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

Previous Article
Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics
DCDS Home
This Issue
Next Article
Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation

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
Related Papers

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 and Continuous Dynamical Systems, 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, 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.

[1]	Genady Ya. Grabarnik, Misha Guysinsky. Livšic theorem for banach rings. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4379-4390. doi: 10.3934/dcds.2017187
[2]	João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
[3]	Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017
[4]	David Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. Journal of Modern Dynamics, 2007, 1 (1) : 61-92. doi: 10.3934/jmd.2007.1.61
[5]	Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185
[6]	Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
[7]	Daniel Guan. Modification and the cohomology groups of compact solvmanifolds. Electronic Research Announcements, 2007, 13: 74-81.
[8]	Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075
[9]	Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245
[10]	Huai-Dong Cao and Jian Zhou. On quantum de Rham cohomology theory. Electronic Research Announcements, 1999, 5: 24-34.
[11]	Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271
[12]	Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[13]	Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765
[14]	Matthieu Porte. Linear response for Dirac observables of Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1799-1819. doi: 10.3934/dcds.2019078
[15]	Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1
[16]	Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403
[17]	H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50
[18]	Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471
[19]	Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193
[20]	Nikolaos Karaliolios. Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups. Journal of Modern Dynamics, 2017, 11: 125-142. doi: 10.3934/jmd.2017006

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Rafael de la Llave A. Windsor
on Google Scholar
Rafael de la Llave A. Windsor

[Back to Top]