We show that most homogeneous Anosov actions of higher rank Abelian groups are locally C∞-rigid (up to an automorphism). This result is the main part in the proof of local C∞-rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil–manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper “non–stationary” generalization of the classical theory of normal forms for local contractions.

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We study the left action α of a Cartan subgroup on the space X = G/Γ, where Γ is a lattice in a simple split connected Lie group G of rank n> 1. Let µ be an α-invariant measure on X. We give several conditions using entropy and conditional measures each of which characterizes the Haar measure on X. Furthermore, we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure. The main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures.

We consider the restriction α0,G of the Weyl chamber flow on SL(n, R)/Γ (where Γ is a cocompact lattice) to a closed subgroup G isomorphic to Z k × R l, k + l ≥ 2 of the group D+ of positive diagonal matrices which contains a lattice in a twodimensional plane in general position. We prove that any C 2 small smooth perturbation of the action α0,G is differentiably conjugate to a standard perturbation which arises from a perturbation of the embedding Z k × R l → D+. We introduce a new method in rigidity of actions of higher rank abelian groups based on the study of combinatorial structure of the the web of Lyapunov (unipotent) foliations. Insights from the classical algebraic K-theory play a crucial role in establishing stability properties of that web. The method has applications to other classes of partially hyperbolic algebraic actions and is complementary to the other new analytic method which we introduced in the first paper of this series.

Abstract. We present a certain version of the “non–stationary ” normal forms theory for extensions of topological dynamical systems (homeomorphisms of compact metrizable spaces) by smooth (C ∞ ) contractions of R n. The central concept is a notion of a sub–resonance relation which is an appropriate generalization of the notion of resonance between the eigenvalues of a matrix which plays a similar role in the local normal forms theory going back to Poincaré and developed in the modern form for C ∞ maps by S. Sternberg and K.-T. Chen. Applicability of these concepts depends on the narrow band condition, a certain collection of inequalities between the relative rates of contraction in the fibers. One of the ways to formulate our first conclusion (the sub–resonance normal form theorem) is to say that there is a continuous invariant family of geometric structures in the fibers whose automorphism groups are certain finite–dimensional Lie groups. Our central result is the joint normal form for the centralizer for an extension satisfying the narrow band condition. While our non–stationary normal forms are rather close to the normal forms in a neighborhood of an invariant manifold, studied in the literature, the centralizer theorem seems to be new even in the classical local case. The principal situation where our analysis applies is a smooth system on a compact manifold with an invariant contracting foliation. In this case

We investigate rigidity of measurable structure for higher rank abelian algebraic actions.

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We prove that every smooth action α of Z k, k ≥ 2, on the (k+ 1)dimensional torus whose elements are homotopic to corresponding elements of an action α0 by hyperbolic linear maps preserves an absolutely continuous measure. This is the first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data. We also show that both ergodic and geometric properties of such a measure are very close to the corresponding properties of the Lebesgue measure with respect to the linear action α0.

ABSTRACT. Every C 2 action α of Z k, k ≥ 2, on the (k+ 1)-dimensional torus whose elements are homotopic to the corresponding elements of an action α0 by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between α and α0. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for α for which the eigenvalues for α and α0 coincide. We describe some nontrivial examples of actions of this kind. 1. PRELIMINARIES 1.1. Introduction. This paper constitutes a direct continuation of [3]. We shall use the terminology and results from [3] with only occasional references. Let α0 be a Z k Cartan action on T k+1 and let α be a smooth Z k action whose elements are homotopic to the corresponding elements of α0. 1 “Smooth ” in our context means C 2 although most of the arguments hold for C 1+ε actions for any

Abstract. We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on T N. 1.