Topological entropy in totally disconnected locally compact groups

Abstract

Let $G$ be a topological group, let $\varphi$ be a continuous endomorphism of $G$ and let $H$ be a closed $\varphi$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$h_{top}(\varphi )=h_{top}(\varphi {\upharpoonright }_H)+h_{top}(\bar{\varphi }),$$ where $\bar{\varphi }:G/H\to G/H$ is the map induced by $\varphi$. 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 $\varphi H=H$ and $\ker (\varphi )\le H$. As an application we give a dynamical interpretation of the scale $s(\varphi )$, by showing that $\log s(\varphi )$ is the topological entropy of a suitable map induced by $\varphi$. Finally, we give necessary and sufficient conditions for the equality $\log s(\varphi )=h_{top}(\varphi )$ to hold.

18 pages

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