Degrees of freedom of a time series (Q1602032)

Let \((M,f,\mu)\) be given dynamics, that is \(M\) is a \(d\)-dimensional smooth submanifold, \(f:M\to M\) is a measurable dynamics, \(\mu\) is an ergodic measure for \(f\). The authors show that if \(\mu\) is \(\alpha\)-exact dimensional with \(\alpha> d-1\), and \(f\) and \(M\) are sufficiently smooth, then one can recover the number \(d\) of degrees of freedom of the dynamics from the observation of an orbit of the system. The authors implement an algorithm with this purpose, and show how the resulting estimate of \(d\) may be used in the computation of the Lyapunov spectrum of the dynamics.