Mathematische Zeitschrift

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

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

  • F. BeguinEmail author
  • C. Bonatti
  • B. Yu


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”.



The authors would like to thank the referee for carefully reading this paper and providing many helpful suggestions and comments. In particular, his suggestion to discuss non-orientable case makes the main result of this paper more general.


  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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fenley, S.: Anosov flows in 3-manifolds. Ann. Math. 139(1), 79–115 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fenley, S.R.: Quasigeodesic Anosov flows and homotopic properties of flow lines. J. Differ. Geom. 41(2), 479–514 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Foulon, P., Hasselblatt, B.: Contact Anosov flows on hyperbolic 3-manifolds. Geom. Topol. 17(2), 1225–1252 (2013)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

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© 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

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