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Automorphisms of compact groups

Published online by Cambridge University Press:  19 September 2008

Bruce Kitchens  and
Klaus Schmidt
Bruce Kitchens
Affiliation:
Mathematical Sciences Department, IBM T. J. Watson Research Center, Yorktown Heights, NY, USA
Klaus Schmidt
Affiliation:
Mathematics Institute, University of Warwick, Coventry, UK
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Abstract

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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 4 , December 1989 , pp. 691 - 735
Copyright
Copyright © Cambridge University Press 1989

References

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