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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)
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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 

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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