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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: [email protected])
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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

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