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A family of counterexamples in ergodic theory

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Abstract

LetT α be the translationx↦x+α (mod 1) of [0, 1), α irrational. LetT be the Lebesgue measure-preserving automorphism ofX=[0, 3/2) defined byTx = x + 1 forx∈[0, 1/2),Tx=T α(x−1) forx∈[1,3/2) andTx = T α x forx∈[1/2, 1), i.e.T isT α with a tower of height one built over [0, 1/2). If α is poorly approximable by rationals (there does not exist {p n /q n } with |α−p n /q n |=o(q n −2)) and λ is a measure onX k all of whose one-dimensional marginals are Lebesgue and which is ⊗ k i − 1 T 1 invariant and ergodic (l>0) then λ is a product of off-diagonal measures. This property suffices for many purposes of counterexample construction. A connection is established with the POD (proximal orbit dense) condition in topological dynamics.

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Authors and Affiliations

  1. Department of Mathematics, Ohio State University, 43210, Columbus, OH, USA

    Andrés del Junco

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  1. Andrés del Junco
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Research supported in part by NSF contract MCS-8003038.

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del Junco, A. A family of counterexamples in ergodic theory. Israel J. Math. 44, 160–188 (1983). https://doi.org/10.1007/BF02760618

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  • DOI: https://doi.org/10.1007/BF02760618

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