Related papers: Effective equidistribution of the horocycle flow o…

We study equidistribution of certain subsets of periodic orbits for subshifts of finite type. Our results solely rely on the growth of these subsets. As a consequence, effective equidistribution results are obtained for both hyperbolic…

In this work, we study the asymptotic distribution of the non discrete orbits of a finitely generated group acting linearly on $\R^2$. To do this, we establish new equidistribution results for the horocyclic flow on the unitary tangent…

Here we study the behaviour of the horocyclic orbit of a vector on the unit tangent bundle of a geometrically infinite surface with variable negative curvature, when the corresponding geodesic ray is almost minimizing and the injectivity…

We consider a minimal compact lamination by hyperbolic surfaces. We prove that if it admits a leaf whose holonomy covering is not topologically trivial, then the horocycle flow on its unitary tangent bundle is minimal.

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the…

A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh…

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent…

We prove effective equidistribution theorems for (weighted) packets of closed periodic orbits for Anosov flows. In particular, for the case of contact Anosov flows on three-dimensional manifolds, we show that the Bowen packets…

Let X be a complete hyperbolic surface of finite area. We establish that the intersection points of closed geodesics with length <T are equidistributed on X as T goes to infinity.

We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for…

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this…

Given a simple closed curve $\gamma$ on a connected, oriented, closed surface $S$ of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on $S$ having a simple closed geodesic…

We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of…

For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the quantum ergodicity theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric…

We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…

We show that the horocycle flows of open tight hyperbolic surfaces do not admit minimal sets.

We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).

We show that the horocycle flow associated with a foliation on a compact manifold by hyperbolic surfaces is minimal under certain conditions.

We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca.

We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.