Time-delayed embeddings of dynamical systems withself-intersections – a probabilistic approach

Abstract: Takens’ embedding theorem deals with reconstructing a
dynamical system T: X->X from observations. It states that if k >
2dim(X), then a typical observable on X can be used to faithfully
reconstruct the dynamics via its model in R^k, which is obtained by the
time-delayed embedding x -> (h(x), h(Tx), …, h(T^k-1 x)). I will
present basics of a probabilistic version of this theory, which allows
for a reconstruction in a lower dimension, at the cost of allowing
intersections to occur along sets of measure zero (with respect to a
given reference measure). I will also discuss applications to the
conjectures posed by Schroer, Sauer, Ott and Yorke regarding predictions
of the future values of observed time-series in the presence of
self-intersections. The talk is based on joint works with Krzysztof
BaraƄski and Jonatan Gutman.