Prevalence of delay embeddings with a fixed observation function

[摘要] 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. (c) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2020-12-15

[关键词] Delay coordinates;Embedding;Prevalence