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A link between topological entropy and Lyapunov exponents

Published online by Cambridge University Press:  20 June 2017

THIAGO CATALAN
THIAGO CATALAN*
Affiliation:
Faculdade de Matemática, FAMAT/UFU, Av. João Naves de Avila, 2121, 38.408-100, Uberlândia, MG, Brazil email tcatalan@ufu.br
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Abstract

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 620 - 637
Copyright
© Cambridge University Press, 2017 

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References

Abdenur, F., Bonatti, C. and Crovisier, S.. Nonuniform hyperbolicity for C 1 -generic diffeomorphisms. Israel J. Math. 183 (2011), 1–60.Google Scholar
Arbieto, A. and Matheus, C.. A pasting lemma and some applications for conservative systems. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1399–1417.Google Scholar
Arnaud, M.-C., Bonatti, C. and Crovisier, S.. Dynamiques symplectiques génériques. Ergod. Th. & Dynam. Sys. 25 (2005), 1401–1436.Google Scholar
Bonatti, C. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33–104.Google Scholar
Bonatti, C., Diaz, L. and Pujals, E.. A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. 158 (2003), 355–418.Google Scholar
Barrientos, P. and Raibekas, A.. Robust cycles and tangencies of large codimension. Preprint, 2015,arXiv:1509.05325.Google Scholar
Catalan, T. and Horita, V.. $C^{1}$ -genericity of symplectic diffeomorphisms and lower bounds for topological entropy. Dyn. Syst. (2017), http://dx.doi.org/10.1080/14689367.2017.1278744.Google Scholar
Catalan, T. and Tahzibi, A.. A lower bound for topological entropy of generic non Anosov symplectic diffeomorphisms. Ergod. Th. & Dynam. Sys. 34 (2014), 1503–1524.Google Scholar
Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301–308.Google Scholar
Hayashi, S.. Connecting invariant manifolds and the solution of the C 1 stability conjectures for flows. Ann. of Math. (2) 145(1) (1997), 81–137.Google Scholar
Horita, V. and Tahzibi, A.. Partial hyperbolicity for symplectic diffeomorphisms. Ann. Inst. H. Poincaré 23 (2006), 641–661.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.Google Scholar
Katok, A. and Hasselblat, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995.Google Scholar
Newhouse, S. E.. Topological entropy and Hausdorff dimension for area preserving diffeomorphisms of surfaces. Astérisque 51 (1978), 323–334.Google Scholar
Newhouse, S. E.. Continuity properties of entropy. Ann. of Math. 129(2) (1989), 215–235.Google Scholar
Wen, L. and Xia, Z.. C 1 connecting lemmas. Trans. Amer. Math. Soc. 352(11) (2000), 5213–5230.Google Scholar