Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


On measure expansive diffeomorphisms

Authors: M. J. Pacifico and J. L. Vieitez
Journal: Proc. Amer. Math. Soc. 143 (2015), 811-819
MSC (2010): Primary 37C29, 37D30
Published electronically: October 17, 2014
MathSciNet review: 3283667
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f: M \to M$ be a diffeomorphism defined on a compact boundaryless $ d$-dimensional manifold $ M$, $ d\geq 2$. In this note we show that diffeomorphisms in a residual subset far from homoclinic tangencies are measure expansive. We also show that surface diffeomorphisms presenting homoclinic tangencies can be $ C^1$-approximated by non-measure expansive diffeomorphisms.

References [Enhancements On Off] (What's this?)

  • [Bo] Rufus Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc. 164 (1972), 323-331. MR 0285689 (44 #2907)
  • [Bo1] R. Bowen, Equilibrium states and the ergodic theory of Anasov diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, 1975. MR 0442989
  • [Bo2] Rufus Bowen, A horseshoe with positive measure, Invent. Math. 29 (1975), no. 3, 203-204. MR 0380890 (52 #1787)
  • [BW] Keith Burns and Amie Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), no. 1, 451-489. MR 2630044 (2011g:37075),
  • [CSY] S. Crovisier, M. Sambarino, D. Yang, Partial Hyperbolicity and Homoclinic Tangencies, arXiv 1103.0869v1, 2011.
  • [DFPV] Lorenzo J. Díaz, Todd Fisher, Maria José Pacifico, and José L. Vieitez, Entropy-expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst. 32 (2012), no. 12, 4195-4207. MR 2966742,
  • [Ft] Albert Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Comm. Math. Phys. 126 (1989), no. 2, 249-262. MR 1027497 (90m:58159)
  • [Hi] Koichi Hiraide, Expansive homeomorphisms with the pseudo-orbit tracing property of $ n$-tori, J. Math. Soc. Japan 41 (1989), no. 3, 357-389. MR 999503 (90h:58062),
  • [Hi2] Koichi Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), no. 1, 117-162. MR 1049828 (91b:58184)
  • [Le] Jorge Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.) 20 (1989), no. 1, 113-133. MR 1129082 (92i:58139),
  • [Mo] C. A. Morales, Measure expansive systems, preprint, IMPA, D083, 2011.
  • [PaVi] María José Pacifico and José L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut. 21 (2008), no. 2, 293-317. MR 2441955 (2009m:37090)
  • [PaVi2] M. J. Pacifico and J. L. Vieitez, Robust entropy expansiveness implies generic domination, Nonlinearity 23 (2010), no. 8, 1971-1990. MR 2669634 (2011k:37047),
  • [PPSV] M. J. Pacifico, E. R. Pujals, M. Sambarino, and J. L. Vieitez, Robustly expansive codimension-one homoclinic classes are hyperbolic, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 179-200. MR 2470632 (2009k:37070),
  • [PPV] M. J. Pacifico, E. R. Pujals, and J. L. Vieitez, Robustly expansive homoclinic classes, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 271-300. MR 2122923 (2006k:37041),
  • [RY] Clark Robinson and Lai Sang Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math. 61 (1980), no. 2, 159-176. MR 590160 (81m:58061),
  • [SV] Martín Sambarino and José L. Vieitez, Robustly expansive homoclinic classes are generically hyperbolic, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1325-1333. MR 2505706 (2010f:37052),
  • [Vi] José L. Vieitez, Lyapunov functions and expansive diffeomorphisms on 3D-manifolds, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 601-632. MR 1898808 (2003b:37049),
  • [Ut] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. MR 0038022 (12,344b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37C29, 37D30

Retrieve articles in all journals with MSC (2010): 37C29, 37D30

Additional Information

M. J. Pacifico
Affiliation: Instituto de Metematica, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil

J. L. Vieitez
Affiliation: Instituto de Matematica, Facultad de Ingenieria, Universidad de la Republica, CC30, CP 11300, Montevideo, Uruguay

Received by editor(s): February 12, 2013
Received by editor(s) in revised form: March 2, 2013, and June 6, 2013
Published electronically: October 17, 2014
Additional Notes: The first author was partially supported by CNPq Brazil, Pronex on Dynamical Systems, FAPERJ, Balzan Research Project of J. Palis
The second author was partially supported by Grupo de Investigación “Sistemas Dinámicos” CSIC (Universidad de la República), SNI-ANII, PEDECIBA, Uruguay
Communicated by: Nimish Shah
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society