Contraction versus Recurrence: An Exponential Separation in Observation-Based Prediction of Deterministic Dynamics

Abstract

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 $\varepsilon$-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 $\varepsilon^{-d}$, 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 $\Theta(\log(1/\varepsilon)/(1-\rho(A_{cl})^2))$ steps, where $\rho(A_{cl})$ 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 $\log(1/\varepsilon)$ with $R^2=1.000$ and in $(1-\rho^2)^{-1}$ with $R^2=0.985$), yielding a measured cost gap of $\sim 10^{9}$ at $\varepsilon=10^{-6}$ for $d\approx 2$. 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 $2\log_{10}N$. On real data the gate admits the Santa Fe laser benchmark ($\hat D_2=2.0$) 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).