# Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

## TL;DR

This paper introduces a generalized concept of cusp excursions in hyperbolic manifolds, analyzing their growth and demonstrating the singularity between hitting and Lebesgue measures on the boundary of hyperbolic space.

## Contribution

It extends the notion of cusp excursions to higher orders and establishes their growth behavior, revealing measure singularity in hyperbolic manifolds.

## Key findings

- Cusp excursions are at most linear for geodesics generic with respect to hitting measure.
- For the highest order, cusp excursions are superlinear for geodesics generic with respect to Lebesgue measure.
- Hitting measure and Lebesgue measure on the boundary are mutually singular.

## Abstract

We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for $k = N-1$, the $k^{th}$ excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space $\mathbb{H}^N$ for any $N \geq 2$ are mutually singular.