Interval Exchange Transformation - Formal Definition

Formal Definition

Let and let be a permutation on . Consider a vector

of positive real numbers (the widths of the subintervals), satisfying

Define a map

called the interval exchange transformation associated to the pair as follows. For

let

and let

Then for, define

T_{\pi,\lambda}(x) = x - a_i + a'_i

if lies in the subinterval . Thus acts on each subinterval of the form by an orientation-preserving isometry, and it rearranges these subintervals so that the subinterval at position is moved to position .

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