AI 中文总结
该研究区分了流的定向与标准两类伪轨跟踪性质，证明其在闭定向曲面上不同，关联标准伪轨跟踪与回复性、熵，还得到曲面流中定向伪轨跟踪结合有限奇点时链回复类为极小的结论。
AI 中文摘要
我们研究流的两类伪轨跟踪性质，二者的区别在于允许的时间重参数化：定向伪轨跟踪允许任意递增的重参数化，而标准伪轨跟踪要求其失真一致接近1。我们证明，这类概念在每个闭定向曲面上的$C^\infty$流中已是不同的。此外，这类例子在带奇点的$C^1$流中是$C^0$-稠密的，因此在欧拉特征非零的闭定向曲面上，它们在所有$C^1$流中是稠密的。随后我们将局部标准伪轨跟踪与回复性、熵关联起来：具有局部标准伪轨跟踪的非平凡链传递集，除非是不可约的几乎异宿集，否则会强制产生正拓扑熵。因此，对于具有标准伪轨跟踪的零熵流，每个非平凡链回复类都具有该形式，且每个非奇异的链回复类都是极小的。对于曲面流，我们进一步证明，定向伪轨跟踪加上有限多个奇点，会强制每个链回复类都是极小的。
英文摘要
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.
Comments53 pages, 7 figures
