Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms

In this article, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of α, for any diffeomorphism f of rotation number α, it is possible to accumulate Rα with a sequence hn f h−1 n , hn being a diffeomorphism. The second result is: for a Baire-dense set of α, given two commuting diffeomorphisms f and g, such that f has α for rotation number, it is possible to approach each of them by commuting diffeomorphisms fn and gn that are differentiably conjugated to rotations. In particular, it implies that if α is in this Baire-dense set, and if β is an irrational number such that (α, β) are not simultaneously Diophantine, then the set of commuting diffeomorphisms ( f , g) with singular conjugacy, and with rotation numbers (α, β) respectively, is C∞-dense in the set of commuting diffeomorphisms with rotation numbers (α, β).