*** Update: this is now published in Ergod. Th. & Dynam. Sys. ***
Jonathan Ben-Artzi and Baptiste Morisse recently submitted a paper entitled Uniform convergence in von Neumann’s ergodic theorem in absence of a spectral gap.
Von Neumann’s ergodic theorem states that “time” averages converge to “spatial” avergaes: given a one-parameter family of unitary maps the average converges to the projection of onto the space of functions invariant under as .
Generally there is no rate. However, if the generator of has a spectral gap, the rate is . In the present paper, it is shown that even in the absence of a spectral gap one can obtain a rate, albeit on a subspace of , and with a rate worse than . This is done by obtaining a suitable estimate for the density of the spectrum near zero (low frequencies).