The Lagrange spectrum of some square-tiled surfaces

Abstract

Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2, R). We consider the closed orbit generated by a specific covering of degree 7 of the standard torus, which is an element of the stratum H(2). We give an explicit formula for the values in the spectrum, in terms of a cocycle over the classical continued fraction. Differently from the classical case of the modular surface, where the lowest part of the Lagrange spectrum is discrete, we find an isolated minimum, and a set with a rich structure right above it.

The beginning of the Lagrange spectrum of certain origamis of genus two

“…Consider the left shift dynamics σ : {a, b} Z → {a, b} Z on the symbolic space Σ := {a, b} Z where a := 1, 4, 2, 4 and b := 1, 3. It was shown in [4…”

“…For the sake of exposition, we divide the rest of this note into five sections: first, we review some results from [4] about the description of the initial portion of L B 7 ; next, we employ the results of Cerqueira, Moreira and the author [2] to deduce the continuity of the Hausdorff dimension of L B 7 ∩ (−∞, t ) as a function of t ∈ (−∞, η 1 ); afterwards, we show that K is a Cantor set; then, we modify an argument of Moreira [7] in order to prove that any φ ∈ K is accumulated by Cantor sets with positive Hausdorff dimensions contained in K; finally, we show that d (t ) is not Hölder continuous near φ 2 .…”

“…On the other hand, it was discovered by Hubert-Lelièvre-Marchese-Ulcigrai [4] that the beginning of the Lagrange spectra of SL(2, R)-orbits of square-tiled surfaces might behave differently from the classical Lagrange spectrum. More precisely, let X be the square-tiled surface of genus two with unit area obtained from seven squares sq(k), 1 ≤ k ≤ 7, in By following the terminology of Hubert-Lelièvre [3], the SL(2, R)-orbit of X is called B 7.…”

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

“…For Lagrange spectra of strata of translation surfaces, the existence of Hall rays was in [20] and the Hurwitz constant was recently found by Boshernitzan and Delecroix [5]. The authors proved in [1] that also for the particularly symmetric class of translation surfaces made of Veech surfaces, the Lagrange spectrum contained a Hall ray and the first values of the Lagrange spectrum a particular example of a square-tiled Veech surface are studied in detail in [19].…”

“…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.…”