On the almost sure spiraling of geodesics in negatively curved manifolds

TL;DR

This paper investigates the long-term behavior of geodesics in negatively curved spaces, establishing statistical laws for their spiraling patterns around various geometric objects and deriving Diophantine approximation results in non-archimedean settings.

Contribution

It introduces Khintchine-type and logarithm law-type results for geodesic spiraling, linking geometric dynamics with number theory in non-archimedean fields.

Findings

Proves statistical laws for geodesic spiraling behavior.

Establishes Diophantine approximation results in non-archimedean fields.

Connects geometric dynamics with number theory in negatively curved spaces.

Abstract

Given a negatively curved geodesic metric space $M$, we study the statistical asymptotic penetration behavior of (locally) geodesic lines of $M$ in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithme 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.