Interval exchange transformation

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

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

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

Any interval exchange transformation ${\displaystyle T_{\pi ,\lambda }}$ is a bijection of ${\displaystyle [0,1]}$ to itself that 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 ${\displaystyle 0<k<n}$ 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 ${\displaystyle n}$.

Interval maps have a topological entropy of zero.[4]

Dyadic odometer ${\displaystyle T}$

Dyadic odometer iterated twice; that is ${\displaystyle T^{2}.}$

Dyadic odometer thrice iterated; that is ${\displaystyle T^{3}.}$

Dyadic odometer iterated four times; that is ${\displaystyle T^{4}.}$

The dyadic odometer can be understood as an interval exchange transformation of a countable number of intervals. The dyadic odometer is most easily written as the transformation

${\displaystyle T\left(1,\dots ,1,0,b_{k+1},b_{k+2},\dots \right)=\left(0,\dots ,0,1,b_{k+1},b_{k+2},\dots \right)}$

defined on the Cantor space ${\displaystyle \{0,1\}^{\mathbb {N} }.}$ The standard mapping from Cantor space into the unit interval is given by

${\displaystyle (b_{0},b_{1},b_{2},\cdots )\mapsto x=\sum _{n=0}^{\infty }b_{n}2^{-n-1}}$

This mapping is a measure-preserving homomorphism from the Cantor set to the unit interval, in that it maps the standard Bernoulli measure on the Cantor set to the Lebesgue measure on the unit interval. A visualization of the odometer and its first three iterates appear on the right.

Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.[5]

• Odometer

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