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Article Dans Une Revue Mathematische Zeitschrift Année : 2016

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

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Résumé

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 $A$ ector 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”.

Dates et versions

hal-01413416 , version 1 (09-12-2016)

Identifiants

Citer

F. Beguin, Christian Bonatti, Bin Yu. A spectral-like decomposition for transitive Anosov flows in dimension three. Mathematische Zeitschrift, 2016, 282 (3-4), pp.889 - 912. ⟨10.1007/s00209-015-1569-6⟩. ⟨hal-01413416⟩
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