Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T21:17:40.069Z Has data issue: false hasContentIssue false
  • English
  • Français

Robust transitivity for endomorphisms

Published online by Cambridge University Press:  06 August 2012

CRISTINA LIZANA  and
ENRIQUE PUJALS
CRISTINA LIZANA
Affiliation:
Departamento de Matemática, Facultad de Ciencias, Universidad de Los Andes, La Hechicera-Mérida, 5101, Venezuela (email: clizana@ula.ve)
ENRIQUE PUJALS
Affiliation:
IMPA, Estrada Dona Castorina 110, CEP 22460-320 Rio de Janeiro, Brazil (email: enrique@impa.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

We address the problem of the conditions under which an endomorphism having a dense orbit is such that a sufficiently close perturbed map also exhibits a dense orbit. For this purpose we give sufficient conditions, covering a large class of examples, for endomorphisms on the $n$-dimensional torus to be robustly transitive: the endomorphism must be volume expanding and any large connected arc must contain a point such that its future orbit belongs to an expanding region.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1082 - 1114
Copyright
Copyright © 2012 Cambridge University Press 

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

[BD96]Bonatti, C. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357–396.Google Scholar
[BDP03]Bonatti, C., Díaz, L. J. and Pujals, E. R.. A $C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158(2) (2003), 355–418.CrossRefGoogle Scholar
[BV00]Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.Google Scholar
[Cas11]Castro, A.. New criteria for hyperbolicity based on periodic sets. Bull. Braz. Math. Soc. (N.S.) 42(3) (2011), 455–483.Google Scholar
[Cro02]Crovisier, S.. Une remarque sur les ensembles hyperboliques localement maximaux. C. R. Math. Acad. Sci. Paris 334(5) (2002), 401–404.Google Scholar
[Fis06]Fisher, T.. Hyperbolic sets that are not locally maximal. Ergod. Th. & Dynam. Sys. 26(5) (2006), 1491–1509.Google Scholar
[Fra71]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.CrossRefGoogle Scholar
[KH95]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995, with a supplementary chapter by Katok and Mendoza.Google Scholar
[Liu91]Liu, P. D.. The $R$-stability of $C^1$ self-mapping orbit spaces. Chinese Ann. Math. Ser. A 12(4) (1991), 415–421.Google Scholar
[Liz10]Lizana, C.. Robust transitivity for endomorphisms. PhD Thesis, IMPA, 2010.Google Scholar
[Man78]Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.CrossRefGoogle Scholar
[NP10]Nassiri, M. and Pujals, E.. Robust transitivity in Hamiltonian dynamics. Ann. Sci. Éc. Norm. Supér. 45(2) (2010), 191–239.CrossRefGoogle Scholar
[Pal68]Palis, J.. On Morse–Smale dynamical systems. Topology 8 (1968), 385–404.Google Scholar
[PS06]Pujals, E. R. and Sambarino, M.. A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26(01) (2006), 281–289.Google Scholar
[Rob76]Robinson, C.. Structural stability of $C^{1}$ diffeomorphisms. J. Differential Equations 22(1) (1976), 28–73.CrossRefGoogle Scholar
[Shu69]Shub, M.. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91(1) (1969), 175–199.Google Scholar
[Shu71]Shub, M.. Topologically transitive diffeomorphisms of $\mathbb {T}^4$. Symposium on Differential Equations and Dynamical Systems (Lecture Notes in Mathematics, 206). Springer, New York, 1971, pp. 39–40.Google Scholar
[Shu87]Shub, M.. Global Stability of Dynamical Systems. Springer, New York, 1987, with the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy.Google Scholar