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Uniform hyperbolicity along periodic orbits


Author: Abbas Fakhari
Journal: Proc. Amer. Math. Soc. 141 (2013), 3107-3118
MSC (2010): Primary 37B20, 37C29, 37C50
DOI: https://doi.org/10.1090/S0002-9939-2013-11553-4
Published electronically: May 3, 2013
MathSciNet review: 3068964
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Abstract: We introduce the notion of uniform hyperbolicity along periodic orbits (UHPO) for homoclinic classes and provide equivalent conditions under which the UHPO property on a $ C^1$-generic homoclinic class implies hyperbolicity. It is shown that for a $ C^1$-generic locally maximal homoclinic class the UHPO property is equivalent to the non-existence of zero Lyapunov exponents. Using the notion of UHPO, we also give new proofs for some recent $ C^1$-dichotomy theorems.


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Additional Information

Abbas Fakhari
Affiliation: Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran
Address at time of publication: School of Mathematics and Computer Sciences, Damghan University, P. O. Box 36715-364, Damghan, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
Email: fakhari@du.ac.ir, abs.fakhari@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2013-11553-4
Keywords: Hyperbolicity along periodic orbits, homoclinic class, dominated splitting, uniform hyperbolicity
Received by editor(s): June 2, 2011
Received by editor(s) in revised form: September 27, 2011, and November 12, 2011
Published electronically: May 3, 2013
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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