Takens' embedding theorem with a continuous observable

TL;DR

This paper generalizes Takens' embedding theorem to continuous observables for dynamical systems on compact metric spaces, showing that a generic continuous function yields an embedding via delay coordinates without dimension restrictions.

Contribution

It extends Takens' theorem to continuous observables in a more general setting, removing the need for finite box-counting dimension assumptions.

Findings

For a generic continuous map, the delay observation map is an embedding.

The result applies to systems with potentially infinite box-counting dimension.

No assumptions on the lower box-counting dimension are required.

Abstract

Let $(X,T)$ be a dynamical system where $X$ is a compact metric space and $T:X\rightarrow X$ is continuous and invertible. Assume the Lebesgue covering dimension of $X$ is $d$. We show that for a generic continuous map $h:X\rightarrow[0,1]$, the $(2d+1)$-delay observation map $x\mapsto\big(h(x),h(Tx),\ldots,h(T^{2d}x)\big)$ is an embedding of $X$ inside $[0,1]^{2d+1}$. This is a generalization of the discrete version of the celebrated Takens embedding theorem, as proven by Sauer, Yorke and Casdagli to the setting of a continuous observable. In particular there is no assumption on the (lower) box-counting dimension of $X$ which may be infinite.