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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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:11 April 2018
Deposited On:23 Jan 2019 16:07
Last Modified:28 Jan 2019 01:20
Publisher:Hebrew University Magnes Press
ISSN:0021-2172
OA Status:Closed
Publisher DOI:https://doi.org/10.1007/s11856-018-1667-3

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