I will continue Jonatan Gutman's talk. It will concern further developments
on the probabilistic Takens embedding theorem.
I will present the following result: let $X \subset \mathbb{R}^N$ be a
compact 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 upper packing dimension of $\mu$
and assume that sets of periodic points have dimensions small enough.
We prove that for a typical polynomial perturbation $\tilde{h}$ of a
given Lipschitz map $h : X \to \mathbb{R}$,
the corresponding $k$-delay coordinate map $\phi : X \to \mathbb{R}^k$
has the following property:
for almost every $x \in X$, measure $\mu$ restricted to $\{ y \in X :
|\phi(x) - \phi(y)| < \epsilon \}$
(and normalized to a probability measure) converges to the point mass in
$x$ as $\epsilon$ goes to zero.
This proves the conjecture of Ott, Sauer, Shroer and Yorke for exact
dimensional measures.
Joint work in progress with Krzysztof Barański and Jonatan Gutman.