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

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$-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.