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, $$egin{eqnarray}h_{ ext{top}}({itphi})=h_{ ext{top}}({itphi} estriction_{H})+h_{ ext{top}}(ar{{itphi}}),end{eqnarray}$$ where ϕ¯:G/H→G/H is the map induced by ϕ. We concentrate on the case when G is totally disconnected locally compact 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)≤H. As an application, we give a dynamical interpretation of the scale s(ϕ) by showing that logs(ϕ) is the topological entropy of a suitable map induced by ϕ. Finally, we give necessary and sufficient conditions for the equality logs(ϕ)=htop(ϕ) to hold.© Cambridge University Press, 2016