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.
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The excursion
In the case
Two situations in Lemma 2.4: first with the center of
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