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Weak mixing properties for non-singular actions

Published online by Cambridge University Press:  01 June 2015

ELI GLASNER  and
BENJAMIN WEISS
ELI GLASNER
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel email glasner@math.tau.ac.il
BENJAMIN WEISS
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel email weiss@math.huji.ac.il
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Abstract

For a general group $G$ we consider various weak mixing properties of non-singular actions. In the case where the action is actually measure preserving all these properties coincide, and our purpose here is to check which implications persist in the non-singular case.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2203 - 2217
Copyright
© Cambridge University Press, 2015 

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References

Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50) . American Mathematical Society, Providence, RI, 1997.Google Scholar
Aaronson, J. and Lemańczyk, M.. Exactness of Rokhlin endomorphisms and weak mixing of Poisson boundaries. Algebraic and Topological Dynamics (Contemporary Mathematics, 385) . American Mathematical Society, Providence, RI, 2005, pp. 77–87.Google Scholar
Aaronson, J., Lin, M. and Weiss, B.. Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products. Israel J. Math. 33(3–4) (1979), 198–224 (1980).Google Scholar
Aaronson, J. and Nadkarni, M.. L ∞ eigenvalues and L 2 spectra of nonsingular transformations. Proc. Lond. Math. Soc. (2) 55 (1987), 538–570.Google Scholar
Bader, U. and Furman, A.. Boundaries, Weyl groups, and superrigidity. Electron. Res. Announc. Math. Sci. 19 (2012), 41–48.Google Scholar
Beck, A.. Eigen operators of ergodic transformations. Trans. Amer. Math. Soc. 94 (1960), 118–129.Google Scholar
Björklund, M.. Five remarks about random walks on groups. Preprint, 2014, arXiv:1406.0763.Google Scholar
Bowels, A., Fidkowski, L., Marinello, A. E. and Silva, C. E.. Double ergodicity of infinite transformations. Illinois J. Math. 45 (2001), 999–1019.Google Scholar
Burger, M. and Monod, N.. Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12(2) (2002), 219–280.Google Scholar
Dixmier, J.. C ∗ -Algebras. North-Holland, Amsterdam, 1982.Google Scholar
Furstenberg, H.. Random Walks and Discrete Subgroups of Lie Groups (Advances in Probability and Related Topics, 1) . Dekker, New York, 1971, pp. 1–63.Google Scholar
Furstenberg, H.. Boundary theory and stochastic processes on homogeneous spaces. Harmonic Analysis on Homogeneous Spaces (Proceedings of Symposia in Pure Mathematics, Vol. XXVI, Williams College, Williamstown, MA, 1972) . American Mathematical Society, Providence, RI, 1973, pp. 193–229.Google Scholar
Furstenberg, H. and Glasner, E.. Stationary dynamical systems. Contemp. Math. 532 (2010), 1–28.CrossRefGoogle Scholar
Furstenberg, H. and Glasner, E.. Recurrence for stationary group actions. From Fourier Analysis and Number Theory to Radon Transforms and Geometry (Developments in Mathematics, 28) . Springer, New York, 2013, pp. 283–291.Google Scholar
Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101) . American Mathematical Society, Providence, RI, 2003, 101.Google Scholar
Glücksam, A.. Ergodic multiplier properties. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2013,arXiv:1306.3669.Google Scholar
Grigoriev, I., Ince, N., Iordan, M. C., Lubin, A. and Silva, C. E.. On 𝜇-compatible metrics and measurable sensitivity. Colloq. Math. 126 (2012), 53–72.CrossRefGoogle Scholar
James, J., Koberda, T., Lindsey, K., Speh, P. and Silva, C. E.. Measurable sensitivity. Proc. Amer. Math. Soc. 136(10) (2008), 3549–3559.Google Scholar
Jaworski, W.. Strongly approximately transitive group actions, the Choquet–Deny theorem, and polynomial growth. Pacific J. Math. 165 (1994), 115–129.Google Scholar
Kaimanovich, V.. The Poisson boundary of covering Markov operators. Israel J. Math. 89 (1995), 77–134.Google Scholar
Kaimanovich, V. A.. Double ergodicity of the Poisson boundary and applications to bounded cohomology. Geom. Funct. Anal. 13(4) (2003), 852–861.Google Scholar
Kaimanovich, V. A. and Vershik, A. M.. Random walks on discrete groups: boundary and entropy. Ann. Probab. 11 (1983), 457–490.CrossRefGoogle Scholar
Krengel, U.. Ergodic theorems (de Gruyter Studies in Mathematics, 6) . Walter de Gruyter, Berlin, 1985. With a supplement by Antoine Brunel.Google Scholar
Mackey, G. H.. The Theory of Unitary Group Representations. University of Chicago Press, Chicago, 1976, (an adapted version of the 1955 ‘Chicago Notes’).Google Scholar
Margulis, G. A.. Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3), 17) . Springer, Berlin, 1991.Google Scholar
Monod, N.. Continuous Bounded Cohomology of Locally Compact Groups (Lecture Notes in Mathematics, 1758) . Springer, Berlin, 2001.CrossRefGoogle Scholar
Monod, N. and Shalom, Y.. Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differential Geom. 67(3) (2004), 395–455.Google Scholar
Moore, C. C.. Groups with finite dimensional irreducible representations. Trans. Amer. Math. Soc. 166 (1972), 401–410.CrossRefGoogle Scholar
Nevo, A. and Zimmer, R. J.. Homogenous projective factors for actions of semi-simple Lie groups. Invent. Math. 138 (1999), 229–252.Google Scholar
Rudolph, D. and Silva, C. E.. Minimal self-joinings for nonsingular transformations. Ergod. Th. & Dynam. Sys. 9 (1989), 759–800.Google Scholar
Tomiyama, J.. Invitation to C ∗ -algebras and Topological Dynamics. World Scientific, Singapore, 1987.Google Scholar
Varadarajan, V. S.. Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191–220.Google Scholar