Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

TL;DR

This paper establishes conditions for minimality and unique ergodicity of certain interval exchange transformations, including Veech 1969 type examples, using combinatorial and recurrence criteria.

Contribution

It provides the first complete characterization of minimality for $ ext{Z}/N ext{Z}$ extensions of rotations, and introduces a general combinatorial criterion for minimality of interval exchanges.

Findings

Solved minimality conditions for prime N extensions, including Veech 1969 cases.

Provided a combinatorial criterion for minimality applicable to general interval exchanges.

Established conditions for unique ergodicity in linearly recurrent cases with marked points.

Abstract

We give conditions for minimality of $\mathbb Z/N\mathbb Z$ extensions of a rotation of angle $\alpha$ with one marked point, solving the problem for any prime $N$: for $N=2$, these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to $\mathbb Z/N\mathbb Z$ extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the initial interval exchange transformation is linearly recurrent and there are one or two marked points.