Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-28T12:07:22.687Z Has data issue: false hasContentIssue false

Automorphisms of compact groups

Published online by Cambridge University Press:  19 September 2008

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[Ab]Abramov, L.M.. The entropy of an automorphism of a solenoidal group. Theory Prob. Appl. 4 (1959), 231236.CrossRefGoogle Scholar
[AP]Adler, R.L. & Palais, R.. Homeomorphic conjugacy of automorphisms of the torus. Proc. Amer. Math. Soc. 16 (1965), 12221225.CrossRefGoogle Scholar
[Ao]Aoki, N.. A simple proof of the Bernoullicity of ergodic automorphisms of compact abelian groups. Israeli Math. 38 (1981), 189198.CrossRefGoogle Scholar
[AD]Aoki, N. & Dateyama, M.. The relationship between algebraic numbers and expansiveness of automorphisms on compact abelian groups. Fundamenta Math. 117 (1983), 2135.CrossRefGoogle Scholar
[Ar]Arov, D.Z.. The computation of the entropy for one class of group endomorphisms. Zap. Mekh.-Matem. Fakulteta Kharkov Matem. 30 (1964), 4869.Google Scholar
[AM]Atiyah, M.F. & MacDonald, I. G.. Introduction to Commutative Algebra. Addison-Wesley: Reading, Mass., 1969.Google Scholar
[Be]Berend, D.. Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc. 289 (1985), 393407.CrossRefGoogle Scholar
[Bo]Bowen, R.. Equilibrium states and the ergodic theory of Anosov difleomorphisms. Lecture Notes in Mathematics. 470, Springer: Berlin-Heidelberg-New York, 1975.CrossRefGoogle Scholar
[Dy]Dye, H.A.. On the ergodic mixing theorem. Trans. Amer. Math. Soc. 118 (1965), 123130.CrossRefGoogle Scholar
[ES]Eilenberg, S. & Steenrod, N.. Foundations of Algebraic Topology. Princeton University Press: Princeton, 1952.CrossRefGoogle Scholar
[Fr]Fried, D.. Finitely presented dynamical systems. Ergod. Th. & Dynam. Sys. 7 (1987), 489507.CrossRefGoogle Scholar
[GS]Grunewald, F.J. & Segal, D.. Decision problems concerning S-arithmetic groups. J. Symbolic Logic 50 (1985), 743772.CrossRefGoogle Scholar
[Ha]Halmos, P.R.. On automorphisms of compact groups. Bull. Amer. Math. Soc. 49 (1943), 619624.CrossRefGoogle Scholar
[Ka]Kaplansky, I.. Groups with representations of bounded degree. Can. J. Math. 1 (1949), 105112.CrossRefGoogle Scholar
[Ki]Kitchens, B.P.. Expansive dynamics on zero-dimensional groups. Ergod. Th. & Dynam. Sys. 7 (1987), 249261.CrossRefGoogle Scholar
[La]Lam, P.-F.. On expansive transformation groups. Trans. Amer. Math. Soc. 105 (1970), 131138.CrossRefGoogle Scholar
[Ln]Lawton, W.. 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.CrossRefGoogle Scholar
[LP]Laxton, R. & Parry, W.. On the periodic points of certain automorphisms and a system of polynomial identities. J. Algebra 6 (1967), 388393.CrossRefGoogle Scholar
[Lc]Ledrappier, F.. Un champ markovien peut etre d''ntropie nulle et melangeant. C.R. Acad. Sc. Paris, Ser. A. 287 (1978), 561562.Google Scholar
[LSW]Lind, D.A., Schmidt, K. & Ward, T.. Mahler measures and entropy for automorphisms of compact groups. In preparation.Google Scholar
[LW]Lind, D.A. & Ward, T.. Automorphisms of solenoids and p-adic entropy. Ergod. Th. & Dynam. Sys. 8 (1988), 411419.CrossRefGoogle Scholar
[MT]Miles, G. & Thomas, R. K.. The breakdown of automorphisms of compact topological groups. In: Studies in Probability and Ergodic Theory, Advances in Mathematics Supplementary Studies, 2, Academic Press: New York-London, 1978, pp. 207218.Google Scholar
[Re]Reddy, W.. The existence of expansive homeomorphisms on manifolds. Duke Math. J. 32 (1965), 494509.CrossRefGoogle Scholar
[Ro]Robinson, R.M.. Undecidability and nonperiodicity for tilings of the plane. Inventiones Math. 12 (1971), 177209.CrossRefGoogle Scholar
[Ru]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: Reading, Mass., 1978.Google Scholar
[SI]Schmidt, K.. Asymptotic properties of unitary representations and mixing. Proc. London Math. Soc. 48 (1984), 445460.CrossRefGoogle Scholar
[S2]Schmidt, K.. Automorphisms of compact abelian groups and affine varieties. Preprint (1987).Google Scholar
[S3]Schmidt, K.. Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pacific J. Math, (to appear).Google Scholar
[W1]Williams, R.F.. A note on unstable homeomorphisms. Proc. Amer. Math. Soc. 6 (1955), 308309.CrossRefGoogle Scholar
[W2]Williams, R.F.. Classification of subshifts of finite type. Annals Math. 98 (1973), 120153. Errata: 99 (1974), 380–381.CrossRefGoogle Scholar
[Ws]Wilson, A.M.. On endomorphisms of a solenoid. Proc. Amer. Math. Soc. 55 (1976), 6974.CrossRefGoogle Scholar
[Wu]Wu, T.S.. Expansive automorphisms of compact groups. Math. Scand. 18 (1966), 1324.CrossRefGoogle Scholar
[Yu]Yuzvinskii, S.A.. Computing the entropy of a group of endomorphisms. Siberian Math. J. 8 (1967), 172178.CrossRefGoogle Scholar