Interval exchange transformation

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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.

[edit] Formal Definition

Let $n > 0$ and let $π$ 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 $\sum_{i=1}^n \lambda_i = 1$ . We define a map $T_{\pi,\lambda}:[0,1]\rightarrow [0,1]$ called the interval exchange transformation associated to the pair $(π,λ)$ 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 $T π,λ (x) = x - a i + a' i$ if $x$ lies in the subinterval $[a i, a i + λ i)$ . Thus $T π,λ$ acts on each subinterval of the form $[a i, a i + λ i)$ by an orientation-preserving isometry, and it rearranges these subintervals so that the subinterval at position $i$ is moved to position $π(i)$ .

[edit] Properties

Any interval exchange transformation $T π,λ$ 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 $π$ ).

The inverse of the interval exchange transformation $T π,λ$ 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 $π = (12)$ (in cycle notation), and if we join up the ends of the interval to make a circle, then $T π,λ$ is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length $λ 1$ is irrational, then $T π,λ$ is uniquely ergodic. Roughly speaking, this means that the orbits of points of $[0,1]$ are uniformly evenly distributed. On the other hand, if $λ 1$ is rational then each point of the interval is periodic, and the period is the denominator of $λ 1$ (written in lowest terms).

If $n > 2$ , and provided $π$ satisfies certain non-degeneracy conditions, a deep theorem due independently to W.Veech and to H.Masur asserts that for almost all choices of $λ$ in the unit simplex $\{(t_1, \dots, t_n) : \sum t_i = 1\}$ the interval exchange transformation $T π,λ$ is again uniquely ergodic. However, for $n \geq 4$ there also exist choices of $(π,λ)$ so that $T π,λ$ is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of $T π,λ$ is finite, and is at most $n$ .