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

Abstract

Let $$\mathcal {F}$$ F and $$\mathcal {K}$$ K be commuting $$C^\infty $$ C ∞ diffeomorphisms of the cylinder $$\mathbb {T}\times \mathbb {R}$$ T × R that are, respectively, close to $$\mathcal {F}_0 (x, y)=(x+\omega (y), y)$$ F 0 ( x , y ) = ( x + ω ( y ) , y ) and $$T_\alpha (x, y)=(x+\alpha , y)$$ T α ( x , y ) = ( x + α , y ) , where $$\omega (y)$$ ω ( y ) is non-degenerate and $$\alpha $$ α is Diophantine. Using the KAM iterative scheme for the group action we show that $$\mathcal {F}$$ F and $$\mathcal {K}$$ K are simultaneously $$C^\infty $$ C ∞ -linearizable if $$\mathcal {F}$$ F has the intersection property (including the exact symplectic maps) and $$\mathcal {K}$$ 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 $$\mathbb {Z}^2$$ Z 2 -actions on the cylinder, generated by commuting twist maps.