Paper detail

Topological entropy in totally disconnected locally compact groups

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