Sets of Dynamical Systems with Various Limit Shadowing Properties

Symplectic diffeomorphisms with limit shadowing

“…In [28], Pilyugin proved that if a diffeomorphism belongs to the C 1 -interior of the set of diffeomorphisms having the limit shadowing property then it is Ω-stable, that is, Axiom A and the no-cycle condition. Moreover, if a diffeomorphism belongs to the C 1 -interior of the set of diffeomorphisms having the limit weak shadowing property, then it is Ω-stable.…”

“…From now on, we introduce the notions of the limit shadowing properties which were studied by [4,19,25,28]. We say that a sequence {x i } i∈Z is a limit pseudo…”

Usual limit shadowable homoclinic classes of generic diffeomorphisms

“…Lemma 2.8 There exists a residual set G 4 ⊂ Diff (M) such that if f ∈ G 4 , and H f (p) is a locally maximal homoclinic class of p which satisfies the usual limit shadowing property, then there exists δ >0 such that no point in H f (p) has a δ-weak eigenvalue.…”

“…Therefore, we study the some kinds of the shadowing property (usual limit shadowing property) and homoclinic classes. As to the research on the usual limit shadowing property, there exist [1,3,4]. We say that f has the usual limit shadowing property on Λ if for any sequence ξ = {x…”

Volume-Preserving Diffeomorphisms with Various Limit Shadowing

“…We will study some type of shadowing properties which are called s-limit shadowing, limit shadowing and weak limit shadowing property. Firstly, the s-limit shadowing property was studied in [Barwell et al, 2012] and [Pilyugin, 2007]. In fact, Pilyugin showed that if a diffeomorphism is structurally stable then the diffeomorphism has the s-limit shadowing property (see [Pilyugin, 1999, Lemma 5]).…”