Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

Abstract

<jats:p>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.</jats:p>

On Good Approximations and the Bowen–series Expansion

“…We follow [6,Section 3], which is based on [1,Section 2.4] and [2,Section 2]. The original construction is the Markov map in [3], which is orbit equivalent to the action of a given finitely generated Fuchsian group of the first kind.…”

“…that is, the n + 1 arcs above share ξ W as common end point (see also [2,Section 2.4] and [1, Section 4.3]). A sequence (a n ) n∈N is called cuspidal if any initial factor (a 0 , .…”

“…This result is used in [6] to approximate the dimension of sets of badly approximable points by the dimension of dynamically defined regular Cantor sets. The study of the higher part of the Markov and Lagrange spectra is also a natural application, in the spirit of [1,2,11].…”

“…of a real number α, where the symbols W r for r ≥ 1 are cuspidal words which belong to a countable alphabet W (definitions are in Sections 2 and 3). Cuspidal words W ∈ W, introduced in [1,2], label a subset of elements {G W : W ∈ W} of Γ, which generalise the role played in the theory of classical continued fractions by the matrices 1 a 2k+1 0 1 and 1 0 a 2k 1 with a 2k , a 2k+1 ∈ N * for any k ∈ N.…”

On the measure of products from the middle-third Cantor set

“…A first use of gap conditions appears in [7] and a more recent application in [3]. A gap condition is used for the product of Cantor sets in [10], and for Lipschitz perturbations of S(•, •) in Theorem 1.12 in [1]. Similar ideas are used also in § 2.2 and § 2.3 of this paper, inspired by [2].…”