publication . Preprint . Part of book or chapter of book . 2015

Takens' embedding theorem with a continuous observable

Yonatan Gutman;
Open Access English
  • Published: 20 Oct 2015
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.
Subjects
free text keywords: Mathematics - Dynamical Systems, Mathematical Physics, 37C45, 54H20
Funded by
EC| UNIVERSALITY
Project
UNIVERSALITY
Universality in Topological Dynamics
  • Funder: European Commission (EC)
  • Project Code: 334564
  • Funding stream: FP7 | SP3 | PEOPLE
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publication . Preprint . Part of book or chapter of book . 2015

Takens' embedding theorem with a continuous observable

Yonatan Gutman;