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arXiv · 2608.12165

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

Sakshi Jain·Piotr Oprocha·Elias Rego
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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.

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Sakshi Jain, Piotr Oprocha, Elias Rego. 2026-08-12. Shadowing in the presence of singularities: oriented versus standard shadowing, entropy and the structure of recurrent sets. https://arxiv.org/abs/2608.12165

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