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).

[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.

[1]

Genady Ya. Grabarnik, Misha Guysinsky. Livšic theorem for banach rings. Discrete & Continuous Dynamical Systems - A, 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]

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

[4]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[5]

Daniel Guan. Modification and the cohomology groups of compact solvmanifolds. Electronic Research Announcements, 2007, 13: 74-81.

[6]

Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245

[7]

Huai-Dong Cao and Jian Zhou. On quantum de Rham cohomology theory. Electronic Research Announcements, 1999, 5: 24-34.

[8]

Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075

[9]

Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271

[10]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403

[11]

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

[12]

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

[13]

Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765

[14]

Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471

[15]

Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193

[16]

Nikolaos Karaliolios. Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups. Journal of Modern Dynamics, 2017, 11: 125-142. doi: 10.3934/jmd.2017006

[17]

Andrey Gogolev. Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori. Journal of Modern Dynamics, 2008, 2 (4) : 645-700. doi: 10.3934/jmd.2008.2.645

[18]

Andrey Gogolev, Misha Guysinsky. $C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1/2) : 183-200. doi: 10.3934/dcds.2008.22.183

[19]

Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817

[20]

Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]