论文标题
自相似群体和全体形态动力学：重新归一化，集成性和频谱
Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum
论文作者
论文摘要
在本文中，我们从动力学和代数几何观点上探索了几个自相似组（Grigorchuk，Lamplighter和河内组）的Schreier图上的Laplacian的光谱测量。对于这些图，经典的Schur重归其化转换作用于适当的光谱参数，作为两个变量中的有理图。我们表明，所讨论的光谱可以通过在相应的理性图下（导致我们达到光谱电流的概念）来解释通过某些代数曲线的一系列迭代回调的切片的渐近分布。我们遵循一个动力标准，以实现频谱的离散性。在离散频谱的情况下，给出了有限尺度近似值与极限光谱度量的精确收敛速率。对于正在考虑的三组，相应的有理图恰好是在一个变量中的多项式上进行的。我们揭示了这种整合性现象的代数几何特性。
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 phenomenon.
