Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-29T03:03:39.485Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms

Published online by Cambridge University Press:  23 June 2009

ANDREY GOGOLEV
ANDREY GOGOLEV*
Affiliation:
Mathematics Department, Penn State University, University Park, State College, PA 16802, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show by means of a counterexample that a C1+Lip diffeomorphism Hölder conjugate to an Anosov diffeomorphism is not necessarily Anosov. Also we include a result from the 2006 PhD thesis of Fisher: a C1+Lip diffeomorphism Hölder conjugate to an Anosov diffeomorphism is Anosov itself provided that Hölder exponents of the conjugacy and its inverse are sufficiently large.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 441 - 456
Copyright
Copyright © Cambridge University Press 2009

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]Bonatti, Ch., Diaz, L. and Vuillemin, F.. Topologically transverse nonhyperbolic diffeomorphisms at the boundary of the stable ones. Bol. Soc. Bras. de Mat. 29 (1998), 99–144.CrossRefGoogle Scholar
[2]Carvalho, M.. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete Continuous Dynam. Syst. A 4 (1998), 765–782.CrossRefGoogle Scholar
[3]Enrich, H.. A heteroclinic bifurction of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 18 (1998), 567–608.CrossRefGoogle Scholar
[4]Fisher, T.. PhD Thesis. Available online at http://www.etda.libraries.psu.edu/theses/approved/WorldWideFiles/ETD-1174/fisher-thesis-draft.pdf. PennState, 2006.Google Scholar
[5]Katok, A.. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2) 110(3) (1979), 529–547.CrossRefGoogle Scholar
[6]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[7]Katok, A. and Nitica, V.. Differentiable rigidity of higher rank abelian group actions. (2008) in preparation.Google Scholar
[8]Lewowicz, J.. Lyapunov functions and topological stability. J. Differential Equations 38 (1980), 192–209.CrossRefGoogle Scholar
[9]Mañé, R.. Quasi-Anosov diffeomorphisms and hyperbolic manifolds. Trans. Amer. Math. Soc. 229 (1977), 351–370.CrossRefGoogle Scholar
[10]Mather, J.. Characterization of Anosov diffeomorphisms. Nederl. Akad. Wetensch. Proc. Ser. A 30 (1968), 479–483.CrossRefGoogle Scholar
[11]Walters, P.. Anosov diffeomorphisms are topologically stable. Topology 9 (1970), 71–78.CrossRefGoogle Scholar