Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-09T13:50:49.882Z 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)
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 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.

Information

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

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