Probabilistic delay embedding and prediction

26.01.2022 16:15 – 18:15

Let $X \subset \R^N$ be a Borel set, $\mu$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \N$ 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 $h : X \to \R$, 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 (1991). We use this theorem in order to establish
an almost-sure predictable reconstruction conjecture by Shroer, Sauer, Ott and Yorke (1998) for ergodic measures.
This is joint work with Krzysztof Barański and Adam Śpiewak.

NB: exceptional date

Lieu

Salle 1-15, Séminaire "Mathématique physique"

entrée libre