A probabilistic Takens theorem

Barański, Krzysztof; Gutman, Yonatan; Śpiewak, Adam

doi:10.1088/1361-6544/ab8fb8

Cited by 13 publications

(12 citation statements)

References 33 publications

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“…The proof is based on the embedding theorem for dim B with Hölder inverse [13,Thm. 4.3] (see [16] for an almost sure embedding theorem for Hausdorff dimension). See [1] for the proof and examples showing that one cannot change the constant 2 1−α to t 1−α for t < 2 and inf L>0 cannot be omitted.…”

Section: Resultsmentioning

confidence: 99%

New Uniform Bounds for Almost Lossless Analog Compression

Gutman

Śpiewak

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Wu and Verdú developed a theory of almost lossless analog compression, where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a (typically infinite-entropy) stationary stochastic process. In this work we consider all stationary stochastic processes with trajectories in a prescribed set S ⊂ [0, 1] Z of (bi)infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension. An essential tool is the recent Lindenstrauss-Tsukamoto variational principle expressing metric mean dimension in terms of rate-distortion functions.

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Section: Resultsmentioning

confidence: 99%

New Uniform Bounds for Almost Lossless Analog Compression

Gutman

Śpiewak

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…To prove Theorem 1.7, we will use our previous result from [BGŚ20], which we recall below, using the notion of prevalence described in Subsection 2.2. This is a probabilistic version of the Takens delay embedding theorem, asserting that under suitable conditions on k, there is a prevalent set of Lipschitz observables, which give rise to an almost surely injective k-delay coordinate map.…”

Section: Probabilistic Takens Delay Embedding Theoremmentioning

confidence: 99%

“…Extensions of Takens' work were obtained in several categories, e.g. in [SYC91, Sta99, Cab00, Rob05, Gut16, GQS18, SBDH97, SBDH03, NV20] (see also [Rob11,BGŚ20] for a more detailed overview). A common feature of these results is that the minimal number of measurements sufficient for the exact reconstruction of the system is k ≈ 2 dim X, where dim X is the dimension of the phase space X.…”

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confidence: 99%

“…It is known that Takens-type theorems require some bounds on the size of sets of T -periodic points of low periods. Following [BGŚ20], we assume dim H (µ| Perp(T ) ) < p for p = 1, . .…”

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confidence: 99%

“…Note that the assumption on the dimension of µ restricted to the set of p-periodic points can be omitted if there are only finitely many periodic points of given period. By the Kupka-Smale theorem (see [PdM82, Chapter 3, Theorem 3.6]), the latter condition is generic (in the Baire category sense) in the space of C r -diffeomorphisms, r ≥ 1, of a compact manifold, equipped with the uniform C r -topology (see [BGŚ20] for more details). Therefore, we immediately obtain the SSOY predictability conjecture for generic smooth C r -diffeomorphisms, with information dimension replaced by the Hausdorff one.…”

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confidence: 99%

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On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay embeddings

Barański,

Gutman,

Śpiewak

2021

Preprint

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Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay embedding theorem can be improved in a probabilistic context. More precisely, their conjecture states that if µ is a natural measure for a smooth diffeomorphism of a Riemannian manifold and k is greater than the information dimension of µ, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for a predictable reconstruction of µ-almost every initial point of the original system. This reduces by half the number of required measurements, compared to the standard (deterministic) setup. We prove the conjecture for ergodic measures and show that it holds for a generic smooth diffeomorphism, if the information dimension is replaced by the Hausdorff one. To this aim, we prove a general version of predictable embedding theorem for injective Lipschitz maps on compact sets and arbitrary Borel probability measures. We also construct an example of a C ∞ -smooth diffeomorphism with a natural measure, for which the conjecture does not hold in its original formulation.

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Equivariant embedding of finite-dimensional dynamical systems

Gutman,

Levin,

Meyerovitch

2024

Math. Ann.

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We prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into $$([0,1]^r)^G$$ ( [ 0 , 1 ] r ) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than $$\frac{rN}{2}$$ rN 2 . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset $$F\subset G$$ F ⊂ G so that for a generic continuous map $$h:X\rightarrow [0,1]^{r}$$ h : X → [ 0 , 1 ] r , the map $$ h^{F}:X\rightarrow ([0,1]^{r})^{F}$$ h F : X → ( [ 0 , 1 ] r ) F given by $$x\mapsto (f(gx))_{g\in F}$$ x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

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A probabilistic Takens theorem

Abstract: Let X ⊂ R N be a Borel set, μ a Borel probability measure on X and T : X → X a locally Lipschitz and injective map. Fix … Show more

Cited by 13 publications

References 33 publications

New Uniform Bounds for Almost Lossless Analog Compression

New Uniform Bounds for Almost Lossless Analog Compression

On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay embeddings

Equivariant embedding of finite-dimensional dynamical systems

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