Consistent Lyapunov exponent estimation for one-dimensional dynamical systems (Q1775946)

Consider a sample \(x_0,\dots,x_T\) from a trajectory \((x_t)_{t\in \mathbb N}\) of the dynamical system \(([0,1], B, S,\mu)\), where \(B\) is the completion of the Borel \(\sigma\)-algebra of \([0,1]\) with respect to probabilistic measure \(\mu\), and let \( S [0,1]\rightarrow [0,1]\) be a measurable map such that the Lyapunov exponent \(\lambda=\int_{[0,1]} \log | S^{'}| d\mu\) is well defined. The author considers the problem of estimating \(\lambda\) for the unknown map \(S\) using the observed sample \(x_0,\dots,x_T\). He proves the consistency of a nearest neighbour estimator of the Lyapunov exponent for a rather general class of one-dimensional ergodic dynamical systems and demonstrates that this estimator has good practical properties via a simulation experiment.