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July  2010, 4(3): 571-584. doi: 10.3934/jmd.2010.4.571

Lipschitz continuous invariant forms for algebraic Anosov systems

Patrick Foulon 1, and  Boris Hasselblatt 2,

1. 

Institut de Recherche Mathematique Avancée, UMR 7501 du Centre National de la Recherche Scientifique, 7 Rue René Descartes, 67084, Strasbourg Cedex, France
2. 

Department of Mathematics, Tufts University, Medford, MA 02155, United States

Received  May 2010 Revised  September 2010 Published  October 2010

We prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz continuous invariant $1$-form. This has corollaries for rigidity of time-changes, and we give a particular application to geometric rigidity of quasiconformal Anosov flows.
   Several features of the reasoning are interesting; namely, the use of exterior calculus for Lipschitz continuous forms, the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.
Keywords: Lipschitz regularity, Anosov flow, invariant forms, smooth rigidity..
Mathematics Subject Classification: Primary: 37D20, 37D40; Secondary: 53C24, 53D2.
Citation: Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571
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.  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), 33-74.  Google Scholar

[3]

S. Dubrovskiy, Stokes Theorem for Lipschitz forms on a smooth manifold,, , ().   Google Scholar

[4]

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

[5]

P. Foulon and B. Hasselblatt, Zygmund strong foliations, Israel Journal of Mathematics, 138 (2003), 157-169. doi: doi:10.1007/BF02783424.  Google Scholar

[6]

Y. Fang, P. Foulon and B. Hasselblatt, Zygmund strong foliations in higher dimension, Journal of Modern Dynamics, 4 (2010), 549-569. Google Scholar

[7]

V. M. Goldshtein, V. I. Kuzminov and I. A. Shvedov, Differential forms on a Lipschitz manifold, Sibirsk. Mat. Zh., 23 (1982), 16-30.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," Interscience Publishers, New York - London, 1963.  Google Scholar

[12]

W. F. Pfeffer, "Derivation and Integration," Cambridge Tracts in Mathematics, Cambridge University Press, 2001.  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, 1990), Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 395-412.  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), 33-74.  Google Scholar

[3]

S. Dubrovskiy, Stokes Theorem for Lipschitz forms on a smooth manifold,, , ().   Google Scholar

[4]

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

[5]

P. Foulon and B. Hasselblatt, Zygmund strong foliations, Israel Journal of Mathematics, 138 (2003), 157-169. doi: doi:10.1007/BF02783424.  Google Scholar

[6]

Y. Fang, P. Foulon and B. Hasselblatt, Zygmund strong foliations in higher dimension, Journal of Modern Dynamics, 4 (2010), 549-569. Google Scholar

[7]

V. M. Goldshtein, V. I. Kuzminov and I. A. Shvedov, Differential forms on a Lipschitz manifold, Sibirsk. Mat. Zh., 23 (1982), 16-30.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," Interscience Publishers, New York - London, 1963.  Google Scholar

[12]

W. F. Pfeffer, "Derivation and Integration," Cambridge Tracts in Mathematics, Cambridge University Press, 2001.  Google Scholar

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