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

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

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.

Keywords: Random walks, cusp excursion, harmonic measure, hitting measure.
Mathematics Subject Classification: Primary: 37D40, 60J50; Secondary: 60G50.
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:
[1]

S. Blachère, P. Haïssinsky and P. Mathieu, Mesures harmoniques et mesures quasiconformes sur les groupes hyperboliques, Ann. Sci. École Norm. Sup. (4), 44 (2011), 683-721.  doi: 10.24033/asens.2153.  Google Scholar

[2]

A. Boulanger, P. Mathieu, C. Sert and A. Sisto, Large deviations for random walks on hyperbolic spaces, preprint, arXiv: 2008.02709. Google Scholar

[3]

C. Connell and R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, Geom. Funct. Anal., 17 (2007), 707-769.  doi: 10.1007/s00039-007-0608-9.  Google Scholar

[4]

B. Deroin, V. Kleptsyn and A. Navas, On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J., 9 (2009), 263-303.  doi: 10.17323/1609-4514-2009-9-2-263-303.  Google Scholar

[5]

B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal., 8 (1998), 810-840.  doi: 10.1007/s000390050075.  Google Scholar

[6]

H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428.  doi: 10.1090/S0002-9947-1963-0163345-0.  Google Scholar

[7]

H. Furstenberg, Random Walks and Discrete Subgroups of Lie Groups, Advances in Probability and Related Topics, 1, Dekker, New York, 1971, 1–63.  Google Scholar

[8]

V. Gadre, J. Maher and G. Tiozzo, Word length statistics and Lyapunov exponents for Fuchsian groups with cusps, New York J. Math., 21 (2015), 511-531.   Google Scholar

[9]

V. Gadre, J. Maher and G. Tiozzo, Word length statistics for Teichmüller geodesics and singularity of harmonic measure, Comment. Math. Helv., 92 (2017), 1-36.  doi: 10.4171/CMH/404.  Google Scholar

[10]

I. Gekhtman, V. Gerasimov, L. Potyagailo and W. Yang, Martin boundary covers Floyd boundary, Invent. Math., 223 (2021), 759-809.  doi: 10.1007/s00222-020-01015-z.  Google Scholar

[11]

I. Gekhtman and G. Tiozzo, Entropy and drift for Gibbs measures on geometrically finite manifolds, Trans. Amer. Math. Soc., 373 (2020), 2949-2980.  doi: 10.1090/tran/8036.  Google Scholar

[12]

S. Gouëzel, Exponential bounds for random walks on hyperbolic spaces without moment conditions, preprint, 2021. Available from: https://perso.univ-rennes1.fr/sebastien.gouezel/articles/linear.pdf Google Scholar

[13]

Y. Guivarc'h and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continuous fractions, Ann. Sci. École Norm. Sup. (4), 26 (1993), 23-50.  doi: 10.24033/asens.1666.  Google Scholar

[14]

V. A. Kaimanovich, The Poisson boundary of hyperbolic groups, C. R. Acad. Sci. Paris, Sér. I Math., 318 (1994), 59-64.   Google Scholar

[15]

V. A. Kaimanovich and V. Le Prince, Matrix random products with singular harmonic measure, Geom. Dedicata, 150 (2011), 257-279.  doi: 10.1007/s10711-010-9504-9.  Google Scholar

[16]

J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510.  doi: 10.1111/j.2517-6161.1968.tb00749.x.  Google Scholar

[17]

T. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom., 19 (1984), 299-323.  doi: 10.4310/jdg/1214438681.  Google Scholar

[18]

J. Maher and G. Tiozzo, Random walks on weakly hyperbolic groups, J. Reine Angew. Math., 742 (2018), 187-239.  doi: 10.1515/crelle-2015-0076.  Google Scholar

[19]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.  Google Scholar

[20]

M. Sunderland, Linear progress with exponential decay in weakly hyperbolic groups, Groups Geom. Dyn., 14 (2020), 539-566.  doi: 10.4171/GGD/554.  Google Scholar

show all references

References:
[1]

S. Blachère, P. Haïssinsky and P. Mathieu, Mesures harmoniques et mesures quasiconformes sur les groupes hyperboliques, Ann. Sci. École Norm. Sup. (4), 44 (2011), 683-721.  doi: 10.24033/asens.2153.  Google Scholar

[2]

A. Boulanger, P. Mathieu, C. Sert and A. Sisto, Large deviations for random walks on hyperbolic spaces, preprint, arXiv: 2008.02709. Google Scholar

[3]

C. Connell and R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, Geom. Funct. Anal., 17 (2007), 707-769.  doi: 10.1007/s00039-007-0608-9.  Google Scholar

[4]

B. Deroin, V. Kleptsyn and A. Navas, On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J., 9 (2009), 263-303.  doi: 10.17323/1609-4514-2009-9-2-263-303.  Google Scholar

[5]

B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal., 8 (1998), 810-840.  doi: 10.1007/s000390050075.  Google Scholar

[6]

H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc., 108 (1963), 377-428.  doi: 10.1090/S0002-9947-1963-0163345-0.  Google Scholar

[7]

H. Furstenberg, Random Walks and Discrete Subgroups of Lie Groups, Advances in Probability and Related Topics, 1, Dekker, New York, 1971, 1–63.  Google Scholar

[8]

V. Gadre, J. Maher and G. Tiozzo, Word length statistics and Lyapunov exponents for Fuchsian groups with cusps, New York J. Math., 21 (2015), 511-531.   Google Scholar

[9]

V. Gadre, J. Maher and G. Tiozzo, Word length statistics for Teichmüller geodesics and singularity of harmonic measure, Comment. Math. Helv., 92 (2017), 1-36.  doi: 10.4171/CMH/404.  Google Scholar

[10]

I. Gekhtman, V. Gerasimov, L. Potyagailo and W. Yang, Martin boundary covers Floyd boundary, Invent. Math., 223 (2021), 759-809.  doi: 10.1007/s00222-020-01015-z.  Google Scholar

[11]

I. Gekhtman and G. Tiozzo, Entropy and drift for Gibbs measures on geometrically finite manifolds, Trans. Amer. Math. Soc., 373 (2020), 2949-2980.  doi: 10.1090/tran/8036.  Google Scholar

[12]

S. Gouëzel, Exponential bounds for random walks on hyperbolic spaces without moment conditions, preprint, 2021. Available from: https://perso.univ-rennes1.fr/sebastien.gouezel/articles/linear.pdf Google Scholar

[13]

Y. Guivarc'h and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continuous fractions, Ann. Sci. École Norm. Sup. (4), 26 (1993), 23-50.  doi: 10.24033/asens.1666.  Google Scholar

[14]

V. A. Kaimanovich, The Poisson boundary of hyperbolic groups, C. R. Acad. Sci. Paris, Sér. I Math., 318 (1994), 59-64.   Google Scholar

[15]

V. A. Kaimanovich and V. Le Prince, Matrix random products with singular harmonic measure, Geom. Dedicata, 150 (2011), 257-279.  doi: 10.1007/s10711-010-9504-9.  Google Scholar

[16]

J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510.  doi: 10.1111/j.2517-6161.1968.tb00749.x.  Google Scholar

[17]

T. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom., 19 (1984), 299-323.  doi: 10.4310/jdg/1214438681.  Google Scholar

[18]

J. Maher and G. Tiozzo, Random walks on weakly hyperbolic groups, J. Reine Angew. Math., 742 (2018), 187-239.  doi: 10.1515/crelle-2015-0076.  Google Scholar

[19]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.  Google Scholar

[20]

M. Sunderland, Linear progress with exponential decay in weakly hyperbolic groups, Groups Geom. Dyn., 14 (2020), 539-566.  doi: 10.4171/GGD/554.  Google Scholar

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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