# On the renormalizations of circle homeomorphisms with several break points

## TL;DR

This paper investigates the renormalization behavior of circle homeomorphisms with multiple break points, showing under certain conditions that their renormalizations can be approximated by piecewise smooth or affine maps.

## Contribution

It establishes the approximation of Rauzy-Veech renormalizations of such homeomorphisms by piecewise smooth functions, extending understanding of their dynamical structure.

## Key findings

- Renormalizations are approximated by piecewise M"obius functions in $C^{1+L_1}$-norm.
- Under certain conditions, renormalizations are approximated by piecewise affine maps.
- Results apply to maps with trivial product of break sizes, leading to affine approximations.

## Abstract

Let $f$ be an orientation preserving homeomorphisms on the circle with several break points, that is, its derivative $Df$ has jump discontinuities at these points. We study Rauzy-Veech renormalizations of piecewise smooth circle homeomorphisms, by considering such maps as generalized interval exchange maps with genus one. Suppose that $Df$ is absolutely continuous on the each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove that, under certain combinatorial assumptions on $f$, renormalizations $R^{n}(f)$ are approximated by piecewise M\"{o}bus functions in $C^{1+L_{1}}$-norm, that means, $R^{n}(f)$ are approximated in $C^{1}$-norm and $D^{2}R^{n}(f)$ are approximated in $L_{1}$-norm. In particular, if $f$ has trivial product of size of breaks, then the renormalizations are approximated by piecewise affine interval exchange maps.