Déformations de flots d'Anosov et de groupes fuchsiens

Ghys, Étienne

Annales de l'Institut Fourier, Tome 42 (1992), p. 209-247 / Harvested from Numdam

Résumé

Nous étudions les flots d’Anosov sur les variétés compactes de dimension 3 pour lesquels les distributions stable et instable faibles sont de classe $C^ \infty $. Nous classons tous ces flots lorsqu’ils préservent le volume puis nous construisons une famille d’exemples qui ne préservent pas le volume. Nous classons aussi ces flots sous une hypothèse de “pincement”. En application, nous décrivons les déformations des groupes fuchsiens dans le groupe des difféomorphismes du cercle.

Abstract

We study Anosov flows on compact 3-manifolds for which weak stable and unstable distributions are $C^\infty $. We classify these flows if they are volume preserving and we construct a family of examples which are not volume preserving. We also classify these flows under a “pinching" assumption. As an application, we describe deformations of Fuchsian groups in the group of diffeomorphisms of the circle.

Détail

Publié le : 1992-01-01

MR 93j:58111 | Zbl 0759.58036 | 5 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1290

Bibliographie

[An] D.V. Anosov, Geodesic flow on compact manifolds of negative curvature, Proc. Steklov Math. Inst. A.M.S. Translations, 1969.

[Ar] P. Armandariz, Codimension one Anosov flows on manifolds with solvable fundamental group, Thèse Univ. Ispapalapa, Mexico.

[Av] A. Avez, Anosov diffeomorphisms, in Proc. Int. Symp. on Topological dynamics, Benjamin, 1968, 17-51. | MR 38 #4646 | Zbl 0203.26101

[BeFoL] Y. Benoist, P. Foulon, F. Labourie, Flots d'Anosov à distributions stable et instable différentiables, prépublication, 1990. | MR 92b:58165 | Zbl 0754.58027

[Bo] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Maths. Springer n° 470, 1975. | MR 56 #1364 | Zbl 0308.28010

[FeKat] R. Feres, A. Katok, Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows, Erg. Th. & Dyn. Sys. 9, (1989), 427-432. | MR 90k:58167 | Zbl 0667.58050

[Fe] R. Feres, Geodesic flows on manifolds of negative curvature with smooth horospheric foliations, preprint, 1990. | Zbl 0729.58039

[Fri] D. Fried, Transitive Anosov flows and Pseudo-Anosov maps, Topology, 22, n° 3 (1983), 299-303. | MR 84j:58095 | Zbl 0516.58035

[FraWi] J. Franks, R. Williams, Anomalous Anosov flows, Lecture Notes in Maths., Springer n° 819, 158-174. | Zbl 0463.58021

[GhSe] E. Ghys, V. Sergiescu, Stabilité et conjugaison différentiable pour certains feuilletages, Topology, 19 (1980), 179-197. | MR 81k:57022 | Zbl 0478.57017

[GhTs] E. Ghys, T. Tsuboi, Différentiabilité des conjugaisons entre systèmes dynamiques de dimension 1, Ann. Inst. Fourier, 38 (1) (1988), 215-244. | Numdam | MR 89i:58119 | Zbl 0633.58018

[Gh1] E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergod. Th. & Dynam. Sys., 4 (1984), 67-80. | MR 86b:58098 | Zbl 0527.58030

[Gh2] E. Ghys, Actions localement libres du groupe affine, Invent. Math., 82 (1985), 479-526. | MR 87f:58084 | Zbl 0577.57010

[Gh3] E. Ghys, Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Scien. Ec. Norm. Sup., 20 (1987), 251-270. | Numdam | MR 89h:58153 | Zbl 0663.58025

[Go] S. Goodman, Dehn surgery on Anosov flows, Geometric dynamics, Lecture Notes in Maths., Springer n° 1007, 300-307. | MR 1691596 | Zbl 0532.58021

[Gr] M. Gromov, Three remarks on geodesic dynamics and fundamental group, texte non publié, S.U.N.Y., vers 1977. | Zbl 01736568

[HuKat] S. Hurder, A. Katok, Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Pub. I.H.E.S., 72 (1990), 5-61. | Numdam | Zbl 0725.58034

[HanTh] M. Handel, W. Thurston, Anosov flows on new 3-manifolds, Inv. Math., vol. 59 (1980), 95-103. | MR 81i:58032 | Zbl 0435.58019

[Hae] A. Haefliger, Groupoïdes d'holonomie et classifiants, Astérique, 116 (1984), 70-97. | MR 86c:57026a | Zbl 0562.57012

[Has] B. Hasselblatt, Bootstrapping regularity of the Anosov splitting, to appear in Proc. A.M.S. | Zbl 0790.58029

[Kan] M. Kanai, Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations, Ergodic Theory & Dynam. Sys., 8 (1988), 215-240. | MR 89k:58230 | Zbl 0634.58020

[O] J.-P. Otal, Le spectre marqué des surfaces à courbure négative, Annals of Maths., 131 (1990), 151-162. | MR 91c:58026 | Zbl 0699.58018

[PaPo] W. Parry, M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, n° 187-188, S.M.F., 1990. | MR 92f:58141 | Zbl 0726.58003

[PlTh] J. Plante, W. Thurston, Anosov flows and the fundamental group, Topology, 11 (1972), 147-150. | MR 45 #4455 | Zbl 0246.58014

[Pa] W. Parry, Synchronisation of canonical measures for hyperbolic attractors, Commun. Math. Phys., 106 (1986), 267-275. | MR 88b:58088 | Zbl 0618.58026

[Pl] J. Plante, Anosov flows, transversely affine foliations and a conjecture of Verjovsky, J. London Math. Soc., (2) 23 (1981), 359-362. | Zbl 0465.58020

[RVa] F. Raymond, T. Vasquez, 3-manifolds whose universal coverings are Lie groups, Topology and its App., vol. 12 (1981), 161-179. | MR 82i:57011 | Zbl 0468.57009

[Sa] R. Sacksteder, Foliations and pseudogroups, Ann. of Math., 87 (1965), 79-102. | MR 30 #4268 | Zbl 0136.20903

[St] S. Sternberg, Local Cn-transformations of the real line, Duke Math. J., 24, 94-102. | MR 21 #1371 | Zbl 0077.06201

[Su] D. Sullivan, Discrete conformal groups and measurable dynamics, Bull. Amer. Math. Soc., 6 (1982), 53-73. | MR 83c:58066 | Zbl 0489.58027

[Th1] W. Thurston, Foliations on 3-manifolds which are circle bundles, Ph. D. Thesis, Berkeley, 1972.

[Th2] W. Thurston, The geometry and topology of 3-manifolds, Princeton Lecture Notes, 1976.

[To] P. Tomter, Anosov flows on infrahomogeneous spaces, Proc. Symp. Pure Maths., 14 (1970), 299-328. | MR 43 #5552 | Zbl 0207.54502

[Ve] A. Verjovsky, Codimension one Anosov flows, Bol. Soc. Matem. Mex., 19 (1974). | MR 55 #4282 | Zbl 0323.58014

[Wo] J. Wolf, Spaces of constant curvature, Publish or Perish. | Zbl 0162.53304