On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

Abstract

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^{\infty }$ diffeomorphisms of the cylinder $\mathbb{T}\times \mathbb{R}$ that are, respectively, close to ${\mathcal{F}}_0(x,y)=(x+\omega (y),y)$ and $T_{\alpha }(x,y)=(x+\alpha ,y)$, where $\omega (y)$ is non-degenerate and $\alpha$ is Diophantine. Using the KAM iterative scheme for the group action we show that $\mathcal{F}$ and $\mathcal{K}$ are simultaneously $C^{\infty }$-linearizable if $\mathcal{F}$ has the intersection property (including the exact symplectic maps) and $\mathcal{K}$ satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of ${\mathbb{Z}}^2$-actions on the cylinder, generated by commuting twist maps.

35 pages, 1 figure

References in corpus (1)

• Global rigidity for totally nonsymplectic Anosov Z^k actions

Cited by in corpus (1)

• Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step