Revista Matemática Iberoamericana


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Published online first: 2020-01-03
DOI: 10.4171/rmi/1160

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

Paulo Varandas

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.

Keywords: Cohomological equation, fiber-bunched cocycles, centralizers

Varandas Paulo: On the cohomology class of fiber-bunched cocycles on semi simple Lie groups. Rev. Mat. Iberoam. Electronically published on January 3, 2020. doi: 10.4171/rmi/1160 (to appear in print)