2010, 4(3): 549-569. doi: 10.3934/jmd.2010.4.549

Zygmund strong foliations in higher dimension

1. 

Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

2. 

Institut de Recherche Mathematique Avancée, UMR 7501 du Centre National de la Recherche Scientifique, 7 Rue René Descartes, 67084, Strasbourg Cedex

3. 

Department of Mathematics, Tufts University, Medford, MA 02155

Received  May 2010 Revised  June 2010 Published  October 2010

For a compact Riemannian manifold $M$, $k\ge2$ and a uniformly quasiconformal transversely symplectic $C^k$ Anosov flow $\varphi$:$\R\times M\to M$ we define the longitudinal KAM-cocycle and use it to prove a rigidity result: $E^u\oplus E^s$ is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that $E^u\oplus E^s$ is Lipschitz-continuous. Results proved elsewhere then imply that the flow is smoothly conjugate to an algebraic one.
Citation: Yong Fang, Patrick Foulon, Boris Hasselblatt. Zygmund strong foliations in higher dimension. Journal of Modern Dynamics, 2010, 4 (3) : 549-569. doi: 10.3934/jmd.2010.4.549
References:
[1]

N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows,, Mathematics Research Letters, 12 (2005), 719.

[2]

D. DeLatte, Nonstationary normal forms and cocycle invariants,, Random and Computational Dynamics, 1 (1995), 229.

[3]

Y. Fang, On the rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 27 (2007), 1773. doi: doi:10.1017/S0143385707000326.

[4]

Y. Fang, Smooth rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 24 (2004), 1937. doi: doi:10.1017/S0143385704000264.

[5]

Y. Fang, Thermodynamic invariants of Anosov flows and rigidity,, Discrete Contin. Dyn. Syst., 24 (2009), 1185. doi: doi:10.3934/dcds.2009.24.1185.

[6]

R. Feres and A. Katok, Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows,, Ergodic Theory and Dynamical Systems, 9 (1989), 427. doi: doi:10.1017/S0143385700005071.

[7]

P. Foulon and B. Hasselblatt, Zygmund strong foliations,, Israel Journal of Mathematics, 138 (2003), 157. doi: doi:10.1007/BF02783424.

[8]

P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems,, Journal of Modern Dynamics, 4 (2010), 571.

[9]

M. Guysinsky, The theory of nonstationary normal forms,, Ergod. Theory and Dyn. Syst., 22 (2002), 845. doi: doi:10.1017/S0143385702000421.

[10]

M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations,, Math. Research Letters, 5 (1998), 149.

[11]

J. S. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles,, Bulletin de la Société Mathématique de France, 29 (1901), 224.

[12]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory and Dynamical Systems, 14 (1994), 645.

[13]

U. Hamenstädt, Invariant two-forms for geodesic flows,, Mathematische Annalen, 101 (1995), 677. doi: doi:10.1007/BF01446654.

[14]

B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239.

[15]

S. Hurder and A. Katok, Differentiability, rigidity, and Godbillon-Vey classes for Anosov flows,, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 5. doi: doi:10.1007/BF02699130.

[16]

M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows,, Japan. J. Math., 19 (1993), 1.

[17]

A. Katok and B. Hasselblatt, "Introduction To The Modern Theory Of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).

[18]

A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75 (1991), 203.

[19]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157.

[20]

G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307. doi: doi:10.1017/S0305004105008613.

[21]

G. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics,, Discrete Contin. Dynam. Systems, 5 (1999), 639. doi: doi:10.3934/dcds.1999.5.639.

[22]

G. Paternain and W. J. Merry, Stability of Anosov Hamiltonian structures,, , ().

[23]

V. Sadovskaya, On uniformly quasiconformal Anosov systems,, Mathematical Research Letters, 12 (2005), 425.

[24]

C. Yue, Quasiconformality in the geodesic flow of negatively curved manifolds,, Geom. Funct. Anal., 6 (1996), 740. doi: doi:10.1007/BF02247120.

[25]

A. S. Zygmund, "Trigonometric Series,", Cambridge University Press, (1959).

show all references

References:
[1]

N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows,, Mathematics Research Letters, 12 (2005), 719.

[2]

D. DeLatte, Nonstationary normal forms and cocycle invariants,, Random and Computational Dynamics, 1 (1995), 229.

[3]

Y. Fang, On the rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 27 (2007), 1773. doi: doi:10.1017/S0143385707000326.

[4]

Y. Fang, Smooth rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 24 (2004), 1937. doi: doi:10.1017/S0143385704000264.

[5]

Y. Fang, Thermodynamic invariants of Anosov flows and rigidity,, Discrete Contin. Dyn. Syst., 24 (2009), 1185. doi: doi:10.3934/dcds.2009.24.1185.

[6]

R. Feres and A. Katok, Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows,, Ergodic Theory and Dynamical Systems, 9 (1989), 427. doi: doi:10.1017/S0143385700005071.

[7]

P. Foulon and B. Hasselblatt, Zygmund strong foliations,, Israel Journal of Mathematics, 138 (2003), 157. doi: doi:10.1007/BF02783424.

[8]

P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems,, Journal of Modern Dynamics, 4 (2010), 571.

[9]

M. Guysinsky, The theory of nonstationary normal forms,, Ergod. Theory and Dyn. Syst., 22 (2002), 845. doi: doi:10.1017/S0143385702000421.

[10]

M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations,, Math. Research Letters, 5 (1998), 149.

[11]

J. S. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles,, Bulletin de la Société Mathématique de France, 29 (1901), 224.

[12]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory and Dynamical Systems, 14 (1994), 645.

[13]

U. Hamenstädt, Invariant two-forms for geodesic flows,, Mathematische Annalen, 101 (1995), 677. doi: doi:10.1007/BF01446654.

[14]

B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239.

[15]

S. Hurder and A. Katok, Differentiability, rigidity, and Godbillon-Vey classes for Anosov flows,, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 5. doi: doi:10.1007/BF02699130.

[16]

M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows,, Japan. J. Math., 19 (1993), 1.

[17]

A. Katok and B. Hasselblatt, "Introduction To The Modern Theory Of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).

[18]

A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75 (1991), 203.

[19]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157.

[20]

G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307. doi: doi:10.1017/S0305004105008613.

[21]

G. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics,, Discrete Contin. Dynam. Systems, 5 (1999), 639. doi: doi:10.3934/dcds.1999.5.639.

[22]

G. Paternain and W. J. Merry, Stability of Anosov Hamiltonian structures,, , ().

[23]

V. Sadovskaya, On uniformly quasiconformal Anosov systems,, Mathematical Research Letters, 12 (2005), 425.

[24]

C. Yue, Quasiconformality in the geodesic flow of negatively curved manifolds,, Geom. Funct. Anal., 6 (1996), 740. doi: doi:10.1007/BF02247120.

[25]

A. S. Zygmund, "Trigonometric Series,", Cambridge University Press, (1959).

[1]

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

[2]

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

[3]

Boris Hasselblatt. Critical regularity of invariant foliations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931

[4]

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

[5]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123

[6]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80

[7]

Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377

[8]

Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873

[9]

A. Kononenko. Twisted cocycles and rigidity problems. Electronic Research Announcements, 1995, 1: 26-34.

[10]

Jean Dolbeault, Maria J. Esteban, Gaspard Jankowiak. Onofri inequalities and rigidity results. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3059-3078. doi: 10.3934/dcds.2017131

[11]

Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207

[12]

David Constantine. 2-Frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature. Journal of Modern Dynamics, 2008, 2 (4) : 719-740. doi: 10.3934/jmd.2008.2.719

[13]

Plamen Stefanov and Gunther Uhlmann. Recent progress on the boundary rigidity problem. Electronic Research Announcements, 2005, 11: 64-70.

[14]

Ralf Spatzier. On the work of Rodriguez Hertz on rigidity in dynamics. Journal of Modern Dynamics, 2016, 10: 191-207. doi: 10.3934/jmd.2016.10.191

[15]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271

[16]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

[17]

Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98.

[18]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639

[19]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3/4) : 499-548. doi: 10.3934/jmd.2014.8.499

[20]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

2016 Impact Factor: 0.706

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]