05].

Fix $\nu\in\mathcal{M}^{+}(f)$. Denote by $\pi\colon Z^{\star}_{\nu}\to\mathbb{P}^{k}$ the projection associating to any orbit $\hat{z}=\{z_{n}\}_{n\in\mathbb{Z}}$ its element $z_{0}$. For $x\in\pi(Z^{\star}_{\nu}(\epsilon))$, $\kappa>0$, and $N\in\mathbb{N}$, we consider (when well-defined) the neighbourhood $U=U(N,x,\kappa,\epsilon)$ of $x$ satisfying

$$f^{N}(U)=B(f^{N}(x),\kappa\,e^{-NM\epsilon})$$

where $e^{M}$ is a bound for the expansion of $f$ and we require that $f^{N}|_{U}$ is injective. It follows from the above result by Berteloot-Dupont-Molino, and by further estimates that we develop in Section 2, that there exist some $r(\epsilon)$ and $n(\epsilon)$ such that, for all $x\in\pi(Z^{\star}_{\nu}(\epsilon))$, $0<\kappa<r(\epsilon)$, and $N\geq n(\epsilon)$ the sets $U(N,x,\kappa,\epsilon)$ are indeed well-defined and approximately ellipses, of controlled geometry. We see these sets $U(N,x,\kappa,\epsilon)$ as a suitable version of the Bowen balls $B_{n}(x,\kappa)$ in any dimension. Let us set

$$\delta_{x}(\epsilon,\kappa,N)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\log\nu(U(N,x,\kappa,\epsilon))}{\log{\rm Vol}(U(N,x,\kappa,\epsilon))},$$

where $\rm Vol$ denotes the volume with respect to the Fubini-Study metric. As a first step (which corresponds to Step (1) above) towards proving Theorem 1.1, we show that every $\nu\in\mathcal{M}^{+}(f)$ is exact (volume-)dimensional; namely, for $\nu$-almost every $x$, we have

$$\limsup_{\epsilon\to 0}\limsup_{\kappa\to 0}\limsup_{N\to\infty}\delta_{x}(\epsilon,\kappa,N)=\liminf_{\epsilon\to 0}\liminf_{\kappa\to 0}\liminf_{N\to\infty}\delta_{x}(\epsilon,\kappa,N)=\frac{h_{\nu}(f)}{2L_{\nu}(f)};$$

see Theorem 3.2 and Corollary 3.4. We adapt here the approach of Mañé [Mañ88] in higher dimensions, thanks to the distortion estimates developed in Section 2.

Once the local dimension of every $\nu\in\mathcal{M}^{+}(f)$ is well-defined as above, we give a global interpretation of this quantity by defining a volume dimension for these measures. The idea is to use the sets $U(N,x,\kappa,\epsilon)$ to cover the “slice” $X\cap\pi(Z^{\star}_{\nu}(\epsilon))$ of every set $X\subseteq Z^{\star}_{\nu}$. More precisely, for every $X\subseteq\pi(Z^{\star}_{\nu})$ and $\epsilon>0$, setting $X^{\epsilon}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=X\cap\pi(Z^{\star}_{\nu}(\epsilon))$, we define the quantity $\operatorname{VD}^{\epsilon}_{\nu}(X^{\epsilon})$ as

$$\operatorname{VD}^{\epsilon}_{\nu}(X^{\epsilon})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup\left\{\alpha:\Lambda^{\epsilon}_{\alpha}(X^{\epsilon})=\infty\right\}=\inf\left\{\alpha:\Lambda^{\epsilon}_{\alpha}(X^{\epsilon})=0\right\},$$

where

$$\Lambda^{\epsilon}_{\alpha}(X^{\epsilon})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{\kappa\to 0}\lim_{N^{\star}\to\infty}\inf_{\{U_{i}\}}\sum_{i\geq 1}\operatorname{Vol}(U_{i})^{\alpha}.$$

Here the infimum is taken over the covers consisting of sets $U_{i}$ of the form $U_{i}=U(N_{i},x,\kappa,\epsilon)$, for some $x\in\pi(Z_{\nu}(\epsilon))$ and $N_{i}\geq N^{\star}$. The volume dimensions of $X$ and $\nu$ are then respectively defined as

$$\operatorname{VD}_{\nu}(X)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\limsup_{\epsilon\to 0}\operatorname{VD}_{\nu}^{\epsilon}(X^{\epsilon})\quad\mbox{ and }\quad\operatorname{VD}(\nu)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\inf\left\{\operatorname{VD}_{\nu}(X)\colon X\subseteq\pi(Z_{\nu}),\nu(X)=1\right\},$$

and the $\limsup_{\epsilon\to 0}$ is actually a limit; see Section 4.2. We prove in Proposition 4.26 a version of Young’s criterion [You82, Proposition 2.1], relating the local volume dimensions $\delta_{x}$ with the volume dimensions $\operatorname{VD}_{\nu}(X)$ and $\operatorname{VD}(\nu)$. This corresponds to Step (2) above and, together with the exact volume-dimensionality of $\nu$ proved in the first step, completes the proof of Theorem 1.1.

The paper is organized as follows. In Section 2, we derive from the distortion theorem [BDM08, BD19] the estimates that we will need, and we introduce the volume-conformal measures and the pressure function $t\mapsto P_{J}^{+}(t)$. We prove the exact dimensionality of every $\nu\in\mathcal{M}^{+}(f)$ in Section 3. In Section 4, we define and study the volume dimensions of sets and measures. We conclude the proof of Theorem 1.1 and prove Theorems 1.2 and 1.3 in Section 5.

The authors would like to thank the University of Lille and the University of Oklahoma for the warm welcome and for the excellent work conditions.

This project has received funding from the French government through the Programme Investissement d’Avenir (LabEx CEMPI ANR-11-LABX-0007-01, ANR QuaSiDy ANR-21-CE40-0016, ANR PADAWAN ANR-21-CE40-0012-01) managed by the Agence Nationale de la Recherche.

Let $f:\mathbb{P}^{k}\to\mathbb{P}^{k}$ be a holomorphic endomorphism and $\nu$ an ergodic $f$-invariant probability measure. By Oseledets’ theorem [Ose68], one can associate to $\nu$ its Lyapunov exponents $\chi_{\min}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\chi_{l}<\ldots<\chi_{1}$, where $1\leq l\leq k$. For $\nu$-almost every $x\in\mathbb{P}^{k}$, there exists a stratification in complex linear subspaces $\{0\}=:{(L_{l+1})_{x}}\subset{(L_{l})_{x}}\subset\ldots\subset(L_{1})_{x}=T_{x}\mathbb{P}^{k}$ of the complex tangent space $T_{x}\mathbb{P}^{k}$ such that $Df_{x}(L_{j})_{x}=(L_{j})_{f(x)}$ and $\lim_{n\to\infty}n^{-1}\log||Df_{x}^{n}v||=\chi_{j}$ for all $v\in(L_{j})_{x}\setminus(L_{j+1})_{x}$ for all $1\leq j\leq l$.

Let us first assume that all the $\chi_{j}$’s are distinct, i.e., that we have $l=k$ and $\chi_{\min}=\chi_{k}<\ldots<\chi_{1}$. Then, $(L_{j})_{x}$ has dimension $k-j+1$ for all $1\leq j\leq k$. We denote by $\mathcal{O}$ a full measure subset of the support of $\nu$ given by Oseledets’ theorem. Take $x\in\mathcal{O}$. Fix a basis $(\ell_{j})_{x}$ of the complex tangent space $T_{x}\mathbb{P}^{k}$ with the property that $(L_{j})_{x}$ is equal to the span of $\{(\ell_{j})_{x},\dots,(\ell_{k})_{x}\}$. Denote by $\{e_{j}\}_{j=1}^{k}$ the standard basis of $\mathbb{R}^{k}\subset\mathbb{C}^{k}$. For every $r_{1},\dots,r_{k}\in\mathbb{R}$ (sufficiently small), we denote by $\mathcal{E}_{x}(r_{1},\dots,r_{k})$ the image of the unit ball $\mathbb{B}^{k}\subset\mathbb{C}^{k}$ (in a given local chart at $x$) under the composition $e\circ\Phi:\mathbb{C}^{k}\to\mathbb{P}^{k}$, where $e\colon T_{x}\mathbb{P}^{k}\to\mathbb{P}^{k}$ is the standard exponential map and $\Phi\colon\mathbb{C}^{k}\to T_{x}\mathbb{P}^{k}\simeq\mathbb{C}^{k}$ is a linear map such that $\Phi\big{(}(e_{j})_{x}\big{)}=r_{j}(\ell_{j})_{x}$.

If, for all $1\leq j\leq k$, the argument $r_{j}$ of $\mathcal{E}_{x}$ is a function $\phi(j)$ depending on $j$, we write

$$\mathcal{E}_{x}(\phi(j))\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{E}_{x}(\phi(1),\dots,\phi(k))$$

for brevity. In particular, we will often have $\chi_{\min}>0$ and take $\phi(j)$ of the form $\phi(j)=c_{1}e^{-n(\chi_{j}\pm c_{2}\epsilon)}$ for some $n\in\mathbb{N}$, $0<\epsilon\ll\chi_{\min}$ sufficiently small, and some positive constants $c_{1}$ (independent of $j$ and $n$) and $c_{2}$ (independent of $j$, $n$, and $\epsilon$). We will call the sets $\mathcal{E}_{x}$ dynamical ellipses in this case.

If $l<k$, i.e., some Lyapunov exponent $\chi_{j}$ has multiplicity larger than 1, the above construction generalizes by taking into account the corresponding $r_{j}$ with the same multiplicity. Namely, we assign the same $r_{j}$ to all the directions associated to the same Lyapunov exponent which has multiplicity larger than 1.

Let $X\subseteq\mathbb{P}^{k}$ be a closed invariant set for $f$. We denote by $\mathcal{M}_{X}^{+}(f)$ the set of all ergodic $f$-invariant measures supported on $X$ with strictly positive Lyapunov exponents. We drop the index $X$ if $X=\mathbb{P}^{k}$. We say that $X$ is uniformly expanding if there exist $\eta>1$ and $C>0$ such that $||Df_{x}^{n}(v)||>C\eta^{n}||v||$ for every $x\in X$, $v\in T_{x}\mathbb{P}^{k}$, and $n\in\mathbb{N}$. We say that $f$ is hyperbolic if $J(f)$ is uniformly expanding.

We will consider the Fubini-Study metric on $\mathbb{P}^{k}$. We will denote by $\operatorname{dist}$ the corresponding distance, and by $B(x,r)$ the open ball centred at $x$ and of radius $r$. For an open set $V\subset\mathbb{P}^{k}$, we denote by $\operatorname{Vol}(V)$ the volume of $V$ with respect to the Fubini-Study metric. Given a holomorphic map $g\colon V\to\mathbb{P}^{k}$, we denote by $\operatorname{Jac}g(x)$ the Jacobian of $g$ at $x\in V$, i.e., the determinant of the differential $Dg_{x}$.

We also fix the positive constant $M>0$ defined as

(2.1)

$$M\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\log\sup_{x\in\mathbb{P}^{k}}\sup_{v\in\mathbb{C}^{k}}\frac{||Df_{x}(v)||}{||v||}$$

and observe that $e^{M}$ dominates the Lipschitz constant of $f$. In particular, we have $\operatorname{dist}(f(x_{1}),f(x_{2}))\leq e^{M}\operatorname{dist}(x_{1},x_{2})$ for every $x_{1},x_{2}\in\mathbb{P}^{k}$, and $f(B(x,r))\subseteq B(f(x),e^{M}r)$ for every $x\in\mathbb{P}^{k}$ and $r>0$.

We fix in this subsection a holomorphic endomorphism $f:\mathbb{P}^{k}\to\mathbb{P}^{k}$ of algebraic degree $d\geq 2$ and a measure $\nu\in\mathcal{M}^{+}(f)$. All the objects and the constants that we introduce in this subsection depend on $f$ and $\nu$. We denote by $\chi_{1}>\ldots>\chi_{l}=\chi_{\min}>0$ the (distinct) Lyapunov exponents of $\nu$, by $k_{1},\dots,k_{l}$ their respective multiplicities, and by $L_{\nu}=L_{\nu}(f)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{j=1}^{l}k_{j}\chi_{j}$ their sum. Recall that we have $L_{\nu}=\int\log|\operatorname{Jac}f|d\nu$ by Birkhoff’s ergodic theorem.

Consider the orbit space of $f$

$$O\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left\{\hat{x}=\{x_{n}\}_{n\in\mathbb{Z}}\in(\mathbb{P}^{k})^{\mathbb{Z}}\colon x_{n+1}=f(x_{n})\quad\forall n\in\mathbb{Z}\right\}$$

and the right shift map $T\colon O\to O$ defined as $T(\hat{x})=\{x_{n+1}\}_{n\in\mathbb{Z}}$ for $\hat{x}=\{x_{n}\}_{n\in\mathbb{Z}}$. Given $\eta>0$, a function $\phi\colon O\to(0,1]$ is said to be $\eta$-slow if for any $\hat{x}\in O$ we have

$$e^{-\eta}\phi(\hat{x})\leq\phi(T(\hat{x}))\leq e^{\eta}\phi(\hat{x}).$$

We now recall the construction of the lift $\hat{\nu}$ of $\nu$ to $O$; see [CFS12, Section 10.4] and [PU10, Section 2.7]. For $n\in\mathbb{Z}$, we let $\pi_{n}\colon O\to\mathbb{P}^{k}$ be the projection map defined by $\pi_{n}(\hat{x})=x_{n}$, where $\hat{x}=\{x_{n}\}_{n\in\mathbb{Z}}$. We write $\pi\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi_{0}$ for brevity. Observe that $\pi_{n}\circ T=f\circ\pi_{n}$ for all $n\in\mathbb{Z}$.

Consider the $\sigma$-algebra $\hat{\mathcal{B}}$ on $O$ generated by the sets of the form

$$A_{n,B}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi_{n}^{-1}(B)=\left\{\hat{x}\colon x_{n}\in B\right\}$$

with $n\leq 0$ and $B\subseteq\mathbb{P}^{k}$ a Borel set. For all such sets $A_{n,B}$, set

$$\hat{\nu}(A_{n,B})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nu(B).$$

Then, by the invariance of $\nu$ and the fact that $x_{n}\in B$ if and only if $x_{n-m}\in f^{-m}(B)$ with $m\geq 0$, we see that $\hat{\nu}$ is well-defined on the sets $A_{n,B}$ as it satisfies $\hat{\nu}(A_{n,B})=\hat{\nu}(A_{n-m,B})$ for all $m\geq 0$. Similarly, for all $m\geq 0$ and Borel sets $B_{0},\dots,B_{-m}\subseteq\mathbb{P}^{k}$, we have

$$\hat{\nu}\big{(}\{\hat{x}\colon x_{0}\in B_{0},\ldots,x_{-m}\in B_{-m}\}\big{)}=\nu\left(f^{-m}(B_{0})\cap f^{-m+1}(B_{1})\cap\ldots\cap B_{-m}\right).$$

We can then extend $\hat{\nu}$ to a probability measure on $\hat{\mathcal{B}}$, that we still denote by $\hat{\nu}$. By construction, $\hat{\nu}$ is $T$-invariant and satisfies $\pi_{*}(\hat{\nu})=\nu$. As $\nu$ is ergodic, one can prove that $\hat{\nu}$ is also ergodic.

Recall that the critical set $C(f)$ of $f$ is the set of points $x\in\mathbb{P}^{k}$ at which the differential $Df_{x}$ is not invertible. As all the Lyapunov exponents of $\nu$ are finite, and their sum is equal to $\int\log|Df|d\nu$, we have in particular $\nu(C(f))=0$. Set

$$Z\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\big{\{}\hat{x}\in O:x_{n}\notin C(f)\quad\forall n\in\mathbb{Z}\big{\}}.$$

Then the set $Z$ is $T$-invariant and satisfies $\hat{\nu}(Z)=1$. For every $\hat{x}\in Z$, we denote by $f_{\hat{x}}^{-n}$ the inverse branch of $f^{n}$ defined in a neighbourhood of $x_{0}$ and such that $f^{-n}_{\hat{x}}(x_{0})=x_{-n}$.

The following result is stated in [BD19, Theorem A] (see also [BDM08, Theorem 1.4]) in the case where $\nu$ is the measure of maximal entropy of $f$. The same statement and proof hold for any measure in $\nu\in\mathcal{M}^{+}(f)$, as stated at the end of the Introduction – and used in later sections – of the same paper.

In particular, for every $n\in\mathbb{N}$ and $\hat{x}$ as in the statement, the inverse branch $f_{\hat{x}}^{-n}$ is well-defined on the ball $B(x_{0},r_{\hat{x}})$.

It follows from the definition that, as $\alpha\to 0$, the sets $Z_{\nu,\alpha}$ increase to $Z_{\nu}$. In particular, we have $\hat{\nu}(Z_{\nu,\alpha})\to 1$ as $\alpha\to 0$.