The measure of quadratic Julia-Lavaurs sets with a bounded type virtual Siegel disk is zero

某些具有Siegel盘的有理映射的Julia集的局部连通性

Local connectivity of Julia sets of some rational maps with Siegel disks

本文证明有理映射长迭代在有界型Siegel盘边界附近扩张，借此将Petersen关于二次Siegel多项式Julia集局部连通性的结果推广至一般情形，且证明无需使用puzzles。

关于扰动Siegel盘的面积率

On the area rate of perturbed Siegel disks

本文研究了扰动Siegel盘的面积率，证明了在构造具有正面积的二次Julia集时，Buff和Chéritat对旋转数添加的限制性条件可以被删除，从而能够构造出更多此类Julia集。

On computational complexity of Siegel Julia sets

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.

具有Siegel盘的一类超越整函数Julia集的局部连通性

Local connectivity of Julia sets of some transcendental entire functions with Siegel disks

本文利用拟Blaschke乘积长迭代的弱扩张性质，证明了一类具有有界型Siegel盘的超越整函数Julia集局部连通，并给出正弦函数实例及含渐近值情形的存在性结果。

三次多项式参数空间中的抛物内爆

Parabolic Implosion in the Parameter Space of Cubic Polynomials

本文研究三次多项式参数空间中抛物内爆引起的分岔轨迹丰富化的拓扑结构，并将其与 P. Lavaurs 系统研究过的二次多项式 Julia 集丰富化建立联系。

Quadratic Julia Sets with Positive Area

We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite) renormalizations.

有界类型塞格尔多项式的刚性

Rigidity of bounded-type Siegel polynomials

本文研究了具有有界类型塞格尔盘的多项式刚性，证明了其朱利亚集的局部连通性和拟共形刚性，并验证了组合刚性猜想。

有界型三次Siegel多项式的刚性

Rigidity of bounded type cubic Siegel polynomials

本文证明有界型旋转数的不可重整化三次Siegel多项式的组合等价蕴含共形等价，并由此验证了Blokh-Oversteegen-Ptacek-Timorin关于Zakeri单参数切片中不可重整化映射轨迹拓扑结构的猜想。

Julia sets as buried Julia components

Let $f$ be a rational map with degree $d\geq 2$ whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map $g$ such that $g$ contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of $f$ on the Julia set if and only if $f$ does not have parabolic basins and Siegel disks. If such $g$ exists, then the degree can be chosen such that $\text{deg}(g)\leq 7d-2$. In particular, if $f$ is a polynomial, then $g$ can be chosen such that $\text{deg}(g)\leq 4d+4$. Moreover, some quartic and cubic rational maps…

Julia集为广义Sierpiński垫圈的有理映射

Rational maps whose Julia sets are generalized Sierpiński gaskets

本文证明了包含Siegel盘、Cremer点或无穷次重正规化的有理映射也能产生广义Sierpiński垫圈Julia集，由此确立了正面积垫圈Julia集的存在性，并给出了判定其存在性的准则。

Constructing Non-Computable Julia Sets

We completely characterize the conformal radii of Siegel disks in the family $$P_θ(z)=e^{2πiθ}z+z^2,$$ corresponding to {\bf computable} parameters $θ$. As a consequence, we constructively produce quadratic polynomials with {\bf non-computable} Julia sets.

超越整函数的谢尔宾斯基地毯 Julia 集

Sierpiński carpet Julia sets of transcendental entire functions

本文证明谢尔宾斯基地毯参数在有限型超越整函数族的双曲分支点集中稠密，并构造了具有此类 Julia 集且含吸引盆、抛物盆、Siegel 盘或游荡域的具体函数。

Siegel Disks and Periodic Rays of Entire Functions

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk such that f restricts to a homeomorphism of the boundary. We show that the Siegel disk is bounded. Using a result of Herman, we deduce that if additionally the rotation number of the Siegel disk is Diophantine, then its boundary contains a critical point of f. Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk $U$ whose boundary contains no singular values, then the condition that f is a homeomorphism of the boundary of U is automat…

Area and Hausdorff dimension of Sierpiński carpet Julia sets

We prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $s\in(1,2)$, we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly $s$.

On Fibers and Renormalization of Julia Sets and Multibrot Sets

We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of fibers of Mandelbrot and Multibrot sets is preserved under tuning maps and other (partial) homeomorphisms. Similarly, we show for unicritical polynomials that triviality of fibers of …

Local connectivity of the Mandelbrot set at some satellite parameters of bounded type

We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.

复Hénon映射的Julia集的可计算性：吸引与中性周期轨道的作用

Computability of Julia sets for complex Hénon maps: The role of attracting and neutral cycles

该研究将复Hénon映射Julia集的可计算性从双曲情形扩展到含吸引盆地的非双曲情形，基于鞍点稳定流形的算法实现，同时指出中性动力学下高维存在部分类似一维的可计算性二分法。

Modeling core parts of Zakeri slices I

The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices. We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and agrees with th…

Locally connected models for Julia sets

Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a \emph{finest monotone map $\ph$ onto a locally connected continuum $J_{\sim_P}$}, i.e. a monotone map $\ph:J\to J_{\sim_P}$ such that for any other monotone map $ψ:J\to J'$ there exists a monotone map $h$ with $ψ=h\circ \ph$. Then we extend $\ph$ onto the complex plane $\C$ (keeping the same notation) and show that $\ph$ monotonically semiconjugates $P|_{\C}$ to a \emph{topological polynomial $g:\C\to \C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proo…

Siegel多项式与抛物二次多项式的Mating

Mating Siegel and parabolic quadratic polynomials

本文证明了有界型无理数和有理数对应的二次多项式共形可配对且在共轭下唯一，并据此证明了一类带抛物点的Siegel有理映射的Julia集局部连通。

Complementary components to the cubic Principal Hyperbolic Domain

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_λ$ with the slice $\mathcal{F}_λ$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $λ$ at $0$. We show that any bounded domain $\mathcal{W}$ of $\mathcal{F}_λ\setminus\mathcal{P}_λ$ consists of $J$-stable polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel capture} type (then $f\in \mathcal{W}$ has an invariant Siegel domain $U$ around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and $f^k(V)=U$ for some $k>0$) or of \e…

关于具有中性不动点的二次多项式的后临界集

On postcritical sets of quadratic polynomials with a neutral fixed point

本文基于 Dudko 和 Lyubich 的伪 Siegel 盘理论，实现了具有中性不动点的二次多项式后临界集在一般情形下的控制，推广了此前仅限于特定旋转数类型的研究成果。