Zygmund strong foliations in higher dimension

Previous Article
Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds
JMD Home
This Issue
Next Article
Lipschitz continuous invariant forms for algebraic Anosov systems

July 2010, 4(3): 549-569. doi: 10.3934/jmd.2010.4.549

Zygmund strong foliations in higher dimension

Yong Fang ^1, , Patrick Foulon ^2, and Boris Hasselblatt ^3,

1.	Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France
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 Revised June 2010 Published October 2010

Abstract
Related Papers

For a compact Riemannian manifold $M$, $k\ge2$ and a uniformly quasiconformal transversely symplectic $C^k$ Anosov flow $\varphi$:$\R\times M\to M$ we define the longitudinal KAM-cocycle and use it to prove a rigidity result: $E^u\oplus E^s$ is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that $E^u\oplus E^s$ is Lipschitz-continuous. Results proved elsewhere then imply that the flow is smoothly conjugate to an algebraic one.

Keywords: Zygmund regularity, quasiconformal, geometric rigidity, smooth rigidity, Anosov flow, foliations., strong invariant subbundles.

Mathematics Subject Classification: Primary: 37D20, 37D40; Secondary: 53C24, 53D2.

Citation: 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

References:

[1]	N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows,, Mathematics Research Letters, 12 (2005), 719. Google Scholar
[2]	D. DeLatte, Nonstationary normal forms and cocycle invariants,, Random and Computational Dynamics, 1 (1995), 229. Google Scholar
[3]	Y. Fang, On the rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 27 (2007), 1773. doi: doi:10.1017/S0143385707000326. Google Scholar
[4]	Y. Fang, Smooth rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 24 (2004), 1937. doi: doi:10.1017/S0143385704000264. Google Scholar
[5]	Y. Fang, Thermodynamic invariants of Anosov flows and rigidity,, Discrete Contin. Dyn. Syst., 24 (2009), 1185. doi: doi:10.3934/dcds.2009.24.1185. 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, 9 (1989), 427. doi: doi:10.1017/S0143385700005071. Google Scholar
[7]	P. Foulon and B. Hasselblatt, Zygmund strong foliations,, Israel Journal of Mathematics, 138 (2003), 157. doi: doi:10.1007/BF02783424. Google Scholar
[8]	P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems,, Journal of Modern Dynamics, 4 (2010), 571. Google Scholar
[9]	M. Guysinsky, The theory of nonstationary normal forms,, Ergod. Theory and Dyn. Syst., 22 (2002), 845. doi: doi:10.1017/S0143385702000421. Google Scholar
[10]	M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations,, Math. Research Letters, 5 (1998), 149. Google Scholar
[11]	J. S. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles,, Bulletin de la Société Mathématique de France, 29 (1901), 224. Google Scholar
[12]	B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory and Dynamical Systems, 14 (1994), 645. Google Scholar
[13]	U. Hamenstädt, Invariant two-forms for geodesic flows,, Mathematische Annalen, 101 (1995), 677. doi: doi:10.1007/BF01446654. Google Scholar
[14]	B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239. Google Scholar
[15]	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. doi: doi:10.1007/BF02699130. Google Scholar
[16]	M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows,, Japan. J. Math., 19 (1993), 1. Google Scholar
[17]	A. Katok and B. Hasselblatt, "Introduction To The Modern Theory Of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995). Google Scholar
[18]	A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75 (1991), 203. Google Scholar
[19]	R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157. Google Scholar
[20]	G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307. doi: doi:10.1017/S0305004105008613. Google Scholar
[21]	G. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics,, Discrete Contin. Dynam. Systems, 5 (1999), 639. doi: doi:10.3934/dcds.1999.5.639. Google Scholar
[22]	G. Paternain and W. J. Merry, Stability of Anosov Hamiltonian structures,, , (). Google Scholar
[23]	V. Sadovskaya, On uniformly quasiconformal Anosov systems,, Mathematical Research Letters, 12 (2005), 425. Google Scholar
[24]	C. Yue, Quasiconformality in the geodesic flow of negatively curved manifolds,, Geom. Funct. Anal., 6 (1996), 740. doi: doi:10.1007/BF02247120. Google Scholar
[25]	A. S. Zygmund, "Trigonometric Series,", Cambridge University Press, (1959). Google Scholar

show all references

References:

[1]	N. Dairbekov and G. Paternain, Longitudinal KAM cocycles and action spectra of magnetic flows,, Mathematics Research Letters, 12 (2005), 719. Google Scholar
[2]	D. DeLatte, Nonstationary normal forms and cocycle invariants,, Random and Computational Dynamics, 1 (1995), 229. Google Scholar
[3]	Y. Fang, On the rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 27 (2007), 1773. doi: doi:10.1017/S0143385707000326. Google Scholar
[4]	Y. Fang, Smooth rigidity of quasiconformal Anosov flows,, Ergodic Theory and Dynamical Systems, 24 (2004), 1937. doi: doi:10.1017/S0143385704000264. Google Scholar
[5]	Y. Fang, Thermodynamic invariants of Anosov flows and rigidity,, Discrete Contin. Dyn. Syst., 24 (2009), 1185. doi: doi:10.3934/dcds.2009.24.1185. 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, 9 (1989), 427. doi: doi:10.1017/S0143385700005071. Google Scholar
[7]	P. Foulon and B. Hasselblatt, Zygmund strong foliations,, Israel Journal of Mathematics, 138 (2003), 157. doi: doi:10.1007/BF02783424. Google Scholar
[8]	P. Foulon and B. Hasselblatt, Lipschitz continuous invariant forms for algebraic Anosov systems,, Journal of Modern Dynamics, 4 (2010), 571. Google Scholar
[9]	M. Guysinsky, The theory of nonstationary normal forms,, Ergod. Theory and Dyn. Syst., 22 (2002), 845. doi: doi:10.1017/S0143385702000421. Google Scholar
[10]	M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations,, Math. Research Letters, 5 (1998), 149. Google Scholar
[11]	J. S. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles,, Bulletin de la Société Mathématique de France, 29 (1901), 224. Google Scholar
[12]	B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations,, Ergodic Theory and Dynamical Systems, 14 (1994), 645. Google Scholar
[13]	U. Hamenstädt, Invariant two-forms for geodesic flows,, Mathematische Annalen, 101 (1995), 677. doi: doi:10.1007/BF01446654. Google Scholar
[14]	B. Hasselblatt, Hyperbolic dynamics,, in, 1A (2002), 239. Google Scholar
[15]	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. doi: doi:10.1007/BF02699130. Google Scholar
[16]	M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows,, Japan. J. Math., 19 (1993), 1. Google Scholar
[17]	A. Katok and B. Hasselblatt, "Introduction To The Modern Theory Of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995). Google Scholar
[18]	A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75 (1991), 203. Google Scholar
[19]	R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions,, Discrete Contin. Dynam. Systems, 5 (1999), 157. Google Scholar
[20]	G. P. Paternain, The longitudinal KAM-cocycle of a magnetic flow,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 307. doi: doi:10.1017/S0305004105008613. Google Scholar
[21]	G. Paternain, On two noteworthy deformations of negatively curved Riemannian metrics,, Discrete Contin. Dynam. Systems, 5 (1999), 639. doi: doi:10.3934/dcds.1999.5.639. Google Scholar
[22]	G. Paternain and W. J. Merry, Stability of Anosov Hamiltonian structures,, , (). Google Scholar
[23]	V. Sadovskaya, On uniformly quasiconformal Anosov systems,, Mathematical Research Letters, 12 (2005), 425. Google Scholar
[24]	C. Yue, Quasiconformality in the geodesic flow of negatively curved manifolds,, Geom. Funct. Anal., 6 (1996), 740. doi: doi:10.1007/BF02247120. Google Scholar
[25]	A. S. Zygmund, "Trigonometric Series,", Cambridge University Press, (1959). Google Scholar

[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]	Boris Hasselblatt. Critical regularity of invariant foliations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	A. Kononenko. Twisted cocycles and rigidity problems. Electronic Research Announcements, 1995, 1: 26-34.
[10]	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
[11]	Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075
[12]	Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[13]	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
[14]	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
[15]	Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98.
[16]	Plamen Stefanov and Gunther Uhlmann. Recent progress on the boundary rigidity problem. Electronic Research Announcements, 2005, 11: 64-70.
[17]	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
[18]	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
[19]	Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68
[20]	Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences