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

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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

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

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

Abstract
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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.

Keywords: Anosov diffeomorphisms, Cohomology equations, rigidity., diffeomorphism groups, Livšic theory.

Mathematics Subject Classification: Primary: 58F15, 22E65, 58D05; Secondary: 37D2.

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).
[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 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.
[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.
[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 higher-rank 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 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.
[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.
[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 higher-rank 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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