Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications

2026/08/03 by Son N. T. Tu, Jianlu Zhang
Mathematics · #math.AP #math.DS #msc:35B27 #msc:35B40 #msc:37C40 #msc:37J51 #msc:47A35 #msc:49L25

paper · pdf

40 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

In this paper, we establish quantitative Denjoy--Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. By means of suitable Sobolev embeddings, these estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton--Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

Citations

• Quantitative homogenization of Hamilton--Jacobi equations on perforated domains with Dirichlet boundary conditions
• Quantitative homogenization of convex Hamilton-Jacobi equations with u/ε-periodic Hamiltonians
• Quantitative homogenization of convex Hamilton-Jacobi equations with Neumann type boundary conditions
• Rate of convergence for homogenization of nonlinear weakly coupled Hamilton-Jacobi systems
• Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates
• Quantitative homogenization of state-constraint Hamilton--Jacobi equations on perforated domains and applications
• Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with Multiscales
• Optimal convergence rate for periodic homogenization of convex Hamilton-Jacobi equations
• Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle
• Effective fronts of polytope shapes
• Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps
• Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting
• An introduction to the qualitative and quantitative theory of homogenization
• Linear response, or else
• Convergence Rates and Hölder Estimates in Almost-Periodic Homogenization of Elliptic Systems
• Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
• The perturbed test function method for viscosity solutions of nonlinear PDE