Lyapunov spectrum of square-tiled cyclic covers

Alex Eskin; Maxim Kontsevich; Anton Zorich

doi:10.3934/jmd.2011.5.319

Article Contents

2011, Volume 5, Issue 2: 319-353. Doi: 10.3934/jmd.2011.5.319

This issue Previous Article Square-tiled cyclic covers Next Article A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle

Lyapunov spectrum of square-tiled cyclic covers

1.
Department of Mathematics, University of Chicago, Chicago, IL 60637
2.
IHES, le Bois Marie, 35, route de Chartres, 91440 Buressur-Yvette
3.
IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex

Received: June 30, 2010

Revised: April 30, 2011

Published: June 2011

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Abstract

A cyclic cover over $CP^1$ branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmüller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.

Keywords:

Teichmüller geodesic flow,
moduli space of quadratic differentials,
Lyapunov exponent,
Hodge norm,
cyclic cover.

Mathematics Subject Classification: Primary: 30F30, 32G15, 32G20, 57M50; Secondary: 14D07, 37D25.

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