与分形群相关的可积混沌系统
Integrable and Chaotic Systems Associated with Fractal Groups
摘要：
分形群（也称为自相似群）是第一作者在上世纪80年代发现的一类群，目的是解决数学中的一些著名问题，包括冯·诺依曼提出的关于非基本顺从性的问题（与巴拿赫-塔斯基公园周围的研究有关）约翰·米尔诺关于多项式和指数之间的中间增长群的存在性问题。分形群出现在数学的各个领域，包括随机游动理论、全纯动力学、自动机理论、算子代数等。它们与混沌理论、准晶体、分形和随机薛定谔算子有关。其中一个重要的发展是它们与多维动力学、算子束的联合谱理论和Laplace算子的谱理论的关系。本文给出了一个快速访问这些主题，提供计算和分析的多维有理映射所产生的Schur补在一些重要的例子，包括第一组的中间增长和它的扩群，包含讨论的二分法“可积混沌”在所考虑的模型，并提出了一个可能的概率方法来研究所讨论的问题。
英文摘要：
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.
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