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Lower bounds for Lyapunov exponents of flat bundles on curves

Alex Eskin, Maxim Kontsevich, Martin Möller and Anton Zorich

Geometry & Topology 22 (2018) 2299–2338

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank-k holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.

Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg for Calabi–Yau-type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.

To the memory of Jean-Christophe Yoccoz

Lyapunov exponents, hypergeometric differential equations, Hodge bundles, parabolic structure
Mathematical Subject Classification 2010
Primary: 37D25
Received: 12 October 2016
Accepted: 14 July 2017
Published: 5 April 2018
Proposed: Benson Farb
Seconded: Ian Agol, Anna Wienhard
Alex Eskin
Department of Mathematics
University of Chicago
Chicago, IL
United States
Maxim Kontsevich
Institut des Hautes Études Scientifiques
le Bois Marie
Martin Möller
Institut für Mathematik
Goethe-Universität Frankfurt
Frankfurt am Main
Anton Zorich
Center for Advanced Studies
Skolkovo Institute of Science and Technology
Institut de Mathématiques de Jussieu
Université Paris 6