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

Published 23 Jul 2018 in math.DS | (1807.09159v1)

Abstract: 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.}