Interval exchange transformation

Graph of interval exchange transformation (in black) with $\lambda =(1/15,2/15,3/15,4/15,5/15)$ and $\pi =(3,5,2,4,1)$. In blue, the orbit generated starting from $1/2$.

In mathematics, an interval exchange transformation^[1] 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.

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 with the pair $(\pi ,\lambda )$ as follows. For $1\leq i\leq n$ let

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

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 a translation, 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 preserves the Lebesgue measure. It is continuous except at a finite number of points.

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 (namely there is no integer $0<k<n$ such that $\pi (\{1,\dots ,k\})=\{1,\dots ,k\}$), a deep theorem which was a conjecture of M.Keane and due independently to William A. Veech^[2] and to Howard Masur ^[3] 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 $n$.

Generalizations

Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.^[4]

Notes

1. Keane, Michael (1975), "Interval exchange transformations", Mathematische Zeitschrift, 141: 25–31, doi:10.1007/BF01236981, MR 0357739.
2. Veech, William A. (1982), "Gauss measures for transformations on the space of interval exchange maps", Annals of Mathematics, Second Series, 115 (1): 201–242, doi:10.2307/1971391, MR 644019.
3. Masur, Howard (1982), "Interval exchange transformations and measured foliations", Annals of Mathematics, Second Series, 115 (1): 169–200, doi:10.2307/1971341, MR 644018.
4. Piecewise isometries – an emerging area of dynamical systems, Arek Goetz

References

• Artur Avila and Giovanni Forni, Weak mixing for interval exchange transformations and translation flows, arXiv:math/0406326v1, http://arxiv.org/abs/math.DS/0406326