On the cohomology class of fiber-bunched cocycles on semi simple Lie groups
Summary: 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.
scientific article; zbMATH DE number 7318478
scientific article