# A uniform ergodic theorem for degenerate flows on the annulus

## TL;DR

This paper establishes a uniform ergodic theorem with explicit convergence rates for degenerate flows on the plane, inspired by the nonlinear pendulum, even without a spectral gap.

## Contribution

It introduces a functional space where time averages converge uniformly for flows with slow trajectories, providing explicit convergence rates.

## Key findings

- Uniform convergence of time averages on a specific functional space
- Explicit rate of convergence established
- Spectral density estimate near zero for the flow generator

## 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 $L^2$) 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.