Nonabelian Rauzy-Veech Renormalization

Abstract

For any Lie group G a renormalization map R on the space of simple G-extensions of Interval Exchange Transformations is constructed. R is applied to prove weak mixing and cohomological non-equivalence of typical G-extensions over IETs, when G is a compact connected Lie group. This extends a result of Avila and Forni for U(1) to any compact connected Lie group. This is a first result on ergodic theory of nonabelian extensions over IETs.