The Lagrange spectrum of a Veech surface has a Hall ray

Abstract

We study Lagrange spectra of Veech translation surfaces, which are a generalization of the classical Lagrange spectrum. We show that any such Lagrange spectrum contains a Hall ray. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics in the corresponding Teichmüller disk and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets.

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

“…Let us first highlight, for its intrinsic interest, a result which will follow as a special case of the more general Theorem 1.8 that we will state in Section 1.6. As evidenced by the brief history in the previous section, the existence of a Hall ray was known for dynamical Markoff and Lagrange spectra dimension greater than 3 [32] and for Markoff spectra of Riemann surfaces [35], as well as for Lagrange spectra in other dynamical contexts, such as [20,1]. Thus, our work deals with the only case that was surprisingly still open in the constant curvature case, namely Lagrange spectra in dimension 2.…”

“…These type of Lagrange spectra are also called dynamical Lagrange spectra in the literature. Dynamical spectra were in particular studied in the seminal works by [27,32,17] and have seen a recent surge of interest, see for example [1,5,7,10,19,20,26]. If the surface X has only one cusp at infinity, height(·) is an example of a proper function on X.…”

“…To study Lagrange spectra with respect to a proper function in presence of several cusps, it is key that through this coding one can see (large) excursions into each cusp. To control these excursions, we use decomposition into cuspidal words, a notion which was introduced in our previous work [1]. A delicate point we show is that by (locally) bounding the lengths of cuspidal words one is able to estimate the penetration into all cusps (see Lemma 5.1), in order to then be able to achieve Lagrange values by prescribing larger excursions in a given chosen cusp.…”

“…We end this section with two Lemmas that give a combinatorial description of cuspidal words, by showing that cuspidal words are obtained by repeating parabolic words (defined below), which are in one to one correspondence with cusps (see Corollary 2.5). The Lemmas were essentially proved in [1] (see Lemmas 4.8 and 4.9 in [1]). For completeness, we include their easy proofs in Appendix A. for any k = 0, .…”

“…Any gap B of K N is the countable union B = ∪ k∈Z A k of a collection of adjacent deleted arcs in D N , where by adjacent we mean that max A k = min A k+1 for any z ∈ Z; the length |A k | ∂D shrinks exponentially as |k| → ∞. An explicit description of all arcs A k fitting together in the same gap is given in § 7.2 in [1]. The explicit description is not needed in this paper: we just need to know that if we set G N ⊂ ∂D to be the union G N := A∈D A of all deleted arcs A ∈ D N , the set G N , described as a countable union of closed arcs, is an open set and it coincides precisely with the union of all gaps of B N , in other words…”