  Shroer, 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 information
dimension of $\mu$, then $k$ time-delayed measurements of a one-dimensional
observable $h$ 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 ergodic measures and show that it holds for
a generic smooth diffeomorphism, if the information dimension is replaced by
the Hausdorff one. To this aim, we prove a general version of predictable
embedding theorem for injective Lipschitz maps on compact sets and arbitrary
Borel probability measures. We also construct an example of a $C^\infty$-smooth
diffeomorphism with a natural measure, for which the conjecture does not hold
in its original formulation.
