Mathematische Zeitschrift

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

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

Article

Abstract

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”.

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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

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