SHADOWABLE CHAIN TRANSITIVE SETS OF C¹-GENERIC DIFFEOMORPHISMS

Abstract

Abstract. We prove that a locally maximal chain transitive set of a C 1 -generic diffeomorphism is hyperbolic if and only if it is shadowable.

Homoclinic classes with shadowing

“…A similar result for locally maximal chain transitive sets was proved in [3]. More precisely, it is proved that C 1 -generically, every locally maximal chain transitive set is hyperbolic if it is shadowable.…”

Shadowing, expansiveness and specification for C1-conservative systems

“…We point out that a manifestation of the phenomenon of genericity of non shadowing away from hyperbolicity was considered in [1,2,23] for dissipative systems. With respect to the analog results for non expansive ones we refer the results [3,38].…”

Average Shadowing Property With Non Uniformly Hyperbolicity on Periodic Points

“…There is a residual set R 1 ⊂ Dif f 1 (M ) such that every f ∈ R 1 satisfies the (2) any ergodic invariant measure µ of f is the limit of a sequence of ergodic invariant measures supported by periodic points [6].…”