Skip to Main content Skip to Navigation

Lyapunov exponents and variations of Hodge structures

Abstract : This thesis is organized around two main themes : on one hand (chapter 1 to 3) we study the Lyapunov exponents associated to a flat bundle on a complex curve, their application to wind-tree models and links with variation of Hodge structures on the bundle endowed with them. On the other hand (chapter 4 and 5) we introduce dilatation surfaces, and study their symmetries and dynamics.In chapter 1, a result binds diffusion rates of wind-tree models and a Lyapunov exponent of some affine invariant spaces in strata of quadratic differentials. This theorem enables us to compute numerically these diffusion rates for a large familly of models and hence to observe the influence of the shape of the obstacles on the speed of the flow. Chapter 2 is devoted to the proof of a conjecture on Lyapunov exponents behaviour for strata of a given genus and large number of poles when the number of zeros is bounded. It confirms an intuition explained in the previous chapter that diffusion rate on periodic wind-tree models with obstacles with a large number of angles is close to zero. At last, in chapter 3, we consider Lyapunov exponents in the more general setting of flat bundles endowed with a variation of Hodge structure on the sphere minus three points. This example coming from hypergeometric equations mimics the structure of a Hodge bundle on a moduli space parametrized by the sphere. We investigate the relation between these exponents and algebraic numbers like parabolic degrees of holomorphic subbundles.In chapter 4 we consider Veech groups of dilatation surfaces and give a complete topological classification of them. We also convey our intuition of this object and claim a conjecture on the existence of cylinders on each surface. In chapter 5 we describe the dynamics of a genus 2 example in every directions. We show the existence and genericity of directions corresponding to cylinders and we describe an infinite set of directions for which the geodesic flow accumulates on a Cantor set
Document type :
Complete list of metadatas

Cited literature [70 references]  Display  Hide  Download
Contributor : Abes Star :  Contact
Submitted on : Friday, March 22, 2019 - 9:55:10 AM
Last modification on : Friday, August 21, 2020 - 5:29:04 AM
Long-term archiving on: : Sunday, June 23, 2019 - 1:19:12 PM


Version validated by the jury (STAR)


  • HAL Id : tel-02076368, version 1


Charles Fougeron. Lyapunov exponents and variations of Hodge structures. General Mathematics [math.GM]. Université Sorbonne Paris Cité, 2017. English. ⟨NNT : 2017USPCC160⟩. ⟨tel-02076368⟩



Record views


Files downloads