Fast Computation of Birkhoff Average along a Quasi-periodic Orbit and its Applications

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05360
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Hitotsubashi University
• Principal Investigator: SAIKI Yoshitaka 一橋大学, 大学院経営管理研究科, 教授 (20433740)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 力学系 / 準周期軌道 / バーコフ平均 / 高速計算

Outline of Final Research Achievements

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages of a function f along an ergodic trajectory of a function T converges to the space average. Convergence of the time average to the space average is slow. We introduce a modified average by giving very small weights to the "end" terms. When (x_n) is a trajectory on a quasiperiodic torus and f and T are infinitely differentiable, we show that our weighted Birkhoff averages converge "super" fast, 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 study 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.

Free Research Field

応用数学

Academic Significance and Societal Importance of the Research Achievements

回転数、リアプノフ数などをはじめとしてバーコフ平均は力学系の軌道に関するさまざまな量に関わっている。軌道長Nのバーコフ平均の収束スピードは一般に1/Nのオーダーであり実際に計算で準周期性の判断をすることは困難であった。しかし、研究代表者らは、準周期軌道上のバーコフ平均に対しては、理論的には1/(Nに関する任意の多項式)よりも速く収束する重み付きバーコフ平均を提案してその応用可能性を示した。