Quenched and annealed equilibrium states for random Ruelle expanding maps and applications

TL;DR

This paper investigates the spectral properties and equilibrium states of random expanding maps on Polish spaces, demonstrating their limiting behavior, exponential decay of correlations, and continuity of measures, with applications to non-autonomous systems.

Contribution

It extends existing results by analyzing quenched and annealed equilibrium states for non-compact spaces, including their continuity and applications to various dynamical systems.

Findings

Existence of limiting behavior for transfer operators.

Equilibrium states exhibit exponential decay of correlations.

Quenched measures vary H"older continuously.

Abstract

In this paper we describe the spectral properties of semigroups of expanding maps acting on Polish spaces, considering both sequences of transfer operators along infinite compositions of dynamics and integrated transfer operators. We prove that there exists a limiting behaviour for such transfer operators, and that these semigroup actions admit equilibrium states with exponential decay of correlations and several limit theorems. The reformulation of these results in terms of quenched and annealed equilibrium states extend results by Baladi (1997) and Carvalho, Rodrigues & Varandas (2017), where the randomness is driven by a random walk and the phase space is assumed to be compact. Furthermore, we prove that the quenched equilibrium measures vary H\"older continuously and that the annealed equilibrium states can be recovered from the latter. Finally, we give some applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and on the boundary of equilibria.