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Lyapunov spectrum of invariant subbundles of the Hodge bundle

Published online by Cambridge University Press:  05 December 2012

GIOVANNI FORNI ,
CARLOS MATHEUS  and
ANTON ZORICH
GIOVANNI FORNI
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA (email: gforni@math.umd.edu)
CARLOS MATHEUS
Affiliation:
Université Paris 13, Sorbonne Paris Cité, LAGA CNRS (UMR 7539), F-93430, Villetaneuse, France (email: matheus@impa.br, matheus@math.univ-paris13.fr)
ANTON ZORICH
Affiliation:
Institut de Mathématiques de Jessieu and Institut Universitaire de France, Université Paris 7, France (email: zorich@math.jussieu.fr)
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Abstract

We study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

Type
Review Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 353 - 408
Copyright
©2012 Cambridge University Press 

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