Circle diffeomorphisms: quasi-reducibility and commuting diffeomorphisms

Abstract

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 R α with a sequence h n f h −1 n , h n 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 f n and g n 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).

On the automorphism group of foliations with geometric transverse structures

“…) by a theorem of Benhenda [1] (whose generalization for any Liouville number has been recently announced by Avila and Krikorian). As a consequence we could choose the diffeomorphism h 0 provided by Corollary 2.14 to have a dense orbit in D ∞ ρ(h 0 ) (S 1 ).…”

Estimates of the linearization of circle diffeomorphisms

“…First, we estimate C 61 . Since 5C 62 ≤ (C 59 (k − 1, 0)) 2 , we estimate C 59 (γ, 0) for 0 ≤ γ ≤ k − 1. By combining the constants appearing in lemma 5.9, we get:…”

“…These estimates have natural applications to the global study of circle diffeomorphisms with Liouville rotation number: in [2], they allow to show the following results: 1) Given a diffeomorphism f of rotation number α, for a Baire-dense set of α, it is possible to accumulate R α with a sequence h n f h −1 n , h n being a diffeomorphism. 2) Given two commuting diffeomorphisms f and g, with the rotation number α of f belonging to a specified Baire-dense set, it is possible to approach each of them by commuting diffeomorphisms f n and g n that are differentiably conjugated to rotations.…”

“…M n m n (x) 2 In particular, we can take For r > 2, we prove lemma 4.4 by induction. Suppose the lemma is proved up to r ≥ 2.…”

Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

“…It deals with the question of the connectedness of the space of commuting diffeomorphisms of a 1-dimensional manifold, which was raised by Rosenberg in the 70's (he was particularly interested in the local path-connectedness of the space of pairs of commuting circle diffeomorphisms). Although the nowadays classical theorems concerning linearization of circle diffeomorphisms somewhat point in this direction ( [29]; see also [3,9]), there is only a couple of definitive results regarding Rosenberg's question, and these are very recent. Indeed, the list reduces to:…”