In this paper we study the problem of knowing when the centralizer of a vector field is “small”. We obtain several criteria that imply different types of “small” centralizers, namely collinear, quasi-trivial and trivial . There are two types of results in the paper: general dynamical criteria that imply one of the “small” centralizers above; and genericity results about the centralizer. Some of our general criteria imply that the centralizer is trivial in the following settings: non-uniformly hyperbolic conservative $$C^2$$ C 2 flows; transitive separating $$C^1$$ C 1 flows; Kinematic expansive $$C^3$$ C 3 flows on 3 manifolds whose singularities are all hyperbolic. For genericity results, we obtain that $$C^1$$ C 1 -generically the centralizer is quasi-trivial, and in many situations we can show that it is actually trivial.

中文翻译：

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关于向量场的中心化：平凡性和通用性结果的标准

在本文中，我们研究了知道向量场的中心化器何时“小”的问题。我们获得了几个标准，这些标准意味着不同类型的“小”扶正器，即共线、准平凡和平凡。论文中有两种类型的结果：暗示上述“小”集中器之一的一般动态标准；和关于扶正器的通用性结果。我们的一些一般标准意味着中心化器在以下设置中是微不足道的：非均匀双曲线保守 $$C^2$$ C 2 流；传递分离 $$C^1$$ C 1 流；运动膨胀 $$C^3$$ C 3 在奇点都是双曲线的 3 个流形上流动。对于一般性结果，我们得到 $$C^1$$C 1 -一般地，中心化器是准平凡的，并且在许多情况下我们可以证明它实际上是平凡的。