Rauzy-Veech Induction for Infinite-Type IETs

Abstract

We consider infinite-type IETs arising from elementary examples of finite-area translation surfaces of infinite genus such as the Baker's surface. We call such IETs tail-reversing and we show that for any tail-reversing permutation the subset of the simplex of lengths $\hat{I}”$ for which the corresponding infinite-type IET is uniquely ergodic contains a dense $G_{\hat{I}}´$ set with respect to the ${\ell }^1$-topology. To this end, we generalize Rauzy-Veech induction to a large class of infinite-type IETs, where we prove a minimality criterion as a generalization of Keane's criterion in the finite setting. We then restrict ourselves to tail-reversing IETs and obtain our genericity result through a combinatorial analysis of their infinite-type Rauzy diagrams. Moreover, we derive an explicit condition for a tail-reversing IET to be uniquely ergodic by studying the diameter of its induction matrices.