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”.



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.


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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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