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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 30332-0160, United States |
2. | Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240 |
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, Dordrecht, 1997. |
[2] |
H. Bercovici and V. Niţică, A Banach algebra version of the Livsic theorem, Discrete Contin. Dynam. Systems, 4 (1998), 523-534.
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), 329-360.
doi: 10.1512/iumj.2003.52.2407. |
[4] |
B. Kalinin, Livsic theorem for matrix cocycles, Annals of Mathematics, to appear. |
[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), 537-611.
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), 157-184. |
[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), 1055-1100.
doi: 10.1017/S014338570900039X. |
[8] |
M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152.
doi: 10.1090/S0002-9904-1969-12184-1. |
[9] |
A. N. Livšic, Certain properties of the homology of $Y$-systems, Mat. Zametki, 10 (1971), 555-564. |
[10] |
A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320. |
[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), 751-810. |
[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), 325-333. |
show all references
References:
[1] |
A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Kluwer Academic Publishers Group, Dordrecht, 1997. |
[2] |
H. Bercovici and V. Niţică, A Banach algebra version of the Livsic theorem, Discrete Contin. Dynam. Systems, 4 (1998), 523-534.
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), 329-360.
doi: 10.1512/iumj.2003.52.2407. |
[4] |
B. Kalinin, Livsic theorem for matrix cocycles, Annals of Mathematics, to appear. |
[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), 537-611.
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), 157-184. |
[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), 1055-1100.
doi: 10.1017/S014338570900039X. |
[8] |
M. W. Hirsch and C. C. Pugh, Stable manifolds for hyperbolic sets, Bull. Amer. Math. Soc., 75 (1969), 149-152.
doi: 10.1090/S0002-9904-1969-12184-1. |
[9] |
A. N. Livšic, Certain properties of the homology of $Y$-systems, Mat. Zametki, 10 (1971), 555-564. |
[10] |
A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1296-1320. |
[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), 751-810. |
[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), 325-333. |
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