关于向量场的中心化子：平凡性的判据和一般性结果
On the centralizer of vector fields: criteria of triviality and genericity results
摘要：
本文研究了紧致连通黎曼流形$M^d$上的一个典型向量场是否有一个“小”中心化子的问题。在$C^1$的情况下，我们给出了两个标准，其中之一是$C^1$ -generic，它保证$C^1$ -generic向量场的中心化子确实很小，即\textit{collinear}。另一个标准规定$C^1$ \textit{separating}流具有共线的$C^1$ -中心化器。当所有的奇点都是双曲型时，我们证明了共线性性质实际上可以提升到一个更强的性质，称为\textit{quasi-triviality}。特别地，$C^1$ -类属向量场的$C^1$ -中心化子是拟平凡的。在一定条件下，我们得到了$C^1$ -一般向量场的中心化子的平凡性，它包括$C^1$ -一般公理A（或截公理A）向量场和具有可数个链递归类的$C^1$ -一般向量场.对于充分经常的向量场，我们也得到了各种标准，确保中心化是\textit{trivial}（尽可能小），我们表明，在更高的正则性，共线性和平凡的$C^d$ -中心化是一个通用的向量场在$C^d$拓扑的等价属性。我们还得到，在非一致双曲的情况下，与正则$C^2$，$C^1$ -中心化子是平凡的。
英文摘要：
In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold $M^d$ has a `small' centralizer. In the $C^1$ case, we give two criteria, one of which is $C^1$-generic, which guarantees that the centralizer of a $C^1$-generic vector field is indeed small, namely \textit{collinear}. The other criterion states that a $C^1$ \textit{separating} flow has a collinear $C^1$-centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as \textit{quasi-triviality}. In particular, the $C^1$-centralizer of a $C^1$-generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a $C^1$-generic vector field, which includes $C^1$-generic Axiom A (or sectional Axiom A) vector fields and $C^1$-generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is \textit{trivial} (as small as it can be), and we show that in higher regularity, collinearity and triviality of the $C^d$-centralizer are equivalent properties for a generic vector field in the $C^d$ topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity $C^2$, the $C^1$-centralizer is trivial.
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