Déformations de flots d'Anosov et de groupes fuchsiens
Annales de l'Institut Fourier, Tome 42 (1992) no. 1-2, pp. 209-247.

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 . 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.

We study Anosov flows on compact 3-manifolds for which weak stable and unstable distributions are C . 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.

@article{AIF_1992__42_1-2_209_0,
     author = {Ghys, \'Etienne},
     title = {D\'eformations de flots {d'Anosov} et de groupes fuchsiens},
     journal = {Annales de l'Institut Fourier},
     pages = {209--247},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {1-2},
     year = {1992},
     doi = {10.5802/aif.1290},
     zbl = {0759.58036},
     mrnumber = {93j:58111},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1290/}
}
TY  - JOUR
AU  - Ghys, Étienne
TI  - Déformations de flots d'Anosov et de groupes fuchsiens
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 209
EP  - 247
VL  - 42
IS  - 1-2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1290/
DO  - 10.5802/aif.1290
LA  - fr
ID  - AIF_1992__42_1-2_209_0
ER  - 
%0 Journal Article
%A Ghys, Étienne
%T Déformations de flots d'Anosov et de groupes fuchsiens
%J Annales de l'Institut Fourier
%D 1992
%P 209-247
%V 42
%N 1-2
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1290/
%R 10.5802/aif.1290
%G fr
%F AIF_1992__42_1-2_209_0
Ghys, Étienne. Déformations de flots d'Anosov et de groupes fuchsiens. Annales de l'Institut Fourier, Tome 42 (1992) no. 1-2, pp. 209-247. doi : 10.5802/aif.1290. https://aif.centre-mersenne.org/articles/10.5802/aif.1290/

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

[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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

Cité par Sources :