Let F and K be commuting C∞ diffeomorphisms of the cylinder T× R that are, respectively, close to F(x, y) = (x+ ω(y) , y) and Tα(x, y) = (x+ α, y) , where ω(y) is non-degenerate and α is Diophantine. Using the KAM iterative scheme for the group action we show that F and K are simultaneously C∞-linearizable if F has the intersection property (including the exact symplectic maps) and 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 Z2-actions on the cylinder, generated by commuting twist maps.

We generalize results of Moser [17] on the circle to T-d: we show that a smooth sufficiently small perturbation of a Z(m) action, m >= 2, on the torus Td by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets of diffeomorphisms isotopic to identity and we answer the question Moser posed in [17] by proving the existence of a continuum of m-tuples of simultaneously Diophantine vectors such that every element of the induced Z(m) action is Liouville.

We study rigidity properties of ABC group actions on the three torus $\mathbb T^3$, by affine transformations. The linear part of such an action is an ABC subgroup of $SL(3,\mathbb Z)$. We investigate when such a linear ABC action on $\mathbb T^3$ can be extended to an affine action that has no identity factors. For such actions, we show KAM rigidity; the main reason for the existence of the conjugacy is KAM rigidity of the parabolic $\mathbb Z^2$ action inside the ABC group action. The main novelty in the opposite case is that we introduce and prove a new type of local rigidity phenomenon, which we label fiberwise KAM rigidity. Even though such affine actions are never KAM rigid, we show that all perturbations of specific form are conjugate to the initial action. We classify fiberwise perturbations of such actions. One important new ingredient is that we use the whole non-commutative action. The method of proof is the KAM iterative method. A detailed analysis of non-commutative group relations is required. Moreover, the systems we consider can be parabolic or partially hyperbolic.

We study rigidity properties of ABC group actions on the three torus $\mathbb T^3$, by affine transformations. The linear part of such an action is an ABC subgroup of $SL(3,\mathbb Z)$. We investigate when such a linear ABC action on $\mathbb T^3$ can be extended to an affine action that has no identity factors. For such actions, we show KAM rigidity; the main reason for the existence of the conjugacy is KAM rigidity of the parabolic $\mathbb Z^2$ action inside the ABC group action. The main novelty in the opposite case is that we introduce and prove a new type of local rigidity phenomenon, which we label fiberwise KAM rigidity. Even though such affine actions are never KAM rigid, we show that all perturbations of specific form are conjugate to the initial action. We classify fiberwise perturbations of such actions. One important new ingredient is that we use the whole non-commutative action. The method of proof is the KAM iterative method. A detailed analysis of non-commutative group relations is required. Moreover, the systems we consider can be parabolic or partially hyperbolic.