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Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups
1.  School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 303320160, United States 
2.  Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 381523240 
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.
References:
[1] 
A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Kluwer Academic Publishers Group, (1997). 
[2] 
H. Bercovici and V. Niţică, A Banach algebra version of the Livsic theorem,, Discrete Contin. Dynam. Systems, 4 (1998), 523. doi: 10.3934/dcds.1998.4.523. 
[3] 
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters,, Indiana Univ. Math. J., 52 (2003), 329. doi: 10.1512/iumj.2003.52.2407. 
[4] 
B. Kalinin, Livsic theorem for matrix cocycles,, Annals of Mathematics, (). 
[5] 
R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation,, Ann. of Math. (2), 123 (1986), 537. doi: 10.2307/1971334. 
[6] 
R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157. 
[7] 
R. de la Llave and A. Windsor, Livšic theorems for noncommutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems,, Ergodic Theory Dynam. Systems, 30 (2010), 1055. doi: 10.1017/S014338570900039X. 
[8] 
M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets,, Bull. Amer. Math. Soc., 75 (1969), 149. doi: 10.1090/S000299041969121841. 
[9] 
A. N. Livšic, Certain properties of the homology of $Y$systems,, Mat. Zametki, 10 (1971), 555. 
[10] 
A. N. Livšic, Cohomology of dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296. 
[11] 
V. Niţică and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higherrank lattices,, Duke Math. J., 79 (1995), 751. 
[12] 
V. Niţică and A. Török, Regularity results for the solutions of the Livsic cohomology equation with values in diffeomorphism groups,, Ergodic Theory Dynam. Systems, 16 (1996), 325. 
show all references
References:
[1] 
A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Kluwer Academic Publishers Group, (1997). 
[2] 
H. Bercovici and V. Niţică, A Banach algebra version of the Livsic theorem,, Discrete Contin. Dynam. Systems, 4 (1998), 523. doi: 10.3934/dcds.1998.4.523. 
[3] 
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters,, Indiana Univ. Math. J., 52 (2003), 329. doi: 10.1512/iumj.2003.52.2407. 
[4] 
B. Kalinin, Livsic theorem for matrix cocycles,, Annals of Mathematics, (). 
[5] 
R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation,, Ann. of Math. (2), 123 (1986), 537. doi: 10.2307/1971334. 
[6] 
R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157. 
[7] 
R. de la Llave and A. Windsor, Livšic theorems for noncommutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems,, Ergodic Theory Dynam. Systems, 30 (2010), 1055. doi: 10.1017/S014338570900039X. 
[8] 
M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets,, Bull. Amer. Math. Soc., 75 (1969), 149. doi: 10.1090/S000299041969121841. 
[9] 
A. N. Livšic, Certain properties of the homology of $Y$systems,, Mat. Zametki, 10 (1971), 555. 
[10] 
A. N. Livšic, Cohomology of dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296. 
[11] 
V. Niţică and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higherrank lattices,, Duke Math. J., 79 (1995), 751. 
[12] 
V. Niţică and A. Török, Regularity results for the solutions of the Livsic cohomology equation with values in diffeomorphism groups,, Ergodic Theory Dynam. Systems, 16 (1996), 325. 
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