无发散向量场的稳定性
Stability properties of divergence-free vector fields
摘要：
一个无发散向量场满足星性质，如果任何无发散向量场在某个C1-邻域内都有奇点且都有双曲周期轨道.本文证明了任何定义在黎曼流形上的无发散向量场满足星性质是Anosov的。还证明了一个C1-结构稳定的无发散向量场可以用一个Anosov无发散向量场来近似.证明了任意无发散向量场都可以用Anosov无发散向量场C1-逼近，或者用具有异维循环的无发散向量场C1-逼近.
英文摘要：
A divergence-free vector field satisfies the star property if any divergence-free vector field in some C1-neighborhood has all singularities and all periodic orbits hyperbolic. In this paper we prove that any divergence-free vector field defined on a Riemannian manifold and satisfying the star property is Anosov. It is also shown that a C1-structurally stable divergencefree vector field can be approximated by an Anosov divergence-free vector field. Moreover, we prove that any divergence-free vector field can be C1-approximated by an Anosov divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle.
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