Super convergence of ergodic averages for quasiperiodic orbits

Abstract

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 , 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, weighting the early and late points of the trajectory much less than those near the midpoint . 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.