Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T02:46:48.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Transitive Anosov flows and Axiom-A diffeomorphisms

Published online by Cambridge University Press:  01 June 2009

CHRISTIAN BONATTI  and
NANCY GUELMAN
CHRISTIAN BONATTI
Affiliation:
Université de Bourgogne, Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, BP 47 870, 21078, Dijon Cedex, France (email: bonatti@u-bourgogne.fr)
NANCY GUELMAN
Affiliation:
IMERL, Facultad de Ingeniería, Universidad de La República, C.C. 30, Montevideo, Uruguay (email: nguelman@fing.edu.uy)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let M be a smooth compact Riemannian manifold without boundary, and ϕ:M×ℝ→M a transitive Anosov flow. We prove that if the time-one map of ϕ is C1-approximated by Axiom-A diffeomorphisms with more than one attractor, then ϕ is topologically equivalent to the suspension of an Anosov diffeomorphism.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 817 - 848
Copyright
Copyright © Cambridge University Press 2008

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]Abraham, R. and Smale, S.. Nongenericity of Ω-stability. Proc. Sympos. Pure Math. 14 (1970), 5–8.CrossRefGoogle Scholar
[2]Barbot, T.. Plane affine geometry and Anosov flows. Ann. Sci. École Norm. Sup. (4) 34 (2001), 871–889.CrossRefGoogle Scholar
[3]Bonatti, C. and Diaz, L.. Persistent non-hyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143 (1995), 357–396.CrossRefGoogle Scholar
[4]Bonatti, C., Diaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics III). Springer, Berlin, 2005.Google Scholar
[5]Bonatti, C. and Langevin, R.. Difféomorphismes de Smale des surfaces. Astérisque 250 (1998), viii+235 pp.Google Scholar
[6]de Melo, W. and van Strien, S.. One Dimensional Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Band 25). Springer, Berlin, 1993.CrossRefGoogle Scholar
[7]Fenley, S.. The structure of branching in Anosov flows of 3-manifolds. Comment. Math. Helv. 73 (1998), 259–297.CrossRefGoogle Scholar
[8]Franks, J.. Anosov diffeomorphisms. Proc. Sympos. Pure Math. 14 (1970), 61–93.CrossRefGoogle Scholar
[9]Franks, J. and Williams, R.. Anomalous Anosov Flows (Global Theory and Dynamical Systems, SLN 819). Springer, Berlin, 1980.CrossRefGoogle Scholar
[10]Guelman, N.. On the approximation of time one map of Anosov flows by Axiom A diffeomorphisms. Bull. Braz. Math. Soc. (N.S.) 33 (2002), 75–97.CrossRefGoogle Scholar
[11]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
[12]Katok, A. and Hasselblat, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[13]Lewowicz, J.. Dinámica de Homeomorfismos Expansivos (Monografías del IMCA, 36). Pontificia Universidad Católica del Perú, Lima, 2003, iv+65 pp.Google Scholar
[14]Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.CrossRefGoogle Scholar
[15]Newhouse, S.. On codimension one Anosov diffeomorphisms. Amer. J. Math. 92 (1970), 761–770.CrossRefGoogle Scholar
[16]Palis, J. and Pugh, C.. Fifty Problems in Dynamical Systems (Lecture Notes in Mathematics, 468). Springer, Berlin, 1975.CrossRefGoogle Scholar
[17]Palis, J. and Takens, F.. Hyperbolicity and Sensitive-chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993.Google Scholar
[18]Schwartzman, S.. Asymptotic cycles. Ann. of Math. (2) 66 (1957), 270–284.CrossRefGoogle Scholar
[19]Shub, M.. Topological Transitive Diffeomorphism on T 4 (Lectures Notes in Mathematics, 206). 1971.Google Scholar
[20]Shub, M.. Global Stability of Dynamical Systems. Springer, Berlin, 1987, p. 39.CrossRefGoogle Scholar
[21]Simić, S.. Volume preserving codimension one Anosov flows in dimensions greater than three are suspensions. Preprint.Google Scholar
[22]Simon, R.. A 3 dimensional Abraham–Smale example. Proc. Amer. Math. Soc. 34 (1972), 629–630.Google Scholar
[23]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[24]Verjovsky, A.. Codimension one Anosov flows. Bol. Soc. Mat. Mexicana 19 (1974), 49–77.Google Scholar
[25]Verjovsky, A.. Sistemas de Anosov (Monografías del IMCA, 9). Instituto de Matemática y Ciencias Afines, IMCA, Lima, 1999, iv+65 pp.Google Scholar