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The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller...

We construct an explicit family of arithmetic Teichmüller curves ${𝒞}_{2k}$, $k\in \mathbb{N}$, supporting $\mathrm{SL}(2,ℝ)$-invariant probabilities ${\mu }_{2k}$ such that the associated $\mathrm{SL}(2,ℝ)$-representation on $L^2({𝒞}_{2k},{\mu }_{2k})$ has complementary series for every $k\ge 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves ${𝒞}_{2k}$ has arbitrarily slow rate of exponential mixing.

We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale’s Axiom A property is $C^1$-dense among the systems in this family, despite the existence of $C^2$-open subsets (closely related to the so-called Newhouse phenomena) where Smale’s Axiom A is violated. In particular, this provides some evidence towards Smale’s conjecture that Axiom A is a $C^1$-dense property among surface diffeomorphisms. The basic...

Given an origami (square-tiled surface) $M$ with automorphism group $\Gamma$, we compute the decomposition of the first homology group of $M$ into isotypic $\Gamma$-submodules. Through the action of the affine group of $M$ on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.