A topological delay embedding theorem for infinite-dimensional dynamical systems

Abstract

A time delay reconstruction theorem inspired by that of Takens (1981 Springer Lecture Notes in Mathematics vol 898, pp 366–81) is shown to hold for finite-dimensional subsets of infinite-dimensional spaces, thereby generalizing previous results which were valid only for subsets of finite-dimensional spaces.Let be a subset of a Hilbert space H with upper box-counting dimension and ‘thickness exponent’ τ, which is invariant under a Lipschitz map Φ. Take an integer k > (2 + τ)d, and suppose that , the set of all p-periodic points of Φ, satisfies for all p = 1, …, k. Then a prevalent set of Lipschitz observation functions make the k-fold observation map one-to-one between and its image. The same result is true if is a subset of a Banach space provided that k > 2(1 + τ)d and .The result follows from a version of the Takens theorem for Hölder continuous maps adapted from Sauer et al (1991 J. Stat. Phys. 65 529–47), and makes use of an embedding theorem for finite-dimensional sets due to Hunt and Kaloshin (1999 Nonlinearity 12 1263–75).