Annales scientifiques de l'Ecole normale supérieure

Horocycle flows on abelian covers of surfaces of negative curvature

Roberto Castorrini, Davide Ravotti — Wed, 22 Jul 2026 13:45:33 +0200

We consider the unit speed parametrization of the horocycle flow on infinite Abelian covers of compact surfaces of negative curvature. We prove an asymptotic result for the ergodic integrals of sufficiently regular functions. Our approach, which does not rely on symbolic dynamics, is based on a general Fourier decomposition for Abelian covers. This enables us to extend the transfer operator techniques to the case of geodesic flow in an infinite volume setting, a development that, to the best of the authors' knowledge, is relatively novel. In the case of constant curvature, where the unit speed and the uniformly contracting parametrizations of horocycles coincide, we recover a result by Ledrappier and Sarig. Finally, as a byproduct result, we obtain a power deviation estimate for the horocycle ergodic averages on compact surfaces, without requiring any pinching condition as in previous results.

Arithmetic dynamics of random polynomials

Niki Myrto Mavraki, Pierre Le Boudec — Wed, 22 Jul 2026 13:46:51 +0200

We investigate from a statistical perspective the arithmetic properties of the dynamics of polynomials of fixed degree and defined over the field of rational numbers. To start with, ordering their affine conjugacy classes by height, we show that their average number of rational preperiodic points is equal to zero, thereby proving a strong average version of the uniform boundedness conjecture of Morton and Silverman. Next, inspired by the analogy with the successive minima of a lattice we define the dynamical successive minima of a polynomial. Noting that these quantities are invariant under the action by conjugacy of the affine group we study their average behaviour using the aforementioned ordering by height. In particular, we prove an optimal statistical version of the dynamical Lang conjecture on the canonical height of rational non-preperiodic points.

Prismatic cohomology relative to $\delta$-rings

Benjamin Antieau, Achim Krause, Thomas Nikolaus — Wed, 22 Jul 2026 13:48:07 +0200

We develop prismatic and syntomic cohomology relative to a δ-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying δ-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context.