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On the cohomology class of fiber-bunched cocycles on semi simple Lie groups. (English) Zbl 1462.37061

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.

MSC:

37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C20 Generic properties, structural stability of dynamical systems
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
20D06 Simple groups: alternating groups and groups of Lie type
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