Complete periodicity of Prym eigenforms

Erwan Lanneau (1) , Duc-Manh Nguyen (2)

1 IF - Institut Fourier
2 Équipe Géométrie

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.

Mots clés: Prym eigenform; moduli space; translation surface

Domaines: Topologie géométrique [math.GT]; Systèmes dynamiques [math.DS]

Erwan Lanneau, Duc-Manh Nguyen. Complete periodicity of Prym eigenforms. Annales Scientifiques de l'École Normale Supérieure, 2016, 49 (1), pp.87-30. ⟨hal-01258865⟩