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

Abstract

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: 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”.