Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Adam Śpiewak

TL;DR

This work extends Takens-type embedding theory into a probabilistic framework that permits self-intersections on a zero-measure set. By combining prevalence-based perturbations with Hausdorff-dimension thresholds, the authors prove that time-delayed embeddings $\phi_{h,k}$ exhibit strong regularity properties—injectivity on a full-measure set with a pointwise Lipschitz inverse, and, when $k>\dim M$, local diffeomorphism almost everywhere. These regularity results yield practical consequences: almost-sure, pointwise control of prediction errors and the ability to approximate the original system’s Lyapunov exponents from the embedded dynamics. The paper also provides a non-dynamical probabilistic Whitney-type embedding result, reinforcing the broader relevance to geometric embedding problems. Overall, the approach links fractal dimension criteria, random perturbations, and dynamical features to advance rigorous guarantees for time-delayed modeling of complex systems.

Abstract

We study regularity of the time-delayed coordinate maps \[φ_{h,k}(x) = (h(x), h(Tx), \ldots, h(T^{k-1}x))\] 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 $φ_{h,k}$ 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 $k \geq \dim M$ and $k > \dim_H(\text{supp} μ)$, then for a typical observable, $φ_{h,k}$ is injective on a full-measure set with a pointwise Lipschitz inverse. If moreover $k > \dim M$, then $φ_{h,k}$ 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.

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

TL;DR

This work extends Takens-type embedding theory into a probabilistic framework that permits self-intersections on a zero-measure set. By combining prevalence-based perturbations with Hausdorff-dimension thresholds, the authors prove that time-delayed embeddings exhibit strong regularity properties—injectivity on a full-measure set with a pointwise Lipschitz inverse, and, when , local diffeomorphism almost everywhere. These regularity results yield practical consequences: almost-sure, pointwise control of prediction errors and the ability to approximate the original system’s Lyapunov exponents from the embedded dynamics. The paper also provides a non-dynamical probabilistic Whitney-type embedding result, reinforcing the broader relevance to geometric embedding problems. Overall, the approach links fractal dimension criteria, random perturbations, and dynamical features to advance rigorous guarantees for time-delayed modeling of complex systems.

Abstract

We study regularity of the time-delayed coordinate maps for a diffeomorphism of a compact manifold and smooth observables on . Takens' embedding theorem shows that if , then is an embedding for typical . We consider the probabilistic case, where for a given probability measure on one allows self-intersections in the time-delayed embedding to occur along a zero-measure set. We show that if and , then for a typical observable, is injective on a full-measure set with a pointwise Lipschitz inverse. If moreover , then is a local diffeomorphism at almost every point. As an application, we show that if , 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.
Paper Structure (14 sections, 15 theorems, 49 equations)

This paper contains 14 sections, 15 theorems, 49 equations.

Table of Contents

  1. Introduction and results
  2. Introduction
  3. Results
  4. Preliminaries
  5. Hausdorff measure and dimension
  6. Smooth manifolds
  7. Prevalence
  8. Proof of Theorem \ref{['thm: projections main']}
  9. Proof of Theorem \ref{['thm: takens main']}
  10. Global almost sure continuity
  11. Almost sure local immersion
  12. Applications
  13. Bounds on the prediction errors
  14. Preservation of the Lyapunov exponents

Key Result

Theorem 1.1

Let $M$ be a compact, Riemannian $C^r$ manifold. Let $T : M \to M$ be a $C^r$ diffeomorphism. Let $X \subset M$ be a compact set and $\mu$ be a Borel probability measure on $X$. Fix $k \in \mathbb N$ and assume that $\mu \left( \bigcup \limits_{p=1}^{k-1} \mathop{\mathrm{Per}}\nolimits_p(T) \right)

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm: projections detail']}
  • ...and 16 more