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  • Ergodic Theory and Dynamical Systems, Volume 9, Issue 4
  • December 1989, pp. 691-735

Automorphisms of compact groups

  • Bruce Kitchens (a1) and Klaus Schmidt (a2)
  • DOI: http://dx.doi.org/10.1017/S0143385700005290
  • Published online: 01 September 2008
Abstract
Abstract

We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.

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