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We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics that is there is an open and dense $f^t$-invariant subset $U\subset\mathcal{M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except the direction of the flow) and is a Bernoulli flow while on the boundary $\partial U$, which has positive volume all Lyapunov exponents of the system are zero., Ya. P. was partially supported by NSF grant DMS-1400027. The authors would like to thank Banff International Research Station, where part of the work was done, for their hospitality

We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure., The published version of this paper contains an error in the proof of Lemma 6.6, which is corrected here (the lemma remains correct as stated). arXiv admin note: text overlap with arXiv:1803.10374

Given a non-conformal repeller $$\Lambda $$ of a $$C^{1+\gamma }$$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok’s approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure.

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.

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods., Comment: 54 pages, significant restructuring of exposition and reorganization of proofs for clarity. Main results remain the same

We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+\alpha}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in nonuniformly hyperbolic sets which is of independent interest., Comment: Identical to v2 except that some blue text marking previous edits has been reverted to black

Some principal contributions of Ya. Sinai to hyperbolic theory of dynamical systems, focusing mainly on constructions of Markov partitions and of Sinai–Ruelle–Bowen measures, are discussed. Some further developments in these directions stemming from Sinai’s work, are described.

We introduce a class of continuous maps $f$ of a compact topological space $X$ admitting inducing schemes of hyperbolic type and describe the associated tower constructions. We then establish a thermodynamic formalism, i.e., we describe a class of real-valued potential functions $\varphi$ on $X$ such that $f$ possess a unique equilibrium measure $\mu_\varphi$, associated to each $\varphi$, which minimizes the free energy among the measures that are liftable to the tower. We also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem. We then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable. Our results extend those obtained in [PS05, PS08] for inducing schemes of expanding types and apply to certain multidimensional maps. Applications include obtaining the thermodynamic formalism for Young's diffeomorphisms, the H\'enon family at the first birfucation and the Katok map. In particular for the Katok map we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with $0\le t, Comment: 42 pages, 1 figure. To appear in Trans. Amer. Math. Soc

In order to characterize the complexity of a system with zero entropy we introduce the notions of scaled topological and metric entropies. We allow asymptotic rates of the general form \(e^{\alpha a(n)}\) determined by an arbitrary monotonically increasing “scaling” sequence \(a(n)\). This covers the standard case of exponential scale corresponding to \(a(n)=n\) as well as the cases of zero and infinite entropy. We describe some basic properties of the scaled entropy including the inverse variational principle for the scaled metric entropy. Furthermore, we present some examples from symbolic and smooth dynamics that illustrate that systems with zero entropy may still exhibit various levels of complexity.

We prove several new versions of the Hadamard-Perron Theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard-Perron Theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of "effective hyperbolicity" and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well-behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information., Comment: 45 pages

We provide a general mechanism for obtaining uniform information from pointwise data. A sample result is that if a diffeomorphism of a compact Riemannian manifold has pointwise expanding and contracting continuous invariant cone families,then the diffeomorphism is an Anosov diffeomorphism, Oberwolfach Preprints;2009,06

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.

For systems with zero Lyapunov exponents we introduce and study the notion of scaled Lyapunov exponents which is used to characterize the sub-exponential separation of nearby trajectories. We briefly discuss the abstract theory of such exponents and discuss some examples.

Author(s): Gorodetski, A; Pesin, Y | Abstract: We show that the space of hyperbolic ergodic measures of a given index supported on an isolated homoclinic class is path connected and entropy dense provided that any two hyperbolic periodic points in this class are ho-moclinically related. As a corollary we obtain that the closure of this space is also path connected.

We show that there exists a C∞ volume preserving topologically transitive diffeomorphism of a compact smooth Riemannian manifold which is ergodic (indeed is Bernoulli) on an open and dense subset \({\mathcal{G}}\) of not full volume and has zero Lyapunov exponent on the complement of \({\mathcal{G}}\) .

We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

This paper is a nontechnical survey and aims to illustrate Dolgo- pyat'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 par- tially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an al- ternative approach to the Pugh-Shub stable ergodicity theory for both con- servative and dissipative systems. On the other hand, ideas of mixed hyper- bolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.

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.

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.

We effect thermodynamical formalism for the non-uniformly hyperbolic $C^\infty$ map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\varphi_t=-t\log |df|_{E^u(x)}|$ for any $t\in(t_0,\infty)$, $t\neq 1$ where $E^u(x)$ denotes the unstable direction. We show that $t_0$ tends to $-\infty$ as the size of the perturbation tends to zero. Finally, we establish exponential decay of correlations as well as the Central Limit Theorem for the equilibrium measures associated to $\varphi_t$ for all values of $t\in (t_0, 1)$., Comment: 31 pages to appear in Ergod. Th. & Dy.nam. Sys

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems., Comment: 30 pages

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also establish a more general version of this result for general foliations with compact leaves.

Given a uniformly expanding map of two intervals we describe a large class of potentials admitting unique equilibrium measures. This class includes all Holder continuous potentials but goes far beyond them. We also construct a family of continuous but not Holder continuous potentials for which we observe phase transitions. This provides a version of the example in (9) for uniformly expanding maps.

We consider a discrete version of the Brusselator Model of the famous Belousov-Zhabotinsky reaction in chemistry. The original model is a reaction-diffusion equation and its discrete version is a coupled map lattice. We study the dynamics of the local map, which is a smooth map of the plane. We discuss the set of trajectories that escape to infinity as well as analyze the set of bounded trajectories – the Julia set of the system.

A number of models are surveyed which appear in physics, biology, chemistry, and other areas and which are described by a reaction-diffusion equation. The corresponding coupled map lattice (CML) system is obtained by discretizing this equation. These CMLs are classified by the type of the dynamics of the local map. Several different types of behavior are observed: Morse-Smale type systems, systems with attractors, and systems with Smale horseshoes.

We develop the multifractal analysis of conformal axiom A flows. This includes the study of the Hausdorff dimension of basic sets of the flow, the description of the dimension spectra for pointwise dimension and for Lyapunov exponents and the multifractal decomposition associated with these spectra. The main tool of study is the thermodynamic formalism for hyperbolic flows by Bowen and Ruelle. Examples include suspensions over axiom A conformal diffeomorphisms, Anosov flows, and in particular, geodesic flows on compact smooth surfaces of negative curvature.

We consider coupled map lattices of hyperbolic type, i.e., chains of weakly interacting hyperbolic sets (attractors) over multi-dimensional lattices. We describe the thermodynamic formalism of the underlying spin lattice system and then prove existence, uniqueness, mixing properties, and exponential decay of correlations of equilibrium measures for a class of Holder continuous potential functions with a sufficiently small Holder constant. We also study finite-dimensional approximations of equilibrium measures in terms of lattice systems (ℤ-approximations) and lattice spin systems (ℤd-approximations). We apply our results to establish existence, uniqueness, and mixing property of SRB-measures as well as obtain the entropy formula.

In this paper we establish the complete multifractal formalism for equilibrium measures for Holder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Holder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.