Topological entropy in totally disconnected locally compact groups

TL;DR

This paper investigates the additivity of topological entropy for continuous endomorphisms of totally disconnected locally compact groups, providing conditions under which entropy behaves additively and relating it to the scale function.

Contribution

It establishes when topological entropy is additive in this setting and links the scale function to entropy, offering new insights into the dynamics of such groups.

Findings

Additivity of topological entropy holds under specific conditions.

Logarithm of the scale function equals the topological entropy in certain cases.

Provides necessary and sufficient conditions for entropy and scale function equality.

Abstract

Let $G$ be a topological group, let $\phi$ be a continuous endomorphism of $G$ and let $H$ be a closed $\phi$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$h_{top}(\phi)=h_{top}(\phi\restriction_H)+h_{top}(\bar\phi)\,,$$ where $\bar\phi:G/H\to G/H$ is the map induced by $\phi$. We concentrate on the case when $G$ is locally compact totally disconnected and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\phi H=H$ and $\ker(\phi)\leq H$. As an application we give a dynamical interpretation of the scale $s(\phi)$, by showing that $\log s(\phi)$ is the topological entropy of a suitable map induced by $\phi$. Finally, we give necessary and sufficient conditions for the equality $\log s(\phi)=h_{top}(\phi)$ to hold.