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Invariant Measures for Actions of Higher Rank Abelian Groups
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
"... The first part of the paper begins with an introduction into Anosov actions of Z k and R k and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed ..."
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Cited by 36 (20 self)
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The first part of the paper begins with an introduction into Anosov actions of Z k and R k and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed proof of a modified version of the main theorem from [KS3] for actions by toral automorphisms of with applications to rigidity of the measurable structure of such actions with respect to Lebesque measure. In the second part principal technical tools for studying nonuniformly hyperbolic actions of Z k and R k are introduced and developed. These include Lyapunov characteristic exponents, nonstationary normal forms and Lyapunov Hoelder structures. At the end new rigidity results for Z² actions on threedimensional manifolds are outlined.
Rigidity of measurable structure for Z d actions by automorphisms of a torus
 Comm. Math. Helv
"... Abstract. We show that for certain classes of actions of Z d, d ≥ 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants ..."
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Cited by 27 (11 self)
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Abstract. We show that for certain classes of actions of Z d, d ≥ 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z dactions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure–theoretic invariant.
Rigidity Of Weakly Hyperbolic Actions Of Higher Real Rank Semisimple Lie Groups And Their Lattices
 Th. & Dyn. Syst
"... . Under some weak hyperbolicity conditions, we establish C 0  and C 1  local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher ran ..."
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Cited by 22 (2 self)
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. Under some weak hyperbolicity conditions, we establish C 0  and C 1  local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C 0 local rigidity for weakly hyperbolic standard actions follows from a cocycle C 0 local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C 1 local rigidity is deduced from the C 0 local rigidity following a procedure outlined by Katok and Spatzier. Using similar consideration, we also establish C 0 global rigidity of ...
LOCAL RIGIDITY OF PARTIALLY HYPERBOLIC ACTIONS. II. THE GEOMETRIC METHOD AND RESTRICTIONS OF WEYL CHAMBER FLOWS ON . . .
"... 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 ..."
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Cited by 22 (11 self)
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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 Ktheory 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.
Invariant measures on G/Γ for split simple Lie groups G
 G, COMM. PURE APPL. MATH
, 2004
"... 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 αinvariant measure on X. We give several conditions using entropy and conditional measures each of which characterizes the Haar measure on X. ..."
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Cited by 18 (5 self)
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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 αinvariant 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.
Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations
 Math. Res. Letters
, 1998
"... 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 ..."
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Cited by 17 (8 self)
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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
Nonabelian Cohomology of Abelian Anosov Actions
"... We develop a new technique for calculating the first cohomology of certain classes of actions of higherrank abelian groups (Z k and R k , k 2) with values in a linear Lie group. In this paper we consider the discretetime case. Our results apply to cocycles of different regularity, from Holde ..."
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Cited by 17 (11 self)
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We develop a new technique for calculating the first cohomology of certain classes of actions of higherrank abelian groups (Z k and R k , k 2) with values in a linear Lie group. In this paper we consider the discretetime case. Our results apply to cocycles of different regularity, from Holder to smooth and realanalytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.
G.A.Margulis, Local rigidity for affine actions of higher rank Lie groups and their lattices, preprint, 54 pages, available at http://comet.lehman.cuny.edu/fisher/, submitted Annals Math
"... Abstract. Let J be a semisimple Lie group with all simple factors of real rank at least two. Let Γ < J be a lattice. We prove a very general local rigidity result about actions of J or Γ. This shows that almost all socalled “standard actions ” are locally rigid. As a special case, we see that any a ..."
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Cited by 15 (8 self)
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Abstract. Let J be a semisimple Lie group with all simple factors of real rank at least two. Let Γ < J be a lattice. We prove a very general local rigidity result about actions of J or Γ. This shows that almost all socalled “standard actions ” are locally rigid. As a special case, we see that any action of Γ by toral automorphisms is locally rigid. More generally, given a manifold M on which Γ acts isometrically and a torus T n on which it acts by automorphisms, we show that the diagonal action on T n ×M is locally rigid. This paper is the culmination of a series of papers and depends heavily on our work in [FM1, FM2]. The reader willing to accept the main results of those papers as “black boxes ” should be able to read the present paper without referring to them. 1.
On the classification of Cartan actions
 Geometric And Functional Analysis
"... Abstract. We study higher rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by R k with k ≥ 3 whose oneparameter groups act transitively as well as nondegenerate totally nonsymplectic Z kactions for k ≥ 3. 1. ..."
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Cited by 14 (6 self)
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Abstract. We study higher rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by R k with k ≥ 3 whose oneparameter groups act transitively as well as nondegenerate totally nonsymplectic Z kactions for k ≥ 3. 1.