Abstract
When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g ∈ G, in the compact space X X. Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d G (x, y) ≔ sup{d(gx, gy): g ∈ G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Akin, Enveloping linear maps, in Topological dynamics and applications, Contemporary Mathematics vol. 215, a volume in honor of R. Ellis, 1998, pp. 121–131.
E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic, in Convergence in Ergodic Theory and Probability, Walter de Gruyter & Co. 1996, pp. 25–40.
E. Akin, J. Auslander and K. Berg, Almost equicontinuity and the enveloping semigroup, in Topological dynamics and applications, Contemporary Mathematics vol. 215, a volume in honor of R. Ellis, 1998, pp. 75–81.
J. Auslander, Minimal Flows and their Extensions, Mathematics Studies 153, Notas de Matemática, 1988.
J. Auslander and J. Yorke, Interval maps, factors of maps, and chaos, The Tohoku Mathematical Journal 32 (1980), 177–188.
J. Bourgain, D. H. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions, American Journal of Mathematics 100 (1978), 845–886.
R. D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikod’ym Property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, 1983.
R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific Technical, 1993.
T. Downarowicz, Weakly almost periodic flows and hidden eigenvalues, in Topological dynamics and applications, Contemporary Mathematics, vol. 215, a volume in honor of R. Ellis, 1998, pp. 101–120.
R. Ellis, Distal transformation groups, Pacific Journal of Mathematics 8, (1957), 401–405.
R. Ellis, Locally compact transformation groups, Duke Mathematical Journal 24, (1957), 119–126.
R. Ellis, Equicontinuity and almost periodic functions, Proceedings of the American Mathematical Society 10 (1959), 637–643.
R. Ellis, The enveloping semigroup of projective flows, Ergodic Theory and Dynamical Systems 13 (1993), 635–660.
R. Ellis and M. Nerurkar, Weakly almost periodic flows, Transactions of the American Mathematical Society 313 (1989), 103–119.
R. Engelking, General Topology, Revised and Completed Edition, Heldermann Verlag, Berlin, 1989.
M. Fabian, Gateaux Differentiability of Convex Functions and Topology. Weak Asplund spaces, Canadian Math. Soc. Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, New York, 1997.
E. Glasner, Structure theory as a tool in topological dynamics, in Descriptive set theory and dynamical systems, LMS Lecture note Series vol. 277, Cambridge University Press, Cambridge, 2000, pp. 173–209.
E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003.
E. Glasner, On tame dynamical systems, Colloquium Mathematicum 105 (2006), 283–295.
E. Glasner, Enveloping semigroups in topological dynamics, Topology and its Applications 154 (2007), 2344–2363.
E. Glasner, The structure of tame minimal dynamical systems, Ergodic Theory and Dynamical Systems 27 (2007), 1819–1837.
E. Glasner and M. Megrelishvili, Hereditarily non-sensitive dynamical systems and linear representations, Colloquium Mathematicum 104 (2006), 223–283.
E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), 1067–1075.
E. Glasner and B. Weiss, Locally equicontinuous dynamical systems, Collloq. Math. 84/85, Part 2, (2000), 345–361.
W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergodic Theory and Dynamical Systems 26 (2006), 1549–1567.
A. S. Kechris, Classical Sescriptive Set Theory, Springer-Verlag, Graduate texts in mathematics 156, 1991.
D. Kerr and H. Li, Independence in topological and C*-dynamics, Mathematische Annalen 338 (2007), 869–926.
M. Megrelishvili, Fragmentability and representations of flows, Topology Proceedings 27 (2003), 497–544. See also: http://www.math.biu.ac.il/~megereli.
M. Megrelishvili, Topological transformation groups: selected topics, in book: Second edition of Open Problems in Topology (Elliott Pearl, editor), to appear.
E. Michael and I. Namioka, Barely continuous functions, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 24 (1976), 889–892.
I. Namioka, Separate continuity and joint continuity, Pacific Journal of Mathematics 51 (1974), 515–531.
I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), 258–281.
J. Saint Raymond, Jeux topologiques et espaces de Namioka, Proceedings of the American Mathematical Society 87 (1983), 499–504.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Glasner, E., Megrelishvili, M. & Uspenskij, V.V. On metrizable enveloping semigroups. Isr. J. Math. 164, 317–332 (2008). https://doi.org/10.1007/s11856-008-0032-3
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11856-008-0032-3