Inverse Shadowing and Weak Inverse Shadowing Property

Abstract

In this paper we show that an -stable diffeomorphism has the weak inverse shadowing property with respect to classes of continuous method and and some of the -stable diffeomorphisms have weak inverse shadowing property with respect to classes . In addition we study relation between minimality and weak inverse shadowing property with respect to class and relation between expansivity and inverse shadowing property with respect to class

On inverse shadowing

“…Meanwhile in [21], Kloeden and Ombach prove that a structurally stable homeomorphism on a compact space has inverse shadowing with respect to the class of methods induced by homeomorphism. Further results in this direction can be found in [8,19,22]. We remark that inverse shadowing, particularly with regard to the class of methods induced by homeomorphism, is closely related to Lewowicz's notion of persistency [24] (see also [35]).…”

“…(NB. In the statement of [19,Theorem 3] the authors assume the phase space is compact, this assumption does not appear necessary for their proof nor ours.) Recall first that a system (X, f ) is said to be minimal if, for A ⊆ X closed, f (A) = A implies that A = X or A = ∅.…”

“…In [19,Theorem 4] the authors show a homeomorphism with T 0 -inverse shadowing is not expansive. Whilst not explicitly stated there, we remark that this result assumes that for any ε there exist distinct points x, y ∈ X with d(x, y) < ε.…”

“…In [19,Theorem 3] the authors show that a chain transitive homeomorphism on a compact metric space is minimal if and only if it has the weak inverse shadowing property with respect to T 0 . With Theorem 3.3 this result, as well as the following analogous one, becomes much more elementary.…”

“…Such classes have been studied in a variety of different settings for example [18,19,20,22,29,33]. Of particular interest has been its relationship to structural stability.…”