Integrable and Chaotic Systems Associated with Fractal Groups

TL;DR

This paper explores the connection between fractal groups and complex systems, analyzing their integrable and chaotic behaviors, and introduces a probabilistic approach to understanding their spectral and dynamical properties.

Contribution

It provides a comprehensive analysis of multi-dimensional rational maps related to fractal groups and discusses the integrable-chaotic dichotomy with new computational insights.

Findings

Analysis of rational maps in fractal groups

Discussion of integrable vs. chaotic dynamics

Probabilistic approach to spectral problems

Abstract

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.