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

What the math gives to ML

The paper provides a sharp distinction between oriented shadowing, which permits arbitrary increasing time reparametrizations, and standard shadowing, which requires all elapsed-time distortions to remain uniformly close to one. Its key transferable mechanism is a quantitative test for temporal as well as geometric robustness: trajectories can remain spatially close while accumulating incompatible crossing times near slow regions or singularities. The paper also gives a constructive entropy criterion: a recurrent set with local standard shadowing has positive topological entropy when two nearby almost-returning orbit segments remain separated at an intermediate time. These mechanisms can be transferred to continuous-depth networks and recurrent latent dynamics as diagnostics or regularizers for time-warp robustness and long-horizon predictability.