Super convergence of ergodic averages for quasiperiodic orbits

TL;DR

This paper introduces a weighted averaging method that accelerates convergence in quasiperiodic systems, enabling efficient numerical computation of key dynamical invariants like rotation numbers and conjugacies.

Contribution

It presents a novel weighted averaging technique that significantly speeds up convergence of Birkhoff averages in quasiperiodic systems, improving numerical analysis methods.

Findings

Weighted averages converge faster than uniform averages in quasiperiodic systems.

The method enables efficient computation of rotation numbers and conjugacies.

Faster convergence occurs when the function is sufficiently differentiable.

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.