Shadowing and expansivity in subspaces

Abstract

Abstract. We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.

On the shadowing and limit shadowing properties

“…This theorem generalizes a result of [4] proving that every equicontinuous homeomorphism of a totally disconnected space (e.g. odometer) satisfies the shadowing property and the limit shadowing property.…”

“…Also in [25], it is proved that for circle homeomorphisms, the shadowing property always implies the limit shadowing property, and the same implication holds true for c-expansive maps including expansive homeomorphisms (see [4,5,17]). It is rather difficult to construct a continuous map satisfying the shadowing property but not the limit shadowing property, but in [12], such an example is given, while the equivalence of the two shadowing properties is proved for a certain class of interval maps.…”

Preservation of shadowing in discrete dynamical systems

“…. Then f has s-limit shadowing and limit shadowing (see [1]) but f n does not have limit shadowing (and consequently it does not have s-limit shadowing) for any n ≥ 3.…”

“…Various other notions of shadowing have since been studied including, for example, ergodic, thick and Ramsey shadowing [8,9,14,17,32], limit shadowing [1,24,38], s-limit shadowing [1,24,27], orbital shadowing [23,38,36], and inverse shadowing [12,26].…”

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]).…”