Publications


Parabolic degrees and Lyapunov exponents for hypergeometric local systems

arXiv:1701.08387

Consider the flat bundle on $\mathbb CP^1 - \{0,1,\infty\}$ corresponding to solutions of the hypergeometric differential equation $$\prod_{i=1}^n (D - \alpha_i) - z \prod_{j=1}^n (D - \beta_j) = 0$$ where $D = z \frac {d}{dz}$. For $\alpha_i$ and $\beta_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathbb CP^1 - \{0,1,\infty\}$, we associate n Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge structure and study the dependence of the Lyapunov exponents in terms of these degrees by means of numerical simulations.


Cascades in the dynamics of affine interval exchange transformations (with A. Boulanger and S. Ghazouani)

arXiv:1701.02332

We describe in this article the dynamics of a 1-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface, the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This is done by analysing the dynamics of the Veech group of this surface, combined with an original use of the Rauzy induction in the context of affine interval exchange transformations. Finally, this analysis allows us to completely compute the Veech group of the Disco surface.


Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

arXiv:1611.07728

We show an upper bound for the sum of positive Lyapunov exponents of any Teichmrüller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary it stands for all Teichmüller curves in these strata and $SL(2, \mathbb R)$ invariant subspaces defined over $\mathbb Q$. This solves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with large number of poles in the situation when at least one zero has large multiplicity.


Affine surfaces and their Veech groups (with E. Duryev and S. Ghazouani)

arXiv:1609.02130

We introduce a class of objects which we call affine surfaces. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalise the notion of Veech group to affine surfaces, and we prove a structure result about these Veech groups.