2.1 Families of quasi-invariant measures

Let X be a locally compact metric space and $f:X\to X$ a bi-measurable map. From now on, by a measure on X, we mean a Radon measure and denote the set of such measures by $\mathscr {Rad}(X)$ . The set of continuous functions with compact support on X is denoted by $\mathcal {C}_c(X)$ . The set $\mathcal {C}_c(X)$ is equipped with the uniform topology, while $\mathscr {Rad}(X)$ is equipped with the vague one.

Notation: If $\phi $ is measurable, we write $f\phi :=\phi \circ f$ . If $\mu \in \mathscr {Rad}(X)$ , the measure $f\mu =f_{\ast }\mu $ is the unique measure $\nu $ such that

$$\begin{align*}\forall \phi\in\mathcal{C}_c(X),\quad \int f\phi \operatorname{d}\hspace{-2pt}\mu =\int \phi \operatorname{d}\hspace{-2pt}\nu. \end{align*}$$

With the above, it is direct to verify that for $\mu ,\nu \in \mathscr {Rad}(X)$ , $\mu =\rho \nu $ , it holds that $f\mu =f^{-1}\rho f\nu $ . Recall that $\mu \in \mathscr {Rad}(X)$ is quasi-invariant if $f^{-1}\mu \sim \mu $ ; in this case, we can write $f^{-1}\mu =\mathtt {h}_{\mu } \mu $ , where $\mathtt {h}_{\mu }=\frac {d f^{-1}\mu }{d \mu }$ . Similarly, $f^{-n}\mu =\mathtt {h}_{\mu }^{(n)} \mu $ .

As it can be deduced directly from the uniqueness of the Radon-Nikodym derivative, the family $\{\mathtt {h}_{\mu }^{(n)}\}_{n\in \mathbb Z}$ is a multiplicative cocycle over f with generator $\mathtt {h}_{\mu }$ .

Multiplicative cocycles

A family $\{h^{(n)}:X\to \mathbb R_{\geq 0}\}_{n\in \mathbb Z}$ is a multiplicative cocycle over f if

$$\begin{align*}\forall n,m \in \mathbb Z, x\in M\Rightarrow h^{(n+m)}(x)=h^{(n)}(f^mx)\cdot h^{(m)}(x). \end{align*}$$

Given $h:X\to \mathbb R_{>0}$ , it generates a multiplicative cocycle over f (and in this case, we call h the generator of the cocycle) by defining

$$\begin{align*}h^{(n)}(x)=\begin{cases} \prod_{k=0}^{n-1}h(f^kx)& k>0\\ 1 & k=0\\ \prod_{k=n}^{-1} h^{-1}(f^kx) &k<0. \end{cases} \end{align*}$$

Observe that if $\{h^{(n)}\}_{n\in \mathbb Z}$ is a positive multiplicative cocycle over f (with generator h), then $\{a^{(n)}:=\log h^{(n)}\}_{n\in \mathbb Z}$ is an additive cocycle (with generator $\log h$ ):

$$\begin{align*}\forall n,m \in \mathbb Z, x\in M\Rightarrow a^{(n+m)}(x)=a^{(n)}(f^mx)+a^{(m)}(x). \end{align*}$$

It will be useful to introduce the following.

Definition 2.1. Let $M,\Lambda $ be sets. We say that M is a bundle over $\Lambda $ if can be written as

$$\begin{align*}M=\bigsqcup_{i\in\Lambda} M_i \end{align*}$$

where $M_i$ is a locally compact metric space, for every $i\in \Lambda $ . We say that $f:M\to M$ is a bundle isomorphism if there exists a bijection $q:\Lambda \to \Lambda $ such that $f|:M_i\to M_{q(i)}$ is a homeomorphism, for every $i\in \Lambda $ .

Remark 2.1. In the definition above we are not assuming, a priori, any topology on M, and therefore, we do not require any global continuity for bundle morphisms. In the cases of interest, M will be manifold and each $M_i$ an immersed (but not necessarily embedded) sub-manifold.

For example, if M is a (Riemannian) manifold and $\mathcal {F}=\{W(x)\}_{x\in M}$ a foliation on M with smooth leaves, then the metric on M induces corresponding metrics, which in turn define a natural topology on each leaf, and we can write $M=\bigsqcup _{L\in \Lambda } L$ , where $\Lambda =M/\mathcal {F}$ . Thus, M is a bundle over $\Lambda $ . Furthermore, if $f:M\to M$ is a diffeomorphism and $\mathcal {F}$ is f-invariant, then f induces a bijection $q:\Lambda \to \Lambda , q(L)=f(L)$ , and thus, fis a bundle equivalence.

Consider the bundles over $\Lambda $ given as

$$ \begin{align*} &\mathrm{Con}:=\bigsqcup_{i\in\Lambda} \mathcal{C}_c(M_i)\\ &\mathrm{Rad}:=\bigsqcup_{i\in\Lambda} \mathscr{Rad}(M_i) \end{align*} $$

and consider also the space of sections of $\mathrm {Rad}$ ,

$$\begin{align*}\mathrm{Meas}:=\{\nu:\Lambda\rightarrow \mathrm{Rad}:\nu_i:=\nu(i)\in \mathscr{Rad}(M_i)\forall\ i\in\Lambda\}. \end{align*}$$

The natural topology of $\mathrm {Meas}$ is the weak topology determined by $\mathrm {Con}$ ; it is a locally convex topology. Let $\mathrm {Con}^+:=\{\phi \in \mathrm {Con},\phi \geq 0, \phi \not \equiv 0\}$ .

Lemma 2.2. Suppose that $\mathcal {A}\subset \mathrm {Meas}$ satisfies

1. 1. for every $\psi \in \mathrm {Con}$ , there exists $c(\psi )>0$ with the property that for every $\mu \in \mathcal {A}, \mu (\psi )\leq c(\psi )$ .

2. 2. For every $\psi \in \mathrm {Con}^+$ , there exists $d(\psi )>0$ with the property that for every $\mu \in \mathcal {A}, \mu (\psi )\geq d(\psi )$ .

Then, $\operatorname {\mathrm {cl}}{\mathcal {A}}\subset \mathrm {Meas}$ is compact, and furthermore, it does not contain the zero section.

Proof. This is a direct consequence of Tychonoff’s theorem.

From now on, we assume that f is a bundle equivalence.

Definition 2.2. $\nu \in \mathrm {Meas}$ is quasi-invariant if for every $i\in \Lambda $ , $f^{-1}\nu _{qi}\sim \nu _{i}$ with positive continuous Radon-Nikodym derivative $\rho _i=\frac {\operatorname {d}\hspace {-2pt} f^{-1}\nu _{qi}}{\operatorname {d}\hspace {-2pt}\nu _i}$ . In this case, $\rho :M\to \mathbb R_{>0}$ given by $\rho |M_i=\rho _i$ will be called the Jacobian of $\nu $ .

One verifies directly that if $\nu $ is quasi-invariant, then $\rho $ generates a (multiplicative) cocycle over q, and we write $\tilde {\rho }=\log \rho $ for the corresponding additive cocycle.

Now suppose that $\varphi :M\to \mathbb R$ is a function such that $\varphi |M_i$ is continuous for each i, and $\nu \in \mathrm {Meas}$ is quasi-invariant: we are interested in finding $\mu \in \mathrm {Meas}$ quasi-invariant with Jacobian $e^{-\varphi }$ . Consider $\nu ^n=e^{S_n\varphi }\cdot f^{-n}\nu $ , i.e.

(2.1) $$ \begin{align} \nu^n_i=e^{S_n\varphi}\cdot f^{-n}\nu_{q^n(i)}=e^{S_n\varphi+S_n\tilde{\rho}}\cdot\nu_i \end{align} $$

and let $\mathcal {C}^{+}$ be the positive cone generated by $\{\nu ^n\}_{n\geq 0}$ ,

(2.2) $$ \begin{align} \mathcal{C}^{+}=\left\{\sum_{j=1}^ka_i\nu^{n_j}:a_j\geq 0,k\in\mathbb{N}\right\}. \end{align} $$

Theorem 2.3. Suppose that $M=\bigsqcup _{i\in \Lambda } M_i$ is a bundle over $\Lambda $ , $f:M\to M$ is a bundle equivalence, $\varphi :M\to \mathbb R$ is a function such that its restriction to each $M_i$ is continuous, and $\nu \in \mathrm {Meas}$ is quasi-invariant. Assume that $\{\nu ^n=e^{S_n\varphi }\cdot f^{-n}\nu \}_{n\geq 0}$ satisfies:

• • if $\phi \in \mathrm {Con}^+$ , then $\nu ^n(\phi )>0$ for all n.

• • Given $\phi , \psi \in \mathrm {Con}$ with $\phi \not \equiv 0$ non-negative, there exists $e(\phi ,\psi )>0$ such that for every $n\geq 0$ , it holds that

  $$\begin{align*}\frac{\nu^n(\psi)}{\nu^n(\phi)}\leq e(\phi,\psi). \end{align*}$$

Then, there exist $P\in \mathbb R$ and a quasi-invariant $\mu \in {\overline {\mathcal {C}^{+}}}$ with Jacobian $e^{P-\varphi }$ . Moreover, $\mu $ has full support on each $M_i$ .

Proof. Define $S:\mathrm {Meas}\rightarrow \mathrm {Meas}$ by

$$\begin{align*}S(\mu)_i=e^{\varphi}\cdot f^{-1}\mu_{q(i)}.\\[-15pt] \end{align*}$$

Then, S is continuous and sends $\mathcal {C}^{+}$ to itself. We compute

(2.3) $$ \begin{align} S(\nu^n)_i=e^{\varphi}\cdot f^{-1}\nu_{q(i)}^n= e^{\varphi}\cdot f^{-1}\left(e^{S_n\varphi}f^{-n}\nu_{q^{n+1}(i)}\right)=e^{\varphi+S_n\varphi\circ f}\cdot f^{-n-1}\nu_{q^{n+1}(i)}=\nu^{n+1}_i.\\[-15pt]\nonumber \end{align} $$

Fix a non-negative, non-identically zero function $\phi _0\in \mathrm {Con}$ and let $\hat {\nu }^n:=\frac {\nu ^n}{\nu ^n(\phi _0)}$ ; consider the set

(2.4) $$ \begin{align} \mathcal{X}&=\operatorname{\mathrm{cl}}{\left\{\sum_{i=1}^ka_i\hat{\nu}^{n_i}:a_i\geq 0,k\in\mathbb{N},\sum_{i=1}^ka_i=1\right\}} \end{align} $$

(2.5)

By Lemma 2.2 and the assumed hypotheses, $\mathcal {X}$ is a compact convex subset of $\mathrm {Meas}$ , and by Equation (2.3), the normalized operator $\widetilde {S}$ ν:= SνSν(ϕ ₀) sends $\mathcal {X}$ to itself. Hence, by the Schauder-Tychonoff fix point theorem, there exists $\mu \in \mathcal {X}$ such that $\widetilde {S}(\mu )=\mu $ . In other words, $\psi \in \mathrm {Con}$ implies

(2.6) $$ \begin{align} \mu_{q(i)}(\psi\circ f^{-1}e^{\varphi\circ f^{-1}})=e^P \cdot \mu_{i}(\psi)\quad (\Rightarrow f^{-1}\mu_{q(i)}=e^{P-\varphi}\cdot \mu_i),\\[-15pt]\nonumber \end{align} $$

where $e^P=S\mu (\psi _0)>0$ .

Observe that if $\psi \in \mathrm {Con}$ is non-negative and non-identically zero, then for some constant $c(\psi )>0$ , it holds that $\hat {\nu }^n(\psi )>0$ for all $n\geq 0$ , and thus, the same is valid for $\mu $ . Therefore, $\mu $ has full support on each $M_i$ , and the proof of the Theorem is complete.

2.2 Quasi-invariant measures along leaves of foliations: the transverse measure

We will now apply the construction of the previous part to the following setting.

1. H-1 M is a compact (closed) smooth Riemannian manifold, and $\mathcal {F}=\{F(x)\}_{x\in M}$ is continuous foliations with smooth leaves. We consider the bundle structure of $M=\bigsqcup _{F\in \Lambda } F$ over $\Lambda =M/\mathcal {F}$ and denote $\mathrm {Con}(\mathcal {F}),\mathrm {Meas}(\mathcal {F})$ the corresponding associated bundles.

2. H-2 f is a diffeomorphism of M of differentiability class $\mathcal {C}^2$ .

3. H-3 $\mathcal {F}$ is f-invariant; the induced map by f on $\Lambda $ is denoted by q.

4. H-4 There exists a complementary continuous foliation $\mathcal {G}=\{G(x)\}_{x\in M}$ of $\mathcal {F}$ with smooth leaves, and such that

   1. (a) $\mathcal {G}$ is minimal: each leaf $G(x)$ is dense.

   2. (b) $\mathcal {G}$ is f-invariant.

   3. (c) $\mathcal {G}$ is contracting under f: if $d_{\mathcal {G}}$ denotes the intrinsic distance (on the corresponding leaf of $\mathcal {G}$ ), we have

      $$\begin{align*}\forall x,y\in M, y\in G(x)\Rightarrow d_{\mathcal{G}}(fx,fy)\leq \lambda d_{\mathcal{G}}(x,y)\\[-15pt] \end{align*}$$

      for some $0<\lambda <1$ .

The idea is that if B is a foliated box corresponding to $\mathcal {F}$ , we will use the (measurable) partition $\mathcal {F}|B$ to play the role of the quotient space where we define our transverse measure. The existence of a complementary contracting foliation is used (crucially) to ensure the compactness property necessary for Lemma 2.2.

Notation: If $A\subset M, \epsilon>0$ , let

$$\begin{align*}G(A,\epsilon):=\{y\in M:\exists x\in A/ d_{\mathcal{G}}(x,y)<\epsilon\}, \end{align*}$$

and if $A=\{x\}$ , we write $G(x,\epsilon )=G(\{x\},\epsilon )$ .

Observe that given $\psi \in \mathrm {Con}(\mathcal {F}),\delta>0$ , there exists $\gamma =\gamma (\psi ,\delta )>0$ such that if $x\in G(\operatorname {\mathrm {supp}}(\psi ),\gamma )$ , then

• • the transverse foliation defines an homeomorphism onto its image $\mathrm {hol}^{\scriptstyle \mathcal {G}}|:\operatorname {\mathrm {supp}}(\psi )\to F(x)$ .

• • $\psi _{F(x)}:=\psi \circ \mathrm {hol}^{\scriptstyle \mathcal {G}}$ is continuous, of compact support, and $\delta $ -equivalent to $\psi $ .

We seek to find a quasi-invariant $\nu \in \mathrm {Meas}(\mathcal {F})$ with regular Radon-Nikodym derivative. A natural class is obtained by fixing a Riemannian metric on M and considering ${\mathrm {Leb}}\in \mathrm {Meas}(\mathcal {F})$ given by

$$\begin{align*}F\in\Lambda\Rightarrow {\mathrm{Leb}}_F=\text{ induced Lebesgue measure on the leaf }F. \end{align*}$$

By the basic change of variables theorem, ${\mathrm {Leb}}$ is quasi-invariant and its Jacobian is $\rho _F=|\det Df|F|$ . Due to compactness of M, we have that

$$\begin{align*}\rho\text{ is uniformly bounded from below: }\rho(x)\geq C_{\rho}>0\ \forall x\in M, \end{align*}$$

and since $f\in \mathscr {Diff}^{1+\alpha }(M)$ , we get that $\rho $ (and therefore, $\tilde {\rho }=\log \rho $ ) is $\alpha $ -Hölder.

We will use the minimality of $\mathcal {G}$ to compare sets (functions) defined on different leaves of $\mathcal {F}$ . It is necessary, however, to control how families of measures $\nu =\{\nu _F\}_{F\in \mathcal {F}}$ vary with the transverse holonomy given by $\mathcal {G}$ . By transversality between $\mathcal {F}$ and $\mathcal {G}$ and compactness of M, it holds that given $\delta _0>0$ , there exists $\delta _1>0$ , so that for every $x,y\in M, y\in G(x,\frac {\delta _0}{2})$ ,

$$\begin{align*}F(y,\delta_1)\text{ is }\delta_0\text{-equivalent to some subset of } F(x).\\[-16pt] \end{align*}$$

Definition 2.4. $\nu \in \mathrm {Meas}(\mathcal {F})$ is absolutely continuous if for every (locally defined) holonomy transport $\mathrm {hol}^{\scriptstyle \mathcal {G}}:F_{i_1}\to F_{i_2}$ , $\mathrm {hol}^{\scriptstyle \mathcal {G}}\nu _{F_{i_1}}\sim \nu _{F_{i_2}}$ . It is strongly absolutely continuous if there exist $\delta _0>0$ and $J:\{(x,y,z)\in M\times M\times M, y\in G(x,\delta _0), z\in F(y,\delta _1)\}\to \mathbb R$ continuous such that if $B=F(y,\delta _1), A=\mathrm {hol}^{\scriptstyle \mathcal {G}}(B) \subset F(x)$ , then

$$\begin{align*}\mathrm{hol}^{\scriptstyle \mathcal{G}}\nu_{F(x)}|A=J(x,y,\cdot)\nu_{F(y)}|B.\\[-16pt] \end{align*}$$

For strongly absolutely continuous sections, we have the following.

Proposition 2.4.2. Assume that $\nu $ is strongly absolutely continuous and of full support on every $F\in \mathcal {F}$ . Then, given $\epsilon>0$ , there exists $\delta>0$ such that for every $\psi _1,\psi _2\in \mathrm {Con}(\mathcal {F})$ that are $\delta $ -equivalent and non-identically zero, it holds that

Proof. We start by noting that since $\nu $ is strongly absolutely continuous, given $\epsilon>0$ , there exists $\delta $ such that if $A,B$ are $\delta $ -equivalent (with nontrivial measure), then

(2.7) $$ \begin{align} \Big|\frac{\nu(A)}{\nu(B)}-1\Big|<\epsilon.\\[-16pt]\nonumber \end{align} $$

The remaining part of the argument is explained in [Reference Margulis32]; for convenience of the reader, we repeat it here.

Claim. Given $\epsilon>0$ , there exists $\delta>0$ such that for every $\psi _1,\psi _2\in \mathrm {Con}(\mathcal {F})$ that are $\delta $ -equivalent, non-identically zero and non-negative, it holds that

$$\begin{align*}\left|1-\frac{\int \psi_2 \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}}}{{\int \psi_1 \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}}}\right|<\delta.\\[-16pt] \end{align*}$$

Proof. Given $\epsilon>0$ , let $\delta>0$ , as above, and consider $\psi _1, \psi _2$ under the hypotheses of the claim. Let $A \subset F_{i_1}, B \subset F_{i_2}$ be the supports of $\psi _1, \psi _2$ . Then, by the Radon-Nikodym theorem, we can write

$$\begin{align*}\int \psi_1 d\nu_{F_{i_1}}=\int \psi_2\cdot j \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}},\\[-16pt] \end{align*}$$

where, by (2.7), $|1-j(z)|<\delta $ for every $z\in B$ . Using that $\psi _1, \psi _2$ are non-negative, we deduce

$$ \begin{align*} (1-\delta)\cdot \int \psi_2 \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}}\leq \int \psi_1 \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}} \leq (1+\delta)\cdot \int \psi_2 \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}}\\[-16pt] \end{align*} $$

which implies the claim.

Given $\psi _1,\psi _2\in \mathrm {Con}(\mathcal {F})$ that are $\delta $ -equivalent and non-identically zero, we decompose into their positive and negative parts, $\psi _1=\psi _1^{+}-\psi _1^{-}, \psi _1^{-}, \psi _2=\psi _2^{+}-\psi _2^{-}$ , and use the previous claim:

$$ \begin{align*} \left|\int \psi_1 \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}-\int \psi_2 \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}}\right|&\leq \left|\int \psi_1^+ \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}-\int \psi_2^+ \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}}\right|+ \left|\int \psi_1^- \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}-\int \psi_2^- \operatorname{d}\hspace{-2pt}\nu_{F_{i_2}}\right|\\ &\leq\delta \int \psi_1^+ \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}+\delta \int \psi_1^- \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}=\delta \int |\psi_1| \operatorname{d}\hspace{-2pt}\nu_{F_{i_1}}. \end{align*} $$

The proof is complete.

In what follows, we will be interested in quasi-invariant sections $\nu \in \mathrm {Meas}(\mathcal {F})$ . We introduce the following definition.

Fix $\nu \in \mathrm {Meas}(\mathcal {F})$ appropriate for f and consider $\varphi :M\to \mathbb R$ a Hölder continuous potential. Let $\rho $ be the Jacobian of $\nu $ , $\hat {\rho }=\log \rho $ and $\hat {\varphi }=\varphi +\hat {\rho }$ ; this is a $(C_{\hat {\varphi }},\theta )$ -Hölder function. For $n\in \mathbb N$ , take $\nu ^{n}=e^{S_n\varphi }\cdot f^{-n}\nu =e^{S_n\hat {\varphi }}\nu $ (cf. Equation (2.1)).

Lemma 2.6. There exists a constant $D_1>0$ and $\ell :\mathbb R_{>0}\to \mathbb R_{>0}$ with the following property. For every $\delta>0$ , if $\psi _1,\psi _2\in \mathrm {Con}^+$ are $\delta $ -equivalent and $n\geq 0$ , then it holds that

$$\begin{align*}\nu^n(\psi_1)\leq \ell(\delta)\cdot e^{D_1\delta^{\theta}}\nu^n(\psi_2). \end{align*}$$

Furthermore, $\ell (\delta )\xrightarrow [\delta \to 0]{}1$ .

Proof. Define $D_1:=C_{\hat {\varphi }}\sum _{i=0}^{\infty } \lambda ^{i\theta }$ ; since $\lambda <1$ , $D<+\infty $ . Assume that $\psi _1,\psi _2$ are $\delta $ -equivalent, $\psi _2\circ \mathrm {hol}^{\scriptstyle \mathcal {G}} =\psi _1$ . Observe that for every $x\in \operatorname {\mathrm {supp}}(\psi _1)$ , for every $n\geq 0$ , it holds that

It then follows that

$$ \begin{align*} \nu^n(\psi_1)&=\int e^{S_n\hat{\varphi}} \psi_2\circ \mathrm{hol}^{\scriptstyle \mathcal{G}} \operatorname{d}\hspace{-2pt}\nu=\int e^{S_n\hat{\varphi}-S_n\hat{\varphi}\circ \mathrm{hol}^{\scriptstyle \mathcal{G}}} \psi_2\circ \mathrm{hol}^{\scriptstyle \mathcal{G}} e^{S_n\hat{\varphi}\circ \mathrm{hol}^{\scriptstyle \mathcal{G}}} \operatorname{d}\hspace{-2pt}\nu\\ &\leq e^{D_1\delta^{\theta}} \int \psi_2 e^{S_n\hat{\varphi}} \operatorname{d}\hspace{-2pt}\mathrm{hol}^{\scriptstyle \mathcal{G}}\nu\leq \ell(\delta)e^{D_1\delta^{\theta}} \nu^n(\psi_2), \end{align*} $$

where $\ell $ is an upper bound of $\frac {d \mathrm {hol}^{\scriptstyle \mathcal {G}}\nu }{d\nu }$ , and therefore converges to $1$ as $\delta \to 0$ .

Lemma 2.7. Let $A_1\subset F_{i_1},A_2\subset F_{i_2}$ be relatively open and pre-compact. Then, there exist $\hat {e}(A_1,A_2)>0$ such that for every $n\geq 0$ , it holds that

$$\begin{align*}\frac{1}{\hat{e}(A_1,A_2)}\leq \frac{\nu^n(A_1)}{\nu^n(A_2)}\leq \hat{e}(A_1,A_2). \end{align*}$$

Proof. Let us start noting that since $\mathcal {G}$ is minimal (on M compact), we have the following property: for any $A\subset F, F\in \mathcal {F}$ there exist $\delta (A),r(A)>0$ such that for every $x\in M$ , one can find $B_x\subset A$ that is $\delta (A)$ -equivalent to $F(x,r(A))$ .

By relative compactness of $\operatorname {\mathrm {cl}}{A_1}$ , the closure of $A_1$ inside $F_{i_1}$ , it follows that $\operatorname {\mathrm {cl}}{A_1}\subset \cup _1^m F(x_j,r(A_2))$ , where each plaque $F(x_j,r(A_2))$ is $\delta (A_2)$ -equivalent to some $B_j\subset A_2$ .

Fix two of these sets $E_1=F(x_j,r(A_2)), E_2=B_j$ and denote by $\mathrm {hol}^{\scriptstyle \mathcal {G}}$ the holonomy transport such that . By regularity of $\nu _{F_{i_2}}$ we can find a non-decreasing sequence $(\phi _k)_k\subset \mathcal {C}_c(F_{i_2})$ satisfying

1. 1. $k\in \mathbb {N}\Rightarrow 0\leq \phi _k\leq 1, \operatorname {\mathrm {supp}}(\phi _k)\subset \operatorname {\mathrm {cl}}{A_2}$ .

2. 2. .

3. 3. $\nu ^n(A)=\lim _k\uparrow \nu ^n(\phi _k)$ .

Define $\psi _k=\phi _k\circ \mathrm {hol}^{\scriptstyle \mathcal {G}}$ and apply the previous lemma to deduce that for every n,

$$ \begin{align*} \frac{\nu^n(E_1)}{\nu^n(E_2)}=\frac{\lim_k \nu^n(\psi_k)}{\sup_k \nu^n(\phi_k)}\leq \sup_k \frac{\psi_k}{\phi_k}\leq \ell(\delta(A_2))e^{D_1\delta(A_2)^{\theta}}:=e_0. \end{align*} $$

Therefore,

$$ \begin{align*} \frac{\nu^n(A_1)}{\nu^n(A_2)}\leq \frac{\nu^n(F(x_j,r(A_2))}{\nu^n(A_2)} \leq m\cdot\max_{j} \frac{\nu^n(F(x_j,r(A_2)))}{\nu^n(B_j)}\leq m\cdot e_0. \end{align*} $$

From here, the conclusion follows.

Corollary 2.8. Fix $\psi \in \mathrm {Con}^+(\mathcal {F})$ . Then, for every $\phi \in \mathrm {Con}(\mathcal {F})$ , there exists $\hat {e}(\phi ,\psi )>0$ such that for every $n\geq 0$ , it holds that

$$\begin{align*}\frac{\nu^n(\phi)}{\nu^n(\psi)}\leq \hat{e}(\phi,\varphi). \end{align*}$$

Proof. This is essentially Lemma $2.4$ of [Reference Margulis32]. For $r>0$ , let $A_r:=\psi ^{-1}(r,\infty )$ and note that since $\psi \in \mathrm {Con}$ is non-negative and non-identically zero, there exists $r>0$ such that $A_r$ is relatively open and pre-compact inside the leaf of $\mathcal {F}$ that contains $\operatorname {\mathrm {supp}}(\psi )$ . Take A open, relatively compact containing $\operatorname {\mathrm {supp}}(\phi )$ and use the previous Lemma to deduce

We have thus shown that in the assumed conditions $H-1$ to $H-4$ , there exists an appropriate $\nu $ (Remark 2.5), and therefore, by Corollary 2.8, we get that the hypotheses of Theorem 2.3 are satisfied. Thus, we conclude the existence of $P\in \mathbb R$ and $\mu \in \mathrm {Meas}(\mathcal {F})$ such that

1. 1. $\mu $ is in the closure of the positive cone generated by $\{\nu ^n\}_{n\geq 0}$ (and thus has full support on each leaf of $\mathcal {F}$ ),

2. 2. $\mu $ is quasi-invariant, with Jacobian $e^{P-\varphi }$ .

Strong absolute continuity of the quasi-invariant section

It turns out that the quasi-invariance of $\mu $ implies that it is strongly absolutely continuous. We keep working with $\nu , \mu $ as in previous part; that is,

• • $\nu $ is appropriate for f,

• • $\mu $ is quasi-invariant with Jacobian $e^{P-\varphi }$ and in the closure of the positive cone generated by $\{\nu ^n\}_{n\geq 0}$ .

Denote for $x\in M$ , $\mu _x=\mu _{F(x)}$ . By quasi-invariance, we get

(2.8) $$ \begin{align} \forall n\in\mathbb N,\quad f^{-n}\mu_{f^nx}=e^{nP-S_n\varphi}\mu_x. \end{align} $$

Lemma 2.9. Given $\epsilon>0$ , there exists $\delta>0$ such that for every $\psi _1,\psi _2\in \mathrm {Con}^+$ that are $\delta $ -equivalent, then

$$\begin{align*}\left|\frac{\mu(\psi_1)}{\mu(\psi_2)}-1\right|<\epsilon. \end{align*}$$

Proof. By Lemma 2.6, for every $n\in \mathbb N$ , we have $\hat {\nu }^n(\psi _1)\leq \ell (\delta )\cdot e^{D_1\delta ^{\theta }}\hat {\nu }^n(\psi _2)$ ; the inequality extends to convex combinations of $\{\hat {\nu }^n\}_n$ and therefore, to elements of $\mathcal {X}$ (cf. Equation (2.4)). In particular, $\mu $ satisfies the claim.

We now fix $x_0, y_0$ in the same leaf of $\mathcal {G}$ and consider $\mathrm {hol}^{\scriptstyle \mathcal {G}}:A(x_0)\subset F(x_0)\to B(y_0)\subset F(y_0)$ the corresponding Poincaré map. For $x\in A(x_0)$ , define

(2.9) $$ \begin{align} \mathrm{Jac}(x)=\prod_{j=0}^{\infty}\frac{e^{\varphi\circ f^j(\mathrm{hol}^{\scriptstyle \mathcal{G}}x)}}{e^{\varphi\circ f^j(x)}}=\lim_n e^{S_n\varphi(\mathrm{hol}^{\scriptstyle \mathcal{G}}x)-S_n\varphi(x)}. \end{align} $$

Since $\varphi $ is Hölder and $\mathcal {G}$ is contracting, the previous formula defines a continuous function $\mathrm {Jac}:A(x_0)\rightarrow \mathbb R$ .

Proposition 2.9.1. It holds that $(\mathrm {hol}^{\scriptstyle \mathcal {G}})^{-1}\mu _{y_0}=\mathrm {Jac} \cdot \mu _{x_0}$ .

Proof. Denote $h=\mathrm {hol}^{\scriptstyle \mathcal {G}}$ , and for $n\geq 0$ , let $h_n$ be the Poincaré map defined on a neighborhood of $f^nx$ inside $f^n(A(x_0))$ . As f preserves $\mathcal {F},\mathcal {G}$ , $h=f^{-n}\circ h_n\circ f^{n}$ and therefore,

$$ \begin{align*} h^{-1}\mu_{y_0}&=(f^{-n}\circ h_n^{-1})e^{S_n\varphi\circ f^{-n}-nP}\mu_{f^ny_0}=f^{-n}\left(e^{S_n\varphi\circ h\circ f^{-n}-nP}h_n^{-1}\mu_{f^ny_0}\right)\\ &=f^{-n}\left(e^{S_n\varphi\circ h\circ f^{-n}-nP}\left(\mu_{f^nx_0}+\left(h_n^{-1}\mu_{f^ny_0}-\mu_{f^nx_0}\right)\right)\right)\\ &=e^{S_n\varphi\circ h-S_n\varphi}\mu_{x_0}+f^{-n}e^{S_n\varphi\circ h\circ f^{-n}-nP}\left(h_n^{-1}\mu_{f^ny_0}-\mu_{f^nx_0}\right). \end{align*} $$

By the lemma above, given $\epsilon>0$ , there exists $n_0$ such that for $n\geq n_0$ , it holds: for any $\psi \in \mathrm {Con}^+$ supported in $f^n(A(x_0))$ ,

which in turn implies

Thus, we can write $h^{-1}\mu _{y_0}=J_n\cdot \mu _{x_0}+\upsilon _n$ , where $\upsilon _n$ converges to zero in the vague topology, and $\{J_n\}_n$ is a sequence of continuous functions that converges uniformly to $\mathrm {Jac}$ . This concludes the proof.

Remark 2.10. Using that $\mathcal {G}$ is contracting under f and $\varphi $ is Hölder, one verifies directly that .

3.1 The conditional measures: potentials constant along the center and SRBs

We are interested in constructing $\mu ^u\in \mathrm {Meas}(\mathcal {W}^u)$ satisfying analogous properties as $\mu ^{cu}$ . Our previous construction, however, does not apply in this case since the transverse foliation to $\mathcal {W}^u$ is not contracting. To bypass this problem, we assume some condition on $\varphi $ , namely that it is either

• • constant on center leaves (c-constant case), or

• • $\varphi =-\log \det Df|E^u$ (SRB case).

In the SRB case, we define

$$\begin{align*}\mu^u_{x}=\text{Lebesgue measure on }W^u(x) \end{align*}$$

by the change of variables theorem $f^{-1}\mu ^u_{fx}=\det Df|E^u\cdot \mu ^u_{x}=e^{-\varphi }\mu ^u_{x}$ . For the c-constant case, one deduces by compactness of M the existence of some ${\mathrm {c}_{\scriptscriptstyle {cen}}}>0$ such that for every $x\in M$ , there is a well-defined projection $\pi _x^c:W^c(W^u(x),{\mathrm {c}_{\scriptscriptstyle {cen}}})\to W^u(x)$ by sliding along local center plaques. Given x, we define a measure on $W^u(x)$ by setting

(3.3) $$ \begin{align} \mu^u_{x}=\pi^c_x\cdot\mu^{cu}_{x}. \end{align} $$

Note that $\mu ^u\in \mathrm {Meas}(\mathcal {W}^u)$ is quasi-invariant with Jacobian $e^{P-\varphi }$ ; indeed, by using that $\varphi $ is constant on center leaves and that f is an isometry on these, we deduce

(3.4) $$ \begin{align} f^{-1}\mu^u_{fx}=f^{-1}\pi^c_{fx}\mu^{cu}_{fx}=\pi^c_{x}f^{-1}\mu^{cu}_{fx}=\pi^c_{x}e^{P-\varphi}\mu^{cu}_{x}=e^{P-\varphi}\pi^c_x\mu^{cu}_{x}=e^{P-\varphi}\mu^u_{x}. \end{align} $$

Arguing with $\mu ^{cs}$ instead of $\mu ^{cu}$ , we can analogously construct a family of measures $\mu ^s=\{\mu ^{cs}_x\}_{x\in M}$ on stable leaves. An important case of c-constant potential is when $\varphi \equiv 0$ ; in this case, we denote $\mathscr {m}^{\ast }\in \mathrm {Meas}(\mathcal {F}^{\ast }),\ \ast \in \{s,u,cs,cu\}$ the corresponding sections.

It follows that

(3.5) $$ \begin{align}f^{-1}{\mathscr{m}^{cu}_{fx}}=e^{h}{\mathscr{m}^{cu}_x} \end{align} $$

(3.6) $$ \begin{align}f^{-1}{\mathscr{m}^{u}_{fx}}=e^{h}{\mathscr{m}^{u}_x} \end{align} $$

for some $h\in \mathbb R$ .

Proof. By part $3$ of Proposition 3.1.1, it holds that $\mathrm {Jac}^{s}_{x_0,y_0}\equiv 1$ , thus implying the first part.

Recall the construction of $\mathscr {m}^{cu}$ on page 13, with $\nu ={\mathrm {Leb}}$ and $\nu ^n=f^{-n}\nu $ and a fixed normalized $\psi _0\in \mathrm {Con}^+(\mathcal {W}^{cu})$ .

Thus, we have for every $n\geq 0$ ,

$$ \begin{align*} f^{-1}\nu^n(\psi_0)\geq \min \{\det Df|E^{cu}\}\nu^{n}(\psi_0)\Rightarrow f^{-1}\hat{\nu}^n(\psi_0)\geq \min \{\det Df|E^{cu}\}. \end{align*} $$

This implies the same inequality for every member of $\mathcal {X}$ , and in particular,

$$ \begin{align*} e^h=\mathscr{m}^{cu}(\psi_0\circ f^{-1})\geq \min \{\det Df|E^{cu}\}>1.\\[-35pt] \end{align*} $$

We return to the general case; we will use the family $\{\mu ^u_{x}\}_x$ to define some new measures $\{\nu ^u_x\}_x$ that will serve as conditional measures on local unstable manifolds for the equilibrium state.

For each $x\in M, y\in W^u(x)$ , let

(3.7) $$ \begin{align} \Delta_x^u(y):=\prod_{k=1}^{\infty}\frac{e^{\varphi\circ f^{-k}(y)}}{e^{\varphi\circ f^{-k}(x)}}, \end{align} $$

and note that $\Delta _x^u:W^u(x)\to \mathbb R$ is continuous. Define

(3.8) $$ \begin{align} \nu^u_x=\Delta_x^u\ \mu^u_{x}. \end{align} $$

One verifies easily that $\nu ^u_{x}$ is a Radon measure on $W^u(x)$ which is positive on open sets, and furthermore,

(3.9) $$ \begin{align} f^{-1}\nu^u_{fx}=\Delta_{fx}^u\circ f\cdot f^{-1}\mu^u_{fx}=\Delta_{fx}^u\circ f \cdot e^{P-\varphi}\mu^u_{x} =e^{P-\varphi(x)}\nu^u_{x}. \end{align} $$

For $y\in W^u(x)$ , we have $\nu ^u_y=c(y,x)\cdot \nu ^u_{x}$ with $c(y,x)>0$ . Hence, $\{\nu ^u_y\}_{y\in W^u(x)}$ defines a projective class of measures $[\nu ^u_x]$ . In particular, if $\xi $ is a $\mu ^{cu}_{x}$ -measurable partition subordinated to $\mathcal {W}^u$ , we have that the probability measures

(3.10) $$ \begin{align} \nu^{\xi}_x:=\nu^u_{x}(\ \cdot\ |\xi(x)) \end{align} $$

satisfy $\nu ^{\xi }_x=\nu ^{\xi }_y$ for $y\in \xi (x)$ .

Before moving on, we establish the following lemma that will allow us to compare different unstable measures. Let $\epsilon>0$ be so that any set of diameter less than $2\epsilon $ is contained in a foliation chart of $\mathcal {W}^u$ and consider the compact space

$$\begin{align*}N=\{(x_0,y_0,z)\in M^3: y_0\in\operatorname{\mathrm{cl}}{W^u(x_0,\epsilon/2)}, z\in \operatorname{\mathrm{cl}}{W^c(x_0,\epsilon/2)}\}. \end{align*}$$

For $(x_0,y_0,z)\in N$ , define

(3.11) $$ \begin{align} T_{x_0,y_0}(z)= \prod_{k=1}^{+\infty}\frac{e^{\varphi\circ f^{-k}(\mathrm{hol}^u_{x_0,y_0}(z))}}{e^{\varphi\circ f^{-k}(z)}}. \end{align} $$

Proof. The first follows directly since $c(y,x)=\Delta _y^u(x)$ . For the second, for every $k\geq 0$ , consider the continuous function $T_k:N\to \mathbb R$ defined as

$$\begin{align*}T_k(x_0,y_0,z)= \sum_{i=1}^{k}\varphi\circ f^{-i}(\mathrm{hol}^u_{x_0,y_0}(z))-\varphi\circ f^{-i}(z) \end{align*}$$

and note that by the Weierstrass’ test, $T_k$ converges uniformly to

$$\begin{align*}\sum_{k=1}^{+\infty}\varphi\circ f^{-k}(\mathrm{hol}^u_{x_0,y_0} z)-\varphi\circ f^{-k}(z), \end{align*}$$

which implies that $\log T$ is continuous. Thus, T is continuous.

3.2 The equilibrium state

It is seldom the case that for a given invariant measure $\mu $ for f, the partition by unstable manifolds is $\mu $ - measurable. Therefore, we will have to work in sets B where $\mathcal {W}^u|B$ defines a sufficiently nice partition. Since we are trying to use the family $\{\mu ^{cs}_{x}\}_{x\in M}$ as a substitute for the transverse measure of the equilibrium state, we will have to deal with the technical difficulty of having to show coherence between different choices $B,B'$ ; this will be bypassed using our control on the behavior of $\{\mu ^{cs}_{x}\}_{x\in M}$ under unstable (local) holonomies. We will now make some geometrical preparations to deal with this problem.

Dynamical boxes

To take advantage of the local product structure between $\mathcal {W}^{cs}$ and $\mathcal {W}^u$ , we introduce some new notation. Due to the fact that the invariant bundles are mutually perpendicular, there exists ${\mathrm {c}_{\scriptscriptstyle {lps}}}>0$ such that for any $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {lps}}}$ it holds that for every $x,y\in M, d(x,y)<\epsilon $ implies

(3.12) $$ \begin{align} &\#W^s(x,2\epsilon)\cap W^{cu}(y,2\epsilon)=1 \end{align} $$

(3.13) $$ \begin{align} &\#W^u(x,2\epsilon)\cap W^{cs}(y,2\epsilon)=1. \end{align} $$

Definition 3.3. For $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {lps}}}$ and $x\in M$ , we consider the sets

(3.14) $$ \begin{align} P^{cs}(x,\epsilon)&:=W^c(W^s(x,\epsilon),\epsilon)\phantom{(x,y)(x,y)x,y} \end{align} $$

(3.15) $$ \begin{align} P^{cu}(x,\epsilon)&:=W^c(W^u(x,\epsilon),\epsilon)\phantom{(x,y)(x,y)x,y} \end{align} $$

(3.16) $$ \begin{align} B(x,\epsilon)&:=\bigcup_{{y\in W^u(x,\epsilon)}}\ \mathrm{hol}^u_{x,y}(P^{cs}(x,\epsilon)) \end{align} $$

where $\mathrm {hol}^u_{x,y}:W^{cs}(x)\rightarrow W^{cs}(y)$ is the locally defined Poincaré map sending x to y.

Sets $B(x,\epsilon )$ constructed in this fashion will be referred to as dynamical boxes, while $\mathrm {hol}^u_{x,y}(P^{cs}(x,\epsilon ))$ will be referred to as the $cs$ -plaques of $B(x,\epsilon )$ .

Shrinking ${\mathrm {c}_{\scriptscriptstyle {lps}}}$ , if necessary, can guarantee also that for $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {lps}}}$ , for every $x,x'\in M$ ,

• • $P^{cu}(x,\epsilon )\cap P^{cu}(x',\epsilon )\neq \emptyset \Rightarrow P^{cu}(x,\epsilon )\cap P^{cu}(x',\epsilon )\subset P^{cu}(x",3\epsilon )$ for some $x"\in M$ .

• • $B(x,\epsilon )\cap B(x',\epsilon )\neq \emptyset \Rightarrow B(x,\epsilon )\cap B(x',\epsilon )\subset B(x",5\epsilon )$ for some $x"\in M$ .

To understand better the structure of dynamical boxes, we note the following.

Lemma 3.4. Fix $x\in M$ and for $y\in W^s(y)$ , consider the locally defined holonomy $\mathrm {hol}^s_{x,y}:W^{cu}(x)\to W^{cu}(y)$ . Then, $\mathrm {hol}^s$ sends center leaves into center leaves and $\mathrm {hol}^s|W^c(x)$ is an isometry (similar for unstable holonomies).

Proof. Let $\delta>0$ be sufficiently small so that $\mathrm {hol}^s_{x,y}:A=W^{cu}(x,\delta )\to B\subset W^{cu}(y)$ is a well-defined homeomorphism. Since f preserves $\mathcal {W}^{cu},\mathcal {W}^c,\mathcal {W}^s$ , it follows that

$$\begin{align*}\mathrm{hol}^s_{x,y}=f^{-n}\circ \mathrm{hol}^s_{f^nx,f^ny}\circ f^n\quad \forall n\geq 0\end{align*}$$

where $\mathrm {hol}^s_{f^nx,f^ny}:f^n(A)\to f^n(B)\subset W^c(f^ny)$ . Using that $f|\mathcal {W}^c$ is an isometry and $f|\mathcal {W}^u$ expands distances, it follows that $\mathrm {hol}^s$ sends $W^c(x,\delta )$ to $W^c(y)$ and that the differentiable map $\mathrm {hol}^s_{f^nx,f^ny}|W^c(f^nx)$ uniformly approaches the identity for large n, hence the claim.

We remark that, in general, $\mathrm {hol}^s$ does not send $\mathcal {W}^u$ to itself. See Figure 1.

Figure 1

Comparison between $P^{cu}(x,\epsilon ),P^{cu}(y,\epsilon )$ for $y\in W^s(x)$ . The stable holonomy sends each center plaque $W^c(x',\epsilon )\in P^{cu}(x,\epsilon )$ to a center plaque $W^c(y',\epsilon )\in P^{cu}(y,\epsilon )$ , but the image of unstable plaques in $P^{cu}(x,\epsilon )$ are only Hölder submanifolds in $P^{cu}(y,\epsilon )$ . Nevertheless, $\mathrm {hol}^s_{x,y}(W^u(x,\epsilon ))\to W^u(x,\epsilon )$ in the $\mathcal {C}^0$ topology as $y\to x$ .

Corollary 3.5. For every $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {lps}}}$ and $x\in M$ , it holds that

$$\begin{align*}B(x,\epsilon)=\bigcup_{{y\in W^s(x,\epsilon)}}\ \ \mathrm{hol}^s_{y,x}(P^{cu}(x,\epsilon)), \end{align*}$$

where $\mathrm {hol}^s_{x,y}:W^{cu}(x)\rightarrow W^{cu}(y)$ is the locally defined Poincaré map sending x to y.

Proof. This is a direct consequence of Lemma 3.4 above, plus the fact that $\mathcal {W}^c$ sub-foliates $\mathcal {W}^{cu},\mathcal {W}^{cs}$ .

In view of this Corollary, sets $\mathrm {hol}^s_{x,y}(P^{cu}(x,\epsilon ))$ will be referred to as the $cu$ -plaques of $B(x,\epsilon )$ . Another consequence is the following.

Corollary 3.6. For every $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {lps}}}$ and $x\in M$ , $B(x,\epsilon ) \subset M$ is open.

Proof. Fix $x\in M, \epsilon>0$ and define $b:W^c(x,\epsilon )\times W^s(x,\epsilon )\to W^{cs}(x)$ by

$$\begin{align*}b(y,z)=W^s(y,2\epsilon)\cap W^c(z,2\epsilon). \end{align*}$$

This map is continuous, injective and by Lemma 3.4, $\Im (b)=P^{cs}(x,\epsilon )$ . The invariance of domain theorem now implies that $P^{cs}(x,\epsilon )$ is an open subset of $W^{cs}(x)$ .

Analogously, define $\tilde {b}: P^{cs}(x,\epsilon )\times W^u(x,\epsilon )\to M$ ,

$$\begin{align*}\tilde b(y,z)=\mathrm{hol}^u_{x,z}(y); \end{align*}$$

$\tilde b$ is continuous, injective and, by definition, $B(x,\epsilon )=\Im (\tilde b)$ . Another application of the invariance of domain theorem implies that $B(x,\epsilon )$ is open.

The equilibrium state

We now construct the equilibrium state. Fix $B=B(x_0,\epsilon )$ a dynamical box with $0<\epsilon <{\mathrm {c}_{\scriptscriptstyle {lps}}}/2$ . By the choice of $\epsilon $ , the set $W^u(x_0,B)=W^u(x_0,2\epsilon )\cap B$ consists of a unique unstable plaque. Similar considerations hold for the other foliations.

If W is a $cs$ -plaque of B and $x\in B$ , denote $x_w=W^u(x,B)\cap W$ . For $U\subset B$ open sub-box, define the function $\alpha _{W,B}^U:W\rightarrow \mathbb R$ by

(3.17) $$ \begin{align} \alpha_{W,B}^U(w)=\nu^u_{w}(U\cap W^u(w,B)). \end{align} $$

Lemma 3.7. $\alpha _{W,B}^U$ is upper semi-continuous, and hence, measurable.

Proof. The SRB case is direct, so we will consider only when $\varphi $ is constant along center leaves.

We use the following notation: for $\gamma>0$ and C contained in a $cu$ -plaque $Z\subset B$ , we write

$$\begin{align*}\mathrm{Col}_{\gamma}(C):=\{y\in Z:\exists y'\in C\text{ s.t. }d_{\mathcal{W}^{cu}}(y,y')<\gamma\}. \end{align*}$$

Consider a sequence $(y_m)_m\subset W$ , $y_m\xrightarrow [m\mapsto \infty ]{}y$ . Let

$$ \begin{align*} A=W^u(y,B)\cap U,\\ A_m=W^u(y_m,B)\cap U. \end{align*} $$

By definition of $\mu ^u_y$ (3.3), $\mu _y^u(A)=\mu ^{cu}_{y}(W^c(A,{\mathrm {c}_{\scriptscriptstyle {cen}}}))$ , and since $\mu ^{cu}_{y}$ is a Radon measure,

$$\begin{align*}\mu_y^u(A)=\inf_{\gamma>0} \mu^{cu}_{y}\left(\mathrm{Col}_{\gamma}(W^c(A,{\mathrm{c}_{\scriptscriptstyle{cen}}}))\right). \end{align*}$$

Similar considerations apply to $\mu _{y_m}^u(A_m)$ and to the measures $\nu ^u_y, \nu ^u_{y_m}$ ; in particular, if

$$\begin{align*}A_{\gamma}=\mathrm{Col}_{\gamma}(W^c(A,{\mathrm{c}_{\scriptscriptstyle{cen}}})\cap W^u(y)), \end{align*}$$

then $A_{\gamma }$ is an open neighborhood of A inside $W^u(y)$ , and

$$\begin{align*}\nu^u_y(A)=\inf_{\gamma>0} \int_{W^c(A_{\gamma},{\mathrm{c}_{\scriptscriptstyle{cen}}})} \Delta_y^u\circ \pi^c_y \operatorname{d}\hspace{-2pt} \mu^{cu}_{y}. \end{align*}$$

Fix $0<\gamma <{\mathrm {c}_{\scriptscriptstyle {lps}}}-{\mathrm {c}_{\scriptscriptstyle {cen}}}$ . By Lemma 3.3, the function $y'\mapsto \Delta _{y_w'}(y')$ is positive and uniformly continuous on B. Hence, given $\tau>0$ , there exists $0<\rho <\tau $ such that $y'\in B$ , d(y, y′) < ρ ⇒ |Δ _{y w} (y)Δ _{y w ′} (y′)−1| < τ. Due to continuity of the unstable and center unstable foliations, there exists $0<\rho '<\rho $ so that if $y'\in B, d(y,y')<\rho '$ , then

• • the local stable holonomy between $W^{cu}(y',{\mathrm {c}_{\scriptscriptstyle {cen}}}+\gamma )$ and $W^{cu}(y)$ is well-defined, with $\mathrm {Jac}^{s}_{y,y'}(z)<1+\tau $ , for every $z\in \mathrm {hol}^s_{y,y'}(W^{cu}(y,{\mathrm {c}_{\scriptscriptstyle {cen}}}+\gamma ))$ (cf. Remark 2.10).

• • $W^c(W^u(y',B)\cap U,{\mathrm {c}_{\scriptscriptstyle {cen}}})$ is $\rho $ -equivalent to some subset $E_{y'}\subset \mathrm {Col}_{\gamma }(W^c(A,{\mathrm {c}_{\scriptscriptstyle {cen}}}))$ .

Using continuity of the product of functions, we can further assume

• • $|\Delta ^u_{y'}\circ \pi ^c_{y'}\circ \mathrm {hol}^s_{y',y}(z)-\Delta ^u_y\circ \pi ^c_y(z)|<\tau $ for every $z\in E_{y'}$ .

Take $m_0$ so that $m\geq m_0$ implies $d(y_m,y)<\rho '$ and write $E_m=E_{y_m}$ ,

$$\begin{align*}g_m:=\mathrm{hol}^s_{y_m,y}| : E_m\rightarrow W^c(A_m,{\mathrm{c}_{\scriptscriptstyle{cen}}}). \end{align*}$$

Then,

$$ \begin{align*} \nu^{u}_{y_m}(A_m)&=\int_{W^c(A_m,{\mathrm{c}_{\scriptscriptstyle{cen}}})}\Delta^{u}_{y_m}\circ \pi^c_{y_m}\operatorname{d}\hspace{-2pt}\mu^{cu}_{y_m}=\int_{g_m(E_m)}\Delta^{u}_{y_m}\circ \pi^c_{y_m}\operatorname{d}\hspace{-2pt}\mu^{cu}_{y_m}\\ &=\int_{E_m}\Delta^{u}_{y_m}\circ \pi^c_{y_m}\circ g_m\cdot \mathrm{Jac}^{s}_{y,y_m}\operatorname{d}\hspace{-2pt}\mu^{cu}_{y}\leq (1+\tau)\int_{E_m}\Delta^{u}_{y_m}\circ \pi^c_{y_m}\circ g_m \operatorname{d}\hspace{-2pt}\mu^{cu}_{y}\\ &<(1+\tau)\left(\int_{E_m}\Delta^{u}_{y}\circ \pi^c_{y}\operatorname{d}\hspace{-2pt}\mu^{cu}_{y}+\tau\cdot\mu^{cu}_{y}(E_m)\right) \\ &\leq(1+\tau)\left(\int_{W^c(A_{\gamma},{\mathrm{c}_{\scriptscriptstyle{cen}}})}\Delta^{u}_{y}\circ \pi^c_{y}\operatorname{d}\hspace{-2pt}\mu^{cu}_{y}+\tau\cdot\mu^{cu}_{y}(\mathrm{Col}_{\gamma}(W^c(A,{\mathrm{c}_{\scriptscriptstyle{cen}}}))\right). \end{align*} $$

Hence,

$$ \begin{align*} &\limsup_m \alpha_{W,B}^U(y_m)=\limsup_m \nu^{u}_{y_m}(A_m)\\&\quad \leq (1+\tau)\left(\int_{W^c(A_{\gamma},{\mathrm{c}_{\scriptscriptstyle{cen}}})}\Delta^{u}_{y}\circ \pi^c_{y}\operatorname{d}\hspace{-2pt}\mu^{cu}_{y}+\tau\cdot\mu^{cu}_{y}(\mathrm{Col}_{\gamma}(W^c(A,{\mathrm{c}_{\scriptscriptstyle{cen}}}))\right). \end{align*} $$

Taking limit as $\tau \mapsto 0$ , we obtain

as we wanted to show.

Standard arguments of measure theory, together with the previous Lemma, permit us to define a Borel measure $\mathrm {m}^{W,B}$ on B by the condition

$$ \begin{align*} U\subset B \text{ open sub-box }\Rightarrow \mathrm{m}^{W,B}(U):=\int \alpha_{W,B}^U(w)\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w), \end{align*} $$

where $w_0\in W$ .

Let $W'$ be another $cs$ -plaque of $\mathcal {W}^{cs}|B$ and consider $\mathrm {hol}^u_{w_0,w_0'}:W\rightarrow W'$ the corresponding Poincaré map. Fix $U\subset B$ open, and to simplify the notation, write $U_w^u:= U\cap W^u(w,B)$ . We compute

$$ \begin{align*} \mathrm{m}^{W',B}(U)&=\int_{W'} \nu^u_{w'}(U^u_{w'})\mu^{cs}_{w_0'}(\operatorname{d}\hspace{-2pt} w')= \int_{W} \nu^u_{\mathrm{hol}^u_{w_0,w_0'}(w)}(U^u_{\mathrm{hol}^u_{w_0,w_0'}(w)}) \mathrm{Jac}^{u}_{w_0,w_0'}(w)\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w)\\ &=\int_{W} \left(\nu^u_{\mathrm{hol}^u_{w_0,w_0'}(w)}(U^u_{w})\prod_{k=1}^{+\infty}\frac{e^{\varphi\circ f^{-k}\mathrm{hol}^u_{w_0,w_0'}(w)}}{e^{\varphi\circ f^{-k}(w)}}\right)\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w)\\ &=\int_W \nu^u_w(U_w)\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w)=\mathrm{m}^{W,B}(U). \end{align*} $$

Hence, $\mathrm {m}^{W',B}=\mathrm {m}^{W,B}$ : we write $\mathrm {m}^B=\mathrm {m}^{W,B}$ for any plaque $W\in \mathcal {W}^{cs}|B$ . The following is now clear.

Lemma 3.8. Consider $0<\epsilon <\epsilon '<\frac {{\mathrm {c}_{\scriptscriptstyle {lps}}}}{2}$ and $x,x'\in M$ such that $B=B(x,\epsilon )\subset B(x',\epsilon ')$ . Then, $\mathrm {m}^B=\mathrm {m}^{B'}|B$ .

Take a finite covering $\mathscr {B}=\{B_1=B(x_1,{\mathrm {c}_{\scriptscriptstyle {cov}}}),\ldots , B_R=B(x_R,{\mathrm {c}_{\scriptscriptstyle {cov}}})\}$ of M with $0<{\mathrm {c}_{\scriptscriptstyle {cov}}}<\frac {{\mathrm {c}_{\scriptscriptstyle {lps}}}}{5}$ and define the measure $\mathrm {m}$ by the condition: $A\subset B_i$ is Borel $\Rightarrow \mathrm {m}(A):=\mathrm {m}^{B_i}(A)$ . If $A\subset B_i\cap B_j$ , then by our choice of ${\mathrm {c}_{\scriptscriptstyle {cov}}}$ , there exists $x\in M$ with $A\subset B(x,\epsilon _0)$ . Thus, by the previous lemma, $\mathrm {m}^{B_i}(A)=\mathrm {m}^{B_j}(A)$ . Hence, $\mathrm {m}$ is a well-defined Borel measure on M. With no loss of generality, we assume further that for every i, there exists $x_i'$ such that $f(B_i)\subset B(x_i',{\mathrm {c}_{\scriptscriptstyle {lps}}})$ .

Proposition 3.8.1. The measure $\mathrm {m}$ is f-invariant.

Proof. Fix $\mathrm {m}^{B_i}=\mathrm {m}^{W_i,B_i}$ , let $B'=B(x_i',{\mathrm {c}_{\scriptscriptstyle {lps}}})$ so that $f(B_i)\subset B(x_i',{\mathrm {c}_{\scriptscriptstyle {lps}}})$ and denote $W'$ the $cs$ -plaque of $B'$ containing $f(W_i)$ . Consider also $U\subset f(B_i)$ as an open sub-box of $B'$ . Then, $f^{-1}U \subset B_i$ is an open sub-box, and for $w\in B_i$ , $(f^{-1}U)^u_w=f^{-1}(U^u_{fw})$ . Therefore,

$$\begin{align*}f\mathrm{m}^{W_i,B_i}(U)=\mathrm{m}^{W_i,B_i}(f^{-1}U)=\int_{W_i} \nu^u_w(f^{-1}U_{fw}^u)\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w). \end{align*}$$

The quasi-invariance of $\nu ^u_w(f^{-1}U_{fw}^u)$ , formula (3.9) permits us to write

$$\begin{align*}\nu^u_w\left(f^{-1}(U_{fw}^u)\right)=e^{\varphi(w)-P}\nu^u_{fw}(U_{fw}^u). \end{align*}$$

Thus,

$$ \begin{align*} f\mathrm{m}^{W_i,B_i}(U)=\int_{W_i} e^{\varphi(w)-P}\nu^u_{fw}(U_{fw}^u)\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w)=e^{-P}\int_{W_i} e^{\alpha^{W',U}(fw)\varphi(w)}\mu^{cs}_{w_0}(\operatorname{d}\hspace{-2pt} w)\end{align*} $$

which by 2. of Proposition 3.1.2, can be written as

$$ \begin{align*} e^{P'-P}\int_{fW_i} \alpha^{W',U}(z)\mu^{cs}_{fw_0}(\operatorname{d}\hspace{-2pt} z)=e^{P'-P}\int_{W'} \alpha^{W',U}(z)\mu^{cs}_{fw_0}(\operatorname{d}\hspace{-2pt} z)=e^{P'-P}\mathrm{m}^{W',B'}(U), \end{align*} $$

where in the last equality we have used that $U \subset f(B_i)$ . Since U is an arbitrary open sub-box, we have proven that $f\mathrm {m}^{W_i,B_i}=e^{P'-P}\mathrm {m}^{W',B'}$ . This implies $f\mathrm {m}=e^{P'-P}\mathrm {m}$ . The measure $\mathrm {m}$ is finite on compact sets and $fM=M$ . Thus, $P=P'$ and $\mathrm {m}$ is f-invariant.

Corollary 3.9. It holds that $P=P'$ .

Definition 3.4. We denote $\mathrm {m}_{\scriptstyle \varphi }$ the probability measure $\frac {\mathrm {m}}{\mathrm {m}(M)}$ .

Remark 3.10. Assuming that existence of $\{\mu ^s_x\}_{x\in M}$ satisfying $f^{-1}\mu ^s_{fx}=e^{\varphi -P'}\mu ^s_{x}$ (for example, in the $c-$ constant case), we can argue similarly and construct another invariant measure $\tilde {\mathrm {m}}_{\scriptstyle \varphi }$ which is given locally as

$$\begin{align*}\tilde{\mathrm{m}}_{\scriptstyle\varphi}(U)=\int_V \nu_v^s(U_v^s)\mu^{cu}_{z_0}(\operatorname{d}\hspace{-2pt} v), \end{align*}$$

where $V=$ is a $cu$ -plaque of some dynamical box containing U, $z_0\in V$ .

Convention: The family $\mathscr {B}=\{B_1,\ldots B_R\}$ , together with corresponding center stable plaques $W_1,\ldots W_R$ , is considered to be fixed.

Directly from the construction, we get:

Corollary 3.11. If $\xi $ is a $\mathrm {m}_{\scriptstyle \varphi }$ measurable partition subordinated to $\mathcal {W}^u$ , then for $\mathrm {m}_{\scriptstyle \varphi }$ -almost every x, it holds that

$$\begin{align*}(\mathrm{m}_{\scriptstyle \varphi})_x^{\xi}=\frac{\nu^u_x(\cdot|\xi(x))}{\nu^u_x(\xi(x))}. \end{align*}$$

Assuming the existence of $\tilde {\mathrm {m}}_{\scriptstyle \varphi }$ , similarly we get that if $\xi $ is a $\tilde {\mathrm {m}}_{\scriptstyle \varphi }$ -measurable partition subordinated to $\mathcal {W}^s$ , then for $\tilde {\mathrm {m}}_{\scriptstyle \varphi }$ -almost every x, it holds that

$$\begin{align*}(\tilde{\mathrm{m}}_{\scriptstyle\varphi})_x^{\xi}=\frac{\nu^s_x(\cdot|\xi(x))}{\nu^s_x(\xi(x))}. \end{align*}$$

Remark 3.12. We will prove later that $\mathrm {m}_{\scriptstyle \varphi }=\tilde {\mathrm {m}}_{\scriptstyle \varphi }$ , provided that these exist.

3.3 Product structure and the Gibb’s property

We start establishing the following.

Proposition 3.12.1 (Product structure of $\mathrm {m}_{\scriptstyle \varphi }$ ).

There exists $0<{\mathrm {c}_{\scriptscriptstyle {psm}}}\leq {\mathrm {c}_{\scriptscriptstyle {cov}}}$ such that for every $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {psm}}}$ , there exists $c(\epsilon )>0$ satisfying for every $x\in M$ ,

$$\begin{align*}\frac{1}{c}\leq \frac{\mathrm{m}_{\scriptstyle \varphi}(B(x,\epsilon))}{\mu^u_{x}(W^u(x,B(x,\epsilon)))\cdot\mu^{cs}_{x}{(P^{cs}(x,\epsilon)})}\leq c. \end{align*}$$

The proof follows directly by the definition of $\mathrm {m}_{\scriptstyle \varphi }$ with the following lemma.

Lemma 3.13. There exists $0<{\mathrm {c}_{\scriptscriptstyle {psm}}}\leq {\mathrm {c}_{\scriptscriptstyle {cov}}}$ such that for $0<\epsilon \leq {\mathrm {c}_{\scriptscriptstyle {psm}}}$ , we can find $c(\epsilon )>0$ such that for all $x,x'\in M$ , the following holds.

$$ \begin{align*} &(\ast)\quad c(\epsilon)^{-1}\leq \frac{\mu^{cu}_{x}(W^{cu}(x,\epsilon))}{\mu^{cu}_{x'}(W^{cu}(x',\epsilon))}, \frac{\mu^{cs}_{x}(W^{cs}(x,\epsilon))}{\mu^{cs}_{x'}(W^{cs}(x',\epsilon))}\leq c(\epsilon)\\ &(\ast\ast)\quad c(\epsilon)^{-1}\leq \frac{\mu^u_{x}(W^u(x,\epsilon))}{\mu^u_{x'}(W^u(x',\epsilon))} \leq c(\epsilon)\\ &(\ast\ast\ast)\quad c(\epsilon)^{-1}\leq \frac{\nu^u_{x}(W^u(x,\epsilon))}{\nu^u_{x'}(W^u(x',\epsilon))} \leq c(\epsilon) \end{align*} $$

Proof. Let ${\mathrm {c}_{\scriptscriptstyle {psm}}}$ be the Lebesgue number associated to the covering $\mathscr {B}$ . We start with $(\ast )$ ; it suffices to consider the first set of inequalities. Since $\mu ^{cu}_x$ is a Radon measure of full support, we have the following: for $\epsilon>0$ fixed, there exists $\gamma>0$ and $c_0(x,\epsilon )>0$ such that

$$\begin{align*}x'\in W^{cu}(x,\gamma)\Rightarrow c_0(\epsilon,x)^{-1}\leq \frac{\mu^{cu}_{x}(W^{cu}(x,\epsilon))}{\mu^{cu}_{x'}(W^{cu}(x',\epsilon))}\leq c_0(\epsilon,x). \end{align*}$$

By Lemma 3.3, it is no loss of generality to assume that the same holds for $x'\in D(x,\gamma )$ , and hence, due to the compactness of M, we get a uniform $c(\epsilon )$ independent of x.

For $(\ast \ast )$ , the claim is immediate for the SRB case, so we just need to consider when $\varphi $ is constant along centrals, but this follows from as in the previous part since $\mu ^u_{x}(A)=\mu ^{cu}_{x}(W^c(A,{\mathrm {c}_{\scriptscriptstyle {cen}}}))$ .

Finally, $(\ast \ast \ast )$ is consequence of $(\ast \ast )$ , together with Lemma 3.3.

Remark 3.14. Arguing similarly and redefining $c,{\mathrm {c}_{\scriptscriptstyle {psm}}}$ if necessary, we can guarantee that for every $x\in M$ ,

$$\begin{align*}\frac{1}{c(\epsilon)}\leq \frac{\tilde{\mathrm{m}}_{\scriptstyle\varphi}(B(x,\epsilon))}{\mu^s_{x}(W^s(x,B(x,\epsilon)))\cdot\mu^{cu}_{x}{(P^{cu}(x,\epsilon)})}\leq c(\epsilon). \end{align*}$$

Next, we establish a Gibbs’s type property for $\mathrm {m}_{\scriptstyle \varphi }$ (resp. $\tilde {\mathrm {m}}_{\scriptstyle \varphi }$ ) that will allow us to show that it is an equilibrium state for the potential $\varphi $ . By reducing ${\mathrm {c}_{\scriptscriptstyle {psm}}}$ if necessary, we can assume that for every $0<\epsilon <{\mathrm {c}_{\scriptscriptstyle {psm}}}$ , for every $\epsilon $ -ball $D\subset M$ , the distance between $x,y\in D$ is given by

$$ \begin{align*} d(x,y)=\inf\{&\mathrm{length}(\alpha):\alpha:[0,1]\rightarrow D \text{ piecewise }\mathcal{C}^1,\alpha(0)=x,\alpha(1)=y, \alpha'\in E^{\ast}\ \ast=s,c,u\}. \end{align*} $$

With this metric and using that f is a center isometry, we have

(3.18)

We note the following classical remark.

Proof. We will prove only the second part, as the arguments for the first are included in the proof of this case. Let $y_s:=W^s(y,\epsilon )\cap f^{-n}W^{cu}(f^nx,\epsilon )$ . Points in the same stable leaf are exponentially contracted and $\varphi $ is Hölder. Thus,

$$\begin{align*}|S_n\varphi(x)-S_n\varphi(y)|\leq |S_n\varphi(x)-S_n\varphi(y_s)|+C_1(\epsilon) \end{align*}$$

for some constant $C_1(\epsilon )$ that converges to zero as $\epsilon $ tends to zero. Since $f^ny_s\in W^{cu}(f^nx,\epsilon )$ , the point $z_u:=W^u(f^ny,\epsilon )\cap W^c(f^nx,\epsilon )$ is well-defined. Observe then that the distance $d(f^{-j}z_u,f^{n-j}y_s)$ is exponentially contracted under $f^{-1}$ , and thus, for some other similar constant $C_2(\epsilon )$ , we have

$$\begin{align*}|S_n\varphi(x)-S_n\varphi(y)\leq |S_n\varphi(x)-S_n\varphi(f^{-n}z_u)|+C_2(\epsilon). \end{align*}$$

Since f is a center isometry and $\varphi $ is Hölder, it follows that

$$\begin{align*}|S_n\varphi(x)-S_n\varphi(f^{-n}z_u)|\begin{cases} =0 & c\text{-constant case}\\ \leq C_{\varphi}n\epsilon^{\theta}:=C_{3}(\epsilon) & \text{ in general}. \end{cases} \end{align*}$$

Defining $K(\epsilon ):=C_{1}(\epsilon )+C_{2}(\epsilon )+C_{3}(\epsilon )$ , we finish the proof.

Therefore, in general, the Birkhoff’s sums for points in the same $(\epsilon ,n)$ -Bowen ball differ by a factor that grows linearly with n. This is not that convenient; to circumvent this, the authors in [Reference Climenhaga, Pesin and Zelerowicz14] work with a type of potential that satisfies for sufficiently small $\epsilon $ ,

$$\begin{align*}\exists\ Q_{\varphi}>0: \forall x,y\in M, y\in W^{cs}(x,\epsilon), \forall n\in\mathbb N \Rightarrow |S_n\varphi(x)-S_n\varphi(y)|<Q_{\varphi}. \end{align*}$$

They call this the $cs$ -Bowen property for $\varphi $ , and they prove that the geometric potential satisfies this condition. In fact, inspection of their arguments shows that the following holds.

Proof. The proof is similar to the proof of Theorem $5.2$ of the above cited article. For convenience of the reader, we rephrase these arguments in our setting.

Since $\mu ^u$ is strongly absolutely continuous, given $\tau>0$ , there exists $\rho>0$ so that for every $x,y\in M$ , it holds that if $A\subset W^u(x), B \subset W^u(y)$ are $\rho $ -equivalent, then $\mathrm {hol}^{\scriptstyle cs}_{x,y}\mu ^u_y|B=J(x,y,\cdot )\mu ^u_{x}|A$ , and $|J(x,y,z)-1|<\tau $ for every $z\in A$ . Take $\epsilon>0$ sufficiently small so that $y\in W^{cs}(x,\rho /2)$ implies that $W^u(y,\epsilon )$ is $\rho $ -equivalent to some subset of $W^u(x)$ . Recalling that $D^u(y,\epsilon ,n)=f^{-n}W^u(f^ny,\epsilon )$ and since f is a center isometry, it follows that if $y\in W^{cs}(x,\rho /2)$ , then for every $n\geq 0$ , $D^u(y,\epsilon ,n)$ is $\rho $ -equivalent to some subset of $J(x,\epsilon ,n) \subset W^u(x)$ , and $f^nJ(x,\epsilon ,n) \rho $ -equivalent to $W^u(f^ny,\epsilon )$ . By reducing $\rho $ if necessary, we can guarantee that $f^n(J(x,\epsilon ,n)) \subset W^u(f^nx,2\epsilon )$ .

Quasi-invariance of $\mu ^u$ and Lemma 3.15 now imply

$$ \begin{align*} \frac{\mu^u_{f^ny}(W^u(f^ny,\epsilon))}{\mu^u_{x}{f^n(J(x,\epsilon,n))}}&=\frac{\int_{D^u(y,\epsilon,n)} e^{-S_n\varphi(z)}\operatorname{d}\hspace{-2pt}\mu^u_{y}(z)}{\int_{J(x,\epsilon,n)} e^{-S_n\varphi(w)}\operatorname{d}\hspace{-2pt}\mu^u_{x}(w)}\\ &=e^{S_n\varphi(x)-S_n\varphi(y)}\frac{\int_{D^u(y,\epsilon,n)} e^{S_n\varphi(y)-S_n\varphi(z)}\operatorname{d}\hspace{-2pt}\mu^u_{y}(z)}{\int_{J(x,\epsilon,n)} e^{S_n\varphi(x)-S_n\varphi(w)}\operatorname{d}\hspace{-2pt}\mu^u_{x}(w)}\\ &\begin{cases} \leq e^{S_n\varphi(x)-S_n\varphi(y)+2K(\epsilon)}\frac{\mu^u_{y}(D^u(y,\epsilon,n))}{\mu^u_{x}(J(x,\epsilon,n))}\\ \geq e^{S_n\varphi(x)-S_n\varphi(y)-2K(\epsilon)}\frac{\mu^u_{y}(D^u(y,\epsilon,n))}{\mu^u_{x}(J(x,\epsilon,n))}, \end{cases}\\[-15pt] \end{align*} $$

and therefore,

$$\begin{align*}\frac{1-\tau}{1+\tau}e^{-2K(\epsilon)}\leq e^{S_n\varphi(x)-S_n\varphi(y)}\leq \frac{1+\tau}{1-\tau}e^{2K(\epsilon)},\\[-15pt] \end{align*}$$

for every $n\geq 0$ .

Now we can deduce:

Corollary 3.18 (Gibbs-like property of $\mathrm {m}_{\scriptstyle \varphi }$ ).

Given $\epsilon>0$ , there exists $d(\epsilon )>0$ such that for every $n\in \mathbb {N},x\in M$ , it holds that

$$ \begin{align*} d^{-1}(\epsilon)\leq &\frac{\nu^u_{x}(D^u(x,\epsilon,n))}{e^{S_n\varphi(x)-nP}} \leq d(\epsilon)\\ d^{-1}(\epsilon)e^{-K(\epsilon)n}\leq &\frac{\mathrm{m}_{\scriptstyle \varphi}(D(x,\epsilon,n))}{e^{S_n\varphi(x)-nP}} \leq d(\epsilon)e^{K(\epsilon)n},\\[-15pt] \end{align*} $$

where $K(\epsilon )$ is the constant given in Lemma 3.15.

If $\varphi $ does not depend on centrals, or if $\varphi $ is the geometric potential, then in the second line of inequalities, we can substitute $nK(\epsilon )$ by $K(\epsilon )$ .

Proof. To simplify the notation, we will write $a\simeq b$ if the quotient $\frac {a}{b}$ is bounded above and below with constants only depending on $\epsilon $ . The first part is a direct consequence of Lemma 3.13 and (3.9). For the second part, we consider $B_i$ so that $D(x,\epsilon ,n)\subset B_i$ : by the considerations made above Lemma 3.8, it is no loss of generality to assume that $W_i=W^{cs}(x,B)$ . Thus, we have $\mathrm {m}^{W_i,B_i}(D(x,\epsilon ,n))=\int _{W^{cs}(x,\epsilon )} \nu ^u_{y_w}(D^u(y,\epsilon ,n))\mu ^{cs}_{x}(\operatorname {d}\hspace {-2pt} y)$ , and therefore,

$$ \begin{align*} \mathrm{m}^{W_i,B_i}(D(x,\epsilon,n))&\simeq\int_{W^{cs}(x,\epsilon)} e^{S_n\varphi(y)-nP}\mu^{cs}_{x}(\operatorname{d}\hspace{-2pt} y)\quad &\text{(by the first part)}\\ &\simeq e^{S_n\varphi(x)-nP\mathscr{Pr}(M) nK}\mu^{cs}_{x}(W^{cs}(x,\epsilon))\quad&\text{(by Lemma 3.15)}\\ &\simeq e^{S_n\varphi(x)-nP\mathscr{Pr}(M) nK} \quad &\text{(by Lemma 3.13)}.\\[-15pt] \end{align*} $$

The last claim follows from Lemmas 3.15 and 3.16.

We are ready to show that $\mathrm {m}_{\scriptstyle \varphi }$ is an equilibrium state.

Proof. Fix $0<\epsilon <{\mathrm {c}_{\scriptscriptstyle {psm}}}$ small. By the previous Corollary, and since $\mathrm {m}_{\scriptstyle \varphi }$ is invariant, one has

$$\begin{align*}\frac{\log d(\epsilon)}{n}-K(\epsilon) \leq P-(\int \log \mathrm{m}_{\scriptstyle \varphi}(D(x,\epsilon,n)) \operatorname{d}\hspace{-2pt}\mathrm{m}_{\scriptstyle \varphi} +\int \varphi \operatorname{d}\hspace{-2pt}\mathrm{m}_{\scriptstyle \varphi}) \leq \frac{\log d(\epsilon)}{n}+K(\epsilon) \end{align*}$$

which, by the Brin-Katok formula [Reference Brin and Katok8], gives

$$\begin{align*}-K(\epsilon)\leq P-(h_{\mathrm{m}_{\scriptstyle \varphi}}(f;A)+\int \varphi \operatorname{d}\hspace{-2pt}\mathrm{m}_{\scriptstyle \varphi})\leq K(\epsilon)\Rightarrow P_{\mathrm{m}_{\scriptstyle \varphi}}(\varphi)=P. \end{align*}$$

Next, we consider a maximally $(n,2\epsilon )$ -separated set $E_n$ : then, for $\{D(x,\epsilon ,n)\}_{x\in E_n}$ are pairwise disjoint and $M=\bigcup _{x\in E_n} D(x,2\epsilon ,n)$ . Take $A^n=\{A^n_x:x\in E_n\}$ partition by Borel sets so that for every $x\in E_n$ , $D(x,\epsilon ,n)\subset A_x^n\subset D(x,2\epsilon ,n)$ , and computeFootnote ⁴

$$ \begin{align*} S(2\epsilon,n)&\leq \sum_{x\in E_n} e^{S_n\varphi(x)}\leq \sum_{x\in E_n}\mathrm{m}_{\scriptstyle \varphi}(D(x,\epsilon,n)) e^{nP+nK(\epsilon)}\leq e^{nP+nK(\epsilon)}\sum_{x\in E_n}\mathrm{m}_{\scriptstyle \varphi}(A_x^n)\\ &= e^{nP+nK(\epsilon)}. \end{align*} $$

It follows that $P_{\scriptstyle \mathit {top}}(\varphi )\leq P$ , and therefore, by the variational principle, that $P=P_{\scriptstyle \mathit {top}}(\varphi )= P_{\mathrm {m}_{\scriptstyle \varphi }}(f)$ .

Since $P=0$ by construction in the SRB case, we get:

Increasing partitions

A $\mathrm {m}$ -measurable partition $\xi $ is called increasing if it refines its image ( $f\xi <\xi $ ). If $\xi $ is a partition, recall that we denote by $\mathcal {B}_{\xi }(M)$ the sub- $\sigma $ algebra of $\mathscr {B}_{\scriptstyle \mathrm {M}}$ that consists of $\xi $ -saturated sets. The mesh of $\xi $ is the supremum of $\{\operatorname {\mathrm {diam}}(A):A\in \xi \}$ .

In our restricted setting, a SLY partition automatically satisfies the following additional properties:

1. 1. $\bigvee _{n\geq 0} f^{-n}\xi =\mathscr {B}_{\scriptstyle \mathrm {M}}$ .

2. 2. $\bigwedge _{n\geq 0} f^n\xi =\mathcal {B}^u$

where $\mathcal {B}^u$ is the $\sigma $ -algebra of $\mathcal {W}^u$ saturated sets. A key result is the following.

Theorem 4.3. There exist SLY partitions of arbitrarily small mesh. If $\xi $ is a SLY partition, then

$$\begin{align*}h_{\mathrm{m}}(f)=h_{\mathrm{m}}(f,\xi)=H_{\mathrm{m}}(f^{-1}\xi|\xi). \end{align*}$$

The existence part is essentially due to Sinai [Reference Sinai47] (compare Section 3 in [Reference Ledrappier and Strelcyn28] and Lemma $2.4.2$ in [Reference Ledrappier and Young29], where the construction is done in more generality). The second part is a deep result of Ledrappier and Young (Corollary 5.3 in [Reference Ledrappier and Young29]). We remark that although in this last article the result is presented for ergodic measures, it extends easily to non-ergodic ones by using the ergodic decomposition. We have tacitly used that the strong unstable manifolds coincide with the Pesin’s unstable manifolds, as consequence of the fact that f does not have any positive Lyapunov exponents along center directions, independently of the measure.

4.1 Equilibrium states have conditionals equivalent to $\mu ^u$

Now, we fix $\xi $ a SLY partition for the measure $\mathrm {m}$ and denote by $\mathrm {m}_x=\mathrm {m}^{\xi }_x$ the conditional measure of $\mathrm {m}$ on $\xi (x)$ . The main result of [Reference Ledrappier and Young29] admits the following generalization in our setting.

Theorem 4.4. If $\mathrm {m}$ is any equilibrium state corresponding to $(f,\varphi )$ , then

$$\begin{align*}\mathrm{m}_x\sim\mu^u_{x} \text{ on }\xi(x)\quad\text{ for } \mathrm{m}\operatorname{-}a.e.(x). \end{align*}$$

To give a self-contained presentation, we include the complete proof instead of just citing the relevant parts of Ledrappier-Young’s work. Assume for the moment that the conclusion of the theorem was true. Then, for $\mathrm {m}\operatorname {-}a.e.(x)$ , we have

$$\begin{align*}\operatorname{d}\hspace{-2pt} \mathrm{m}_x=\rho_x \operatorname{d}\hspace{-2pt}\mu^u_{x}, \end{align*}$$

where $\rho _x$ is a non-negative function such that $\int _{\xi (x)} \rho _x \operatorname {d}\hspace {-2pt}\mu ^u_{x}=1$ . Using that the partition $\xi $ is measurable, we can assemble all $\{\rho _x\}$ into one measurable function $\rho :M\rightarrow \mathbb R$ . Since , we have

$$\begin{align*}f\mathrm{m}_{x}^{f^{-1}\xi}=\mathrm{m}_{fx} (=\mathrm{m}_{fx}^{\xi})\quad \mathrm{m}\operatorname{-}a.e.(x). \end{align*}$$

However, $f^{-1}\xi>\xi $ . Thus, for any $A\in \mathscr {B}_{\scriptstyle \mathrm {M}}$ , we can write

$$ \begin{align*} &\frac{1}{\mathrm{m}_x(f^{-1}\xi(x))}\int_{A\cap f^{-1}\xi(x)}\rho \operatorname{d}\hspace{-2pt}\mu^u_{x}=\frac{1}{\mathrm{m}_x(f^{-1}\xi(x))}\mathrm{m}_x(A\cap f^{-1}\xi(x))=\mathrm{m}_{x}^{f^{-1}\xi}(A)\\ &=\mathrm{m}_{f_x}(fA\cap \xi(fx))=\int_{f(A\cap f^{-1}\xi(x))}\rho \operatorname{d}\hspace{-2pt}\mu^u_{fx}=\int_{A\cap f^{-1}\xi(x)}\rho\circ f e^{P-\varphi}\operatorname{d}\hspace{-2pt}\mu^u_{x}. \end{align*} $$

Hence, by uniqueness of the conditional measures $\mathrm {m}\operatorname {-}a.e.(x)$ , it holds that

$$\begin{align*}\frac{\rho(y)}{\mathrm{m}_x(f^{-1}\xi(x))}=\rho(fy)\cdot e^{P-\varphi(y)}\quad \text{ for }\mu_x\operatorname{-}a.e.(y)\in f^{-1}\xi(x). \end{align*}$$

As the denominator appearing in the previous equation is $f^{-1}\xi $ - measurable, we finally conclude that the function

$$\begin{align*}x\mapsto \frac{\rho(x)}{\rho(f^{-1}x)}e^{P-\varphi(f^{-1}x)} \end{align*}$$

is $\xi $ - measurable (i.e. constant on the atoms of $\xi $ ). By a simple inductive argument, one is led to conclude that $\rho $ satisfies

(4.3) $$ \begin{align} y\in \xi(x)\Rightarrow \frac{\rho(y)}{\rho(x)}=\prod_{k=1}^{\infty}\frac{e^{\varphi\circ f^{-k}(y)}}{e^{\varphi\circ f^{-k}(x)}}=\Delta_x^u(y). \end{align} $$

The proof is straightforward using that for every $x\in M$ , the set $W^u(x)$ is an immersed submanifold and the fact that $\varphi $ is globally Hölder.

From (4.3), we deduce that $\rho $ should have the form

(4.4) $$ \begin{align} \rho(y)=\frac{\Delta_x^u(y)}{L(x)},\quad y\in \xi(x) \end{align} $$

with $L(x)=\int _{\xi (x)} \Delta _x^u(y)d\mu ^u_{x}(y)$ .

We now define a measure $\tilde {\mathrm {m}}$ by requiring $\tilde {\mathrm {m}}=\mathrm {m}$ on $\mathcal {B}_{\xi }$ and such its conditionals on $\xi $ are given by d m~ _x = Δ _x ^u L(x) d μ _x = ρd μ _x (cf. (3.10)). The proof of Theorem 4.4 will be achieved by establishing that $\mathrm {m}=\tilde {\mathrm {m}}$ , and for this, it suffices to show that $\mathrm {m}=\tilde {\mathrm {m}}$ on every $\mathcal {B}_{f^{-n}\xi },n\geq 0$ , since the previous $\sigma $ - algebras generate $\mathscr {B}_{\scriptstyle \mathrm {M}}$ .

We start by noticing the following: for $q(x)=\tilde {\mathrm {m}}_{x}(f^{-1}\xi (x))$ , we have

$$ \begin{align*} q(x)&=\frac{1}{L(x)}\int_{f^{-1}(\xi(fx))} \Delta_x(f^{-1}fy) \frac{e^{P-\varphi(y)}}{e^{P-\varphi(f^{-1}fy)}}\operatorname{d}\hspace{-2pt}\mu^u_{x}\\ &=\frac{1}{L(x)}\int_{\xi(fx)} J_x^u(f^{-1}z) e^{\varphi(f^{-1}z)-P}\operatorname{d}\hspace{-2pt}\mu^u_{fx}(z)=\frac{L(fx)}{L(x)}e^{\varphi(x)-P}\leq 1. \end{align*} $$

Thus, $\frac {L(fx)}{L(x)}\leq e^{P-\varphi (x)}\Rightarrow \log \frac {L\circ f(x)}{L(x)}\leq P-\varphi (x)$ , and therefore, $\max \{\log \frac {L\circ f}{L},0\}$ is in $\mathscr {L}^{1}(\mathrm {m})$ . A classic lemma of measure theory (cf. Proposition $2.2$ in [Reference Ledrappier and Strelcyn28]) then implies

$$\begin{align*}\int \log \frac{L\circ f}{L} \operatorname{d}\hspace{-2pt}\mathrm{m}=0\Rightarrow \int -\log q(x)\operatorname{d}\hspace{-2pt}\mathrm{m}(x)=P-\int\varphi \operatorname{d}\hspace{-2pt}\mathrm{m}. \end{align*}$$

We have shown the following.

Lemma 4.6. It holds that

$$\begin{align*}P- \int\varphi \operatorname{d}\hspace{-2pt}\mathrm{m} =\int -\log\tilde{\mathrm{m}}_{x}(f^{-1}\xi(x))\operatorname{d}\hspace{-2pt}\mathrm{m}(x)=\int -\log\left(\int_{f^{-1}\xi(x)} \rho(y)\operatorname{d}\hspace{-2pt}\mu^u_{x}(y)\right) \operatorname{d}\hspace{-2pt}\mathrm{m}(x). \end{align*}$$

By definition, the measures $\mathrm {m}$ and $\tilde {\mathrm {m}}$ coincide on $\mathcal {B}_{\eta }$ ; to prove that these two measures coincide, it suffices to establish that $\mathrm {m}$ and $\tilde {\mathrm {m}}$ coincide on $\mathcal {B}_{f^{-1}\xi }$ provided that $\mathrm {m}$ is an equilibrium state, and argue by induction.

Denote by $g_{f^{-1}\xi }=\frac {d\tilde {\mathrm {m}}}{d\mathrm {m}}\big |_{f^{-1}\xi }$ , and observe that since the partition $f^{-1}\xi $ restricted to each atom $\xi (x)$ is countable, we have for $\mathrm {m}\operatorname {-}a.e.(x)$

$$\begin{align*}g_{f^{-1}\xi}(x)=\frac{\tilde{\mathrm{m}}_{x}(f^{-1}\xi(x))}{m_x(f^{-1}\xi(x))}. \end{align*}$$

To prove that $\mathrm {m}=\tilde {\mathrm {m}}$ on $\mathcal {B}_{f^{-1}\xi }$ , we will show that $g_{f^{-1}\xi }(x)\equiv 1$ for $\mathrm {m}\operatorname {-}a.e.(x)$ . Observe that for $\mathrm {m}\operatorname {-}a.e.(x)$ , one has $\mathrm {m}_x(f^{-1}\xi (x))>0$ . Hence, the function $g_{f^{-1}\xi }(x)$ is well-defined for $\mathrm {m}\operatorname {-}a.e.(x)$ , but not necessarily for $\tilde {\mathrm {m}}\operatorname {-}a.e.(x)$ . This point is not emphasized in [Reference Ledrappier and Young29], and some minor extra considerations are required to complete the argumentFootnote ⁵ . For the sake of exactness, we will be precise in what follows.

For an atom $\xi (x)$ , we write $f^{-1}\xi |\xi (x)=\{A^x_j\}_{j=1}^{\infty }$ . We compute

(4.5)

(4.6)

Now, we use our hypothesis that $\mathrm {m}$ is an equilibrium state: by Theorem 4.3 and Lemma 4.6,

$$\begin{align*}\int-\log \mathrm{m}_x(f^{-1}\xi(x))\operatorname{d}\hspace{-2pt}\mathrm{m}(x)=H_{\mathrm{m}}(f^{-1}\xi|\xi)=P-\int \varphi \operatorname{d}\hspace{-2pt}\mathrm{m}=\int -\log \tilde{\mathrm{m}}_x(f^{-1}\xi(x))\operatorname{d}\hspace{-2pt}\mathrm{m}(x) \end{align*}$$

and thus, ∫ g _{f ⁻¹ ξ} (x)d m(x) = 0. Using Jensen’s inequality and (4.6), we conclude

$$ \begin{align*} 0=\int \log g_{f^{-1}\xi}(x) \operatorname{d}\hspace{-2pt}\mathrm{m}(x)\leq \log\int g_{f^{-1}\xi}(x) \operatorname{d}\hspace{-2pt}\mathrm{m}(x) \leq \log 1=0. \end{align*} $$

Strict convexity of the logarithm function then implies

$$\begin{align*}g_{f^{-1}\xi}(x) =1\quad \mathrm{m}\operatorname{-}a.e. \end{align*}$$

We conclude that the chain of inequalities appearing are equalities, and in particular, for equation (4.5). As a result, we have $\mu _x(A_j^x)>0\Rightarrow \mathrm {m}_x(A^x_j)>0$ , and this in turn implies that $\mathrm {m}\operatorname {-}a.e.(x)$ . It holds that $\mathrm {m}_x=\tilde {\mathrm {m}}_x$ on $f^{-1}\xi |\xi (x)$ since these measures coincide on all the atoms and not only on the ones of $\mathrm {m}_x$ positive measure. Consequently, $\mathrm {m}=\tilde {\mathrm {m}}$ on $\mathcal {B}_{f^{-1}\xi }$ , and this finishes the general inductive step necessary to prove Theorem 4.4.

Corollary 4.7. Suppose that is an equilibrium state for the potential $\varphi $ and $\xi $ is a SLY partition as above. Then, for $\mathrm {m}\operatorname {-}a.e.(x)$ , the conditional measures of $\mathrm {m}$ with respect to $\xi $ are given by

$$\begin{align*}\mathrm{m}^{\xi}_x=\tilde{\mathrm{m}}^{\xi}_x=\frac{\Delta_x^u}{L(x)}d\mu^u_{x}|\xi(x)\quad L(x)=\int_{\xi(x)}\Delta_x^u\operatorname{d}\hspace{-2pt}\mu^u_{x}. \end{align*}$$

In particular, the disintegration can be extended to every leaf of $\mathcal {W}^u$ and .

We remark that above we only used the existence of an equilibrium state with conditionals satisfying the quasi-invariance condition $3$ of Theorem A.

4.2 Conditional measures determine equilibrium states

We are interested now in the following converse of Theorem 4.4.

Proof. The proof uses similar techniques to the ones used above and is based in [Reference Ledrappier and Strelcyn28]. Consider $\xi $ a SLY partition for $\mathrm {m}$ and define the measure $\tilde {\mathrm {m}}$ on M by requiring that for $A\subset M$ Borel,

(4.7) $$ \begin{align} \tilde{\mathrm{m}}(A)=\int \mu^u_{x}(A\cap \xi(x)) \operatorname{d}\hspace{-2pt} m(x). \end{align} $$

Observe that $\tilde {\mathrm {m}}$ is not necessarily finite, but only $\sigma $ -finite. By hypothesis, $\operatorname {d}\hspace {-2pt}\mathrm {m}_x=\rho \operatorname {d}\hspace {-2pt}\mu _x$ . Thus, $\mathrm {m}<<\tilde {\mathrm {m}}$ , and moreover, $\mathrm {m}\operatorname {-}a.e.(x)$ . It holds that

(4.8) $$ \begin{align} \frac{\operatorname{d}\hspace{-2pt}\mathrm{m}}{\operatorname{d}\hspace{-2pt}\tilde{\mathrm{m}}}=\rho \quad \tilde{\mathrm{m}}\operatorname{-}a.e.(x) \end{align} $$

(see Proposition $4.1$ in [Reference Ledrappier and Strelcyn28]). Note that $\mathrm {m}^{f^{-1}\xi }_x=\frac {\mathrm {m}_x(\cdot |f^{-1}\xi )}{\mathrm {m}_x(f^{-1}\xi (x))}$ ; by an analogous computation as the one carried in the proof of Lemma 4.6, we get

$$\begin{align*}\mathrm{m}_x(f^{-1}\xi(x))=\frac{\rho\circ f(x)}{\rho(x)}e^{\varphi(x)-P}. \end{align*}$$

Finally, we compute

$$ \begin{align*} 0\leq h_{\mathrm{m}}(f)&=H(f^{-1}\xi|\xi)=\int-\log \mathrm{m}_x(f^{-1}\xi(x))\operatorname{d}\hspace{-2pt}\mathrm{m}(x)\\ &=P-\int \varphi d\mathrm{m} +\int -\log\frac{\rho\circ f(x)}{\rho(x)} \operatorname{d}\hspace{-2pt}\mathrm{m}(x) \end{align*} $$

and since $\varphi $ is integrable (bounded), the function $-\log ^{-} \frac {\rho \circ f}{\rho }$ is integrable as well. Using again Proposition $2.2$ of [Reference Ledrappier and Strelcyn28], we conclude that $\log \frac {\rho \circ f}{\rho }$ has zero integral. Hence,

$$\begin{align*}h_{\mathrm{m}}(f)=P-\int \varphi \operatorname{d}\hspace{-2pt}\mathrm{m} \end{align*}$$

and $\mathrm {m}$ is an equilibrium state. The proof is complete.