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

Abdumajid Begmatov, Kleyber Cunha

Published 2018-07-23Version 1

We study renormalizations of piecewise smooth homeomorphisms on the circle, 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 $f_{1}$ and $f_{2}$, corresponding renormalizations approach to each other in $C^{1+L_{1}}$-norm.