Lyapunov exponents, holomorphic flat bundles and de Rham moduli space
- PDF / 586,836 Bytes
- 71 Pages / 429.408 x 636.768 pts Page_size
- 19 Downloads / 192 Views
LYAPUNOV EXPONENTS, HOLOMORPHIC FLAT BUNDLES AND DE RHAM MODULI SPACE
BY
Matteo Costantini Institut f¨ ur Mathematik, Universit¨ at Bonn Endenicher Allee 60, 53115 Bonn, Germany e-mail: [email protected]
ABSTRACT
We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, M¨ oller and Zorich showing that the sum of the first k exponents is greater than or equal to the sum of the degree of any rank k holomorphic subbundle of the flat bundle and the asymptotic degree of its equivariant developing map. We also show that this inequality is an equality if the base curve is compact. We moreover relate the asymptotic degree to the dynamical degree defined by Daniel and Deroin. We then use the previous results to study properties of Lyapunov exponents on variations of Hodge structures and on Shatz strata of the de Rham moduli space. In particular, we show that the top Lyapunov exponent function is unbounded on the maximal Shatz stratum, the oper locus. In the final part of the work we specialize to the rank two case, generalizing a result of Deroin and Dujardin about Lyapunov exponents of holonomies of projective structures.
Received December 6, 2018 and in revised form Novembner 4, 2019
1
2
M. COSTANTINI
Isr. J. Math.
1. Introduction Lyapunov exponents are characteristic numbers describing the behavior of a cocycle over a dynamical system. If the cocycle satisfies an integrability property, Oseledets’ theorem states that there is a decomposition of the underlying vector bundle such that the norm of vectors in each component grows with different speed along the flow. The different possible growth rates are called Lyapunov exponents. An interesting instance of a dynamical system is given by playing billiards on tables of polygonal shape with angles that are rational multiples of π. Lyapunov exponents describe the diffusion rate of the trajectories of the ball. Even in this special case, Lyapunov exponents are very hard to compute using standard ergodic theoretic tools. There are two remarkable facts that allowed one to get a hold onto these invariants. The first is that the Lyapunov exponents given as diffusion rates of trajectories on a billiard given by a flat surface (X, ω) are the same as those of a completely different dynamical system, where the Lyapunov exponents are defined as the asymptotic growth rate of the Hodge norm of vectors in the variation of Hodge structures over the flow in the affine invariant manifold SL2 (R)(X, ω). The second key tool used for computing Lyapunov exponents makes use of algebraic geometry. It is in [EKZ14] where Eskin, Kontsevich and Zorich proved that the sum of positive Lyapunov exponents of the Kontsevich–Zorich cocycle over an affine invariant manifold can be computed by computing the normalized degree of the Hodge bundle restricted to the affine invariant manifold. Starting from billiards, algebraic geometry was used to investigate Lyapunov exponents in more general setti
Data Loading...
