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

Pages: 985 - 1005, Volume 13, Issue 4, November 2005

doi:10.3934/dcds.2005.13.985       Abstract        Full Text (279.5K)       Related Articles

Danijela Damjanović - Erwin Schroedinger Institute, Boltzmanngasse 9, A-1090 Vienna, Austria (email)
Anatole Katok - Department of Mathematics, Penn State University, University Park, State College, PA 16802, United States (email)

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 $SL_n$.

Keywords:  Cocycles, rigidity, Weyl chamber flow, partial hyperbolicity.
Mathematics Subject Classification:  37D40, 37C85.

Received: November 2004;      Revised: May 2005;      Published: August 2005.