Gibbs Measures.- General Thermodynamic Formalism.- Axiom a Diffeomorphisms.- Ergodic Theory of Axiom a Diffeomorphisms.

https://www.cpht.polytechnique.fr/sites/default/files/Bowen_LN_Math_470_second_ed_v2013.pdf

Equilibrium states and the ergodic theory of Anosov diffeomorphisms

Introduction. Mappings from an interval to itself provide the simplest possible examples of smooth dynamical systems. Such mappings have been widely studied in recent years since they occur in quite varied applications, and since their surprisingly complex behavior may provide a useful introduction to the study of higher dimensional situations. The present paper sets up an effective calculus for describing the qualitative behavior of the successive iterates of a piecewise monotone mapping. Let I be a closed interval of real numbers. By definition, a continuous mapping f from I to itself is p!ecewise monotone if I can be subdivided into finitely many subintervals I I, .... I Z on which f is alternately strictly increasing or strictly decreasing. Each such maximal interval on which f is monotone is called a lap of f, and £ = Z(f) points c I ..... cz_ I at which f called the turninq points of f. is the lap number. The separating has a local minimum or maximum are

On iterated maps of the interval

/pdf/topological-entropy-for-noncompact-sets-501l4vhdvt.pdf

Topological entropy for noncompact sets

In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

/pdf/entropy-properties-of-rational-endomorphisms-of-the-riemann-26krxdpagr.pdf

Entropy properties of rational endomorphisms of the Riemann sphere

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of . We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), where .

T. N. T. Goodman

Papers

Relating Topological Entropy and Measure Entropy