Super convergence of ergodic averages for quasiperiodic orbits

Abstract

By definition, a map quasiperiodic on a set $X$ if the map is conjugate to a pure rotation. Suppose we have a trajectory $(x_n)$ that we suspect is quasiperiodic. How do we determine if it is? In this paper we show how to compute the conjugacy map using only knowledge of $(x_n)$. Our main tool is a variant of Birkhoff averages. The Birkhoff Ergodic Theorem asserts that time averages of a function evaluated along a trajectory of length $N$ converge to the space average, the integral of $f$, as $N\to\infty$, for ergodic dynamical systems. But that convergence can be slow. Instead of uniform averages that assign equal weights to points along the trajectory, we use an average with a non-uniform distribution of weights, weighing the early and late points of the trajectory much less than those near the midpoint $N/2$. We show that in quasiperiodic dynamical systems, our weighted averages converge far faster provided $f$ is sufficiently differentiable. This result can be applied to obtain efficient numerical computation of rotation numbers, invariant densities and conjugacies of quasiperiodic systems.