Thermodynamic formalism for general iterated function systems with measures

TL;DR

This paper develops a thermodynamic formalism for iterated function systems with measures, analyzing spectral properties, variational principles, and equilibrium states, extending the theoretical framework for such systems.

Contribution

It introduces a new thermodynamic formalism for IFSm, including spectral analysis, variational formulations, and existence and uniqueness of equilibrium states.

Findings

Spectral properties of transfer and Markov operators analyzed

Variational formulas for topological entropy and pressure established

Existence and uniqueness of equilibrium states proved

Abstract

This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSm given by a potential. A definition of equilibrium state is then natural and we prove its existence for any continuous potential. We show, in this setting, a uniqueness result for the equilibrium state requiring only the G\^ateaux differentiability of the pressure functional.