April  2011, 5(2): 319-353. doi: 10.3934/jmd.2011.5.319

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, France

3. 

IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex

Received  July 2010 Revised  May 2011 Published  July 2011

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.
Citation: Alex Eskin, Maxim Kontsevich, Anton Zorich. Lyapunov spectrum of square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 319-353. doi: 10.3934/jmd.2011.5.319
References:
[1]

I. Bouw, "Tame Covers of Curves: P-Ranks and Fundamental Groups,", Ph.D thesis, (1998). Google Scholar

[2]

I. Bouw, The $p$-rank of ramified covers of curves,, Compositio Math., 126 (2001), 295. doi: 10.1023/A:1017513122376. Google Scholar

[3]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar

[4]

D. Chen, Covers of the projective line and the moduli space of quadratic differentials,, (2010), (2010), 1. Google Scholar

[5]

A. Elkin, The rank of the Cartier operator on cyclic covers of the projective line,, Jour. of Algebra, 327 (2011), 1. doi: 10.1016/j.jalgebra.2010.09.046. Google Scholar

[6]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., (). Google Scholar

[7]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar

[8]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, 1B (2006), 549. doi: 10.1016/S1874-575X(06)80033-0. Google Scholar

[9]

G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle,, J. Mod. Dyn., 5 (2011), 355. doi: 10.3934/jmd.2011.5.355. Google Scholar

[10]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, (2008), (2008), 1. Google Scholar

[11]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar

[12]

S. Grushevsky and I. Krichever, The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces,, in, XIV (2009), 111. Google Scholar

[13]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1996), 318. Google Scholar

[14]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Inventiones Math., 153 (2003), 631. doi: 10.1007/s00222-003-0303-x. Google Scholar

[15]

J. K. Koo, On holomorphic differentials of some algebraic function field of one variable over $\mathbbC$,, Bull. Aust. Math. Soc., 43 (1991), 399. doi: 10.1017/S0004972700029245. Google Scholar

[16]

C. McMullen, Braid groups and Hodge theory,, to appear in Math. Ann., (). Google Scholar

[17]

C. Peters, A criterion for flatness of Hodge bundles over curves and Geometric Applications,, Math. Ann., 268 (1984), 1. doi: 10.1007/BF01463870. Google Scholar

[18]

G. Schmithüsen, Examples for Veech groups of origamis,, in, 397 (2006), 193. Google Scholar

[19]

M. Schmoll, Moduli spaces of branched covers of Veech surfaces I: D-symmetric differentials,, (2006), (2006), 1. Google Scholar

[20]

M. Schmoll, Veech groups for holonomy free torus covers,, preprint, (2010). Google Scholar

[21]

, J. Smillie,, in preparation., (). Google Scholar

[22]

A. Wright, Abelian square-tiled surfaces,, preprint, (2011). Google Scholar

show all references

References:
[1]

I. Bouw, "Tame Covers of Curves: P-Ranks and Fundamental Groups,", Ph.D thesis, (1998). Google Scholar

[2]

I. Bouw, The $p$-rank of ramified covers of curves,, Compositio Math., 126 (2001), 295. doi: 10.1023/A:1017513122376. Google Scholar

[3]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar

[4]

D. Chen, Covers of the projective line and the moduli space of quadratic differentials,, (2010), (2010), 1. Google Scholar

[5]

A. Elkin, The rank of the Cartier operator on cyclic covers of the projective line,, Jour. of Algebra, 327 (2011), 1. doi: 10.1016/j.jalgebra.2010.09.046. Google Scholar

[6]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., (). Google Scholar

[7]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150. Google Scholar

[8]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, 1B (2006), 549. doi: 10.1016/S1874-575X(06)80033-0. Google Scholar

[9]

G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle,, J. Mod. Dyn., 5 (2011), 355. doi: 10.3934/jmd.2011.5.355. Google Scholar

[10]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum,, (2008), (2008), 1. Google Scholar

[11]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar

[12]

S. Grushevsky and I. Krichever, The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces,, in, XIV (2009), 111. Google Scholar

[13]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1996), 318. Google Scholar

[14]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Inventiones Math., 153 (2003), 631. doi: 10.1007/s00222-003-0303-x. Google Scholar

[15]

J. K. Koo, On holomorphic differentials of some algebraic function field of one variable over $\mathbbC$,, Bull. Aust. Math. Soc., 43 (1991), 399. doi: 10.1017/S0004972700029245. Google Scholar

[16]

C. McMullen, Braid groups and Hodge theory,, to appear in Math. Ann., (). Google Scholar

[17]

C. Peters, A criterion for flatness of Hodge bundles over curves and Geometric Applications,, Math. Ann., 268 (1984), 1. doi: 10.1007/BF01463870. Google Scholar

[18]

G. Schmithüsen, Examples for Veech groups of origamis,, in, 397 (2006), 193. Google Scholar

[19]

M. Schmoll, Moduli spaces of branched covers of Veech surfaces I: D-symmetric differentials,, (2006), (2006), 1. Google Scholar

[20]

M. Schmoll, Veech groups for holonomy free torus covers,, preprint, (2010). Google Scholar

[21]

, J. Smillie,, in preparation., (). Google Scholar

[22]

A. Wright, Abelian square-tiled surfaces,, preprint, (2011). Google Scholar

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