A probabilistic Takens theorem

Krzysztof Barański,Adam Śpiewak,Yonatan Gutman

doi:10.1088/1361-6544/ab8fb8

Krzysztof Barański, Adam Śpiewak + Show 1 more

Open Access

https://doi.org/10.1088/1361-6544/ab8fb8

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Abstract

Let be a Borel set, μ a Borel probability measure on X and T : X → X a locally Lipschitz and injective map. Fix strictly greater than the (Hausdorff) dimension of X and assume that the set of p-periodic points of T has dimension smaller than p for p = 1, …, k − 1. We prove that for a typical polynomial perturbation of a given locally Lipschitz function , the k-delay coordinate map is injective on a set of full μ-measure. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required coordinates from 2 dim X to dim X and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

Highlights

Consider an experimentalist observing a physical system modeled by a discrete time dynamical system (X, T ), where T : X → X is the evolution rule and the phase space X is a subset of the Euclidean space RN
It is natural to ask, to what extent the original system can be reconstructed from such sequences of measurements and what is the minimal number k, referred to as the number of delay-coordinates, required for a reliable reconstruction
Note that one cannot expect a reliable reconstruction of the system based on the measurements of a given observable h, as it may fail to distinguish the states of the system

Summary

Introduction

Consider an experimentalist observing a physical system modeled by a discrete time dynamical system (X, T ), where T : X → X is the evolution rule and the phase space X is a subset of the Euclidean space RN. The upper box-counting dimension of X can be smaller than the dimension of any smooth manifold containing X, so Theorem 1.1 may require fewer delay-coordinates than its smooth counterpart in [Tak81] As it was noted above, usually an experimentalist may perform only a finite number of observations h(xj), . In this paper we are interested in the question of reconstruction of the system in presence of such process Speaking, this corresponds to fixing a probability measure μ on X and asking whether the delay-coordinate map φTα is injective almost surely with respect to μ. As in the probabilistic setting one can work with the Hausdorff dimension, we consider a set X ⊂ R2 similar to the one provided by Kan, which cannot be embedded linearly into R, but when endowed with a natural probability measure, almost every linear transformation L : R2 → R is injective on a set of full measure.

Preliminaries

Probabilistic embedding theorem

Probabilistic Takens delay embedding theorem

Examples

A modified Kan example

Measure with dimH μ < dim MB μ