Self-Similar Groups and Holomorphic Dynamics: Renormalization, Integrability, and Spectrum

Q3 Mathematics

Arnold Mathematical Journal Pub Date : 2023-01-05 DOI:10.1007/s40598-022-00223-0

N.-B. Dang, R. Grigorchuk, M. Lyubich

引用次数: 6

Abstract

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 atomic 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.

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自相似群与全纯动力学：重整化、可积性和谱

本文从动力学和代数几何的角度探讨了几个自相似群(Grigorchuk、Lamplighter和Hanoi群)在Schreier图上的拉普拉斯谱测度。对于这些图，经典的舒尔重整化变换作用于适当的谱参数作为两个变量的有理映射。我们表明，所讨论的谱可以被解释为切片的渐近分布，通过在相应的有理映射下的某些代数曲线的迭代回调线(导致我们得到谱电流的概念)。接着给出了谱离散性的动力学判据。对于原子光谱，给出了极限光谱测度的有限尺度近似的精确收敛速率。对于所考虑的三组，相应的有理映射恰好是在一个变量的多项式上编织的。我们揭示了这种可积现象的代数几何性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.