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.