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Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups

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  • 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 cohomology 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 $\eta \in C$k+α$(M,$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 $\varphi\in C$k+α$(M,$Diff$^1(N))$ solving

    $ \varphi_{f(x)} = \eta_x \circ \varphi_x$

    then in fact $\varphi \in C$k+α$(M,$Diff$^r(N))$. The existence of this solutions for some range of regularities is studied in the literature.

    Mathematics Subject Classification: Primary: 58F15, 22E65, 58D05; Secondary: 37D20.


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