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A topological delay embedding theorem for infinite-dimensional dynamical systems. (English) Zbl 1084.37063

Summary: A time delay reconstruction theorem inspired by that of F. Takens [Dynamical systems and turbulence, Proc. Symp., Coventry 1980, Lect. Notes Math. 898, 366–381(1981; Zbl 0513.58032)] 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 \({\mathcal{A}}\) be a subset of a Hilbert space \(H\) with upper box-counting dimension \(d({\mathcal{A}})=d\) and ‘thickness exponent’ \(\tau\), which is invariant under a Lipschitz map \(\Phi\). Take an integer \(k > (2 + \tau)d\), and suppose that \({\mathcal{A}}_p\), the set of all \(p\)-periodic points of \(\Phi\), satisfies \(d({\mathcal{A}}_p)<p/(2+\tau)\) for all \(p = 1,\dots, k\). Then a prevalent set of Lipschitz observation functions \(h:H\rightarrow {\mathbb{R}}\) make the \(k\)-fold observation map
\[ u\mapsto[h(u),h(\Phi(u)),h(\Phi^{k-1}(u))] \]
one-to-one between \(\mathcal{A}\) and its image. The same result is true if \({\mathcal{A}}\) is a subset of a Banach space provided that \(k > 2(1 + \tau)d\) and \(d({\mathcal{A}}_p)<p/(2+2\tau)\).
The result follows from a version of the Takens theorem for Hölder continuous maps adapted from T. Sauer, J. A. Yorke and M. Casdagli [J. Stat. Phys. 65, 579–616 (1991; Zbl 0943.37505)], and makes use of an embedding theorem for finite-dimensional sets due to B. R. Hunt and V. Y. Kaloshin [Nonlinearity 12, No. 5, 1263–1275 (1999; Zbl 0932.28006)].

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
35Q30 Navier-Stokes equations
76F20 Dynamical systems approach to turbulence
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