海森堡里程计的结构和谱
The structure and spectrum of Heisenberg odometers
摘要：
在最近的工作中Cortez和Petite定义了离散的、非线性生成的和剩余的有限群G的里程计作用。在本文中，我们重点讨论的情况下，G是离散海森堡群。当海森堡群的有限指数子群被看作Z^3的子集时，我们基于它们的几何证明了它们的一个结构定理。我们提供了一个完整的分类海森堡里程计的基础上，其定义的子组的结构，我们提供了每个类的例子。Mackey已经证明了所有这样的作用都有离散谱，也就是说，与动力系统相关的酉算子允许分解成群G的有限维的不可约表示。在这里，我们提供了一个明确的证明，这一事实的一般G里程表。我们的证明使我们能够明确地定义那些出现在海森堡里程计的谱分解中的海森堡群的表示，作为定义子群的函数。在此过程中，我们还提供了充要条件的Z ^d里程表是一个产品里程表定义的科尔特斯。
英文摘要：
In recent work Cortez and Petite defined odometer actions of discrete, finitely generated and residually finite groups G. In this paper we focus on the case where G is the discrete Heisenberg group. We prove a structure theorem for finite index subgroups of the Heisenberg group based on their geometry when they are considered as subsets of Z^3. We provide a complete classification of Heisenberg odometers based on the structure of their defining subgroups and we provide examples of each class. Mackey has shown that all such actions have discrete spectrum, i.e. that the unitary operator associated to the dynamical system admits a decomposition into finite dimensional, irreducible representations of the group G. Here we provide an explicit proof of this fact for general G odometers. Our proof allows us to define explicitly those representations of the Heisenberg group which appear in the spectral decomposition of a Heisenberg odometer, as a function of the defining subgroups. Along the way we also provide necessary and sufficient conditions for a Z^d odometer to be a product odometer as defined by Cortez.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
