Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms

Mostapha Benhenda (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of $\alpha$, for any diffeomorphism $f$ of rotation number $\alpha$, it is possible to accumulate $R_\alpha$ with a sequence $h_n f h_n^{-1}$, $h_n$ being a diffeomorphism. The second result is: for a Baire-dense set of $\alpha$, given two commuting diffeomorphisms $f$ and $g$, such that $f$ has $\alpha$ 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 for $\alpha$ in this Baire-dense set, and if $\beta$ is an irrational number such that $(\alpha,\beta)$ are not simultaneously Diophantine, the set of commuting diffeomorphisms $(f,g)$ with singular conjugacy, and with rotation numbers $(\alpha,\beta)$ respectively, is $C^\infty$-dense in the set of commuting diffeomorphisms with rotation numbers $(\alpha,\beta)$.

Mots clés: circle diffeomorphisms; commuting diffeomorphisms; Liouville rotation number

Domaines: Systèmes dynamiques [math.DS]

Mostapha Benhenda. Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms. 2011. ⟨hal-00628298v4⟩