Mixing and observation for Markov operator cocycles

TL;DR

This paper extends classical concepts of mixing and exactness to Markov operator cocycles in random dynamical systems, providing new criteria and equivalences that deepen understanding of their long-term behavior.

Contribution

It introduces six definitions of mixing for Markov operator cocycles, extends Lin's criterion for exactness, and establishes equivalences in asymptotically periodic cases.

Findings

Six fundamental mixing definitions for Markov operator cocycles

Extension of Lin's criterion for exactness to cocycles

Equivalence between mixing, exactness, and stability in asymptotically periodic cocycles

Abstract

We consider generalized definitions of mixing and exactness for random dynamical systems in terms of Markov operator cocycles. We first give six fundamental definitions of mixing for Markov operator cocycles in view of observations of the randomness in environments, and show that they can be reduced into two different groups. Secondly, we give the definition of exactness for Markov operator cocycles and show that Lin's criterion for exactness can be naturally extended to the case of Markov operator cocycles. Finally, in the class of asymptotically periodic Markov operator cocycles, we show the Lasota-Mackey type equivalence between mixing, exactness and asymptotic stability.