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

We study regularity of the time-delayed coordinate maps ϕh,k(x)=(h(x),h(Tx),…,h(Tk−1x)) for a diffeomorphism T of a compact manifold M and smooth observables h on M. Takens’ embedding theorem shows that if k>2dimM, then ϕh,k is an embedding for typical h. We consider the probabilistic case, where for a given probability measure μ on M one allows self-intersections in the time-delayed embedding to occur along a zero-measure set. We show that if k≥dimM and k>dimH(suppμ), then for a typical observable, ϕh,k is injective on a full-measure set with a pointwise Lipschitz inverse. If moreover k>dimM, then ϕh,k is a local diffeomorphism at almost every point. As an application, we show that if k>dimM, 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 nondynamical analogue of the main result, which can be seen as a probabilistic version of Whitney’s embedding theorem.