Cascades in the dynamics of affine interval exchange transformations

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], 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 study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.

VEECH GROUPS FOR HOLONOMY-FREE TORUS COVERS

For fixed coprime k, l ∈ ℕ and each pair (w, z) ∈ ℂ2we define an infinite cyclic cover Σk,l(w, z) → 𝕋, called a k-l-surface or k-l-cover. We show that [Formula: see text] classifies k-l-covers up to isomorphism away from a rather small set. The diagonal action of SL2(ℤ) on ℂ2descends to [Formula: see text], reflecting the SL2(ℤ)-action on the family of k-l-surfaces equipped with a translation structure. The moduli space of holonomy free k-l-surfaces is a compact SL2(ℤ) invariant subspace [Formula: see text] containing all k-l-surfaces with a lattice stabilizer with respect to the SL2(ℤ) action. We calculate the stabilizer, the Veech group, explicitly and represent k-l-covers branched over two points by a generalized class of staircase surfaces. Finally we study SL2(ℤ)-equivariant translation maps from the Hurwitz space of k-(d - k)-covers to Hurwitz spaces of ℤ/d-covers branched over two points.