Absolutely Continuous Invariant measures for non-autonomous dynamical systems

TL;DR

This paper extends classical theorems to non-autonomous dynamical systems, demonstrating conditions under which absolutely continuous invariant measures persist in the limit of converging sequences of maps.

Contribution

It generalizes the Krylov-Bogoliubov and Straube's theorems to non-autonomous systems, establishing the existence of acims for the limit map.

Findings

Generalized classical theorems to non-autonomous systems

Proved existence of acims for limit maps under certain conditions

Extended understanding of invariant measures in non-autonomous dynamics

Abstract

We consider the non autonomous dynamical system $\{\tau_{n}\},$ where $\tau_{n}$ is a continuous map $X\rightarrow X,$ and $X$ is a compact metric space. We assume that $\{\tau_{n}\}$ converges uniformly to $\tau .$ The inheritance of chaotic properties as well as topological entropy by $\tau $ from the sequence $\{\tau_{n}\}$ has been studied in \cite{Can1, Can2, Li,Ste,Zhu}. In \cite{You} the generalization of SRB\ measures to non-autonomous systems has been considered. In this paper we study absolutely continuous invariant measures (acim) for non autonomous systems. After generalizing the Krylov-Bogoliubov Theorem \cite{KB} and Straube's Theorem \cite{Str} to the non autonomous setting, we prove that under certain conditions the limit map $\tau $ of a non autonomous sequence of maps $\{\tau_n\}$ with acims has an acim.