Relative slow entropy

TL;DR

This paper extends the concept of slow entropy to a relative setting conditioned on a factor, providing new tools for classifying measure-preserving systems and exploring rigidity properties.

Contribution

It introduces a relative slow entropy framework, generalizing existing theories and connecting it to rigidity and classification of systems.

Findings

Developed a relative slow entropy definition with desirable properties.

Proved a classification result for isometric systems using relative slow entropy.

Explored the relationship between rigidity and relative slow entropy.

Abstract

In 1997, Katok--Thouvenot and Ferenczi independently introduced a notion of ``slow entropy'' as a way to quantitatively compare measure-preserving systems with zero entropy. We develop a relative version of this theory for a measure-preserving system conditioned on a given factor. Our new definition inherits many desirable properties that make it a natural generalization of both the Katok--Thouvenot/Ferenczi theory and the classical conditional Kolmogorov--Sinai entropy. As an application, we prove a relative version of a result of Ferenczi that classifies isometric systems in terms of their slow entropy. We also introduce a new definition for the notion of a rigid extension and investigate its relationship to relative slow entropy.