Prevalence of Delay Embeddings with a Fixed Observation Function

Abstract

Let $x_{j+1}=\varphi (x_j)$, $x_j\in {\mathbb{R}}^d$, be a dynamical system with $\varphi$ being a diffeomorphism. Although the state vector $x_j$ is often unobservable, the dynamics can be recovered from the delay vector $\left(o(x_1),\dots ,o(x_D)\right)$, where $o$ is the scalar-valued observation function and $D$ is the embedding dimension. The delay map is an embedding for generic $o$, and more strongly, the embedding property is prevalent. We consider the situation where the observation function is fixed at $o={\pi }_1$, with ${\pi }_1$ being the projection to the first coordinate. However, we allow polynomial perturbations to be applied directly to the diffeomorphism $\varphi$, thus mimicking the way dynamical systems are parametrized. We prove that the delay map is an embedding with probability one with respect to the perturbations. Our proof introduces a new technique for proving prevalence using the concept of Lebesgue points.