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

Danijela Damjanović; Anatole Katok

doi:10.3934/dcds.2005.13.985

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2005, Volume 13, Issue 4: 985-1005. Doi: 10.3934/dcds.2005.13.985

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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
2.
Department of Mathematics, Penn State University, University Park, State College, PA 16802

Received: October 31, 2004

Revised: April 30, 2005

Published: June 2005

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Abstract

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 $SL_n$.

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Mathematics Subject Classification: 37D40, 37C85.

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