On the renormalizations of circle homeomorphisms with several break points

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 Möbus 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.

28 pages. arXiv admin note: text overlap with arXiv:1510.03202 by other authors

References in corpus (1)

• On the convergence of renormalizations of piecewise smooth homeomorphisms on the circle