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 $SL_n$.
Mathematics Subject Classification:
Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems - A,
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures.
Ergodicity and partial hyperbolicity on Seifert manifolds.
Journal of Modern Dynamics,
2020, 16: 331-348.
Hua Qiu, Zheng-An Yao.
The regularized Boussinesq equations with partial dissipations in dimension two.
Electronic Research Archive,
A brief and personal history of stochastic partial differential equations.
Discrete & Continuous Dynamical Systems - A,
Yueyang Zheng, Jingtao Shi.
A stackelberg game of backward stochastic differential equations with partial information.
Mathematical Control & Related Fields,