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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
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