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×2 and ×3 invariant measures and entropy

Published online by Cambridge University Press:  19 September 2008

Daniel J. Rudolph
Daniel J. Rudolph
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
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Abstract

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Let p and q be relatively prime natural numbers. Define T0 and S0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).

Let μ be a borel measure invariant for both T0 and S0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T0 and S0 have entropy zero. Equivalently, both T0 and S0 are μ-almost surely invertible.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 395 - 406
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[F]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation. Math. Sys. Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[I-T]Tulcea, A. Ionescu. Contributions to information theory for abstract alphabets. Arkiv Math. 4 (1960), 235–247.Google Scholar
[K]Krengel, U.. Ergodic Theorems. De Gruyter Studies in Math. 6, de Gruyter-Berlin: New York, 1985.Google Scholar
[L]Lyons, R.. On measures simultaneously 2- and 3-invariant. Preprint.Google Scholar