# American Institute of Mathematical Sciences

2021, 17: 183-211. doi: 10.3934/jmd.2021006

## Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

 1 Department of Mathematics, University of Toronto, 40 St George St, Toronto ON M5S 2E5, Canada 2 Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

Received  August 09, 2019 Revised  November 18, 2020 Published  April 2021

Fund Project: Partially supported by NSERC and the Alfred P. Sloan Foundation

We generalize the notion of cusp excursion of geodesic rays by introducing for any $k\geq 1$ the $k^\text{th}$ 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 $k^\text{th}$ 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.

Citation: Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006
##### References:

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##### References:
The excursion $E(\gamma, H)$ of the geodesic segment $\gamma$ in the horoball $H$ is the length of the thickly drawn arc of the horoball
In the case $N = 2$, the excursion can be calculated as in Lemma 2.2
Two situations in Lemma 2.4: first with the center of $C_H$ to the left of $\gamma$ and $\gamma'$, then with the center of $C_H$ between $\gamma$ and $\gamma'$
Setting for the definition of excursion (Definition 3.1)
The three cases in the proof of Lemma 3.5. From left to right, the three horoballs correspond to cases (ⅱ), (ⅰ), and (ⅲ)
Setting of Proposition 3.8
Setting of Lemma 4.2
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