Suspensions of homeomorphisms with the two-sided limit shadowing property

Abstract

In this paper, we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyse 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 satisfying the two-sided limit shadowing property satisfy even the specification property. 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 [B. Carvalho, D. Kwietniak. On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl. 420 (2014), 801–813]). 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: One-parameter flows with the pseudo orbit tracing property. Mh. Math. 98 (1984) 219–253]) do not satisfy the two-sided limit shadowing property.