## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 85, Number 2 (2010), 271-314.

Dates
First available in Project Euclid: 20 October 2010

https://projecteuclid.org/euclid.jdg/1287580966

Digital Object Identifier
doi:10.4310/jdg/1287580966

Mathematical Reviews number (MathSciNet)
MR2732978

Zentralblatt MATH identifier
1229.53050

#### Citation

Hersonsky, Sa’ar; Paulin, Frédéric. On the almost sure spiraling of geodesics in negatively curved manifolds. J. Differential Geom. 85 (2010), no. 2, 271--314. doi:10.4310/jdg/1287580966. https://projecteuclid.org/euclid.jdg/1287580966