Mathematics – Dynamical Systems
Scientific paper
2008-09-08
Mathematics
Dynamical Systems
Scientific paper
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 $\varphi \in C^{\reg}(M,\Diff^r(N))$.
la Llave Rafael de
Windsor Alistair
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