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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: [email protected])
NANCY GUELMAN
Affiliation:
IMERL, Facultad de Ingeniería, Universidad de La República, C.C. 30, Montevideo, Uruguay (email: [email protected])
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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

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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