Interval exchange transformation

Example

Let $d \ge 2$, let $\lambda_1, \dots, \lambda_d > 0$ be lengths with sum $\abs{\lambda} = \lambda_1 + \cdots + \lambda_d$, and let $\pi$ be a permutation of $\set{1, \dots, d}$. Cut $I = [0, \abs{\lambda})$ into the intervals $I_j = [\lambda_1 + \cdots + \lambda_{j-1}, \lambda_1 + \cdots + \lambda_j)$, for $j = 1, \dots, d$. The interval exchange transformation $T \colon I \to I$ translates each $I_j$ so that the intervals appear in the new order $I_{\pi^{-1}(1)}, I_{\pi^{-1}(2)}, \dots, I_{\pi^{-1}(d)}$ from left to right. Explicitly, for $x \in I_j$, $$T(x) = x - \sum_{i < j} \lambda_i + \sum_{\pi(i) < \pi(j)} \lambda_i.$$

Check that $T$ is a measure-preserving transformation for Lebesgue measure. Indeed, $T$ is a bijection of $I$, and it is a translation on each of the finitely many intervals $I_j$. Translations preserve length, so for every Borel set $E \subseteq I$, $\abs{T^{-1}E} = \sum_j \abs{T^{-1}E \cap I_j} = \sum_j \abs{E \cap T(I_j)} = \abs{E}$.

Concrete case: $d = 2$ and $\pi$ the transposition, with $0 < \alpha < 1$ and lengths $\lambda_1 = 1 - \alpha$ and $\lambda_2 = \alpha$. Then $T(x) = x + \alpha$ on $[0, 1 - \alpha)$ and $T(x) = x - (1 - \alpha)$ on $[1 - \alpha, 1)$, so $T(x) = x + \alpha \bmod 1$, the rotation of the circle. Interval exchanges are thus a generalization of circle rotations.

For an irreducible permutation $\pi$ and Lebesgue almost every length vector, the interval exchange is uniquely ergodic. Masur and Veech proved this independently in 1982.