Thermodynamic Formalism for Random Non-uniformly Expanding Maps

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

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The thermodynamic formalism for expanding maps

Communications in Mathematical Physics ◽

10.1007/bf01217908 ◽

1989 ◽

Vol 125 (2) ◽

pp. 239-262 ◽

Cited By ~ 117

Author(s):

David Ruelle

Keyword(s):

Thermodynamic Formalism ◽

Expanding Maps

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Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.43 ◽

2021 ◽

pp. 1-43

Author(s):

DOMINIC VECONI

Keyword(s):

Thermodynamic Formalism ◽

Equilibrium States ◽

Maximal Entropy ◽

Srb Measure ◽

The Central Limit Theorem ◽

Exponential Decay Of Correlations ◽

Surface Homeomorphisms ◽

Young Towers ◽

Uniqueness Of Equilibrium ◽

Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

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ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

Reviews in Mathematical Physics ◽

10.1142/s0129055x10004168 ◽

2010 ◽

Vol 22 (10) ◽

pp. 1147-1179 ◽

Cited By ~ 12

Author(s):

LUIS BARREIRA

Keyword(s):

Variational Principle ◽

Existence And Uniqueness ◽

Gibbs Measures ◽

Thermodynamic Formalism ◽

Topological Pressure ◽

Present Stage ◽

The Past ◽

Recent Developments ◽

General Sequences ◽

Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

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