Renormalizing an infinite rational IET

TL;DR

This paper investigates a specific interval exchange transformation created by cutting and reversing subintervals, revealing that it is mostly periodic except on a zero-dimensional Cantor set where it behaves like a 2-adic odometer.

Contribution

It introduces a new class of infinite rational IETs and characterizes their dynamics, showing periodicity and near conjugacy to a 2-adic odometer on a Cantor set.

Findings

Transformation is periodic outside a zero Hausdorff dimension Cantor set.

On the Cantor set, the dynamics are nearly conjugate to the 2-adic odometer.

The study provides insight into the structure of infinite rational IETs.

Abstract

We study an interval exchange transformation of [0,1] formed by cutting the interval at the points 1/n and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the 2-adic odometer.