Given a (transitive or non-transitive) Anosov vector field
X on a closed three dimensional manifold
M, one may try to decompose (
M,
X) 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 (M, X) (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”.
