Dynamically defined measures and equilibrium states

TL;DR

This paper develops a new technique for defining measures dynamically using Caratheodory's method, linking it to equilibrium states, and applies it to contractive Markov systems to obtain equilibrium states.

Contribution

It introduces a novel approach to measure construction and connects it to equilibrium states, extending the theory to contractive Markov systems.

Findings

Established a relation between dynamically defined measures and equilibrium states.

Applied the technique to obtain equilibrium states for contractive Markov systems.

Extended the mathematical framework for analyzing dynamical systems.

Abstract

A technique of dynamically defined measures is developed and its relation to the theory of equilibrium states is shown. The technique uses Caratheodory's method and the outer measure introduced in (I. Werner, Math. Proc. Camb. Phil. Soc. 140 (2) (2006) 333-347). As an application, equilibrium states for contractive Markov systems (I. Werner, J. London Math. Soc. 71 (2005), no. 1, 236-258) are obtained.