Orbital Shadowing for -Generic Volume-Preserving Diffeomorphisms

Abstract

We show that -generically, if a volume-preserving diffeomorphism has the orbital shadowing property, then the diffeomorphism is Anosov.

Orbital shadowing property on chain transitive sets for generic diffeomorphisms

“…Lee and Lee [10] proved that a volume preserving diffeomorphism f has the C 1 robustly orbital shadowing property then it is Anosov. Moreover, we can find similar results [12,13,14,15]. We say that the set Λ is transitive if there is a point x ∈ Λ such that ω(x) = Λ, where ω(x) is the omega limit set of x.…”

“…From the property, a general shadowing property was introduced by [20] which is called the orbital shadowing property. For the orbital shadowing property, many results published by the various view points (see [10,13,14,15,16,17,19]). We say that f has the orbital shadowing property on Λ if for any > 0 there is δ > 0 such that for any δ pseudo orbit ξ = {x i } ∈Z ⊂ Λ such that there is a point y ∈ M such that Orb(y) ⊂ B (ξ) and ξ ⊂ B (Orb(y)).…”

Symplectic diffeomorphisms with limit shadowing

“…The shadowing property is very closely related to the stability theory. For this reason, various shadowing properties (shadowing [3,29,30], limit shadowing [4,6,8,14,18,19], inverse shadowing [9,13,26], orbital shadowing [10,16,27], average shadowing [11,12,21,24], ergodic shadowing [5,17,20], periodic shadowing [15], etc.) have been studied by many researchers.…”