Entropy for Group Endomorphisms and Homogeneous Spaces

Abstract

Abstract. Topological entropy há(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare hd(T) with measure theoretic entropy h(T); in particular h(T) = hd(T) for Haar measure and affine maps Ton compact metrizable groups. A particular case of this yields the wellknown formula for h(T) when T is a toral automorphism.

Invariance entropy for outputs

“…[3, Lemma 2.1]), the topological entropy h top (Φ) equals the topological entropy of the time-one-mapΦ 1 (x) = eĀx. Recall from Bowen [2] (cf. also Katok and Hasselblatt [6] or Robinson [11]) that the topological entropy of a linear map Ψ on R d can be defined in the following way: For a compact set K ⊂ R d , numbers n ∈ N and ε > 0 an (n, ε, K, Ψ )-spanning set is a subset R ⊂ K such that for all x ∈ K there is…”

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“…In this paper we will use the Bowen-Dinaburg's definitions of the topological entropy (see e.g. [6]) for systems on compact metric spaces, which agree with Adler-Konheim-McAndrew's one for systems on topological metrizable spaces. Let (X, ρ) be a compact metric space and let f : X → X be a map.…”

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“…By now, it is still one of the important invariants in dynamical systems. Since then, Bowen also gave a definition of topological entropy on a metric space for an uniformly continuous map, by spanning sets and separated sets (see [4]), and proved that the definition of topological entropy is equivalent to its definition by using open covers when the space is compact. Therefore, these have given the topological entropy a more clear and direct dynamical meaning.…”