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

TL;DR

This paper establishes a canonical spectral-like decomposition for transitive and non-transitive Anosov flows on 3-manifolds, identifying minimal invariant sets analogous to basic pieces in spectral decomposition.

Contribution

It introduces a finite, canonical collection of transverse tori that decompose the manifold into minimal invariant sets, extending spectral decomposition concepts to 3D Anosov flows.

Findings

Existence of a finite collection of transverse tori with minimal invariant sets

The invariant sets are canonical and minimal for the flow

Almost uniqueness of the decomposition up to topological equivalence

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 two-tori transverse to $X$. We prove that one can find a finite collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel incompressible tori transverse to $X$, such that the maximal invariant sets $\Lambda_1,\dots,\Lambda_m$ of the connected components $V_1,\dots,V_m$ of $M-(T_1\cup\dots\cup T_n)$ satisfy the following properties: 1, each $\Lambda_i$ is a compact invariant locally maximal transitive set for $X$, 2, 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 $\{T_1,\dots,T_n\}$), 3, the $\Lambda_i$'s are the smallest possible: for every (possibly infinite) collection $\{S_i\}_{i\in I}$ of tori 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".