  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.

Dynamical Systems;General Topology;Group Theory