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Distal systems in topological dynamics and ergodic theory

Part of: Ergodic theory Topological dynamics

Published online by Cambridge University Press:  01 August 2022

NIKOLAI EDEKO Open the ORCID record for NIKOLAI EDEKO [Opens in a new window]  and
HENRIK KREIDLER Open the ORCID record for HENRIK KREIDLER [Opens in a new window]
NIKOLAI EDEKO
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland (e-mail: nikolai.edeko@math.uzh.ch)
HENRIK KREIDLER*
Affiliation:
Fachgruppe für Mathematik und Informatik, Bergische Universität Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany
*
e-mail: kreidler@uni-wuppertal.de
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Abstract

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg–Zimmer tower of a measurably distal system and showing that at each step there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.

Keywords

ergodic theorytopological dynamicsdistal systemstopological models

MSC classification

Primary: 37A05: Measure-preserving transformations 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 8 , August 2023 , pp. 2651 - 2672
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Auslander, J.. Minimal Flows and their Extensions. Elsevier, Amsterdam, 1988.Google Scholar
Auslander, J.. Characterizations of distal and equicontinuous extensions. Ergodic Theory and Dynamical Systems. Proceedings of the Ergodic Theory Workshops at University of North Carolina at Chapel Hill, 2011–2012. Ed. Assani, I.. De Gruyter, Berlin, 2013, pp. 59–66.Google Scholar
Berglund, J. F., Junghenn, H. and Milnes, P.. Analysis on Semigroups: Function Spaces, Compactifications, Representations . Wiley, New York, 1989.Google Scholar
Bronšteǐn, I. U.. Extensions of Minimal Transformation Groups. Sijthoff & Noordhoff, Alphen aan den Rijn, 1979.CrossRefGoogle Scholar
Derrien, J.-M.. On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups. Ann. Inst. Henri Poincaré Probab. Stat. 36 (2000), 291–300.CrossRefGoogle Scholar
Dupré, M. J. and Gillette, R. M.. Banach Bundles, Banach Modules and Automorphisms of C*-Algebras. Pitman, Boston, MA, 1983.Google Scholar
de Vries, J.. Elements of Topological Dynamics. Springer, Dordrecht, 1993.CrossRefGoogle Scholar
Edeko, N.. On the isomorphism problem for non-minimal transformations with discrete spectrum. Discrete Contin. Dyn. Syst. Ser. A 39(10) (2019), 6001–6021.10.3934/dcds.2019262CrossRefGoogle Scholar
Eisner, T., Farkas, B., Haase, M. and Nagel, R.. Operator Theoretic Aspects of Ergodic Theory. Springer, Cham, 2015.CrossRefGoogle Scholar
Edeko, N., Haase, M. and Kreidler, H.. A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem. Preprint, 2021, arXiv:2104.04865.Google Scholar
Edeko, N. and Kreidler, H.. Uniform enveloping semigroupoids for groupoid actions. J. Anal. Math., to appear.Google Scholar
Ellis, R.. Topological dynamics and ergodic theory. Ergod. Th. & Dynam. Sys. 7 (1987), 25–47.CrossRefGoogle Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory: With a View towards Number Theory. Springer, London, 2011.CrossRefGoogle Scholar
Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.CrossRefGoogle Scholar
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.CrossRefGoogle Scholar
Gierz, G.. Bundles of Topological Vector Spaces and Their Duality. Springer, Berlin, 1982.CrossRefGoogle Scholar
Gutman, Y. and Lian, Z.. Strictly ergodic distal models and a new approach to the Host–Kra factors. Preprint, 2019, arXiv:1909.11349v2.Google Scholar
Glasner, E.. Ergodic Theory via Joinings. American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Glasner, E. and Weiss, B.. On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems. Vol. 1B. Ed. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2006, pp. 597–648.Google Scholar
Huang, W., Shao, S. and Xiangdong, Y.. Pointwise convergence of multiple ergodic averages and strictly ergodic models. J. Anal. Math. 139 (2019), 265–305.CrossRefGoogle Scholar
Jamneshan, A. and Tao, T.. An uncountable Mackey–Zimmer theorem. Studia Math. 266 (2022), 241–289.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
Lindenstrauss, E.. Measurable distal and topological distal systems. Ergod. Th. & Dynam. Sys. 19(4) (1999), 1063–1076.CrossRefGoogle Scholar
Parry, W.. Zero entropy of distal and related transformations. Topological Dynamics: An International Symposium. Ed. Auslander, J. and Gottschalk, W. H.. W. A. Benjamin, New York, 1968, pp. 383–389.Google Scholar
Parry, W.. A note on cocycles in ergodic theory. Compos. Math. 28 (1974), 343–350.Google Scholar
Tao, T.. Poincaré’s Legacies, Part I. American Mathematical Society, Providence, RI, 2009.Google Scholar
Zimmer, R. J.. Extensions of ergodic group actions. Illinois J. Math. 20 (1976), 373–409.CrossRefGoogle Scholar