具有二维中心的遍历环面自同构的扰动的拓扑中心化子
Topological centralizers of perturbations of ergodic toral automorphisms with two dimensional center
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中文总结 AI 辅助
该研究针对具有二维中心的遍历环面自同构的C²²邻域内的微分同胚，证明了二分性结果，明确了其可达性与拓扑共轭的等价条件，给出了其中心化子的结构。
中文摘要 AI 辅助
设A是环面T^N的一个遍历线性自同构，其中心空间的维数为2。我们证明了一个二分性结果：对于每一个属于U的f（其中U是A在T^N上保体积微分同胚构成的集合内的C²²邻域），两种情形中恰好有一种成立。要么f具有可达性，要么f通过一个同痕于恒等映射的同胚与A拓扑共轭，该同胚同时将每一个与f交换的同胚共轭为环面的仿射自同构。我们不假设A的特征多项式是不可约的。这两种情形相互排斥，因为与A拓扑共轭的部分双曲微分同胚永远不会是可达的，因此在U内，可达性的失效等价于与A的拓扑共轭。在第二种情形下，f在环面同胚群中的中心化子同构于与A交换的仿射变换构成的群。当A的特征多项式不可约时，我们证明该群是由A的一个特征值生成的数域中某个序的单位群的有限扩张，因此是秩为r₁ + r₂ - 1的几乎自由阿贝尔群。允许特征多项式可约，此时相同的描述仍然成立，只是线性部分的范围是GL(N,Z)中A的中心化子。我们的方法利用了格在中心空间上投影的一个基本密度性质，该性质替代了不可约性，并且我们为每一个遍历自同构建立了该性质，对中心空间的维数无任何限制。
英文摘要
Let $A$ be an ergodic linear automorphism of the torus $\T^N$ whose center space has dimension two. We prove a dichotomy result that for every $f \in \mathcal{U}$, where $\mathcal{U}$ is the $C^{22}$ neighborhood of $A$ inside volume preserving diffeomorphisms on $\T^N$, exactly one of two alternatives holds. Either $f$ has the accessibility property, or $f$ is topologically conjugate to $A$ by a homeomorphism homotopic to the identity which simultaneously conjugates every homeomorphism commuting with $f$ to an affine automorphism of the torus. No irreducibility of the characteristic polynomial of $A$ is assumed. The alternatives exclude each other because a partially hyperbolic diffeomorphism topologically conjugate to $A$ is never accessible, so within $\mathcal{U}$ the failure of accessibility is equivalent to topological conjugacy to $A$. In the second case the centralizer of $f$ in the group of homeomorphisms of the torus is isomorphic to the group of affine transformations commuting with $A$. When the characteristic polynomial of $A$ is irreducible we show that this group is a finite extension of the unit group of an order in the number field generated by an eigenvalue of $A$, hence virtually free abelian of rank $r_1 + r_2 - 1$. The characteristic polynomial is allowed to be reducible, in which case the same description holds with the linear parts ranging over the centralizer of $A$ in $\GL(N,\Z)$. Our method uses an elementary density property of the projection of the lattice to the center space, which replaces irreducibility and which we establish for every ergodic automorphism with no restriction on the dimension of the center space.
