Sub-additivity of measure-theoretic entropies of commuting transformations on Banach space

TL;DR

This paper proves the sub-additivity of measure-theoretic entropies for commuting smooth transformations on Banach spaces and extends finite-dimensional results to infinite-dimensional systems under mild conditions.

Contribution

It extends Hu's finite-dimensional results on commuting diffeomorphisms to infinite-dimensional Banach space systems, establishing entropy sub-additivity and conditions for equality.

Findings

Proves sub-additivity of measure-theoretic entropies for commuting transformations.

Provides conditions for equality of entropies.

Extends finite-dimensional entropy results to infinite-dimensional Banach spaces.

Abstract

This paper considers two commuting smooth transformations on a Banach space, and proves the sub-additivity of the measure theoretic entropies under mild conditions. Furthermore, some additional conditions are given for the equality of the entropies. This extends Hu's work about commuting diffeomorphisms in a finite dimensional space (Huyi Hu, 1993, Ergod. Th. Dynam. Sys., \textbf{13}: 73-100) to the case of systems on an infinite dimensional Banach space.