Duke Mathematical Journal

Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle

Simion Filip

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Abstract

We describe all the situations in which the Kontsevich–Zorich (KZ) cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, we find this only occurs when the cocycle satisfies additional geometric constraints. We also show that the connected components of the Zariski closure of the monodromy must be from a specific list, and the representations in which they can occur are described. The number of zero exponents of the KZ cocycle is then as small as possible, given its monodromy.

Article information

Source
Duke Math. J., Volume 166, Number 4 (2017), 657-706.

Dates
Received: 11 May 2015
Revised: 25 May 2016
First available in Project Euclid: 31 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1477918664

Digital Object Identifier
doi:10.1215/00127094-3715806

Mathematical Reviews number (MathSciNet)
MR3619303

Zentralblatt MATH identifier
1370.37066

Subjects
Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 14D07: Variation of Hodge structures [See also 32G20]

Keywords
Lyapunov exponents Kontsevich–Zorich cocycle flat surfaces translation surfaces monodromy variations of Hodge structure

Citation

Filip, Simion. Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle. Duke Math. J. 166 (2017), no. 4, 657--706. doi:10.1215/00127094-3715806. https://projecteuclid.org/euclid.dmj/1477918664


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