On the almost sure spiraling of geodesics in negatively curved manifolds

Abstract

Given a negatively curved geodesic metric space $M$, we study the almost sure asymptotic penetration behavior of (locally) geodesic lines of $M$ into small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objets. As a consequence in the tree setting, we obtain Diophantine approximation results of elements of non-archimedian local fields by quadratic irrational ones.