Jesús Aponte | Searcharxiv

arXiv2019

Suspensions of homeomorphisms with the two-sided limit shadowing property

In this paper we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyze some of the results known for the case of homeomorphisms in the case of continuous flows and observe that some differences appear in this scenario. We prove that the suspension flow of a homeomorphism satisfying the two-sided limit shadowing property also satisfies it. This gives a lot of examples of flows satisfying this property, however it enlighten an important difference between the case of flows and homeomorphisms: there are flows satisfying the two-sided limit shadowing property that are not topologically mixing, while homeomorphisms satifying the two-sided limit shadowing property satisfy even the specification property. 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 \cite{CK}). 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. Finally, we prove that singular suspension flows (in the sense of Komuro \cite{K}) do not satisfy the two-sided limit shadowing property.

math.DS↗

arXiv2017

Shadowable Points for flows

A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a $G_δ$ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. We study the relations between shadowable points of a homeomorphism and the shadowable points of its suspension flow. We characterize the set of forward shadowable points for transitive flows and chain transitive flows. We prove that the geometric Lorenz attractor does not have shadowable points. We show that in the presence of shadowable points chain transitive flows are transitive and that transitivity is a necessary condition for chain recurrent flows with shadowable points whenever the phase space is connected. Finally, as an application these results we give concise proofs of some well known theorems establishing that flows with POTP admitting some kind of recurrence are minimal. These results extends those presented in [10].

math.DS↗