Topological entropy in totally disconnected locally compact groups

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 (sic) H) + h(top) ((phi) over bar), where (phi) over bar : G/H -> G/H is the map induced by phi. 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 phi H = H and ker(phi) <= 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.