# The Lagrange spectrum of some square-tiled surfaces

Hubert, Pascal; Lelièvre, Samuel; Marchese, Luca; Ulcigrai, Corinna (2018). The Lagrange spectrum of some square-tiled surfaces. Israel Journal of Mathematics, 225(2):553-607.

## 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.

## 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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 11 April 2018 23 Jan 2019 16:07 28 Jan 2019 01:20 Hebrew University Magnes Press 0021-2172 Closed https://doi.org/10.1007/s11856-018-1667-3

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