Jouni Parkkonen

Prescribing the behaviour of geodesics in negative curvature

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.

Counting common perpendicular arcs in negative curvature

Let

Skinning measures in negative curvature and equidistribution of equidistant submanifolds

D^-

On the representation of integers by indefinite binary Hermitian forms

and

Geometric limits of cyclic groups and the shape of Schottky space

D^+

A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as

Conformal Dehn surgery and the shape of Maskit’s embedding

t\to+\infty

Appendix: Diophantine Approximation on Hyperbolic Surfaces

for the number of common perpendiculars of length at most

Spiraling spectra of geodesic lines in negatively curved manifolds

t

Équidistribution, comptage et approximation par irrationnels quadratiques

from