Learning about Reality from Observation

Takens, Ruelle, Eckmann, Sano and Sawada launched an investigation of images of attractors of dynamical systems. Let A be a compact invariant set for a map f on Rn and let φ : Rn → Rm where n > m be a “typical” smooth map. When can we say that A and φ(A) are similar, based only on knowledge of the images in Rm of trajectories in A? For example, under what conditions on φ(A) (and the induced dynamics thereon) are A and φ(A) homeomorphic? Are their Lyapunov exponents the same? Or, more precisely, which of their Lyapunov exponents are the same? This paper addresses these questions with respect to both the general class of smooth mappings φ and the subclass of delay coordinate mappings. In answering these questions, a fundamental problem arises about an arbitrary compact set A in Rn. For x ∈ A, what is the smallest integer d such that there is a C1 manifold of dimension d that contains all points of A that lie in some neighborhood of x? We define a tangent space TxA in a natural way and show that the answer is d = dim(TxA). As a consequence we obtain a Platonic version of the Whitney embedding theorem.