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 Ĺšpiewak (Institute of
Mathematics of the Polish Academy of Sciences).

Meeting ID: 852 4277 3200
Passcode: 103121