Takens' embedding theorem with a continuous observable

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.