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 (M, X) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets of the connected components of satisfy the following properties:each is a compact invariant locally maximal transitive set for X; the collection is canonically attached to the pair (M, X) (i.e. it can be defined independently of the collection of tori and Klein bottles ); the 's are the smallest possible: for every (possibly infinite) collection of tori and Klein bottles transverse to X, the 's are contained in the maximal invariant set of .