On homeomorphisms with the two-sided limit shadowing property

Abstract

We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitivity. We show that those two properties allow to characterize topological transitivity and mixing in a class of expansive homeomorphisms and hence they characterize transitive (mixing) shifts of finite type.

Suspensions of homeomorphisms with the two-sided limit shadowing property

“…In topological dynamics, an important place is given to the shadowing theory, where many variants of pseudo-orbit tracing properties are discussed, mainly considering different notions of pseudo-orbits and shadowing points. Among them there is the limit shadowing property which has been given much attention recently (see [5], [6], [7], [8], [10], [11], [15], [16], [18], [19] and others). It deals with pseudoorbits indexed by positive integers and with one-step errors converging to zero in the future, usually called limit pseudo-orbits, and with orbits shadowing them in the limit (see Section 2 for precise definitions).…”

“…There are no homeomorphisms on the circle satisfying the two-sided limit shadowing property but we exhibit examples of flows on the circle satisfying it. It can happen that a suspension flow has the two-sided limit shadowing property but the base homeomorphism does not, though it is proved that it must satisfy a strictly weaker property called two-sided limit shadowing with a gap (as in [10]). We define a similar notion of two-sided limit shadowing with a gap for flows and prove that these notions are actually equivalent in the case of flows.…”

Periodic shadowing and standard shadowing property

“…(1) ergodic shadowing, (2) shadowing and chain mixing, (3) shadowing and topologically mixing, (4) pseudo-orbital specification, (5) periodic shadowing and chain mixing.…”

Average Chain Transitivity and the Almost Average Shadowing Property

“…There are now various variants of this concept which exist in the literature, for instance, one can refer [2,3,4,5,10] and some equivalences are obtained for expansive homeomorphisms having the shadowing property on compact metric spaces [7,8,12].…”