Renormalization for piecewise smooth homeomorphisms on the circle

TL;DR

This paper investigates the renormalization process for piecewise smooth circle homeomorphisms, demonstrating convergence to affine maps under specific conditions, thus advancing understanding of interval exchange transformations.

Contribution

It establishes the convergence of renormalizations for certain circle homeomorphisms to affine maps, linking Rauzy-Veech renormalizations with generalized interval exchanges.

Findings

Renormalizations converge to piecewise affine maps.

Zero mean nonlinearity is a key condition.

Results connect circle homeomorphisms with interval exchange maps.

Abstract

In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converges to the set of piecewise affine interval exchange maps.