Cusp excursion in hyperbolic manifolds and singularity of harmonic measure
We generalize the notion of cusp excursion of geodesic rays by introducing for any <inline-formula><tex-math id="M1">\begin{document}$ k\geq 1 $\end{document}</tex-math></inline-formula> the <i><inline-formula><tex-math id="M2">\begin{doc...
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| Main Authors: | , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
2021
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| In: |
Journal of modern dynamics
Year: 2021, Volume: 17, Pages: 183-211 |
| ISSN: | 1930-532X |
| DOI: | 10.3934/jmd.2021006 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.3934/jmd.2021006 Verlag, lizenzpflichtig, Volltext: https://www.aimsciences.org/article/doi/10.3934/jmd.2021006 
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| Author Notes: | Anja Randecker and Giulio Tiozzo |
| Summary: | We generalize the notion of cusp excursion of geodesic rays by introducing for any <inline-formula><tex-math id="M1">\begin{document}$ k\geq 1 $\end{document}</tex-math></inline-formula> the <i><inline-formula><tex-math id="M2">\begin{document}$ k^\text{th} $\end{document}</tex-math></inline-formula> excursion</i> in the cusps of a hyperbolic <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula>-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for <inline-formula><tex-math id="M4">\begin{document}$ k = N-1 $\end{document}</tex-math></inline-formula>, the <inline-formula><tex-math id="M5">\begin{document}$ k^\text{th} $\end{document}</tex-math></inline-formula> excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{H}^N $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M7">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula> are mutually singular. |
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| Item Description: | Gesehen am 16.06.2021 |
| Physical Description: | Online Resource |
| ISSN: | 1930-532X |
| DOI: | 10.3934/jmd.2021006 |