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Lifting mixing properties by Rokhlin cocycles

Published online by Cambridge University Press:  08 November 2011

MARIUSZ LEMAŃCZYK and
FRANÇOIS PARREAU
MARIUSZ LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 12/18 Chopin Street, 87–100 Toruń, Poland (email: mlem@mat.uni.torun.pl)
FRANÇOIS PARREAU
Affiliation:
Laboratoire d’Analyse, Géométrie, et Applications, UMR 7539, Université Paris 13 et CNRS, 99, av. J.-B. Clément, 93430 Villetaneuse, France (email: parreau@math.univ-paris13.fr)
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Abstract

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We study the problem of lifting various mixing properties from a base automorphism T∈Aut(X,ℬ,μ) to skew products of the form Tφ,𝒮, where φ:X→G is a cocycle with values in a locally compact Abelian group G, 𝒮=(Sg)g∈G is a measurable representation of G in Aut(Y,𝒞,ν) and Tφ,𝒮 acts on the product space (X×Y,ℬ⊗𝒞,μ⊗ν)by \[ \tfs (x,y)=(Tx,S_{\va (x)}(y)). \] It is also shown that whenever T is ergodic (mildly mixing, mixing) but Tφ,𝒮 is not ergodic (is not mildly mixing, not mixing), then, on a non-trivial factor 𝒜⊂𝒞 of 𝒮, the corresponding Rokhlin cocycle x↦Sφ(x) ∣𝒜 is a coboundary (a quasi-coboundary).

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012, pp. 763 - 784
Copyright
Copyright © Cambridge University Press 2011

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