Takens’ embedding theorem for infinite-dimensional dynamical systems

Takens’ time delay embedding theorem is shown to hold for finitedimensional subsets of infinite-dimensional spaces, thereby generalising previous results which were only valid for subsets of finite-dimensional spaces: Let X be a subset of a Hilbert space with upper box-counting dimension d and ‘thickness exponent’ τ , which is invariant under a Lipschitz map Φ. Suppose that E , the set of all fixed points of Φ, is not too large, in particular that df(E) < 1/2. Then for every k > (2+τ)d such that Φ has no periodic orbits of period 2, . . . , k, a prevalent set of Lipschitz observation functions f : H → R makes the k-fold time delay map u 7→ [h(u), h(Φ(u)), h(Φk−1(u))] one-to-one between X and its image. The same result is true if X is a subset of a Banach space provided that k > 2(1 + τ)d. The result follows from a version of Takens’ theorem for Hölder continuous maps adapted from Sauer, Yorke, & Casdagli (J. Stat. Phys. (1991) 65 529–547), and makes use of an embedding theorem for finite-dimensional sets due to Hunt & Kaloshin (Nonlinearity (1999) 12 1263–1275). Submitted to: Nonlinearity An infinite-dimensional Takens theorem 2