89 research outputs found
Rigidity of real-analytic actions of on \T^n: A case of realization of Zimmer program
We prove that any real-analytic action of with standard
homotopy data that preserves an ergodic measure whose support is not
contained in a ball, is analytically conjugate on an open invariant set to the
standard linear action on the complement to a finite union of periodic orbits
Local Rigidity of Partially Hyperbolic Actions: Solution of the General Problem via KAM Method
We consider a broad class of partially hyperbolic algebraic actions of
higher-rank abelian groups. Those actions appear as restrictions of full Cartan
actions on homogeneous spaces of Lie groups and their factors by compact
subgroups of the centralizer. The common property of those actions is that
hyperbolic directions generate the whole tangent space. For these actions we
prove differentiable rigidity for perturbations of sufficiently high
regularity. The method of proof is KAM type iteration scheme. The principal
difference with previous work that used similar methods is very general nature
of our proofs: the only tool from analysis on groups is exponential decay of
matrix coefficients and no specific information about unitary representations
is required
Invariant measures and the set of exceptions to Littlewood's conjecture
We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic
under the action of the group A of positive diagonal matrices with positive
entropy. We apply this to prove that the set of exceptions to Littlewood's
conjecture has Hausdorff dimension zero.Comment: 48 page
Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature
We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χ_μ > 0 such that the Lyapunov exponents are −2χ_μ, −χ_μ, 0, χ_μ, and 2χ_μ, μ-almost everywhere.
As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature
Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups
We prove absolute continuity of "high entropy" hyperbolic invariant measures
for smooth actions of higher rank abelian groups assuming that there are no
proportional Lyapunov exponents. For actions on tori and infranilmanifolds
existence of an absolutely continuous invariant measure of this kind is
obtained for actions whose elements are homotopic to those of an action by
hyperbolic automorphisms with no multiple or proportional Lyapunov exponents.
In the latter case a form of rigidity is proved for certain natural classes of
cocycles over the action.Comment: 28 page
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