无理中性吸引子的维数悖论

双临界无理无关吸引子的重整化算术几何模型

Arithmetic Geometric Model for the Renormalisation of Bi-critical Irrationally Indifferent Attractors

本文提出了一种用于双临界无理无关吸引子重整化的算术几何模型，结合旋转数和临界点角度，构建了局部动力学拓扑模型并分析了最大不变集的拓扑结构。

Linearizability of cubic polynomials at irrationally indifferent fixed points

This paper has been withdrawn by the authors due to a more research needed to estimate $h^{*}_{n}(b)$ which really should be written as $h^{*}_{n}(b,A)$ for $b\in Γ(t)$.

无理中性吸引子重整化的算术几何模型

Arithmetic geometric model for the renormalisation of irrationally indifferent attractors

本研究针对无理中性不动点的重整化问题，构建了融合旋转数精细算术性质的几何模型，进而建立对应全纯映射附近动力学的拓扑模型，并阐明了其最大不变集的拓扑与映射动力学特性。

无理中性吸引子的拓扑

Topology of irrationally indifferent attractors

本研究针对带无理中性不动点的全纯系统的后临界集，基于旋转数的算术性质证明其拓扑存在三分性，揭示了西格尔圆盘内闭不变曲线随旋转数变化的退化机制。

Linearizability of Saturated Polynomials

Brjuno and Rüssmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this is sharp for all rational functions of degree at least 2, i.e., that non-Möbius rational functions cannot have Siegel disks with non-Brjuno rotation numbers. We prove that Douady's conjecture holds for the class of polynomials for which the number of infinite tails of critical orbits in the Julia set equals the number of irrationally indifferent cycles. As a …

Biaccessiblility in quadratic Julia sets II: The Siegel and Cremer cases

Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $α$. Let $z$ be a biaccessible point in the Julia set of $f$. Then: 1. In the Siegel case, the orbit of $z$ must eventually hit the critical point of $f$. 2. In the Cremer case, the orbit of $z$ must eventually hit the fixed point $α$. Siegel polynomials with biaccessible critical point certainly exist, but in the Cremer case it is possible that biaccessible points can never exist. As a corollary, we conclude that the set of biaccessible points in the Julia set of a Siegel or Cremer quadratic polynomial has Bro…