A uniform ergodic theorem for degenerate flows on the annulus

2019/02/18 by Jonathan Ben‐Artzi, Ben-Artzi, Jonathan, Baptiste Morisse +1
Mathematics · Physics and Astronomy · #35B40 #35P20 #37A30 (primary) #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1902.06681

openalex publication_date 2019/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by the well-known phase-space portrait of the nonlinear pendulum, the purpose of this paper is to obtain convergence rates in the ergodic theorem for flows in the plane that have arbitrarily slow trajectories. Considering bounded periodic trajectories near the heteroclinic orbits, it is shown that despite lacking a spectral gap, there exists a functional space (which is a strict subset of L²) on which time averages converge uniformly to spatial averages (with an explicit rate). The main ingredient of the proof is an estimate of the density of the spectrum of the generator of the flow near zero.

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