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.

[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.

[3]

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

[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. 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 {\bf 9} (1989), 9 (1989), 427. doi: 10.1017/S0143385700005071.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239. 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.

[14]

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

[15]

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

[16]

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

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.

[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.

[3]

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

[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. 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 {\bf 9} (1989), 9 (1989), 427. doi: 10.1017/S0143385700005071.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239. 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.

[14]

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

[15]

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

[16]

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

[1]

Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471

[2]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[3]

Yong Fang. Rigidity of Hamenstädt metrics of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1271-1278. doi: 10.3934/dcds.2016.36.1271

[4]

Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123

[5]

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

[6]

Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377

[7]

Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873

[8]

A. Kononenko. Twisted cocycles and rigidity problems. Electronic Research Announcements, 1995, 1: 26-34.

[9]

Jean Dolbeault, Maria J. Esteban, Gaspard Jankowiak. Onofri inequalities and rigidity results. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3059-3078. doi: 10.3934/dcds.2017131

[10]

Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207

[11]

David Constantine. 2-Frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature. Journal of Modern Dynamics, 2008, 2 (4) : 719-740. doi: 10.3934/jmd.2008.2.719

[12]

Plamen Stefanov and Gunther Uhlmann. Recent progress on the boundary rigidity problem. Electronic Research Announcements, 2005, 11: 64-70.

[13]

Ralf Spatzier. On the work of Rodriguez Hertz on rigidity in dynamics. Journal of Modern Dynamics, 2016, 10: 191-207. doi: 10.3934/jmd.2016.10.191

[14]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271

[15]

Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

[16]

Bassam Fayad, Raphaël Krikorian. Rigidity results for quasiperiodic SL(2, R)-cocycles. Journal of Modern Dynamics, 2009, 3 (4) : 479-510. doi: 10.3934/jmd.2009.3.479

[17]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499

[18]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[19]

Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004

[20]

Agnieszka Badeńska. Measure rigidity for some transcendental meromorphic functions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2375-2402. doi: 10.3934/dcds.2012.32.2375

2016 Impact Factor: 0.483

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]