Smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on diffeomorphism groups

de la Llave, Rafael; Windsor, Alistair

Smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on diffeomorphism groups

Authors

Description

We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $η\in C^{\reg}(M,\Diff^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < α\leq \Lip$. We show that if there exists a $φ\in C^{\reg}(M,\Diff^1(N))$ solving \begin{equation*} φ_{f(x)} = η_x \circ φ_x \end{equation*} then in fact $φ\in C^{\reg}(M,\Diff^r(N))$.