The complete family of Arnoux-Yoccoz surfaces

TL;DR

This paper studies the family of Arnoux-Yoccoz translation surfaces across all genera, analyzing their geometric properties, convergence to an infinite genus surface, and the dynamics of the associated infinite interval exchange map.

Contribution

It provides a comprehensive analysis of the Arnoux-Yoccoz surfaces, including triangulations, convergence behavior, and the affine group of the infinite genus limit surface.

Findings

Surfaces converge to an infinite genus, finite area surface.

The infinite interval exchange map has a well-defined dynamics.

The affine group of the limit surface has a cyclic subgroup of index 2.

Abstract

The family of translation surfaces $(X_g,\omega_g)$ 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 $(X_g,\omega_g)$ converge to a surface $(X_\infty,\omega_\infty)$ of infinite genus and finite area. We study the exchange on infinitely many intervals that arises from the vertical flow on $(X_\infty,\omega_\infty)$ and compute the affine group of $(X_\infty,\omega_\infty)$, which has an index $2$ cyclic subgroup generated by a hyperbolic element.