# Rigidity properties of certain discrete solvable group actions on tori

Abstract: The main results of this thesis provide smooth classification of large classes of perturbations of certain multidimensional time dynamical systems: they show a form of preservation of differentiable structure under small perturbations. In our work we obtain results for systems with very different dynamical properties. In Paper A we study elliptic dynamical systems, in Paper B parabolic, and in Paper C partially hyperbolic dynamical systems. Nevertheless, in all three papers we rely on the same general method of KAM of construction of conjugacies via successive iterations. Also, the groups whose actions we study in Paper A and Paper B are abelian, while in Paper C the acting groups are solvable and non-abelian. Linearization of specific non-commutative relations in the acting group was not used in combination with KAM theory before. This demonstrates the power of the method and also reveals mechanisms needed for its application to other group actions.Paper A generalizes results obtained by Moser in \cite{Mos90} to tori of any dimension. The local rigidity result is obtained under simultaneously Diophantine condition on rotation vectors of generators. We also answer a question raised by Moser about higher-dimensional Diophantine metric approximations, which implies that the local rigidity result for $\mathbb Z^m$ actions, for $m \geq 2$, is strictly stronger then the one for the $\mathbb Z$ action. Namely, there exists a continuum of $\mathbb Z^m$ actions by Liouville rotations which are simultaneously Diophantine. Paper B studies simultaneous linearization of certain commuting nearly integrable diffeomorphisms on the cylinder. To obtain a simultaneous linearization result, this time it is not enough to assume only the proper arithmetic condition on rotation number for a certain generator of this action, but in turn we need two more conditions. One condition is the intersection property for one of the generators of our action, introduced by Moser in his famous paper about twist maps, and the second condition is the existence of Lipschitz semi-conjugacy between the second generator and the rotation map. Both of these two conditions are in fact necessary since otherwise we construct counterexamples to simultaneous linearization. Further, we compare conditions in our results with those used by Fayad and Krikorian in \cite{FK09} and conclude that their result cannot be used to obtain the results in this paper. Paper C studies rigidity properties of ABC group actions on the torus $\mathbb T^3$, by affine transformations. The linear part of such an action is an ABC subgroup of $SL(3,\mathbb Z)$. We study affine actions whose linear part contains a $\mathbb Z^2$ subgroup generated by parabolic matrices. The first step is to investigate when a linear ABC action on $\mathbb T^3$ can be extended to an affine action. The analysis in this paper shows that there are generally two types of linear ABC actions on $\mathbb T^3$. Those that belong to a large family of affine actions, and those that do not and which we call stiff actions. Stiff actions have a factor that cannot be affinely perturbed. We prove conditional local rigidity statement, namely KAM rigidity, for a full measure set of affine actions with a non-stiff linear part. More surprisingly, for the actions that are stiff, we prove a new form of local rigidity, which we label fiberwise KAM rigidity, where we show that even a stiff action can have some local rigidity properties. We classify fiberwise perturbations of such actions. Namely, we can consider stiff actions as extensions, where in the base we have the identity map, and the main dynamics happens in the fibers. Fiberwise KAM rigidity means that fiberwise perturbations are conjugate to the initial action. The method of proof is the KAM iterative method, where one solves the linearized problem approximately, shows that the approximate solution has some good estimates, and uses it as the base of KAM iteration. One important new ingredient is that we use the whole non-commutative action, that is, we use linearization of even non-commutative relations to deal with cohomological obstructions. Another new ingredient is to obtain a posteriori estimates for obstructions that come from leafwise averages of the perturbation. Application of KAM method to stiff actions, without any Diophantine assumptions, shows the power of the KAM method. The method is applied to the ABC action and the study of the non-commutative part is necessary since without it there is not enough cohomological information that would lead to an approximate solution of the linearized problem.

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