Periodic data rigidity for cocycles and hyperbolic automorphisms

We study cohomology of Holder continuous linear cocycles over a hyperbolic dynamical system and regularity of conjugacy between Anosov systems. For cocycles $A$ and $B$ with conjugate periodic data, we establish Holder cohomology under various conditions: the periodic data of $B$ has narrow spectrum and the periodic data conjugacy $C(p)$ is Holder continuous at a periodic point; $B$ is constant and the cocycles are measurably cohomologous; $B$ is constant and diagonalizable over $\mathbb C$ and either its Lyapunov spaces are at most two-dimensional or $C(p)$ is in a bounded set. We also prove that a topological conjugacy between a weakly irreducible hyperbolic automorphism $L$ and an Anosov diffeomorphism $f$ of $\mathbb T^d$ is smooth if their derivative cocycles $L$ and $Df$ are conjugate. Using this and our results on cohomology of cocycles we obtain global periodic data rigidity results for weakly irreducible hyperbolic automorphisms. In the argument we also establish differentiability of stable holonomies in low regularity setting.