Predicting dynamical systems with too few time-delay measurements: error estimates

We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with an invariant Borel probability measure μ of Hausdorff dimension d, one needs at least d measurements of a typical (prevalent) Lipschitz observable for μ-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of μ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. Furthermore, we establish a lower bound postulated in the Schroer–Sauer–Ott–Yorke prediction error conjecture from 1998, after necessary modifications (whereas the upper estimates were obtained in our previous work). To this aim, we prove a general theorem on the dimensions of conditional measures of μ with respect to time-delay coordinate maps.