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HYPERBOLICITY OF HOMOCLINIC CLASSES OF $C^{1}$ VECTOR FIELDS

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  21 November 2014

KEONHEE LEE ,
MANSEOB LEE  and
SEUNGHEE LEE
KEONHEE LEE
Affiliation:
Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea email khlee@cnu.ac.kr
MANSEOB LEE*
Affiliation:
Department of Mathematics, Mokwon University, Daejeon 302-729, Korea email lmsds@mokwon.ac.kr
SEUNGHEE LEE
Affiliation:
Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea email shlee@cnu.ac.kr
*
✉lmsds@mokwon.ac.kr
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Abstract

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Let ${\it\gamma}$ be a hyperbolic closed orbit of a $C^{1}$ vector field $X$ on a compact $C^{\infty }$ manifold $M$ and let $H_{X}({\it\gamma})$ be the homoclinic class of $X$ containing ${\it\gamma}$. In this paper, we prove that if a $C^{1}$-persistently expansive homoclinic class $H_{X}({\it\gamma})$ has the shadowing property, then $H_{X}({\it\gamma})$ is hyperbolic.

Keywords

persistent expansivenesshomoclinic classesvector fieldsshadowing property

MSC classification

Primary: 37D05: Hyperbolic orbits and sets
Secondary: 37C10: Vector fields, flows, ordinary differential equations 37C29: Homoclinic and heteroclinic orbits
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 3 , June 2015 , pp. 375 - 389
Copyright
© 2014 Australian Mathematical Publishing Association Inc. 

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