On the regularity of time-delayed embeddings with self-intersections

TL;DR

This paper investigates the regularity and injectivity of time-delayed embeddings for dynamical systems, showing under what conditions they are almost surely invertible and can approximate Lyapunov exponents, extending classical embedding theorems.

Contribution

It provides probabilistic conditions ensuring that time-delayed embeddings are injective and locally diffeomorphic for typical observables, with applications to Lyapunov exponent approximation.

Findings

Time-delayed embeddings are injective on a full-measure set when k ≥ dim M and k > Hausdorff dimension of support.

For k > dim M, embeddings are local diffeomorphisms at almost every point.

Lyapunov exponents can be approximated arbitrarily well using time-delayed models for k > dim M.

Abstract

We study regularity of the time-delayed coordinate maps \[\phi_{h,k}(x) = (h(x), h(Tx), \ldots, h(T^{k-1}x))\] for a diffeomorphism $T$ of a compact manifold $M$ and smooth observables $h$ on $M$. Takens' embedding theorem shows that if $k > 2\dim M$, then $\phi_{h,k}$ is an embedding for typical $h$. We consider the probabilistic case, where for a given probability measure $\mu$ on $M$ one allows self-intersections in the time-delayed embedding to occur along a zero-measure set. We show that if $k \geq \dim M$ and $k > \dim_H(\text{supp} \mu)$, then for a typical observable, $\phi_{h,k}$ is injective on a full-measure set with a pointwise Lipschitz inverse. If moreover $k > \dim M$, then $\phi_{h,k}$ is a local diffeomorphism at almost every point. As an application, we show that if $k > \dim M$, then the Lyapunov exponents of the original system can be approximated with arbitrary precision by almost every orbit in the time-delayed model of the system. We also give almost sure pointwise bounds on the prediction error and provide a non-dynamical analogue of the main result, which can be seen as a probabilistic version of Whitney's embedding theorem.