Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:11:52.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Constraints on dynamics preserving certain hyperbolic sets

Published online by Cambridge University Press:  26 May 2010

AARON W. BROWN
AARON W. BROWN*
Affiliation:
Tufts University, Medford, MA 02155, USA (email: aaron.brown@tufts.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish two results under which the topology of a compact hyperbolic set constrains ambient dynamics. First, if Λ is a transitive, codimension-one, expanding attractor for some diffeomorphism, then Λ is a union of transitive, codimension-one attractors (or contracting repellers) for any diffeomorphism such that Λ is hyperbolic. Secondly, if Λ is a locally maximal nonwandering set for a surface diffeomorphism, then Λ is locally maximal for any diffeomorphism for which Λ is hyperbolic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 719 - 739
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Fisher, T.. Hyperbolic sets with nonempty interior. Discrete Contin. Dyn. Syst. 15(2) (2006), 433–446.CrossRefGoogle Scholar
[2]Fisher, T.. The topology of hyperbolic attractors on compact surfaces. Ergod. Th. & Dynam. Syst. 26 (2006), 1511–1520.CrossRefGoogle Scholar
[3]Franks, J.. Anosov diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV). American Mathematical Society, Providence, RI, 1970, pp. 61–93.CrossRefGoogle Scholar
[4]Grines, V. and Zhuzhoma, E.. On structurally stable diffeomorphisms with codimension one expanding attractors. Trans. Amer. Math. Soc. 357 (2005), 617–667.CrossRefGoogle Scholar
[5]Hirsch, M. W. and Pugh, C.. Stable manifolds and hyperbolic sets. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV). American Mathematical Society, Providence, RI, 1970,pp. 133–163.CrossRefGoogle Scholar
[6]Hurewicz, W. and Wallman, H.. Dimension Theory (Princeton Mathematical Series, 4). Princeton University Press, Princeton, NJ, 1941.Google Scholar
[7]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[8]Katok, A. and Spatzier, R. J.. Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Tr. Mat. Inst. Steklova 216 (1997), 292–319.Google Scholar
[9]Kollmer, H.. On hyperbolic attractors of codimension one. Geometry and Topology (Lecture Notes in Mathematics, 597). Springer, Berlin, 1977, pp. 330–334.CrossRefGoogle Scholar
[10]Newhouse, S. E.. On codimension one Anosov diffeomorphisms. Amer. J. Math. 92(3) (1970), 761–770.CrossRefGoogle Scholar
[11]Plykin, R. V.. Hyperbolic attractors of diffeomorphisms. Russian Math. Surveys 35(3) (1980), 109–121.CrossRefGoogle Scholar
[12]Plykin, R. V.. Hyperbolic attractors of diffeomorphisms (the nonorientable case). Russian Math. Surveys 35(4) (1980), 186–187.CrossRefGoogle Scholar
[13]Plykin, R. V.. The geometry of hyperbolic attractors of smooth cascades. Russian Math. Surveys 39(6) (1984), 85–131.CrossRefGoogle Scholar
[14]Ruelle, D. and Sullivan, D.. Currents, flows and diffeomorphisms. Topology 14 (1975), 319–327.CrossRefGoogle Scholar
[15]Sinaĭ, J. G.. Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2 (1968), 61–82.CrossRefGoogle Scholar
[16]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar