Research ExpositionArnold Mathematical Journal

Received: 4 February 2016 / Accepted: 20 July 2016

Building Thermodynamics for Non-uniformly Hyperbolic Maps

Vaughn Climenhaga climenha@math.uh.edu
Department of Mathematics, University of Houston, Houston TX 77204 USA, Yakov Pesin pesin@math.psu.edu
Department of Mathematics, Pennsylvania State University, University Park PA 16802 USA

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 non-uniform hyperbolicity. The first of these approaches involves Markov models such as Young towers, countable-state Markov shifts, and inducing schemes. The other two are less fully developed but have seen significant progress in the last few years: these involve coarse-graining techniques (expansivity and specification) and geometric arguments involving push-forward of densities on admissible manifolds.

Keywords

Mathematics Subject Classification

1. 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], [Sinai1968], [Sinai1972], 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 $ \ph\colon X\to \RR$, which we will refer to as a potential . Denote by $ \MMM(f)$ the space of all $ f$-invariant Borel probability measures on X. Given $ \mu\in \MMM(f)$, the free energy of the system with respect to $ \mu$ is \begin{eqnarray*} E_\mu(\ph) := -\left(h_\mu(f) + \int_X \ph\,d\mu\right), \end{eqnarray*} where $ h_\mu(f)$ is the Kolmogorov-Sinai (measure-theoretic) entropy of $ (X,f,\mu)$. Optimizing over all invariant measures gives the topological pressure \begin{eqnarray*} P(\ph) := -\inf_{\mu\in \MMM(f)} E_\mu(\ph) = \sup_{\mu\in \MMM(f)} \left(h_\mu(f) + \int_X \ph\,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 $ \MMM^e(f) \subset \MMM(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 $ \eps> 0$ and $ n\in \NN$, a set $ E\subset X$ is $ (n,\eps)$- separated if points in $ E$ can be distinguished at a scale $ \eps$ 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\eps$. Then one has

\begin{eqnarray}\label{eqn:Pspsgrowth} P(\ph)=\lim_{\eps\to 0}\limsup_{n\to\infty}\frac1n\log\sup_{\substack{E\subset X \\ (n,\eps)\text{-sep.}}}\sum_{x\in E} e^{S_n\ph(x)}, \end{eqnarray}

(1.1)

where

\begin{eqnarray}\label{ind:poten} S_n\ph(x):=\sum_{k=0}^{n-1}\ph(f^kx). \end{eqnarray}

(1.2)

The sum in (1.1) is a partition sum that quantifies "weighted orbit complexity at spatial scale $ \eps$ and time scale $ n$"; $ P(\ph)$ represents the growth rate of this complexity as time increases. In the particular case $ \ph=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 non-uniformly 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, coarse-graining techniques (based on expansivity and specification) in Sect. 5, and push-forward (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 two-sided 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 $ \ZZ$ 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
(2) Decay of correlations. Let $ \mathcal{H}$ be a class of square-integrable test functions $ X\to\RR$ and define \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$ \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$ \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 \begin{eqnarray*} \mu\left\{x : \frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\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 non-uniformly 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], [Rey-Bellet 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 Then for any Hölder continuous potential $ \ph$, the following are true:

(1) Existence: there is an equilibrium measure $ \mu_\ph$.
(2) Uniqueness: $ \mu_\ph$ is the only equilibrium measure for $ \ph$.
(3) Ergodic and statistical properties:
1. (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^n|A,\mu_\ph)$ has the Bernoulli property;
2. (b) exponential decay of correlations: there are $ A,n$ as above such that $ (f^n|A,\mu_\ph)$ has EDC with respect to the class of Hölder continuous functions.
3. (c) the Central Limit Theorem: $ \mu_\ph$ satisfies the CLT with respect to the class of Hölder continuous functions.

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 $ 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 two-sided infinite sequences $ \omega=(\omega_i) \in \{1,\dots, p\}^\ZZ$ which are admissible with respect to $ A$; that is, for which $ a_{\omega_i\omega_{i+1}}=1$ for all $ i\in \ZZ$.

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 well-defined.
(2) $ R_i=\overline{\text{int}\,R_i}$ ⁴ 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 \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 2-torus. 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 [Sinai1968]. Furthermore, in [Sinai1972] 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 \ZZ$ (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 one-to-one 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 $ \ph=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 \NN$, 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 \begin{eqnarray*} \mu[\omega_1 \cdots \omega_n] = \pi_{\omega_1} P_{\omega_1\omega_2} \cdots P_{\omega_{n-1}\omega_n}. \end{eqnarray*}

Theorem 1.3 was adapted to non-zero 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 two-sided SFT $ \Sigma_A$ with its one-sided version $ \Sigma_A^+$, and define the transfer operator associated to $ \ph$ on $ C(\Sigma_A^+)$ by ⁵ \begin{eqnarray*} (\LLL_\ph f)(x) = \sum_{\sigma y=x} e^{\ph(y)} f(y). \end{eqnarray*}
(2) Show that $ \LLL_\ph$ 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 $ \LLL_\ph$, and an eigenmeasure $ \nu\in \MMM(\Sigma_A^+)$ for the dual $ \LLL_\ph^*$.
(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].

[]

Figure 1 The pressure function for a typical hyperbolic sets; b a hyperbolic attractor; c a non-uniformly hyperbolic map with a phase transition.

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 $ \ph_t(x):= -t\log \abs{\Jac(df|E^u(x))}$ for $ t\in\RR$. Since the subspaces $ E^u(x)$ depend Hölder continuously on $ x$, the potential $ \ph_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(\ph_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 $ \htop(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 two-dimensional case its root is the Hausdorff dimension of $ \Lambda\cap V^u(x)$ ⁶ 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 \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 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

\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 \ph_1\,d\mu$, and hence $ h_\mu(f)+\int\ph_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 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 $ \RRR$ of local unstable manifolds $ V^u$ and a measure $ \eta$ on $ \RRR$ such that $ \mu_1$ can be written as

\begin{eqnarray}\label{eqn:ac} \mu_1(E) = \int_\RRR \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 \begin{eqnarray*} G_\mu := \left\{x\in M \mid \frac 1n \sum_{k=0}^{n-1} \varphi(f^k(x)) \to \int\ph\,d\mu \text{ for all continuous } \ph\colon M\to \RR \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 $ \ph_1$.

1.3. Non-uniformly Hyperbolic Maps.

1.3.1. Definition of Non-uniform Hyperbolicity

1.3.2. Possibility of Phase Transitions and Non-hyperbolic Behavior.

A general theory of thermodynamic formalism for non-uniformly hyperbolic maps is far from being complete, although certain examples here are well-understood. They include one-dimensional maps, where the pressure function $ P(t)=P(\ph_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 non-differentiability 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 $ \ph_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 non-differentiable at $ t=1$, and $ \ph_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 The measure $ \mu_1$ is Bernoulli and decay of correlations is polynomial (in particular, subexponential).
Non-hyperbolic behavior for $ t> 1$: for every $ t\in (1,\infty)$, the unique equilibrium state for $ \ph_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 one-dimensional maps (e.g., unimodal and multimodal maps) and rather specific higher-dimensional examples (e.g., polynomial and rational maps and (piecewise) non-uniformly 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 Rivera-Letelier2007], [Pesin and Senti2008], [Bruin and Todd2008], [Bruin and Todd2009], [Dobbs2009], [Iommi and Todd2010], [Przytycki and Rivera-Letelier2011], [Li and Rivera-Letelier2014a], [Li and Rivera-Letelier2014b]), as well as the comprehensive and far-reaching 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 multi-dimensional non-uniformly 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 non-uniform 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$, \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 $ \MMM^e(f,\Gamma)\subset \MMM^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 $ \ph$ 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 $ \ph_t$ is not a continuous function. ¹⁰ Consider the hyperbolic pressure defined by using only hyperbolic measures:

\begin{eqnarray}\label{eqn:PG} P_\Gamma(\ph) := -\inf_{\mu\in \MMM^e(f,\Gamma)} E_\mu(\ph). \end{eqnarray}

(1.5)

Say that $ \mu_\ph$ is a hyperbolic equilibrium measure if $ -E_{\mu_\ph}(\ph)=P_\Gamma(\ph)$. For the Manneville–Pomeau example above, we have $ P_\Gamma(\ph_t)=P(\ph_t)$ for every $ t\in\RR$, and the equilibrium measure $ \mu_t$ is the unique hyperbolic equilibrium measure for every $ t\le 1$, ¹¹ 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 $ \MMM^e(f,\Gamma,h)$ of all measures in $ \MMM^e(f,\Gamma)$ whose entropies are greater than $ h$; restricting our attention to measures from this class gives the restricted pressure ¹²

\begin{eqnarray}\label{eqn:PGh} P_\Gamma^h(\ph) := -\inf_{\mu\in\MMM^e(f,\Gamma,h)} E_\mu(\ph). \end{eqnarray}

(1.6)

For the Manneville–Pomeau example, we have for every $ t\in \RR$, ¹³ \begin{eqnarray*} \lim_{h\to 0} P_\Gamma^h(\ph_t) = P_\Gamma(\ph_t) = P(\ph_t). \end{eqnarray*} In addition to the use of $ \mu_t^h$ to approximate non-hyperbolic 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 This concept originated in the work of Buzzi on piecewise invertible continuous maps of compact metric spaces [Buzzi1999], ¹⁵ but it is reasonable to consider it in other situations. ¹⁶

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 $ \MMM^e(f,\Gamma,\psi,p)$ of all measures in $ \MMM^e(f,\Gamma)$ for which $ -E_\mu(\psi)> p$. Optimizing $ E_\mu(\ph)$ 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(\ph)$ 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 non-uniformly hyperbolic setting is hopeless in general; we cannot expect to have finite Markov partitions. ¹⁷ 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 non-uniformly hyperbolic setting; this is discussed in Sect. 5. This is similar to the symbolic approach in that one uses a "coarse-graining" 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 specification-based 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 So far the geometric approach can be used to establish existence of SRB measures for uniformly hyperbolic and some non-uniformly 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 non-uniformly 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 non-uniformly hyperbolic systems.

2. Markov Models for Non-uniformly Hyperbolic Maps I: Young Diffeomorphisms

3. Markov Models for Non-uniformly Hyperbolic Maps II: Countable State Markov Shifts

4. Markov Models for Non-uniformly 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 countable-state 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 non-invertible maps (e.g., non-uniformly expanding maps) while the latter were introduced in [Pesin et al.2016b] and are used to model invertible maps (e.g,, non-uniformly 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 $ \htop(f)< \infty$. An inducing scheme of hyperbolic type for $ f$ consists of a countable collection of disjoint Borel sets $ S=\{J\}$ and a positive integer-valued 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 $ F|J:=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 bi-infinite 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
1. (a) $ x_n\in\overline{J_{a_n}}\quad\mbox{ and }\quad f^{\tau(J_{a_n})}(x_n)=x_{n+1}$;
2. (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^\ZZ\to S^\ZZ$ we consider the set \begin{eqnarray*} \check{S}:=\{\underline{a}\in S^\ZZ \mid x_n(\underline{a})\in J_{a_n}\text{ for all } n\in\mathbb{Z}\}. \end{eqnarray*} For any $ \underline{a}\in S^\mathbb{Z}{\setminus}\check{S}$ there exists $ n\in\mathbb{Z}$ such that $ \pi\circ\sigma^n(\underline{a})\in\overline{J_{a_n}}{\setminus} J_{a_n}$. In particular, if all $ J\in S$ are closed then $ S^\mathbb{Z}{\setminus}\check{S}=\emptyset$; however, this need not always be the case. It follows from ( I1 ) and ( I2 ) that the map $ \pi$ has the following properties:

(1) $ \pi$ is well defined, continuous and for all $ \underline{a}\in S^\mathbb{Z}$ one has $ \pi\circ\sigma(\underline{a})=f^{\tau(J)}\circ\pi(\underline{a})$ where $ J\in S$ is such that $ \pi(\underline{a})\in \bar{J}$;
(2) $ \pi$ is one-to-one on $ \check S$ and $ \pi(\check S)=W$;
(3) if $ \pi(\underline{a})=\pi(\underline{b})$ for some $ \underline{a}, \underline{b}\in\check{S}$ then $ a_n=b_n$ for all $ n\ge 0$.

Proving the existence and uniqueness of equilibrium measures requires some additional condition on the inducing scheme $ \{S,\tau\}$:

( 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 two-sided 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 $ \varphi-P_L(\varphi)$ by \begin{eqnarray*} \varphi^+:=\overline{\varphi-P_L(\varphi)}=\overline{\varphi}-P_L(\varphi)\tau \end{eqnarray*} and let $ \Phi^+:=\varphi^+\circ\pi$.

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) ).

The following result describes ergodic properties of equilibrium measures. assume that $ \nu_{\varphi^+}$ has exponential tails (see (2.2) ): there exist $ C> 0$ and $ 0< \theta< 1$ such that for all $ n> 0$, \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$, \begin{eqnarray*} \nu_{\varphi^+}(\{x\in W: \tau(x)\ge n\})\le C\theta^n; \end{eqnarray*} (compare to (2.2) );
the tower satisfies the arithmetic condition. ³¹

Then $ (f,\mu_\varphi)$ has exponential decay of correlations and satisfies the CLT with respect to the class of functions whose induced functions on $ W$ are bounded locally Hölder continuous.

We describe some verifiable conditions on the potential function $ \varphi$ under which the assumptions of Theorem 4.1 hold:

( P2 ) there exist $ C> 0$ and $ 0< r< 1$ such that for any $ n\ge 1$ \begin{eqnarray*} V_n(\phi):=V_n(\Phi)\le C r^n, \end{eqnarray*} where \begin{eqnarray*} V_n(\Phi) := \sup_{[b_{-n+1}, \cdots, b_{n-1}]}\sup_{\underline{a},\underline{a}'\in [b_{-n+1}, \cdots, b_{n-1}]}\{|\Phi(\underline{a})-\Phi(\underline{a}')|\} \end{eqnarray*} is the $ n$ variation of $ \Phi$;
( P3 ) $ \sum_{J\in S}\,\sup_{x\in J}\exp {\bar{\varphi}}(x)< \infty; $
( P4 ) there exists $ \epsilon> 0$ such that \begin{eqnarray*} \sum_{J\in S}\tau(J)\sup_{x\in J}\exp(\varphi^+(x)+\epsilon\tau(x))< \infty. \end{eqnarray*}

The following result is a corollary of Theorems 4.1 and 4.2.

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. Coarse-Graining, 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 $ \eps> 0$ and $ x\in X$, the set

\begin{eqnarray}\label{eqn:nespsset} \Gamma_\eps(x) := \left\{y\in X \mid d(f^nx,f^ny) < \eps \text{ for all } n\in \ZZ \right\} \end{eqnarray}

(5.1)

contains all points whose trajectory stays within $ \eps$ of the trajectory of $ x$ for all time. The map $ f$ is expansive if there is $ \eps> 0$ such that $ \Gamma_\eps(x) = \{x\}$ for every $ x\in X$; that is, if any two distinct trajectories eventually separate at scale $ \eps$. 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 $ \ph$; indeed, the standard proof of the variational principle ([Walters1982], Theorem 8.6) gives a construction of such a measure. The idea is that one "coarse-grains" the system at scale $ \eps$ and builds a measure that is appropriately distributed over all trajectories that separate by $ \eps$ 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 $ \eps> 0$ there is $ \tau\in \NN$ such that any collection of finite-length orbit segments can be $ \eps$-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^{n-1}(x)$ and write \begin{eqnarray*} B_n(x,\eps) = \{y\in X \mid d(f^kx,f^ky) \leq \eps \text{ for all } 0\leq k < n\} \end{eqnarray*} for the Bowen ball of points that shadow $ (x,n)$ to within $ \eps$ 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,\eps)$, then $ f^{n_1 + \tau}(y)\in B_{n_2}(x_2,\eps)$, and in general

\begin{eqnarray}\label{eqn:spec} f^{\sum_{i=0}^{j-1} (n_i + \tau)}(y) \in B_{n_j}(x,\eps) \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 $ \ph\colon X\to \RR$ satisfies the Bowen property if there is $ K\in \RR$ such that $ |S_n\ph(x) - S_n\ph(y)| < K$ whenever $ y\in B_n(x,\eps)$, where $ S_n\ph(x) = \sum_{j=0}^{n-1} \ph(f^jx)$. The following theorem summarizes the classical results due to Bowen on systems with specification [Bowen1974]. ³²

Theorem 5.1.

If $ (X,f)$ is an expansive system with specification and $ \ph$ 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 $ \ph$ is Hölder continuous.

5.2. Non-uniform 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 $ \eps> 0$, consider the 'non-expansive set' $ \NE(\eps) = \{x\in X \mid \Gamma_\eps(x) \neq \{x\}\}$, where $ \Gamma_\eps(x)$ is as in (5.1) . Note that $ (X,f)$ is expansive if and only if $ \NE(\eps)=\emptyset$. The pressure of obstructions to expansivity is ³³

\begin{eqnarray}\label{eqn:pexp} \Pexp(\ph) = \lim_{\eps\to 0} \sup_{\mu \in \MMM^e(f)} \left\{h_\mu(f) + \int \ph\,d\mu \mid \mu(\NE(\eps))=1\right\}. \end{eqnarray}

(5.3)

In particular, expansive systems have $ \Pexp(\ph)=-\infty$. It follows from the results in [Climenhaga and Thompson2016] that the condition $ \Pexp(\ph) < P(\ph)$ 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 $ \GGG \subset X\times \NN$ (and similarly for the Bowen property). One must also require $ \GGG$ to be large enough, which in this case means that there are collections of orbit segments $ \PPP,\SSS\subset X\times \NN$ that have small pressure compared to the whole system, but are sufficient to generate $ X\times \NN$ from $ \GGG$ by adding prefixes from $ \PPP$ and suffixes from $ \SSS$. ³⁴

Let us make this more precise. A decomposition of the space of orbit segments consists of $ \PPP, \GGG, \SSS \subset X\times \NN$ and functions $ p,g,s\colon X\times \NN\to \NN\cup \{0\}$ such that $ (p+g+s)(x,n) = n$ and \begin{align*} (x,p(x,n))&\in \PPP,\\ (f^{p(x,n)}(x),g(x,n)) &\in \GGG, \\ (f^{(p+g)(x,n)}(x),s(x,n))& \in \SSS. \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 $ \ph\colon X\to \RR$ a continuous function. Suppose that $ \Pexp(\ph)< P(\ph)$ and $ X\times \NN$ admits a decomposition $ (\PPP,\GGG,\SSS)$ with the following properties:

( I ) $ \GGG$ has specification at every scale;
( II ) $ \ph$ has the Bowen property on $ \GGG$;
( III ) $ P(\PPP\cup \SSS,\ph) < P(\ph)$.

Then $ f$ has a unique equilibrium measure $ \mu_\ph$.

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 3-dimensional 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 $ \ph\colon \mathbb{T}^3\to \RR$, 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(df|E^{cu})|$ has a unique equilibrium state for every $ t\in (t_0,t_1)$, and $ \ph_1$ is the unique SRB measure.

A related second example is the Bonatti–Viana example introduced in [Bonatti and Viana2000]. Here one takes a 4-dimensional 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 non-uniformity in both the stable and unstable directions makes tower constructions more difficult, and no Gibbs-Markov-Young 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 semi-conjugacy between the examples and the original toral automorphisms, and in particular do not generalize to equilibrium states corresponding to non-zero 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 If $ M$ is a smooth rank 1 manifold with nonpositive curvature, then its geodesic flow is non-uniformly 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 non-zero potentials. Using non-uniform 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 A decomposition of the space of orbit segments $ X\times \NN$ is obtained by fixing a threshold $ \gamma> 0$ and taking \begin{align*} \PPP &= \SSS = \{(x,n) \mid G(x,n) < \gamma\}, \\ \GGG &= \{(x,n) \mid G(x,k) \geq \gamma, G(f^kx,n-k)\geq \gamma \text{ for all } 0\leq k \leq n \}. \end{align*} Indeed, given any $ (x,n)\in X\times \NN$, one can take $ p$ and $ s$ to be maximal such that $ (x,p) \in \PPP$ and $ (f^{n-s}x,s)\in \SSS$, and use additivity of $ G$ along orbit segments to argue that $ (f^px,n-p-s)\in \GGG$, which yields a decomposition $ X\times \NN = \PPP\GGG\SSS$. Then one makes the following arguments to apply Theorem 5.2.

Assumption (1) above leads to hyperbolic estimates along trajectories in $ \GGG$, which can be used to prove specification for $ \GGG$ (condition ( I ) in Theorem 5.2) and the Bowen property on $ \GGG$ for Hölder continuous potentials (condition ( II )).
Assumption (2) gives the pressure estimate $ P(\PPP\cup \SSS,\ph)< P(\ph)$ from ( III ).
The expansion estimates along $ \GGG$ and the pressure estimates on $ \PPP$ and $ \SSS$ also yield $ \Pexp(\ph)< P(\ph)$.

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 non-uniform 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 coarse-graining (expansivity and specification), we turn our attention now to a third approach, which is in some sense more natural and more simple-minded. 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 non-uniform 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 non-invariant measure $ m$, average it under the dynamics to produce the sequence

\begin{eqnarray}\label{evol} \mu_n = \frac1n\sum_{k=0}^{n-1}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

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 $ \RRR$ 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 $ \Mac$ be the set of all (not necessarily invariant) probability measures $ \mu$ on the manifold $ M$ that can be expressed as

\begin{eqnarray}\label{eqn:Mac} \mu(E)=\int_\RRR\int_{W\cap E} \rho(x)\,dm_W(x)\,d\zeta(W,\rho) \end{eqnarray}

(6.2)

for some measure $ \zeta$ on $ \RRR$; in other words, $ \mu$ admits a decomposition (in the sense of Fubini's theorem) along unstable local manifolds, in which conditional measures are leaf-volumes. Then one can show that $ \Mac\cap\MMM(f)$ is precisely the set of SRB measures for $ f$. Moreover, Lebesgue measure $ m$ is in $ \Mac$ and thus since images of unstable manifolds can be decomposed into small pieces of unstable manifolds, we have $ f_*^km\in\Mac$ for all $ k\in \NN$, so the averaged measures given by (6.1) are in $ \Mac$ as well.

In order to pass to the limit and obtain $ \mu\in\Mac$ one needs a little more control. Fixing $ K> 0$, let $ \RRR_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 $ \Mac_K$ using (6.2) with $ \RRR_K$ in place of $ \RRR$, one can show that $ \Mac_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 \Mac_K$ for all $ n$ by invariance, and by compactness, $ \mu=\lim \mu_{n_j}\in\Mac_K\cap\MMM(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 centre-unstable direction $ E^{cu}$ is only non-uniformly expanding more care must be taken with the above approach because $ \Mac_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^k|E^u(f^{n-k}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

\begin{eqnarray}\label{eqn:nun} \nu_n := \frac 1n \sum_{k=0}^{n-1} f_*^k(m|H_k) \end{eqnarray}

(6.3)

are all contained in $ \Mac_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\MMM(f)$ has some ergodic component in $ \Mac_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}^{n-1}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

\begin{eqnarray}\label{evol1} \nu_n(x)=\frac1n\sum_{k=0}^{n-1}\,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 Sinai1983] 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 leaf-volume 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 central-unstable 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 non-uniform. To extend this approach to settings where the geometry is non-uniform (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 $ \Mac$, and the sequences of measures (6.4) using admissible manifolds in place of unstable manifolds.

6.1.3. Non-uniform Geometry: Effective Hyperbolicity

The difficulties encountered in the geometrically non-uniform 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 \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 \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

\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 $ \ba> 0$, let \begin{eqnarray*} \rho_{\ba}(x) := \ulim_{n\to\infty} \frac 1n \#\{0\leq k< n \mid \alpha(f^kx) < \ba\} \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 $ \ba> 0$ with $ \rho_{\ba}(x) = 0$ for every $ x$; however, for a system with non-uniform geometry it may be the case that every $ \ba> 0$ has points with $ \rho_{\ba}(x) > 0$.

With the above notions in mind, we consider the following set of points: \begin{eqnarray*} S' = \left\{x\in S \mid \llim_{n\to\infty} \frac 1n \sum_{k=0}^{n-1} \lambda(f^kx) > 0 \text{ and } \lim_{\ba\to 0} \rho_{\ba}(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 $ \Mac_K$.

In this more general setting the definition of $ \Mac_K$ is significantly more involved. Broadly speaking, in the definition of $ \RRR$ 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 \begin{eqnarray*} \sum_{j=k}^{n-1}\lambda(f^jx)\ge\chi(n-k) \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 $ \RRR$; most notably, one must fix $ n\in \NN$ and then consider only admissible manifolds $ W$ for which \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 $ \Mac_{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 $ \Mac_{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}^{n-1}f_*^k\in\Mac_{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 $ \Mac$; 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 $ \LLL$.

The central issue in studying $ \LLL$ is the problem of finding a Banach space $ \BBB$ (of functions) on which $ \LLL$ acts "with good spectral properties". Generally speaking this means that 1 is a simple eigenvalue of $ \LLL$ (so there is a unique fixed point $ h=\LLL h$, which corresponds to the unique absolutely continuous invariant measure) and the rest of the spectrum of $ \LLL$ 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 $ \LLL h$. (The precise meaning of this statement depends on the particular choice of Banach space $ \BBB$, 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 non-trivial 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 two-sided SFT $ \Sigma \subset A^\ZZ$ by its one-sided version $ \Sigma^+ \subset A^\NN$. As described after Theorem 1.3, the transfer operator $ \LLL$ has an eigenfunction $ h\in C(\Sigma^+)$, and its dual $ \LLL^*$ 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 $ \BBB$ 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 $ \LLL$ 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 non-uniformly 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 One might also hope to adapt the functional analytic approach from Sect. 6.2 into the non-uniformly 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

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 two-dimensional diffeomorphisms and three-dimensional 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 Lebesgue-a.e. in $ U$, with the property that there is $ \chi> 0$ for which \begin{eqnarray*} S'=\left\{x\in U\mid\llim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\lambda(f^kx)> \chi\text{ and } \lim_{\ba\to 0}\rho_{\ba}(x)=0\right\} \end{eqnarray*} has full Lebesgue measure in $ U$. Consider for each $ N\in \NN$ the set \begin{eqnarray*} X_N=\left\{x\in U\mid\sum_{k=0}^{n-1}\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.

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Abstract

Keywords

Mathematics Subject Classification

1. Introduction

1.1. The General Setting

1.2. Uniformly Hyperbolic Maps (Sinai, Ruelle, Bowen)

1.2.1. General Thermodynamic Results

Theorem 1.1.

Proposition 1.2.

Theorem 1.3.

1.2.2. Thermodynamic Formalism for the Geometric $ t$-Potential

1.2.3. Hyperbolic Attractors

1.3. Non-uniformly Hyperbolic Maps.

1.3.1. Definition of Non-uniform Hyperbolicity

1.3.2. Possibility of Phase Transitions and Non-hyperbolic Behavior.

1.3.3. Different Types of Equilibrium Measures.

1.3.4. Outline of the Paper

2. Markov Models for Non-uniformly Hyperbolic Maps I: Young Diffeomorphisms

2.1. Earlier Results: One-dimensional and Rational Maps

2.2. Young Towers and Gibbs–Markov–Young Structures.

2.2.1. Tower Constructions in Dynamical Systems

2.2.2. Young Diffeomorphisms

2.2.3. SRB Measures for Young Diffeomorphisms.

Theorem 2.1.

Theorem 2.2.

2.2.4. Thermodynamics of Young Diffeomorphisms for the Geometric $ t$-Potential

Theorem 2.3.

Remark.

2.3. Examples of Young Diffeomorphisms.

2.3.1. A Hénon-like Diffeomorphism at the First Bifurcation

Theorem 2.4.

2.3.2. The Katok Map.

Theorem 2.5.