Topological mean dimension of induced systems

TL;DR

This paper investigates the topological mean dimension of induced systems, showing that for systems with positive entropy, the induced measure space has infinite mean dimension and analyzing entropy divergence rates.

Contribution

It establishes that induced transformations on measure spaces have infinite topological mean dimension for systems with positive entropy and provides entropy divergence estimates.

Findings

Induced systems have infinite topological mean dimension when original systems have positive entropy.

Entropy divergence rate increases as the scale approaches zero.

Provides quantitative estimates of entropy divergence in Wasserstein distance.

Abstract

For a topological system with positive topological entropy, we show that the induced transformation on the set of probability measures endowed with the weak-$*$ topology has infinite topological mean dimension. We also estimate the rate of divergence of the entropy with respect to the Wasserstein distance when the scale goes to zero.