2010, 17: 80-89. doi: 10.3934/era.2010.17.80

Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

1. 

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

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 Published  October 2010

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin, and we prove new results of this smooth rigidity type for a class of flows.
    For a transversely symplectic uniformly quasiconformal $C^2$ Anosov flow on a compact Riemannian manifold we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form.
    We obtain a pertinent geometric rigidity result: Uniformly quasiconformal magnetic flows are geodesic flows of hyperbolic metrics.
    Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.
Citation: 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
References:
[1]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions de Liapounov différentiables. I.,, Hyperbolic behaviour of dynamical systems (Paris, 53 (1990), 395.   Google Scholar

[2]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions stable et instable différentiables,, Journal of the American Mathematical Society, 5 (1992), 33.  doi: 10.2307/2152750.  Google Scholar

[3]

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

[4]

S. Dubrovskiy, Stokes Theorem for Lipschitz forms on a smooth manifold,, \arXiv{0805.4144v1}, ().   Google Scholar

[5]

Y. Fang, On the rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 27 (2007), 1773.  doi: 10.1017/S0143385707000326.  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 {\bf 9} (1989), 9 (1989), 427.  doi: 10.1017/S0143385700005071.  Google Scholar

[7]

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

[8]

Y. Fang, P. Foulon and B. Hasselblatt, Zygmund foliations in higher dimension,, Journal of Modern Dynamics, 4 (2010), 549.   Google Scholar

[9]

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

[10]

V. M. Goldshtein, V. I. Kuzminov and I. A. Shvedov, Differential forms on a Lipschitz manifold,, Sibirsk. Mat. Zh., 23 (1982), 16.   Google Scholar

[11]

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

[12]

B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239.  doi: 10.1016/S1874-575X(02)80005-4.  Google Scholar

[13]

S. Hurder and Anatole Katok, Differentiability, rigidity, and Godbillon-Vey classes for Anosov flows,, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 5.   Google Scholar

[14]

A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,, Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[15]

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

[16]

A. S. Zygmund, Trigonometric series,, Cambridge University Press, (1959).   Google Scholar

show all references

References:
[1]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions de Liapounov différentiables. I.,, Hyperbolic behaviour of dynamical systems (Paris, 53 (1990), 395.   Google Scholar

[2]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions stable et instable différentiables,, Journal of the American Mathematical Society, 5 (1992), 33.  doi: 10.2307/2152750.  Google Scholar

[3]

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

[4]

S. Dubrovskiy, Stokes Theorem for Lipschitz forms on a smooth manifold,, \arXiv{0805.4144v1}, ().   Google Scholar

[5]

Y. Fang, On the rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 27 (2007), 1773.  doi: 10.1017/S0143385707000326.  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 {\bf 9} (1989), 9 (1989), 427.  doi: 10.1017/S0143385700005071.  Google Scholar

[7]

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

[8]

Y. Fang, P. Foulon and B. Hasselblatt, Zygmund foliations in higher dimension,, Journal of Modern Dynamics, 4 (2010), 549.   Google Scholar

[9]

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

[10]

V. M. Goldshtein, V. I. Kuzminov and I. A. Shvedov, Differential forms on a Lipschitz manifold,, Sibirsk. Mat. Zh., 23 (1982), 16.   Google Scholar

[11]

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

[12]

B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239.  doi: 10.1016/S1874-575X(02)80005-4.  Google Scholar

[13]

S. Hurder and Anatole Katok, Differentiability, rigidity, and Godbillon-Vey classes for Anosov flows,, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 72 (1990), 5.   Google Scholar

[14]

A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,, Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[15]

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

[16]

A. S. Zygmund, Trigonometric series,, Cambridge University Press, (1959).   Google Scholar

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