# 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).

# Research journey to Lyon

Thanks to his LMS Scheme 3 grant worth £1200, Baptiste enjoyed a two weeks stay in Lyon from November 10 to 25. Baptiste worked with his collaborator Francesco Fanelli on a new project that aims to explore the question of the inviscid limit for some hydrodynamical equations – as the Euler equation with variable density – in the wake of the important work of Jacob Bedrossian and Nader Masmoudi on Euler equations. This project lies mathematically at the intersection of Baptiste’s line of research on Gevrey regularity and Francesco’s knowledge on hydrodynamical systems.

Baptiste was also invited to the Analysis Seminar of Besançon and gave a talk on weakly hyperbolic systems. The audience was very interested by this subject and raised many interesting questions. It may be the sign that Baptiste will go back to Besançon in the near future!

# New team member: Baptiste Morisse

Baptiste Morisse, who recently graduated from Université Paris Diderot (Paris 7), joined the project on 1 September 2017. Baptiste, who was a student of Benjamin Texier, brings expertise in well-posedness theory for first-order quasilinear operators, in particular in the context of Gevrey spaces. Baptiste will be in Cardiff for the next three years.

Welcome Baptiste!