Interval Exchange Transformations: Ergodicity & Rauzy Induction

Interval exchange transformations (IETs): Partition [0,1) into k sub-intervals and rearrange them by a permutation π. The map T: x ↦ x + c_i mod 1 on each interval is piecewise isometric (measure-preserving).

Keane's theorem (1975): If lengths {λ_i} satisfy no rational relation and the permutation is irreducible (no proper π-invariant prefix), then T is minimal (all orbits dense). Masur (1982) and Veech (1982) proved ergodicity holds for almost all λ (with respect to Lebesgue measure).

Rauzy induction is an accelerated return map to a shorter interval — an analog of the Euclidean algorithm for IETs. It defines the Rauzy-Veech cocycle whose Lyapunov exponents control the Birkhoff sum deviations. IETs are deeply connected to the dynamics of billiards in rational polygons and Teichmüller geodesic flows.