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Algebraic entropy on strongly compactly covered groups. (English) Zbl 1429.37006
The authors consider the algebraic entropy of continuous endomorphisms on what they call strongly compactly covered groups; these are topological groups in which every compact set is contained in some compact open normal subgroup. The entropy is defined as \(h_{\mathrm{alg}}(\phi)=\sup_U H_{\mathrm{alg}}(\phi,U)\), where \(U\) varies over the compact neighbourhoods of the neutral element. The quantity \(H_{\mathrm{alg}}(\phi,U)\) is defined as \[ \limsup_{n\to\infty}\frac{\log\mu(T_n(\phi,U))}{n} \] where \(\mu\) is (some) Haar measure and \(T_n(\phi,U)=U\cdot\phi[U]\cdot\cdots\cdot\phi^{n-1}[U]\). The authors provide basic properties of this invariant, a limit-free formula, equality with the topological entropy of the dual endomorphism and an addition theorem for coordinate-wise products of endomorphisms.
Reviewer: K. P. Hart (Delft)

MSC:
37B02 Dynamics in general topological spaces
37B40 Topological entropy
54C70 Entropy in general topology
28D20 Entropy and other invariants
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