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Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

Part of: Riemann surfaces Dynamical systems with hyperbolic behavior Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  11 February 2019

MAURO ARTIGIANI Open the ORCID record for MAURO ARTIGIANI [Opens in a new window] ,
LUCA MARCHESE  and
CORINNA ULCIGRAI
MAURO ARTIGIANI
Affiliation:
Universidad de los Andes, Cra 1 #18A – 12, Bogotá, 111711, Colombia email m.artigiani@uniandes.edu.co
LUCA MARCHESE
Affiliation:
Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539, 99 Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France email marchese@math.univ-paris13.fr
CORINNA ULCIGRAI
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK email corinna.ulcigrai@bristol.ac.uk
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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.

Keywords

Lagrange spectrumHall rayFuchsian groupsboundary expansionssum of Cantor sets

MSC classification

Primary: 11J06: Markov and Lagrange spectra and generalizations 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 30F35: Fuchsian groups and automorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2017 - 2072
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Copyright
© Cambridge University Press, 2019

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