Proximal Relations and Maximal Equicontinuous Factors for Non-autonomous Dynamical Systems

What the math gives to ML

The paper provides a transfer mechanism for time-varying neural dynamics: under uniform convergence of the maps, collective convergence, and uniform rigidity, proximality relations of a non-autonomous system can be inherited from an autonomous limit system. Its most useful asset is a principled distinction between finite-time closeness and long-horizon orbit equivalence, together with a construction of the smallest closed relation invariant under every time-dependent generator. This can be used to design or monitor recurrent or state-space networks whose weights, normalization, or controllers change during training or inference while preserving a stable autonomous quotient. The key falsifiable prediction is that long-horizon pairwise orbit behavior should match the limiting autonomous model only after generator-discrepancy and rigidity diagnostics cross measurable thresholds.