3 research outputs found
Integrable and Chaotic Systems Associated with Fractal Groups
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
Schr\"odinger 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.Comment: 48 pages, 15 figure