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

Zygmund strong foliations in higher dimension

Yong Fang 1, , Patrick Foulon 2, and  Boris Hasselblatt 3,

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.
Keywords: Zygmund regularity, quasiconformal, geometric rigidity, smooth rigidity, Anosov flow, foliations., strong invariant subbundles.
Mathematics Subject Classification: Primary: 37D20, 37D40; Secondary: 53C24, 53D2.
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), 719-729.  Google Scholar

[2]

D. DeLatte, Nonstationary normal forms and cocycle invariants, Random and Computational Dynamics, 1 (1992/93), 229-259; On normal forms in Hamiltonian dynamics, a new approach to some convergence questions, Ergodic Theory and Dynamical Systems, 15 (1995), 49-66.  Google Scholar

[3]

Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory and Dynamical Systems, 27 (2007), 1773-1802. doi: doi:10.1017/S0143385707000326.  Google Scholar

[4]

Y. Fang, Smooth rigidity of quasiconformal Anosov flows, Ergodic Theory and Dynamical Systems, 24 (2004), 1937-1959. doi: doi:10.1017/S0143385704000264.  Google Scholar

[5]

Y. Fang, Thermodynamic invariants of Anosov flows and rigidity, Discrete Contin. Dyn. Syst., 24 (2009), 1185-1204. doi: doi:10.3934/dcds.2009.24.1185.  Google Scholar

[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), 427-432. doi: doi:10.1017/S0143385700005071.  Google Scholar

[7]

P. Foulon and B. Hasselblatt, Zygmund strong foliations, Israel Journal of Mathematics, 138 (2003), 157-188. doi: doi:10.1007/BF02783424.  Google Scholar

[8]

P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems, Journal of Modern Dynamics, 4 (2010), 571-584. Google Scholar

[9]

M. Guysinsky, The theory of nonstationary normal forms, Ergod. Theory and Dyn. Syst., 22 (2002), 845-862. doi: doi:10.1017/S0143385702000421.  Google Scholar

[10]

M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations, Math. Research Letters, 5 (1998), 149-163.  Google Scholar

[11]

J. S. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles, Bulletin de la Société Mathématique de France, 29 (1901), 224-228. Google Scholar

[12]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory and Dynamical Systems, 14 (1994), 645-666.  Google Scholar

[13]

U. Hamenstädt, Invariant two-forms for geodesic flows, Mathematische Annalen, 101 (1995), 677-698. doi: doi:10.1007/BF01446654.  Google Scholar

[14]

B. Hasselblatt, Hyperbolic dynamics, in "Handbook of Dynamical Systems," 1A, North Holland, (2002), 239-319.  Google Scholar

[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), 5-61. doi: doi:10.1007/BF02699130.  Google Scholar

[16]

M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30.  Google Scholar

[17]

A. Katok and B. Hasselblatt, "Introduction To The Modern Theory Of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.  Google Scholar

[18]

A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math., 75 (1991), 203-241.  Google Scholar

[19]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.  Google Scholar

[20]

G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307-316. doi: doi:10.1017/S0305004105008613.  Google Scholar

[21]

G. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics, Discrete Contin. Dynam. Systems, 5 (1999), 639-650. doi: doi:10.3934/dcds.1999.5.639.  Google Scholar

[22]

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

[23]

V. Sadovskaya, On uniformly quasiconformal Anosov systems, Mathematical Research Letters, 12 (2005), 425-441.  Google Scholar

[24]

C. Yue, Quasiconformality in the geodesic flow of negatively curved manifolds, Geom. Funct. Anal., 6 (1996), 740-750. doi: doi:10.1007/BF02247120.  Google Scholar

[25]

A. S. Zygmund, "Trigonometric Series," Cambridge University Press, 1959 (and 1968, 1979, 1988), revised version of "Trigonometrical Series," Monografje Matematyczne, Tom V., Warszawa-Lwow, 1935.  Google Scholar

show all references

References:
[1]

N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows, Mathematics Research Letters, 12 (2005), 719-729.  Google Scholar

[2]

D. DeLatte, Nonstationary normal forms and cocycle invariants, Random and Computational Dynamics, 1 (1992/93), 229-259; On normal forms in Hamiltonian dynamics, a new approach to some convergence questions, Ergodic Theory and Dynamical Systems, 15 (1995), 49-66.  Google Scholar

[3]

Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory and Dynamical Systems, 27 (2007), 1773-1802. doi: doi:10.1017/S0143385707000326.  Google Scholar

[4]

Y. Fang, Smooth rigidity of quasiconformal Anosov flows, Ergodic Theory and Dynamical Systems, 24 (2004), 1937-1959. doi: doi:10.1017/S0143385704000264.  Google Scholar

[5]

Y. Fang, Thermodynamic invariants of Anosov flows and rigidity, Discrete Contin. Dyn. Syst., 24 (2009), 1185-1204. doi: doi:10.3934/dcds.2009.24.1185.  Google Scholar

[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), 427-432. doi: doi:10.1017/S0143385700005071.  Google Scholar

[7]

P. Foulon and B. Hasselblatt, Zygmund strong foliations, Israel Journal of Mathematics, 138 (2003), 157-188. doi: doi:10.1007/BF02783424.  Google Scholar

[8]

P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems, Journal of Modern Dynamics, 4 (2010), 571-584. Google Scholar

[9]

M. Guysinsky, The theory of nonstationary normal forms, Ergod. Theory and Dyn. Syst., 22 (2002), 845-862. doi: doi:10.1017/S0143385702000421.  Google Scholar

[10]

M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations, Math. Research Letters, 5 (1998), 149-163.  Google Scholar

[11]

J. S. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles, Bulletin de la Société Mathématique de France, 29 (1901), 224-228. Google Scholar

[12]

B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory and Dynamical Systems, 14 (1994), 645-666.  Google Scholar

[13]

U. Hamenstädt, Invariant two-forms for geodesic flows, Mathematische Annalen, 101 (1995), 677-698. doi: doi:10.1007/BF01446654.  Google Scholar

[14]

B. Hasselblatt, Hyperbolic dynamics, in "Handbook of Dynamical Systems," 1A, North Holland, (2002), 239-319.  Google Scholar

[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), 5-61. doi: doi:10.1007/BF02699130.  Google Scholar

[16]

M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math., 19 (1993), 1-30.  Google Scholar

[17]

A. Katok and B. Hasselblatt, "Introduction To The Modern Theory Of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.  Google Scholar

[18]

A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math., 75 (1991), 203-241.  Google Scholar

[19]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.  Google Scholar

[20]

G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307-316. doi: doi:10.1017/S0305004105008613.  Google Scholar

[21]

G. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics, Discrete Contin. Dynam. Systems, 5 (1999), 639-650. doi: doi:10.3934/dcds.1999.5.639.  Google Scholar

[22]

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

[23]

V. Sadovskaya, On uniformly quasiconformal Anosov systems, Mathematical Research Letters, 12 (2005), 425-441.  Google Scholar

[24]

C. Yue, Quasiconformality in the geodesic flow of negatively curved manifolds, Geom. Funct. Anal., 6 (1996), 740-750. doi: doi:10.1007/BF02247120.  Google Scholar

[25]

A. S. Zygmund, "Trigonometric Series," Cambridge University Press, 1959 (and 1968, 1979, 1988), revised version of "Trigonometrical Series," Monografje Matematyczne, Tom V., Warszawa-Lwow, 1935.  Google Scholar

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