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

TL;DR

This paper investigates how solutions to cohomology equations over Anosov systems depend smoothly on parameters, establishing optimal regularity results with applications to rigidity, dynamical systems, and geometry.

Contribution

It proves that solutions to cohomology equations over Anosov diffeomorphisms depend smoothly on parameters, with optimal regularity results for equations valued in diffeomorphism groups.

Findings

Solutions depend smoothly on parameters as smoothly as the data

Optimal regularity results for solutions in diffeomorphism groups

Applications to rigidity theory and dynamical systems

Abstract

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 $\eta \in C^{\reg}(M,\Diff^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C^{\reg}(M,\Diff^1(N))$ solving \begin{equation*} \varphi_{f(x)} = \eta_x \circ \varphi_x \end{equation*} then in fact $\varphi \in C^{\reg}(M,\Diff^r(N))$.