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Interval exchange transformation

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

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.

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Formal definition

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

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

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

${\displaystyle a_{i}=\sum _{1\leq j

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

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

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 ${\displaystyle \pi (i)}$.

Properties

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

The inverse of the interval exchange transformation ${\displaystyle T_{\pi ,\lambda }}$ 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).

If ${\displaystyle n>2}$, and provided ${\displaystyle \pi }$ satisfies certain non-degeneracy conditions (namely there is no integer ${\displaystyle 0 such that ${\displaystyle \pi (\{1,\dots ,k\})=\{1,\dots ,k\}}$), 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 ${\displaystyle \lambda }$ in the unit simplex ${\displaystyle \{(t_{1},\dots ,t_{n}):\sum t_{i}=1\}}$ the interval exchange transformation ${\displaystyle T_{\pi ,\lambda }}$ is again uniquely ergodic. However, for ${\displaystyle n\geq 4}$ there also exist choices of ${\displaystyle (\pi ,\lambda )}$ so that ${\displaystyle T_{\pi ,\lambda }}$ is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of ${\displaystyle T_{\pi ,\lambda }}$ is finite, and is at most ${\displaystyle n}$.

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 0644019.
3. ^ Masur, Howard (1982), "Interval exchange transformations and measured foliations", Annals of Mathematics, Second Series, 115 (1): 169–200, doi:10.2307/1971341, MR 0644018.
4. ^