  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.
