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 f Show more