Nonabelian Rauzy-Veech Renormalization

TL;DR

This paper develops a renormalization map for simple G-extensions of Interval Exchange Transformations over any Lie group G, demonstrating weak mixing and cohomological non-equivalence for typical extensions when G is compact and connected.

Contribution

It extends ergodic theory results from U(1) to all compact connected Lie groups, introducing a new renormalization approach for nonabelian extensions over IETs.

Findings

Proves weak mixing for typical G-extensions

Shows cohomological non-equivalence of extensions

Extends results from abelian to nonabelian Lie groups

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.