Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T16:56:15.931Z Has data issue: false hasContentIssue false
  • English
  • Français

A topological lens for a measure-preserving system

Published online by Cambridge University Press:  02 February 2010

E. GLASNER,
M. LEMAŃCZYK and
B. WEISS
E. GLASNER
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel (email: glasner@math.tau.ac.il)
M. LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland (email: mlem@mat.uni.torun.pl)
B. WEISS
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel (email: weiss@math.huji.ac.il)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011, pp. 49 - 75
Copyright
Copyright © Cambridge University Press 2009

References

[1]Glasner, E. and Weiss, B.. On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems, Vol. 1B. Elsevier B.V., Amsterdam, 2006, pp. 597–648.CrossRefGoogle Scholar
[2]Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G.. Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar
[3]Danilenko, A.. On simplicity concepts for ergodic actions. J. Anal. Math. 102 (2007), 77–117.CrossRefGoogle Scholar
[4]Ellis, R. and Nerurkar, M.. Weakly almost periodic flows. Trans. Amer. Math. Soc. 313 (1989), 103–119.CrossRefGoogle Scholar
[5]Fathi, A.. Le groupe de transformations de [0,1] qui preservent la measure de Lebesgue est un groupe simple. Israel J. Math. 29 (1978), 302–308.CrossRefGoogle Scholar
[6]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[7]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[8]Glasner, E. and King, J.. A zero–one law for dynamical properties. Topological Dynamics and Applications (Minneapolis, MN, 1995) (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 231–242.CrossRefGoogle Scholar
[9]Glasner, E. and Weiss, B.. Sensitive dependence on initial conditions. Nonlinearity 6 (1993), 1067–1075.CrossRefGoogle Scholar
[10]del Junco, A. and Rudolph, D. J.. On ergodic actions whose self-joinings are graphs. Ergod. Th. & Dynam. Sys. 7 (1987), 531–557.CrossRefGoogle Scholar
[11]Kechris, A. S. and Rosendal, C.. Turbulence, amalgamation and generic automorphisms of homogeneous structures. Proc. Lond. Math. Soc. (3) 94 (2007), 302–350.CrossRefGoogle Scholar
[12]Kieffer, J. C.. A simple development of the Thouvenot relative isomorphism theory. Ann. Probab. 12 (1984), 204–211.CrossRefGoogle Scholar
[13]Kǔshnirenko, A. G.. Metric invariants of entropy type. Uspekhi Mat. Nauk 22(5) (1967), 57–65, 137, in Russian.Google Scholar
[14]Lemańczyk, M., Parreau, F. and Thouvenot, J.-P.. Gaussian automorphisms whose ergodic self-joinings are Gaussian. Fund. Math. 164 (2000), 253–293.CrossRefGoogle Scholar
[15]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.CrossRefGoogle Scholar
[16]Ornstein, D.. Ergodic Theory, Randomness, and Dynamical Systems (James K. Whittemore Lectures in Mathematics given at Yale University, Yale Mathematical Monographs, 5). Yale University Press, New Haven, CT, 1974.Google Scholar
[17]Thouvenot, J.-P.. Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systm̀es dont l’un est un schéma de Bernoulli. Israel J. Math. 21 (1975), 177–207.CrossRefGoogle Scholar
[18]Zimmer, R. J.. Extensions of ergodic group actions. Illinois J. Math. 20 (1976), 373–409.CrossRefGoogle Scholar
[19]Zimmer, R. J.. Ergodic actions with generalized discrete spectrum. Illinois J. Math. 20 (1976), 555–588.CrossRefGoogle Scholar