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 $k \in \mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and
assume that the set of $p$-periodic points
has dimension smaller than $p$ for $p=1, \ldots, k-1$. We prove that for
a typical polynomial perturbation $\tilde{h}$ 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, 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 measurements from $2\dim X$ to $\dim X$ and
using Hausdorff dimension
instead of the box-counting one. Finally we discuss related conjectures
in the physics literature.

Joint work with Krzysztof Barański and Adam Śpiewak.