Building Thermodynamics for Nonuniformly Hyperbolic Maps
Department of Mathematics, University of Houston, Houston, TX 77204, USA
V. Climenhaga is partially supported by NSF Grant DMS1362838. Yakov Pesinpesin@math.psu.edu
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Y. Pesin is partially supported by NSF Grant DMS1400027
Abstract
We briefly survey the theory of thermodynamic formalism for uniformly hyperbolic systems, and then describe several recent approaches to the problem of extending this theory to nonuniform hyperbolicity. The first of these approaches involves Markov models such as Young towers, countablestate Markov shifts, and inducing schemes. The other two are less fully developed but have seen significant progress in the last few years: these involve coarsegraining techniques (expansivity and specification) and geometric arguments involving pushforward of densities on admissible manifolds.
Keywords
Mathematics Subject Classification
Primary 37D25, 37D351 Introduction
1.1 The General Setting
Thermodynamic formalism, i.e., the formalism of equilibrium statistical physics, was adapted to the study of dynamical systems in the classical works of [Ruelle1972]; [Ruelle1978], [Sinaĭ1968]; [Sinaĭ1972], and [Bowen1970]; [Bowen1974]; [Bowen2008]. It provides an ample collection of methods for constructing invariant measures with strong statistical properties. In particular, this includes constructing a certain “physical” measure known as the SRB measure (for Sinai–Ruelle–Bowen).
The general ideas can be given as follows. Let $ (X,d)$ be a compact metric space and $ f \colon X \to X$ a continuous map of finite topological entropy. Fix a continuous function $ \varphi\colon X\to \mathbb{R}$, which we will refer to as a potential. Denote by $ \mathcal{M}(f)$ the space of all $ f$invariant Borel probability measures on X. Given $ \mu\in \mathcal{M}(f)$, the free energy of the system with respect to $ \mu$ is
$\displaystyle \begin{eqnarray*} E_\mu(\varphi) := \left(h_\mu(f) + \int_X \varphi\,d\mu\right), \end{eqnarray*}$ 
where $ h_\mu(f)$ is the Kolmogorov–Sinai (measuretheoretic) entropy of $ (X,f,\mu)$. Optimizing over all invariant measures gives the topological pressure
$\displaystyle \begin{eqnarray*} P(\varphi) := \inf_{\mu\in \mathcal{M}(f)} E_\mu(\varphi) = \sup_{\mu\in \mathcal{M}(f)} \left(h_\mu(f) + \int_X \varphi\,d\mu\right), \end{eqnarray*}$ 
and a measure achieving this extremum is called an equilibrium measure (or equilibrium state). Note that it suffices to take the infimum (supremum) over the space $ \mathcal{M}^e(f) \subset \mathcal{M}(f)$ of ergodic measures.
The variational principle relates the definition of pressure as an extremum over invariant measures to an alternate definition in terms of growth rates. Given $ \varepsilon> 0$ and $ n\in \mathbb{N}$, a set $ E\subset X$ is $ (n,\varepsilon)$separated if points in $ E$ can be distinguished at a scale $ \varepsilon$ within $ n$ iterates; more precisely, if for every $ x,y\in E$ with $ x\ne y$, there is $ 0\le k\le n$ such that $ d(f^kx,f^ky)\ge\varepsilon$. Then one has
$\displaystyle \begin{eqnarray*}\label{eqn:Pspsgrowth} P(\varphi)=\lim_{\varepsilon\to 0}\limsup_{n\to\infty}\frac1n\log\sup_{\substack{E\subset X \\ (n,\varepsilon)\text{sep.}}}\sum_{x\in E} e^{S_n\varphi(x)}, \end{eqnarray*}$  (1.1) 
where
$\displaystyle \begin{eqnarray*}\label{ind:poten} S_n\varphi(x):=\sum_{k=0}^{n1}\varphi(f^kx). \end{eqnarray*}$  (1.2) 
The sum in (1.1) is a partition sum that quantifies “weighted orbit complexity at spatial scale $ \varepsilon$ and time scale $ n$”; $ P(\varphi)$ represents the growth rate of this complexity as time increases. In the particular case $ \varphi=0$, the value $ P(0)$ is the topological entropy $ h_{\text{top}}(f)$ of the map $ f$.
Thermodynamic formalism is most useful when the system possesses some degree of hyperbolic behavior, so that orbit complexity increases exponentially. The most complete results are available when $ f$ is uniformly hyperbolic; we discuss these in Sect. 1.2. In this article we focus on nonuniformly hyperbolic systems, and we discuss the general picture in Sect. 1.3. Our emphasis will be on general techniques rather than on specific examples. In particular, we discuss Markov models (including Young towers) in Sects. 2–4, coarsegraining techniques (based on expansivity and specification) in Sect. 5, and pushforward (geometric) approaches (based on newly introduced standard pairs approach) in Sect. 6.
1.2 Uniformly Hyperbolic Maps (Sinai, Ruelle, Bowen)
1.2.1 General Thermodynamic Results
We refer the reader to ([Katok and Hasselblatt1995]; [Brin and Stuck2002]) for fundamentals of uniform hyperbolicity theory and to ([Bowen2008]; [Parry and Pollicott1990]) for a complete description of thermodynamic formalism for uniformly hyperbolic systems. Consider a compact smooth Riemannian manifold $ M$ and a $ C^1$ diffeomorphism $ f\colon M\to M$. A compact invariant set $ \Lambda\subset M$ is called hyperbolic if for every $ x\in\Lambda$ the tangent space $ T_xM$ admits an invariant splitting $ T_xM=E^s(x)\oplus E^u(x)$ into stable and unstable subspaces with uniform contraction and expansion: this means that there are numbers $ c> 0$ and $ 0< \lambda< 1$ such that for every $ x\in\Lambda$:

(1)
$ \ df^nv\\le c\lambda^n\ v\$ for $ v\in E^s(x)$ and $ n\ge0$;

(2)
$ \ df^{n}v\\le c\lambda^n\ v\$ for $ v\in E^u(x)$ and $ n\ge0$.
One can show that the subspaces $ E^s$ and $ E^u$ depend Hölder continuously on $ x$; in particular, there is $ k> 0$ such that $ \angle(E^s(x),E^u(x))\ge k$ for every $ x\in\Lambda$.
Moving from the tangent bundle to the manifold itself, for every $ x\in\Lambda$ one can construct local stable $ V^s(x)$ and unstable $ V^u(x)$ manifolds (also called leaves) through $ x$ which are tangent to $ E^s(x)$ and $ E^u(x)$ respectively and depend Hölder continuously on $ x$ ([Katok and Hasselblatt1995], Sect. 6.2). In particular, there is $ \varepsilon> 0$ such that for any $ x,y\in\Lambda$ for which $ d(x,y)\le\varepsilon$ one has that the intersection $ V^s(x)\cap V^u(y)$ consists of a single point (here $ d(x,y)$ denotes the distance between points $ x$ and $ y$ induced by the Riemannian metric on $ M$). We denote this point by $ [x,y]$.
A hyperbolic set $ \Lambda$ is called locally maximal if there is a neighborhood $ U$ of $ \Lambda$ such that for any invariant set $ \Lambda'\subset U$ we have that $ \Lambda'\subset\Lambda$. In other words, $ \Lambda=\bigcap_{n\in\mathbb{Z}}\,f^n(U)$. One can show that a hyperbolic set $ \Lambda$ is locally maximal if and only if for any $ x,y\in\Lambda$ which are sufficiently close to each other, the point $ [x,y]$ lies in $ \Lambda$ ([Katok and Hasselblatt1995], Sect. 6.4).
Given a locally maximal hyperbolic set and a Hölder continuous potential function, thermodynamic formalism produces unique equilibrium measures with strong ergodic properties: before stating the theorem we recall some notions from ergodic theory for the reader’s convenience. Let $ (X,\mu)$ be a Lebesgue space with a probability measure $ \mu$ and $ T\colon X\to X$ an invertible measurable transformation that preserves $ \mu$.

(1)
The Bernoulli property. Let $ Y$ be a finite set and $ \nu$ a probability measure on $ Y$ (that is, a probability vector). One can associate to $ (Y,\nu)$ the twosided Bernoulli shift $ \sigma\colon Y^{\mathbb{Z}}\to Y^{\mathbb{Z}}$ defined by $ (\sigma y)_n=y_{n+1}$, $ n\in\mathbb{Z}$; this preserves the measure $ \kappa$ given as the direct product of $ \mathbb{Z}$ copies of $ \nu$. We say that $ (T,\mu)$ is a Bernoulli automorphism (or “has the Bernoulli property”) if $ (T,\mu)$ is metrically isomorphic to the Bernoulli shift $ (\sigma,\kappa)$ associated to some Lebesgue space $ (Y,\nu)$ and we also say that $ \mu$ is a Bernoulli measure.^{1}^{1}More generally, one can take $ (Y,\nu)$ to be a Lebesgue space, so $ \nu$ is metrically isomorphic to Lebesgue measure on an interval together with at most countably many atoms. For all the cases we discuss, it suffices to take $ Y$ finite.

(2)
Decay of correlations. Let $ \mathcal{H}$ be a class of squareintegrable test functions $ X\to\mathbb{R}$ and define
$\displaystyle \begin{eqnarray*} \text{Cor}_n(h_1,h_2):=\left\int h_1(T^n(x))h_2(x)\,d\mu \int h_1(x)\,d\mu \int h_2(x)\,d\mu\right . \end{eqnarray*}$ We say that $ (T,\mu)$ has

–
exponential decay of correlations (EDC) with respect to $ \mathcal{H}$ if there is $ 0< \theta< 1$ satisfying: for every $ h_1, h_2\in\mathcal{H}$ there is $ K=K(h_1,h_2)> 0$ such that for every $ n> 0$
$\displaystyle \begin{eqnarray*} \text{Cor}_n(h_1,h_2)\le K \theta^{n}; \end{eqnarray*}$ 
–
polynomial decay of correlations (PDC) with respect to $ \mathcal{H}$ if there is $ \alpha> 0$ satisfying: for every $ h_1, h_2\in\mathcal{H}$ there is $ K=K(h_1,h_2)> 0$ such that for every $ n> 0$
$\displaystyle \begin{eqnarray*} \text{Cor}_n(h_1,h_2)\le Kn^\alpha. \end{eqnarray*}$

–

(3)
The Central Limit Theorem. Say that a measurable function $ h$ is cohomologous to a constant if there is a measurable function $ g$ and a constant $ c$ such that $ h=g\circ T  g + c$ almost everywhere. We say that the transformation $ T$ satisfies the Central Limit Theorem (CLT) for functions in a class $ \mathcal{H}$ if for any $ h\in\mathcal{H}$ that is not cohomologous to a constant, there exists $ \gamma> 0$ such that
$\displaystyle \begin{eqnarray*} \mu\left\{x : \frac{1}{\sqrt{n}}\sum_{i=0}^{n1}\left(h(T^i(x))\int h\,d\mu\right)< t\right\}\rightarrow\frac{1}{\gamma\sqrt{2\pi}}\int_{\infty}^t e^{\tau^2/2\gamma^2}\,d\tau. \end{eqnarray*}$
Before stating the formal result, we point out that uniformly hyperbolic systems (and many nonuniformly hyperbolic ones) satisfy various other statistical properties, which we do not discuss in detail in this survey. These include large deviations principles ([Orey and Pelikan1988]; [Young1990]; [Kifer1990]; [Pfister and Sullivan2005]; [Melbourne and Nicol2008]; [ReyBellet and Young2008]; [Climenhaga et al.2013]), Borel–Cantelli lemmas ([Chernov and Kleinbock2001]; [Dolgopyat2004]; [Kim2007]; [Gouëzel2007]; [Gupta et al.2010]; [Haydn et al.2013]), the almost sure invariant principle ([Denker and Philipp1984]; [Melbourne and Nicol2005]; [Melbourne and Nicol2009]), and many more besides.
Theorem 1.1.
Let $ \Lambda$ be a locally maximal hyperbolic set for $ f$, and assume that $ f\Lambda$ is topologically transitive.^{2}^{2}This means that there is a point $ x\in\Lambda$ whose trajectory is everywhere dense, i.e., $ \Lambda=\overline{\{f^nx : n\in \mathbb{Z}\}}$. An equivalent definition is that for any two nonempty open sets $ U$ and $ V$ there is $ n\in\mathbb{Z}$ such that $ f^n(U)\cap V\ne\emptyset$. Then for any Hölder continuous potential $ \varphi$, the following are true:

(1)
Existence: there is an equilibrium measure $ \mu_\varphi$.

(2)
Uniqueness: $ \mu_\varphi$ is the only equilibrium measure for $ \varphi$.

(3)
Ergodic and statistical properties:

(a)
the Bernoulli property: there is $ A\subset\Lambda$ and $ n> 0$ such that the sets $ f^k(A)$, $ 0\le k< n$ are (essentially) disjoint and cover $ \Lambda$, $ f^n(A)=A$, and $ (f^nA,\mu_\varphi)$ has the Bernoulli property;

(b)
exponential decay of correlations: there are $ A,n$ as above such that $ (f^nA,\mu_\varphi)$ has EDC with respect to the class of Hölder continuous functions.

(c)
the Central Limit Theorem: $ \mu_\varphi$ satisfies the CLT with respect to the class of Hölder continuous functions.

(a)
The proof of Theorem 1.1 uses the fact that $ f\Lambda$ can be represented by a subshift of finite type via a Markov partition. Recall that a $ p\times p$ transition matrix^{3}^{3}That is, a matrix whose entries $ a_{ij}$ are each equal to 0 or 1. $ A$ determines a subshift of finite type (SFT) $ (\Sigma_A,\sigma)$ as the (left) shift $ \sigma(\omega)_i=\omega_{i+1}$ on the space $ \Sigma_A$ of twosided infinite sequences $ \omega=(\omega_i) \in \{1,\dots, p\}^\mathbb{Z}$ which are admissible with respect to $ A$; that is, for which $ a_{\omega_i\omega_{i+1}}=1$ for all $ i\in \mathbb{Z}$.
Recall also that a finite partition $ \mathcal{R}=\{R_1,\dots,R_p\}$ of $ \Lambda$ is a Markov partition if the following are true.

(1)
The diameter $ diam\mathcal{R}=\max_{1\le i\le p}diam R_i$ is sufficiently small; this guarantees that $ \mathcal{R}$ is generating so the coding map $ \pi\colon \Sigma_A \to X$ introduced below is welldefined.

(2)
$ R_i=\overline{\text{int}\,R_i}$^{4}^{4}Here $ \text{int}\,R_i$ means the interior of the set $ R_i$ in the relative topology. and for any $ 1\le i,j\le p$, $ i\ne j$ we have that $ \text{int}\,R_i\cap\text{int}\,R_j=\emptyset$; this guarantees that the coding map is injective away from the boundaries.

(3)
Each set $ R_i$ is a rectangle, i.e., for any $ x,y\in R_i$ we have that $ z=[x,y]\in R_i$; this is the local product structure (or hyperbolic product structure) of the partition elements.

(4)
The Markov property: for each $ x\in\Lambda$, if $ x\in R_i$ and $ f(x)\in R_j$ for some $ 1\le i,j\le p$, then
$\displaystyle \begin{align*} f(V^s(x)\cap R_i)&\subset V^s(f(x))\cap R_j,\\ f^{1}(V^u(f(x))\cap R_j)&\subset V^u(x)\cap R_i. \end{align*}$
The first construction of Markov partitions was obtained by [Adler and Weiss1967]; [Adler and Weiss1970], and independently by [Berg1967], in the particular case of hyperbolic automorphisms of the 2torus. They observed that the map allowed a symbolic representation by a subshift of finite type and that this can be used to study its ergodic properties. Sinai realized that existence of Markov partitions is a rather general phenomenon and he constructed Markov partitions for general Anosov diffeomorphisms, see [Sinaĭ1968]. Furthermore, in [Sinaĭ1972] he observed the analogy between the symbolic models of Anosov diffeomorphisms and lattice gas models in physics—the starting point in developing the thermodynamic formalism. Finally, in the more general setting of locally maximal hyperbolic sets Markov partitions were constructed by [Bowen1970].
Markov partitions allow one to obtain a symbolic representation of the map $ f\Lambda$ by subshifts of finite type. More precisely, let $ \mathcal{R}=\{R_1,\dots,R_p\}$ be a finite Markov partition of $ \Lambda$. Consider the subshift of finite type $ (\Sigma_A,\sigma)$ with the transition matrix $ A$ whose entries are given by $ a_{ij}=1$ if $ f(\mathrm{int}\, R_i)\cap \mathrm{int}\, R_j\ne\emptyset$ and $ a_{ij}=0$ otherwise. One can show that for every $ \omega=(\omega_i)\in\Sigma_A$ the intersection $ \bigcap_{i\in\mathbb{Z}}f^{i}(R_{\omega_i}) $ is not empty and consists of a single point $ \pi(\omega)$. This defines the coding map $ \pi\colon\Sigma_A\to\Lambda$, which is characterized by the fact that $ f^i(\pi(\omega)) \in R_{\omega_i}$ for all $ i\in \mathbb{Z}$ (thus $ \omega$ “codes” the orbit of $ \pi(\omega)$).
Proposition 1.2.
The map $ \pi$ has the following properties:

(1)
$ \pi$ is Hölder continuous;

(2)
$ \pi$ is a conjugacy between the shift $ \sigma$ and the map $ f\Lambda$, i.e., $ (f\Lambda)\circ\pi=\pi\circ\sigma$;

(3)
$ \pi$ is onetoone on the set $ \Sigma'\subset\Sigma$ which consists of points $ \omega$ for which the trajectory of the point $ \pi(\omega)$ never hits the boundary of the Markov partition.
Consider a Hölder continuous potential $ \varphi$ on $ \Lambda$. By Proposition 1.2, the function $ \tilde\varphi$ on $ \Sigma_A$ given by $ \tilde\varphi(\omega)=\varphi(\pi(\omega))$ is Hölder continuous. Thus in order to prove Theorem 1.1 it suffices to study thermodynamic formalism for Hölder continuous potentials for SFTs. The starting point for this theory is the following result of [Parry1964], which uses Perron–Frobenius theory to deal with the case $ \varphi=0$. The corresponding equilibrium measure is the measure of maximal entropy (MME) for which $ h_\mu(f)=h_{\text{top}}(f)$.
Theorem 1.3.
Let $ A$ be a transition matrix such that $ A^n> 0$ for some $ n\in \mathbb{N}$, and let $ \Sigma_A$ be the corresponding SFT.

(1)
The topological entropy of $ \Sigma_A$ is $ \log\lambda$, where $ \lambda> 1$ is the maximal eigenvalue of $ A$ guaranteed by the Perron–Frobenius theorem.

(2)
Let $ v$ be a positive right eigenvector for $ (A,\lambda)$ (so $ Av = \lambda v$); then the matrix $ P$ given by $ P_{ij} = A_{ij}\frac{ v_j}{\lambda v_i}$ is stochastic (its rows are probability vectors), so it defines transition probabilities for a Markov chain .

(3)
Let $ h$ be a positive left eigenvector for $ (A,\lambda)$, normalized so that $ \pi_i = h_i v_i$ defines a probability vector $ \pi$. Then $ \pi$ is the unique probability vector with $ \pi P = \pi$, and the unique MME for $ \Sigma_A$ is the Markov measure defined by
$\displaystyle \begin{eqnarray*} \mu[\omega_1 \cdots \omega_n] = \pi_{\omega_1} P_{\omega_1\omega_2} \cdots P_{\omega_{n1}\omega_n}. \end{eqnarray*}$
Theorem 1.3 was adapted to nonzero potentials by [Ruelle1968]; [Ruelle1976], replacing the transition matrix with a transfer operator. Ruelle’s version of the Perron–Frobenius theorem for this transfer operator is at the heart of the classical results in thermodynamic formalism for SFTs, and hence, for uniformly hyperbolic systems. Roughly speaking the idea is the following.

(1)
Replace the twosided SFT $ \Sigma_A$ with its onesided version $ \Sigma_A^+$, and define the transfer operator associated to $ \varphi$ on $ C(\Sigma_A^+)$ by^{5}^{5}It is instructive to consider the case $ \varphi=0$ and write down the action of $ \mathcal{L}_0$ on the (finitedimensional) space of functions constant on 1cylinders, where the action is given by the (transpose of the) transition matrix $ A$.
$\displaystyle \begin{eqnarray*} (\mathcal{L}_\varphi f)(x) = \sum_{\sigma y=x} e^{\varphi(y)} f(y). \end{eqnarray*}$ 
(2)
Show that $ \mathcal{L}_\varphi$ has a largest eigenvalue $ \lambda$ and that the rest of the spectrum lies inside a disc with radius $ {< }\lambda$ (the spectral gap property).

(3)
Instead of the left and right eigenvalues $ h$ and $ v$, find a positive eigenfunction $ h\in C(\Sigma_A^+)$ for $ \mathcal{L}_\varphi$, and an eigenmeasure $ \nu\in \mathcal{M}(\Sigma_A^+)$ for the dual $ \mathcal{L}_\varphi^*$.

(4)
Obtain the unique equilibrium state as $ d\mu = h\,d\nu$.
We stress that this result (and hence Theorem 1.1) may not hold if the the potential function fails to be Hölder continuous, see [Hofbauer1977], [Sarig2001a], [Pesin and Zhang2006].
1.2.2 Thermodynamic Formalism for the Geometric $ t$Potential
Returning from SFTs to the setting of uniformly hyperbolic smooth systems, the most significant potential function is the geometric $ t$potential: a family of potential functions $ \varphi_t(x):= t\log \abs{Jac(dfE^u(x))}$ for $ t\in\mathbb{R}$. Since the subspaces $ E^u(x)$ depend Hölder continuously on $ x$, the potential $ \varphi_t$ is Hölder continuous for each $ t$ whenever $ f$ is $ C^{1+\alpha}$; in particular, it admits a unique equilibrium measure $ \mu_t$. Furthermore, the pressure function $ P(t) := P(\varphi_t)$ is well defined for all $ t$, is convex, decreasing, and real analytic in $ t$, as in Fig. 1a.
There are certain values of $ t$ that are particularly important.

•
When $ t=0$, we obtain the topological entropy $ {h}_{{\rm top}}(f)$ as $ P(0)$, and the unique MME as $ \mu_0$.

•
Since $ P$ is strictly decreasing and has $ P(0) > 0$ and $ P(t)\to \infty$ as $ t\to\infty$, there is a unique number $ t_0> 0$ for which $ P(t_0)=0$. The equation $ P(t)=0$ is called Bowen’s equation. In the twodimensional case its root is the Hausdorff dimension of $ \Lambda\cap V^u(x)$^{6}^{6}The value of the Hausdorff dimension does not depend on $ x$. and the equilibrium measure $ \mu_{t_0}$ achieves this Hausdorff dimension (i.e., is the measure of maximal dimension) ([Bowen1979]; [Ruelle1982]; [McCluskey and Manning1983]).
To further study the properties of the pressure function (and $ t_0$ in particular) we recall the notion of the Lyapunov exponent. Given $ x\in\Lambda$ and $ v\in T_xM$, define the Lyapunov exponent
$\displaystyle \begin{eqnarray*} \chi(x,v)=\limsup_{n\to\infty}\,\frac1n\log\df^nv\. \end{eqnarray*}$ 
For every $ x\in\Lambda$ the function $ \chi(x,\cdot)$ takes on finitely many values $ \chi_1(x)\le\cdots\le\chi_d(x)$ where $ d=\dim M$. The functions $ \chi_i(x)$ are Borel and are invariant under $ f$; in particular, if $ \mu$ is an ergodic measure, then $ \chi_i(x)=\chi_i(\mu)$ is constant almost everywhere for each $ i=1,\dots, d$, and the numbers $ \chi_i(\mu)$ are called the Lyapunov exponent of the measure $ \mu$. If none of these numbers is equal to zero, $ \mu$ is called a hyperbolic measure;^{7}^{7}It is assumed that some of these numbers are positive while others are negative. note that when $ \Lambda$ is a hyperbolic set for $ f$, every invariant measure supported on $ \Lambda$ is hyperbolic. The Margulis–Ruelle inequality (see [Ruelle1979]; [Barreira and Pesin2013]) says that
$\displaystyle \begin{eqnarray*}\label{mri} h_\mu(f)\le\sum_{i:\chi_i(\mu)\ge 0}\chi_i(\mu) \end{eqnarray*}$  (1.3) 
and in particular implies that $ t_0\le 1$, since the sum in (1.3) is equal to $ \int \varphi_1\,d\mu$, and hence $ h_\mu(f)+\int\varphi_1\,d\mu \le 0$ for every ergodic $ \mu$.
1.2.3 Hyperbolic Attractors
We consider the particular case when $ \Lambda$ is a topological attractor for $ f$. This means that there is a neighborhood $ U\supset\Lambda$ such that $ \overline{f(U)}\subset U$ and $ \Lambda=\bigcap_{n\ge 0}f^n(U)$. It is not difficult to see that for every $ x\in\Lambda$, the local unstable manifold $ V^u(x)$ is contained in $ \Lambda$;^{8}^{8}Indeed, for any $ y\in V^u(x)$ the trajectory of $ y$, $ \{f^n(y)\}_{n\in\mathbb{Z}}$ lies in $ U$ and hence, must belong to $ \Lambda$ since it is locally maximal. the same is true for the global unstable manifold through $ x$. Therefore, the attractor contains all the global unstable manifolds of its points. On the other hand the intersection of $ \Lambda$ with stable manifolds of its points is usually a Cantor set.
In the case when $ \Lambda$ is a hyperbolic attractor we have that $ t_0=1$ (see [Bowen2008]), so $ P(t)$ is as in Fig. 1b. The equilibrium state $ \mu_1$ is a hyperbolic ergodic measure for which the Margulis–Ruelle inequality (1.3) becomes equality. By [Ledrappier and Young1985], this implies that $ \mu_1$ has absolutely continuous conditional measures along unstable manifolds; that is, there is a collection $ \mathcal{R}$ of local unstable manifolds $ V^u$ and a measure $ \eta$ on $ \mathcal{R}$ such that $ \mu_1$ can be written as
$\displaystyle \begin{eqnarray*}\label{eqn:ac} \mu_1(E) = \int_\mathcal{R} \mu_{V^u}(E) \,d\eta(V^u) \end{eqnarray*}$  (1.4) 
where the measures $ \mu_{V^u}$ are absolutely continuous with respect to the leaf volumes $ m_{V^u}$. A hyperbolic measure $ \mu$ satisfying (1.4) is said to be a Sinai–Ruelle–Bowen (SRB) measure, and it can be shown that such measures are physical: the set of generic points
$\displaystyle \begin{eqnarray*} G_\mu := \left\{x\in M \mid \frac 1n \sum_{k=0}^{n1} \varphi(f^k(x)) \to \int\varphi\,d\mu \text{ for all continuous } \varphi\colon M\to \mathbb{R} \right\} \end{eqnarray*}$ 
has positive volume, and so $ \mu$ is the appropriate invariant measure for studying “physically relevant” trajectories. The discussion above shows that when $ \Lambda$ is a hyperbolic attractor, SRB measures are precisely the equilibrium states for the geometric potential $ \varphi_1$.
1.3 Nonuniformly Hyperbolic Maps
1.3.1 Definition of Nonuniform Hyperbolicity
A $ C^{1+\alpha}$ diffeomorphism $ f$ of a compact smooth Riemannian manifold $ M$ is nonuniformly hyperbolic on an invariant Borel subset $ S\subset M$ if there are a measurable $ df$invariant decomposition of the tangent space $ T_xM=E^s(x)\oplus E^u(x)$ for every $ x\in S$ and measurable $ f$invariant functions $ \varepsilon(x)> 0$ and $ 0< \lambda(x)< 1$ such that for every $ 0< \varepsilon\le\varepsilon(x)$ one can find measurable functions $ c(x)> 0$ and $ k(x)> 0$ satisfying for every $ x\in S$:

(1)
$ \ df^nv\\le c(x)\lambda(x)^n\ v\$ for $ v\in E^s(x)$, $ n\ge0$;

(2)
$ \ df^{n}v\\le c(x)\lambda(x)^n\ v\$ for $ v\in E^u(x)$, $ n\ge0$;

(3)
$ \angle(E^s(x),E^u(x))\ge k(x)$;

(4)
$ c(f^m(x))\le e^{\varepsilon m}c(x)$, $ k(f^m(x))\ge e^{\varepsilon m}k(x)$, $ m\in{\mathbb{Z}}$.
The last property means that the estimates in (1) and (2) can deteriorate but with subexponential rate.
If $ \mu$ is an invariant measure for $ f$ with $ \mu(S)=1$, then by the Multiplicative Ergodic theorem, if for almost every $ x\in S$ the Lyapunov exponents at $ x$ are all nonzero, i.e., $ \mu$ is a hyperbolic measure, then $ f$ is nonuniformly hyperbolic on $ S$.
1.3.2 Possibility of Phase Transitions and Nonhyperbolic Behavior
A general theory of thermodynamic formalism for nonuniformly hyperbolic maps is far from being complete, although certain examples here are wellunderstood. They include onedimensional maps, where the pressure function $ P(t)=P(\varphi_t)$ associated with the family of geometric potentials may behave as in the uniformly hyperbolic case, or may exhibit new phenomena such as phase transitions (points of nondifferentiability where there is more than one equilibrium measure). The latter is illustrated in Fig. 1c and is most thoroughly studied for the Manneville–Pomeau map $ x\mapsto x+x^{1+\alpha} \pmod 1$, where $ \alpha\in (0,1)$ controls the degree of intermittency at the neutral fixed point. In this example one has the following behavior ([Pianigiani1980]; [Thaler1980]; [Thaler1983]; [Lopes1993]; [Pollicott and Weiss1999]; [Liverani et al.1999]; [Young1999]; [Sarig2002]; [Hu2004]).

•
Hyperbolic behavior for $ t< 1$: the pressure function $ P(t)$ is real analytic and decreasing on $ (\infty,1)$, and for every $ t$ in this range, the geometric $ t$potential $ \varphi_t$ has a unique equilibrium measure $ \mu_t$, which is Bernoulli, has EDC, and satisfies the CLT with respect to the class of Hölder continuous potentials. In a nutshell, for $ t\in (\infty,1)$, the thermodynamics of this system is just as in the case of uniform hyperbolicity.

•
Phase transition at $ t=1$: the pressure function $ P(t)$ is nondifferentiable at $ t=1$, and $ \varphi_1$ has two ergodic equilibrium measures. One of these is the absolutely continuous invariant probability measure $ \mu_1$ (which plays the role of SRB measure), and the other is the point mass $ \delta_0$ on the neutral fixed point.^{9}^{9}For $ \alpha\in(0,1)$ the measure $ \mu_1$ is finite but for $ \alpha\ge 1$, a new phenomenon occurs: the intermittent behavior becomes strong enough that while there is still an absolutely continuous invariant measure, it is infinite. At the same time, the pressure function for $ \alpha\ge 1$ becomes differentiable at $ t=1$, and the measure $ \delta_0$ becomes the unique equilibrium measure. The measure $ \mu_1$ is Bernoulli and decay of correlations is polynomial (in particular, subexponential).

•
Nonhyperbolic behavior for $ t> 1$: for every $ t\in (1,\infty)$, the unique equilibrium state for $ \varphi_t$ is the point mass $ \delta_0$, which has zero entropy and zero Lyapunov exponent.
Similar results for the geometric $ t$potential are available for other classes of onedimensional maps (e.g., unimodal and multimodal maps) and rather specific higherdimensional examples (e.g., polynomial and rational maps and (piecewise) nonuniformly expanding maps); in some of these examples phase transitions occur while others are without phase transitions. As a small sample of the recent literature on the topic, we mention only ([Bruin and Keller1998]; [Makarov and Smirnov2000]; [Oliveira2003]; [Alves et al.2005]; [Przytycki and RiveraLetelier2007]; [Pesin and Senti2008]; [Bruin and Todd2008]; [Bruin and Todd2009]; [Dobbs2009]; [Iommi and Todd2010]; [Przytycki and RiveraLetelier2011]; [Li and RiveraLetelier2014a]; [Li and RiveraLetelier2014b]), as well as the comprehensive and farreaching discussion of thermodynamics for interval maps with critical points in [Dobbs and Todd2015].
Our goal in the rest of this paper is not to discuss these results, which rely on the specific structure of the examples being studied (or on the absence of a contracting direction); rather, we want to discuss the recently developed techniques for studying multidimensional nonuniformly hyperbolic systems, with particular emphasis on recent results that have the potential to be applied very generally, although they do not yet give as complete a picture as the one outlined above. These general results have been obtained in the last few years and represent an actively evolving area of research.
1.3.3 Different Types of Equilibrium Measures
Before describing the general methods, we recall some basic notions from nonuniform hyperbolicity; see [Barreira and Pesin2007] for more complete definitions and properties. Let $ M$ be a compact smooth manifold and $ f\colon M\to M$ a $ C^{1+\alpha}$ diffeomorphism. Recall that a point $ x\in M$ is called Lyapunov–Perron regular if for any basis $ \{v_1,\dots, v_p\}$ of $ T_xM$,
$\displaystyle \begin{eqnarray*} \liminf_{n\to\pm\infty}\frac1n\log V(n)= \limsup_{n\to\pm\infty}\frac1n\log V(n) =\sum_{i=1}^p\,\chi_i(x,v_i), \end{eqnarray*}$ 
where $ V(n)$ is the volume of the parallelepiped built on the vectors $ \{df^nv_1,\dots, { } df^nv_p\}$.
Let $ \mathcal{R}$ be the set of all Lyapunov–Perron regular points. The Multiplicative Ergodic theorem claims that this set has full measure with respect to any invariant measure. Consider now the set $ \Gamma\subset\mathcal{R}$ of points for which all Lyapunov exponents are nonzero, and let $ \mathcal{M}^e(f,\Gamma)\subset \mathcal{M}^e(f)$ be the set of all ergodic measures that give full weight to the set $ \Gamma$; these are hyperbolic measures and they form the class of measures where it is reasonable to attempt to recover some of the theory of uniformly hyperbolic systems.
Let $ \varphi$ be a measurable potential function; note that we cannot a priori assume more than measurability if we wish to include the family of geometric potentials, since in general the unstable subspace varies discontinuously and so $ \varphi_t$ is not a continuous function.^{10}^{10}On the other hand, for surface diffeomorphisms [Sarig2013] constructed Markov partitions with countably many partition elements (see Sect. 3 below), and showed [Sarig2011] that the function $ \varphi_t$ can be lifted to a function on the symbolic space that is globally welldefined and is Hölder continuous. This can be used to study equilibrium measures for this function. Consider the hyperbolic pressure defined by using only hyperbolic measures:
$\displaystyle \begin{eqnarray*}\label{eqn:PG} P_\Gamma(\varphi) := \inf_{\mu\in \mathcal{M}^e(f,\Gamma)} E_\mu(\varphi). \end{eqnarray*}$  (1.5) 
Say that $ \mu_\varphi$ is a hyperbolic equilibrium measure if $ E_{\mu_\varphi}(\varphi)=P_\Gamma(\varphi)$. For the Manneville–Pomeau example above, we have $ P_\Gamma(\varphi_t)=P(\varphi_t)$ for every $ t\in\mathbb{R}$, and the equilibrium measure $ \mu_t$ is the unique hyperbolic equilibrium measure for every $ t\le 1$,^{11}^{11}Note that for $ t=1$ it is no longer the unique equilibrium measure, but it is the only hyperbolic one. while for $ t> 1$ there is no hyperbolic equilibrium measure, since $ \delta_0$ has zero Lyapunov exponent.
One could also fix a threshold $ h> 0$ and consider the set $ \mathcal{M}^e(f,\Gamma,h)$ of all measures in $ \mathcal{M}^e(f,\Gamma)$ whose entropies are greater than $ h$; restricting our attention to measures from this class gives the restricted pressure^{12}^{12}Because $ \mathcal{M}^e(f,\Gamma,h)$ is not compact, the existence of an optimizing measure in (1.6) becomes a more subtle issue. Although it may happen that the value of $ P_\Gamma^h(\varphi)$ is achieved by a measure $ \mu$ whose entropy may not be greater than $ h$, the restriction to measures in the class $ \mathcal{M}^e(f,\Gamma,h)$ is often made to ensure a certain “liftability” condition, which may still be satisfied by $ \mu$; see Theorem 2.3 and the discussion in that section.
$\displaystyle \begin{eqnarray*}\label{eqn:PGh} P_\Gamma^h(\varphi) := \inf_{\mu\in\mathcal{M}^e(f,\Gamma,h)} E_\mu(\varphi). \end{eqnarray*}$  (1.6) 
For the Manneville–Pomeau example, we have for every $ t\in \mathbb{R}$,^{13}^{13}This is reminiscent of the use of Katok horseshoes to approximate (with respect to entropy) an arbitrary system with a uniformly hyperbolic one [Katok1980], which was recently generalized to pressure by [SánchezSalas2015].
$\displaystyle \begin{eqnarray*} \lim_{h\to 0} P_\Gamma^h(\varphi_t) = P_\Gamma(\varphi_t) = P(\varphi_t). \end{eqnarray*}$ 
In addition to the use of $ \mu_t^h$ to approximate nonhyperbolic measures by hyperbolic ones, the above approach is also useful when one can identify a (not necessarily invariant) subset $ \mathcal{A}\subset X$ of “bad” points away from which the dynamics exhibits good hyperbolic behavior; then putting $ h> h_{\text{top}}(f,\mathcal{A})$ guarantees that we consider only measures to which $ \mathcal{A}$ is invisible.^{14}^{14}Note that since $ \mathcal{A}$ is not assumed to be invariant, one should use the definition of the topological entropy based on the Carathéodory construction of dimensionlike characteristics for dynamical systems ([Bowen1973]; [Pesin1997]). This concept originated in the work of Buzzi on piecewise invertible continuous maps of compact metric spaces [Buzzi1999],^{15}^{15}An important goal there was to study the notion of $ h$isomorphism, which asks for two systems to have (measuretheoretically) conjugate subsystems that carry all ergodic measures with large enough entropy, even if the whole systems are not conjugate. but it is reasonable to consider it in other situations.^{16}^{16}For example, if the set $ \mathcal{A}$ is an elliptic island and the potential function is sufficiently large on $ \mathcal{A}$, then the equilibrium measure may be a zero entropy measure sitting outside the set with nonzero Lyapunov exponents. Putting any positive threshold removes this measure from consideration.
One could also impose a threshold in other ways. For example, one could fix a reference potential $ \psi$ and a threshold $ p < P(\psi)$, then restrict attention to the set $ \mathcal{M}^e(f,\Gamma,\psi,p)$ of all measures in $ \mathcal{M}^e(f,\Gamma)$ for which $ E_\mu(\psi)> p$. Optimizing $ E_\mu(\varphi)$ over this restricted set of measures gives another notion of thresholded equilibrium states that may be useful; again, it is often natural to take $ p=P_S(\varphi)$ as the topological pressure of $ f$ on a (not necessarily invariant) subset $ S\subset M$ of bad points. Another approach would be to consider only measures whose Lyapunov exponents are sufficiently large; it may be that this is a more natural approach in certain settings. We stress that while restricting the class of invariant measures using thresholds for the topological pressure or Lyapunov exponents seem to be natural it is yet to be shown to be a working tool in effecting thermodynamic formalism.
1.3.4 Outline of the Paper
A direct application of the uniformly hyperbolic approach in the nonuniformly hyperbolic setting is hopeless in general; we cannot expect to have finite Markov partitions.^{17}^{17}Indeed, if a map possesses a Markov partition, then its topological entropy is the logarithm of an algebraic number, which should certainly not be expected in general. On the other hand, in the presence of a hyperbolic invariant measure $ \mu$ of positive entropy, there are horseshoes with finite Markov partitions whose entropy approximates the entropy of $ \mu$ [Katok1980], but these have zero $ \mu$measure. However, in many cases it is possible to use the symbolic approach by finding a countable Markov partition, or the related tools of a Young tower or a more general inducing scheme; these are discussed in Sects. 2–4. This approach is challenging to apply completely, but can help establish existence and uniqueness of equilibrium measures and study their statistical properties including decay of correlations and the CLT.
A second approach is to avoid the issue of building a Markov partition by adapting Bowen’s specification property to the nonuniformly hyperbolic setting; this is discussed in Sect. 5. This is similar to the symbolic approach in that one uses a “coarsegraining” of the system to make counting arguments borrowed from statistical physics, but sidesteps the issue of producing a Markov structure. The price paid for this added flexibility is that while existence and uniqueness can be obtained with specificationbased techniques, there does not seem to be a direct way to obtain strong statistical properties without first establishing some sort of Markov structure.
A third approach, which we discuss in Sect. 6, is geometric and is based on pushing forward the leaf volume on unstable manifolds by the dynamics. More generally, one can work with approximations to unstable manifolds by admissible manifolds and use measures which have positive densities with respect to the leaf volume as reference measures. Such pairs of admissible manifolds and densities are called standard and working with them has proven to be quite a useful technique in various problems in dynamics.^{18}^{18}This notion was introduced by [Chernov and Dolgopyat2009]. So far the geometric approach can be used to establish existence of SRB measures for uniformly hyperbolic and some nonuniformly hyperbolic attractors and one can also use a version of this method to construct equilibrium measures for uniformly hyperbolic sets, see Sect. 6; the questions of uniqueness and statistical properties using this approach as well as construction of equilibrium measures for nonuniformly hyperbolic systems are still open.
In the remainder of this paper we describe the three approaches just listed in more detail, and discuss their application to open problems in the thermodynamics of nonuniformly hyperbolic systems.
2 Markov Models for Nonuniformly Hyperbolic Maps I: Young Diffeomorphisms
2.1 Earlier Results: Onedimensional and Rational Maps
In one form or another, the use of Markov models with countably many states to study nonuniformly hyperbolic systems dates back to the late 1970s and early 1980s, when [Hofbauer1979]; [Hofbauer1981a]; [Hofbauer1981b] used a countablestate Markov model to study equilibrium states for piecewise monotonic interval maps. Indeed, such models for $ \beta$transformations were studied already in 1973 by [Takahashi1973].
In [Jakobson1981] Jakobson initiated the study of thermodynamics of unimodal interval maps by constructing absolutely continuous invariant measures (acim) for the family of quadratic maps $ f_a(x) = 1ax^2$ whenever $ a\in\Delta$, where $ \Delta$ is a set of parameters with positive Lebesgue measure. First we discuss in Sect. 2.2 the extensions of Jakobson’s result to study SRB measures by what have become known as Young towers. Then in Sect. 3 we discuss the study of general equilibrium states in the setting of topological Markov chains with countably many states, which generalizes the SFT theory from Sect. 1.2. Finally, in Sect. 4 we discuss the use of inducing schemes to apply this theory to the thermodynamics of smooth examples.
2.2 Young Towers and Gibbs–Markov–Young Structures
2.2.1 Tower Constructions in Dynamical Systems
Roughly speaking, a tower construction begins with a base set $ \Lambda$, a map $ G\colon\Lambda\to\Lambda$, and a height function $ R\colon\Lambda\to\mathbb{N}$. Then the tower is constructed as $ \tilde\Lambda:=\{(z,n)\in\Lambda\times \{0,1,2,\dots\}:n< R(z)\}$, and a map $ g\colon\tilde\Lambda\to\tilde\Lambda$ is defined by $ g(z,n)=(z,n+1)$ whenever $ n+1< R(z)$, and $ g(z,R(z)1)=(F(z),0)$. Typically one requires that the dynamics of the return map $ G$ can be coded by a full shift, or a Markov shift on a countable set of states. To study a dynamical system $ f\colon X\to X$ using a tower, one defines a coding map $ \pi\colon\tilde \Lambda\to X$ such that $ f\circ\pi=\pi\circ g$; this coding map is usually not surjective (the tower does not cover the entire space), and so we will ultimately need to give some “largeness” condition on the tower. It is important to distinguish between the case when $ \pi(\Lambda)$ is disjoint from $ \pi(\tilde\Lambda{\setminus}\Lambda)$, so that the height $ R$ is the first return time to the base $ \pi(\Lambda)$, and the case when $ R$ is not the first return time.
Tower constructions for which the height of the tower is the first return time to the base of the tower are classical objects in ergodic theory and were considered in works of Kakutani, Rokhlin, and others. Towers for which the height of the tower is not the first return time appeared in the paper by [Neveu1969] under the name of temps d’arret and in the context of dynamical systems in the paper by [Schweiger1975]; [Schweiger1979] under the name jump transformation (which are associated with some fibered systems; see also the paper by [Aaronson et al.1993] for some general results on ergodic properties of Markov fibered systems and jump transformations).
A tower construction is implicitly present in Jakobson’s proof of existence of physical measures for quadratic maps. The first significant use of the tower approach beyond the onedimensional setting came in the study of the Hénon map
$\displaystyle \begin{eqnarray*}\label{henon} f_{a,b}(x,y) = (1ax^2 + y, bx), \end{eqnarray*}$  (2.1) 
which for $ b\approx 0$ can be viewed as a twodimensional extension of a unimodal map with parameter $ a$. Building on their alternate proof of Jakobson’s theorem in [Benedicks and Carleson1985], Benedicks and Carleson showed in ([Benedicks and Carleson1991]) that when $ b$ is sufficiently close to 0, there is a set $ \Delta_b\subset \mathbb{R}$ of positive Lebesgue measure such that $ f_{a,b}$ has a topologically transitive attractor for every $ a\in \Delta_b$. Soon afterwards, Benedicks and Young established existence of an SRB measure for these examples ([Benedicks and Young1993]); their approach also gives exponential decay of correlations and the CLT ([Benedicks and Young1995]).
The general structure behind these results was developed in [Young1998] and has come to be known as a Young tower,^{19}^{19}It is worth mentioning that a major achievement of [Young1998] was to establish exponential decay of correlations for billiards with convex scatterers, which is an example of a uniformly hyperbolic system with discontinuities; we will not discuss such examples further in this paper. or a Gibbs–Markov–Young structure. The principal feature of a Young tower is that the induced map on the base of the tower is conjugate to the full shift on the space of twosided sequences over countable alphabet. This allows one to use some recent results on thermodynamics of this symbolic map to establish existence and uniqueness of equilibrium measures for the original map and study their ergodic properties.
2.2.2 Young Diffeomorphisms
A $ C^{1+\alpha}$ diffeomorphism $ f$ of a compact smooth manifold $ M$ is called Young diffeomorphism if it admits a Young tower. This tower has a particular structure which is characterized as follows:

The base $ \Lambda$ of the tower has hyperbolic product structure which is generated by continuous families $ {\bf V}^u=\{V^u\}$ and $ {\bf V}^s=\{V^s\}$ of local unstable and stable manifolds.

The induced map has the Markov property, is uniformly hyperbolic and has uniform bounded distortion.

The intersection of at least one unstable manifold with the base of the tower has positive leaf volume^{20}^{20}It follows that every local unstable manifold intersects the base in a set of positive leaf volume. and the integral of the height of the tower against leaf volume is finite.
In particular, the tower codes a positive volume part of the system (but not necessarily all trajectories) by a countable state Markov shift.
A formal description of the Young tower is as follows. There are two continuous families $ {\bf V}^u=\{V^u\}$ and $ {\bf V}^s=\{V^s\}$ of local unstable and stable manifolds, respectively, with the property that each $ V^s$ meets each $ V^u$ transversely in a single point and $ \Lambda=(\bigcup V^u)\cap (\bigcup V^s)$; a union of some of the manifolds $ V^u$ is called a $ u$set, a union of some of the manifolds $ V^s$ is called an $ s$set. One asks for $ \Lambda$ to have the following properties; here $ C,\eta> 0$ and $ \beta\in (0,1)$ are constants.

(P1) Positive measure: each $ V^u\cap\Lambda$ has positive leaf volume $ m_{V^u}$.

(P2) Markov structure: there are (countably many) pairwise disjoint $ s$sets $ \Lambda_i^s\subset\Lambda$ and numbers $ R_i\in\mathbb{N}$ such that

$ \Lambda{\setminus}\bigcup_i\Lambda_i^s$ is $ m_{V^u}$null for all $ V^u$;

$ \Lambda_i^u=f^{R_i}(\Lambda_i^s)$ is a $ u$set in $ \Lambda$;

for every $ x\in\Lambda_i^s$,
$\displaystyle \begin{align*} f^{R_i}(V^s(x))&\subset V^s(f^{R_i}(x)), \\ f^{R_i}(V^u(x))&\supset V^u(f^{R_i}(x)), \\ f^{R_i}(V^s(f^{R_i}(x))\cap\Lambda_i^u)&=V^s(x)\cap\Lambda, \\ f^{R_i}(V^u(x)\cap\Lambda_i^s)&= V^u(f^{R_i}(x))\cap\Lambda; \end{align*}$


(P3) Defining the recurrence (induced) time $ R\colon\bigcup_i\Lambda_i^s\to\Lambda$ by $ R\Lambda_i^s=R_i$ and the induced map $ F(x) = f^{R(x)}(x)$, we have that for all $ n\ge 1$

Forward contraction on $ V^s$: if $ x,y$ are in the same leaf $ V^s$, then $ d(F^nx,F^ny)\le C\beta^nd(x,y)$.

Backward contraction on $ V^u$: if $ x,y$ are in the same leaf $ V^u$ and the same $ s$set $ \Lambda_i^s$, then $ d(F^{n}x,F^{n}y)\le C\beta^nd(Fx,Fy)$.

Bounded distortion: if $ x,y$ are in the same leaf $ V^u$ and the same $ s$set $ \Lambda_i^s$ then
$\displaystyle \begin{eqnarray*} \log\frac{\det dF^u(x)}{\det dF^u(y)}\le C d(Fx,Fy)^\eta. \end{eqnarray*}$

Our description of Young tower follows [Pesin et al.2016b] and differs from the original description in [Young1998]. Most importantly, we do not require that the map $ f$ contracts distances along local stable manifolds uniformly with an exponential rate and neither does the inverse map $ f^{1}$ along local unstable manifolds but that this requirement holds with respect to the induced map $ F$ (see (P3)). We stress that in constructing SRB and equilibrium measures on Young towers and studying their ergodic properties these extra requirements on the maps $ f$ and $ f^{1}$ are not needed and that there are examples in which the map $ f$ contracts distances along local stable manifolds uniformly with a polynomial rate, see Sect. 2.3.2.
2.2.3 SRB Measures for Young Diffeomorphisms
Once a tower structure has been found, the strength of the conclusions one can draw depends on the rate of decay of the tail of the tower; that is, the speed with which $ m_{V^u}\{x\in V^u\mid R(x) > T\}\to 0$ as $ T\to\infty$ for $ V^u\in {\bf V}^u$. We say that with respect to the measure $ m_{V^u}$ the tower has

•
integrable tails if
$\displaystyle \begin{eqnarray*} \int R\,dm_{V^u}< \infty; \end{eqnarray*}$ 
•
exponential tails if for some $ C,a> 0$ and $ T\ge 1$,
$\displaystyle \begin{eqnarray*}\label{expspstails} m_{V^u}\{x\mid R(x)> T\} < Ce^{aT}; \end{eqnarray*}$ (2.2) 
•
polynomial tails if for some $ C,a> 0$ and $ T\ge 1$,
$\displaystyle \begin{eqnarray*} m_{V^u}\{x\mid R(x)> T\} < CT^{an}. \end{eqnarray*}$
Theorem 2.1.
([Young1998]) Let $ f$ be a $ C^{1+\alpha}$ diffeomorphism of a compact manifold $ M$ admitting a Young tower. Assume that

(1)
there is local unstable manifold $ V^u$ such that
$\displaystyle \begin{eqnarray*}\label{fullmeasure} m_{V^u}\left(\bigcup_{i\ge 1}\bar{\Lambda}_i{\setminus}\Lambda_i\right)=0; \end{eqnarray*}$ (2.3) 
(2)
the tower has integrable tails.
Then $ f$ has an SRB measure $ \mu$.
To describe ergodic properties of the SRB measure one needs an extra condition. We say that the tower satisfies the arithmetic condition if the greatest common denominator of of the set of integers $ \{R_i\}$ is one.^{21}^{21}The tower $ \tilde\Lambda$ admits a natural countable Markov partition (see [Young1998]) and the arithmetic condition is equivalent to the requirement that the corresponding Markov shift is topologically mixing.
Theorem 2.2.
([Young1998]) Let $ f$ be a $ C^{1+\alpha}$ diffeomorphism of a compact manifold $ M$ admitting a Young tower. Assume that the tower satisfies (2.3), the arithmetic condition and has exponential (respectively, polynomial) tails. Then $ (f,\mu)$ is Bernoulli, has exponential (respectively, polynomial) decay of correlations and satisfies the CLT with respect to the class of functions which are Hölder continuous on $ \Lambda$.
Note that even without the arithmetic condition one still obtains the “exponential decay up to a period” result stated earlier in Theorem 1.1 ((b)).
In [Young1999], Young gave an extension of the results from [Young1998] that applies in a more abstract setting, giving existence of an invariant measure that is absolutely continuous with respect to some reference measure (not necessarily Lebesgue). She also provided a condition on the height of the tower that guarantees a polynomial upper bound for the decay of correlations. The corresponding polynomial lower bound (showing that Young’s bound is optimal) was obtained by [Sarig2002] and [Gouëzel2004].
The flexibility in the reference measure makes Young’s result suitable for studying existence, uniqueness and ergodic properties of equilibrium measures other than SRB measures (although this was not done in [Young1999]). In particular, this is used in the proof of Statement 2 of Theorem 2.3 below; we discuss such questions more in Sects. 3, 4.
Just as the Hénon maps can be studied as a “small” twodimensional extension of the unimodal maps, Theorems 2.1 and 2.2 can be applied to more general ‘strongly dissipative’ maps that are obtained as ‘small’ twodimensional extensions of onedimensional maps; this is carried out in [Wang and Young2001]; [Wang and Young2008].
Aside from such strongly dissipative maps, Young towers have been constructed for some partially hyperbolic maps where the center direction is nonuniformly contracting ([Castro2004]) or expanding ([Alves and Pinheiro2010]; [Alves and Li2015]); the latter papers are built on earlier results for nonuniformly expanding maps where one does not need to worry about the stable direction ([Alves et al.2005]; [Gouëzel2006]). In both cases existence (and uniqueness) of an SRB measure was proved first ([Bonatti and Viana2000]; [Alves et al.2000]) via other methods closer to the pushforward geometric approach that we discuss in Sect. 6, so the achievement of the tower construction was to establish exponential decay of correlations and the CLT. These results only cover the SRB measure and do not consider more general equilibrium states.
2.2.4 Thermodynamics of Young Diffeomorphisms for the Geometric $ t$Potential
Let $ f$ be a $ C^{1+\alpha}$ Young diffeomorphism of a compact smooth manifold $ M$. Consider the set $ \Lambda$ with hyperbolic product structure. Let $ \Lambda_i^s$ be the collections of $ s$sets and $ R_i$ the corresponding inducing times. Set
$\displaystyle \begin{eqnarray*} Y=\bigcup_{k\ge 0}\,f^k(\Lambda). \end{eqnarray*}$ 
This is a forward invariant set for $ f$. For every $ y\in Y$ the tangent space at $ y$ admits an invariant splitting $ T_yM=E^s(y)\oplus E^u(y)$ into stable and unstable subspaces. Thus we can consider the geometric $ t$potential $ \varphi_t(y)$ which is well defined for $ y\in Y$ and is a Borel (but not necessarily continuous) function for every $ t\in\mathbb{R}$. We consider the class $ \mathcal{M}(f,Y)$ of all invariant measures $ \mu$ supported on $ Y$, i.e., for which $ \mu(Y)=1$. It follows that $ \mu(\Lambda)> 0$, so that $ \mu$ ‘charges’ the base of the Young tower. Further, given a number $ h> 0$, we denote by $ \mathcal{M}(f,Y, h)$ the class of invariant measures $ \mu\in\mathcal{M}(f,Y)$ for which $ h_\mu(f)> h$.
The following result describes existence, uniqueness, and ergodic properties of equilibrium measures. Given $ n> 0$, denote by
$\displaystyle \begin{eqnarray*} S_n:=\text{Card}\{\Lambda_i^s\colon R_i=n\}. \end{eqnarray*}$ 
Theorem 2.3.
(see [Pesin et al.2016b]; [Melbourne and Terhesiu2014]) Assume that the Young tower satisfies:

(1)
for all large $ n$
$\displaystyle \begin{eqnarray*}\label{growth} S_n\le e^{hn}, \end{eqnarray*}$ (2.4) where $ 0< h< h_{\mu_1}(f)$ is a constant and $ \mu_1$ is the SRB measure for $ f$;

(2)
the set $ \bigcup_{i\ge 1}(\bar{\Lambda}_i{\setminus}\Lambda_i)$ supports no invariant measure that gives positive weight to any open set. ^{22}^{22} This condition is stronger than the corresponding condition ( 2.3 ).
Then there is $ t_0< 0$ such that for $ t_0\le t< 1$ there exists a measure $ \mu_t$ which is a unique equilibrium measure for $ \varphi_t$ among all liftable measures (see the remark below). If in addition, the tower satisfies the arithmetic condition,^{23}^{23}This requirement should be added to Theorem 4.5, Statement 2 of Theorem 4.7 and Statement 3 of Theorem 7.1 in [Pesin et al.2016b]. then $ (f,\mu_t)$ is Bernoulli, has exponential decay of correlations and satisfies the CLT with respect to a class of potential functions which contains all Hölder continuous functions on $ Y$.
Remark.

1.
The requirement ( 2.4 ) means that the number of $ s$sets in the base of the tower can grow exponentially but with rate slower than the metric entropy of the SRB measure. This is a strong requirement on the Young tower, but it is known to hold in some examples, see Sect. 2.3 below.
 2.

3.
We stress that the measures $ \mu_t$ are equilibrium measures within the class of measures that can be lifted to the tower: recall that an invariant measure $ \mu$ supported on $ Y$ is called liftable if there is a measure $ \nu$ supported on $ \Lambda$ and invariant under the induced map $ F$ such that the number
$\displaystyle \begin{eqnarray*}\label{qnu} Q_\nu=\int_\Lambda R\, d\nu \end{eqnarray*}$ (2.5) is finite, and for any measurable set $ E\subset Y$,
$\displaystyle \begin{eqnarray*}\label{lift} \mu(E)={\mathcal L}(\nu)(E):=\frac{1}{Q_\nu}\sum_{i\ge 0}\sum_{k=0}^{R_i1}\nu(f^{k}(E)\cap \Lambda_i^s). \end{eqnarray*}$ (2.6) In particular, $ \mu_t={\mathcal{L}}(\nu_t)$ for some measure $ \nu_t$ which is an equilibrium (and indeed, Gibbs) measure for the induced map $ F$.
Under the condition 2.4 every measure with entropy $ {> }h$ is liftable. In general, it is shown in [ [Zweimüller2005] ] that if $ R\in L^1(Y,\mu)$ then $ \mu$ is liftable. In particular, if the return time $ R$ is the first return time to the base of the tower, then every measure that charges the base of the tower is liftable.

4.
The proof of exponential decay of correlations and the CLT is based on showing the exponential tails property of the measure $ \nu_t$ ^{24}^{24} See ( 2.2 ) where one should replace the leaf volume with the measure $ \nu_t$. (see [ [Pesin et al.2016b] ] , Theorem 4.5) and then applying results from [ [Melbourne and Terhesiu2014] ] . ^{25}^{25} In [ [Melbourne and Terhesiu2014] ] the authors considered only expanding maps and Young towers with polynomial tails, however, their results can easily be extended to invertible maps and Young towers with exponential tails.

5.
For a $ C^{1+\alpha}$ diffeomorphism $ f$ there may exist several Young towers with bases $ \Lambda_k$, $ k=1,\dots,m$, such that the corresponding sets $ Y_k$ are disjoint. For each $ k$, Theorem 2.3 gives a number $ t_{0k}< 0$ and for every $ t_{0k}< t< 1$ the equilibrium measure $ \mu_{tk}$ for the geometric potential $ \varphi_t$. This measure is unique within the class of measures $ \mu$ for which $ \mu(Y_k)=1$ and $ h_\mu(f)> h$ where $ 0< h< h_{\mu_1}(f)$. ^{26}^{26} Note that both $ h$ and $ h_{\mu_1}(f)$ do not depend on $ k$. Setting $ t_0=\max_{1\le k\le m}t_{0k}$, for every $ t_0< t< 1$ we obtain the measure $ \mu_t$ such that $ \mu_tY_k=\mu_{tk}$. If for every measure $ \mu$ with $ h_\mu(f)> h$, we have that $ \mu(Y_k)> 0$ for some $ 1\le k\le m$, then the measure $ \mu_t$ is the unique equilibrium measure for $ \varphi_t$ within the class of invariant measures with large entropy. This is the case in the two examples described in Sect. 2.3 .

6.
It is known that $ t=1$ can be a phase transition, that is the pressure function $ P(t)$ is not differentiable and there are more than one equilibrium measures for $ \varphi_1$. However, it is not known whether phase transitions can occur for $ t< t_0$.

7.
Theorem 2.3 is a corollary of a more general result establishing thermodynamics for maps admitting inducing schemes of hyperbolic type, see Theorem 4.1 .
2.3 Examples of Young Diffeomorphisms
We describe two examples of Young diffeomorphisms for which Theorem 2.3 applies.
2.3.1 A Hénonlike Diffeomorphism at the First Bifurcation
The first example is Hénonlike diffeomorphisms of the plane at the first bifurcation parameter. For parameters $ a,b$ consider the Hénon map $ f_{a,b}$ given by (2.1). It is shown in [Bedford and Smillie2004], [Bedford et al.2006], [Cao et al.2008] that for each $ 0< b\ll 1$ there exists a uniquely defined parameter $ a^*=a^*(b)$ such that the nonwandering set for $ f_{a,b}$ is a uniformly hyperbolic horseshoe for $ a > a^*$ and the parameter $ a^*$ is the first parameter value for which a homoclinic tangency between certain stable and unstable manifolds appears.
Theorem 2.4.
([Senti and Takahasi2013]; [Senti and Takahasi2016], Theorem A) For any bounded open interval $ I\subset(1,+\infty)$ there exists $ 0< b_0\ll 1$ such that if $ 0\le b< b_0$ then

(1)
the map $ f_{a^*(b), b}$ is a Young diffeomorphism;

(2)
there exists a unique equilibrium measure for the geometric $ t$potential and for all $ t\in I$.
2.3.2 The Katok Map
We describe the Katok map ([Katok1979]) (see also [Barreira and Pesin2013]), which can be thought of as an invertible and twodimensional analogue of the Manneville–Pomeau map. Consider the automorphism of the 2torus given by the matrix $ T=( \begin{array}{ll} 2 &\quad 1\\ 1 &\quad 1\end{array})$ and then choose $ 0< \alpha< 1$ and a function $ \psi:[0,1]\mapsto[0,1]$ satisfying:

•
$ \psi$ is of class $ C^\infty$ except at zero;

•
$ \psi(u)=1$ for $ u\ge r_0$ and some $ 0< r_0< 1$;

•
$ \psi'(u)> 0$ for every $ 0< u< r_0$;

•
$ \psi(u)=(ur_0)^\alpha$ for $ 0\le u\le\frac{r_0}{2}$.
Let $ D_r=\{(s_1,s_2): {s_1}^2+{s_2}^2\le r^2\}$ where $ (s_1,s_2)$ is the coordinate system obtained from the eigendirections of $ T$. Consider the system of differential equations in $ D_{r_0}$
$\displaystyle \begin{eqnarray*}\label{batata10} \dot{s}_1= s_1\log\lambda,\quad \dot{s}_2=s_2\log\lambda, \end{eqnarray*}$  (2.7) 
where $ \lambda> 1$ is the eigenvalue of $ T$. Observe that $ T$ is the time1 map of the flow generated by the system of equations (2.7).
We slow down trajectories of (2.7) by perturbing it in $ D_{r_0}$ as follows:
$\displaystyle \begin{eqnarray*} \dot{s}_1=s_1\psi({s_1}^2+{s_2}^2)\log\lambda, \quad \dot{s}_2= s_2\psi({s_1}^2+{s_2}^2)\log\lambda. \end{eqnarray*}$ 
This generates a local flow, whose time1 map we denote by $ g$. The choices of $ \psi$ and $ r_0$ guarantee that the domain of $ g$ contains $ D_{r_0}$. Furthermore, $ g$ is of class $ C^\infty$ in $ D_{r_0}$ except at the origin and it coincides with $ T$ in some neighborhood of the boundary $ \partial D_{r_0}$. Therefore, the map
$\displaystyle \begin{eqnarray*} G(x)=\begin{cases} T(x) & \text{if} \quad x\in\mathbb{T}^2{\setminus} D_{r_0},\\ g(x) & \text{if} \quad x\in D_{r_0} \end{cases} \end{eqnarray*}$ 
defines a homeomorphism of the torus, which is a $ C^\infty$ diffeomorphism everywhere except at the origin.
The map $ G$ preserves the probability measure $ d\nu=\kappa_0^{1}\kappa\,dm$ where $ m$ is the area and the density $ \kappa$ is defined by
$\displaystyle \begin{eqnarray*} \kappa(s_1,s_2):=\begin{cases} (\psi({s_1}^2+{s_2}^2))^{1} &\text{if}\,\, (s_1,s_2)\in D_{r_0},\\ 1 & \text{otherwise} \end{cases} \end{eqnarray*}$ 
and
$\displaystyle \begin{eqnarray*} \kappa_0:=\int_{\mathbb{T}^2}\kappa\,dm. \end{eqnarray*}$ 
We further perturb the map $ G$ by a coordinate change $ \phi$ in $ \mathbb{T}^2$ to obtain an areapreserving $ C^\infty$ diffeomorphism. To achieve this, define a map $ \phi$ in $ D_{r_0}$ by the formula
$\displaystyle \begin{eqnarray*}\label{mapshi} \phi(s_1,s_2):=\frac{1}{\sqrt{\kappa_0({s_1}^2+{s_2}^2)}} \bigg(\int_0^{{s_1}^2+{s_2}^2}\frac{du}{\psi(u)}\bigg)^{1/2} (s_1,s_2) \end{eqnarray*}$  (2.8) 
and set $ \phi=\text{Id}$ in $ \mathbb{T}^2{\setminus} D_{r_0}$. Clearly, $ \phi$ is a homeomorphism and is a $ C^\infty$ diffeomorphism outside the origin. One can show that $ \phi$ transfers the measure $ \nu$ into the area and that the map $ f=\phi\,\circ\, G\,\circ\,\phi^{1}$ is a $ C^\infty$ diffeomorphism. This is the Katok map ([Katok1979]; [Barreira and Pesin2013]). One can show that the map $ f$ has nonzero Lyapunov exponents almost everywhere.^{27}^{27}However, there are trajectories with zero Lyapunov exponents, for example the origin is a neutral fixed point.
Theorem 2.5.
(see [Pesin et al.2016a]) The following statements hold:

(1)
the Katok map $ f$ is a Young diffeomorphism; moreover,

there are finitely many disjoint sets $ \Lambda_k$ that are bases of Young towers for which the corresponding sets $ Y_k$ cover the whole torus except for the origin;

every invariant measure $ \mu$ except for the Dirac measure at the origin $ \delta_0$ can be lifted to one of the towers.


(2)
For any $ t_0< 0$ one can find a small $ r_0=r_0(t_0)$ such that if the construction is carried out with this value of $ r_0$, then for every $ t_0< t< 1$

there exists a unique equilibrium ergodic measure $ \mu_t$ associated to the geometric potential $ \varphi_t$;

$ (f,\mu_t)$ has exponential decay of correlations and satisfies the CLT with respect to a class of functions which includes all Hölder continuous functions on the torus;

the pressure function $ P_t$ is real analytic on $ (t_0, 1)$.


(3)
For $ t=1$ there exist two equilibrium measures associated to $ \varphi_1$, namely the Dirac measure at the origin $ \delta_0$ and the Lebesgue measure.

(4)
For $ t> 1$, $ \delta_0$ is the unique equilibrium measure associated to $ \varphi_t$.
3 Markov Models for Nonuniformly Hyperbolic Maps II: Countable State Markov Shifts
The thermodynamic formalism for SFTs rested on the Ruelle’s version of the Perron–Frobenius theorem for finitestate topological Markov chains. For the class of twostep potential functions $ \varphi(x) = \varphi(x_0,x_1)$, which includes the zero potential $ \varphi=0$, the extension of this theory to countablestate Markov shifts dates back to work of [VereJones1962]; [VereJones1967], [Gurevič1969]; [Gurevič1970]; [Gurevič1984], and [Gurevich and Savchenko1998]; we discuss this in Sect. 3.1. For more general potential functions a sufficiently complete picture is primarily due to Sarig, and we discuss these in Sect. 3.2.
3.1 Recurrence Properties for Random Walks
Recall the form of Theorem 1.3 on existence of a unique MME for SFTs:

(1)
the largest eigenvalue $ \lambda$ of the transition matrix $ A$ determines the topological entropy;

(2)
the right eigenvector $ v = (v_i)$ for $ \lambda$ determines a Markov chain whose transition probabilities are given by a stochastic matrix $ P_{ij} = A_{ij} \frac {v_j}{\lambda v_i}$;

(3)
$ P$ has a unique stationary vector $ \pi$ (which can be written in terms of left and right eigenvectors for $ (A,\lambda)$), which determines a Markov measure that is the unique MME.
In the countablestate setting, existence of eigenvectors and stationary vectors is a more subtle question (although once these are found, the proof of uniqueness goes through just as in the finitestate case). The general story is wellillustrated by just considering the last step above: suppose we are given a stochastic matrix $ P_{ij}$ with countably many entries. This corresponds to a directed graph $ G$ with countably many vertices, whose edges are given weights as follows: the weight of the edge from $ i$ to $ j$ is $ P_{ij}$. Then one can consider the Markov chain described by $ P$ as a random walk on $ G$.
Existence of a stationary vector $ \pi=(\pi_i)$ with $ \pi P=\pi$ is determined by the recurrence properties of the shift [VereJones1962]; [VereJones1967]. Suppose we start our random walk at a vertex $ a$; one can show that the probability that we return to $ a$ infinitely many times is either 0 or 1. If the probability of returning infinitely many times is 1, then the walk is recurrent. Recurrence is necessary in order to have a stationary probability vector $ \pi$, but it is not sufficient; one must distinguish between the case when our expected return time is finite (positive recurrence) and when it is infinite (null recurrence). If the walk is positive recurrent then there is a stationary probability vector $ \pi$; if it is null recurrent then one can still find a vector $ \pi$ such that $ \pi P=\pi$, but one has $ \sum_i\pi_i=\infty$, so $ \pi$ cannot be normalized to a probability vector.
In fact, the trichotomy between transience, null recurrence, and positive recurrence is the key to generalizing all of Theorem 1.3 to the countablestate case [Pesin2014]. The recurrence conditions can be formulated in terms of the number of loops in the graph $ G$. Fixing a vertex $ a$, let $ Z_n^*$ be the number of simple loops of length $ n$ based at $ a$ (first returns to $ a$) and $ Z_n$ be the number of all loops of length $ n$ based at $ a$ (including loops which return more than once).^{28}^{28}In the next section when we consider nonzero potentials, we will have to count the loops with weights coming from the potential.

(1)
The supremum of the metric entropies is equal to the Gurevich entropy $ h_G :=\lim\frac1n\log Z_n$ (the limit exists if the graph is aperiodic; otherwise one should take the upper limit).

(2)
The shift $ \Sigma_A$ is recurrent if $ \sum_n e^{nh_G} Z_n = \infty$, and transient if the sum is finite.^{29}^{29}For some intuition behind this definition, it may be helpful to consider again a countablestate random walk: writing $ \mathbb{P}(n)$ for the probability of returning to the original vertex at time $ n$, we recall that by the Borel–Cantelli lemma, the walk is recurrent (infinitely many returns a.s.) if $ \sum_n \mathbb{P}(n)=\infty$, and transient (finitely many returns a.s.) if the sum is finite. The eigenvectors $ h$ and $ v$ for $ (A,\lambda)$ exist if and only if $ \Sigma_A$ is recurrent.

(3)
Among recurrent shifts, one must distinguish between positive recurrence, when $ \sum_n n e^{nh_G} Z_n^* < \infty$, and null recurrence, when the sum diverges. Writing $ \pi_i = h_i v_i$, one has $ \sum \pi_i < \infty$ if $ \Sigma_A$ is positive recurrent (hence, $ \pi$ can be normalized), and $ \sum \pi_i = \infty$ if it is null recurrent. One can also characterize positive recurrent shifts as those for which $ e^{nh_G} Z_n$ is bounded away from 0 and $ \infty$, which immediately implies divergence of the sum $ \sum_n e^{nh_G} Z_n$, while null recurrent shifts are those for which $ \varliminf_n e^{nh_G} Z_n = 0$ but the sum still diverges.
It is instructive to note that once a distinguished vertex $ a$ is fixed as the starting point of the loops, one can view the first return map to $ [a]$ as a Young tower, and then the summability condition in positive recurrence is equivalent to the condition that the tails of the tower are integrable, which was the existence criterion in Theorem 2.1.
3.2 Nonzero Potentials
In discussing the extension to nonzero potentials on countablestate topological Markov chains, we will follow the notation, terminology, and results of [Sarig1999]; [Sarig2001b]; [Sarig2001a], although the contributions of [Gurevič1984], [Gurevich and Savchenko1998], [Mauldin and Urbański1996]; [Mauldin and Urbański2001], [Aaronson and Denker2001], and of [Fiebig et al.2002] should also be mentioned. Sarig adapted transience, null recurrent, and positive recurrence for nonzero potential functions. The summability criterion for positive recurrence is exactly as above, except that now $ Z_n$ represents the total weight of all loops of length $ n$ and $ Z_n^*$ represents the total weight of simple loops of length $ n$ where weight is computed with respect to the potential function; more precisely
$\displaystyle \begin{eqnarray*} Z_n=Z_n(\varphi,a)=\sum_{\sigma^n(x)=x}\exp(\Phi_n(x))1_{[a]}(x) \end{eqnarray*}$ 
and
$\displaystyle \begin{eqnarray*} Z_n^*=Z_n^*(\varphi,a)=\sum_{\sigma^n(x)=x}\exp(\Phi_n(x))1_{[\varphi_a=n]}(x), \end{eqnarray*}$ 
where $ \Phi_n(x)=\sum_{k=0}^{n1}\varphi(f^kx)$. Furthermore, the Gurevich entropy $ h_G(\sigma)$ is replaced with the GurevichSarig pressure $ P_{GS}(\sigma,\varphi)$, which is the exponential growth rate of $ Z_n$, i.e.,
$\displaystyle \begin{eqnarray*} P_{GS}(\sigma,\varphi)=\lim_{n\to\infty}\frac1n \log Z_n. \end{eqnarray*}$ 
For Markov shifts with finite topological entropy, [Buzzi and Sarig2003] proved that an equilibrium measure exists if and only if the shift is positive recurrent. A good summary of the theory can be found in [Sarig2015]. For our purposes the main result is the following.
Theorem 3.1.
Let $ \Sigma$ be a topologically mixing countablestate Markov shift with finite topological entropy, and let $ \varphi\colon X\to\mathbb{R}$ be a Hölder continuous^{30}^{30}In fact Theorem 3.1 holds for the more general class of potentials with summable variations, but Hölder continuity is needed for the statistical properties mentioned below. function such that $ P_{GS}(\varphi)< \infty$. Then $ \varphi$ is positive recurrent if and only if there are $ \lambda> 0$, a positive continuous function $ h$, and a conservative measure $ \nu$ (i.e., a measure that allows no nontrivial wandering sets) which is finite and positive on cylinders, such that $ \mathcal{L}_\varphi h=\lambda h$, $ \mathcal{L}_\varphi^*\nu=\lambda\nu$, and $ \int h\,d\nu=1$. In this case the following are true.

(1)
$ P_{GS}(\varphi) = \log \lambda$, and $ d\mu = h\,d\nu$ defines a $ \sigma$invariant measure.

(2)
If $ h(\mu)< \infty$, then $ \mu$ is the unique equilibrium state for $ \varphi$.

(3)
For every cylinder $ [w]\subset \Sigma$, we have $ \lambda^{n} \nu[w]^{1} \mathcal{L}_\varphi^n \boldsymbol{1}_{[w]} \to h$ uniformly on compact subsets.
The statistical properties of $ \mu$ depend on the rate of convergence in the last item of Theorem 3.1, which in turn depends on how quickly $ Z_n^* e^{nP_{GS}}$ goes to 0. If it goes to zero with polynomial rate then the corresponding tower (obtained by inducing on a single state) has polynomial tails, and the equilibrium state has polynomial decay of correlations. If it goes to zero with exponential speed—that is, if $ Z_n^*$ has smaller exponential growth rate than $ Z_n$—then the tower has exponential tails and correlations decay exponentially. In this case the shift is called strong positive recurrent; see [Cyr and Sarig2009] for a summary of the results in this case.
3.3 CountableState Markov Partitions for Smooth Systems
Using Pesin theory, Sarig recently carried out a version of the construction of Markov partitions for nonuniformly hyperbolic diffeomorphisms in two dimensions. Recall that for a uniformly hyperbolic diffeomorphism $ f\colon M\to M$, one obtains an SFT $ \Sigma$ and a coding map $ \pi\colon \Sigma\to M$ such that

•
$ \pi$ is Hölder continuous and has $ f\circ \pi = \pi \circ \sigma$;

•
$ \pi$ is onto and is 1–1 on a residual set $ \Sigma' \subset \Sigma$ that has full measure for every equilibrium state of a Hölder potential on $ \Sigma$.
In nonuniform hyperbolicity one must replace the SFT with a countablestate Markov shift, and also weaken some of the conclusions.
Theorem 3.2.
[Sarig2013] Let $ M$ be a compact smooth surface and $ f\colon M\to M$ a $ C^{1+\alpha}$ diffeomorphism of positive topological entropy. Fix a threshold $ 0< \chi < {h}_{{\rm top}}(f)$. Then there is a countablestate topological Markov shift $ \Sigma_\chi$ and a coding map $ \pi_\chi \colon \Sigma_\chi \to M$ such that

•
$ \pi_\chi$ is Hölder continuous and has $ f\circ \pi_\chi = \pi_\chi \circ \sigma$;

•
if $ \mu$ is an ergodic $ f$invariant measure on $ M$ with $ h_\mu(f) > \chi$, then $ \mu(\pi(\Sigma_\chi))=1$, and moreover there is an ergodic $ \sigma$invariant measure $ {\hat{\mu}}$ on $ \Sigma_\chi$ such that $ (\pi_\chi)_* {\hat{\mu}} = \mu$ and $ h_{{\hat{\mu}}}(\sigma) = h_\mu(f)$.
Observe that Theorem 3.2 echoes our recurring theme that in nonuniform hyperbolicity, to obtain ‘good’ hyperbolictype results one often needs to ignore a ‘smallentropy’ part of the system. In fact the key property of the threshold $ \chi$ is that by the Margulis–Ruelle inequality, any ergodic measure with $ h_\mu(f)> \chi$ must have positive Lyapunov exponent at least $ \chi$. Thus for a higherdimensional generalization of Theorem 3.2, one should expect that the natural condition would be on the Lyapunov exponents, rather than the entropy.
The analogous result to Theorem 3.2 for threedimensional flows was proved by [Lima and Sarig2014]. In both cases this can be used to deduce Bernoullicity up to finite rotations of ergodic positive entropy equilibrium states ([Sarig2011]; [Ledrappier et al.2016]). However, these general results do not give any information on the recurrence properties of the countable state shift, or the tail of the resulting tower, and in particular they do not provide a mechanism for verifying decay of correlations and the CLT. This is of no surprise, since at this level of generality, one should not expect to get exponential decay (or any other particular rate).
4 Markov Models for Nonuniformly Hyperbolic Maps III: Inducing Schemes of Hyperbolic Type
The study of SRB measures via Young towers generalizes to the study of equilibrium states via inducing schemes, which use the tower approach to model (a large part of) the system by a countablestate Markov shift, and then apply the thermodynamic results from Sect. 3. The concept of an inducing scheme in dynamics is quite broad and applies to systems which may be invertible or not, smooth or not differentiable. Every inducing scheme generates a symbolic representation by a tower which is well adapted to constructing equilibrium measures for an appropriate class of potential functions using the formalism of countable state Markov shifts. The projection of these measures from the tower are natural candidates for the equilibrium measures for the original system.
In order to use this symbolic approach to establish existence and to study equilibrium states, some care must be taken to deal with the liftability problem as only measures that can be lifted to the tower can be ‘seen’ by the tower.
One may consider inducing schemes of expanding type, or of hyperbolic type. The former were introduced in [Pesin and Senti2008] and apply to study thermodynamics of noninvertible maps (e.g., nonuniformly expanding maps) while the latter were introduced in [Pesin et al.2016b] and are used to model invertible maps (e.g,, nonuniformly hyperbolic maps). In this paper we only consider inducing schemes of hyperbolic types and we follow [Pesin et al.2016b].
Let $ f\colon X\to X$ be a homeomorphism of a compact metric space $ (X, d)$. We assume that $ f$ has finite topological entropy $ {h}_{{\rm top}}(f)< \infty$. An inducing scheme of hyperbolic type for $ f$ consists of a countable collection of disjoint Borel sets $ S=\{J\}$ and a positive integervalued function $ \tau\colon S\to\mathbb{N}$; the inducing domain of the inducing scheme $ \{S,\tau\}$ is $ W=\bigcup_{J\in S}J$, and the inducing time $ \tau\colon X\to\mathbb{N}$ is defined by $ \tau(x)=\tau(J)$ for $ x\in J$ and $ \tau(x)=0$ otherwise. We require several conditions.
(I1) For any $ J\in S$ we have $ f^{\tau(J)}(J)\subset W$ and $ \bigcup_{J\in S}f^{\tau(J)}(J)=W$. Moreover, $ f^{\tau(J)}J$ can be extended to a homeomorphism of a neighborhood of $ J$.
This condition allows one to define the induced map $ F\colon W\to W$ by setting $ FJ:=f^{\tau(J)}J$ for each $ J\in S$. If $ \tau$ is the first return time to $ W$, then all images $ f^{\tau(J)}(J)$ are disjoint. However, in general the sets $ f^{\tau(J)}(J)$ corresponding to different $ J\in S$ may overlap. In this case the map $ F$ may not be invertible.
(I2) For every biinfinite sequence $ \underline{a}=(a_n)_{n\in\mathbb{Z}}\in S^{\mathbb{Z}}$ there exists a unique sequence $ \underline{x}=\underline{x}(\underline{a})=(x_n=x_n(\underline{a}))_{n\in\mathbb{Z}}$ such that (a) $ x_n\in\overline{J_{a_n}}\quad\mbox{ and }\quad f^{\tau(J_{a_n})}(x_n)=x_{n+1}$; (b) if $ x_n(\underline{a})=x_n(\underline{b})$ for all $ n\le 0$ then $ \underline{a}=\underline{b}$. This condition allows one to define the coding map $ \pi\colon S^{\mathbb{Z}}\to \bigcup\overline{J}$ by $ \pi(\underline{a}):=x_0(\underline{a})$. Within the full shift $ \sigma\colon S^\mathbb{Z}\to S^\mathbb{Z}$ we consider the set
$\displaystyle \begin{eqnarray*} \check{S}:=\{\underline{a}\in S^\mathbb{Z} \mid x_n(\underline{a})\in J_{a_n}\text{ for all } n\in\mathbb{Z}\}. \end{eqnarray*}$ 
(I3) The set $ S^\mathbb{Z}{\setminus}\check S$ supports no (ergodic) $ \sigma$invariant measure which gives positive weight to any open subset. This condition is designed to ensure that every equilibrium measure for the shift is supported on $ \check{S}$ and its projection by $ \pi$ is thus supported on $ W$ and is $ F$invariant. This projection is a natural candidate for the equilibrium measure for $ F$. Set $ Y= \{f^k(x)\mid x\in W, 0\le k\le \tau(x)1\}$. Note that $ Y$ is forward invariant under $ f$. This can be thought of as the region of $ X$ that is ‘swept out’ as $ W$ is carried forward under the dynamics of $ f$; in particular, it contains all trajectories that intersect the base $ W$. Let $ \varphi$ be a potential function. Existence of an equilibrium measure for $ \varphi$ is obtained by first studying the problem for the induced system $ (F,W)$ and the induced potential $ \overline{\varphi}\colon W\to\mathbb{R}$ defined by (1.2). The study of existence and uniqueness of equilibrium measures for the induced system $ (F,W)$ is carried out by conjugating the induced system to the twosided full shift over the countable alphabet $ S$. This requires that the potential function $ \Phi:={\bar{\varphi}}\circ \pi$ be well defined on $ S^\mathbb{Z}$. To this end we require that (P1) the induced potential $ \overline{\varphi}$ can be extended by continuity to a function on $ \bar{J}$ for every $ J\in S$. Denote the potential induced by the normalized potential $ \varphiP_L(\varphi)$ by
$\displaystyle \begin{eqnarray*} \varphi^+:=\overline{\varphiP_L(\varphi)}=\overline{\varphi}P_L(\varphi)\tau \end{eqnarray*}$ 
Theorem 4.1.
(see [Pesin et al.2016b]) Let $ \{S,\tau\}$ be an inducing scheme of hyperbolic type satisfying Conditions (I1)–(I3) and $ \varphi$ a potential satisfying Condition (P1). Assume that • $ \Phi$ has strongly summable variations; • $ P_{GS}(\Phi)< \infty$ and $ P_{GS}(\Phi^+)< \infty$; • $ \sup_{a\in S^\mathbb{Z}}\,\Phi^+(a)< \infty$. Then (1) There exists a $ \sigma$invariant ergodic measure $ \nu_{\Phi^+}$ for $ \Phi^+$; (2) If $ h_{\nu_{\Phi^+}}(\sigma)< \infty$, then $ \nu_{\Phi^+}$ is the unique equilibrium measure for $ \Phi^+$; (3) If $ h_{\nu_{\Phi^+}}(\sigma)< \infty$, then the measure $ \nu_{\varphi^+}:=\pi_*\nu_{\Phi^+}$ is a unique $ F$invariant ergodic equilibrium measure for $ \varphi^+$; (4) If $ P_{GS}(\Phi^+)=0$ and $ Q_{\nu_{\varphi^+}}< \infty$, then $ \mu_\varphi={\mathcal L}(\nu_{\varphi^+})$ is the unique equilibrium ergodic measure in the class $ \mathcal{M}_L(f, Y)$ of liftable measures (see (2.5) and (2.6)).$\displaystyle \begin{eqnarray*} \nu_{\varphi^+}(\{x\in W: \tau(x)\ge n\})\le C\theta^n. \end{eqnarray*}$ 
Theorem 4.2.
(see [Pesin et al.2016b]) Under the conditions of Theorem 4.1 assume that • the induced function $ \overline\varphi$ on $ W$ is locally Hölder continuous; • the tower has exponential tails with respect to the measure $ \nu_{\varphi^+}$ that is there exist $ C> 0$ and $ 0< \theta< 1$ such that for all $ n> 0$,$\displaystyle \begin{eqnarray*} \nu_{\varphi^+}(\{x\in W: \tau(x)\ge n\})\le C\theta^n; \end{eqnarray*}$ 
(P2) there exist $ C> 0$ and $ 0< r< 1$ such that for any $ n\ge 1$
$\displaystyle \begin{eqnarray*} V_n(\phi):=V_n(\Phi)\le C r^n, \end{eqnarray*}$ 
$\displaystyle \begin{eqnarray*} V_n(\Phi) := \sup_{[b_{n+1}, \cdots, b_{n1}]}\sup_{\underline{a},\underline{a}'\in [b_{n+1}, \cdots, b_{n1}]}\{\Phi(\underline{a})\Phi(\underline{a}')\} \end{eqnarray*}$ 
$\displaystyle \begin{eqnarray*} \sum_{J\in S}\tau(J)\sup_{x\in J}\exp(\varphi^+(x)+\epsilon\tau(x))< \infty. \end{eqnarray*}$ 
Theorem 4.3.
(see [Pesin et al.2016b]) Let $ \{S,\tau\}$ be an inducing scheme of hyperbolic type satisfying Conditions (I1)–(I3). Assume that the potential function $ \varphi$ satisfies Conditions (P1)–(P4). Then (1) there exists a unique equilibrium measure $ \mu_\varphi$ for $ \varphi$ among all measures in $ \mathcal{M}_L(f, Y)$; the measure $ \mu_\varphi$ is ergodic; (2) if $ \nu_{\varphi^+}=\mathcal{L}^{1}(\mu_\varphi)$ has exponential tail and the tower satisfies the arithmetic condition, then $ (f,\mu_\varphi)$ has exponential decay of correlations and satisfies the CLT with respect to a class of functions whose corresponding induced functions on $ W$ (see (1.2)) are bounded locally Hölder continuous functions.5 CoarseGraining, Expansivity, and Specification
5.1 Uniform Expansivity and Specification
Let $ X$ be a compact metric space and $ f\colon X\to X$ a homeomorphism; given $ \varepsilon> 0$ and $ x\in X$, the set
$\displaystyle \begin{eqnarray*}\label{eqn:nespsset} \Gamma_\varepsilon(x) := \left\{y\in X \mid d(f^nx,f^ny) < \varepsilon \text{ for all } n\in \mathbb{Z} \right\} \end{eqnarray*}$  (5.1) 
contains all points whose trajectory stays within $ \varepsilon$ of the trajectory of $ x$ for all time. The map $ f$ is expansive if there is $ \varepsilon> 0$ such that $ \Gamma_\varepsilon(x) = \{x\}$ for every $ x\in X$; that is, if any two distinct trajectories eventually separate at scale $ \varepsilon$. Uniformly hyperbolic systems can easily be shown to be expansive, and expansivity is a sufficient condition for existence of an equilibrium measure for any continuous potential $ \varphi$; indeed, the standard proof of the variational principle ([Walters1982], Theorem 8.6) gives a construction of such a measure. The idea is that one “coarsegrains” the system at scale $ \varepsilon$ and builds a measure that is appropriately distributed over all trajectories that separate by $ \varepsilon$ within $ n$ iterates; sending $ n\to\infty$ and using expansivity one guarantees that this measure is an equilibrium state.
To show that this equilibrium state is unique, Bowen used the following specification property of uniformly hyperbolic systems: for every $ \varepsilon> 0$ there is $ \tau\in \mathbb{N}$ such that any collection of finitelength orbit segments can be $ \varepsilon$shadowed by a single orbit that takes $ \tau$ iterates to transition from one segment to the next. More precisely, if we associate $ (x,n) \in X\times N$ to the orbit segment $ x,f(x),\dots, f^{n1}(x)$ and write
$\displaystyle \begin{eqnarray*} B_n(x,\varepsilon) = \{y\in X \mid d(f^kx,f^ky) \leq \varepsilon \text{ for all } 0\leq k < n\} \end{eqnarray*}$ 
for the Bowen ball of points that shadow $ (x,n)$ to within $ \varepsilon$ for those $ n$ iterates, then specification requires that for every $ (x_1,n_1),\dots, (x_k,n_k)$ there is $ y\in X$ such that $ y\in B_{n_1}(x_1,\varepsilon)$, then $ f^{n_1 + \tau}(y)\in B_{n_2}(x_2,\varepsilon)$, and in general
$\displaystyle \begin{eqnarray*}\label{eqn:spec} f^{\sum_{i=0}^{j1} (n_i + \tau)}(y) \in B_{n_j}(x,\varepsilon) \text{ for all } 1\leq j\leq k. \end{eqnarray*}$  (5.2) 
Mixing Axiom A systems satisfy specification; this is a consequence of the mixing property together with the shadowing lemma.
A continuous potential $ \varphi\colon X\to \mathbb{R}$ satisfies the Bowen property if there is $ K\in \mathbb{R}$ such that $ S_n\varphi(x)  S_n\varphi(y) < K$ whenever $ y\in B_n(x,\varepsilon)$, where $ S_n\varphi(x) = \sum_{j=0}^{n1} \varphi(f^jx)$. The following theorem summarizes the classical results due to Bowen on systems with specification [Bowen1974].^{32}^{32}In fact, Bowen required the slightly stronger property that the shadowing point $ y$ in (5.2) be periodic, but this is only necessary for the part of his results dealing with periodic orbits, which we omit here.
Theorem 5.1.
If $ (X,f)$ is an expansive system with specification and $ \varphi$ is a potential with the Bowen property, then there is a unique equilibrium measure $ \mu$. This includes the case when $ f\Lambda$ is topologically mixing and uniformly hyperbolic, and $ \varphi$ is Hölder continuous.
5.2 Nonuniform Expansivity and Specification
Various weaker versions of the specification property have been introduced in the literature. The one which is most relevant for our purposes first appeared in [Climenhaga and Thompson2012] for MMEs in the symbolic setting, and was developed in [Climenhaga and Thompson2013]; [Climenhaga and Thompson2014]; [Climenhaga and Thompson2016] to a version that applies to smooth maps and flows.
Given $ \varepsilon> 0$, consider the ‘nonexpansive set’ $ {\rm NE}(\varepsilon) = \{x\in X \mid \Gamma_\varepsilon(x) \neq \{x\}\}$, where $ \Gamma_\varepsilon(x)$ is as in (5.1). Note that $ (X,f)$ is expansive if and only if $ {\rm NE}(\varepsilon)=\emptyset$. The pressure of obstructions to expansivity is^{33}^{33}The idea of ignoring measures sitting on $ {\rm NE}(\varepsilon)$ was introduced earlier by [Buzzi and Fisher2013].
$\displaystyle \begin{eqnarray*}\label{eqn:pexp} {P}_{{\rm exp}}^bar{\perp}(\varphi) = \lim_{\varepsilon\to 0} \sup_{\mu \in \mathcal{M}^e(f)} \left\{h_\mu(f) + \int \varphi\,d\mu \mid \mu({\rm NE}(\varepsilon))=1\right\}. \end{eqnarray*}$  (5.3) 
In particular, expansive systems have $ {P}_{{\rm exp}}^bar{\perp}(\varphi)=\infty$. It follows from the results in [Climenhaga and Thompson2016] that the condition $ {P}_{{\rm exp}}^bar{\perp}(\varphi) < P(\varphi)$ is enough for existence of an equilibrium measure. For uniqueness, we need to weaken the notion of specification. The idea behind this is to only require the specification property (5.2) to hold for a certain ‘good’ collection of orbit segments $ \mathcal{G} \subset X\times \mathbb{N}$ (and similarly for the Bowen property). One must also require $ \mathcal{G}$ to be large enough, which in this case means that there are collections of orbit segments $ \mathcal{P},\mathcal{S}\subset X\times \mathbb{N}$ that have small pressure compared to the whole system, but are sufficient to generate $ X\times \mathbb{N}$ from $ \mathcal{G}$ by adding prefixes from $ \mathcal{P}$ and suffixes from $ \mathcal{S}$.^{34}^{34}One can sum up the situation by saying that “the pressure of obstructions to specification is small”. A related idea of studying shift spaces for which “the entropy of constraints is small” appeared in [Buzzi2005], where Buzzi studied shifts of quasifinite type. For more details on the relationship between the two notions, see [Climenhaga2015], especially Theorem 1.4.
Let us make this more precise. A decomposition of the space of orbit segments consists of $ \mathcal{P}, \mathcal{G}, \mathcal{S} \subset X\times \mathbb{N}$ and functions $ p,g,s\colon X\times \mathbb{N}\to \mathbb{N}\cup \{0\}$ such that $ (p+g+s)(x,n) = n$ and
$\displaystyle \begin{align*} (x,p(x,n))&\in \mathcal{P},\\ (f^{p(x,n)}(x),g(x,n)) &\in \mathcal{G}, \\ (f^{(p+g)(x,n)}(x),s(x,n))& \in \mathcal{S}. \end{align*}$ 
The following is ([Climenhaga and Thompson2016], Theorem 5.5).
Theorem 5.2.
Let $ X$ be a compact metric space, $ f\colon X\to X$ a homeomorphism, and $ \varphi\colon X\to \mathbb{R}$ a continuous function. Suppose that $ {P}_{{\rm exp}}^bar{\perp}(\varphi)< P(\varphi)$ and $ X\times \mathbb{N}$ admits a decomposition $ (\mathcal{P},\mathcal{G},\mathcal{S})$ with the following properties:

(I) $ \mathcal{G}$ has specification at every scale;

(II) $ \varphi$ has the Bowen property on $ \mathcal{G}$;

(III) $ P(\mathcal{P}\cup \mathcal{S},\varphi) < P(\varphi)$.
Then $ f$ has a unique equilibrium measure $ \mu_\varphi$.
We describe two examples for which Theorem 5.2 applies. One of them is the Mañé example [Mañé1978], which was introduced as an example of a robustly transitive diffeomorphism that is not Anosov. This “derived from Anosov” example is obtained by taking a 3dimensional hyperbolic toral automorphism with one unstable direction and performing a pitchfork bifurcation in $ E^{cs}$ near the fixed point so that $ E^c$ becomes weakly expanding in that neighborhood. One obtains a partially hyperbolic diffeomorphism with a splitting $ E^s\oplus E^c\oplus E^u$ such that $ E^c$ “contracts on average” with respect to the Lebesgue measure; this falls under the results in [Castro2004] mentioned above, and its inverse map (for which $ E^c$ “expands on average”) is covered by [Alves and Pinheiro2010], [Alves and Li2015].
Now given any Hölder continuous potential $ \varphi\colon \mathbb{T}^3\to \mathbb{R}$, it is shown in [Climenhaga et al.2015] that there is a $ C^1$open class of Mañé examples for which this potential has a unique equilibrium state. In particular, when $ f$ is $ C^2$, there is an interval $ (t_0,t_1) \supset [0,1]$ such that the geometric $ t$potential $ t\log\det(dfE^{cu})$ has a unique equilibrium state for every $ t\in (t_0,t_1)$, and $ \varphi_1$ is the unique SRB measure.
A related second example is the Bonatti–Viana example introduced in [Bonatti and Viana2000]. Here one takes a 4dimensional hyperbolic toral automorphism with $ \dim E^s=\dim E^u=2$, and makes two perturbations, one in the $ E^s$direction and another one in the $ E^u$direction. The first perturbation creates a pitchfork bifurcation as above in $ E^s$ and then “mixes up” the two directions in $ E^s$ so that there is no invariant subbundle of $ E^s$; the second perturbation does a similar thing to $ E^u$. One obtains a map with a dominated splitting $ E^{cs}\oplus E^{cu}$ that has no uniformly hyperbolic subbundles.
The same approach as above works for the Bonatti–Viana examples, which have a dominated splitting but are not partially hyperbolic; see [Climenhaga et al.2015]. In this case the presence of nonuniformity in both the stable and unstable directions makes tower constructions more difficult, and no GibbsMarkovYoung structure has been built for these examples. Earlier results on thermodynamics of these examples (and the Mañé examples) were given in [Buzzi et al.2012], [Buzzi and Fisher2013], which proved existence of a unique MME. These results make strong use of the semiconjugacy between the examples and the original toral automorphisms, and in particular do not generalize to equilibrium states corresponding to nonzero potentials.
Finally, the flow version of Theorem 5.2 can be applied to geodesic flow in nonpositive curvature. Geodesic flow in negative curvature is one of the classical examples of an Anosov flow [Anosov1969], and in particular it has unique equilibrium states with strong statistical properties.^{35}^{35}Although the issue of decay of correlations is more subtle because it is a flow, not a map; see [Dolgopyat1998], among others. If $ M$ is a smooth rank 1 manifold with nonpositive curvature, then its geodesic flow is nonuniformly hyperbolic. Bernoullicity of the regular component of the Liouville measure was shown by [Pesin1977]. It was shown by [Knieper1998] that there is a unique measure of maximal entropy; his proof uses powerful geometric tools and does not seem to generalize to nonzero potentials. Using nonuniform specification, Knieper’s result can be extended to the geometric $ t$potential for $ t\approx 0$, and when $ \dim M = 2$, it works for any $ t\in (\infty,1)$, showing that the pressure function is differentiable on this interval and we recover the same picture as for Manneville–Pomeau [Burns et al.2016].
In each of the above examples, the basic idea is as follows: one identifies a “bad set” $ B\subset X$ with the properties that

(1)
the system has uniformly hyperbolic properties outside of $ B$;

(2)
trajectories spending all (or almost all) of their time in $ B$ carry small pressure relative to the whole system.
For the Mañé and Bonatti–Viana examples, $ B$ is the neighborhood where the perturbation is carried out; for the geodesic flow, $ B$ is a small neighborhood of the singular set.
Given an orbit segment $ (x,n)$, let $ G(x,n) = \frac 1n\#\{0\leq k< n \mid f^kx \notin B\}$ be the proportion of time that the orbit segment spends in the “good” part of the system.^{36}^{36}For flows one should make the obvious modifications, replacing $ \mathbb{N}$ by $ [0,\infty)$ and cardinality with Lebesgue measure. A decomposition of the space of orbit segments $ X\times \mathbb{N}$ is obtained by fixing a threshold $ \gamma> 0$ and taking
$\displaystyle \begin{align*} \mathcal{P} &= \mathcal{S} = \{(x,n) \mid G(x,n) < \gamma\}, \\ \mathcal{G} &= \{(x,n) \mid G(x,k) \geq \gamma, G(f^kx,nk)\geq \gamma \text{ for all } 0\leq k \leq n \}. \end{align*}$ 
Indeed, given any $ (x,n)\in X\times \mathbb{N}$, one can take $ p$ and $ s$ to be maximal such that $ (x,p) \in \mathcal{P}$ and $ (f^{ns}x,s)\in \mathcal{S}$, and use additivity of $ G$ along orbit segments to argue that $ (f^px,nps)\in \mathcal{G}$, which yields a decomposition $ X\times \mathbb{N} = \mathcal{P}\mathcal{G}\mathcal{S}$. Then one makes the following arguments to apply Theorem 5.2.

•
Assumption (1) above leads to hyperbolic estimates along trajectories in $ \mathcal{G}$, which can be used to prove specification for $ \mathcal{G}$ (condition (I) in Theorem 5.2) and the Bowen property on $ \mathcal{G}$ for Hölder continuous potentials (condition (II)).

•
Assumption (2) gives the pressure estimate $ P(\mathcal{P}\cup \mathcal{S},\varphi)< P(\varphi)$ from (III).

•
The expansion estimates along $ \mathcal{G}$ and the pressure estimates on $ \mathcal{P}$ and $ \mathcal{S}$ also yield $ {P}_{{\rm exp}}^bar{\perp}(\varphi)< P(\varphi)$.
This approach establishes existence and uniqueness, and yields some statistical properties such as large deviations estimates. However, it does not yet give stronger statistical results such as a rate of decay of correlations, or the CLT. In the setting when $ X$ is a shift space with nonuniform specification, results along these lines have recently been established [Climenhaga2015] by using conditions (I)–(III) (or closely related ones) to build a tower with exponential tails, but it is not yet clear how this result extends to the smooth setting.
6 The Geometric Approach
6.1 Geometric Construction of SRB Measures
6.1.1 Idea of Construction
Having discussed constructions of SRB and equilibrium measures via Markov dynamics (SFTs and Young towers) and via coarsegraining (expansivity and specification), we turn our attention now to a third approach, which is in some sense more natural and more simpleminded. The first two approaches addressed not just existence but also questions of uniqueness and statistical properties; the price to be paid for these stronger results is that the construction of a tower (or even the verification of nonuniform specification) may be difficult in many examples. The approach that we now describe is best suited to prove existence, rather than uniqueness or statistical properties, but has the advantage that it seems easier to verify.
We start by discussing SRB measure, which for dissipative systems plays the role of Lebesgue measure in conservative systems and is the most natural measure from the physical point of view. So in trying to find an SRB measure, it is natural to start with Lebesgue measure itself; while it may not be invariant, we will follow the standard Bogolubov–Krylov procedure of taking a noninvariant measure $ m$, average it under the dynamics to produce the sequence
$\displaystyle \begin{eqnarray*}\label{evol} \mu_n = \frac1n\sum_{k=0}^{n1}f_*^km, \end{eqnarray*}$  (6.1) 
and pass to a weak*convergent subsequence $ \mu_{n_j} \to \mu$; then $ \mu$ is $ f$invariant. If we do this starting with Lebesgue measure as our reference measure $ m$, then it is reasonable to expect that the limiting measure will have something to do with Lebesgue, and may even be an SRB measure.^{37}^{37}In general, though, the measure $ \mu$ may be quite trivial—just consider the point mass at an attracting fixed point.
At an intuitive level, this approach is consistent with Viana’s conjecture ([Viana1998]) that nonzero Lyapunov exponents imply existence of an SRB, since this should be exactly the setting in which the iterates of Lebesgue spread out along the unstable manifolds and converge in average to a measure that is absolutely continuous in the unstable direction. Now we describe how it can be made precise.
6.1.2 Uniform Geometry: Uniform and Partial Hyperbolicity
In the uniformly hyperbolic setting, this approach can be carried out as follows. Let $ \mathcal{R}$ be the set of all standard pairs $ (W,\rho)$, where $ W$ is a small piece of unstable manifold and $ \rho\colon W\to (0,\infty)$ is integrable with respect to $ m_W$, the leaf volume on $ W$. Let $ \mathcal{M}^{\rm ac}$ be the set of all (not necessarily invariant) probability measures $ \mu$ on the manifold $ M$ that can be expressed as
$\displaystyle \begin{eqnarray*}\label{eqn:Mac} \mu(E)=\int_\mathcal{R}\int_{W\cap E} \rho(x)\,dm_W(x)\,d\zeta(W,\rho) \end{eqnarray*}$  (6.2) 
for some measure $ \zeta$ on $ \mathcal{R}$; in other words, $ \mu$ admits a decomposition (in the sense of Fubini’s theorem) along unstable local manifolds, in which conditional measures are leafvolumes. Then one can show that $ \mathcal{M}^{\rm ac}\cap\mathcal{M}(f)$ is precisely the set of SRB measures for $ f$. Moreover, Lebesgue measure $ m$ is in $ \mathcal{M}^{\rm ac}$ and thus since images of unstable manifolds can be decomposed into small pieces of unstable manifolds, we have $ f_*^km\in\mathcal{M}^{\rm ac}$ for all $ k\in \mathbb{N}$, so the averaged measures given by (6.1) are in $ \mathcal{M}^{\rm ac}$ as well.
In order to pass to the limit and obtain $ \mu\in\mathcal{M}^{\rm ac}$ one needs a little more control. Fixing $ K> 0$, let $ \mathcal{R}_K$ be the set of all standard pairs $ (W,\rho)$ such that $ W$ has size at least $ 1/K$, and $ \rho\colon W\to [1/K,K]$ is Hölder continuous with constant $ K$. Then defining $ \mathcal{M}^{\rm ac}_K$ using (6.2) with $ \mathcal{R}_K$ in place of $ \mathcal{R}$, one can show that $ \mathcal{M}^{\rm ac}_K$ is weak* compact and is $ f_*$invariant for large enough $ K$. This is basically a consequence of the Arzelà–Ascoli theorem and the fact that $ f$ uniformly expands unstable manifolds; in particular it relies strongly on the uniform hyperbolicity assumption. Then $ \mu_n\in \mathcal{M}^{\rm ac}_K$ for all $ n$ by invariance, and by compactness, $ \mu=\lim \mu_{n_j}\in\mathcal{M}^{\rm ac}_K\cap\mathcal{M}(f)$ is an SRB measure. Thus we have the following statement.
Theorem 6.1.
Let $ \Lambda$ be a hyperbolic attractor for $ f$ and assume that $ f\Lambda$ is topologically transitive. If the reference measure $ m$ is the restriction of the Lebesgue measure to a neighborhood of $ \Lambda$, then the sequence of measures (6.1) converges and the limit measure is the unique SRB measure for $ f$.
Now consider the setting where $ f$ is partially hyperbolic, i.e., for every point $ x$ the tangent space splits $ T_xM=E^s(x)\oplus E^c(x)\oplus E^u(x)$ into stable, central, and unstable subspaces respectively with uniform contraction along $ E^s(x)$, uniform expansion along $ E^u(x)$, and possible contractions and/or expansions along $ E^c(x)$ with rates which are weaker than the corresponding rates along $ E^s(x)$ and $ E^u(x)$.
In the situation where the centreunstable direction $ E^{cu}$ is only nonuniformly expanding more care must be taken with the above approach because $ \mathcal{M}^{\rm ac}_K$ may no longer be $ f_*$invariant: even if $ W$ is a “sufficiently large” local unstable manifold, its image $ f(W)$ may be smaller than $ 1/K$, and similarly the Hölder constant of the density $ \rho$ can get worse under the action of $ f_*$ if $ W$ is contracted by $ f$.
The solution is to use hyperbolic times, which were introduced by [Alves2000]. Roughly speaking, a time $ n$ is hyperbolic for a point $ x$ if $ df^kE^u(f^{nk}x)$ is uniformly expanding for every $ 0\leq k\leq n$. If $ W$ is a local unstable manifold around $ x$ and $ n$ is a hyperbolic time for $ x$, then $ f^n(W)$ contains a large neighborhood of $ f^n(x)$, and the density $ \rho$ behaves well under $ f_*^n$. Thus from the point of view of the construction above, the key property of hyperbolic times is that if $ H_n$ is the set of all points $ x$ for which $ n$ is a hyperbolic time, then the measures
$\displaystyle \begin{eqnarray*}\label{eqn:nun} \nu_n := \frac 1n \sum_{k=0}^{n1} f_*^k(mH_k) \end{eqnarray*}$  (6.3) 
are all contained in $ \mathcal{M}^{\rm ac}_K$ (after rescaling to obtain a probability measure). As long as $ \nu_n \not\to 0$, one concludes that $ \mu=\lim\nu_{n_k}\in\mathcal{M}(f)$ has some ergodic component in $ \mathcal{M}^{\rm ac}_K$, which must be an SRB measure. To get the lower bound on the total weight of $ \nu_n$, one needs a lower bound on $ \frac1n\sum_{k=0}^{n1}m(H_k)$, which can be obtained using Pliss’ lemma as long as a positive Lebesgue measure set of points have positive Lyapunov exponents along $ E^{cu}$.
One can also construct the SRB measure by beginning “within the attractor”: instead of using Lebesgue measure on $ M$ as the starting point for the sequence (6.1), one can let $ m^u$ be leaf volume along a local unstable manifold and then consider the sequence
$\displaystyle \begin{eqnarray*}\label{evol1} \nu_n(x)=\frac1n\sum_{k=0}^{n1}\,f_*^km^u(x). \end{eqnarray*}$  (6.4) 
If the attractor $ \Lambda$ is uniformly hyperbolic, the sequence of measures 6.4 converges and the limit measure is a unique SRB measure for $ f$. In the partially hyperbolic setting, it was shown by [Pesin and Sinaĭ1983] that every limit measure $ \nu$ of the sequence of measures 6.4 is a $ u$measure on $ \Lambda$: that is, the conditional measures it generates on local unstable manifolds are absolutely continuous with respect to the leafvolume on these manifolds. What prevents $ \nu$ from being an SRB measure in general is that the Lyapunov exponents in the central direction can be positive or zero.
Several results are available that establish existence (and in some cases uniqueness) of SRB measures under some additional requirements on the action of the system along the central direction $ E^c$ or centralunstable direction $ E^{cu}$. For example the case of systems with mostly contracting central directions was carried out in [Bonatti and Viana2000], [Burns et al.2008] and with mostly expanding central directions in [Alves et al.2000]. A more general case of systems whose central direction is weakly expanding was studied in [Alves et al.2014].
In these settings one at least has a dominated splitting, which gives the system various uniform geometric properties, even if the dynamics is nonuniform. To extend this approach to settings where the geometry is nonuniform (no dominated splitting, stable and unstable directions vary discontinuously) some new tools are needed. An important observation (which holds in the uniform case as well) is that for many purposes we can replace $ V^u(x)$ itself with a local manifold passing through $ x$ that is $ C^1$close to $ V^u(x)$. Such a manifold is called admissible, and in the next section will develop the machinery of standard pairs, the class of measures $ \mathcal{M}^{\rm ac}$, and the sequences of measures (6.4) using admissible manifolds in place of unstable manifolds.
6.1.3 Nonuniform Geometry: Effective Hyperbolicity
The difficulties encountered in the geometrically nonuniform setting can be overcome by the machinery of ‘effective hyperbolicity’ from [Climenhaga and Pesin2016], [Climenhaga et al.2016]. This approach has the advantage that the requirements on the system appear weaker, and much closer to the Viana conjecture. The drawback of this approach is that it is currently out of reach to use it to establish exponential (or even polynomial) decay of correlations and the CLT.
Let $ U$ be a neighborhood of the attractor $ \Lambda$ for a $ C^{1+\epsilon}$ diffeomorphism, and consider a forward invariant set $ S\subset U$ on which there are two measurable cone families $ K^s(x)=K^s(x,E^s(x),\theta)$ and $ K^u(x)=K^u(x,E^s(x),\theta)$ that are

•
invariant: $ \overline{Df(K^u(x))}\subset K^u(fx)$ and $ \overline{Df^{1}(K^s(fx))} \subset K^s(x)$;

•
transverse: $ T_x M=E^s(x) \oplus E^u(x)$.
Given $ x\in S$, consider the expansion and contraction coefficients
$\displaystyle \begin{align*} \lambda^u(x) &:= \inf\{\log \Df(v)\ \mid v\in K^u(x), \v\ = 1\}, \\ \lambda^s(x) &:= \sup\{\log \Df(v)\ \mid v\in K^s(x), \v\ = 1\}, \end{align*}$ 
and the defect from domination
$\displaystyle \begin{eqnarray*} \Delta(x) := \max(0,\tfrac 1\epsilon(\lambda^s(x)  \lambda^u(x)); \end{eqnarray*}$ 
note that $ \Delta(x)=0$ whenever $ \lambda^u(x)> \lambda^s(x)$, so the defect only comes into play when the stable cone expands more than the unstable cone. The coefficient of effective hyperbolicity is
$\displaystyle \begin{eqnarray*}\label{eqn:lambdaspsx} \lambda(x) := \min(\lambda^u(x)  \Delta(x),  \lambda^s(x)); \end{eqnarray*}$  (6.5) 
thus $ \lambda(x)> 0$ whenever the system “behaves hyperbolically” at $ x$, while $ \lambda(x)\leq 0$ when one of the following happens:

•
some stable vectors expand (so $ \lambda^s(x)< 0$); or

•
some unstable vectors contract (so $ \lambda^u(x)< 0$); or

•
the defect from domination is greater than the expansion in the unstable cone (so $ \lambda^u(x)\Delta(x) < 0$).
Let $ \alpha(x)$ be the angle between the cones $ K^s(x)$ and $ K^u(x)$, and given a threshold $ \bar{\alpha}> 0$, let
$\displaystyle \begin{eqnarray*} \rho_{\bar{\alpha}}(x) := \varlimsup_{n\to\infty} \frac 1n \#\{0\leq k< n \mid \alpha(f^kx) < \bar{\alpha}\} \end{eqnarray*}$ 
be the asymptotic upper bound on how often the angle drops below that threshold. Notice that in the case of a dominated splitting, $ \alpha(x)$ is uniformly bounded away from 0, so there is $ \bar{\alpha}> 0$ with $ \rho_{\bar{\alpha}}(x) = 0$ for every $ x$; however, for a system with nonuniform geometry it may be the case that every $ \bar{\alpha}> 0$ has points with $ \rho_{\bar{\alpha}}(x) > 0$.
With the above notions in mind, we consider the following set of points:
$\displaystyle \begin{eqnarray*} S' = \left\{x\in S \mid \varliminf_{n\to\infty} \frac 1n \sum_{k=0}^{n1} \lambda(f^kx) > 0 \text{ and } \lim_{\bar{\alpha}\to 0} \rho_{\bar{\alpha}}(x) = 0 \right\}. \end{eqnarray*}$ 
Thus $ S'$ contains points for which the average asymptotic rate of effective hyperbolicity is positive, and for which the asymptotic frequency with which the angle between the cones degenerates can be made arbitrarily small. Then we have the following result, which is a step towards Viana’s conjecture.
Theorem 6.2.
([Climenhaga et al.2016]) If $ S'$ has positive volume then $ f$ has an SRB measure.
6.1.4 Idea of Proof
The construction of an SRB measure in the setting of Theorem 6.2 follows the same averaging idea as in Sects. 6.1.1, 6.1.2: if $ \mu_n$ is the sequence of measures given by (6.1), then one wish to show that a uniformly large part of $ \mu_n$ lies in the set of “uniformly absolutely continuous” measures $ \mathcal{M}^{\rm ac}_K$.
In this more general setting the definition of $ \mathcal{M}^{\rm ac}_K$ is significantly more involved. Broadly speaking, in the definition of $ \mathcal{R}$ we must replace unstable manifolds $ W$ with admissible manifolds; an admissible manifold $ W$ through a point $ x\in S$ is a smooth submanifold such that $ T_xW \subset K^u(x)$ and $ W$ is the graph of a function $ \psi\colon B^u(r)\subset E^u(x) \to E^s(x)$, such that $ D\psi$ is uniformly bounded and is uniformly Hölder continuous. The Hölder constant for $ D\psi$ can be thought of as the “curvature” of $ W$.
When the geometry is uniform as in the previous setting, the image of an admissible manifold $ W$ is itself admissible; this is essentially the classical Hadamard–Perron theorem. In the more general case this is no longer true; although $ f^n(W)$ contains an admissible manifold, its size and curvature may vary with time $ n$, with the size becoming arbitrarily small and the curvature arbitrarily large. In this setting a version of the Hadamard–Perron theorem was proved in [Climenhaga and Pesin2016] that gives good bounds on $ f^n(W)$ when $ n$ is an effective hyperbolic time for $ x\in W$; that is, when
$\displaystyle \begin{eqnarray*} \sum_{j=k}^{n1}\lambda(f^jx)\ge\chi(nk) \end{eqnarray*}$ 
for every $ 0\le k < n$, where $ \chi> 0$ is a fixed rate of effective hyperbolicity.
The set of effective hyperbolic times is a subset of the set of hyperbolic times; the extra conditions in the definition of effective hyperbolic time guarantee that we can control the dynamics of $ f$ on the manifold itself, not just the dynamics of $ df$ on the tangent bundle. In the uniform geometry setting from earlier, this extension came for free for hyperbolic times.
With the notion of effective hyperbolic times, the approach outlined in Sects. 6.1.1, 6.1.2 can be carried out. One must add some more conditions to the collection $ \mathcal{R}$; most notably, one must fix $ n\in \mathbb{N}$ and then consider only admissible manifolds $ W$ for which
$\displaystyle \begin{eqnarray*} d(f^{k}(x),f^{k}(y))\le Ce^{\chi k}d(x,y)\quad \text{ for all} \quad 0\le k\le n \quad \text{ and} \quad x,y\in W, \end{eqnarray*}$ 
and then define $ \mathcal{M}^{\rm ac}_{K,n}$ using only this class of admissible manifolds. In addition to size of $ W$ and regularity of $ \rho$, the constant $ K$ must also be chosen to govern the curvature of $ W$, but we omit details here. The point is that the set $ \mathcal{M}^{\rm ac}_{K,n}$ is compact, but not $ f_*$invariant, and so the proof of Theorem 6.2 can be completed via the following steps.

(1)
Writing $ H_n$ for the set of points with $ n$ as an effective hyperbolic time, use Pliss’ lemma and the assumption that $ S'$ has positive volume to show that $ H_n$ has positive Lebesgue measure on average.

(2)
Use the effective Hadamard–Perron theorem from [Climenhaga and Pesin2016] to show that $ \nu_n:=\frac 1n\sum_{k=0}^{n1}f_*^k\in\mathcal{M}^{\rm ac}_{K,n}$, and use the bound from the previous step to get a lower bound on the total weight of $ \nu_n$.

(3)
Write $ \mu_n=\nu_n+\zeta_n$ and argue from general principles that if $ \mu_{n_k}\to\mu$, then $ \mu$ has an ergodic component in $ \mathcal{M}^{\rm ac}$; moreover, this ergodic component is hyperbolic and $ f$invariant, so it is an SRB measure.
6.2 Constructing Equilibrium Measures
A natural next step is to extend the above procedure to study general equilibrium states, and not just SRB measures. The direct analogue of the previous section has not yet been fully developed, and we describe instead a related approach that is also based on studying how densities transform under the dynamics.
First consider the case of a piecewise expanding interval map, and the question of finding an SRB measure. In this case there is no stable direction, and so we do not have to keep track of the “shape” of unstable manifolds, or admissible manifolds; indeed, a local unstable manifold is just a small piece of the interval, and an SRB measure is just an invariant measure that is absolutely continuous with respect to Lebesgue. Thus the entire problem is reduced to the following question: given a (not necessarily invariant) absolutely continuous measure $ \mu\ll m$, how is the density function of its image $ f_*\mu$ related to the density function of $ \mu$? One ends up defining a transfer operator $ \mathcal{L}$ with the property that if $ d\mu = h\,dx$, then $ d(f_*\mu)=(\mathcal{L} h)\,dx$. Questions about the existence of an absolutely continuous invariant measure, and its statistical properties, can be reduced to questions about the transfer operator $ \mathcal{L}$.
The central issue in studying $ \mathcal{L}$ is the problem of finding a Banach space $ \mathcal{B}$ (of functions) on which $ \mathcal{L}$ acts “with good spectral properties”. Generally speaking this means that 1 is a simple eigenvalue of $ \mathcal{L}$ (so there is a unique fixed point $ h=\mathcal{L} h$, which corresponds to the unique absolutely continuous invariant measure) and the rest of the spectrum of $ \mathcal{L}$ lies inside a disc of radius $ r< 1$, which guarantees exponential decay of correlations and other statistical properties.
For piecewise expanding interval maps, this was accomplished by [Lasota and Yorke1974], and the approach can be adapted to equilibrium states for other potential functions by considering a transfer operator that depends on the potential in an appropriate way. A thorough account of this approach is given in [Baladi2000].
The mechanism that drives this approach is that the expansion of the dynamics acts to “smooth out” the density function; irregularities in the function $ h$ are made milder by passing to $ \mathcal{L} h$. (The precise meaning of this statement depends on the particular choice of Banach space $ \mathcal{B}$, and is encoded by the Lasota–Yorke inequality, which we do not pursue further here.) But this means that one runs into problems when going from expanding interval maps to hyperbolic diffeomorphisms, where there is a nontrivial stable direction; the contracting dynamics in the stable direction make irregularities in the function worse!
In the classical approach to uniformly hyperbolic systems, this was dealt with by passing to a symbolic coding by an SFT (as described after Theorem 1.1) and then replacing the twosided SFT $ \Sigma \subset A^\mathbb{Z}$ by its onesided version $ \Sigma^+ \subset A^\mathbb{N}$. As described after Theorem 1.3, the transfer operator $ \mathcal{L}$ has an eigenfunction $ h\in C(\Sigma^+)$, and its dual $ \mathcal{L}^*$ has an eigenmeasure $ \nu \in C(\Sigma^+)^*$; combining them gives the equilibrium state $ d\mu = h\,d\nu$. Note that positive indices of an element of $ \Sigma$ code the future of a trajectory, while negative indices code the past, and so dynamically, passing from $ \Sigma$ to $ \Sigma^+$ can be interpreted as “forgetting the past”. Geometrically, this means that we conflate points lying on the same local stable manifold; taking a quotient in the stable direction eliminates the problem described in the previous paragraph, where contraction in the stable direction exacerbates irregularities in the density function.
More recent work has shown that this problem can be addressed without the use of symbolic dynamics. The key is to consider a Banach space $ \mathcal{B}$ whose elements are not functions, but are rather objects that behave like functions in the unstable direction, and like distributions in the stable direction. For SRB measures, this was carried out in [Blank et al.2002], [Gouëzel and Liverani2006], [Baladi and Tsujii2007]. A further generalization to equilibrium states for other potential functions was given in [Gui and Li2008]; as with expanding interval maps, this requires working with a transfer operator $ \mathcal{L}$ that depends on the potential. Moreover, instead of distributions along the stable direction, one must consider a certain class of “generalized differential forms”. We refer the reader to ([Gui and Li2008], Sect. 7) for a comparison of this approach to equilibrium states and other related approaches, including the technique of “standard pairs”.
It remains an open problem to extend this approach to the nonuniformly hyperbolic setting.
6.3 Ergodic Properties
An important open question is to study uniqueness and statistical properties of the SRB measure produced in Theorem 6.2, or of any equilibrium states that may be produced by an analogous result for other potentials. One potential approach is to study the standard pairs $ (W,\rho)$ and derive statistical properties via coupling techniques, as was done by Chernov and Dolgopyat in another setting ([Chernov and Dolgopyat2009]).^{38}^{38}Coupling techniques are also at the heart of Young’s tower results for subexponential mixing rates ([Young1999]). One might also hope to adapt the functional analytic approach from Sect. 6.2 into the nonuniformly hyperbolic setting and obtain statistical properties this way. For now, though, we only mention results on Bernoullicity and hyperbolic product structure.
6.3.1 SRB Measures
By a result of [Ledrappier1984], a hyperbolic SRB measure has at most countably many ergodic components and every hyperbolic SRB measure is Bernoulli up to a finite period. It follows that there may exist at most countably many ergodic SRB measures on $ \Lambda$. One way to ensure uniqueness of SRB measures is to show that its every ergodic component is open (mod 0) in the topology of $ \Lambda$ and that $ f\Lambda$ is topologically transitive.
6.3.2 Equilibrium Measures
Let $ \mu$ be a hyperbolic ergodic measure for a $ C^{1+\alpha}$ diffeomorphism $ f$. Given $ \ell> 0$, consider the regular set $ \Gamma_\ell$, which consists of points $ x\in \Gamma$ whose local stable $ V^s(x)$ and unstable $ V^u(x)$ manifolds have size at least $ 1/\ell$. For $ x\in \Gamma_\ell$ and some sufficiently small $ r> 0$ let $ R_\ell(x,r)=\bigcup_{y\in A^u(x)}V^s(y)$ be a rectangle at $ x$, where $ A^u(x)$ is the set of points of intersection of $ V^u(x)$ with local stable manifolds $ V^s(z)$ for $ z\in\Gamma_\ell\cap B(x,r)$. We denote by

•
$ \pi\colon V^u(z_1)\to V^u(z_2)$ with $ z_1,z_2\in R_\ell(x,r)\cap \Gamma_\ell$ the holonomy map generated by local stable manifolds;

•
$ \mu^u(z)$ the conditional measure generated by $ \mu$ on local unstable manifolds $ V^u(z)$.
Say that $ \mu$ has a direct product structure if the holonomy map is absolutely continuous with the Jacobian uniformly bounded away from 0 and $ \infty$ on $ R_\ell(x,r)$.
Conjecture 6.3.
If $ \mu$ is a hyperbolic ergodic equilibrium measure for the geometric $ t$potential for a $ C^{1+\alpha}$ diffeomorphism $ f$, then $ \mu$ has a direct product structure.
If true, this would imply that $ \mu$ has some “nice” ergodic properties; for example, it has at most countably many ergodic components. Similar results have recently been established (using the symbolic approach) for twodimensional diffeomorphisms and threedimensional flows ([Sarig2011]; [Ledrappier et al.2016]).
We conclude with a conjecture on the relationship between effective hyperbolicity (from Sect. 6.1) and decay of correlations. Suppose that $ \Lambda$ is an attractor with trapping region $ U$, and that we have invariant measurable transverse cone families defined Lebesguea.e. in $ U$, with the property that there is $ \chi> 0$ for which
$\displaystyle \begin{eqnarray*} S'=\left\{x\in U\mid\varliminf_{n\to\infty}\frac1n\sum_{k=0}^{n1}\lambda(f^kx)> \chi\text{ and } \lim_{\bar{\alpha}\to 0}\rho_{\bar{\alpha}}(x)=0\right\} \end{eqnarray*}$ 
has full Lebesgue measure in $ U$. Consider for each $ N\in \mathbb{N}$ the set
$\displaystyle \begin{eqnarray*} X_N=\left\{x\in U\mid\sum_{k=0}^{n1}\lambda(f^kx)> \chi n\text{ for all } n> N\right\}, \end{eqnarray*}$ 
and note that the assumption on $ S'$ guarantees that $ m(U{\setminus} X_N)\to 0$ as $ N\to\infty$.
Conjecture 6.4.
If $ m(U{\setminus} X_N)$ decays exponentially in $ N$, then the SRB measure $ \mu$ produced by Theorem 6.2 has exponential decay of correlations.
Some support for this conjecture is provided by the fact that the analogous result for partially hyperbolic attractors with mostly expanding central direction was proved in [Alves and Li2015].
Acknowledgements
We are grateful to the anonymous referee for a very careful reading and for many thoughtful and insightful comments that have improved this paper. We are also grateful to ICERM and the Erwin Schrödinger Institute, where parts of this work were completed, for their hospitality.References
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