Interval exchange transformation

In mathematics, an interval exchange transformation is a kind of dynamical system that generalises the idea of a 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.

Formal definition

Let $n>0$ and let $\pi$ be a permutation on $1,\dots ,n$. Consider a vector

$\lambda =(\lambda _{1},\dots ,\lambda _{n})$

of positive real numbers (the widths of the subintervals), satisfying

$\sum _{i=1}^{n}\lambda _{i}=1.$

Define a map

$T_{\pi ,\lambda }:[0,1]\rightarrow [0,1],$

called the interval exchange transformation associated to the pair $(\pi ,\lambda )$ as follows. For

$1\leq i\leq n$

let

$a_{i}=\sum _{1\leq j<i}\lambda _{j}$ and let

$a'_{i}=\sum _{1\leq j<\pi (i)}\lambda _{\pi ^{-1}(j)}.$

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

$T_{\pi ,\lambda }(x)=x-a_{i}+a'_{i}$

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

Properties

Any interval exchange transformation $T_{\pi ,\lambda }$ is a bijection of $[0,1]$ to itself which preserves Lebesgue measure. It is not usually continuous at each point $a_{i}$ (but this depends on the permutation $\pi$).

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

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

If $n>2$, and provided $\pi$ satisfies certain non-degeneracy conditions, a deep theorem due independently to W.Veech and to H.Masur asserts that for almost all choices of $\lambda$ in the unit simplex $\{(t_{1},\dots ,t_{n}):\sum t_{i}=1\}$ the interval exchange transformation $T_{\pi ,\lambda }$ is again uniquely ergodic. However, for $n\geq 4$ there also exist choices of $(\pi ,\lambda )$ so that $T_{\pi ,\lambda }$ is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of $T_{\pi ,\lambda }$ is finite, and is at most $n$.