Random Expansive Measures

TL;DR

This paper extends the concepts of expansivity and related properties from deterministic to random dynamical systems, establishing key relationships with entropy and invariant measures.

Contribution

It introduces the notion of random expansive measures, proves their connection to positive entropy, and shows the existence of invariant measures with expansive properties.

Findings

Positive relative topological entropy implies zero measure for $w$-stable classes.

Existence of an invariant measure that is also expansive.

Relation established between random expansive measures and random countably-expansive systems.

Abstract

The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the $w$-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.