publication . Preprint . 2015

Topological entropy of continuous actions of compactly generated groups

Schneider, Friedrich Martin;
Open Access English
  • Published: 13 Feb 2015
Abstract
We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It is shown that any continuous action of a compactly generated topological group on a compact Hausdorff space with vanishing topological entropy is amenable. Given an arbitrary compactly generated locally compact Hausdorff topological group $G$, we consider the canonical action of $G$ on the closed unit ball of $L^{1}(G)' \cong L^{\infty}(G)$ endowed with the corresponding weak-$^{\ast}$ topology. We prove that this action has vanishing topological entropy if and only if $G$ is compact. Furthermore, we show that the consider...
Subjects
free text keywords: Mathematics - Group Theory, Mathematics - Dynamical Systems, Mathematics - General Topology
Download from
19 references, page 1 of 2

John Wiley & Sons, Inc., New York, 1989 (cit. on p. 2).

Nathanial P. Brown and Narutaka Ozawa. C∗-algebras and finite-dimensional approximations. Vol. 88. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008 (cit. on p. 3). [OpenAIRE]

Andrzej Bi´s and Mariusz Urban´ski. “Some remarks on topological entropy of a semigroup of continuous maps”. In: Cubo 8.2 (2006), pp. 63-71 (cit. on pp. 1, 4).

Andrzej Bi´s and Pawel G. Walczak. “Entropy of distal groups, pseudogroups, foliations and laminations”. In: Ann. Polon. Math. 100.1 (2011), pp. 45-54 (cit.

on pp. 1, 4).

A. Deitmar and S. Echterhoff. Principles of harmonic analysis. Universitext. New York: Springer, 2009, pp. xvi+333 (cit. on pp. 7, 8).

Tomasz Downarowicz. Entropy in dynamical systems. Vol. 18. New Mathematical Monographs. Cambridge University Press, Cambridge, 2011 (cit. on p. 1).

E´tienne Ghys, R´emi Langevin and Pawel G. Walczak. “Entropie g´eom´etrique des feuilletages”. In: Acta Math. 160.1-2 (1988), pp. 105-142 (cit. on pp. 1, 4).

Philip Hall. “On representatives of subsets”. In: Journal of the London Mathematical Society 10 (1935), pp. 26-30 (cit. on p. 5).

Karl H. Hofmann and Sidney A. Morris. The Lie theory of connected pro-Lie groups. Vol. 2. EMS Tracts in Mathematics. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. European Mathematical Society (EMS), Zu¨rich, 2007 (cit. on p. 2).

E. Hewitt and K. A. Ross. Abstract harmonic analysis. Vol. I. Second. Vol. 115.

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Structure of topological groups, integration theory, group representations. Berlin: Springer-Verlag, 1979, pp. ix+519 (cit. on p. 7).

Alan L. T. Paterson. Amenability. Vol. 29. Mathematical Surveys and Monographs.

American Mathematical Society, Providence, RI, 1988 (cit. on p. 3).

Volker Runde. Lectures on amenability. Vol. 1774. Lecture Notes in Mathematics.

19 references, page 1 of 2
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue