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.

区间交换变换通常是唯一的遍历映射，因此具有均匀分布的轨道。它们的均匀度可以根据星差来衡量。到目前为止，很少有具有低差异轨道的区间交换变换的例子，并且仅对于\(n=2,3\)区间，有一些标准可以完全表征这些区间交换变换。在本文中，表明具有低差异轨道是映射组合下的共轭类不变量。在一定程度上，这种方法使我们能够区分具有低差异轨道的区间交换变换和没有轨道的区间交换变换。对于\(n=4\)区间，分类几乎完成，唯一的例外情况是具有单向不变性\(\rho = (4,3,2,1)\)。详细讨论了这个特定的单项不变量。