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

TL;DR

This paper proves a probabilistic version of the Shroer-Sauer-Ott-Yorke conjecture, showing fewer measurements suffice for predictable time-delay embeddings under certain conditions, and provides a counterexample to the original conjecture.

Contribution

It establishes the conjecture for ergodic measures and generic smooth diffeomorphisms, replacing information dimension with Hausdorff dimension, and proves a general predictable embedding theorem.

Findings

Proved the conjecture for ergodic measures.

Showed the conjecture holds for generic smooth diffeomorphisms with Hausdorff dimension.

Constructed a counterexample where the original conjecture fails.

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 $\mu$ 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.