Exponential convergence of weighted Birkhoff average

TL;DR

This paper proves that weighted Birkhoff averages for irrational rotations on tori can achieve exponential or polynomial convergence rates, depending on the system's properties, with universality demonstrated in quasiperiodic and almost periodic cases.

Contribution

It establishes the first proof of universal exponential convergence for quasiperiodic systems and polynomial convergence for almost periodic systems under analyticity.

Findings

Exponential convergence achieved for certain irrational rotations.

Polynomial convergence demonstrated in almost periodic systems.

Universality of convergence rates proven in specified dynamical systems.

Abstract

In this paper, we consider the polynomial and exponential convergence rate of weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.