In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. They arise naturally in the study of polygonal billiards and in area-preserving flows.

Let ${\displaystyle n>0}$ and let ${\displaystyle \pi }$ be a permutation on ${\displaystyle 1,\dots ,n}$. Consider a vector ${\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{n})}$ of positive real numbers (the widths of the subintervals), satisfying

Define a map ${\displaystyle T_{\pi ,\lambda }:[0,1]\rightarrow [0,1],}$ called the interval exchange transformation associated with the pair ${\displaystyle (\pi ,\lambda )}$ as follows. For ${\displaystyle 1\leq i\leq n}$ let

Then for ${\displaystyle x\in [0,1]}$, define

if ${\displaystyle x}$ lies in the subinterval ${\displaystyle [a_{i},a_{i}+\lambda _{i})}$. Thus ${\displaystyle T_{\pi ,\lambda }}$ acts on each subinterval of the form ${\displaystyle [a_{i},a_{i}+\lambda _{i})}$ by a translation, and it rearranges these subintervals so that the subinterval at position ${\displaystyle i}$ is moved to position ${\displaystyle \pi (i)}$.

Any interval exchange transformation ${\displaystyle T_{\pi ,\lambda }}$ is a bijection of ${\displaystyle [0,1]}$ to itself that preserves the Lebesgue measure. It is continuous except at a finite number of points.

The inverse of the interval exchange transformation ${\displaystyle T_{\pi ,\lambda }}$ is again an interval exchange transformation. In fact, it is the transformation ${\displaystyle T_{\pi ^{-1},\lambda '}}$ where ${\displaystyle \lambda '_{i}=\lambda _{\pi ^{-1}(i)}}$ for all ${\displaystyle 1\leq i\leq n}$.

If ${\displaystyle n=2}$ and ${\displaystyle \pi =(12)}$ (in cycle notation), and if we join up the ends of the interval to make a circle, then ${\displaystyle T_{\pi ,\lambda }}$ is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length ${\displaystyle \lambda _{1}}$ is irrational, then ${\displaystyle T_{\pi ,\lambda }}$ is uniquely ergodic. Roughly speaking, this means that the orbits of points of ${\displaystyle [0,1]}$ are uniformly evenly distributed. On the other hand, if ${\displaystyle \lambda _{1}}$ is rational then each point of the interval is periodic, and the period is the denominator of ${\displaystyle \lambda _{1}}$ (written in lowest terms).