A probabilistic Takens theorem

Takens theorem (1981) states that for a $C^2$ manifold $M$ of dimension $d$, the delay map $x \mapsto (h(x), h(Tx),..., h(T^{k-1} x))$ is an embedding for a generic pair of diffeomorphism $T:M \to M$ and smooth map $h: M \to \mathbb{R}$, as long as $k>2d$. I will present a survey on results in this direction. I will also prove a probabilistic version of Takens theorem, where a probability measure $\mu$ is given and one is interested in injectivity of the delay map on a set of full measure. It turns out that in such a setting, it suffices to take the number of measurements $k$ to be greater than the Hausdorff dimension of $\mu$. This is based on a joint work with Krzysztof Barański and Yonatan Gutman.