Schroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay-embedding theorem can be improved in a probabilistic context. More precisely, their conjecture states that if $\mu$ is a natural measure for a smooth diffeomorphism of a Riemannian manifold and $k$ is greater than the dimension of $\mu$, then $k$ time-delayed measurements of a one-dimensional observable are generically sufficient for a predictable reconstruction of $\mu$-almost every initial point of the original system. This reduces by half the number of required measurements, compared to the standard (deterministic) setup. We prove the conjecture for all Lipschitz systems (also non-invertible) on compact sets with an arbitrary Borel probability measure and estimate the decay rate of the measure of the set of points where the prediction is subpar. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space. This is a joint work with Yonatan Gutman and Adam Śpiewak (Institute of Mathematics of the Polish Academy of Sciences).
Meeting ID: 852 4277 3200 Passcode: 103121