The Lagrange spectrum of some square-tiled surfaces

TL;DR

This paper computes the Lagrange spectrum for a specific square-tiled surface, revealing unique spectral features such as an isolated minimum and a complex structure above it, expanding understanding of translation surface dynamics.

Contribution

It provides an explicit formula for the Lagrange spectrum of a particular square-tiled surface, highlighting differences from classical spectra on modular surfaces.

Findings

Identifies an isolated minimum in the spectrum.

Describes a complex structure above the minimum.

Provides an explicit formula involving a cocycle.

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.