Absolutely continuous invariant measures for random non-uniformly expanding maps

TL;DR

This paper establishes the existence and finiteness of absolutely continuous invariant measures for random non-uniformly expanding dynamical systems, including higher-dimensional cases, using a novel construction method that handles critical points.

Contribution

It introduces a new approach to constructing absolutely continuous invariant measures for random systems with asymptotic expansion, covering both one-dimensional and higher-dimensional cases.

Findings

Existence of at most countably many invariant measures for one-dimensional systems.

Finiteness of ergodic measures when expansion rate is bounded away from zero.

Extension of results to higher-dimensional systems with asymptotic expansion.

Abstract

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical or singular points for the random maps.