2026/07/16 by Pavel Popovich
#math.DS
Given a scalar observable of an ergodic dynamical system with a low-dimensional attractor, two families of methods reconstruct and predict the underlying state: recurrence-based methods (the method of analogues and its descendants), which wait for the trajectory to return to an ε-neighborhood of a previously observed state, and observer-based methods, which fit a converging state estimator on the delay reconstruction. We formalize and empirically verify an exponential separation between the two: the expected cost of recurrence scales as ε-d, where d is the pointwise dimension of the invariant measure (a consequence of the Kac lemma and quantitative Poincare recurrence), whereas a detectable linear observer converges in Θ(log(1/ε)/(1-ρ(Acl)2)) steps, where ρ(Acl) is the closed-loop spectral radius of the Riccati fixed point. Both laws are verified numerically (return-time exponent -1.8 on the Lorenz attractor against the theoretical -2.05; observer cost linear in log(1/ε) with R2=1.000 and in (1-ρ2)-1 with R2=0.985), yielding a measured cost gap of ∼ 109 at ε=10-6 for d≈ 2. We complement the theorem with an admission protocol (the Kac-Riccati gate) deciding whether a signal lies inside the theorem's class, via surrogate-data prediction gating; it also explains the folklore of "universal" fractal dimensions as a dataset-size artifact bounded by 2log10N. On real data the gate admits the Santa Fe laser benchmark ( D2=2.0) and refuses the monthly sunspot series, reproducing the settled resolution of historical low-dimensionality claims. All results reproduce from a single verification script (17/17 checks).