Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(5): 1059-1079

Online first article July 27, 2021      Printed September 1, 2021

https://doi.org/10.4134/JKMS.j190083

Copyright © The Korean Mathematical Society.

Eventual shadowing for chain transitive sets of $C^1$ generic dynamical systems

Manseob Lee

Mokwon University

Abstract

We show that given any chain transitive set of a $C^1$ generic diffeomorphism $f$, if a diffeomorphism $f$ has the eventual shadowing property on the locally maximal chain transitive set, then it is hyperbolic. Moreover, given any chain transitive set of a $C^1$ generic vector field $X$, if a vector field $X$ has the eventual shadowing property on the locally maximal chain transitive set, then the chain transitive set does not contain a singular point and it is hyperbolic. We apply our results to conservative systems (volume-preserving diffeomorphisms and divergence-free vector fields).

Keywords: Shadowing, eventual shadowing, chain transitive, locally maximal, generic, hyperbolic

MSC numbers: Primary 37C50; Secondary 37D20