Omri Sarig

Existence of Gibbs measures for countable Markov shifts

We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbanski (2001) who showed that this condition is sufficient.

Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams.

Thermodynamic formalism for null recurrent potentials

We extend Ruelle’s Perron-Frobenius theorem to the case of Hölder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(ϕ) denote the Gurevic pressure of ϕ and letLϕ be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatLϕ*ν=eP(ϕ)ν,Lϕh=eP(ϕ)h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofLϕk, and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.

Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics

This chapter discusses smooth ergodic theory and nonuniformly hyperbolic dynamics. Smooth ergodic theory studies topological and ergodic properties of smooth dynamical systems with nonzero Lyapunov exponents. There are two classes of hyperbolic invariant measures on compact manifolds for which one can obtain a sufficiently complete description of its ergodic properties. They are: smooth measures, that is, measures that are equivalent to the Riemannian volume with the Radon-Nikodym derivative bounded from above and bounded away from zero, and Sinai-Ruelle-Bowen measures. Nonuniform hyperbolicity conditions can be expressed in terms of the Lyapunov exponents—that is, a dynamical system is nonuniformly hyperbolic if it admits an invariant measure with nonzero Lyapunov exponents almost everywhere. This provides an efficient tool in verifying the nonuniform hyperbolicity conditions and determines the importance of the nonuniform hyperbolicity theory in applications. The nonuniform hyperbolicity theory covers an enormous area of dynamics, such as nonuniformly hyperbolic one-dimensional transformations, random dynamical systems with nonzero Lyapunov exponents, billiards and related systems (for example, systems of hard balls), and numerical computation of Lyapunov exponents.

Invariant measures and asymptotics for some skew products

For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures. These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic ergodic theorems, and bounded rational ergodicity.

Aperiodicity of cocycles and conditional local limit theorems

We establish conditions for aperiodicity of cocycles (in the sense of [12]), obtaining, via a study of perturbations of transfer operators, conditional local limit theorems and exactness of skew-products. Our results apply to a large class of Markov and non-Markov interval maps, including beta transformations. This allows us to establish various stochastic properties of beta expansions.

Symbolic dynamics for surface diffeomorphisms with positive entropy

Part 1. Chains as pseudo–orbits 8 2. Pesin charts 8 2.1. Non-uniform hyperbolicity 8 2.2. Lyapunov change of coordinates 8 2.3. Pesin Charts 10 2.4. Distortion compensating bounds 11 2.5. NUHχ (f) 11 3. Overlapping charts 11 3.1. The overlap condition 12 3.2. The form of f in overlapping charts 14 3.3. Coarse graining 16 4. e–chains and an infinite-to-one Markov extension of f 18 4.1. Double charts and e–chains 18 4.2. Admissible manifolds and the graph transform 20 4.3. A Markov extension 23 4.4. The relevant part of the extension 27

On an example with a non-analytic topological pressure

Abstract We construct a function φ = φ ( x 0 , x 1 ) on a topologically mixing countable Markov shift such that the set of values β >0 for which the pressure function P top ( βφ ) is not analytic in a neighbourhood of β has positive Lebesgue measure. The construction is based on renewal theoretic ideas.

Exchangeable measures for subshifts

Abstract Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω , if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finettis theorem characterizes these measures when Ω = S N . We apply the ergodic theory of equivalence relations to study the case Ω ≠ S N , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [ 0 , 1 ] (a non-Markovian constraint).

Ergodic properties of equilibrium measures for smooth three dimensional flows

Let