A topological delay embedding theorem for infinite-dimensional dynamical systems

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 A be a subset of a Hilbert space H with upper box-counting dimension d(A) = d and ‘thickness exponent’ τ , which is invariant under a Lipschitz map . Take an integer k > (2 + τ)d , and suppose that Ap, the set of all p-periodic points of , satisfies d(Ap) < p/(2 + τ) for all p = 1, . . . , k. Then a prevalent set of Lipschitz observation functions h : H → R make the k-fold observation map u → [h(u), h( (u)), h( k−1(u))], one-to-one between A and its image. The same result is true if A is a subset of a Banach space provided that k > 2(1 + τ)d and d(Ap) < p/(2 + 2τ). 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). Mathematics Subject Classification: 37L30, 35B41, 35Q30, 76F20