  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.
