---
abstract: |
    Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $\mu$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,\mu)$. Our class $\UU$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $\omega: (\Sigma , s , \nu) \to
    (\mathbb{P}^k,f,\mu)$. We obtain the invariance principle for an observable $\psi$ on $(\mathbb{P}^k,f,\mu)$ by applying Philipp-Stout’s theorem for $\chi = \psi
    \circ \omega$ on $(\Sigma , s , \nu)$. The invariance principle implies the Central Limit Theorem as well as several statistical properties for the class $\UU$. As an application, we give a *direct* proof of the absolute continuity of the measure $\mu$ when it satisfies Pesin’s formula. This approach relies on the Central Limit Theorem for the unbounded observable $\log
    \textsf{Jac} \,  f \in \UU$.
author:
- 'C. Dupont'
date:  
title: 'Bernoulli coding map and almost sure invariance principle for endomorphisms of ${\mathbb {P}}^k$'
---

Introduction {#intro}
============

Let $f : {\mathbb {P}}^k \to {\mathbb {P}}^k$ be an holomorphic endomorphism of algebraic degree $d
\geq 2$. Its equilibrium measure $\mu$ is the limit of the probability measures $d_t^{-n} (f^n)^* \eta^k$, where $d_t:= d^k$ is the topological degree of $f$ and $\eta^k$ is the standard volume form on ${\mathbb {P}}^k$. We refer to the survey article of Sibony [@Sib] for an introduction to the dynamical systems $({\mathbb {P}}^k,f,\mu)$. Fornaess-Sibony proved that $\mu$ is mixing [@FS] and Briend-Duval that $\mu$ is the unique measure of maximal entropy [@BrDu2].\
Przytycki-Urbański-Zdunik [@PUZ] introduced coding techniques for $({\mathbb {P}}^1,f,\mu)$. This allowed them to prove the Almost Sure Invariance Principle (ASIP) for Hölder and singular observables, like $\log \abs{f'}$. In the present article, we extend the coding techniques to $({\mathbb {P}}^k,f,\mu)$ and obtain the ASIP for observables which allow analytic singularities. As an application, we obtain a direct proof of the absolute continuity of $\mu$ when it satisfies Pesin’s formula. We review our results in subsections \[baba\] - \[jaja\], subsection \[relt\] is devoted to related results.

Bernoulli coding maps {#baba}
---------------------

Let us endow $\Sigma
:=  \{ 1, \ldots, d_t \}^{\mathbb {N}}$ with the natural product measure $\nu:= \otimes_{n = 0}^\infty \bar
\nu$, where $\bar \nu$ is equidistributed on $\{ 1, \ldots, d_t \}$. We denote by ${{\tilde \alpha}}$ the elements of $\Sigma$ and by $s$ the left shift acting on $\Sigma$. Let $\JJ$ be the support of $\mu$. The following theorem yields coding maps ${\omega}: \Sigma \to \JJ$ up to zero measure sets. The set $\SS \subset {\mathbb {P}}^k$ will be defined in section \[GCT\], it has zero Lebesgue measure.

[A:]{} Let $z \in {\mathbb {P}}^k \setminus \SS$. There exist an $s$-invariant set $\Sigma'
\subset \Sigma$ of full $\nu$-measure and an $f$-invariant set $\JJ'
\subset \JJ$ of full $\mu$-measure satisfying the following properties. For any ${{\tilde \alpha}}\in
\Sigma'$, the point ${\omega}({{\tilde \alpha}}):= \lim_{n \to \infty} z_n({{\tilde \alpha}}) \in \JJ'$ is well defined. We have ${\omega}_* \nu  = \mu$ and the following diagram commutes: $$\xymatrix{
           \Sigma'  \ar[rr]^{s}  \ar[d]_{\omega}&    & \Sigma'  \ar[d]^{\omega}\\
           \JJ'  \ar[rr]^{f}       & &  \JJ'        }$$ Moreover there exist $\theta , {\epsilon}> 0$, $n_z \geq 1$ and $\tilde n : \Sigma' \to {\mathbb {N}}$ larger than $n_z$ such that:

1.  $d(z_n({{\tilde \alpha}}),{\omega}({{\tilde \alpha}}) ) \leq \tilde c_{\epsilon}\, d^{-{\epsilon}n}$ for every ${{\tilde \alpha}}\in \Sigma'$ and $n \geq \tilde n ({{\tilde \alpha}})$,

2.  $\nu ( \{ \tilde n  \leq q \} )  \geq 1 - c_\theta \, d^{-\theta q}$ for every $q \geq n_z$.

We note that $\Sigma'$, $\JJ'$ and ${\omega}$ depend on $z \in {\mathbb {P}}^k \setminus \SS$, but ${\omega}_* \nu = \mu$ holds true for any such $z$. Observe also that ${\omega}$ is not necessarily injective. The proof of theorem A (see section \[GCT\]) is based on the construction of a geometric coding tree, following the approach of Przytycki-Urbański-Zdunik [@PUZ] for $({\mathbb {P}}^1,f,\mu)$. The point $z$ is the *root* of the tree, and the set $\{ z_n({{\tilde \alpha}}) , {{\tilde \alpha}}\in \Sigma \}$ is a suitable enumeration of the $d_t^{n+1}$ points of $f^{-(n+1)}(z)$, these are *vertices* of the tree. The convergence of $(z_n({{\tilde \alpha}}))_n$ for a generic ${{\tilde \alpha}}\in \Sigma$ is obtained by constructing $d_t$ good paths joining $z$ to $w \in f^{-1}(z)$, whose inverse images decrease exponentially. In the context of $({\mathbb {P}}^1,f,\mu)$, that property was obtained in [@PUZ] by using Koebe distortion theorem. The difficulty in higher dimensions is to substitute this argument. We establish for that purpose a quantified version of a theorem of Briend-Duval (see section \[BD\]).

The class $\UU$ and approximation by cylinders {#byby}
----------------------------------------------

[:]{} An observable $\psi : {\mathbb {P}}^k \to {\mathbb {R}}\cup \{ -
  \infty \}$ belongs to the class $\UU$ if:

- $e^\psi$ is $h$-Hölder for some $h > 0$,

- $\NN_\psi:= \{ \psi = -\infty \}$ is a (possibly empty) proper algebraic set of ${\mathbb {P}}^k$,

- $\psi \geq \log d(\cdot,\NN_\psi)^\rho$ for some $\rho >0$.

For instance, the Hölder functions are in $\UU$, as well as the unbounded function $\log {{\textsf {Jac} \,  }}f$. We will show that $\UU \subset
L^p(\mu)$ for any $1 \leq p < + \infty$ (see subsection \[cccd\]).

[B:]{} Let $\psi \in \UU$ be a $\mu$-centered observable and ${\omega}$ be a coding map provided by theorem A. Let $\chi:= \psi \circ
  {\omega}$ and $1 \leq p < + \infty$. We denote by ${\mathbb {E}}( \chi \vert
  \CC_n  )$ the conditional expectation of $\chi$ with respect to the $(n+1)$-cylinders.

1.  there exist $\hat c_p, \lambda_p > 0$ such that $\norm { \chi   - {\mathbb {E}}( \chi \vert
      \CC_n  )  }_p \leq \hat c_p \, e^{-n\lambda_p}$ for every $n \geq 0$.

2.  $R_j(\chi):= \int_\Sigma \chi \cdot
    \chi \circ s^j \, d\nu$ satisfies $\abs{R_j(\chi)} \leq 2 \norm {\chi}_2  \hat c_2 \,
      e^{-(j-1) \lambda_2}$ for every $j \geq 1$.

The proof occupies section \[estiho\], it is based on the regularity properties of ${\omega}$ (namely the points 1, 2 of theorem A) and on the fact that $\mu$ is a Monge-Ampère mass with Hölder potentials. Theorem B allows us to prove theorem C below.

Almost Sure Invariance Principle {#bibi}
--------------------------------

Let $\psi \in L^2(\mu)$ be a $\mu$-centered observable and $S_n(\psi):= \sum_{j=0}^{n-1} \psi \circ f^j$. We say that $\psi$ satisfies the Almost Sure Invariance Principle (ASIP) if there exist, on an extended probability space, a sequence of random variables $(\SS_n)_{n \geq 0}$ together with a Brownian motion $\WW$ such that for some $\gamma > 0$:

1.  $\SS_n =  \WW(n) + o(n^{1/2 -
      \gamma})$ almost everywhere,

2.  $(S_0(\psi),\ldots,S_n(\psi))$ and $(\SS_0,\ldots,\SS_n)$ have the same distribution for any $n \geq 0$.

We shall denote ${\sigma}$-ASIP to specify the variance of Brownian motion.

[C:]{} For every $\mu$-centered observable $\psi \in \UU$, we have:

1.  ${\sigma}:= \lim_{n \to \infty} {1 \over \sqrt n} \norm{S_n(\psi)}_2$ exists, and ${\sigma}^2 = \int_{{\mathbb {P}}^k} \psi^2 \, d\mu + 2\sum_{j \geq 1} \int_{{\mathbb {P}}^k}
     \psi \cdot \psi \circ f^j \, d\mu$.

2.  If ${\sigma}= 0$, then $\psi = u - u \circ f$ holds $\mu$-a.e. for some $u \in L^2(\mu)$.

3.  If ${\sigma}> 0$, then $\psi$ satisfies the ${\sigma}$-ASIP.

The ASIP implies classical limit theorems related to Brownian motion: the Central Limit Theorem (CLT), the Law of Iterated Logarithm, Kolmogorov integral tests (see [@De], [@PS]). The ASIP also implies the almost sure version of the CLT, meaning that ${1 \over
 \log n} \sum_{k=1}^n {1 \over k} \delta_{ {1 \over
    \sqrt{k}} S_k(\psi)(x)}$ converges $\mu {\textrm{-a.e.}}$ to the normal law $\NN(0,{\sigma}^2)$ (see [@LP], [@CG]).\
Let us outline the proof of theorem C (see section \[CC\]). Let ${\omega}: \Sigma \to {\mathbb {P}}^k$ be a coding map provided by theorem A and $\psi \in \UU$. Since ${\omega}$ satisfies $f
\circ {\omega}= {\omega}\circ s$ and ${\omega}_*\nu=\mu$, we are reduced to prove the assertions for $\chi = \psi \circ {\omega}$ on $(\Sigma,s,\nu)$. The points 1 and 2 follow from theorem B(2) and classical arguments. The point 3 is a consequence of theorem B(1) and Philipp-Stout’s theorem ([@PS], section 7). That result relies on an approximation of the partial sums of $(\chi
\circ s^j)_{j \geq 0}$ by a sequence of martingale differences defined with respect to the increasing filtration $(\CC_n)_{n \geq 0}$.

An application to smooth ergodic theory {#jaja}
---------------------------------------

Let $\chi_1 \leq \ldots \leq \chi_k$ be the Lyapunov exponents of $\mu$. Briend-Duval [@BrDu1] proved that they are larger than or equal to $\log d^{1/2}$. Since $\mu$ has entropy $\log d^k$, Pesin’s formula $h(\mu) = 2 (\chi_1 + \ldots + \chi_k)$ holds if and only if these exponents are minimal. We proved in a previous article that $\mu$ is then absolutely continuous with respect to Lebesgue measure [@Du]. We there followed the classical approach of Sinai-Pesin-Ledrappier, based on the construction of a suitable invariant partition which is dilated and realizes entropy (see [@P1], [@Le]). We propose in section \[appll\] a new proof, based on the CLT for the unbounded $\mu$-centered observable $J:= \log
{{\textsf {Jac} \,  }}f -  2 (\chi_1 + \ldots + \chi_k) \in \UU$. We obtain the following result, where ${\sigma}_J: = \lim_{n \to \infty} {1 \over \sqrt n}
\norm{S_n(J)}_2$.

[D:]{} If the Lyapunov exponents are minimal equal to $\log d^{1/2}$, then ${\sigma}_J = 0$, and $\mu$ is absolutely continuous with respect to Lebesgue measure.

A crucial fact for the proof is that for any holomorphic endomorphism of ${\mathbb {P}}^k$ and any $\mu$-generic point $x \in {\mathbb {P}}^k$, the minimal dilation rate of $f^n$ at $x$ (i.e. $\norm{(d_x
  f^n)^{-1}}^{-1}$) is bounded below by $d^{n/2}$ up to the multiplicative factor $1/n$. In other words, the usual $e^{-n{\epsilon}}$-correction, due to the non-uniform hyperbolicity of $({\mathbb {P}}^k,f,\mu)$, can be replaced here by $1/n$. This was proved by Berteloot-Dupont [@BeDu], using a pluripotential result of Briend-Duval [@BrDu1] and the fact that $\mu$ is a Monge-Ampère mass. In particular, the product of the dilation rates satisfies ${{\textsf {Jac} \,  }}f^n(x) \geq \norm{(d_x f^n)^{-1}}^{-2k}  \geq (d^{n/2}/n)^{2k} = d^{kn}/n^{2k}$. Now if we assume ${\sigma}_J > 0$, then the function $\log {{\textsf {Jac} \,  }}f^n$ would present non trivial oscillations around its mean value $\log d^{kn}$, due to the CLT. More precisely, it would imply $\log {{\textsf {Jac} \,  }}f^n \leq \log d^{kn} - {\sigma}_J \sqrt n$ on a subset of $\mu$-measure $\simeq \int_{-\infty}^{-1}
e^{-u^2/2}$. That contradicts the preceding estimate, hence ${\sigma}_J  = 0$. We deduce the absolute continuity of $\mu$ from the cocycle property $J = u - u
\circ f$ $\mu$-a.e. and a linearization property of the dynamics along typical negative orbits [@BeDu].

Related results {#relt}
---------------

The systems $({\mathbb {P}}^k , f , \mu)$ and $(\Sigma , s
, \nu)$ are actually conjugated by a bimeasurable map up to zero measure subsets, that property was proved by Briend [@Br]. However, the regularity of the conjugacy seems difficult to handle. Let us also mention that finite-to-one coding maps $({\mathbb {P}}^k,f,\mu) \to (\Sigma,s,\nu)$ were constructed by Buzzi [@Bu] by means of suitable partitions of ${\mathbb {P}}^k$.\
The ASIP has been proved for many dynamical systems: for piecewise monotonic maps by Hofbauer-Keller [@HK], for Anosov maps by Denker-Philipp [@DP] and for partially and non-uniformly hyperbolic systems by Dolgopyat [@Do] and Melbourne-Nicol [@MN]. We refer to the survey articles of Chernov [@C] and Denker [@De] for limit theorems and statistical properties concerning dynamical systems.\
The ASIP implies the CLT. Nevertheless, the latter can be directly proved via coding techniques and Ibragimov’s theorem [@I]. That method was employed by Sinai [@Sin] and Ratner [@R] for the geodesic flow in negative curvature, and by Bowen [@Bo] for Anosov maps. In the present article, Ibragimov’s condition is fulfilled by theorem B.\
The Gordin’s theorem provides another method for proving the CLT (see [@G], [@Li]). It relies on an approximation of $(\psi \circ
f^j)_{j \geq 0}$ by a sequence of reverse martingale differences. In our context, this can be done if $\sum_{n \geq 0} \norm{\Lambda^n
   \psi}_2$ (denoted $(\star)$) converges, where $\Lambda$ denotes the Ruelle-Perron-Frobenius operator (we have $\Lambda^n \psi (z) = {1 \over d_t^n}
 \sum_{y \in f^{-n}(z)} \psi(y)$ for every $z \in {\mathbb {P}}^k$). Let us note that the reverse martingale mentioned is defined with respect to the decreasing filtration $(f^{-n}\BB)_{n \geq 0}$, where $\BB$ is the Borel ${\sigma}$-algebra of ${\mathbb {P}}^k$.

The exponential decay of correlations ensures the convergence of $(\star)$. This was proved in the context of $({\mathbb {P}}^k,f,\mu)$ by Fornaess-Sibony [@FS] for $C^2$ observables and by Dinh-Sibony for Hölder observables [@DS2]. Dinh-Nguyen-Sibony have recently extended that property for differences of quasi-plurisubharmonic functions (the so-called *dsh* functions) [@DNS2]. The proof relies on exponential estimates for plurisubharmonic functions with respect to $\mu$. They also obtained in that article a Large Deviations Theorem for bounded dsh and Hölder observables. In [@DNS1], Dinh-Nguyen-Sibony proved the local CLT for $({\mathbb {P}}^1,f,\mu)$ by using the theory of perturbed operators.

Denker-Przytycki-Urbański [@DPU] employed a geometric method to prove the convergence of $(\star)$ for $({\mathbb {P}}^1,f,\mu)$ and Hölder observables. The idea was to compare $\Lambda^n \psi(z)$ to $\Lambda^n
\psi(z')$ by using the contraction of most of the inverse branches of $f^n$. The cornerstone is a precise analysis of the dynamics near the critical points in the support of $\mu$. Cantat-Leborgne [@CL] extended this approach to $({\mathbb {P}}^k,f,\mu)$. A crucial ingredient was a polynomial estimate for the $\mu$-measure of postcritical neighbourhoods (lemma 5.7 of [@CL]). The original proof of that lemma contains a gap, the authors have recently proposed another one. Cantat-Leborgne also established in [@CL] a quantified version of the Briend-Duval theorem. Our version is similar, but we shall give a different proof.\
The systems $({\mathbb {P}}^k,f,\mu)$ whose measure $\mu$ is absolutely continuous with respect to Lebesgue measure were characterized by Berteloot, Dupont and Loeb [@BeDu], [@BL]. In that case, $f$ is semi-conjugated to an affine dilation on a complex torus, these maps are the so-called *Lattès examples*. We note that theorem D characterizes these maps by the minimality of the Lyapunov exponents. Another characterization of Lattès examples involves the *Hausdorff dimension* of $\mu$, defined as the infimum of the Hausdorff dimension of Borel sets with full $\mu$-measure (see Pesin’s book [@P2]): Dinh-Dupont [@DD] proved that $\dim_\HH(\mu) = 2k$ if and only if the exponents are minimal. In the context of $({\mathbb {P}}^1,f,\mu)$, Mañé proved that $\log d = \dim_\HH(\mu) \cdot \chi$, where $\chi$ denotes the Lyapunov exponent of $\mu$. In particular, the function $L:= \log d
- \dim_\HH(\mu) \cdot \log \abs{f'}$ is a $\mu$-centered observable. Zdunik [@Z] proved that ${\sigma}_L = 0$ if and only if $f$ is a Lattès example, a Tchebychev polynomial or a power $z^{\pm d}$. The proof relies on the classification of critically finite fractions with parabolic Thurston’s orbifold.

Generalities {#cons}
============

The holomorphic systems $({\mathbb {P}}^k,f,\mu)$ {#consa}
-------------------------------------------------

We introduce in this section the systems $({\mathbb {P}}^k,f,\mu)$. We refer to the articles [@BrDu1], [@BrDu2], [@FS] and [@Sib] for definitions and properties. Here ${\mathbb {P}}^k$ denotes the complex projective space of dimension $k$. We denote by $\eta$ the Fubini-Study form on ${\mathbb {P}}^k$. This is a $(1,1)$-form defined in homogeneous coordinates by ${i \over 2 \pi} \partial  \bar \partial \log \norm{z}^2$. It induces the standard metric on ${\mathbb {P}}^k$, the volume of ${\mathbb {P}}^k$ with respect to this metric is equal to $1$. The form $\eta$ induces on every complex line $L \subset {\mathbb {P}}^k$ the spherical metric with area $1$. Let $f$ be an holomorphic endomorphism of ${\mathbb {P}}^k$ with algebraic degree $d \geq 2$. It is defined in homogeneous coordinates by $[ P_0 :
  \cdots : P_k ]$ where the $P_i$ are homogeneous polynomials of degree $d$ (without common zero except the origin). The topological degree of $f$ is $d_t:= d^k$. An inverse branch of $f^n$ on $U \subset {\mathbb {P}}^k$ is an injective holomorphic map $g_n$ satisfying $f^n \circ g_n = {{\rm Id}}_U$. We let ${{\textsf {Per}  }}f:= \cup_{n \geq 1} \{ x \in {\mathbb {P}}^k
\, , \, f^n(x) = x \}$, this set is at most countable. Let $\CC$ be the critical set of $f$, $\VV:= \cup_{i = 0}^\infty
f^i(\CC)$ and $\VV_n:= \cup_{i = 1}^ n f^i(\CC)$. The degree of $\VV_n$, denoted $\tau_n$, is equal to $(d + \ldots +
d^n) \deg \CC$ counted with multiplicity.\
The equilibrium measure $\mu$ is defined as the limit of $\mu_{n,z}:=
{1 \over d_t ^n} \sum_{f^n(y) = z} \delta_y$, where $\delta_y$ denotes the Dirac mass at $y$. In that definition, $z$ has to be taken outside a totally invariant algebraic set $\EE \subset \VV$, the so-called exceptional set of $f$. We denote by $\JJ$ the support of $\mu$. The measure $\mu$ is mixing and satisfies $\mu(f(B)) = d_t
\, \mu(B)$ whenever $f$ is injective on $B$. It is the unique measure of maximal entropy (equal to $\log d_t$). The Lyapunov exponents $\chi_1 \leq \ldots \leq \chi_k$ of $\mu$ are larger than or equal to $\log
d^{1/2}$. They satisfy the classical formula $\int_{{\mathbb {P}}^k} \log {{\textsf {Jac} \,  }}f \, d\mu = 2 (
\chi_1 + \ldots +\chi_k )$, where ${{\textsf {Jac} \,  }}f$ is the non-negative $\CC^\infty$ function on ${\mathbb {P}}^k$ satisfying $f^* \eta^k = {{\textsf {Jac} \,  }}f
\cdot \eta^k$. The latter is the real jacobian of $f$, it vanishes on the critical set $\CC$ of $f$.\
The measure $\mu$ can also be defined *via* pluripotential theory: we have $\mu = T^k$, where $T$ is the Green current of $f$. The latter is a closed positive $(1,1)$ current on ${\mathbb {P}}^k$ with Hölder potentials. In particular, for any algebraic subset $A \subset {\mathbb {P}}^k$, there exist $c,{\gamma}> 0$ such that the $r$-neighbourhood of $A$ satisfies $\mu ( A[r] )
\leq c \, r^{\gamma}$ for any $r >
0$ (see [@DS4], Prop. 2.3.7). For any $\delta > 0$ and $\tilde c > 0$, we set $c_\delta:= (1-d^{-\delta})^{-1}$ and $\tilde c_\delta:= \tilde c (1-d^{-\delta})^{-1}$. In the sequel, $c > 0$ is a constant independent of $n$, it may differ from a line to another.

The class $\UU$ {#cccd}
---------------

Let us recall the definition of the class $\UU$ (see subsection \[byby\]).

\[unb\] Let $\UU$ be the set of functions $\psi : {\mathbb {P}}^k \to {\mathbb {R}}\cup \{ -
\infty \}$ satisfying:

- $e^\psi$ is $h$-Hölder on ${\mathbb {P}}^k$ for some $h >0$,

- $\NN_\psi:= \{ \psi = -\infty \}$ is a (possibly empty) proper algebraic set of ${\mathbb {P}}^k$,

- $\psi \geq \log d(\cdot,\NN_\psi)^\rho$ on ${\mathbb {P}}^k$ for some $\rho >0$.

The Hölder functions belong to $\UU$. Examples of unbounded observables are:

1.  the functions $\psi = \log \abs Q - q \log \norm{\cdot}$, where $Q$ is a $q$-homogeneous polynomial on ${\mathbb {C}}^{k+1}$. Here the algebraic subset $\NN_\psi$ is the zero set of $Q$.

2.  the functions $\psi = \log \norm{\Lambda^j d_x f}$ ($1
      \leq j \leq k$), where $\Lambda^j d_x f$ is the $j$-exterior power of the differential $d_x
      f$. In particular, $\log {{\textsf {Jac} \,  }}f \in \UU$ (take $j = k$).

The conditions of definition \[unb\] are easy to verify for these functions, the last one is a consequence of Lojasiewicz’s inequality (see [@Lo], §4.7). We prove below that $\psi \in L^p(\mu)$ for any $\psi \in \UU$ and $1 \leq p < + \infty$. Actually, we establish an estimate for $\int_{\NN_\psi[r]}  \abs{ \psi }^p$, useful to prove theorem B. We recall that $\mu \left( \NN_\psi[r]  \right) \leq c \, r^{\gamma}$ for some $c,{\gamma}> 0$ (see subsection \[consa\]).

\[cuun\] Let $\psi \in \UU$ and $1 \leq p < + \infty$. There exists $\kappa > 0$ such that: $$\label{dec}
\forall \, 0 < r < 1/2 \ \ , \ \ \int_{\NN_\psi[r]}  \abs{ \psi }^p \, d\mu  \leq \kappa \, r^{{\gamma}/2}.$$ In particular $\psi \in L^p(\mu)$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}Let $\psi \in \UU$ and $\NN:= \NN_\psi$. We may assume that $0 \leq e^\psi \leq 1$ by adding some constant to $\psi$. Let $r < 1/2$ and $\QQ_j:= \NN[r/2^j] \setminus \NN[r/2^{j+1}]$. Since $e^\psi \geq (r/2^{j+1})^\rho$ on $\QQ_j$, we obtain: $$\int_{\NN[r]}  \abs {\psi }^p \, d\mu  = \sum_{j \geq 0}
\int_{\QQ_j}  \abs{ \log e^\psi} ^p \, d\mu \leq \sum_{j \geq 0}
\left \vert \, \rho \log \left( {r \over 2^{j+1}} \right) \, \right  \vert ^p \cdot \mu (\QQ_j).$$ The inequalities $\mu (\QQ_j) \leq c (r/2^j)^{\gamma}$ and $\abs{\log {r \over 2^{j+1}}} =  (j+1) \log 2 + \log {1 \over r}
 \leq   (j+2)   \log {1 \over r}$ yield: $$\int_{\NN[r]}  \abs{ \psi }^p \, d\mu \leq  \left[ c \, \rho^p \,  \sum_{j \geq
  0} {(j+2)^p \over 2^{{\gamma}j}} \right]  \left( \log {1 \over r}
\right)^p   r^{\gamma}= M_{\rho,{\gamma}} \cdot \left(  \log {1 \over r} \right)^p  r^{{\gamma}/2} \cdot  r^{{\gamma}/2} .$$ The lemma follows with $\kappa:=   M_{\rho,{\gamma}} \cdot \sup_{ 0 < r < 1/2 } \left( \log {1 \over r} \right)^p r^{{\gamma}/2}$.

The Bernoulli space $(\Sigma,s,\nu)$ {#consaaa}
------------------------------------

We endow $\AA:= \{ 1, \ldots, d_t \}$ with the equidistributed probability measure $\bar \nu$. We set $\Sigma:= \AA^{\mathbb {N}}$, $s : \Sigma \to \Sigma$ the left shift and $\nu:= \otimes_{n =
  0}^\infty \bar \nu$. We denote by ${{\tilde \alpha}}:= ({\alpha}_n)_{n \geq 0}$ the elements of $\Sigma$, by $\CC_n$ the set of cylinders of length $n+1$, and by $\pi_n : \Sigma \to \AA^{n+1}$ the projection $\pi_n
({{\tilde \alpha}}):= ({\alpha}_0 , \ldots , {\alpha}_n )$. For any ${{\tilde \alpha}}\in \Sigma$, we set $C_n({{\tilde \alpha}}): = \pi_n^{-1} ({\alpha}_0 , \ldots , {\alpha}_n
)$. We denote by ${\mathbb {E}}(\chi \vert \CC_n)$ the conditional expectation of $\chi \in L^2(\nu)$ with respect to $\CC_n$. If $\LL = \{ A_1 , \ldots
, A_p \} \subset \CC_n$, we set $\LL^*:= \cup_{1 \leq j \leq p} A_j$.

Almost Sure Invariance Principle {#consb}
--------------------------------

Let $(X,g,m)$ be either $(\Sigma,s,\nu)$ or $({\mathbb {P}}^k,f,\mu)$. For any observable $\varphi \in
L^2(m)$, we set $S_n(\varphi):= \sum_{j = 0}^{n-1} \varphi \circ
g^j$ and $R_j(\varphi):= \int_X \varphi \cdot \varphi \circ g^j \,
dm$. We say that $\varphi$ is $m$-centered if $\int_X \varphi \, dm =
0$ and that $\varphi$ is a cocycle if $\varphi = u - u \circ g$ $m$-a.e. for some $u \in L^2(m)$.\
An observable $\varphi$ on $(X,g,m)$ satisfies the Almost Sure Invariance Principle (ASIP) if there exist on a probability space $(\tilde X , \tilde m)$ a sequence of random variables $(\SS_n)_{n \geq 0}$ and a Brownian motion $\WW$ such that:

1.  $\SS_n =  \WW(n) + o(n^{1/2 -
      \gamma})$ $\tilde m$-a.e. for some $\gamma > 0$,

2.  $(S_0(\psi),\ldots,S_n(\psi))$ and $(\SS_0,\ldots,\SS_n)$ have the same distribution for any $n \geq 0$.

We denote ${\sigma}$-ASIP to specify the variance of Brownian motion. The ${\sigma}$-ASIP implies the ${\sigma}$-Central Limit Theorem (${\sigma}$-CLT), meaning that: $$\forall t \in {\mathbb {R}}\ , \ \lim_{n \to \infty}  m \left( {S_n(\varphi)
  \over {\sigma}\sqrt n} \leq t  \right)  =  {1 \over \sqrt{2\pi}}
\int_{-\infty}^t e^{-u^2/2} \, du .$$

Suppose that ${\omega}: \Sigma \to {\mathbb {P}}^k$ is a coding map provided by theorem A. Since ${\omega}_* \nu = \mu$ and $f \circ {\omega}= {\omega}\circ s$, a $\mu$-centered observable $\psi \in L^{2}(\mu)$ satisfies the ${\sigma}$-ASIP if and only if the $\nu$-centered observable $\chi:= \psi \circ {\omega}\in
L^{2}(\nu)$ satisfies the ${\sigma}$-ASIP.

We shall use Philipp-Stout’s theorem ([@PS], Section 7) to prove the ASIP for $\chi
:= \psi \circ {\omega}$ on the Bernoulli space $(\Sigma,s,\nu)$. The version below comes from the original one by using the $s$-invariance of $\nu$ and the independence of the random process $(\xi_n)_{n \geq 0}$ defined by $\xi_n({{\tilde \alpha}}) = {\alpha}_n$.

[(Philipp-Stout)]{} Let $\chi$ be a $\nu$-centered observable on $\Sigma$ satisfying:

1.  $\chi \in L^{2+\delta} (\nu)$ for some $\delta > 0$,

2.  $\norm{  \chi - {\mathbb {E}}(\chi \vert \CC_n)  }_{2+\delta} \leq c \, \beta ^n$ for some $c > 0$ and $\beta < 1$.

Then the sequence ${1 \over \sqrt n} \norm{S_n(\chi)}_2$ has a limit ${\sigma}$. If ${\sigma}> 0$, then $\chi$ satisfies the ${\sigma}$-ASIP.

Let us compare that result with Ibragimov’s theorem (see [@I], Theorem 2.1), which only requires moments of order $2$ and a summability condition:

[(Ibragimov)]{} Let $\chi$ be a $\nu$-centered observable on $\Sigma$ satisfying: $$\sum_{n \geq 0}
  \norm{  \chi - {\mathbb {E}}(\chi \vert \CC_n)  }_2 < \infty .$$ Then the sequence ${1 \over
    \sqrt n} \norm{S_n(\chi)}_2$ has a limit ${\sigma}$. If ${\sigma}> 0$, then $\chi$ satisfies the ${\sigma}$-CLT.

A quantified version of Briend-Duval theorem {#BD}
============================================

This section is devoted to the proof of theorem \[good\] (see subsection \[cons\]). That result will be crucial to establish theorem A.

Briend-Duval theorem {#cons}
--------------------

We recall that $\VV_l = \cup_{i = 1}^ l f^i(\CC)$, $\VV = \cup_{i = 0}^\infty
f^i(\CC)$ and that $d_t = d^k$ is the topological degree of $f$ (see subsection \[consa\]). We set $\tau_*:= 2 \, {{\rm deg \, }}\VV_1 / (1-1/d)$.

[(Briend-Duval [@BrDu2])]{} Let $\eta > 0$ and $l \geq 1$ be such that $\tau_* / d^l < \eta$. Let $L$ be a complex line in ${\mathbb {P}}^k$ not contained in $\VV$, and $\Delta \Subset \tilde \Delta$ be topological discs in $L \setminus \VV_l$. Then, for any $n \geq
l$, there exist $(1-\eta)d_t^n$ inverse branches $g_n$ on $\Delta$ satisfying: $${{\textsf {diam} \, }}g_n (\Delta)  \leq { \tilde c \, d^{-n/2}   \over \eta^{1/2} \  {{\textsf {mod}  \, }}(\tilde \Delta \setminus \Delta )^{1/2} },$$ where $\tilde c$ is a universal constant, and ${{\textsf {mod}  \, }}(\tilde \Delta \setminus \Delta )$ is the modulus of the annulus $\tilde \Delta \setminus \Delta$.

Let us recall the definition of the modulus (see Ahlfors book [@A], chapters 1 and 2). Let $\Lambda$ denote the family of curves joining the boundary components of $A:= \tilde \Delta
\setminus \Delta$. For any conformal metric $\rho$ on $A$, we respectively denote by ${{\textsf {area}  }}_\rho$ and by $l_\rho$ the area and the length with respect to $\rho$. We denote by ${{\textsf {conf} \,  }}(A)$ the set of conformal metrics giving finite area to $A$. The modulus of the annulus $A$ is then defined by: $${{\textsf {mod}  \, }}( A ):= \sup_{\rho \in {{\textsf {conf} \,  }}(A)} \,   { l_\rho(\Lambda)^2 \over {{\textsf {area}  }}_\rho (A) } ,$$ where $l_\rho(\Lambda):= \inf_{\lambda \in \Lambda} l_\rho
(\lambda)$.

Statement of the quantified version {#cons}
-----------------------------------

We begin with some notations. Let $0 < \theta <1$ and $\theta_n:= [  \theta  n + { \log
    \tau_* \over  \log d } ] +1$. We introduce this integer in view of applying Briend-Duval theorem with $\eta = d^{-\theta n}$ and $l = \theta_n$ (indeed, $\tau_* / d^{\theta_n} < d^{-\theta
    n}$). Since the degree of $\VV_{\theta_n} = \cup_{i = 1}^{\theta_n} f^i(\CC)$ is at most $\tau_{\theta_n} = (d + \ldots + d^{\theta_n}) \deg \CC$, we have $\tau_{\theta_n} < d^{\theta_n}$ up to a multiplicative constant.\
We let $0 < \theta < \theta' < 1$ and consider $n_0 \geq 1$ satisfying: $$\label{nzero}
\forall n \geq n_0  \ \ \ ,  \ \ \ \theta_n < \theta'n \ \ \textrm{ and }  \ \ \ \tau_{\theta_n} < {d}^{\, \theta' n}.$$ Let us recall that $\VV_{\theta_n}[\delta]$ is the $\delta$-neighbourhood of $\VV_{\theta_n}$ in ${\mathbb {P}}^k$. We fix $\theta'/2 < \zeta < 1$ and define $\DD:= \limsup_{n \geq n_0}
\VV_{\theta_n}[d^{-\zeta n}]$.

\[vold\] The set $\DD$ satisfies ${{\rm Vol \, }}(\DD) = 0$.

The proof is postponed to subsection \[volo\]. We now state the quantified version. The constant $\tilde c$ has been introduced in the statement of Briend-Duval’s theorem, and we denote by $L$ the complex line containing $z$ and $w$.

\[good\] There exists ${\epsilon}> 0$ such that for every distinct points $(z,w) \notin \DD \cup \VV$, there exist an injective smooth path ${\gamma}:
[0,1] \to L \setminus \VV$ joining $z$ and $w$, a decreasing family of topological discs $(\Delta_n)_n \subset L$ and an integer $n_{z,w}$ such that for any $n \geq n_{z,w}$:

1.  ${\gamma}[0,1] \subset \Delta_n \subset L \setminus \VV_{\theta_n}$,

2.  there exist $(1-d^{-\theta n }) d_t ^n$ inverse branches of $f^n$ on $\Delta_n$,

3.  these branches satisfy ${{\textsf {diam} \, }}g_n (\Delta_n)  \leq \tilde c \,  d^{- {\epsilon}n }$.

We note that $\theta,{\epsilon}$ and $\tilde c$ do not depend on $(z,w) \in {\mathbb {P}}^k \setminus (\DD \cup \VV)$.

Construction of good paths in the complex line $L \subset {\mathbb {P}}^k$ {#ppaa}
--------------------------------------------------------------------------

Let $(z,w)$ be distinct points in ${\mathbb {P}}^k \setminus (\DD \cup \VV
)$. We identify the complex line $L$ containing $z$ and $w$ with the $2$-dimensional sphere. We recall that the Fubini-Study metric induces on $L$ the standard spherical metric ${{\textsf s }}$ with area $1$. We assume with no loss of generality that $z$ and $w$ are the North and South pole of $L$. Let $E$ be the equator of $L$. For any $y \in E$, we denote by $M_y$ the meridian containing $y$, and by $M_y \{ \delta \}$ the $\delta$-neighbourhood of $M_y$ in $L$ for the spherical metric. The constants $0 < \theta < \theta'
< 2\zeta$ have been defined in subsection \[cons\]. Now we let $0 < \zeta < \zeta' < \zeta'' < 1$ satisfying: $$\label{sop}
\theta' < \zeta'' - \zeta'   \ \textrm{ and } \  \theta + \zeta'' < 1.$$ We may take for $(\theta,\theta',\zeta, \zeta' ,\zeta'')$ suitable multiples of a small $\theta > 0$. The second inequality of (\[sop\]) will be used in next subsection. The integer $n_0$ has been defined in subsection \[cons\].

\[touch\] Let $(z,w)$ be distinct points in ${\mathbb {P}}^k \setminus (\DD \cup \VV )$. With the above notations, there exists a subset $F \subset E$ of full Lebesgue measure satisfying the following properties. For any $y \in F$, there exists $n_{z,w}(y) \geq n_0$ such that:

1.  the meridian $M_y$ does not intersect $\VV$,

2.  the neighbourhood $M_y \{ d^{- \zeta'' n} \}$ does not intersect $\VV_{\theta_n}$ for any $n \geq n_{z,w}(y)$.

Let us now prove proposition \[touch\]. We start with some notations. Let $H^+$ and $H^-$ be the (open) North and South hemispheres of $L$, these sets induce a partition $L = H^+ \sqcup E
\sqcup  H^-$. We denote by ${{\textsf {Leb}  }}$ the Lebesgue measure on $E$ and by $p_1$ (resp. $p_2$) the spherical projection from $z$ (resp. $w$) to $E$. For any $y \in E$ and $\delta > 0$, let $\II(y,\delta)$ be the interval in $E$ centered at $y$ with length $2\delta$. We also denote by $D(c,\delta) \subset L$ the disc with center $c$ and radius $\delta$. We define $p_\kappa(c):= p_1(c)$ if $c \in H^+ \cup E$ and $p_\kappa(c):= p_2(c)$ if $c \in H^-$. The same convention holds for the projection of $D(c,\delta)$ to $E$: we use $p_1$ or $p_2$ depending on $c \in H^+ \cup E$ or $c \in H^-$.\
Let $\{ c_i \, , \, 1 \leq i \leq
l_{\theta_n} \}:= \VV_{\theta_n} \cap L$, where $l_{\theta_n} \leq
{{\rm deg \, }}(\VV_{\theta_n}) \leq \tau_{\theta_n}$. Since the Fubini-Study metric induces ${{\textsf s }}$ on $L$, the set $\LL_{\theta_n}:=
  \cup_{i=1}^{l_{\theta_n}} D(c_i, d^{-\zeta n})$ is a subset of $\VV_{\theta_n}[ d^{-\zeta n} ]$. We recall that $\DD = \limsup_{n \geq n_0}
\VV_{\theta_n}[d^{-\zeta n}]$ and that $(z,w)
\notin \DD$. Thus there exists $n_1 \geq n_0$ depending on $(z,w)$ such that: $$\label{fl}
 \forall n \geq n_1 \ , \ (z , w)  \notin \VV_{\theta_n}[d^{-\zeta n}].$$ In particular $(z,w) \notin
  \LL_{\theta_n}$. Since $\zeta < \zeta' < \zeta''$, we may increase $n_1$ so that $d^{- \zeta' n} + d^{- \zeta'' n} < d^{-\zeta n}$ for any $n \geq n_1$. We have therefore, for $\rho = z$ or $w$: $$\label{evit}
  \forall \, 1 \leq i \leq l_{\theta_n} \ , \  \forall n \geq n_1  \ , \   D(\rho ,d^{-\zeta' n}) \cap
  D(c_i,d^{-\zeta'' n}) = \emptyset.$$ This implies, with $e_i:= p_\kappa(c_i) \in E$ and $c$ a positive constant: $$\label{kkkk}
 \forall \, 1 \leq i \leq l_{\theta_n} \ , \ p_\kappa  \left( D (c_i,d^{-\zeta''n} ) \right) \subset \II_i:= \II(e_i \, , \,  c \,  d^{-\zeta'' n} \cdot d^{\zeta' n}).$$ Hence $\II(\theta_n):= \cup_{i = 1}^{l_{\theta_n}} \II_i$ satisfies ${{\textsf {Leb}\,  }}\II(\theta_n) \leq \tau_{\theta_n}  \cdot c \, d^{-(\zeta''- \zeta') n} \leq c \, d^{(\theta'- (\zeta'' - \zeta'))n}$. Since $\sum_n {{\textsf {Leb}\,  }}\II(\theta_n) < \infty$ (see (\[sop\])), the Borel-Cantelli lemma yields, for every $y$ in a full Lebesgue measure subset $F' \subset E$, an integer $n_{z,w}(y) \geq n_1$ satisfying: $$\label{flf}
y \notin  \bigcup_{n \geq n_{z,w}(y)} \II(\theta_n) .$$ Let us prove the point 2 of proposition \[touch\] (the point 1 will be proved below, $F$ is a subset of $F'$). Let $y \in F'$ and $\II:= \II(y,d^{ - (\zeta'' -
  \zeta') n})$. Since the intervals $\II_i$ defining $\II(\theta_n)$ are centered at $e_i = p_\kappa(c_i)$, the set $p_1^{-1}(\II)$ does not intersect any point $c_i \in H^+ \cup E$. The same property holds for $p_2^{-1} (\II)$ with the $c_i \in H^-$. This implies that $M_y \{
d^{-\zeta'' n} \}$ does not intersect $\VV_{\theta_n} \cap L$ for any $n \geq n_{z,w}(y)$, and yields the point 2.

For the point 1, it suffices to verify that $p_\kappa(\VV \cap L)$ has zero Lebesgue measure. Let $\WW:= \VV \cap L$. Since $(z,w) \in L$ and $(z,w) \notin  \VV = \cup_{i =
  0}^\infty f^i(\CC)$, the complex line $L$ is not an algebraic subset of the hypersurface $f^i(\CC)$ for any $i \geq 0$. In particular, $\WW_i:=  f^i(\CC) \cap L$ is finite for every $i \geq 0$. Hence $\WW = \cup_{i \geq
  0} \WW_i$ satisfies ${{\textsf {Leb}  }}(p_\kappa(\WW)) = 0$. We finally set $F:= F' \setminus
p_\kappa(\WW)$, that completes the proof of proposition \[touch\].

Proof of theorem \[good\] {#bpo}
-------------------------

We set ${\epsilon}:= {1 \over 2} (1 - (\theta + \zeta'')) > 0$ (see (\[sop\])). Let $(z,w)$ be distinct points in ${\mathbb {P}}^k \setminus (\DD \cup \VV )$ and consider some $y \in F$ provided by proposition \[touch\]: the meridian $M_y$ does not intersect $\VV$ and its neighbourhood $M_y \{ d^{- \zeta'' n} \}$ in $L$ does not intersect $\VV_{\theta_n}$ for every $n \geq n_{z,w}(y)$.\
We set $n_{z,w}:= n_{z,w}(y)$ and denote $M:= M_y$ for sake of simplicity. Let ${\gamma}: [0,1] \to L$ be the natural parametrization of $M$. We define $\Delta_n:= M \{ d^{-\zeta'' n} / 2 \}$ and $\tilde \Delta_n:= M \{ d^{- \zeta'' n} \}$. Let us apply Briend-Duval’s theorem with $\eta = d^{-\theta n}$, $l =
\theta_n$ and $\Delta_n  \Subset \ \tilde \Delta_n \subset \ L
\setminus \VV_{\theta_n}$. Since $n > \theta' n \geq \theta_n = l$ and $\tau_* / d^{\theta_n} < d^{-\theta n}$ (see (\[nzero\])), there exist $(1-d^{-\theta  n }) d_t^n$ inverse branches on the disc $\Delta_n$ satisfying: $$\label{kopa}
  {{\textsf {diam} \, }}g_n (\Delta_n)  \leq  \tilde c \,  d^{-n/2}  \, \left(  d^{-\theta n} \, {{\textsf {mod}  \, }}\left[ \tilde \Delta_n \setminus \Delta_n \right] \right)  ^{-1/2} .$$ It remains to bound the modulus of $A_n:= \tilde
\Delta_n \setminus \Delta_n$. Let $\Lambda_n$ be the set of curves joining the boundary components of $A_n$. We denote by ${{\textsf {area}  }}_{{\textsf s }}$ and by $l_{{\textsf s }}$ the area and the length in $L$ with respect to the spherical metric ${{\textsf s }}$. The following estimates hold up to multiplicative constants. We have $l_{{\textsf s }}(\lambda) \geq d^{- \zeta'' n}$ for any $\lambda \in \Lambda_n$, hence $l_{{\textsf s }}(\Lambda_n) = \inf_{\lambda \in \Lambda_n} l_{{\textsf s }}(\lambda) \geq  d^{- \zeta'' n}$. The inequalities ${{\textsf {area}  }}_{{\textsf s }}(A_n) \leq {{\textsf {area}  }}_{{\textsf s }}(\tilde \Delta_n) \leq d^{- \zeta'' n}$ then imply: $$\label{kopb}
{{\textsf {mod}  \, }}( A_n ) =   \sup_{\rho \in {{\textsf {conf} \,  }}A_n}  { l_\rho(\Lambda)^2 \over {{\textsf {area}  }}_\rho (A_n) } \geq  { l_{{\textsf s }}(\Lambda_n)^2 \over {{\textsf {area}  }}_{{\textsf s }}(A_n) } \geq { d^{- 2 \zeta'' n} \over d^{- \zeta'' n} } = d^{- \zeta'' n} .$$ From (\[kopa\]), (\[kopb\]) and ${\epsilon}= {1 \over 2 } (1 - (\theta + \zeta''))$, we deduce that ${{\textsf {diam} \, }}g_n (\Delta_n) \leq \tilde c \, d^{-{\epsilon}n}$. That completes the proof of theorem \[good\].

Volume of neighbourhoods {#volo}
------------------------

This subsection is devoted to the proof of proposition \[vold\]: we want to show ${{\rm Vol \, }}(\DD) = 0$, where $\DD = \bigcap_{n \geq n_0} \bigcup_{p \geq
  n} \VV_{\theta_p}[d^{-\zeta p}]$. We recall that $\VV_{\theta_p}[d^{-\zeta
    p}]$ is the $d^{-\zeta p}$-neighbourhood of $\cup_{i =
  1}^{\theta_p} f^i(\CC)$ and that $\zeta > \theta'/2$. The proof is based on the following lemma (see [@DS4], lemma 2.3.8).

\[vois\] Let $X \subset {\mathbb {P}}^k$ be an algebraic subvariety of dimension $m$ and degree $q$. Then ${{\rm Vol \, }}X[\delta]  \leq q \, \delta^{2(k-m)}$ for any $\delta > 0$, up to a multiplicative constant independent of $X$.

We deduce ${{\rm Vol \, }}(\DD) = 0$ as follows. We set $p \geq n \geq n_0$ and apply lemma \[vois\] with $X = \VV_{\theta_p}$ and $\delta =
d^{-\zeta p}$ (here $k-m = 1$ and $q = {{\rm deg \, }}\VV_{\theta_p} \leq
\tau_{\theta_p}$). We obtain with $\tau_{\theta_p} \leq d^{\theta' p}$ (see (\[nzero\])): ${{\rm Vol \, }}\VV_{\theta_p}[d^{-\zeta p}]
 \leq \tau_{\theta_p} (d^{-\zeta p})^2  \leq d^{- (2\zeta -
   \theta')p}$. Hence: $$\forall n \geq n_0 \ , \ {{\rm Vol \, }}(\DD) \leq  {{\rm Vol \, }}\bigcup_{p \geq n}  \VV_{\theta_p}[d^{-\zeta
    p}] \leq c_{2\zeta - \theta'} \, d ^{- ( 2\zeta - \theta' ) n} .$$ This yields ${{\rm Vol \, }}(\DD) = 0$ when $n$ tends to infinity.

[ [<span style="font-variant:small-caps;">Proof</span> ]{}]{}<span style="font-variant:small-caps;">of lemma \[vois\]</span>: The argument is based on Lelong’s inequality. Let $\EE$ be a *maximal* $\delta$-separated set in $X$ for the ambient metric: this means that $d(a,b) \geq \delta$ for any pair of distinct elements of $\EE$, and that for any $x \in X$ there exists $a \in \EE$ satisfying $d(a,x) < \delta$. Since $X[\delta] \subset \cup_{a \in \EE}
B_a(2\delta)$, we get up to a multiplicative constant: $$\label{seff}
{{\rm Vol \, }}X[\delta]   \leq  (2\delta)^{2k} \, {\textsf {Card }}\EE .$$ We now give an upper bound for ${\textsf {Card }}\EE$. Observe that ${{\rm Vol \, }}X$ is equal to the degree of $X$, and that the balls $(B_a (\delta / 2 ) )_{a \in \EE}$ are mutually disjoint. Thus: $$\label{sefa}
  q = {{\rm Vol \, }}X \geq \sum_{a \in \EE} {{\rm Vol \, }}\left( X \cap B_a (\delta / 2 ) \right).$$ Now Lelong’s inequality asserts that ${{\rm Vol \, }}(X \cap B_a (\delta /
2 )) \geq \delta^{2m}$ for any $a \in \EE$, up to a multiplicative constant. Hence ${\textsf {Card }}\EE \leq q \, \delta^{-2m}$, as desired.

Proof of theorem A {#GCT}
==================

We set $\SS:= \VV \cup \DD \cup f(\DD) \cup
{{\textsf {Per}  }}(f)$, where $\DD$ is defined in subsection \[cons\]. We have ${{\rm Vol \, }}(\SS)=0$ since ${{\rm Vol \, }}(\DD) = 0$. Let us recall the statement of theorem A.

[A:]{} Let $z \in {\mathbb {P}}^k \setminus \SS$. There exist an $s$-invariant set $\Sigma'
\subset \Sigma$ of full $\nu$-measure and an $f$-invariant set $\JJ'
\subset \JJ$ of full $\mu$-measure satisfying the following properties. For any ${{\tilde \alpha}}\in
\Sigma'$, the point ${\omega}({{\tilde \alpha}}):= \lim_{n \to \infty} z_n({{\tilde \alpha}}) \in \JJ'$ is well defined. We have ${\omega}_* \nu  = \mu$ and the following diagram commutes: $$\xymatrix{
           \Sigma'  \ar[rr]^{s}  \ar[d]_{\omega}&    & \Sigma'  \ar[d]^{\omega}\\
           \JJ'  \ar[rr]^{f}       & &  \JJ'        }$$ Moreover there exist $\theta , {\epsilon}> 0$, $n_z \geq 1$ and $\tilde n: \Sigma' \to {\mathbb {N}}$ larger than $n_z$ such that:

1.  $d(z_n({{\tilde \alpha}}),{\omega}({{\tilde \alpha}}) ) \leq \tilde c_{\epsilon}\, d^{-{\epsilon}n}$ for every ${{\tilde \alpha}}\in \Sigma'$ and $n \geq \tilde n ({{\tilde \alpha}})$,

2.  $\nu ( \{ \tilde n  \leq q \} )  \geq 1 - c_\theta \, d^{-\theta q}$ for every $q \geq n_z$.

We shall use theorem \[good\] and the method of coding trees introduced in [@PUZ] for $({\mathbb {P}}^1,f,\mu)$. We recall that $\AA = \{ 1, \ldots, d_t \}$. Let $z \notin \SS$ and $\{ w_{\alpha}\, , \, {\alpha}\in \AA \}:= f^{-1}(z)$. By the very definition of $\SS$, the cardinal of $f^{-1}(z)$ is equal to $d_t$ and $w_{\alpha}\neq z$, $w_{\alpha}\notin \VV \cup \DD$ for every ${\alpha}\in \AA$. We denote by $L_{\alpha}$ the projective line in ${\mathbb {P}}^k$ containing $(z,w_{\alpha})$ and apply theorem \[good\]: let ${\gamma}_{\alpha}$ be an injective smooth path joining $(z, w_{\alpha})$ and $(\Delta_n({\alpha}))_n \subset L_{\alpha}$ be a decreasing sequence of discs containing ${\gamma}_{\alpha}$ provided by that theorem. We set $n_z:= \max \{ n_{z,w_{\alpha}} \, , \, {\alpha}\in \AA \}$.\
Let us fix $\tilde {\alpha}= ({\alpha}_n)_{n \geq 0} \in \Sigma$. We define inductively injective smooth paths ${\gamma}_n({{\tilde \alpha}}) : [0,1] \to {\mathbb {P}}^k \setminus \VV$ and points $z_n({{\tilde \alpha}}) \in {\mathbb {P}}^k \setminus \VV$. We first set ${\gamma}_0 ({{\tilde \alpha}}):=
{\gamma}_{{\alpha}_0}$. This path joins $z =
{\gamma}_0 ({{\tilde \alpha}})(0)$ and $w_{{\alpha}_0} = {\gamma}_0 ({{\tilde \alpha}})(1) =: z_0
({{\tilde \alpha}})$. Assume that the paths ${\gamma}_j ({{\tilde \alpha}})$ and the points $z_j ({{\tilde \alpha}})$ have been defined for $0 \leq j \leq n-1$. We let ${\gamma}_n(\tilde {\alpha})$ to be the lift of ${\gamma}_{{\alpha}_n}$ by $f^n$ with starting point ${\gamma}_n(\tilde
{\alpha})(0) = z_{n-1} ({{\tilde \alpha}})$. This path is well defined since ${\gamma}_{{\alpha}_n}$ does not intersect $\VV$. We finally let $z_n({{\tilde \alpha}}):=
{\gamma}_n ({{\tilde \alpha}})(1)$.\
We note that $z_{n-1}({{\tilde \alpha}})$ and $z_n({{\tilde \alpha}})$ are the endpoints of ${\gamma}_n({{\tilde \alpha}})$ and that $z_n(\Sigma) = f^{-(n+1)}(z)$ has cardinal $d_t^{n+1}$. The reader will easily check the relation $f \circ z_n({{\tilde \alpha}}) = z_{n-1} \circ s ({{\tilde \alpha}})$. Observe also that ${\gamma}_n({{\tilde \alpha}})$ and $z_n({{\tilde \alpha}})$ depend only on $\pi_n({{\tilde \alpha}}) = ({\alpha}_0,\ldots,{\alpha}_n)$. The following lemma is a consequence of theorem \[good\] and the fact that ${\gamma}_{\alpha}[0,1] \subset
\Delta_n({\alpha})$.

\[decr\] For every ${\alpha}\in \AA$ and $n \geq
n_z$, there exist at least $(1-d^{-\theta n})d_t^n$ elements $({\alpha}_0,\ldots,{\alpha}_{n-1}) \in \AA^n$ such that ${{\textsf {diam} \, }}{\gamma}_n ({\alpha}_0,\ldots,{\alpha}_{n-1},{\alpha})  \leq \tilde c \,  d^{- {\epsilon}n }$.

Let $\Omega_n:= \{ \tilde {\alpha}\in \Sigma \, , \, {{\textsf {diam} \, }}{\gamma}_n({{\tilde \alpha}}) > \tilde c \,
d^{-{\epsilon}n} \}$ and $\BB_n$ be the collection of $(n+1)$-cylinders $\{ C_n({{\tilde \alpha}})  \ , \ {{\tilde \alpha}}\in \Omega_n \}$. We have $\Omega_n = \BB_n^*$. Let us also define: $$\Omega(n) \ := \ \bigcup_{p \geq n} \Omega_p \  =   \  \bigcup_{p \geq n} \BB_p^* \, .$$

\[cdi\] For any $n \geq n_z$, we have:

1.  ${\textsf {Card }}(\BB_n) \leq d_t^{n+1}  d^{-\theta n}$.

2.  $\nu(\Omega_n) \leq d^{-\theta n}$, hence $\nu (\Omega(n))  \leq c_\theta \, d^{-\theta n}$.

3.  if ${{\tilde \alpha}}\notin \Omega(n)$, then $d( z_{m-1} ({{\tilde \alpha}}), z_m({{\tilde \alpha}})
      ) \leq \tilde c \, d^{-{\epsilon}m}$ for any $m \geq n$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}We have $\BB_n = \{ C_n({{\tilde \alpha}})  \ , \ {{\textsf {diam} \, }}{\gamma}_n({{\tilde \alpha}}) > \tilde c \,
d^{-{\epsilon}n}   \}$. For every ${\alpha}\in \AA$, we set $\BB_n({\alpha})
\subset \BB_n$ to be the collection of $(n+1)$-cylinders whose last coordinate is equal to ${\alpha}$. The lemma \[decr\] implies that ${\textsf {Card }}(\BB_n({\alpha}))
\leq d_t^n  d^{-\theta n}$ and thus ${\textsf {Card }}(\BB_n) = \sum_{{\alpha}\in \AA} {\textsf {Card }}( \BB_n({\alpha})
) \leq d_t^{n+1} d^{-\theta n}$, which is the point 1. The point 2 follows: $$\nu(\Omega_n)
= \nu(\BB_n^*) = {\textsf {Card }}(\BB_n) / d_t^{n+1} \leq d^{-\theta
  n}.$$ For the point 3, observe that $d(  z_{m-1} ({{\tilde \alpha}}) , z_m({{\tilde \alpha}}) ) \leq
  {{\textsf {diam} \, }}{\gamma}_m({{\tilde \alpha}})$. If ${{\tilde \alpha}}\notin \Omega(n)$, then ${{\tilde \alpha}}\notin
  \Omega_m$ for any $m \geq n$, hence ${{\textsf {diam} \, }}{\gamma}_m({{\tilde \alpha}}) \leq
\tilde c \, d^{-{\epsilon}m}$. [\
]{}

Let $\Omega:= \bigcap_{n \geq n_z} \Omega(n) =  \limsup_{n
  \geq n_z} \Omega_n$. The set $\Sigma'':= \Sigma \setminus
\Omega$ has full $\nu$-measure since $\nu(\Omega) \leq \nu(\Omega(n)) \leq c_\theta d^{-\theta n}$ for any $n
\geq n_z$. For every ${{\tilde \alpha}}\in \Sigma''$, we define $\tilde n ({{\tilde \alpha}})$ to be the least integer $n \geq n_z$ satisfying ${{\tilde \alpha}}\notin
\Omega(n)$. Let $\Theta_q:= \{ \tilde n  \leq q \}$.

\[pushoo\] $ $

1.  ${\omega}({{\tilde \alpha}})=\lim_{n
      \to \infty} z_n({{\tilde \alpha}})$ is well defined for every ${{\tilde \alpha}}\in \Sigma''$.

2.  $d(
      z_n({{\tilde \alpha}}), {\omega}({{\tilde \alpha}}) ) \leq \tilde c_{\epsilon}\, d^{-{\epsilon}n}$ for every $n
      \geq \tilde n ({{\tilde \alpha}})$.

3.  ${\omega}: \Sigma'' \to {\mathbb {P}}^k$ satisfies ${\omega}_*\nu = \mu$.

4.  $\nu ( \Theta_q )  \geq 1 - c_\theta \, d^{-\theta q}$ for any $q \geq n_z$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}The points 1, 2 and 4 come from lemma \[cdi\](3,2) and the definition of $\tilde n ({{\tilde \alpha}})$. Now we prove the point 3. Let us consider the surjective map $z_n : \Sigma'' \to  f^{-(n+1)}(z)$. Since $z_n({{\tilde \alpha}})$ depends only on $\underline {\alpha}:=
({\alpha}_0,\ldots,{\alpha}_n) \in \AA^{n+1}$, the measure ${z_n}_* \nu$ is equal to: $${z_n}_* \nu = \sum_{\underline {\alpha}\in \AA^{n+1}} \nu \left( \Sigma''
\cap C_n( \underline {\alpha}) \right) \, \delta_{z_n(\underline {\alpha})} =
 {1 \over d_t^{n+1}} \, \sum_{f^{n+1}(y) = z} \delta_y = \mu_{n+1,z}.$$ Since $z \notin \SS$ and $\EE \subset \VV \subset
\SS$, the sequence of probability measures $(\mu_{n,z})_n$ converges to $\mu$ (see subsection \[consa\]). Hence it remains to prove ${z_n}_* \nu \to {\omega}_* \nu$, meaning that $\int_{\Sigma''} \varphi \circ z_n \, d\nu \to \int_{\Sigma''}
 \varphi \circ {\omega}\, d\nu$ for every test function $\varphi :
 {\mathbb {P}}^k \to {\mathbb {R}}$. But this follows from point 1 and Lebesgue convergence theorem. [\
]{}

It remains to define $\Sigma' , \JJ'$ and to verify the relation $f \circ {\omega}= {\omega}\circ s$ on $\Sigma '$. The lemma \[pushoo\](3) implies that $\Sigma_*:= {\omega}(\Sigma'')$ satisfies $\mu(\Sigma_*) = \nu({\omega}^{-1}\Sigma_*) \geq \nu
(\Sigma'') = 1$. We define $\JJ':=  \bigcap_{n
  \in {\mathbb {Z}}} f^n(\JJ \cap \Sigma_*) $ and $\Sigma':=   \bigcap_{n \in {\mathbb {Z}}}
s^n(\Sigma'' \cap {\omega}^{-1} \JJ')$. These are invariant subsets of full measure. We obtain $f \circ {\omega}= {\omega}\circ s$ on $\Sigma'$ by taking limits in $f \circ z_n({{\tilde \alpha}}) = z_{n-1} \circ s ({{\tilde \alpha}})$. That completes the proof of theorem A.

Proof of theorem B {#estiho}
==================

Let us recall the statement.

[B:]{} Let $\psi \in \UU$ be a $\mu$-centered observable and ${\omega}$ be a coding map provided by theorem A. Let $\chi:= \psi \circ
  {\omega}$ and $1 \leq p < +\infty$.

1.  there exist $\hat c_p, \lambda_p > 0$ such that $\norm { \chi   - {\mathbb {E}}( \chi \vert
      \CC_n  )  }_p \leq \hat c_p \, e^{-n\lambda_p}$ for every $n \geq 0$.

2.  $R_j(\chi):= \int_\Sigma \chi \cdot
    \chi \circ s^j \, d\nu$ satisfies $\abs{R_j(\chi)} \leq 2 \norm {\chi}_2  \hat c_2 \,
      e^{-(j-1) \lambda_2}$ for every $j \geq 1$.

Proof of theorem B(1)
---------------------

We set $\chi_B:= \chi .  1_B$ for any $B
\subset \Sigma$ and use the following estimates provided by theorem A. We recall that $\Theta_n = \{ \tilde n ({{\tilde \alpha}}) \leq n \}$.

1.  $d( z_n({{\tilde \alpha}}),
    {\omega}({{\tilde \alpha}}) ) \leq \tilde c_{\epsilon}\, d^{-{\epsilon}n}$ for every ${{\tilde \alpha}}\in \Theta_n$,

2.  $\nu(\Theta_n) \geq 1 -  c_\theta  \, d^{- n
      \theta}$ for every $n \geq n_z$.

We will need the following lemma, which is a direct consequence of ($\star$).

\[fcx\] Let ${{\tilde \alpha}}\in \Theta_n$ and ${{\tilde \beta}}\in C_n({{\tilde \alpha}}) \cap \Theta_n$. Then $d({\omega}({{\tilde \alpha}}),{\omega}({{\tilde \beta}})) \leq 2 \, \tilde c_{\epsilon}\, d^{-{\epsilon}n}$.

### The Hölder case {#ccc2}

Let $\psi$ be an $h$-Hölder and $\mu$-centered observable on ${\mathbb {P}}^k$. We set $\chi:= \psi \circ
{\omega}$. The theorem B(1) is a consequence of the following estimates, which hold for every $n \geq n_z$.

\[esb1\] $\norm { \chi_{\Theta_n^c}   - {\mathbb {E}}( \chi_{\Theta_n^c} \vert \CC_n  )
  }_p \leq 2 \norm{\chi}_\infty  \, \left( c_\theta   \, d^{- n  \theta} \right)^{1/p} $.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}The left hand side is less than $2 \norm {\chi_{\Theta_n^c} }_p$ by Jensen inequality. Then the conclusion follows from $(\star \star)$.

\[esb2\] $\norm { \chi_{\Theta_n} - {\mathbb {E}}( \chi_{\Theta_n} \vert \CC_n  )
  }_p \leq c \, d^{-n \tau}$ for some $c, \tau > 0$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}We denote $\varphi:= \chi_{\Theta_n} - {\mathbb {E}}( \chi_{\Theta_n} \vert
\CC_n)$ and estimate $\norm
   {\varphi_{\Theta_n^c}}_p$, $\norm{\varphi_{\Theta_n}}_p$. Since $\varphi_{\Theta_n^c} =  - {\mathbb {E}}( \chi_{\Theta_n} \vert
\CC_n) \cdot 1_{\Theta_n^c}$, we have: $$\norm{\varphi_{\Theta_n^c}}_p \leq \norm { {\mathbb {E}}( \chi_{\Theta_n} \vert
\CC_n)  }_{2p} \cdot \nu(\Theta_n^c) ^{1/2p} \leq \norm{\chi}_{2p} \cdot \left( c_\theta \, d^{- n  \theta} \right)^{1/2p}.$$ We now deal with $\norm{\varphi_{\Theta_n}}_p$. For every ${{\tilde \alpha}}\in
\Theta_n$, let $\nu_{{\tilde \alpha}}$ be the conditional measure of $\nu$ on the cylinder $C_n({{\tilde \alpha}})$. We have for every ${{\tilde \alpha}}\in \Theta_n$: $$\label{lki}
  \varphi_{\Theta_n}({{\tilde \alpha}}) = \int_{C_n({{\tilde \alpha}}) \cap \Theta_n} \left( \chi({{\tilde \alpha}})
-\chi ({{\tilde \beta}}) \right)  d\nu_{{\tilde \alpha}}({{\tilde \beta}})  + \chi({{\tilde \alpha}}) \cdot  \nu_{{\tilde \alpha}}(C_n({{\tilde \alpha}}) \cap \Theta_n^c).$$ We deduce from $\chi =  \psi \circ {\omega}$, lemma \[fcx\] and the fact that $\psi$ is $h$-Hölder: $$\forall {{\tilde \alpha}}\in \Theta_n \ , \ \abs{\varphi_{\Theta_n}({{\tilde \alpha}})} \leq \left( 2 \, \tilde
 c_{\epsilon}\, d^{-n{\epsilon}} \right)^h  + \norm{\chi_{\Theta_n}}_\infty \cdot  \nu_{{\tilde \alpha}}(C_n({{\tilde \alpha}}) \cap \Theta_n^c).$$ Hence we get for every $p \geq 1$ up to a multiplicative constant: $$\forall {{\tilde \alpha}}\in \Theta_n \ , \ \abs{\varphi_{\Theta_n}({{\tilde \alpha}})}^p \leq d^{-nhp
 {\epsilon}} + \norm{\chi_{\Theta_n}}^p_\infty  \cdot  \nu_{{\tilde \alpha}}(C_n({{\tilde \alpha}}) \cap \Theta_n^c).$$ By integrating over $\Theta_n$ and using $(\star \star)$, we deduce: $$\norm{\varphi_{\Theta_n}}_p^p \leq d^{-nhp
 {\epsilon}} +  \norm{\chi}^p_\infty  \cdot  c_\theta d^{- n
 \theta}.$$ That completes the proof of the lemma.

### The general case $\psi \in \UU$ {#estib}

Let $\psi : {\mathbb {P}}^k \to {\mathbb {R}}\cup \{ - \infty \}$ be a $\mu$-centered observable in $\UU$: the function $e^\psi$ is $h$-Hölder and satisfies $\psi \geq \log
 d(\cdot,\NN_\psi)^\rho$ on ${\mathbb {P}}^k$ (see definition \[unb\]). Observe in particular that $\psi$ is bounded from above. We recall that $\NN_\psi[r]$ is the $r$-neighbourhood of $\{ \psi = - \infty\}$ and that $\chi = \psi
 \circ {\omega}$. We consider the following subsets of $\Sigma$: $$\SS_n:=
\Theta_n^c  \setminus \NN_n \ \ \ , \ \ \ \Gamma_n = \Theta_n \setminus \NN_n   \ \ \ , \ \ \      \NN_n:=  {\omega}^{-1} \left( \NN_\psi[d^{-n (h {\epsilon}/ 2 \rho)}]
\right).$$ We shall need the following observations. First, we have $\nu(\NN_n) = \mu ( \NN_\psi[d^{-n (h {\epsilon}/ 2 \rho)}] ) \leq d^{-n {\gamma}(h {\epsilon}/ 2 \rho)}$ up to a multiplicative constant (see subsection \[consa\]). We deduce from $(\star \star)$: $$\label{vatoo}
 \nu({\Gamma}_n^c) = \nu ( \Theta_n^c \cup \NN_n  ) \leq  c_\theta d^{- n
 \theta} + d^{-n {\gamma}(h {\epsilon}/ 2 \rho)} \leq c \, d^{- n \eta }$$ for some $c, \eta > 0$. Second, for every ${{\tilde \alpha}}\in \NN_n^c = \SS_n \cup \Gamma_n$, we have $\chi({{\tilde \alpha}}) \geq \log d({\omega}({{\tilde \alpha}}) ,
  \NN_\psi)^\rho \geq \log d^{-\rho n (h {\epsilon}/ 2 \rho)}$, hence: $$\label{vato}
 \norm{\chi_{ \SS_n \cup \Gamma_n} }_\infty
  \leq n \, (h {\epsilon}\log d ) / 2.$$ The theorem B(1) is now a consequence of the following estimates.

$\norm {\chi_{\NN_n} - {\mathbb {E}}( \chi_{\NN_n} \vert \CC_n ) }_p \leq \left( \kappa \, d^{- n (h {\epsilon}/ 2 \rho)\cdot ({\gamma}/ 2) } \right)^{1/p}$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}The left hand side is less than $2 \norm {
  \chi_{\NN_n} }_p$. Proposition \[cuun\] yields $\norm {\chi_{\NN_n}}_p =  \norm {\psi \circ {\omega}\cdot 1_{\NN_n}}_p  \leq \left( \kappa \,   d^{-n(h {\epsilon}/ 2
  \rho)\cdot({\gamma}/2)} \right)^{1/p}$ for every $n$ such that $d^{-n(h {\epsilon}/ 2
  \rho)} < 1/2$.

$\norm {\chi_{\SS_n} - {\mathbb {E}}( \chi_{\SS_n} \vert \CC_n ) }_p \leq  n \, (h {\epsilon}\log d) \cdot \left( c \, d^{- n  \eta} \right)^{1/p}$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}The left hand side is less than $2\norm {
  \chi_{\SS_n} }_p$. We conclude by using (\[vatoo\]) and (\[vato\]) (observe that $\SS_n \subset \Gamma_n^c$).

$\norm { \chi_{\Gamma_n}   - {\mathbb {E}}( \chi_{\Gamma_n} \vert \CC_n  )
  }_p \leq c \, d^{-n\tau}$ for some $c, \tau > 0$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}We follow the proof of lemma \[esb2\]: we set $\varphi:= \chi_{\Gamma_n} - {\mathbb {E}}( \chi_{\Gamma_n} \vert
\CC_n)$ and estimate $\norm
   {\varphi_{\Gamma_n^c}}_p$, $\norm{\varphi_{\Gamma_n}}_p$. The line (\[vatoo\]) yields: $$\norm{\varphi_{\Gamma_n^c}}_p \leq \norm { {\mathbb {E}}( \chi_{\Gamma_n} \vert
\CC_n)  }_{2p} \cdot \nu(\Gamma_n^c) ^{1/2p} \leq \norm{\chi}_{2p}
\cdot \left( c \, d^{- n \eta} \right)^{1/2p}.$$ Now we deal with $\norm{\varphi_{\Gamma_n}}_p$. We can write as in (\[lki\]): $$\label{popp}
 \forall {{\tilde \alpha}}\in \Gamma_n \ , \ \varphi({{\tilde \alpha}}) = \int_{C_n({{\tilde \alpha}}) \cap \Gamma_n} \left( \chi({{\tilde \alpha}})
-\chi ({{\tilde \beta}}) \right)  d\nu_{{\tilde \alpha}}({{\tilde \beta}})  + \chi({{\tilde \alpha}}) \cdot  \nu_{{\tilde \alpha}}(C_n({{\tilde \alpha}}) \cap \Gamma_n^c).$$ Let ${{\tilde \alpha}}\in \Gamma_n$ and ${{\tilde \beta}}\in C_n({{\tilde \alpha}}) \cap \Gamma_n$. We deduce from $({{\tilde \alpha}},{{\tilde \beta}}) \notin \NN_n$ that $e^\psi \circ {\omega}({{\tilde \alpha}})$ and $e^\psi
 \circ {\omega}({{\tilde \beta}})$ are larger than $d^{-n h {\epsilon}/ 2 }$. This implies: $$\abs{ \chi({{\tilde \alpha}}) -\chi({{\tilde \beta}}) } = \abs{ \log e^\psi \circ {\omega}({{\tilde \alpha}})   -  \log e^\psi \circ {\omega}({{\tilde \beta}})}  \leq d^{n h {\epsilon}/ 2}  \abs{ e^\psi \circ {\omega}({{\tilde \alpha}})   -  e^\psi \circ {\omega}({{\tilde \beta}}) }.$$ Using lemma \[fcx\] and the fact that $e^\psi$ is $h$-Hölder, the last term is less than $d^{n h {\epsilon}/ 2} \cdot
\left( 2 \, \tilde c_{\epsilon}\, d^{-n {\epsilon}}\right)^h$. Then we deduce from (\[popp\]), up to a multiplicative constant: $$\forall {{\tilde \alpha}}\in \Gamma_n \ , \ \abs{\varphi({{\tilde \alpha}})} \leq   d^{-n h {\epsilon}/ 2} + \norm{\chi_{\Gamma_n}}_\infty \cdot  \nu_{{\tilde \alpha}}(C_n({{\tilde \alpha}}) \cap \Gamma_n^c).$$ Taking the $p$-th power, integrating over $\Gamma_n$ and using (\[vatoo\]), (\[vato\]), we obtain up to a multiplicative constant: $$\norm{\varphi_{\Gamma_n}}_p^p \leq d^{-nhp
 {\epsilon}/2} +  \left( n ( h {\epsilon}\log d ) / 2 \right)^p  \cdot  c \, d^{- n
 \eta} .$$ That completes the proof of the lemma.

Proof of theorem B(2) {#corre}
---------------------

Let $\psi \in \UU$ be a $\mu$-centered observable and $\chi
= \psi \circ {\omega}$. Let $j \geq 1$ and $n \geq 0$ to be specified later. We set $\chi_n:= {\mathbb {E}}(\chi \vert \CC_n)$ and write: $$\chi \cdot \chi \circ s^j = (\chi - \chi_n) \cdot \chi \circ s^j + \chi_n \cdot ( \chi \circ s^j - \chi_n \circ s^j ) + \chi_n \cdot \chi_n \circ s^j  .$$ By using the $s$-invariance of $\nu$ and Jensen inequality $\norm{\chi_n}_2 \leq \norm{\chi}_2$, we deduce: $$\label{indp}
  \abs{R_j(\chi)}  = \left \vert  \int_{\Sigma} \chi \cdot \chi \circ s^j \, d\nu \right \vert  \leq 2  \norm{\chi}_2 \norm{\chi - \chi_n}_2 + \left \vert {\int_{\Sigma} \chi_n \cdot \chi_n \circ s^j \, d\nu } \right \vert .$$ The variables $\chi_n$ and $\chi_n
\circ s^j$ respectively depend on $(\xi_0,\ldots,\xi_n)$ and $(\xi_j,\ldots,\xi_{n+j})$, where $\xi_n : \Sigma \to \AA$ is the projection $\xi_n({{\tilde \alpha}}) = {\alpha}_n$. These are independent variables when $n = j-1$, hence $\int_{\Sigma} \chi_n \cdot \chi_n \circ s^j \, d\nu = \int_{\Sigma}
\chi_n \, d\nu  \int_{\Sigma} \chi_n \circ s^j \, d\nu$ in that case. But this product is zero since $\chi$ is $\nu$-centered. The conclusion then follows from (\[indp\]) with $n = j-1$ and theorem B(1).

Proof of theorem C {#CC}
==================

Let us recall the statement.

[C:]{} For every $\mu$-centered observable $\psi \in \UU$, we have:

1.  ${\sigma}:= \lim_{n \to \infty} {1 \over \sqrt n} \norm{S_n(\psi)}_2$ exists, and ${\sigma}^2 = \int_{{\mathbb {P}}^k} \psi^2 \, d\mu + 2\sum_{j \geq 1} \int_{{\mathbb {P}}^k}
     \psi \cdot \psi \circ f^j \, d\mu$.

2.  If ${\sigma}= 0$, then $\psi = u - u \circ f$ $\mu$-a.e. for some $u \in L^2(\mu)$.

3.  If ${\sigma}> 0$, then $\psi$ satisfies the ${\sigma}$-ASIP.

The points 1 and 2 are consequences of classical lemma \[somme\] below, whose condition $\sum_{j \geq 1} j \abs{R_j (\varphi)} < \infty$ is fulfilled by theorem B(2). The point 3 follows from proposition \[cuun\], theorem B(1) and Philipp-Stout’s theorem (see subsection \[consb\]).

\[somme\] Let $(X,g,m)$ be a dynamical system and $\varphi \in L^2(m)$ be a $m$-centered observable. We denote $S_n(\varphi) = \sum_{j = 0}^{n-1}
\varphi \circ g^j$ and $R_j(\varphi) = \int_X \varphi \cdot \varphi
\circ g^j \, dm$. Let ${\sigma}^2:=  R_0(\varphi) + 2\sum_{j \geq 1}
R_j(\varphi)$. If $\sum_{j \geq 1} j \abs{R_j (\varphi)} < \infty$, then ${\sigma}^2$ is finite and we have:

1.  $\norm{S_n(\varphi)}^2_2 = n {\sigma}^2 + O(1)$. In particular, $\lim_{n \to \infty} {1 \over n} \norm{S_n(\varphi)}^2_2 = {\sigma}^2$.

2.  ${\sigma}^2 = 0$ if and only if $\varphi = u - u \circ g$ $m$-a.e. for some $u \in L^2(m)$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}Let $S_n:= S_n(\varphi)$ and $R_j:= R_j (\varphi)$. Since $m$ is $g$-invariant, we have $\norm{S_n}^2_2 = n R_0 + 2 \sum_{j=1}^{n-1} \left( n-j  \right)
R_j$. We deduce for every $n \geq 1$: $$\label{pml}
   \norm{S_n}^2_2 = n \left(  R_0 + 2 \sum_{j = 1}^\infty R_j
\right) +  (- 2) \left( \sum_{j = 1}^{n-1} j R_j +   \sum_{j = n}^\infty n R_j\right) = n {\sigma}^2 + A_n ,$$ where $\abs{A_n} \leq  2 \sum_{j \geq 1} j \abs{R_j}$. That proves the point 1. Let us show the point 2. Suppose ${\sigma}^2 =
0$. In view of (\[pml\]), the function $u_p:= {1 \over p} \sum_{n=1}^p S_n$ satisfies $\norm{u_p}_2 \leq ( 2 \sum_{j \geq 1} j \abs{R_j} )^{1/2}$ for every $p \geq 1$. Let $u:= \lim_{j \to \infty} u_{p_j}$ be a weak cluster point in $L^2(m)$ and observe that: $$\forall j \geq 1 \ , \ u_{p_j} - u_{p_j} \circ g = {1 \over p_j}  \sum_{n=0}^{p_j - 1} \left( \varphi - \varphi \circ g^n \right) = \varphi -  {1 \over p_j}  S_{p_j} .$$ We deduce $\varphi = u - u \circ g$ $m$-a.e. by taking limits in $L^2(m)$ : $\lim_{j \to \infty}
u_{p_j} \circ g = u \circ g$ since $m$ is $g$-invariant, and $\lim_{j \to
  \infty}{1 \over p_j} S_{p_j} = \int_X \varphi \, dm = 0$ by Von Neumann theorem. The reverse implication of the point 2 comes from ${\sigma}^2 = \lim_{n \to \infty} {1 \over n} \norm{S_n(\varphi)}^2_2 =
\lim_{n \to \infty} {1 \over n} \norm{u - u \circ g ^n}^2_2 = 0$.

Proof of theorem D {#appll}
==================

We recall that $J:= \log {{\textsf {Jac} \,  }}f - \int_{{\mathbb {P}}^k} \log {{\textsf {Jac} \,  }}f \, d\mu$, this is an unbounded $\mu$-centered observable in $\UU$. We set ${\sigma}_J:= \lim_n {1 \over \sqrt n} \norm{S_n(J)}_2$, which is well defined by theorem C. We denote by $\chi_1 \leq \ldots \leq \chi_k$ the Lyapunov exponent of $\mu$, they are larger than or equal to $\log d^{1/2}$.

[D:]{} If the Lyapunov exponents of $\mu$ are minimal equal to $\log
d^{1/2}$, then ${\sigma}_J = 0$ and $\mu$ is absolutely continuous with respect to Lebesgue measure.

The first part ${\sigma}_J = 0$ will be proved in subsection \[zeze\]. The second part is a consequence of theorem \[laa\] below (that theorem will be proved in subsection \[mpo\] by using ${\sigma}_J = 0$). In the sequel, the maps $f^n$ and $d_x f^n$ are implicitely written in some fixed charts of ${\mathbb {P}}^k$.

\[laa\] Assume that the Lyapunov exponents are minimal. Then for $\mu$ almost every $x \in {\mathbb {P}}^k$, there exists $\rho(x) > 0$ and a subsequence $(n_j(x))_{j \geq 1}$ such that $f^{n_j} \circ \left(x +
d^{-n_j/2} \cdot {{\rm Id}}_{{\mathbb {C}}^k} \right) : B(\rho(x)) \to {\mathbb {P}}^k$ is injective.

[ [<span style="font-variant:small-caps;">Proof</span> ]{}]{}<span style="font-variant:small-caps;">of the second part of theorem D (abolute continuity):</span> We use the notations of theorem \[laa\]. Let $x \in
{\mathbb {P}}^k$ be a $\mu$-generic point and set $n_j:= n_j(x)$. Since $f^{n_j}$ is injective on the ball $B_j:= B_x ( \rho(x)d^{-n_j/2} )
$ and $\mu$ has constant jacobian $d^k$ (see subsection \[consa\]), we obtain $\mu (B_j) =  \mu (f^{n_j} (B_j)) d^{-k n_j}$. Observe also that ${{\textsf {Leb}  }}(B_j) = \rho(x)^{2k} \left(
d^{-n_j / 2}  \right)^{2k} =  \rho(x)^{2k} d^{-k n_j}$ up to a multiplicative constant. We obtain therefore for $\mu {\textrm{-a.e.}}\, x \in
{\mathbb {P}}^k$: $$\liminf_{r \to 0}  { \mu (B_x(r)) \over {{\textsf {Leb}  }}(B_x(r)) }  \leq \liminf_{j \to \infty} { \mu (B_j) \over {{\textsf {Leb}  }}(B_j) } =  \liminf_{j \to \infty} {\mu ( f^{n_j} (B_j) ) \over \rho(x)^{2k} } \leq {1 \over  \rho(x)^{2k} } < \infty.$$ That proves the absolute continuity of $\mu$ (see [@Mat], theorem 2.12).

Preliminaries {#abvc}
-------------

Observe that $J = \log {{\textsf {Jac} \,  }}f - \log d^k$ when the Lyapunov exponents are equal to $\log d^{1/2}$. Since the jacobian is a multiplicative function, we have in that case: $$\label{jjaa}
 S_n(J) = \sum_{i=0}^{n-1} J \circ f^i(x) = \log  {{\textsf {Jac} \,  }}f^n - \log d^{kn}.$$ The *singular values* $\delta_1 \leq
\ldots \leq \delta_k$ of the linear map $A:= d_x f^n$ are defined as the eigenvalues of $\sqrt{A A^*}$. In particular, there exist unitary matrices $(U,V)$ such that $d_x f^n = U \,
{{\textsf {Diag}  }}(\delta_1,\ldots,\delta_k) \, V$. We have therefore: $$\label{delja}
\delta_1 = \norm {(d_x f^n)^{-1} }^{-1} \ \  \textrm{ and }  \ \  \prod_{i=1}^k {\delta_i}^2 = {{\textsf {Jac} \,  }}f^n(x) \geq \delta_1^{2k}.$$ For any $\rho,\tau > 0$ and $n \geq 1$, we define: $$\BB_n(\rho):= \left \{  x \in {\mathbb {P}}^k \, , \, f^n \circ (x + d_x f^n)^{-1}  : B(\rho) \to {\mathbb {P}}^k \textrm{ is an injective map} \right\} ,$$ $$\RR_n(\tau):= \left\{ x \in {\mathbb {P}}^k \, , \, \norm {(d_x f^n)^{-1} }^{-1}  \geq d^{n/2}/\tau  \right \} .$$ The following estimates were proved by Berteloot-Dupont [@BeDu]. They hold for every system $({\mathbb {P}}^k, f, \mu)$ whose Lyapunov exponents satisfy $\chi_k < 2 \chi_1$.

\[bdu\] There exists ${\alpha}: \, ]0,1] \to {\mathbb {R}}^*_+$ satisfying $\lim_{\rho \to 0} {\alpha}(\rho) = 1$ and for $n \geq
      1$:

1.  $\mu(\BB_n(\rho)) \geq {\alpha}(\rho)$,

2.  $\mu(\BB_n(\rho) \cap \RR_n(\tau)^c ) \leq (\rho \, \tau)^{-2}$.

That result implies the following lemma.

\[eoui\] Let $\rho \in ]0,1]$. There exists $\HH \subset {\mathbb {P}}^k$ satisfying $\mu(\HH)=1$ and: $$\forall x \in \HH \ , \ \exists n(x) \geq 1 \ , \ \forall n \geq
n(x) \ , \ x \notin \BB_n(\rho)  \textrm{ or }  {{\textsf {Jac} \,  }}f^{n}(x) \geq
d^{kn} / n^{2k} .$$

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}We apply proposition \[bdu\](2) with $\tau = n$ to get $\mu (\BB_n(\rho) \cap \RR_n(n)^c) \leq  (\rho \, n)^{-2}$. Since $\sum_{n
  \geq 1} \mu (\BB_n(\rho) \cap \RR_n(n)^c) < \infty$, there exists by Borel-Cantelli lemma a subset $\HH$ of full $\mu$-measure satisfying: $$\forall x \in \HH \ , \ \exists n(x) \geq 1 \ , \ \forall n \geq
n(x) \ , \ x \notin \BB_n(\rho) \textrm{ or }  x \in  \RR_n(n) .$$ But $x \in  \RR_n(n)$ implies by (\[delja\]): ${{\textsf {Jac} \,  }}f^{n}(x) \geq \left( d^{n/2} / n \right)^{2k} = d^{kn} /
n^{2k}$ .

Proof of the first part of theorem D (${\sigma}_J = 0$) {#zeze}
-------------------------------------------------------

Suppose that the exponents are minimal and that ${\sigma}_J  = \lim_n
{1 \over \sqrt n} \norm{S_n(J)}_2 > 0$. Then $J$ satisfies the CLT: if $V
:= {1 \over \sqrt{2\pi}} \int_{- \infty}^{-1}  e^{-u^2/2} \, du$, we get $\mu \left( \GG_n:= \{  {S_n(J) \over \sqrt n } \leq - {\sigma}_J \} \right) \geq V / 2$ for $n$ larger than some $N$ (see subsection \[consb\]).\
Let $\rho > 0$ be such that $\mu(\BB_n(\rho)) > 1 - V / 4$ for every $n \geq 1$. If we set $\FF_n:= \BB_n(\rho) \cap \GG_n$, then $\FF
:= \limsup_{n \geq N} \FF_n$ satisfies $\mu(\FF)
\geq V/4$. Let $x \in \FF
\cap \HH$, where $\HH$ is provided by lemma \[eoui\]. Let $(n_j(x))_{j \geq 1}$ be such that $x \in
\FF_{n_j}$ for every $j \geq 1$. The inclusion $\FF_{n_j}  \subset \GG_{n_j}$ yields $S_{n_j}(J)(x) \leq - {\sigma}_J \sqrt{n_j}$ for every $j \geq 1$. Since $S_{n_j}(J) = \log {{\textsf {Jac} \,  }}f^{n_j} - \log d^{kn_j}$ (the exponents are indeed minimal, see (\[jjaa\])), we deduce: $$\label{ma}
  \forall j \geq 1 \ , \ {{\textsf {Jac} \,  }}f^{n_j}(x) \leq d^{k n_j} e^{- {\sigma}_J \sqrt{n_j}}.$$ But ${{\textsf {Jac} \,  }}f^{n_j}(x) \geq
d^{kn_j} / n_j^{2k}$ for every $n_j \geq n(x)$, following from $x \in \BB_{n_j}(\rho) \cap \HH$ and lemma \[eoui\]. That contradicts (\[ma\]) when $j$ tends to infinity.

Proof of theorem \[laa\] {#mpo}
------------------------

We proved in subsection \[zeze\] that ${\sigma}_J
= 0$. Hence $J = u - u \circ f$ $\mu$-a.e. for some $u
\in L^2(\mu)$ by theorem C. We obtain therefore: $$\label{coc}
  u - u \circ f^n (x) = \sum_{i=0}^{n-1} J \circ f^i(x) =  \log {{\textsf {Jac} \,  }}f^n(x) - \log d^{kn} .$$ Let ${\epsilon}>0$ and $m \geq 1$ such that $\MM:= \{\abs{u} \leq
\log m  \}$ satisfies $\mu(\MM) \geq (1-{\epsilon})^{1/2}$. Since $\mu$ is mixing, $\MM_n:= \MM \cap f^{-n} \MM$ satisfies $\mu (\MM_n)
\geq \mu(\MM)^2 - {\epsilon}\geq 1-2{\epsilon}$ for $n$ larger than some $N'$. Let $\rho$ be small and $\tau$ be large enough such that $\mu(
\BB_n(\rho) \cap \RR_n(\tau)) \geq 1-2{\epsilon}$ for every $n \geq 1$. We define $\TT_n:=
\BB_n(\rho) \cap \RR_n(\tau) \cap \MM_n$ and $\TT:= \limsup_{n \geq
  N'} \TT_n$. Observe that $\mu(\TT) \geq 1-4{\epsilon}$. Let $x \in \TT$ and $(n_j)_j$ (depending on $x$) such that $x \in
\TT_{n_j}$ for every $j \geq 1$. Since $x \in \TT_{n_j} \subset
\BB_{n_j}(\rho)$, the map $f^{n_j} \circ (x + (d_x f^{n_j})^{-1})  : B(\rho)
\to {\mathbb {P}}^k$ is injective.\
Let $\Lambda_{n} = d^{-n/2} \cdot
{{\rm Id}}_{{\mathbb {C}}^k}$. It is enough to prove that $d_x f^{n_j} = (U_j \, P_j \, V_j) \, \Lambda_{n_j}^{-1}$, where $(U_j,V_j)$ are unitary matrices and $P_j$ is a diagonal matrix with entries in $[a,b] \subset {\mathbb {R}}^*_+$ ($(a,b)$ being independent of $j$). Indeed, this implies that $f^{n_j} \circ (x + \Lambda_{n_j})$ is injective on $B(\rho/b)$, completing the proof of theorem \[laa\]. We shall omit the subscript $j$ for simplification, and denote by $\delta_1
\leq \ldots \leq \delta_k$ the singular values of $d_x f^n$. Let $(U,V)$ be unitary matrices such that $d_x f^n = U \,
{{\textsf {Diag}  }}(\delta_1,\ldots,\delta_k) \, V$ (see subsection \[abvc\]). The fact that $x \in \RR_{n}(\tau)$ yields: $$\label{mmm}
\delta_1  = \norm {(d_x f^n)^{-1} }^{-1} \geq d^{n/2}/\tau.$$ Now we give an upper bound for $\delta_k$. Since $x \in \TT_n \subset \MM_n$, we have $(x,f^n(x)) \in \MM = \{ \abs{u} \leq \log m \}$. This implies by (\[coc\]): $$d^{kn/2} / m \, \leq \,  \prod_{i=1}^k \delta_i = {{\textsf {Jac} \,  }}f^n(x) ^{1/2} \, \leq \, d^{kn/2} m.$$ We deduce from (\[mmm\]): $$\delta_k  \, \leq  \, {\delta_1 \ldots \delta_{k-1} \over \delta_1^{k-1}} \delta_k  \, = \, {{{\textsf {Jac} \,  }}f^n(x) ^{1/2} \over  \delta_1^{k-1}} \, \leq  \,  {   d^{kn/2} m \over ( d^{n/2} / \tau )^{k-1}  }  \, =  \,  d^{n/2} \tau^{k-1}  m.$$ Thus ${{\textsf {Diag}  }}(\delta_1,\ldots,\delta_k) =  \Lambda_n^{-1} \, P$, where $P$ is diagonal with entries in $[1/\tau , \tau^{k-1} m]$. We obtain finally $d_x f^n = U \, \Lambda_n^{-1} \, P \, V =  (U \, P \, V) \,
\Lambda_n^{-1}$, as desired.\
*Acknowledgements:* We thank the referees for their careful readings and useful suggestions. Their comments enabled us to improve the exposition of the article.

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[Christophe Dupont]{}

[Université Paris XI]{}\
[CNRS UMR 8628]{}\
[Mathématique, Bâtiment 425]{}\
[91405 Orsay Cedex, France]{}

[christophe.dupont@math.u-psud.fr]{}\
