Cocycle rigidity and splitting for some discrete parabolic actions
Danijela Damjanović James Tanis
Discrete & Continuous Dynamical Systems - A 2014, 34(12): 5211-5227 doi: 10.3934/dcds.2014.34.5211
We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of $\mathbb{Z}^2$ parabolic actions on $(SL(2, \mathbb R)\times SL(2, \mathbb R))/\Gamma$, where the lattice $\Gamma$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.
keywords: splitting Cocycle rigidity discrete parabolic actions.
Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions
Danijela Damjanović Anatole Katok
Discrete & Continuous Dynamical Systems - A 2005, 13(4): 985-1005 doi: 10.3934/dcds.2005.13.985
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
Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions
Danijela Damjanović
Journal of Modern Dynamics 2007, 1(4): 665-688 doi: 10.3934/jmd.2007.1.665
We consider partially hyperbolic abelian algebraic higher-rank actions on compact homogeneous spaces obtained from simple split Lie groups of nonsymplectic type. We show that smooth, real-valued cocycles trivialize as well as small cocycles taking values in groups of diffeomorphisms of compact manifolds and some semisimple Lie groups. In the second part of the paper, we show local differentiable rigidity for such actions.
keywords: partial hyperbolicity abelian actions rigidity. cocycles

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