Thermodynamic Formalism for Iterated Function Systems with Weights

TL;DR

This paper develops a comprehensive thermodynamic formalism for weighted iterated function systems, establishing spectral properties, variational principles, and existence and uniqueness of equilibrium states, generalizing classical dynamical systems results.

Contribution

It introduces a generalized thermodynamic formalism for IFS with weights, including spectral analysis, variational principles, and equilibrium state existence and uniqueness.

Findings

Existence of eigenprobabilities for the Markov operator.

Conditions for eigenvalues to be the spectral radius.

Existence and uniqueness of equilibrium states.

Abstract

This paper introduces an intrinsic theory of Thermodynamic Formalism for Iterated Functions Systems with general positive continuous weights (IFSw).We study the spectral properties of the Transfer and Markov operators and one of our first results is the proof of the existence of at least one eigenprobability for the Markov operator associated to a positive eigenvalue. Sufficient conditions are provided for this eingenvalue to be the spectral radius of the transfer operator and we also prove in this general setting that positive eigenfunctions of the transfer operator are always associated to its spectral radius. We introduce variational formulations for the topological entropy of holonomic measures and the topological pressure of IFSw's with weights 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. We also recover the classical formula relating the powers of the transfer operator and the topological pressure and establish its uniform convergence. In the last section we present some examples and show that the results obtained can be viewed as a generalization of several classical results in Thermodynamic Formalism for ordinary dynamical systems.