# American Institute of Mathematical Sciences

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

 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 $SL_n$.
Citation: Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
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