Quantitative Quasiperiodicity

TL;DR

This paper introduces a weighted Birkhoff average method that converges super fast for quasiperiodic trajectories, enabling highly precise computations of rotation numbers and conjugacies.

Contribution

It presents a novel weighted averaging technique that accelerates convergence in quasiperiodic systems, improving computational accuracy and efficiency.

Findings

Weighted averages converge faster than traditional methods

Achieved 30-digit precision in numerical examples

Effective for 1D and 2D quasiperiodic sets

Abstract

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages, $\Sigma_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic trajectory $(x_n)$ of a function $T$ converges to the space average $\int f d\mu$, where $\mu$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the "end" terms when $n$ is near $0$ or $N$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\infty$, we show that our weighted Birkhoff averages converge "super" fast to $\int f d\mu$, {\em i.e.} with error smaller than every polynomial of $1/N$. Our goal is to show that our weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit precision.