Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

Anja Randecker; Giulio Tiozzo

doi:10.3934/jmd.2021006

Article Contents

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

This volume Previous Article The orbital equivalence of Bernoulli actions and their Sinai factors Next Article A prime system with many self-joinings

Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

Anja Randecker^1,2, and
Giulio Tiozzo^1,

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

Partially supported by NSERC and the Alfred P. Sloan Foundation

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Abstract

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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Mathematics Subject Classification: Primary: 37D40, 60J50; Secondary: 60G50.

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Figure 1. 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

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Figure 2. In the case $ N = 2 $, the excursion can be calculated as in Lemma 2.2

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Figure 3. 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' $

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Figure 4. Setting for the definition of excursion (Definition 3.1)

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Figure 5. The three cases in the proof of Lemma 3.5. From left to right, the three horoballs correspond to cases (ⅱ), (ⅰ), and (ⅲ)

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Figure 6. Setting of Proposition 3.8

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Figure 7. Setting of Lemma 4.2

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