Duke Mathematical Journal
- Duke Math. J.
- Volume 166, Number 4 (2017), 657-706.
Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle
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.
Duke Math. J., Volume 166, Number 4 (2017), 657-706.
Received: 11 May 2015
Revised: 25 May 2016
First available in Project Euclid: 31 October 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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