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

On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay 
embeddings (joint work with Yonatan Gutman and Adam Śpiewak).

Abstract:

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 μ is a natural measure for a 
smooth diffeomorphism of a Riemannian manifold and k is greater than the 
information dimension of μ, then k time-delayed measurements of a 
one-dimensional observable h are generically sufficient for a predictable 
reconstruction of μ-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∞-smooth diffeomorphism with a natural 
measure, for which the conjecture does not hold in its original 
formulation.