We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Le Masson, and Lindenstrauss on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalized averaging operators over disks, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

Fix a finite field. The set of ${PGL}_2$ bundles over ${\mathbb{P}}^1$ is in bijection with the natural numbers, and carries a natural measure assigning to each bundle the inverse of the number of automorphisms. A branched double cover $\pi :C\to {\mathbb{P}}^1$ determines another measure, given by counting the number of line bundles over $C$ whose image on ${\mathbb{P}}^1$ has a given sheaf of endomorphisms. We show the measures induced by a sequence of such hyperelliptic curves tends to the canonical measure on the space of ${PGL}_2$ bundles.

This is a function field analogue of Duke’s theorem on the equidistribution of Heegner points, and can be proven similarly. Our real interest is the corresponding analogue of the “Mixing Conjecture” of Michel and Venkatesh. This amounts to considering measures on the space of pairs of ${PGL}_2$ bundles induced by taking a fixed line bundle $\mathcal{L}$ over $C$, and looking at the distribution of pairs $({\pi }_{\ast }\mathcal{M},{\pi }_{\ast }(\mathcal{L}\otimes \mathcal{M}\left)\right)$. As in the number field setting, ergodic theory classifies limiting measures for sufficiently special $\mathcal{L}$.

The heart of this work is a geometric attack on the general case. We count points on intersections of translates of loci of special divisors in the Jacobian of a hyperelliptic curve. To prove equidistribution, we would require two results. The first, we prove: in high degree, the cohomologies of these loci match the cohomology of the Jacobian. The second, we establish in characteristic zero and conjecture in characteristic $p$: the cohomology of these spaces grows at most exponentially in the genus of the curve $C$.

The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\le 3$ under strong assumptions on the initial data, whereas weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these results to show that weak/renormalized solutions of Vlasov–Poisson are Lagrangian and actually that the concepts of renormalized and Lagrangian solutions are equivalent. As a corollary, we prove that finite-energy solutions in dimension $d\le 4$ are transported by a global flow (in particular, they preserve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.

Let $X$ be an infinite metrizable compact space, and let $\sigma :X\to X$ be a minimal homeomorphism. Suppose that $(X,\sigma )$ has zero mean topological dimension. The associated C∗-algebra $A=\mathrm{C}\left(X\right){⋊}_{\sigma }\mathbb{Z}$ is shown to absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially; that is, $A\cong A\otimes \mathcal{Z}$. This implies that $A$ is classifiable when $(X,\sigma )$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang–Su algebra.