# American Institute of Mathematical Sciences

July  2011, 29(3): 1141-1154. doi: 10.3934/dcds.2011.29.1141

## 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 30332-0160, United States 2 Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240

Received  January 2010 Revised  June 2010 Published  November 2010

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.

Citation: Rafael de la Llave, A. Windsor. Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1141-1154. doi: 10.3934/dcds.2011.29.1141
##### References:
 [1] A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Kluwer Academic Publishers Group, (1997).   Google Scholar [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.  Google Scholar [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.  Google Scholar [4] B. Kalinin, Livsic theorem for matrix cocycles,, Annals of Mathematics, ().   Google Scholar [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.  Google Scholar [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.   Google Scholar [7] R. de la Llave and A. Windsor, Livšic theorems for non-commutative 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.  Google Scholar [8] M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets,, Bull. Amer. Math. Soc., 75 (1969), 149.  doi: 10.1090/S0002-9904-1969-12184-1.  Google Scholar [9] A. N. Livšic, Certain properties of the homology of $Y$-systems,, Mat. Zametki, 10 (1971), 555.   Google Scholar [10] A. N. Livšic, Cohomology of dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296.   Google Scholar [11] V. Niţică and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices,, Duke Math. J., 79 (1995), 751.   Google Scholar [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.   Google Scholar

show all references

##### References:
 [1] A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Kluwer Academic Publishers Group, (1997).   Google Scholar [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.  Google Scholar [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.  Google Scholar [4] B. Kalinin, Livsic theorem for matrix cocycles,, Annals of Mathematics, ().   Google Scholar [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.  Google Scholar [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.   Google Scholar [7] R. de la Llave and A. Windsor, Livšic theorems for non-commutative 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.  Google Scholar [8] M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets,, Bull. Amer. Math. Soc., 75 (1969), 149.  doi: 10.1090/S0002-9904-1969-12184-1.  Google Scholar [9] A. N. Livšic, Certain properties of the homology of $Y$-systems,, Mat. Zametki, 10 (1971), 555.   Google Scholar [10] A. N. Livšic, Cohomology of dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296.   Google Scholar [11] V. Niţică and A. Török, Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices,, Duke Math. J., 79 (1995), 751.   Google Scholar [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.   Google Scholar
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