Expansive subdynamics for algebraic \mathbb{Z}^d-actions
Expansive subdynamics for algebraic \mathbb{Z}^d-actions
复制标题
代数 mathbb{Z}^d-actions 的扩展子动力学
DOI：
10.1017/s014338570100181x
复制
发表时间：
2001
影响因子：
0.9
通讯作者：
T. Ward
中科院分区：
文献类型：
--
作者：
M. Einsiedler;D. Lind;R. Miles;T. Ward
Boyle和Lind利用紧致阿贝尔群的交换自同构所给出的代数{Z}^d作用在低维子空间上的扩张行为，提出了一个研究紧致度量空间上的拓扑作用的一般框架.这种刻画使用了代数簇的对数映象和具有多个交换变量的Laurent多项式环上的Notherian模的定向形式。对于拓扑型{Z}^d-作用，我们引入了两个秩概念，而对于代数\mathbb{Z}^d-作用，则描述了它们之间的相互关系以及它们与Krull维度的关系。对于{R}^d的线性子空间，我们定义了沿子空间同宿为零的点群，并证明了这个群在扩张分支内是常数.
A general framework for investigating topological actions of \mathbb{Z}^d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of \mathbb{R}^d. Here we completely describe this expansive behavior for the class of algebraic \mathbb{Z}^d-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables. We introduce two notions of rank for topological \mathbb{Z}^d-actions, and for algebraic \mathbb{Z}^d-actions describe how they are related to each other and to Krull dimension. For a linear subspace of \mathbb{R}^d we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.