On the Renormalizations of Circle Homeomorphisms with Several Break Points

Abstract

Let f be an orientation preserving homeomorphism 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 of genus one. Suppose that Df is absolutely continuous on 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 Möbius 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 the product of the sizes of breaks of f is trivial, then the renormalizations are approximated by piecewise affine interval exchange maps.