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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)
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