Dense properties of the space of circle diffeomorphisms with a Liouville rotation number (Q2888621)

Let \(\alpha \in (0, 1)\) and denote by \(F_{\alpha }\) the space of all orientation-preserving \(C^{\infty }\)-diffeomorphisms of the unit circle \(S^1\) with rotation number \(\alpha \). It is well known (by a theorem of Denjoy) that if \(\alpha \) is irrational, then for any \(f\in F_{\alpha }\) there exists a unique orientation-preserving homeomorphism \(H_f :S^1 \to S^1\) such that \(H_f (0)=0\) and \(f=H_f \circ R_{\alpha }\circ H_f ^{-1}\), where \(R_{\alpha }\) is the rotation by \(\alpha \). In [Astérisque. 231. Paris: Société Math. de France (1995; Zbl 0836.30001)], \textit{J.-C. Yoccoz} showed that, in this case, the subspace of \(F_{\alpha }\) consisting of all diffeomorphisms for which \(H_f\) are \(C^{\infty }\)-diffeomorphisms is \(C^{\infty }\)-dense in \(F_{\alpha }\).NEWLINENEWLINEIn this paper, the author proves that if \(\alpha \) is a Liouville number, then the same property have some other subspaces of \(F_{\alpha }\) defined by the regularities of \(H_f\) (for instance, the subspace of elements with \(H_f\) being \(C^k\)-diffeomorphisms but not \(C^{k+1}\)-diffeomorphisms). To this end, the method of fast approximation by conjugacy with estimate, developed by \textit{B. Fayad} and \textit{M. Saprykina} in [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 3, 339--364 (2005; Zbl 1090.37001)], is applied.