发表机构
Brown University（布朗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究刚性旋转上扇形重整化操作的算术和组合性质，采用修正连分数框架，分析其性质、自然扩张及时间（半）群概念，展示无理旋转双无限扇形重整化塔的打包方式，为相关全纯映射几何性质研究提供基础工具。
AI 中文摘要
本笔记旨在系统地发展刚性旋转上扇形重整化操作的基本算术和组合性质。我们采用适用于扇形重整化的修正连分数的特定框架并分析其性质。通过允许无限首次返回时间，该框架产生具有通用性质的动态紧致化。我们还讨论了相应的自然扩张并引入了时间（半）群的概念。例如，我们展示了无理旋转的双无限扇形重整化塔如何作为平移级联打包在单个动态平面内。本笔记将作为研究具有无理中性不动点的全纯映射，特别是中性二次多项式的扇形重整化几何性质的基础组合工具。
英文摘要
The goal of this paper is to systematically develop the fundamental arithmetic and combinatorial properties of the sector renormalization operation on rigid rotations. We employ the specific framework of modified continued fractions appropriate for sector renormalization and analyze their properties. By allowing infinite first return times, this framework yields a dynamical compactification of the space of irrationals called the parabolic compactification; we show that it is characterized by some universal properties. We also discuss the corresponding natural extension and introduce the continuant group, the time semigroup, and topological cascades. For example, we demonstrate how a bi-infinite tower of sector renormalizations of irrational rotations can be packaged within a single dynamical plane as a cascade of translations. This will serve as a foundational combinatorial tool for studying the geometric properties of sector renormalizations of holomorphic maps with irrationally indifferent fixed points, particularly neutral quadratic polynomials.
Comments40 pages, 6 figures
