Erwan Lanneau

Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmuller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations on surfaces (orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with Z/2Z linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy-Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows, in particular, to classify the connected components of all exceptional strata.

Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation surfaces satisfying the topological Veech dichotomy. As a corollary we obtain new translation surfaces whose Veech group is infinitely generated and of the first kind.

Parity of the spin structure defined by a quadratic differential.

According to the work of Kontsevich{Zorich, the invariant that classies nonhyperelliptic connected components of the moduli spaces of Abelian dierentials with prescribed singularities, is the parity of the spin structure. We show that for the moduli space of quadratic dierentials, the spin structure is constant on every stratum where it is dened. In particular this disproves the conjecture that it classies the non-hyperelliptic connected components of the strata of quadratic dierentials with prescribed singularities. An explicit formula for the parity of the spin structure is given.

GL+(2, ℝ)–orbits in Prym eigenform loci

This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The proof uses several topological properties of Prym eigenforms, which are proved by the authors in a previous work. In particular the tools and the proof are independent of the recent results of Eskin-Mirzakhani-Mohammadi. As an application we obtain a finiteness result for the number of closed GL^+(2,R)-orbits (not necessarily primitive) in the Prym eigenform locus Prym_D(2,2) for any fixed D that is not a square.

Tell Me a Pseudo-Anosov

Anosov linear homeomorphisms, and more generally Anosov flows, as well as their hyperbolic analogues, have played an important role in the theory of dynamical systems [1, 2, 7].1 Their cousins, the pseudo-Anosov homeomorphisms, which are also interesting and important, seem to be less well known. In contrast to the theory of Anosov flows, for which we know their contours rather well, there are several fundamental questions about pseudo-Anosov homeomorphisms that so far remain widely open.