# New paper: Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap

*** 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 $U_t:\mathcal{H}\to\mathcal{H},\,t\in\mathbb{R},$ the average $\frac{1}{2T}\int_{-T}^TU_tf\,\mathrm{d}t$ converges to the projection of $f$ onto the space of functions invariant under $U_t$ as $T\to+\infty$.

Generally there is no rate. However, if the generator of $U_t$ has a spectral gap, the rate is $T^{-1}$. 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 $\mathcal{H}$, and with a rate worse than $T^{-1}$. This is done by obtaining a suitable estimate for the density of the spectrum near zero (low frequencies).