Mathematische Zeitschrift

, Volume 282, Issue 3–4, pp 889–912 | Cite as

A spectral-like decomposition for transitive Anosov flows in dimension three



Given a (transitive or non-transitive) Anosov vector field X on a closed three dimensional manifold M, one may try to decompose (MX) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection \(\{S_1,\dots ,S_n\}\) of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets \(\Lambda _1,\dots ,\Lambda _m\) of the connected components \(V_1,\dots ,V_m\) of \(M-(S_1\cup \dots \cup S_n)\) satisfy the following properties:
  • each \(\Lambda _i\) is a compact invariant locally maximal transitive set for X;

  • the collection \(\{\Lambda _1,\dots ,\Lambda _m\}\) is canonically attached to the pair (MX) (i.e. it can be defined independently of the collection of tori and Klein bottles \(\{S_1,\dots ,S_n\}\));

  • the \(\Lambda _i\)’s are the smallest possible: for every (possibly infinite) collection \(\{S_i\}_{i\in I}\) of tori and Klein bottles transverse to X, the \(\Lambda _i\)’s are contained in the maximal invariant set of \(M-\cup _i S_i\).

To a certain extent, the sets \(\Lambda _1,\dots ,\Lambda _m\) are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition \(V_1,\dots ,V_m\), equipped with the restriction of the Anosov vector field X, are “almost unique up to topological equivalence”.


  1. 1.
    Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90, 3–210 (1967). (Russian)MathSciNetGoogle Scholar
  2. 2.
    Barbot, T.: Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergod. Theory Dyn. Syst. 15(2), 247–270 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barbot, T.: Mise en position optimale de tores par rapport à un flot d’Anosov. (French) [Optimal positioning of tori with respect to an Anosov flow]. Comment. Math. Helv. 70(1), 113–160 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barbot, T.: Generalizations of the Bonatti–Langevin example of Anosov flow and their classification up to topological equivalence. Commun. Anal. Geom. 6(4), 749–798 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barbot, T.: De l’hyperbolique au globalement hyperbolique. Mémoire pour obtenir l’habilitation à diriger des recherches (2005)Google Scholar
  6. 6.
    Barbot, T., Fenley, S.: Pseudo-Anosov flows in toroidal manifolds. Geom. Topol. 17(4), 1877–1954 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Barbot, T., Fenley, S.: Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds. Ergod. Theory Dyn. Syst. 35(6), 1681–1722 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Béguin, F., Bonatti, C.: Flots de Smale en dimension 3: présentations finies de voisinages invariants d’ensembles selles. (French) [Smale flows in dimension 3: finite presentations of invariant neighborhoods of saddle sets]. Topology 41(1), 119–162 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Béguin, F., Bonatti, C., Yu, B.: Building Anosov flows on three-manifolds. arXiv:1408.3951
  10. 10.
    Bonatti, C., Langevin, R.: Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension. Ergod. Theory Dyn. Syst. 14(4), 633–643 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brunella, M.: Separating the basic sets of a nontransitive Anosov flow. Bull. Lond. Math. Soc. 25(5), 487–490 (1993)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fenley, S.: Anosov flows in 3-manifolds. Ann. Math. 139(1), 79–115 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fenley, S.R.: Quasigeodesic Anosov flows and homotopic properties of flow lines. J. Differ. Geom. 41(2), 479–514 (1995)MathSciNetMATHGoogle Scholar
  14. 14.
    Foulon, P., Hasselblatt, B.: Contact Anosov flows on hyperbolic 3-manifolds. Geom. Topol. 17(2), 1225–1252 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Franks, J., Williams, B.: Anomalous Anosov flows. Global theory of dynamical systems. In: Proceedings of International Conference on Northwestern University, Evanston, 1979. Lecture Notes in Mathematics, vol. 819, pp. 158–174, Springer, Berlin (1980)Google Scholar
  16. 16.
    Ghys, É.: Flots d’Anosov sur les 3-variétés fibrées en cercles. Ergod. Theory Dyn. Syst. 4(1), 67–80 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ghys, É.: Flots d’Anosov dont les feuilletages stables sont différentiables. Ann. Sci. École Norm. Sup. 20(2), 251–270 (1987)MathSciNetGoogle Scholar
  18. 18.
    Goodman, S.: Dehn surgery on Anosov flows. Geometric dynamics (Rio de Janeiro, 1981). Lecture Notes in Mathematics, vol. 1007, pp. 300–307. Springer, Berlin, 1983Google Scholar
  19. 19.
    Handel, M., Thruston, W.P.: Anosov flows on new three manifolds. Invent. Math. 59(2), 95–103 (1980)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Plante, J.F.: Anosov flows, transversely affine foliations, and a conjecture of Verjovsky. J. Lond. Math. Soc. 23(2), 359–362 (1981)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Shub, M.: Stabilité globale des systèmes dynamiques. Astérisque, vol. 56. Socit Mathmatique de France, Paris, pp. iv+211 (1978)Google Scholar
  22. 22.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Laboratoire Analyse, Géométrie, Applications - UMR 7539 du CNRSUniversité Paris 13VilletaneuseFrance
  2. 2.Institut de Mathématiques de Bourgogne - UMR 5584 du CNRSUniversité de BourgogneDijonFrance
  3. 3.Department of MathematicsTongji UniversityShanghaiChina

Personalised recommendations