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Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delayembedding theorem can be improved in a probabilistic context. More precisely,their conjecture states that if $\mu$ is a natural measure for a smoothdiffeomorphism of a Riemannian manifold and $k$ is greater than the informationdimension of $\mu$, then $k$ time-delayed measurements of a one-dimensionalobservable $h$ are generically sufficient for a predictable reconstruction of$\mu$-almost every initial point of the original system. This reduces by halfthe number of required measurements, compared to the standard (deterministic)setup. We prove the conjecture for ergodic measures and show that it holds fora generic smooth diffeomorphism, if the information dimension is replaced bythe Hausdorff one. To this aim, we prove a general version of predictableembedding theorem for injective Lipschitz maps on compact sets and arbitraryBorel probability measures. We also construct an example of a $C^\infty$-smoothdiffeomorphism with a natural measure, for which the conjecture does not holdin its original formulation.

On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay embeddings