In mathematics, an interval exchange transformation 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. They arise naturally in the study of polygonal billiards and in area-preserving flows.
Formal definition Let n > 0 {\displaystyle n>0} and let π {\displaystyle \pi } be a permutation on 1 , … , n {\displaystyle 1,\dots ,n} . Consider a vector λ = ( λ 1 , … , λ n ) {\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{n})} of positive real numbers (the widths of the subintervals), satisfying
∑ i = 1 n λ i = 1. {\displaystyle \sum _{i=1}^{n}\lambda _{i}=1.}
Define a map T π , λ : [ 0 , 1 ] → [ 0 , 1 ] , {\displaystyle T_{\pi ,\lambda }:[0,1]\rightarrow [0,1],} called the interval exchange transformation associated with the pair ( π , λ ) {\displaystyle (\pi ,\lambda )} as follows. For 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} let
a i = ∑ 1 ≤ j < i λ j and a i ′ = ∑ 1 ≤ j < π ( i ) λ π − 1 ( j ) . {\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 x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]} , define
T π , λ ( x ) = x − a i + a i ′ {\displaystyle T_{\pi ,\lambda }(x)=x-a_{i}+a'_{i}}
if x {\displaystyle x} lies in the subinterval [ a i , a i + λ i ) {\displaystyle [a_{i},a_{i}+\lambda _{i})} . Thus T π , λ {\displaystyle T_{\pi ,\lambda }} acts on each subinterval of the form [ a i , a i + λ i ) {\displaystyle [a_{i},a_{i}+\lambda _{i})} by a translation, and it rearranges these subintervals so that the subinterval at position i {\displaystyle i} is moved to position π ( i ) {\displaystyle \pi (i)} .
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