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

Rafael de la Llave; A. Windsor

doi:10.3934/dcds.2011.29.1141

Article Contents

2011, Volume 29, Issue 3: 1141-1154. Doi: 10.3934/dcds.2011.29.1141

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

Rafael de la Llave^1, and
A. Windsor^2,

1.
School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160
2.
Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240

Received: December 31, 2009

Revised: May 31, 2010

Published: August 2011

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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 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.

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

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References

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