The complete family of Arnoux-Yoccoz surfaces

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.

18 pages (incl. references)

References in corpus (1)

• Dynamics on an infinite surface with the lattice property