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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:
  • Published online: 01 September 2008

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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[Ab]L.M. Abramov . The entropy of an automorphism of a solenoidal group. Theory Prob. Appl. 4 (1959), 231236.

[AP]R.L. Adler & R. Palais . Homeomorphic conjugacy of automorphisms of the torus. Proc. Amer. Math. Soc. 16 (1965), 12221225.

[Ao]N. Aoki . A simple proof of the Bernoullicity of ergodic automorphisms of compact abelian groups. Israeli Math. 38 (1981), 189198.

[Be]D. Berend . Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc. 289 (1985), 393407.

[Dy]H.A. Dye . On the ergodic mixing theorem. Trans. Amer. Math. Soc. 118 (1965), 123130.

[ES]S. Eilenberg & N. Steenrod . Foundations of Algebraic Topology. Princeton University Press: Princeton, 1952.

[Ha]P.R. Halmos . On automorphisms of compact groups. Bull. Amer. Math. Soc. 49 (1943), 619624.

[Ka]I. Kaplansky . Groups with representations of bounded degree. Can. J. Math. 1 (1949), 105112.

[Ln]W. Lawton . The structure of compact connected groups which admit an expansive automorphism. In: Recent advances in Topological Dynamics, Lecture Notes in Mathematics. Springer: Berlin- Heidelberg-New York, 1973, pp. 182196.

[LP]R. Laxton & W. Parry . On the periodic points of certain automorphisms and a system of polynomial identities. J. Algebra 6 (1967), 388393.

[Re]W. Reddy . The existence of expansive homeomorphisms on manifolds. Duke Math. J. 32 (1965), 494509.

[Ro]R.M. Robinson . Undecidability and nonperiodicity for tilings of the plane. Inventiones Math. 12 (1971), 177209.

[W1]R.F. Williams . A note on unstable homeomorphisms. Proc. Amer. Math. Soc. 6 (1955), 308309.

[W2]R.F. Williams . Classification of subshifts of finite type. Annals Math. 98 (1973), 120153. Errata: 99 (1974), 380–381.

[Ws]A.M. Wilson . On endomorphisms of a solenoid. Proc. Amer. Math. Soc. 55 (1976), 6974.

[Yu]S.A. Yuzvinskii . Computing the entropy of a group of endomorphisms. Siberian Math. J. 8 (1967), 172178.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
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