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

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.