Takens' embedding theorem with a continuous observable

2015/10/20 by Gutman, Yonatan
#37C45 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1510.05843

Abstract

Let (X,T) be a dynamical system where X is a compact metric space and T:X→ X is continuous and invertible. Assume the Lebesgue covering dimension of X is d. We show that for a generic continuous map h:X→[0,1], the (2d+1)-delay observation map x↦(h(x),h(Tx),…,h(T^2dx)) 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.

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