Discrepancy Properties and Conjugacy Classes of Interval Exchange Transformations

TL;DR

This paper investigates the properties of interval exchange transformations, focusing on their discrepancy and conjugacy classes, and provides classifications for transformations with low-discrepancy orbits, especially for four intervals.

Contribution

It demonstrates that low-discrepancy orbit property is a conjugacy class invariant and offers a near-complete classification for four-interval transformations.

Findings

Low-discrepancy orbits are conjugacy class invariants.

Classification of four-interval transformations is nearly complete.

Exceptional case with monodromy invariant (4,3,2,1) analyzed.

Abstract

Interval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange transformations with low-discrepancy orbits are known so far and only for $n=2,3$ intervals, there are criteria to completely characterize those interval exchange transformations. In this paper, it is shown that having low-discrepancy orbits is a conjugacy class invariant under composition of maps. To a certain extent, this approach allows us to distinguish interval exchange transformations with low-discrepancy orbits from those without. For $n=4$ intervals, the classification is almost complete with the only exceptional case having monodromy invariant $\rho = (4,3,2,1)$. This particular monodromy invariant is discussed in detail.