On the regularity of time-delayed embeddings with self-intersections
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908359557382144 |
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| author | Śpiewak, Adam |
| author_facet | Śpiewak, Adam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the regularity of time-delayed embeddings with self-intersections Śpiewak, Adam Dynamical Systems Differential Geometry Chaotic Dynamics 37C45, 37C40, 37D25, 58D10 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. |
| title | On the regularity of time-delayed embeddings with self-intersections |
| topic | Dynamical Systems Differential Geometry Chaotic Dynamics 37C45, 37C40, 37D25, 58D10 |
| url | https://arxiv.org/abs/2505.06712 |