# Suspensions of homeomorphisms with the two-sided limit shadowing property

## TL;DR

This paper investigates the two-sided limit shadowing property in continuous flows and homeomorphisms, revealing differences between flows and homeomorphisms and providing new examples and theoretical insights.

## Contribution

It proves that suspension flows of homeomorphisms with the property also satisfy it, introduces the concept of shadowing with a gap, and distinguishes flow behaviors from homeomorphisms.

## Key findings

- Suspension flows inherit the two-sided limit shadowing property from their base homeomorphisms.
- Some flows with this property are not topologically mixing, unlike homeomorphisms.
- Certain singular suspension flows do not satisfy the property.

## Abstract

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.