# American Institute of Mathematical Sciences

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, 1990), Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 395-412. [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), 33-74. doi: 10.2307/2152750. [3] N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows, Mathematics Research Letters, 12 (2005), 719-729. [4] S. Dubrovskiy, Stokes Theorem for Lipschitz forms on a smooth manifold, arXiv:0805.4144v1 [5] Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory and Dynamical Systems, 27 (2007), 1773-1802. doi: 10.1017/S0143385707000326. [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: 10.1017/S0143385700005071. [7] P. Foulon and B. Hasselblatt, Zygmund strong foliations, Israel Journal of Mathematics, 138 (2003), 157-188. doi: 10.1007/BF02783424. [8] Y. Fang, P. Foulon and B. Hasselblatt, Zygmund foliations in higher dimension, Journal of Modern Dynamics, 4 (2010), 549-569. [9] P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems, Journal of Modern Dynamics, 4 (2010), 571-584. [10] V. M. Goldshtein, V. I. Kuzminov and I. A. Shvedov, Differential forms on a Lipschitz manifold, Sibirsk. Mat. Zh., 23 (1982), 16-30. [11] U. Hamenstädt, Invariant two-forms for geodesic flows, Mathematische Annalen, 101 (1995), 677-698. doi: 10.1007/BF01446654. [12] B. Hasselblatt, Hyperbolic dynamics, in "Handbook of Dynamical Systems," 1A, North Holland, (2002), 239-319. doi: 10.1016/S1874-575X(02)80005-4. [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), 5-61. [14] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995. [15] G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307-316. doi: 10.1017/S0305004105008613. [16] 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.

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##### References:
 [1] Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions de Liapounov différentiables. I., Hyperbolic behaviour of dynamical systems (Paris, 1990), Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 395-412. [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), 33-74. doi: 10.2307/2152750. [3] N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows, Mathematics Research Letters, 12 (2005), 719-729. [4] S. Dubrovskiy, Stokes Theorem for Lipschitz forms on a smooth manifold, arXiv:0805.4144v1 [5] Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory and Dynamical Systems, 27 (2007), 1773-1802. doi: 10.1017/S0143385707000326. [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: 10.1017/S0143385700005071. [7] P. Foulon and B. Hasselblatt, Zygmund strong foliations, Israel Journal of Mathematics, 138 (2003), 157-188. doi: 10.1007/BF02783424. [8] Y. Fang, P. Foulon and B. Hasselblatt, Zygmund foliations in higher dimension, Journal of Modern Dynamics, 4 (2010), 549-569. [9] P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems, Journal of Modern Dynamics, 4 (2010), 571-584. [10] V. M. Goldshtein, V. I. Kuzminov and I. A. Shvedov, Differential forms on a Lipschitz manifold, Sibirsk. Mat. Zh., 23 (1982), 16-30. [11] U. Hamenstädt, Invariant two-forms for geodesic flows, Mathematische Annalen, 101 (1995), 677-698. doi: 10.1007/BF01446654. [12] B. Hasselblatt, Hyperbolic dynamics, in "Handbook of Dynamical Systems," 1A, North Holland, (2002), 239-319. doi: 10.1016/S1874-575X(02)80005-4. [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), 5-61. [14] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995. [15] G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307-316. doi: 10.1017/S0305004105008613. [16] 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.
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