Quantitative Quasiperiodicity

Abstract

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, $Σ_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$ ergodic trajectory $(x_n)$ of a function $T$ converge to the space average $\int f dμ$, where $μ$ 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 $N-1$. 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 $\int f dμ$ with respect to the number of iterates $N$, i.e. with error decaying faster than $N^{-m}$ for every integer $m$. 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 accuracy.