Equilibrium measures for maps with inducing schemes
Yakov Pesin Samuel Senti
Journal of Modern Dynamics 2008, 2(3): 397-430 doi: 10.3934/jmd.2008.2.397
We introduce a class of continuous maps $f$ of a compact topological space $I$ admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, \ie describe a class of real-valued potential functions $\varphi$ on $I$, which admit a unique equilibrium measure $\mu_\varphi$ minimizing the free energy for a certain class of invariant measures. We also describe ergodic properties of equilibrium measures, including decay of correlation and the Central Limit Theorem. Our results apply to certain maps of the interval with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions $\varphi_t=-t\log|df|$ with $t\in(t_0, t_1)$ for some $t_0<1 < t_1$. In the particular case of $S$-unimodal maps we show that one can choose $t_0<0$ and that the class of measures under consideration consists of all invariant Borel probability measures.
keywords: equilibrium meassures inducing schemes liftability. towers Thermodynamic formalism
On the work of Sarig on countable Markov chains and thermodynamic formalism
Yakov Pesin
Journal of Modern Dynamics 2014, 8(1): 1-14 doi: 10.3934/jmd.2014.8.1
The paper is a nontechnical survey and is aimed to illustrate Sarig's profound contributions to statistical physics and in particular, thermodynamic formalism for countable Markov shifts. I will discuss some of Sarig's work on characterization of existence of Gibbs measures, existence and uniqueness of equilibrium states as well as phase transitions for Markov shifts on a countable set of states.
keywords: Sarig. Brin prize
On the work of Dolgopyat on partial and nonuniform hyperbolicity
Yakov Pesin
Journal of Modern Dynamics 2010, 4(2): 227-241 doi: 10.3934/jmd.2010.4.227
This paper is a nontechnical survey and aims to illustrate Dolgopyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyperbolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.
keywords: Brin prize accessibility SRB-measures. partial hyperbolicity stable ergodicity Lyapunov exponents Dolgopyat
Open problems in the theory of non-uniform hyperbolicity
Yakov Pesin Vaughn Climenhaga
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 589-607 doi: 10.3934/dcds.2010.27.589
This is a survey-type article whose goal is to review some recent developments in studying the genericity problem for non-uniformly hyperbolic dynamical systems with discrete time on compact smooth manifolds. We discuss both cases of systems which are conservative (preserve the Riemannian volume) and dissipative (possess hyperbolic attractors). We also consider the problem of coexistence of hyperbolic and regular behaviour.
keywords: SRB measures $u$-measures. Non-uniform hyperbolicity generic properties mixed hyperbolicity dissipative systems conservative systems
Stable ergodicity for partially hyperbolic attractors with negative central exponents
Keith Burns Dmitry Dolgopyat Yakov Pesin Mark Pollicott
Journal of Modern Dynamics 2008, 2(1): 63-81 doi: 10.3934/jmd.2008.2.63
We establish stable ergodicity of diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the central direction are all negative with respect to invariant SRB-measures.
keywords: Partial hyperbolicity stable ergodicity accessibility Lyapunov exponents SRB-measures.
Pointwise hyperbolicity implies uniform hyperbolicity
Boris Hasselblatt Yakov Pesin Jörg Schmeling
Discrete & Continuous Dynamical Systems - A 2014, 34(7): 2819-2827 doi: 10.3934/dcds.2014.34.2819
We provide a general mechanism for obtaining uniform information from pointwise data. For instance, a diffeomorphism of a compact Riemannian manifold with pointwise expanding and contracting continuous invariant cone families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.
keywords: nonuniform hyperbolicity Uniform hyperbolicity pointwise hyperbolicity.

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