Quantitative quasiperiodicity

Abstract

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff\naverages, $\\Sigma_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$\nergodic trajectory $(x_n)$ of a function $T$ converge to the space average\n$\\int f d\\mu$, where $\\mu$ is the unique invariant probability measure.\nConvergence of the time average to the space average is slow. We introduce a\nmodified average of $f(x_n)$ by giving very small weights to the "end" terms\nwhen $n$ is near $0$ or $N-1$. When $(x_n)$ is a trajectory on a quasiperiodic\ntorus and $f$ and $T$ are $C^\\infty$, we show that our weighted Birkhoff\naverages converge 'super" fast to $\\int f d\\mu$ with respect to the number of\niterates $N$, i.e. with error decaying faster than $N^{-m}$ for every integer\n$m$. Our goal is to show that our weighted Birkhoff average is a powerful\ncomputational tool, and this paper illustrates its use for several examples\nwhere the quasiperiodic set is one or two dimensional. In particular, we\ncompute rotation numbers and conjugacies (i.e. changes of variables) and their\nFourier series, often with 30-digit accuracy.\n