Resumo: Consider a dynamical system (X,T) consisting of a compact set X in the Euclidean space and a transformation T on X. Takens-type time-delay embedding theorems state that for a generic real-valued observable h on X, one can reconstruct uniquely the initial state x of the system from a sequence of values of h(x), h(Tx), ..., h(T^{k-1} x), provided that k is large enough. In the deterministic setting, the number of measurements k has to be at least twice the dimension of the state space X. This was proved in several categories and can be seen as dynamical versions of the classical (non-dynamical) embedding theorems. We provide a probabilistic counterpart of this theory, in which one is interested in reconstructing almost every state x, subject to a given probability measure. We prove that in this setting it suffices to take k greater than the Hausdorff dimension of the considered measure, hence reducing the number of measurements at least twice. Using this, we prove a related conjecture of Shroer, Sauer, Ott and Yorke in the ergodic case (and construct an example showing that the conjecture does not hold in its original formulation). Time permitting, I will also describe a work in progress on quantitative aspects of probabilistic time-delay embeddings. This is based on joint works with Krzysztof Barański and Yonatan Gutman.

Os interessados na palestra podem acessar o link e o calendário do grupo em https://sites.google.com/usp.br/evol-eq-and-dyn-systems