论文标题
与分形组相关的可集成和混乱系统
Integrable and Chaotic Systems Associated with Fractal Groups
论文作者
论文摘要
分形小组（也称为自相似群体）是上个世纪80-S发现的一组群体，目的是解决数学中的一些著名问题，包括提出的问题提出了冯·诺伊曼（Von Neumann）的问题（关于非元素不足的性能（与Banach-Tarski Paradox的研究相关联，与Banach-Tarski Paradox的研究相关联），而John Milnor对expeent的研究和整体的成长构成了互动的群体和互动的成长。分形群在数学的各个领域中都出现，包括随机步行的理论，全态动力学，自动机理论，操作员代数等。它们与混乱，准晶体，分形和随机schrödingeroberators的理论有关系。重要的发展之一是它们与多维动力学，铅笔的关节光谱理论以及在图上的Laplace运算符的光谱理论之间的关系。本文可以快速访问这些主题，对通过Schur补体产生的多维理性图进行计算和分析，这些图在一些重要示例中，包括第一组中级生长及其过度群，其中包含了对二分法“可综合性 - chaotic”的讨论，并提出了一种可能的概率方法来研究讨论的问题。
Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor's question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One of important developments is the relation of them to the multi-dimensional dynamics, theory of joint spectrum of pencil of operators, and spectral theory of Laplace operator on graphs. The paper gives a quick access to these topics, provide calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains discussion of the dichotomy "integrable-chaotic" in the considered model, and suggests a possible probabilistic approach to the study of discussed problems.
