2018/01/10 by Suddhasattwa Das, James A Yorke · 2 citations
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · doi:10.1088/1361-6544/aa99a0
openalex publication_date 2018/01/10 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28
Abstract The Birkhoff ergodic theorem asserts that time averages of a function evaluated along a trajectory of length N converge to the space average, the integral of f , as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>N</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mstyle> </mml:math> , for ergodic dynamical systems. But that convergence can be slow. Instead of uniform averages that assign equal weights to points along the trajectory, we use an average with a non-uniform distribution of weights, weighting the early and late points of the trajectory much less than those near the midpoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mstyle> </mml:math> . We show that in quasiperiodic dynamical systems, our weighted averages converge far faster provided f is sufficiently differentiable. This result can be applied to obtain efficient numerical computation of rotation numbers, invariant densities and conjugacies of quasiperiodic systems.