Observing Infinite-dimensional Dynamical Systems

We study the extent to which properties of infinite-dimensional dynamical systems can be accurately detected by examining observations of such systems. Let H be a separable Hilbert space. Let f : H → H be a map and let A ⊂ H be a compact set satisfying f(A) = A. We prove that for almost every (in the sense of prevalence) continuous observable φ : H → RM , if f induces a map f̄ satisfying f̄ ◦ φ = φ ◦ f on A and if this induced map has certain properties, then the observable φ is one-to-one on A and therefore the dynamics of f on A are topologically conjugate to those of f̄ on φ(A).