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Hyperbolicity versus weak periodic orbits inside homoclinic classes

Published online by Cambridge University Press:  14 March 2017

XIAODONG WANG
Affiliation:
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China School of Mathematical Sciences, Peking University, Beijing, 100871, China Laboratoire de Mathématiques d’Orsay, Université Paris-Sud 11, Orsay, 91405, France email xdwang1987@sjtu.edu.cn, xdwang1987@gmail.com
Corresponding

Abstract

We prove that, for $C^{1}$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from zero, then $H(p)$ must be (uniformly) hyperbolic. This is in the spirit of the works on the stability conjecture, but with a significant difference that the homoclinic class $H(p)$ is not known isolated in advance, hence the ‘weak’ periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an ‘intrinsic’ nature, and the classical proof of the stability conjecture does not work. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.

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Original Article
Copyright
© Cambridge University Press, 2017 

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