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).

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

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