Differentiability, rigidity and Godbillon-Vey classes for Anosov flows
Publications Mathématiques de l'IHÉS, Tome 72 (1990), pp. 5-61. 
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     title = {Differentiability, rigidity and {Godbillon-Vey} classes for {Anosov} flows},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {5--61},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {72},
     year = {1990},
     mrnumber = {92b:58179},
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     language = {en},
     url = {http://archive.numdam.org/item/PMIHES_1990__72__5_0/}
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Hurder, S.; Katok, Anatoly. Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Publications Mathématiques de l'IHÉS, Tome 72 (1990), pp. 5-61. http://archive.numdam.org/item/PMIHES_1990__72__5_0/

[1] D. V. Anosov, Tangent fields of transversal foliations in U-systems, Math. Notes Acad. Sci., USSR, 2 (5) (1967), 818-823. | MR | Zbl

[2] D. V. Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature, Proc. Steklov Inst. Math., vol. 90 (1967), Amer. Math. Soc., 1969. | MR | Zbl

[3] V. I. Arnold, Mathematical Methods in Classical Mechanics, Springer-Verlag, 1980.

[4] A. Avez, Anosov diffeomorphisms, in Proceedings Int. Symp. on Topological Dynamics, Benjamin, 1968, 17-51. | MR | Zbl

[5] R. Bishop and R. Crittenden, Geometry of Manifolds, Academic Press, 1964. | MR | Zbl

[6] R. Bott, On some formulas for the characteristic classes of group actions, in Springer Lect. Notes in Math., 652 (1978), 25-61. | MR | Zbl

[7] C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Progress in Math. Birkhausser, Boston, Basel and Stuttgart, 1985. | MR | Zbl

[8] C. Croke, Rigidity for surfaces of non-positive curvature, Comment. Math. Helv., 65 (1990), 150-169. | MR | Zbl

[9] G. D'Ambra and M. Gromov, Lectures on transformation groups : Geometry and Dynamics, Preprint, Inst. Hautes Etudes Sci., IHES/M/90/1, 1990.

[10] P. Eberlein, When is a geodesic flow of Anosov type? I, Jour. Differential Geom., 8 (1973), 437-463. | MR | Zbl

[11] R. Feres, Geodesic flows on manifolds of negative curvature with smooth horospheric foliations, preprint, 1989.

[12] R. Feres and A. Katok, Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows, Ergodic Theory Dynamical Systems, 9 (1989), 427-432. | MR | Zbl

[13] R. Feres and A. Katok, Anosov flows with smooth foliations and rigidity of geodesic flows in three dimensional manifolds, Ergodic Theory Dynamical Systems, 10 (1990), 657-670. | MR | Zbl

[14] L. Flaminio and A. Katok, Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori, Ergodic Theory Dynamical Systems, 11 (1991), to appear. | MR | Zbl

[15] P. Foulon and F. Labourie, Flots d'Anosov à distributions de Liapunov différentiables, C. R. Acad. Sci. Paris, 309 (1989), 255-260. | MR | Zbl

[16] E. Ghys, Actions localement libres du groupe affine, Invent. Math., 82 (1985), 479-526. | MR | Zbl

[17] E. Ghys, Flots d'Anosov dont les feuilletages stables sont différentiables, Annales Ecole Norm. Sup., 20 (1987), 251-270. | Numdam | MR | Zbl

[18] E. Ghys, Sur l'invariance topologique de la classe de Godbillon-Vey, Annales Inst. Fourier, 37 (4) (1987), 59-76. | Numdam | MR | Zbl

[19] E. Ghys, Codimension-one Anosov flows and suspensions, in R. LABARCA, R. BAMON and J. PALIS, editors, Springer Lect. Notes Math., 1331 (1989), 59-72. | MR | Zbl

[20] E. Ghys, Sur l'invariant de Godbillon-Vey, in Séminaire Bourbaki, mars 1989, Astérisque, 177-178 (1990), 155-181. | Numdam | MR | Zbl

[21] E. Ghys and T. Tsuboi, Différentiabilité des conjugaisons entre systèmes dynamiques de dimension 1, Annales Inst. Fourier, 38 (1) (1988), 215-244. | Numdam | MR | Zbl

[22] C. Godbillon and J. Vey, Un invariant de feuilletages de codimension 1, C. R. Acad. Sci. Paris, 273 (1971), 92-95. | MR | Zbl

[23] P. Grzigorchuk, On three-dimensional Anosov flows with C2-invariant foliations, preprint, 1988.

[24] V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems, Topology, 19 (1980), 291-299. | MR | Zbl

[25] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19 (1980), 301-313. | MR | Zbl

[26] U. Hamenstadt, Entropy-rigidity of locally symmetric spaces of negative curvature, Annals of Math., 131 (1990), 35-51. | MR | Zbl

[27] U. Hamenstadt, Metric and topological entropies of geodesic flows, preprint, 1990.

[28] B. Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, preprint, 1990.

[29] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, in Proc. Symp. Pure Math., Amer. Math. Soc. 14 (1970), 133-164. | MR | Zbl

[30] M. Hirsch and C. Pugh, Smoothness of horocycle foliations, Jour. Differential Geom., 10 (1975), 225-238. | MR | Zbl

[31] S. Hurder, Characteristic classes for C1-foliations, preprint, 1990. | Zbl

[32] S. Hurder, Deformation rigidity for subgroups of SL(n, Z) acting on the n-torus, Bulletin Amer. Math. Soc., 23 (1990), 107-113. | MR | Zbl

[33] S. Hurder, Rigidity for Anosov actions of higher rank lattices, preprint, 1990.

[34] J.-L. Journé, On a regularity problem occurring in connection with Anosov diffeomorphisms, Comm. Math. Phys., 106 (1986), 345-352. | MR | Zbl

[35] J.-L. Journé, A regularity lemma for functions of several variables, Revista Mat. Iber., 4 (2) (1988), 187-193. | MR | Zbl

[36] M. Kanai, Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory Dynamical Systems, 8 (1988), 215-240. | MR | Zbl

[37] M. Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, preprint, 1989.

[38] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci., 51 (1980), 137-173. | Numdam | MR | Zbl

[39] A. Katok, Entropy and closed geodesics, Ergodic Theory Dynamical Systems, 2 (1982), 339-366. | MR | Zbl

[40] A. Katok, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynamical Systems, 8 (1988), 139-152. | MR | Zbl

[41] A. Katok, Canonical cocycles for Anosov flows, to appear.

[42] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, preprint, 1990. | Zbl

[43] G. Knieper and H. Weiss, Smoothness of measure theoretic entropy for Anosov flows, preprint, 1989.

[44] S. Krantz, Lipshitz spaces, smoothness of functions, and approximation theory, Expositions Mathematicae, 3 (1983), 193-260. | MR | Zbl

[45] A. Livsic, Homology properties of U-systems, Math. Notes of U.S.S.R., 10 (1971), 758-763. | MR | Zbl

[46] A. Livsic, Cohomology of dynamical systems, Math. U.S.S.R. Izvestija, 6 (1972), 1278-1301. | MR | Zbl

[47] A. Livsic and J. Sinai, On invariant measures compatible with the smooth structure for transitive U-systems, Soviet Math. Doklady, 13 (6) (1972), 1656-1659. | Zbl

[48] R. De La Llavé, Analytic regularity of solutions of Livsic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems, Ergodic Theory Dynamical Systems, to appear. | Zbl

[49] R. De La Llavé, J. Marco and R. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for Livsic cohomology equation, Annals of Math., 123 (1986), 537-612. | MR | Zbl

[50] R. De La Llavé and R. Moriyon, Invariants for smooth conjugacy of hyperbolic dynamical systems, IV, Comm. Math. Phys., 116 (1988), 185-192. | MR | Zbl

[51] C. Pugh, M. Hirsch and M. Shub, Invariant Manifolds, Springer Lect. Notes Math., n° 583, 1977. | MR | Zbl

[52] J. M. Marco and R. Moriyon, Invariants for smooth conjugacy of hyperbolic dynamical systems, III, Comm. Math. Phys., 112 (1987), 317-333. | MR | Zbl

[53] Y. Mitsumatsu, A relation between the topological invariance of the Godbillon-Vey invariant and the differentiability of Anosov foliations, in Foliations, vol. 5 of Advanced Studies in Pure Math., University of Tokyo Press, 1985. | MR | Zbl

[54] J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed-point, Comm. Pure and Applied Math., IX (1956), 673-692. | MR | Zbl

[55] J.-P. Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Annals of Math., 131 (1990), 151-162. | MR | Zbl

[56] Ya. B. Pesin, Equations for the entropy of a geodesic flow on a compact Riemannian manifold without conjugacy points, Math. Notes U.S.S.R., 24 (1978), 796-805. | Zbl

[57] J. Plante, Anosov flows, American Jour. Math., 94 (1972), 729-755. | MR | Zbl

[58] G. Raby, Invariance de classes de Godbillon-Vey par C1-difféomorphismes, Annales Inst. Fourier, 38 (1) (1988), 205-213. | Numdam | MR | Zbl

[59] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, Princeton University Press, 1970. | MR | Zbl

[60] S. Sternberg, The structure of local diffeomorphisms, III, American Jour. Math., 81 (1959), 578-604. | MR | Zbl

[61] W. Thurston, Noncobordant foliations of S3, Bulletin Amer. Math. Soc., 78 (1972), 511-514. | MR | Zbl

[62] W. Thurston, Foliations and groups of diffeomorphisms, Bulletin Amer. Math. Soc., 80 (1974), 304-307. | MR | Zbl

[63] T. Tsuboi, Area functional and Godbillon-Vey cocycles, preprint, 1989. | Numdam | Zbl

[64] T. Tsuboi, On the foliated products of class C1, Annals of Math., 130 (1989), 227-271. | MR | Zbl

[65] A. Zygmund, Trigonometric Series, Cambridge University Press, 1968. | MR | Zbl