Possible generalized entropy convergence rates

TL;DR

This paper introduces a new invariant called generalized entropy convergence rates for measure-preserving systems, exploring its relation to Shannon entropy convergence and using it to distinguish systems with zero entropy.

Contribution

It defines and analyzes generalized entropy convergence rates, connecting them with Shannon entropy and applying them to classify certain measure-preserving systems.

Findings

Generalized entropy convergence rates relate to Shannon entropy convergence.

Differences in these rates help distinguish aperiodic, ergodic, and rank one systems.

The invariant can identify systems with zero entropy.

Abstract

We consider an isomorphism invariant for measure-preserving systems - types of generalized entropy convergence rates. We show the connections of this invariant with the types of Shannon entropy convergence rates. In the case when they differ we show several facts for aperiodic, completely ergodic and rank one systems. We use this concept to distinguish some measure-preserving systems with zero entropy.