Robust local Hölder rigidity of circle maps with breaks

Abstract

We prove that, for every ε∈(0,1), every two C2+α-smooth (α>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 ρ∈(0,1) and the same size of the break c∈R+\\{1}, are conjugate to each other via a conjugacy which is (1−ε)-Hölder continuous at the break points. An analogous result does not hold for circle diffeomorphisms even when they are analytic.