Topologie locale des méthodes de Newton cubiques: plan dynamique

For a cubic Newton map N, we obtain the following theorems: 1) The boundary of the immediate basin of each fixed critical point is locally connected. 2) The Julia set J(N) is locally connected provided either N has no irrational indifferent periodic point or N has no Siegel disc and the orbit of the non-fixed critical point doesn 't accumulate on the boundary of the fixed immediate basins. In particular, in contrast with Julia sets of polynomials, J(N) can be locally connected even if N has a periodic Cremer point.The proofs rely on the construction of articulated rays which are very special simple arcs landing on J(N).     

Julia集的淹没分支

In this note,it is shown that if a rational function fofdegree≥2 has a nonempty set of buried points ,then for a generic choice of the point z in the Julia set ,z is a buried point ,and if the Julia set is disconnected,it has uncountably many buried components.     

The topology of Julia sets for polynomials

We prove that wandering components of the Julia set of a polynomial are singletons provided each critical point in a wandering Julia component is non-recurrent. This means a conjecture of Branner-Hubbard is true for this kind of polynomials     

Building blocks for Quadratic Julia sets

We obtain results on the structure of the Julia set of a quadratic polynomial with an irrationally indifferent fixed point in the iterative dynamics of . In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set : there exists a nowhere dense subcontinuum such that , is the union of the impressions of a minimally invariant Cantor set of external rays, contains the critical point, and contains both the Cremer point and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and contains no periodic points. In both cases, the Julia set is the closure of a skeleton which is the increasing union of countably many copies of the building block joined along preimages of copies of a critical continuum containing the critical point. In addition, we prove that if is any polynomial of degree with a Siegel disk which contains no critical point on its boundary, then the Julia set is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

     

The Julia sets of quadratic Cremer polynomials

We study the topology of the Julia set of a quadratic Cremer polynomial P. Our main tool is the following topological result. Let be a homeomorphism of a plane domain U and let T⊂U be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x, then T contains a non-repelling fixed point. Given P, two angles θ,γ are K-equivalent if for some angles x₀=θ,…,xn=γ the impressions of xi−1 and xi are non-disjoint, 1?i?n; a class of K-equivalence is called a K-class. We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense Gδ-set of angles each of which is a K-class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent.     

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．     

Weak hyperbolicity on periodic orbits for polynomials

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n^5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.     

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.”     

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 finest monotone map φ onto a locally connected continuumJP_∼, i.e. a monotone map φ:J→JP_∼ such that for any other monotone map ψ:J→J^′ there exists a monotone map h with ψ=h°φ. Then we extend φ onto the complex plane C (keeping the same notation) and show that φ monotonically semiconjugates PC_| to a topological polynomialg:C→C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map φ not to collapse all of J into a point.     

Proof of the Branner-Hubbard conjecture on Cantor Julia sets

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992. This work was supported by the National Natural Science Foundation of China     

Reverse bifurcation and fractal of the compound logistic map

The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot–Julia set of compound logistic map. We generalize the Welstead and Cromer’s periodic scanning technology and using this technology construct a series of Mandelbrot–Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot–Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.     

Indecomposable continua and the Julia sets of polynomials, II

We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree d?2 to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors.We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable.     

Symmetries of the Julia sets of Newton’s method for multiple root

The symmetries of Julia sets of Newton’s method is investigated in this paper. It is shown that the group of symmetries of Julia set of polynomial is a subgroup of that of the corresponding standard, multiple and relax Newton’s method when a nonlinear polynomial is in normal form and the Julia set has finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relax Newton’s method to be horizontal line is obtained.     

Dynamics of McMullen maps

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to the question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.     

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.     

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.     

Harmonic measure on the Julia set for polynomial-like maps

For a generalized polynomial-like mapping we prove the existence of an invariant ergodic measure equivalent to the harmonic measure on the Julia set J( f). We also prove that for polynomial-like mappings the harmonic measure is equivalent to the maximal entropy measure iff f is conformally equivalent to a polynomial. Next, we show that the Hausdorff dimension of harmonic measure on the Julia set of a generalized polynomial-like map is strictly smaller than 1 unless the Julia set is connected. Oblatum 24-IV-1995 & 22-VII-1996     

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.     

Attractors and recurrence for dendrite-critical polynomials

We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.     

Julia集及其Hausdorff维数的连续性

对于d≥2,考虑多项式族Pc=Zd+c,c∈C.Kc={z∈C|{Pcn(z)}n≥0有界}为Pc的填充Julia集,Jc=(?)Kc为其Julia集.HD(Jc)为Jc的Hausdorff维数.设ω(0)为Pc0的临界点0的轨道的聚点集.我们假定Pc0在ω(0)上是扩张的,且O∈Jc0,|c0|>ε>0.如果一序列Cn→c0,则Jcn→Jc0,Kcn→Jc0,在Hausdorff拓扑下.如果存在一常数C1>0和一序列cn→c0,使得d(cn,Jc0)≥C1|cn-c0|1+1/d,则HD(Jcn)→HD(Jc0).这里d(cn,Jc0)为cn与Jc0间距离.