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×2 and ×3 invariant measures and entropy

  • Daniel J. Rudolph (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700005629
  • Published online: 01 September 2008
Abstract
Abstract

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.

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[F]H. Furstenberg . Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation. Math. Sys. Theory 1 (1967), 149.

[I-T]A. Ionescu Tulcea . Contributions to information theory for abstract alphabets. Arkiv Math. 4 (1960), 235247.

[K]U. Krengel . Ergodic Theorems. De Gruyter Studies in Math. 6, de Gruyter-Berlin: New York, 1985.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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