The complete family of Arnoux–Yoccoz surfaces

Abstract

The family of translation surfaces (X g , ω g ) constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus g greater than or equal to 3. We triangulate these surfaces and deduce general properties they share. The surfaces (X g , ω g ) converge to a surface (X ∞, ω ∞) of infinite genus and finite area. We study the exchange on infinitely many intervals that arises from the vertical flow on (X ∞, ω ∞) and compute the affine group of (X ∞, ω ∞), which has an index 2 cyclic subgroup generated by a hyperbolic element.