Equilibrium states for non-transitive random open and closed dynamical systems

TL;DR

This paper establishes a new theoretical framework for random dynamical systems, proving the existence of equilibrium states without transitivity assumptions, and explores applications including Lyapunov exponents and escape rates.

Contribution

It introduces a random Ruelle--Perron--Frobenius theorem and demonstrates the existence of unique equilibrium states for non-transitive random maps.

Findings

Existence of unique random conformal and invariant measures.

Exponential decay of correlations in the systems studied.

Connection between Lyapunov exponents and escape rates.

Abstract

We prove a random Ruelle--Perron--Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.