Let $x_{ j+1}=\phi(x_{ j})$, $x_{ j}\in\mathbb{ R}^{ d}$, be a dynamical system with $\phi$ being a diffeomorphism. Although the state vector $x_{ j}$ is often unobservable, the dynamics can be recovered from the delay vector $\left(o(x_{ 1}),\ldots,o(x_{ D})\right)$, 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 $\phi$, 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.
Categories: math.DS nlin.CD