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The Bridge Theorem for totally disconnected LCA groups. (English) Zbl 1322.37007
The authors use the notion of topological entropy for uniformly continuous self-maps of uniform spaces introduced by [B. M. Hood, J. Lond. Math. Soc., II. Ser. 8, 633–641 (1974; Zbl 0291.54051)] applied to continuous endomorphisms of locally compact abelian groups. On the other hand they consider the notion of algebraic entropy given by [S. Virili, Topology Appl. 159, No. 9, 2546–2556 (2012; Zbl 1243.22007)]. In this terms a subcategory \(\Xi\) of the category of all locally compact abelian groups satisfies the Bridge Theorem if \(h_{\mathrm{top}}(\phi)=h_{\mathrm{alg}}(\hat{\phi})\) for every endomorphism \(\phi:G\rightarrow G\) in \(\Xi\), where \(\hat{\phi}:\hat{G}\rightarrow \hat{G}\) denotes the dual and \(\hat{G}\) denotes the Pontryagin dual group of \(G\). In this paper, the Bridge Theorem is proved for totally disconnected locally compact abelian groups and the result is extended to locally compact abelian groups under additional assumptions.

MSC:
37B40 Topological entropy
22B05 General properties and structure of LCA groups
22D40 Ergodic theory on groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
54H11 Topological groups (topological aspects)
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