A probabilistic Takens theorem

Let $X\subset {\mathbb{R}}^N$ be a Borel set, $\mu$ a Borel probability measure on $X$ and $T:X\to X$ a Lipschitz and injective map. Fix $k \in $\mathbb{N}$$greaterthanthe(Hausdorff)dimensionof$X$ and assume that the set of $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots, k-1$\mathrm{.}Weprovethatforatypicalpolynomialperturbation$\tilde{h}$ of a given Lipschitz map $h:X\to \mathbb{R}$, the $k$-delay coordinate map $x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is injective on a set of full measure $\mu$. 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, B"{o}lcskei, 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 measurements 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.