\(C^1\) smoothness of Liouville arcs in Arnol'd tongues (Q2706600)

Let us consider a family of circle diffeomorphisms whose liftings are given by NEWLINE\[NEWLINEf_{t,\alpha}= x+ t+ \alpha\gamma(x)NEWLINE\]NEWLINE with \(\gamma\) at least \(C^1\), periodic, of average \(0\), and \(|\alpha|\) suitably bounded to ensure that \(\partial_x f>0\). For an irrational \(\rho\) and \(\alpha\) there is a unique \(t(\rho,\alpha)\) characterized by the condition that \(f_{t(\rho,\alpha),\alpha}\) has rotation number \(\rho\).NEWLINENEWLINENEWLINEThe main result of the paper is that for a generic \(\gamma\in C^r\) \((r\geq 2??\)) and a fixed \(\rho\) irrational, \(t(\rho,\alpha)\) is a \(C^1\) function of \(\alpha\). The result is obtained through approximation of \(t(\rho,\alpha)\) by functions which represent the boundary points of frequency-locking regions for rational \(\rho_n\), with \(\rho_n\) converging to \(\rho\). The key technical point is a little known theorem attributed to Douady and Yoccoz concerning the existence and uniqueness of conformal measures for circle diffeomorphisms.NEWLINENEWLINENEWLINEThe result is very good, unfortunately the reviewer felt the representation a bit confusing. In particular, I could not find a definite statement of the main result clarifying the minimal smoothness required to \(\gamma\).