Robust local Hölder rigidity of circle maps with breaks

Abstract

We prove that, for every $\epsilon \in (0,1)$, every two $C^{2+\alpha }$-smooth $(\alpha >0)$ circle diffeomorphisms with a break point, i.e. circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, with the same irrational rotation number $\rho \in (0,1)$ and the same size of the break $c\in {\mathbb{R}}_{+}\backslash \{1\}$, are conjugate to each other via a conjugacy which is $(1-\epsilon )$-Hölder continuous at the break points. An analogous result does not hold for circle diffeomorphisms even when they are analytic.