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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: [email protected])
ALEXANDER FISH
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (email: [email protected])
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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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References

[1]Bourgain, J., Furman, A., Lindenstrauss, E. and Mozes, S.. Invariant measures and stiffness for non-abelian groups of toral automorphisms. C. R. Math. Acad. Sci. Paris 344(12) (2007), 737–742.CrossRefGoogle Scholar
[2]Bourgain, J. and Lindenstrauss, E.. Entropy of quantum limits. Comm. Math. Phys. 233(1) (2003), 153–171.CrossRefGoogle Scholar
[3]Einsiedler, M., Katok, A. and Lindenstrauss, E.. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164(2) (2006), 513–560.CrossRefGoogle Scholar
[4]Einsiedler, M. and Lindenstrauss, E.. Diagonalizable flows on locally homogeneous spaces and number theory. International Congress of Mathematicians, II. Eur. Math. Soc., Zürich, 2006, pp. 1731–1759.Google Scholar
[5]Einsiedler, M., Lindenstrauss, E., Michel, Ph. and Venkatesh, A.. The distribution of periodic torus orbits on homogeneous spaces. Duke Math. J. to appear.Google Scholar
[6]Einsiedler, M., Lindenstrauss, E., Michel, Ph. and Venkatesh, A.. Distribution of periodic torus orbits and Duke’s theorem for cubic fields. Preprint, 2007.Google Scholar
[7]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[8]Johnson, A. S. A.. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers. Israel J. Math. 77(1-2) (1992), 211–240.CrossRefGoogle Scholar
[9]Katznelson, Y.. An Introduction to Harmonic Analysis, 3rd edn(Cambridge Mathematical Library). Cambridge University Press, Cambridge, 2004.CrossRefGoogle Scholar
[10]Lindenstrauss, E.. Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163(1) (2006), 165–219.CrossRefGoogle Scholar
[11]Michel, P. and Venkatesh, A.. Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik. International Congress of Mathematicians, II. Eur. Math. Soc., Zürich, 2006, pp. 421–457.Google Scholar
[12]Rudolph, J. D.. ×2 and ×3 invariant measures and entropy. Ergod. Th. & Dynam. Sys. 10(2) (1990), 395–406.CrossRefGoogle Scholar