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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

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

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.

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) |

\begin{eqnarray}\label{ind:poten} S_n\ph(x):=\sum_{k=0}^{n-1}\ph(f^kx). \end{eqnarray} | (1.2) |

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.

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$.

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. 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. ×
^{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.

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

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}$
Here $ \text{int}\,R_i$ means the interior of the set $ R_i$ in the
relative topology. ×
^{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*}

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)$).

- (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.

- (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*}

- (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
It is
instructive to consider the case $ \ph=0$ and write down the action of
$ \LLL_0$ on the (finite-dimensional) space of functions constant on
1-cylinders, where the action is given by the (transpose of the) transition
matrix $ A$. ×
^{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].

[]

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)$ The value of the Hausdorff dimension does not depend
on $ 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
; It
is assumed that some of these numbers are positive while others are negative.
× ^{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) |

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$; 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. ×
^{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) |

A $ C^{1+\alpha}$ diffeomorphism $ f$ of a compact smooth Riemannian manifold $ M$ is non-uniformly 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}}$.

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 non-uniformly hyperbolic on $ S$.

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. 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. ×
^{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.

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.

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. 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 $ \ph_t$ can be
lifted to a function on the symbolic space that is globally well-defined and is
Hölder continuous. This can be used to study equilibrium measures for
this 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) |

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 Because $ \MMM^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(\ph)$ is achieved by a measure $ \mu$ whose entropy may not
be greater than $ h$, the restriction to measures in the class
$ \MMM^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. × ^{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) |

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.

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. 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. × ^{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. This notion was introduced by [Chernov and Dolgopyat2009]. × ^{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.

In one form or another, the use of Markov models with countably many states to study non-uniformly hyperbolic systems dates back to the late 1970s and early 1980s, when [Hofbauer1979], [Hofbauer1981a], [Hofbauer1981b] used a countable-state 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) = 1-ax^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.

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\NN$. 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 one-dimensional setting came in the study of the Hénon map

\begin{eqnarray}\label{henon} f_{a,b}(x,y) = (1-ax^2 + y, bx), \end{eqnarray} | (2.1) |

The general structure behind these results was developed in [Young1998] and has come to be known as a Young tower
, 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. ×
^{19} 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 two-sided 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.

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 It follows
that every local unstable manifold intersects the base in a set of positive leaf
volume. ×
^{20}and the integral of the height of the tower against leaf volume is finite.

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\NN$ 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$, \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 \begin{eqnarray*} \log\frac{|\det dF^u(x)|}{|\det dF^u(y)|}\le C d(Fx,Fy)^\eta. \end{eqnarray*}

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 \begin{eqnarray*} \int R\,dm_{V^u}< \infty; \end{eqnarray*}
- exponential
tails
if for some $ C,a> 0$ and $ T\ge 1$,
\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$, \begin{eqnarray*} m_{V^u}\{x\mid R(x)> T\} < CT^{-an}. \end{eqnarray*}

- (1) there is local
unstable manifold $ V^u$ such that
\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.

Note that even without the arithmetic condition one still obtains the
"exponential decay up to a period" result stated earlier in Theorem 1.1
(^{2})
.

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" two-dimensional 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' two-dimensional extensions of one-dimensional 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 non-uniformly contracting ([Castro2004]) or expanding ([Alves and Pinheiro2010], [Alves and Li2015]); the latter papers are built on earlier results for non-uniformly 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 push-forward 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.

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 \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\RR$. 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 \begin{eqnarray*} S_n:=\text{Card}\{\Lambda_i^s\colon R_i=n\}. \end{eqnarray*}

- (1) for all large
$ n$
\begin{eqnarray}\label{growth} S_n\le e^{hn}, \end{eqnarray} (2.4) - (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. This condition is stronger
than the corresponding condition
(2.3)
. ×
^{22}

- 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. For $ t=1$, the SRB measure $ \mu_1$ may not have exponential decay of correlations; this is the case for the Manneville–Pomeau map where the decay is polynomial. See Sect. 1.3.2 and also Sect. 2.3 for more details.
- 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
\begin{eqnarray}\label{qnu} Q_\nu=\int_\Lambda R\, d\nu \end{eqnarray} (2.5) \begin{eqnarray}\label{lift} \mu(E)={\mathcal L}(\nu)(E):=\frac{1}{Q_\nu}\sum_{i\ge 0}\sum_{k=0}^{R_i-1}\nu(f^{-k}(E)\cap \Lambda_i^s). \end{eqnarray} (2.6) 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$ See
(2.2)
where one should replace the leaf volume with the measure $ \nu_t$.
×
^{24}(see [Pesin et al.2016b], Theorem 4.5) and then applying results from [Melbourne and Terhesiu2014]. 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. ×^{25} - 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)$. Note that both $ h$ and
$ h_{\mu_1}(f)$ do not depend on $ k$. ×
^{26}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_t|Y_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.

We describe two examples of Young diffeomorphisms for which Theorem 2.3 applies.

The first example is Hénon-like 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 non-wandering 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.

- (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$.

We describe the Katok map ([Katok1979]) (see also [Barreira and Pesin2013]), which can be thought of as an invertible and two-dimensional analogue of the Manneville–Pomeau map. Consider the automorphism of the 2-torus 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}$.

\begin{eqnarray}\label{batata10} \dot{s}_1= s_1\log\lambda,\quad \dot{s}_2=-s_2\log\lambda, \end{eqnarray} | (2.7) |

We slow down trajectories of (2.7) by perturbing it in $ D_{r_0}$ as follows: \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 time-1 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 \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 \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 \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 area-preserving $ C^\infty$ diffeomorphism. To achieve this, define a map $ \phi$ in $ D_{r_0}$ by the formula

\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) |

- (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$.

The thermodynamic formalism for SFTs rested on the Ruelle's version of the Perron–Frobenius theorem for finite-state topological Markov chains. For the class of two-step potential functions $ \ph(x) = \ph(x_0,x_1)$, which includes the zero potential $ \ph=0$, the extension of this theory to countable-state Markov shifts dates back to work of [Vere-Jones1962], [Vere-Jones1967], [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.

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.

Existence of a stationary vector $ \pi=(\pi_i)$ with $ \pi P=\pi$ is determined by the recurrence properties of the shift [Vere-Jones1962], [Vere-Jones1967]. 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
countable-state 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). In the next section when we consider non-zero
potentials, we will have to count the loops with weights coming from the
potential. × ^{28}

- (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. For some intuition behind this definition, it may be
helpful to consider again a countable-state random walk: writing
$ \PP(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 \PP(n)=\infty$, and transient (finitely
many returns a.s.) if the sum is finite. ×
^{29}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 $ \llim_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.

In discussing the extension to non-zero potentials on countable-state 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 non-zero 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 \begin{eqnarray*} Z_n=Z_n(\varphi,a)=\sum_{\sigma^n(x)=x}\exp(\Phi_n(x))1_{[a]}(x) \end{eqnarray*} and \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}^{n-1}\varphi(f^kx)$. Furthermore, the Gurevich entropy $ h_G(\sigma)$ is replaced with the Gurevich-Sarig pressure $ P_{GS}(\sigma,\ph)$, which is the exponential growth rate of $ Z_n$, i.e., \begin{eqnarray*} P_{GS}(\sigma,\ph)=\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.

- (1) $ P_{GS}(\ph) = \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 $ \ph$.
- (3) For every cylinder $ [w]\subset \Sigma$, we have $ \lambda^{-n} \nu[w]^{-1} \LLL_\ph^n \one_{[w]} \to h$ uniformly on compact subsets.

Using Pesin theory, Sarig recently carried out a version of the construction of Markov partitions for non-uniformly 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$.

- $ \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)$.

The analogous result to Theorem 3.2 for three-dimensional 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).

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$.

- ( 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
- (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}$.

- (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$.

- ( I3 ) The set $ S^\mathbb{Z}{\setminus}\check S$ supports no (ergodic) $ \sigma$-invariant measure which gives positive weight to any open subset.

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$.

- $ \Phi$ has strongly summable variations;
- $ P_{GS}(\Phi)< \infty$ and $ P_{GS}(\Phi^+)< \infty$;
- $ \sup_{a\in S^\mathbb{Z}}\,\Phi^+(a)< \infty$.

- (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 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. This
requirement should be added to Theorem 4.6 in [Pesin et al.2016b]. ×
^{31}

- ( 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*}

- (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.

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) |

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) |

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]. 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. ×
^{32}

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 The idea of ignoring measures sitting on $ \NE(\eps)$ was introduced
earlier by [Buzzi and Fisher2013]. × ^{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) |

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).

- ( I ) $ \GGG$ has specification at every scale;
- ( II ) $ \ph$ has the Bowen property on $ \GGG$;
- ( III ) $ P(\PPP\cup \SSS,\ph) < P(\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. Although the issue of
decay of correlations is more subtle because it is a flow, not a map; see [Dolgopyat1998], among others. × ^{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.

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. For flows one should make the obvious modifications,
replacing $ \NN$ by $ [0,\infty)$ and cardinality with Lebesgue
measure. × ^{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)$.

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) |

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.

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) |

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.

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) |

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) |

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.

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)$.

\begin{eqnarray}\label{eqn:lambdaspsx} \lambda(x) := \min(\lambda^u(x) - \Delta(x), - \lambda^s(x)); \end{eqnarray} | (6.5) |

- 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$).

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.

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.

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.

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]). Coupling
techniques are also at the heart of Young's tower results for subexponential
mixing rates ([Young1999]). ×
^{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.

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.

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)$.

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$.

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].

V. Climenhaga is partially supported by NSF Grant DMS-1362838. Y. Pesin is partially supported by NSF Grant DMS-1400027

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1.1. The General Setting

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

2. Markov Models for Non-uniformly Hyperbolic Maps I: Young
Diffeomorphisms
1.2.1. General Thermodynamic Results

1.2.3. Hyperbolic Attractors

1.3. Non-uniformly Hyperbolic Maps
Theorem 1.1 Proposition
1.2 Theorem
1.3 Figure 1

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

2.1. Earlier Results: One-dimensional and Rational
Maps

2.2. Young Towers and Gibbs–Markov–Young
Structures

3. Markov Models for Non-uniformly Hyperbolic Maps II: Countable State
Markov Shifts
2.2.1. Tower Constructions in Dynamical Systems

2.2.2. Young Diffeomorphisms

2.2.3. SRB Measures for Young Diffeomorphisms

2.3. Examples of Young Diffeomorphisms
Theorem 2.1 Theorem
2.2

2.2.4. Thermodynamics of Young Diffeomorphisms for the Geometric
$ t$-Potential
Theorem 2.3

2.3.1. A Hénon-like Diffeomorphism at the First
Bifurcation

Theorem 2.4

2.3.2. The Katok Map
Theorem 2.5

3.1. Recurrence Properties for Random Walks

3.2. Non-zero Potentials

4. Markov Models for Non-uniformly Hyperbolic Maps III: Inducing Schemes
of Hyperbolic Type
Theorem 3.1

3.3. Countable-State Markov Partitions for Smooth
Systems
Theorem 3.2

Theorem 4.1 Theorem
4.2 Theorem
4.3

5. Coarse-Graining, Expansivity, and Specification
5.1. Uniform Expansivity and Specification

6. The Geometric Approach
Theorem 5.1

5.2. Non-uniform Expansivity and Specification
Theorem 5.2

6.1. Geometric Construction of SRB Measures

6.3. Ergodic Properties

6.1.1. Idea of Construction

6.1.2. Uniform Geometry: Uniform and Partial
Hyperbolicity

6.2. Constructing Equilibrium Measures
Theorem 6.1

6.1.3. Non-uniform Geometry: Effective Hyperbolicity
Theorem 6.2

6.1.4. Idea of Proof
6.3.1. SRB Measures

6.3.2. Equilibrium Measures

Conjecture 6.3 Conjecture
6.4