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\begin{document}$$\mathcal {F}$$\end{document} and K\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K}$$\end{document} be commuting C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document} diffeomorphisms of the cylinder T×R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {T}\times \mathbb {R}$$\end{document} that are, respectively, close to F0(x,y)=(x+ω(y),y)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}_0 (x, y)=(x+\omega (y), y)$$\end{document} and Tα(x,y)=(x+α,y)\documentclass[12pt]{minimal}
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\begin{document}$$T_\alpha (x, y)=(x+\alpha , y)$$\end{document}, where ω(y)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (y)$$\end{document} is non-degenerate and α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is Diophantine. Using the KAM iterative scheme for the group action we show that F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} and K\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K}$$\end{document} are simultaneously C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document}-linearizable if F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} has the intersection property (including the exact symplectic maps) and K\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K}$$\end{document} 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 Z2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^2$$\end{document}-actions on the cylinder, generated by commuting twist maps.
