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Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on 𝕋

Published online by Cambridge University Press:  27 February 2009

MANFRED EINSIEDLER  and
ALEXANDER FISH
MANFRED EINSIEDLER
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (email: manfred@math.osu.edu)
ALEXANDER FISH
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (email: manfred@math.osu.edu)
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Abstract

We prove that if a Borel probability measure on the circle group is invariant under the action of a ‘large’ multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 151 - 157
Copyright
Copyright © Cambridge University Press 2009

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