A probabilistic Takens theorem

Let X ⊂ℝN be a Borel set, μ a Borel probability measure on X and T : X → X a locally Lipschitz and injective map. Fix k ∈ N strictly greater than the (Hausdorff) dimension of X and assume that the set of p-periodic points of T has dimension smaller than p for p = 1, ⋯, k - 1. We prove that for a typical polynomial perturbation h of a given locally Lipschitz function h:X ℝ, the k-delay coordinate map x (h(x), h (Tx),⋯, h (Tk-1 x) is injective on a set of full μ-measure. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required coordinates from 2 dim X to dim X and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.