Shadowing in the presence of singularities: oriented versus standard shadowing, entropy and the structure of recurrent sets

Abstract

We study two shadowing properties for flows that differ in the allowed reparametrizations of time: oriented shadowing permits arbitrary increasing reparametrizations, whereas standard shadowing requires their distortion to be uniformly close to one. We prove that these notions are distinct already for $C^\infty$ flows on every closed oriented surface. Moreover, such examples are $C^0$-dense among $C^1$ flows with a singularity and consequently, on closed oriented surfaces with non-zero Euler characteristic, they are dense among all $C^1$ flows. We then relate local standard shadowing to recurrence and entropy. A non-trivial chain-transitive set with local standard shadowing forces positive topological entropy unless it is an irreducible almost heteroclinic set. Consequently, for a zero-entropy flow with standard shadowing, every non-trivial chain-recurrent class has this form, and every non-singular one is minimal. For surface flows, we further prove that oriented shadowing together with finitely many singularities forces every chain-recurrent class to be minimal.