Interval exchange transformation

In mathematics, the interval exchange transformation is a simple ergodic system that generalizes the idea of an ergodic rotation. The system consists of the unit interval cut up into multiple pieces, which are then permuted.

This system clearly preserves the Lebesgue measure on the unit interval.

If the ratios of the sizes of the pieces are not rational numbers, and meet some additional irreducibility criteria, then the orbit of a point is ergodic with respect to Lebesgue measure. That is, the orbit becomes uniformly distributed on the unit interval.