Skip to main content Accessibility help
×
Home

Genericity in topological dynamics

  • MICHAEL HOCHMAN (a1)

Abstract

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.

Copyright

References

Hide All
[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), 439442.
[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.
[3]Akin, E., Hurley, M. and Kennedy, J. A. . Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(783) (2003), viii+130 .
[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. 1627.
[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), 15871620.
[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), 229331.
[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. 94113.
[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. 8189.
[9]Furstenberg, H. . Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 149.
[10]Furstenberg, H. , Keynes, H.  and Shapiro, L. . Prime flows in topological dynamics. Israel J. Math. 14 (1973), 2638.
[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. 231242.
[12]Glasner, E.  and Weiss, B. . The topological Rohlin property and topological entropy. Amer. J. Math. 123(6) (2001), 10551070.
[13]Halmos, P. R. . Approximation theories for measure-preserving transformations. Trans. Amer. Math. Soc. 55 (1944), 118.
[14]Halmos, P. R. . In general a measure-preserving transformation is mixing. Ann. of Math. (2) 45 (1944), 786792.
[15]Halmos, P. R.. Lectures on Ergodic Theory. Chelsea Publishing Co., New York, 1960.
[16]Kechris, A. S. and Rosendal, C.. Turbulence, amalgamation and generic automorphisms of homogeneous structures. Proc. Lond. Math. Soc. (3) 94 (2007), 302350.
[17]King, J. L. . A map with topological minimal self-joinings in the sense of del Junco. Ergod. Th. & Dynam. Sys. 10(4) (1990), 745761.
[18]Mozes, S. . Tilings, substitution systems and dynamical systems generated by them. J. Analyse Math. 53 (1989), 139186.
[19]Oxtoby, J. C. and Ulam, S. M.. Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42 (1941), 874920.
[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.
[21]Rohlin, V.. A ‘general’ measure-preserving transformation is not mixing. Dokl. Akad. Nauk SSSR (N.S.) 60 (1948), 349351.
[22]Rudolph, D.. Residuality and orbit equivalence. Topological Dynamics and Applications (Minneapolis, MN, 1995) (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 243254.
[23]Rudolph, D. J.. An example of a measure preserving map with minimal self-joinings, and applications. J. Analyse Math. 35 (1979), 97122.
[24]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982, pp. ix + 250.
[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. 189196.

Related content

Powered by UNSILO

Genericity in topological dynamics

  • MICHAEL HOCHMAN (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.