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Recurrence in unipotent groups and ergodic nonabelian group extensions

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Abstract

LetT be a measure-preserving and ergodic transformation of a standard probability space (X,S, μ) and letf:X → SUT d (ℝ) be a Borel map into the group of unipotent upper triangulard ×d matrices. We modify an argument in [12] to obtain a sufficient condition for the recurrence of the random walk defined byf, in terms of the asymptotic behaviour of the distributions of the suitably scaled mapsf(n,x)=(fT n−1·fT n−2…fT·f). We give examples of recurrent cocycles with values in the continuous Heisenberg group H1(ℝ)=SUT3(ℝ), and we use a recurrent cocycle to construct an ergodic skew-product extension of an irrational rotation by the discrete Heisenberg group H1(ℤ)=SUT3(ℤ).

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Vienna, Austria

    Gernot Greschonig

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  1. Gernot Greschonig
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The author was partially supported by the FWF research project P16004-MAT.

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Greschonig, G. Recurrence in unipotent groups and ergodic nonabelian group extensions. Isr. J. Math. 147, 245–267 (2005). https://doi.org/10.1007/BF02785367

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  • DOI: https://doi.org/10.1007/BF02785367

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