Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-17T04:15:36.493Z Has data issue: false hasContentIssue false
  • English
  • Français

On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

Published online by Cambridge University Press:  13 October 2009

C. R. E. RAJA
C. R. E. RAJA*
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560 059, India (email: creraja@isibang.ac.in)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1803 - 1816
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Baumgartner, U. and Willis, G.. Contraction groups and scales of automorphisms of totally disconnected locally compact groups. Israel J. Math. 142 (2004), 221–248.CrossRefGoogle Scholar
[2]Berend, D.. Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc. 289(1) (1985), 393–407.Google Scholar
[3]Bergelson, V. and Gorodnik, A.. Ergodicity and mixing of non-commuting epimorphisms. Proc. London Math. Soc. (3) 95 (2007), 329–359.CrossRefGoogle Scholar
[4]Dixmier, J.. C*-Algebras (North-Holland Mathematical Library, 15). North-Holland, Amsterdam, 1977, translated from the French by Francis Jellett.Google Scholar
[5]Ellis, R.. Distal transformation groups. Pacific J. Math. 8 (1958), 401–405.CrossRefGoogle Scholar
[6]Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.Google Scholar
[7]Golod, E. S.. On nil-algebras and finitely approximable p-groups. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273–276.Google Scholar
[8]Hewit, E. and Ross, K. A.. Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups. Integration Theory, Group Representations (Die Grundlehren der mathematischen Wissenschaften, 115). Academic Press, New York, 1963, series now published by Springer, Berlin.Google Scholar
[9]Hewit, E. and Ross, K. A.. Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups (Die Grundlehren der mathematischen Wissenschaften, 152). Springer, Berlin, 1970.Google Scholar
[10]Hofmann, K. H. and Morris, S. A.. The structure of compact groups. A Primer for the Student—A Handbook for the Expert (de Gruyter Studies in Mathematics, 25). Walter de Gruyter, Berlin, 1998.Google Scholar
[11]Jaworksi, W. and Raja, C. R. E.. The Choquet–Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth. New York J. Math. 13 (2007), 159–174.Google Scholar
[12]Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9 (1989), 691–735.Google Scholar
[13]Morris, S. A.. Pontryagin Duality and the Structure of Locally Compact Abelian Groups (London Mathematical Society Lecture Note Series, 29). Cambridge University Press, Cambridge, 1977.Google Scholar
[14]Raja, C. R. E.. Distal actions and ergodic actions on compact groups. New York J. Math. 15 (2009), 301–318.Google Scholar
[15]Rohlin, V. A.. On endomorphisms of compact commutative groups. Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 329–340.Google Scholar
[16]Schmidt, K.. Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496.Google Scholar
[17]Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128). Birkhäuser, Basel, 1995.Google Scholar
[18]Serre, J. P.. Lie Algebras and Lie Groups. 1964 Lectures given at Harvard University, 2nd edn(Lecture Notes in Mathematics, 1500). Springer, Berlin, 1992.Google Scholar
[19]Sharp, R. Y.. Steps in Commutative Algebra (London Mathematical Society Student Texts, 19). Cambridge University Press, Cambridge, 1990.Google Scholar
[20]Weil, A.. Basic Number Theory (Classics in Mathematics). Springer, Berlin, 1995, reprint of the second (1973) edition.Google Scholar