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Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. (English) Zbl 0725.58034
The main geometric objects associated with an Anosov dynamical system on a compact manifold are the invariant stable and unstable foliations. Although each stable and unstable manifold is as smooth as the system itself, it is believed that the foliations that they form have only a moderate degree of regularity for most systems. In the paper under review, the authors analyze the exact degree of regularity for the weak-stable and weak-unstable foliations for a volume-preserving Anosov system of codimension one. The main results relate the regularity of these foliations to cohomology classes associated to the system: the Anosov class, a new invariant of the flow which is introduced in this paper, and the Godbillon-Vey class of the weak-stable foliation, which the authors show that it is a well-defined invariant of the system.
The general outline of this paper is based on the authors’ study of the local obstacles discovered by Anosov for the two-torus [D. V. Anosov, Tr. Mat. Inst. Steklova 90 (1967; Zbl 0163.43604)]. The authors show that the Anosov class has a remarkable application: when this cohomology “obstacle to regularity” vanishes, the foliations must be infinitely smooth (this is the rigidity phenomenon mentioned in the title). This implies that the system is algebraic by results of A. Avez [Topol. Dynamics, Int. Symp. Colorado State Univ. 1967, 17–51 (1968; Zbl 0203.26101)] or E. Ghys [Ann. Sci. Éc. Norm. Supér. (4) 20, 251–270 (1987; Zbl 0663.58025)]. The Godbillon-Vey invariant of the flow has two applications. Firstly, the authors show that there are continuous families of topologically conjugate, codimension-one foliations for which the Godbillon-Vey invariants vary continuously (and are not constant!). Secondly, the Godbillon-Vey invariant of the flow characterizes the geodesic flows for metrics of negative curvature as the flows with maximal value for this invariant, among the geodesic flows for metrics of negative curvature on closed surfaces. The authors use this observation to give a new proof that the harmonic measure at infinity for metrics of variable negative curvature on surfaces is totally singular.
Both of the cohomology invariants introduced in the paper are parameters on the space of low-dimensional, volume-preserving Anosov systems, for which the O-set is parametrized by Teichmüller spaces. The authors conjecture that for other values of these cohomology invariants, the system is determined up to smooth equivalence by a finite set of auxiliary, Teichmüller-like parameters.
Reviewer: D. Savin (Iaşi)

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37C10 Dynamics induced by flows and semiflows
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##### References:
 [1] D. V. Anosov, Tangent fields of transversal foliations in U-systems,Math. Notes Acad. Sci., USSR,2 (5) (1967), 818–823. · Zbl 0171.42201 [2] D. V. Anosov,Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature, Proc. Steklov Inst. Math., vol. 90 (1967), Amer. Math. Soc., 1969. · Zbl 0176.19101 [3] V. I. Arnold,Mathematical Methods in Classical Mechanics, Springer-Verlag, 1980. [4] A. Avez, Anosov diffeomorphisms, inProceedings Int. Symp. on Topological Dynamics, Benjamin, 1968, 17–51. [5] R. Bishop andR. Crittenden,Geometry of Manifolds, Academic Press, 1964. · Zbl 0132.16003 [6] R. Bott, On some formulas for the characteristic classes of group actions, inSpringer Lect. Notes in Math.,652 (1978), 25–61. [7] C. Camacho andA. Lins Neto,Geometric Theory of Foliations, Progress in Math. Birkhausser, Boston, Basel and Stuttgart, 1985. [8] C. Croke, Rigidity for surfaces of non-positive curvature,Comment. Math. Helv.,65 (1990), 150–169. · Zbl 0704.53035 [9] G. D’Ambra andM. 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. · Zbl 0285.58008 [11] R. Feres,Geodesic flows on manifolds of negative curvature with smooth horospheric foliations, preprint, 1989. · Zbl 0727.58035 [12] R. Feres andA. Katok, Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows,Ergodic Theory Dynamical Systems,9 (1989), 427–432. · Zbl 0667.58050 [13] R. Feres andA. Katok, Anosov flows with smooth foliations and rigidity of geodesic flows in three dimensional manifolds,Ergodic Theory Dynamical Systems,10 (1990), 657–670. · Zbl 0729.58039 [14] L. Flaminio andA. Katok, Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori,Ergodic Theory Dynamical Systems,11 (1991), to appear. · Zbl 0725.58033 [15] P. Foulon andF. Labourie, Flots d’Anosov à distributions de Liapunov différentiables,C. R. Acad. Sci. Paris, 309 (1989), 255–260. [16] E. Ghys, Actions localement libres du groupe affine,Invent. Math.,82 (1985), 479–526. · Zbl 0577.57010 [17] E. Ghys, Flots d’Anosov dont les feuilletages stables sont différentiables,Annales Ecole Norm. Sup.,20 (1987), 251–270. · Zbl 0663.58025 [18] E. Ghys, Sur l’invariance topologique de la classe de Godbillon-Vey,Annales Inst. Fourier,37 (4) (1987), 59–76. [19] E. Ghys, Codimension-one Anosov flows and suspensions,in R. Labarca, R. Bamon andJ. Palis, editors,Springer Lect. Notes Math.,1331 (1989), 59–72. [20] E. Ghys, Sur l’invariant de Godbillon-Vey, inSéminaire Bourbaki, mars 1989,Astérisque,177–178 (1990), 155–181. [21] E. Ghys andT. Tsuboi, Différentiabilité des conjugaisons entre systèmes dynamiques de dimension 1,Annales Inst. Fourier,38 (1) (1988), 215–244. · Zbl 0633.58018 [22] C. Godbillon andJ. Vey, Un invariant de feuilletages de codimension 1,C. R. Acad. Sci. Paris,273 (1971), 92–95. · Zbl 0215.24604 [23] P. Grzigorchuk,On three-dimensional Anosov flows with C2-invariant foliations, preprint, 1988. [24] V. Guillemin andD. Kazhdan, On the cohomology of certain dynamical systems,Topology,19 (1980), 291–299. · Zbl 0498.58018 [25] V. Guillemin andD. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,Topology,19 (1980), 301–313. · Zbl 0465.58027 [26] U. Hamenstadt, Entropy-rigidity of locally symmetric spaces of negative curvature,Annals of Math.,131 (1990), 35–51. · Zbl 0699.53049 [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. · Zbl 0821.58032 [29] M. Hirsch andC. Pugh, Stable manifolds and hyperbolic sets, inProc. Symp. Pure Math., Amer. Math. Soc.14 (1970), 133–164. [30] M. Hirsch andC. Pugh, Smoothness of horocycle foliations,Jour. Differential Geom.,10 (1975), 225–238. · Zbl 0312.58008 [31] S. Hurder,Characteristic classes for C1-foliations, preprint, 1990. · Zbl 0687.57010 [32] S. Hurder, Deformation rigidity for subgroups of SL(n,Z) acting on then-torus,Bulletin Amer. Math. Soc.,23 (1990), 107–113. · Zbl 0713.57022 [33] S. Hurder,Rigidity for Anosov actions of higher rank lattices, preprint, 1990. · Zbl 0754.58029 [34] J.-L. Journé, On a regularity problem occurring in connection with Anosov diffeomorphisms,Comm. Math. Phys.,106 (1986), 345–352. · Zbl 0603.58019 [35] J.-L. Journé, A regularity lemma for functions of several variables,Revista Mat. Iber.,4 (2) (1988), 187–193. · Zbl 0699.58008 [36] M. Kanai, Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations,Ergodic Theory Dynamical Systems,8 (1988), 215–240. · Zbl 0634.58020 [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. · Zbl 0445.58015 [39] A. Katok, Entropy and closed geodesics,Ergodic Theory Dynamical Systems,2 (1982), 339–366. · Zbl 0525.58027 [40] A. Katok, Four applications of conformal equivalence to geometry and dynamics,Ergodic Theory Dynamical Systems,8 (1988), 139–152. · Zbl 0668.58042 [41] A. Katok,Canonical cocycles for Anosov flows, to appear. [42] A. Katok andJ. Lewis,Local rigidity for certain groups of toral automorphisms, preprint, 1990. · Zbl 0785.22012 [43] G. Knieper andH. Weiss,Smoothness of measure theoretic entropy for Anosov flows, preprint, 1989. · Zbl 0671.58021 [44] S. Krantz, Lipshitz spaces, smoothness of functions, and approximation theory,Expositiones Mathematicae,3 (1983), 193–260. · Zbl 0518.46018 [45] A. Livsic, Homology properties of U-systems,Math. Notes of U.S.S.R.,10 (1971), 758–763. · Zbl 0235.58010 [46] A. Livsic, Cohomology of dynamical systems,Math. U.S.S.R. Izvestija,6 (1972), 1278–1301. · Zbl 0273.58013 [47] A. Livsic andJ. Sinai, On invariant measures compatible with the smooth structure for transitive U-systems,Soviet Math. Doklady,13 (6) (1972), 1656–1659. · Zbl 0284.58014 [48] R. deLa 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 0883.58024 [49] R. de La Llavé, J. Marco andR. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for Livsic cohomology equation,Annals of Math.,123 (1986), 537–612. · Zbl 0603.58016 [50] R. de La Llavé andR. Moriyon, Invariants for smooth conjugacy of hyperbolic dynamical systems, IV,Comm. Math. Phys.,116 (1988), 185–192. · Zbl 0673.58038 [51] C. Pugh, M. Hirsch andM. Shub,Invariant Manifolds, Springer Lect. Notes Math., no 583, 1977. [52] J. M. Marco andR. Moriyon, Invariants for smooth conjugacy of hyperbolic dynamical systems, III,Comm. Math. Phys.,112 (1987), 317–333. · Zbl 0673.58037 [53] Y. Mitsumatsu, A relation between the topological invariance of the Godbillon-Vey invariant and the differentiability of Anosov foliations, inFoliations, vol. 5 ofAdvanced Studies in Pure Math., University of Tokyo Press, 1985. · Zbl 0653.57018 [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. · Zbl 0072.40801 [55] J.-P. Otal, Le spectre marqué des longueurs des surfaces à courbure négative,Annals of Math.,131 (1990), 151–162. · Zbl 0699.58018 [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 0411.58016 [57] J. Plante, Anosov flows,American Jour. Math.,94 (1972), 729–755. · Zbl 0257.58007 [58] G. Raby, Invariance de classes de Godbillon-Vey par C1-difféomorphismes,Annales Inst. Fourier,38 (1) (1988), 205–213. · Zbl 0596.57018 [59] E. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton, Princeton University Press, 1970. · Zbl 0207.13501 [60] S. Sternberg, The structure of local diffeomorphisms, III,American Jour. Math.,81 (1959), 578–604. · Zbl 0211.56304 [61] W. Thurston, Noncobordant foliations of S3,Bulletin Amer. Math. Soc.,78 (1972), 511–514. · Zbl 0266.57004 [62] W. Thurston, Foliations and groups of diffeomorphisms,Bulletin Amer. Math. Soc.,80 (1974), 304–307. · Zbl 0295.57014 [63] T. Tsuboi,Area functional and Godbillon-Vey cocycles, preprint, 1989. [64] T. Tsuboi, On the foliated products of class C1,Annals of Math.,130 (1989), 227–271. · Zbl 0701.57012 [65] A. Zygmund,Trigonometric Series, Cambridge University Press, 1968. · Zbl 0157.38204
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