On the cohomology class of fiber-bunched cocycles on semi simple Lie groups

Abstract

We study the cohomological equation associated to linear cocycles on semi simple Lie groups $\mathcal{G}$ over hyperbolic dynamics. We give sufficient conditions for the solution of the cohomological equation of fiber-bunched cocycles to be unique and for the Hölder conjugacy class of the cocycle to coincide with $C^{\nu }(M,\mathcal{G})$. In particular, we prove that there exists an open and dense subset of the set $C_b^{\nu }(M,\mathcal{G})$ of fiber-bunched cocycles with trivial centralizer. As a consequence we deduce that the solutions of the {cohomological} equation of fiber-bunched cocycles form a finite abelian subgroup of $C_b^{\nu }(M,\mathcal{G})$ for an open and dense subset of fiber-bunched cocycles in $C_b^{\nu }(M,\mathcal{G})$. Some results on the centralizer of skew-products are also given.