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Topological entropy in totally disconnected locally compact groups. (English) Zbl 1380.37032
Summary: 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, $h_{\text{top}}(\phi)=h_{\text{top}}(\phi\restriction_H)+h_{\text{top}}(\bar{\phi}),$ where $$\bar{\phi}:G/H\rightarrow G/H$$ 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 $$\ker (\phi)\leq H$$. 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 $$\log s(\phi)=h_{\text{top}}(\phi)$$ to hold.

##### MSC:
 37B40 Topological entropy 37B10 Symbolic dynamics 54H20 Topological dynamics (MSC2010)
##### Keywords:
topological group; topological entropy
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##### References:
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