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July  2005, 13(4): 985-1005. doi: 10.3934/dcds.2005.13.985

Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions

Danijela Damjanović 1, and  Anatole Katok 2,

1. 

Erwin Schroedinger Institute, Boltzmanngasse 9, A-1090 Vienna, Austria
2. 

Department of Mathematics, Penn State University, University Park, State College, PA 16802

Received  November 2004 Revised  May 2005 Published  August 2005

We give a proof of cocycle rigidity in Hölder and smooth categories for Cartan actions on $SL(n, \mathbb R)$/$\Gamma$ and $SL(n, \mathbb C)$/$\Gamma$ for $n\ge 3$ and $\Gamma$ cocompact lattice, and for restrictions of those actions to subspaces which contain a two-dimensional plane in general position. This proof does not use harmonic analysis, it relies completely on the structure of stable and unstable foliations of the action. The key new ingredient is the use of the description of generating relations in the group $SL_n$.
Keywords: rigidity, Cocycles, partial hyperbolicity., Weyl chamber flow.
Mathematics Subject Classification: 37D40, 37C8.
Citation: Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
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