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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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