Interval Exchange Transformations and Low-Discrepancy

TL;DR

This paper investigates conditions under which orbits of interval exchange transformations produce low-discrepancy sequences, extending known results from circle rotations to more complex transformations and constructing numerous examples.

Contribution

It provides new conditions for low-discrepancy in 3-interval exchanges and constructs infinitely many such transformations for four or more intervals, expanding the understanding of distribution properties.

Findings

Conditions for low-discrepancy in 3-interval exchanges

Construction of infinitely many low-discrepancy transformations for n ≥ 4

Examples do not coincide with LS-sequences for S ≥ 2

Abstract

In [Mas82] and [Vee78] it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case of $n=2$ intervals corresponds to circle rotation, where conditions for low-discrepancy are well-known. In this paper, we give corresponding conditions in the case $n=3$. Furthermore, we construct infinitely many interval exchange transformations with low-discrepancy orbits for $n \geq 4$. We also show that these examples do not coincide with $LS$-sequences if $S \geq 2$.