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Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces


Artigiani, Mauro; Marchese, Luca; Ulcigrai, Corinna (2020). Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces. Ergodic Theory and Dynamical Systems, 40(8):2017-2072.

Abstract

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

Abstract

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

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Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Physical Sciences > Applied Mathematics
Language:English
Date:1 August 2020
Deposited On:28 Jun 2019 13:45
Last Modified:06 Sep 2020 12:16
Publisher:Cambridge University Press
ISSN:0143-3857
OA Status:Green
Publisher DOI:https://doi.org/10.1017/etds.2018.143

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