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

Lipschitz continuous invariant forms for algebraic Anosov systems

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.
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, 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.   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.  doi: doi:10.1017/S0143385707000326.  Google Scholar

[5]

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

[8]

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

[9]

S. Hurder and A. Katok, Differentiability, rigidity, and Godbillon-Vey classes for Anosov flows,, Pub\-li\-ca\-tions Mathé\-ma\-tiques de l'Institut des Hautes \'Etudes Scientifiques, 72 (1990), 5.  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 (1995).   Google Scholar

[11]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", Interscience Publishers, (1963).   Google Scholar

[12]

W. F. Pfeffer, "Derivation and Integration,", Cambridge Tracts in Mathematics, (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, 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.   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.  doi: doi:10.1017/S0143385707000326.  Google Scholar

[5]

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

[8]

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

[9]

S. Hurder and A. Katok, Differentiability, rigidity, and Godbillon-Vey classes for Anosov flows,, Pub\-li\-ca\-tions Mathé\-ma\-tiques de l'Institut des Hautes \'Etudes Scientifiques, 72 (1990), 5.  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 (1995).   Google Scholar

[11]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", Interscience Publishers, (1963).   Google Scholar

[12]

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

[1]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[2]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[3]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[4]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

[5]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[6]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[7]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409

[8]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[9]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[10]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[11]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[12]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030

[13]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

[14]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408

[15]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[16]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[17]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083

[18]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341

[19]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230

[20]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]