Finitude of physical measures for Markovian random maps

TL;DR

This paper investigates conditions under which Markovian random dynamical systems have finitely many physical measures, extending results from i.i.d. systems to Markov chains, and applies these to specific classes of random maps.

Contribution

It establishes criteria for the finiteness of physical measures in Markov-driven systems using an i.i.d. representation and explores their properties in various dynamical contexts.

Findings

Finiteness of physical measures is guaranteed under certain contraction conditions.

Results extend Bernoulli system findings to Markovian systems.

Application to random diffeomorphisms on the circle and interval.

Abstract

We study the finiteness of physical measures for skew-product transformations $F$ associated with discrete-time random dynamical systems driven by ergodic Markov chains. We develop a framework, using an independent and identically distributed (i.i.d.) representation of the Markov process, that facilitates transferring results from the well-studied Bernoulli (i.i.d.) setting to the Markovian context. Specifically, we establish conditions for the existence of finitely many ergodic, $F$-invariant measures, absolutely continuous with respect to a reference measure, such that their statistical basins of attraction for measurable bounded observables cover the phase space almost everywhere. Furthermore, we investigate a weaker notion, which demands finitely many physical measures (not necessarily absolutely continuous) whose weak$^*$ basins of attraction cover the phase space almost everywhere. We show that for random maps on compact metric spaces driven by Markov chains on finite state spaces, this property holds if the system is mostly contracting, i.e., if all the Markovian invariant measures have negative maximal Lyapunov exponents. This result is applied to random $C^1$ diffeomorphisms of the circle and the interval under conditions based on the absence of invariant probability measures or finite invariant sets, respectively. We also connect our result to the quasi-compactness of the Koopman operator on the space of H\"older continuous functions.