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Ergodic properties of discrete dynamical systems and enveloping semigroups

Published online by Cambridge University Press:  04 August 2014

A. V. ROMANOV
A. V. ROMANOV*
Affiliation:
National Research University Higher School of Economics, Moscow, Russia email [email protected]
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Abstract

For a continuous semicascade on a metrizable compact set ${\rm\Omega}$, we consider the weak$^{\ast }$ convergence of generalized operator ergodic means in $\text{End}\,C^{\ast }({\rm\Omega})$. We discuss conditions under which: every ergodic net contains a convergent sequence; all ergodic nets converge; all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on ${\rm\Omega}$, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three associated algebraic-topological objects: the Ellis semigroup $E$, the Köhler operator semigroup ${\rm\Gamma}\subset \text{End}\,C^{\ast }({\rm\Omega})$, and the semigroup $G=\overline{\text{co}}\,{\rm\Gamma}$. The main results are stated for semicascades with metrizable $E$ and for tame semicascades.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 1 , February 2016 , pp. 198 - 214
Copyright
© Cambridge University Press, 2014 

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References

Bogachev, V. I.. Measure Theory, Vol. I. Springer, Berlin, 2007.CrossRefGoogle Scholar
Bourbaki, N.. Elements of Mathematics: General Topology, Part 2. Hermann, Paris, 1966.Google Scholar
Dunford, N. and Schwartz, J. T.. Linear Operators General Theory, Part I. Interscience Publishers, New York, 1958.Google Scholar
Edwards, R. E.. Functional Analysis Theory and Applications. Holt, Rinehart and Winston, New York, 1965.Google Scholar
Ellis, R.. Lectures on Topological Dynamics. Benjamin, New York, 1969.Google Scholar
Ellis, R. and Nerurkar, M.. Weakly almost periodic flows. Trans. Amer. Math. Soc. 313 (1989), 103–119.CrossRefGoogle Scholar
Glasner, E.. On tame dynamical systems. Colloq. Math. 105 (2006), 283–295.CrossRefGoogle Scholar
Glasner, E.. Enveloping semigroups in topological dynamics. Topology Appl. 154 (2007), 2344–2363.CrossRefGoogle Scholar
Glasner, E. and Megrelishvili, M.. Hereditarily non-sensitive dynamical systems and linear representations. Colloq. Math. 104 (2006), 223–283.CrossRefGoogle Scholar
Glasner, E., Megrelishvili, M. and Uspenskij, V. V.. On metrizable enveloping semigroups. Israel J. Math. 164 (2008), 317–332.CrossRefGoogle Scholar
Huang, W.. Tame systems and scrambled pairs under an abelian group action. Ergod. Th. & Dynam. Sys. 26 (2006), 1549–1567.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.Google Scholar
Kerr, D. and Li, H.. Independence in topological and C ∗ -dynamics. Math. Ann. 338 (2007), 869–926.CrossRefGoogle Scholar
Köhler, A.. Enveloping semigroups for flows. Proc. R. Irish Acad. 95A (1995), 179–191.Google Scholar
Kryloff, N. and Bogoliouboff, N.. La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. of Math. 38 (1937), 65–113.CrossRefGoogle Scholar
Lloyd, S. P.. On the ergodic theorem of Sine. Proc. Amer. Math. Soc. 56 (1976), 121–126.CrossRefGoogle Scholar
Phelps, R. R.. Lectures on Choquet’s Theorem. Van Nostrand, Princeton, NJ, 1966.Google Scholar
Romanov, A. V.. Weak∗ convergence of operator means. Izv. Math. 75 (2011), 1165–1183.CrossRefGoogle Scholar
Romanov, A. V.. Enveloping semigroups and ergodic properties of compact semi-cascades. International Conference on Differential Equations and Dynamical Systems (Suzdal 2012), Abstracts. MIAN, Moscow, 2012, pp. 151–152.Google Scholar
Romanov, A. V.. Ordinary semicascades and their ergodic properties. Funct. Anal. Appl. 47 (2013), 160–163.CrossRefGoogle Scholar