# Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

## TL;DR

This paper establishes a general theorem on the shrinking target problem for geodesic flows on geometrically finite hyperbolic manifolds, extending Sullivan's logarithm law and providing quantitative estimates for hitting times.

## Contribution

It introduces a broad theorem leveraging exponential mixing to analyze shrinking targets, strengthening existing logarithm laws and offering new quantitative results.

## Key findings

- Proves a general shrinking target theorem for geodesic flows.
- Extends Sullivan's logarithm law to various shrinking targets.
- Provides quantitative estimates for geodesic hitting times.

## Abstract

Let $\mathcal{M}$ be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.