Eventual shadowing for chain transitive sets of $C^1$ generic dynamical systems
J. Korean Math. Soc. 2021 Vol. 58, No. 5, 1059-1079
https://doi.org/10.4134/JKMS.j190083
Published online July 27, 2021
Printed September 1, 2021
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
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