Compact families and typical entropy invariants of measure-preserving actions

TL;DR

This paper introduces an entropy invariant for a compact set of measure-preserving actions, showing that typical actions have infinite entropy, and demonstrates that they are not isomorphic to geometric shape exchange transformations.

Contribution

It defines a new entropy invariant that distinguishes typical measure-preserving actions from geometric shape exchange transformations.

Findings

Typical measure-preserving transformations have infinite entropy.

The entropy invariant is zero on a compact set but infinite for typical actions.

Typical actions are not isomorphic to geometric shape exchange transformations.

Abstract

For a compact set of actions, an invariant of Kushnirenko's entropy type is chosen in such a way that on this set it is equal to zero, but will be infinity for typical actions. As a consequence, we show that typical measure-preserving transformations are not isomorphic to geometric shape exchange transformations. This problem arose in connection with the result of Chaika and Davis about the atypical nature of IETs.