Rigidity for circle homeomorphisms with a break-type singularity (Q1595705)

Let \(f\) be a smooth except for one point homeomorphism of a circle \(f\in C^{2+\varepsilon}(S^1\setminus\{x_0\})\), with existing one-sided derivatives at \(x_0\). Let \(c(f)=\sqrt{f'_{-}(x_0)/f'_{+}(x_0)}\) and \(\rho(f)\) be the rotation number. Theorem: If \(f\) and \(h\) are such that \(c(f)=c(h)\not=1\) and \(\rho(f)=\rho(h)\) is irrational with periodic continued fraction expansion then there exists a \(\delta>0\) such that \(f\) and \(g\) are \(C^{1+\delta}\) conjugate. Conjecture. The assumption on periodic expansion can be omitted.