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

:$T\_\{pi,lambda\}:\; [0,1]\; ightarrow\; [0,1]\; ,$

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

:$1\; leq\; i\; leq\; n$

let

: and let

:

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

:$T\_\{pi,lambda\}(x)\; =\; x\; -\; a\_i\; +\; a\text{'}\_i$

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 $T\_\{pi^\{-1\},\; lambda\text{'}\}$ where $lambda\text{'}\_i\; =\; lambda\_\{pi^\{-1\}(i)\}$ 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 $n$.

References