We show the following dichotomy for a linear parabolic Z2-action ρL on the torus with at least one step-2 generator: (i) Any affine Z2-action with linear part ρL has a ℤ-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine Z2-action with linear part ρL is KAM-rigid under volume preserving perturbations.

We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension f0: M → M of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any f ∈ Diff∞(M) sufficiently C1-close to f0 has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then f is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we classify the centralizer of any partially hyperbolic diffeomorphism of a 3-dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually Z × T.

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic Zk (k ≥ 2) actions on the torus. We prove a C∞ local rigidity result for such actions, provided that the smooth perturbations of the actions satisfy the intersection property. We also give a local rigidity result within a class of volume preserving actions. Our method mainly uses a generalization of the Kolmogorov-Arnold-Moser iterative scheme.

In this paper we study Zimmer's conjecture for C-1 actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in SL(n, R), the dimensional bound is sharp.

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.

In this paper we prove a perturbative result for a class of actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth ergodic perturbations of certain algebraic systems-including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle-the smooth centralizer is either virtually Z(l) or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered by Avila, Viana, and the second author: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is equivalent either to Lebesgue or atomic. The second phenomenon, described by the first and third authors, is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation. We employ a variety of techniques, among them a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, measure rigidity via thermodynamic formalism for circle extensions of Anosov diffeomorphisms, partially hyperbolic Livsic theory, and nonstationary normal forms.

We prove that every smooth diffeomorphism group valued cocycle over certain Z(k) Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan Z(k) (k >= 3) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over Z(k) actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

We prove global smooth classification results for Anosov ℤk actions on general compact manifolds, under certain irreduciblity conditions and the presence of sufficiently many Anosov elements. In particular, we remove all the uniform control assumptions which were used in all the previous results towards the Katok-Spatzier global rigidity conjecture on general manifolds. The main idea is to create a new mechanism labelled nonuniform redefining argument, to prove continuity of certain dynamically-defined objects. This leads to uniform control for higher-rank actions and should apply to more general rigidity problems in dynamical systems.

We define globally hypoelliptic smooth R-k actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When k = 1, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous R-k actions (k >= 2) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic R-2 actions on homogeneous spaces G/Gamma, with at least one quasi-unipotent generator, where G = SL(n, R). We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.