Independence, sequence entropy and mean sensitivity for invariant measures

TL;DR

This paper explores the relationships between independence, sequence entropy, and mean sensitivity in measure-preserving systems with countable group actions, establishing key equivalences and properties for ergodic measures.

Contribution

It proves that sequence entropy tuples are IT tuples and shows the equivalence of various sensitive tuples for ergodic measures under amenable group actions.

Findings

Sequence entropy tuples are IT tuples.

For ergodic measures, various sensitive tuples coincide.

Results apply to systems with countable infinite discrete group actions.

Abstract

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered F{\o}lner sequence, and the sensitive in the mean tuples along some tempered F{\o}lner sequence coincide.