Se p 20 02 From Fractal Groups to Fractal Sets (original) (raw)
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Integrable and Chaotic Systems Associated with Fractal Groups
Entropy
Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of 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 important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dim...
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Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum
arXiv: Group Theory, 2020
In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of discrete spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability ph...
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On the spectrum of Hecke type operators related to some fractal groups
Arxiv preprint math/9910102, 1999
Abstract: We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common``finite approximation''method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also``substitutional graphs''. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular ...
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