Equilibrium states and invariant measures for random dynamical systems

TL;DR

This paper investigates conditions under which random dynamical systems with countably many maps admit invariant measures and equilibrium states, providing necessary and sufficient criteria for their existence.

Contribution

It introduces new non-degeneracy and consistency conditions for such systems and establishes a bijective correspondence between invariant measures and equilibrium states.

Findings

Every uniformly continuous Markov system with a dominating chain is consistent.

Necessary and sufficient conditions for invariant measure existence are provided.

A bijection between invariant measures and equilibrium states is established.

Abstract

Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.