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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

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 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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References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Abramov, L. M. and Rokhlin, V. A.. The entropy of a skew product of measure preserving transformation. Amer. Math. Soc. Transl. 48 (1965), 225–245.Google Scholar
[3]Anzai, H.. Ergodic skew product transformation on the torus. Osaka J. Math. 3 (1951), 83–99.Google Scholar
[4]Austin, T. and Lemańczyk, M.. Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles. J. Fixed Point Theory Appl. 6(1) (2009), 115–131.CrossRefGoogle Scholar
[5]Danilenko, A. and Lemańczyk, M.. A class of multipliers for 𝒲⊥. Israel J. Math. 148 (2005), 137–168.CrossRefGoogle Scholar
[6]Foguel, S. R.. The Ergodic Theory of Markov Processes. Van Nostrand, New York, 1969.Google Scholar
[7]Furstenberg, H.. Strict ergodicity and transformations of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
[8]Furstenberg, H.. Disjointness in ergodic theory, minimal sets and diophantine approximation. Math. Syst. Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[9]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[10]Furstenberg, H. and Weiss, B.. The finite multipliers of infinite ergodic transformations. Lecture Notes in Math. 668 (1978), 127–132.CrossRefGoogle Scholar
[11]Glasner, E.. On the multipliers of 𝒲⊥. Ergod. Th. & Dynam. Sys. 14 (1994), 129–140.CrossRefGoogle Scholar
[12]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[13]Glasner, E. and Weiss, B.. Processes disjoint from weak mixing. Trans. Amer. Math. Soc. 316 (1989), 689–703.CrossRefGoogle Scholar
[14]Glasner, S.. Proximal flows. Lecture Notes in Math. 517 (1976).Google Scholar
[15]Hahn, F. and Parry, W.. Some characteristic properties of dynamical systems with quasi-discrete spectra. Math. Systems Theory 2 (1968), 179–190.CrossRefGoogle Scholar
[16]Hamachi, T. and Osikawa, M.. Ergodic groups of automorphisms and Krieger’s theorems. Sem. Math. Sci. 3 (1991).Google Scholar
[17]Host, B., Méla, J.-F. and Parreau, F.. Analyse Harmonique des mesures (Astérisque, 135–136). Soc. Math. France, Paris, 1986.Google Scholar
[18]Host, B., Méla, J.-F. and Parreau, F.. Non-singular transformations and spectral analysis of measures. Bull. Soc. Math. France 119 (1991), 33–90.CrossRefGoogle Scholar
[19]Katok, A. B.. Cocycles, cohomology and combinatorial constructions in ergodic theory. Proc. Symp. Pure Math. 69 (2001), 107–173, in collaboration with E. A. Robinson Jr.CrossRefGoogle Scholar
[20]Katok, A. and Thouvenot, J.-P.. Spectral Properties and Combinatorial Constructions in Ergodic Theory (Handbook of Dynamical Systems, 1B). Elsevier B. V., Amsterdam, 2006, pp. 649–743.Google Scholar
[21]Lemańczyk, M.. Spectral Theory of Dynamical Systems (Encyclopedia of Complexity and System Science, 19). Springer, Berlin, 2009, pp. 8554–8575.Google Scholar
[22]Lemańczyk, M. and Lesigne, E.. Ergodicity of Rokhlin cocycles. J. Anal. Math. 85 (2001), 43–86.CrossRefGoogle Scholar
[23]Lemańczyk, M. and Parreau, F.. Rokhlin extensions and lifting disjointness. Ergod. Th. & Dynam. Sys. 23 (2003), 1525–1550.CrossRefGoogle Scholar
[24]Queffélec, M.. Substitution dynamical systems—spectral analysis. Lecture Notes in Math. 1294 (1987).Google Scholar
[25]Robinson, E. A.. A general condition for lifting theorems. Trans. Amer. Math. Soc. 330 (1992), 725–755.CrossRefGoogle Scholar
[26]Rudolph, D.. Classifying isometric extensions of a Bernoulli shift. Israel J. Math. 34 (1978), 36–60.Google Scholar
[27]Rudolph, D.. k-fold mixing lifts to weakly mixing isometric extensions. Ergod. Th. & Dynam. Sys. 5 (1985), 445–447.CrossRefGoogle Scholar
[28]Rudolph, D.. and cocycle extensions and complementary algebras. Ergod. Th. & Dynam. Sys. 6 (1986), 583–599.CrossRefGoogle Scholar
[29]Ryzhikov, V. V.. Joinings, intertwinings operators, factors, and mixing properties of dynamical systems. Russian Acad. Sci. Izv. Math. 42 (1994), 91–114.Google Scholar
[30]Schmidt, K.. Cocycles of Ergodic Transformation Groups (Lecture Notes in Mathematics, 1). MacMillan Co. of India, 1977.Google Scholar
[31]Zimmer, R.. Extensions of ergodic group actions. Illinois J. Math. 20(3) (1976), 373–409.CrossRefGoogle Scholar