Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T09:24:31.884Z Has data issue: false hasContentIssue false
  • English
  • Français

Genericity in topological dynamics

Published online by Cambridge University Press:  01 February 2008

MICHAEL HOCHMAN
MICHAEL HOCHMAN*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (email: mhochman@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 study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner–King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing and minimal self joinings. The latter two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 125 - 165
Copyright
Copyright © Cambridge University Press 2008

References

[1]Ageev, O. N.. The generic automorphism of a Lebesgue space conjugate to a G-extension for any finite abelian group G. Dokl. Akad. Nauk 374(4) (2000), 439–442.Google Scholar
[2]Akin, E., Glasner, E. and Weiss, B.. Generically there is but one homeomorphism of the cantor set. Preprint, 2006, http://www.arxiv.org/abs/math.DS/0603538. Trans. Amer. Math. Soc. accepted.Google Scholar
[3]Akin, E., Hurley, M. and Kennedy, J. A. . Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(783) (2003), viii+130 .Google Scholar
[4]Alpern, S. . Generic properties of measure-preserving homeomorphisms. Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) (Lecture Notes in Mathematics, 729). Springer, Berlin, 1979, pp. 16–27.Google Scholar
[5]Alpern, S. and Prasad, V. S.. Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms. Ergod. Th. & Dynam. Sys. 22(6) (2002), 1587–1620.CrossRefGoogle Scholar
[6]Bezugly, S., Dooley, A. H. and Kwiatkowski, J.. Topologies on the group of homeomorphisms of a Cantor set. Topol. Methods Nonlinear Anal. 27 (2006), 229–331.Google Scholar
[7]Choksi, J. R. and Prasad, V. S.. Approximation and Baire category theorems in ergodic theory. Measure Theory and Its Applications (Sherbrooke, Quebec, 1982) (Lecture Notes in Mathematics, 1033). Springer, Berlin, 1983, pp. 94–113.CrossRefGoogle Scholar
[8]del Junco, A. . Disjointness of measure-preserving transformations, minimal self-joinings and category. Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979–80) (Progress in Mathematics, 10). Birkhäuser, Boston, MA, 1981, pp. 81–89.Google Scholar
[9]Furstenberg, H. . Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[10]Furstenberg, H. , Keynes, H.  and Shapiro, L. . Prime flows in topological dynamics. Israel J. Math. 14 (1973), 26–38.CrossRefGoogle Scholar
[11]Glasner, E.  and King, J. L. . 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
[12]Glasner, E.  and Weiss, B. . The topological Rohlin property and topological entropy. Amer. J. Math. 123(6) (2001), 1055–1070.CrossRefGoogle Scholar
[13]Halmos, P. R. . Approximation theories for measure-preserving transformations. Trans. Amer. Math. Soc. 55 (1944), 1–18.CrossRefGoogle Scholar
[14]Halmos, P. R. . In general a measure-preserving transformation is mixing. Ann. of Math. (2) 45 (1944), 786–792.CrossRefGoogle Scholar
[15]Halmos, P. R.. Lectures on Ergodic Theory. Chelsea Publishing Co., New York, 1960.Google Scholar
[16]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
[17]King, J. L. . A map with topological minimal self-joinings in the sense of del Junco. Ergod. Th. & Dynam. Sys. 10(4) (1990), 745–761.CrossRefGoogle Scholar
[18]Mozes, S. . Tilings, substitution systems and dynamical systems generated by them. J. Analyse Math. 53 (1989), 139–186.CrossRefGoogle Scholar
[19]Oxtoby, J. C. and Ulam, S. M.. Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42 (1941), 874–920.CrossRefGoogle Scholar
[20]Oxtoby, J. C. . Measure and Category, 2nd edn(Graduate Texts in Mathematics, 2). Springer, New York, 1980, A survey of the analogies between topological and measure spaces.CrossRefGoogle Scholar
[21]Rohlin, V.. A ‘general’ measure-preserving transformation is not mixing. Dokl. Akad. Nauk SSSR (N.S.) 60 (1948), 349–351.Google Scholar
[22]Rudolph, D.. Residuality and orbit equivalence. Topological Dynamics and Applications (Minneapolis, MN, 1995) (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 243–254.CrossRefGoogle Scholar
[23]Rudolph, D. J.. An example of a measure preserving map with minimal self-joinings, and applications. J. Analyse Math. 35 (1979), 97–122.CrossRefGoogle Scholar
[24]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982, pp. ix + 250.CrossRefGoogle Scholar
[25]Weiss, B.. Multiple recurrence and doubly minimal systems. Topological Dynamics and Applications (Minneapolis, MN, 1995) (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 189–196.CrossRefGoogle Scholar