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CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  01 February 2021

MANSEOB LEE Open the ORCID record for MANSEOB LEE [Opens in a new window]  and
LE HUY TIEN Open the ORCID record for LE HUY TIEN [Opens in a new window]
MANSEOB LEE
Affiliation:
Department of Mathematics, Mokwon University, Daejeon, 302-729Korea e-mail: [email protected]
LE HUY TIEN
Affiliation:
Department of Mathematics, VNU Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam e-mail: [email protected]
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Abstract

Let M be a closed n-dimensional smooth Riemannian manifold, and let X be a $C^1$-vector field of $M.$ Let $\gamma $ be a hyperbolic closed orbit of $X.$ In this paper, we show that X has the $C^1$-stably shadowing property on the chain component $C_X(\gamma )$ if and only if $C_X(\gamma )$ is the hyperbolic homoclinic class.

Keywords

shadowingchain componenthomoclinic classstar conditionhyperbolic

MSC classification

Primary: 37D05: Hyperbolic orbits and sets
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D50: Hyperbolic systems with singularities (billiards, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 2 , April 2021 , pp. 243 - 259
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2017R1A2B4001892). The second author is supported by the VNU Project of Vietnam National University (No. QG.15.01).

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