Journal of Modern Dynamics (JMD)

Lyapunov spectrum of square-tiled cyclic covers

Pages: 319 - 353, Issue 2, April 2011      doi:10.3934/jmd.2011.5.319

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Alex Eskin - Department of Mathematics, University of Chicago, Chicago, IL 60637, United States (email)
Maxim Kontsevich - IHES, le Bois Marie, 35, route de Chartres, 91440 Buressur-Yvette, France (email)
Anton Zorich - IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France (email)

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.

Received: July 2010;      Revised: May 2011;      Available Online: July 2011.