Prevalence of delay embeddings with a fixed observation function

Abstract

The Whitney embedding theorem is a basic result of differential topology and it is natural to ask for versions of the embedding theorem for time series data. Embedding theorems for dynamical time series data were claimed by Takens (1981) and proved in a different setting by Sauer, Yorke, and Casdagli (1991). Those results were stated assuming the observation function to be parametrized. A possibly more pertinent setting is to assume the dynamical system to be parametrized, with the observation function fixed, for example, as a projection to a certain coordinate. We prove an embedding theorem in such a setting. Our proof introduces a technique that relies on the notion of Lebesgue points.