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”.