Julia集的拓扑复杂性

研究有理函数及整函数Julia集的拓扑结构,刻画了有理函数Julia集的复杂性,展示了整函数在Fatou集上的动力学性质对其Julia集拓扑复杂性的影响.     

Julia集的生成

该文运用Hausdorff意义下的极限研究了有理动力系统的Julia集, 用新的思路证明了几个关于Julia集的定理. 为计算机作Julia集提供了更多的理论根据.     

牛顿变换Julia集的对称性

主要讨论多项式的牛顿变换Julia集的对称性问题．利用复动力系统理论,证明了多项式P（z）的Julia集的对称群是其牛顿变换Np（z）的Julia集的对称群的子群．获得了Julia集为一水平直线的充分必要条件．     

具有共形迭代函数系统有理函数Julia集的性质

本文研究了几何有限有理函数的复解析动力性质.利用Markov划分与共形迭代函数系统的理论,获得了几何有限有理函数Julia集的性质.如有理函数是几何有限的,且Julia集是连通的,则Julia集的Hausdorff维数为1当且仅当Julia集为一圆周或直线的一段.     

有理函数及整函数Julia集上的淹没点

对有理函数证明了:如果Julia集不连通,Fatou集没有完全不变分支,则Julia集存在淹没分支;对有限型超越整函数证明了:如果Fatou集不连通,则Julia集上存在淹没点集的无界连续统.     

拟多项式的迭代

讨论了更广泛的拟多项式映射,研究了拟多项式的迭代,证明了关于逃逸集,充满 Julia集和Julia集的几个定理.推广了多项式动力系统的相关结果.     

重根牛顿变换的Julia集

作者分析了重根牛顿变换的Julia集理论,并利用迭代法构造了标准牛顿变换、松弛牛顿变换和重根牛顿变换的Julia集．采用实验数学方法,作者得出如下结论:(1)函数f(z)=zα(zβ-1) 的三种牛顿变换Julia集的中心为原点目具有β倍的旋转对称性; (2)三种牛顿变换Julia集的重根吸引域对α具有敏感的依赖性;(3)由于的零点是松弛牛顿变换的中性或斥性不动点,故松弛牛顿变换的Julia集中不存在单根吸引域;(4)由于∞点不是重根牛顿变换的不动点,故重根牛顿变换的Julia集中多为重根和单根吸引域;(5)重根牛顿法受计算误差影响最小,松弛牛顿法次之, 标准牛顿法最大．     

半双曲有理映射的Julia集

本文证明了半双曲有理映射Julia集的局部连通性，推广了Carleson－Jones－Yoccoz关于多项式的结果，同时还考虑了半双曲有理映射Julia集的面积问题．     

线性微分方程的整函数解及其导数的Julia集的公共极限方向（英文）

本文主要研究了线性微分方程解的Julia集的极限方向问题.利用值分布论的方法,在一定条件下,获得了这类方程非平凡解的Julia集的极限方向分布的下界,推广了相关结果.     

超整函数族Julia集的Hausdorff维数

Stallard曾经用一族特殊的整函数说明了:超越整函数的Julia集的Hausdorff维数可以无限接近1.本文证明了该函数族的随机迭代的Julia集的Hausdorff维数也可无限接近于1.另一方面,对任意自然数M及任意实数d∈(1,2),本文给出了M个元素的整函数族其随机迭代的Julia集的Hausdorff维数等于d.     

Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set

We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. By using a more general strategy, we prove that all of these Julia sets fail to be arc-wise connected, and that their critical point is non-accessible. Using the first strategy we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists spiral.”     

Alternate superior Julia sets

Alternate Julia sets have been studied in Picard iterative procedures. The purpose of this paper is to study the quadratic and cubic maps using superior iterates to obtain Julia sets with different alternate structures. Analytically, graphically and computationally it has been shown that alternate superior Julia sets can be connected, disconnected and totally disconnected, and also fattier than the corresponding alternate Julia sets. A few examples have been studied by applying different type of alternate structures.     

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines.Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.     

On a topological closeness of perturbed Julia sets

In the present work we expand our previous work in [1] by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.     

Julia sets that are full of holes

Conclusion  Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Although there are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure bigger than zero have been found. Shishikura’s surprising result (1998) shows that there are other Julia sets of dimension 2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of holes, or porous, provides a bound on the upper box dimension, but this has so far been possible only for special classes of Julia sets. Mean porosity and mean e-porosity, both found in Koskela and Rohde (1997), provide better dimension bounds; nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used in some cases when it is not possible to prove porosity. In the end, we do not know in general which Julia sets are porous and which are not. In fact, forJ _R, little is known about its dimension or measure. There is much left to explore.     

Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

We prove that Collet–Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.     

Preservation of external rays in non-autonomous iteration

We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain properties of the corresponding Julia sets are preserved. In particular, we show that if the sequence is hyperbolic and all the Julia sets are connected, then the whole basin at infinity moves holomorphically. This extends also to the landing points of external rays and the resultant holomorphic motion of the Julia sets coincides with that obtained earlier in [9] using grand orbits. In addition, we have combinatorial rigidity in the sense that if a finite set of external rays separates the Julia set for a particular parameter value, then the rays with the same external angles separate the Julia set for every parameter in the same hyperbolic component.     

Non-autonomous Julia sets with escaping critical points

We consider non-autonomous iteration which is a generalization of standard polynomial iteration where we deal with Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. In this paper, we look at examples where all the critical points escape to infinity. In the classical case, any example of this type must be hyperbolic and there can be only one Fatou component, namely the basin at infinity. This result remains true in the non-autonomous case if we also require that the dynamics on the Julia set be hyperbolic or semi-hyperbolic. However, in general it fails and we exhibit three counterexamples of sequences of quadratic polynomials all of whose critical points escape but which have bounded Fatou components.     

一个非解析复映射的广义Julia集

推广了Michelitsch和Rossler所提出的由一个简单非解析映射所构造Julia集的方法,并由推广的复映射,构造出一系列实数阶的广义Julia集(简称广义J集)． 利用复变函数理论和计算机制图相结合的实验数学的方法,对广义J集的结构和演化进行了研究,结果表明： ①广义J集的几何结构依赖于参数α、R和c; ②广义J集具有对称性和分形特征; ③小数阶广义J集出现了错动和断裂,且其演化过程依赖于相角主值范围的选取.