Exponential mixing via invariant foliations and relatively Anosov homeomorphisms

What the math gives to ML

The paper’s transferable mechanism is a quantitative mixing dichotomy for systems preserving an invariant foliation: correlation decay is governed by the slower of the quotient dynamics and the dynamics within the leaves. This suggests a hierarchical neural dynamical system whose latent state is split into a coarse quotient variable and a leaf or fibre variable, with independently controlled contraction rates. The main engineering prediction is that long-horizon correlations and perturbation memory decay at approximately the slower branch rate, \(\max(\rho_q,\rho_f)\), so accelerating a non-bottleneck branch should have little effect on asymptotic memory. This provides a concrete architecture and falsifiable diagnostic for recurrent networks, state-space models, and learned simulators.