Abstract
AbstractThe topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.
Highlights
The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action
The notion of topological entropy htop for continuous endomorphisms of locally compact groups is obtained by specializing the general notion of topological entropy given by Hood [29] for uniformly continuous selfmaps of uniform spaces, which was inspired by the classical topological entropy of Bowen [4] and Dinaburg [19]
The algebraic entropy was introduced in connection to the topological entropy by means of Pontryagin duality
Summary
The notion of topological entropy htop for continuous endomorphisms of locally compact groups is obtained by specializing the general notion of topological entropy given by Hood [29] for uniformly continuous selfmaps of uniform spaces, which was inspired by the classical topological entropy of Bowen [4] and Dinaburg [19] (see [18, 27] for the details). The same approach based on nets was used by Virili [40] to introduce topological entropy and algebraic entropy for actions on locally compact abelian groups We consider these entropies in the case of semigroup actions on locally compact (abelian) groups by continuous endomorphisms; in this case they depend on the choice of the net s of non-empty nite subsets of the acting semigroup S, so we denote them respectively by hstop and hsalg. They extend in a natural way the topological entropy htop and the algebraic entropy halg of a single continuous endomorphism recalled above, by taking S = N and s =
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