摘要
Riemann球面上复解析动力系统的研究是许多数学家和数学工作者感兴趣的课题,它起源于上世纪初,P. Fatou和G. Julia受Newton迭代法以及Mobius变换群的子群的极限集的启发独立地作了系列研究,形成了Riemann球面上复解析动力系统的研究思想和经典的Fatou-Julia理论.近几十年来,电子计算机飞速发展给研究者提供了有效的工具,使得这一领域的研究方兴未艾.借助于快速计算和精确模拟,研究者可以更直观地观察到Julia集漂亮丰富而复杂的拓扑结构,更准确的把握动力学性质,很多构想因此而实现,并进一步促进这一学科的发展,使之成为复分析的主要研究方向之一.许多国际上著名的数学家如A. Douady, J. H. Hubbard, D. Sullivan, W. Thurston, I. N. Baker和J-C. Yoccoz等均在这一研究领域作出了杰出贡献.
我们根据轨道极限对初始状态的敏感性把Riemann球面划分成两类集合：如果迭代序列{fn}在某点z是Montel意义下的正规族,则称点z是正规的,所有正规点形成的集合为Fatou集,其余集为Julia集,它同时也是所有排斥周期点集的闭包.
关于Fatou集,P. Fatou曾经猜想,对有理函数来说,不存在游荡的Fatou连通分支,后来Sullivan引进了有理函数的拟共形形变证明了这个猜想,并对Fatou分支进行了分类,从而有理函数在Fatou集上的动力系统有了完整的刻画[71].此后,Shishikura利用拟共形手术给出了有理函数Fatou循环域个数的精确的上界估计[68].
借助于Caratheodory定理,若Julia集拥有良好的拓扑性质如局部连通性等则可以继承Fatou域中有序的动力模型,因此,有理函数Julia集局部连通性的研究是很关键的工作,对于探索动力学平面进一步的结构非常重要.许多数学家和数学工作者倾注于这个课题：如多项式情况下Douady-Hubbard的研究[30]; Yoccoz [57]利用拼图技术证明了当二次多项式的所有周期点都是排斥周期点并且函数不是无穷可重整的时对应的Julia是局部连通的;Douady找到了一个无穷可重整的二次多项式,其Julia集不是局部连通的[57]; Roesch和尹永成老师证明了多项式的有界吸性或抛物Fatou分支的边界是一条简单闭曲线,进一步,若边界上没有抛物点和回归的临界点则边界是一个拟圆周[67]Roesch[66]研究了三次多项式求根过程中Newton迭代公式的动力系统,证明除了一些特殊情况外Julia集都是局部连通的；继双曲和次双曲的有理函数之后[32][42],谭蕾和尹永成老师[73]证明了更广的几何有限有理函数Julia集的局部连通性;Carleson-Jones-Yoccoz[15]证明了半双曲多项式的Fatou分支都是John域从而是局部连通的,Mihalache[54]把这一结果推广到半双曲的有理函数.
在研究多项式的局部连通性时最重要的工具是‘'Branner-Hubbard-Yoccoz"拼图,对于有理函数,找到合适的拼图比较困难,有时甚至找不到拼图,人们一般选择某类特殊的有理函数进行研究,到目前为止用拼图方法研究有理函数Julia集局部性的例子有[66]及[63].
作为对最简单的m次单项式z(?)zm的有理扰动,McMullen研究了有理函数Fλ(z)=Zm+λ/zl,λ∈c*=c\{0},1/m+1/l<1. (0.0.1)的Julia集,证明当参数λ充分小时Fλ的Julia集是Cantor环[52].这种类型的Julia集是多项式不具备的,此外,当1/m+1/e≥1时Cantor环的情况也不会发生[27].之后,这族函数的动力学性质引起了很多研究者的兴趣,如Devaney及其合作者[8,9,18-23,25-29],Roesch[65],Steinmetz[69,70]等.他们发现并证明了这族函数Julia集的丰富拓扑性质,并对参数空间结构进行了细致研究.这些文章中绝大多数针对m=e的情况作了讨论,也就是Fλ(z)=zm+λ/zm,λ∈c*,m≥2, (0.0.2)此时函数族具有更加丰富的对称性.
根据Devaney,Look以及Uminsky[27]的研究结果,当自由临界轨道逃逸到无穷远点∞时,(0.0.1)或(0.0.2)中的有理函数Fλ的Julia集要么是Cantor集,要么是Cantor环,要么是Sierpinski曲线(逃逸三分定理).在该函数族中,某些临界有限的情况也对应着Sierpinski曲线的Julia集[18,25].以上均对函数的临界轨道作了限制使其具有相对简单的特性.然而,当临界点轨道比较复杂比如回归的时会怎样?Julia集何时会再度成为一条Sierpinski曲线?在[24]中,Devaney提出了一个开放问题：当Julia集不是Cantor集时,(0.0.2)中函数Fλ无穷远点的直接吸引盆的边界是否一直是一条Jordan曲线?此外,当Julia集是连通的时它是否会局部连通呢?注意到连通性和局部连通性是一个集合成为一条Seirpinski曲线的必要条件.
当参数λ是一个复数且m=e≥3时,Fλ的Fatou集和Julia集类似的拓扑性质在[63]中已作讨论,该文中利用拼图的方法证明了Julia集的局部连通性,但当参数是正实数时不能构造类似的拼图,所以正实参数情形我们需要另作讨论.本文讨论正实参数下(0.0.1)中有理函数Fλ的Julia集和Fatou分支的拓扑性质,并且不局限于m=e.证明了当λ>0时,如果Fλ的Julia集不是一个Cantor集,则无穷远点∞的直接吸引盆Bλ始终是一个Jordan区域,这不仅在正实参数下回答了Devaney对于(0.0.2)的问题,还把结果推广到(0.0.1)的情形.文中也讨论了Bλ进一步的性质.除了边界上有抛物不动点这种情况外,单连通的Bλ都是一个拟圆盘.此外,如果凡的Julia集是连通的则它一定是局部连通的且所有的Fatou分支都是Jordan区域.我们得到关于Fλ的Julia集拓扑更细致全面的刻画,给出Julia集何时为Sierpinski曲线的完整描述.
式(0.0.1)中的有理函数Fλ(z)通过z(?)(1/A)zd半共轭到fn(z)=ηzm(1+1/z)d,1/m+1/l<1. (0.0.3)这里η=Am-1,d=m+e.我们通过研究fη的动力系统来导出Fλ动力系统的相应性质在操作上更加可行,这是由于有理函数fη只有一个自由临界点,该自由临界点是Fλ的d个临界点在z(?)(1/λ)zd下的像.
当fη限制在正实轴上时,记它与直线y=x正好相切时对应的参数为η0；当1/m+1/e<1时,存在唯一的参数η*∈(0,η0)使得临界值的像正好等于fη(x)-x的较大零点.记Bη是∞在fη下的直接吸引盆,我们首先确定Bη何时为Jordan区域.定理1.如果η>η0,那么(?)Bη=J(fη)是一个Cantor集.如果0<η≤η0,那么(?)Bη是一条Jordan曲线.
定理1在实参数情况下回答了Devaney在[24]提出的问题,并把结论推广到了m≠l的情形.事实上当Bη是单连通时除了个别情况外它具有更好的几何性质.定理2.如果0<η<η0,则Bη是一个拟圆盘.如果η=η0,则Bη不是拟圆盘.
对于整个Julia集,我们知道η>η0时J(fη)是Cantor集,当0<η<η*时J(fη)是Cantor环,当η*≤η≤η0时J(fη)是连通的,对于最后一种情况,定理3.当矿η*≤η≤η0时,J(fη)是连通且局部连通的,当η*<η≤η0时,每一个Fatou分支都是一个Jordan区域.
进一步,我们对J(fη)成为Sierpinski曲线给出一个完整的刻画,首先证明1/m+1/l<1时每一个η∈[η*,η0]都对应一个参数c=c(η)∈[-2,1/4]使得J(fη)包含了一个二次多项式pc(z)=z2+c的Julia集的嵌入像.从而得到,定理4.设η*≤η≤η0,则J(fη)是一条Sierpiriski曲线当且仅当满足以下任一条件(ⅰ)η≠η*且c(η)不落在任何Mandelbrot集双曲分支的闭包内;(ⅱ)c(η)落在MMandelbrot集周期大于1的主干双曲分支内；(ⅲ)c(η)是某个周期大于1的主干双曲分支的根.
关于Mandelbrot集的主干双曲分支以及它的根的概念将在第四章中作介绍.
至此我们已经知道Bη在单连通的情况下是Jordan域,其边界是一条Jordan闭曲线,并且除了个别情况(η=η0)外这个单连通的Bη还是拟圆.如果Bη不是单连通的,则必然是无穷连通的,对应着Julia集是Cantor集的情形,此时Bη作为唯一的Fatou分支会有怎样的性质?鉴于拟圆盘是单连通情形下的一致域,我们猜想Fλ的Julia集是Cantor集时,Bλ也是一致域.
为了研究无穷连通的一致域,我们考虑一类典型的集合-自相似集,它是有限个压缩相似映射组成的迭代系统下的吸引子.我们有：定理5.Riemann球面上具有强开集条件的自相似集的余集是一致域.
随后我们研究更广的一类集合-自共形集,它们与自相似集的唯一区别在于迭代函数系统由有限个共形映射组成,同样强开集条件也能够使得其余集为一致域.即：定理6.Riemann球面上具有强开集条件的自共形集的余集是一致域.
在此基础上回到McMullen函数族,当它的Julia集是Cantor集时实际上是满足强开集条件的自共形集,不仅如此,其吸引子Julia集还是不变的.我们有：推论7.对于(0.0.1)中的函数Fλ,若J(Fλ)为Cantor集,则Bλ是一致域.
事实上,我们据此可以证明了一个更普遍的结论：推论8.如果双曲有理函数的Julia集为Cantor集,则其Fatou集是一致域.
The dynamics of the rational maps on the Riemann sphere is a topic that many mathematicians and researchers are interested in, which originated in the beginning of last century when P. Fatou and G. Julia both made series of study, with the ideas derived from the Newton method and the subgroup of the Mobius transformation group, which came into being the typical Fatou-Julia throry about the the complex dynamics on the Riemann sphere. Over the past few decades, the development of the computer provided efficient tools for the researchers and brought this field towards prosperity. With the help of rapid calculation and accurate simulation, the researchers become conscious of the rich and beautiful topological structure of the Julia set and hold the dynamical structure precisely, which promote the development of complex dynamics and make it one of the major research fields of complex analysis. Many international mathematics such as A. Douady, J. H. Hubbard, D. Sullivan, W. Thurston, I. N. Baker and J-C. Yoccoz made notable contributions to it.
We divides the Riemann sphere into two parts from the sensitivity of the limit of the orbit on the initial state:if an iterated sequence{fn} is normal on z in the sense of Montel, then we call z a normal point. All the normal points form the Fatou set with whose complement the Julia set, which is also the closure of all the repulsive periodic orbit.
P. Fatou once made a conjugate about the Fatou set that there is no wandering Fatou component for the rational maps. Sullivan proved this conjugate with the help of quasiconformal deformation and classified the Fatou components. Up to then, the dynamics of rational maps on the Fatou set is completely described. Later, Shishikura made a precise calculation of the upper bound of the number of Fatou cycles [68].
From Caratheodory theorem we know that if the Julia set has good topolog-ical properties such as local connectivity then it can inherit the ordered dynamic model. Thus to study the local connectivity is a key step to discuss further dy-namic structure of the Julia set. In fact, many researchers devoted to this work. In addition to Douady-Hubbard's study [30], Yoccoz [57] proved with the help of 'puzzle'that when all the periodic points are repulsive and the function is not in-finitely renormalized the Julia set of quadratic polynomials are locally connected. Douady found an infinitely renormalized quadratic polynomial whose Julia set is not locally connected [57]; Roesch and Y. C. Yin proved that for polynomials the boundary of a bounded attractive or parabolic Fatou component is a simply closed curve, what's more, if there is neither parabolic point nor recurrent critical point on the boundary then it is a quasicircle [67]. Roesch [66] studied the complex dynamics of Newton formula for cubic polynomials and proved that the Julia sets are locally connected with rare exceptions; After the hyperbolic and subhyperbolic rational maps [32] [42], geometrically finite rational maps, which are a kind of more general rational maps, were studied and proved to have locally connectied Julia set by L. Tan and Y. C. Yin [73]; Carleson-Jones-Yoccoz [15] proved that the Fatou com-ponents of simihyperbolic polynomials are John domains this are locally connected, which is generalized to simihyperbolic rational maps by Mihalache [54].
The so-called'Branner-Hubbard-Yoccoz'puzzles are very useful in the study of the local connectivity of Julia set. However, there are no suitable puzzles for all the rational maps. People usually choose some kind of them to study. Refer to [66] and [63] for the two examples of puzzle.
As a rational singular perturbation of the monomial z→zm, McMullen studied the Julia set of the rational maps Fλ(z)= zm+λ/zl, A∈C*= C\{0},1/m+1/l.(0.0.4) It was shown in [52] that the Julia set of Fλis a Cantor set of circles if the parameterλis sufficiently small, which neither exists for a polynomial nor the case 1/m+1/l≥1
[27]. For the whole family, rich topological structures of Julia sets and the bifurcation locus of this family were recently found by many authors, for example, Devaney and his collaborators [8,9,18-23,25-29], Roesch [65], Steinmetz [69,70]. In most of papers (but not all) listed above, attentions were focused on the case of m= l, that is, the family of rational maps Fλ(z)=zm+λ/zm,λ∈C*, m≥2, (0.0.5) for which there are rich symmetries.
By the famous work of Devaney, Look and Uminsky [27], it is known that when the free critical orbits escape to oo, the Julia set of the rational map Fλin (0.0.4) or (0.0.5) is either a Cantor set, or a Cantor set of circles, or a Sierpinski curve (the escape trichotomy theorem). The Sierpinski curve Julia set was also found for some post critical finite cases in this family [18,25]. In all of these cases, functions are restricted so that the critical orbits have simple behaviors. However, when the critical points have complicated orbits, for example, the critical orbits are recurrent, what will happen? When the Julia set can be a Sierpinski curve again? In [24], Devaney posed an open problem that if the boundary of the attracting basin of the infinity for Fa in (0.0.5) is always a Jordan curve when the Julia set is not a Cantor set? It can be also asked when the Julia set of Fλis locally connected if it is connected. Note that these two properties are the necessary conditions that the Julia set becomes a Seirpinski curve.
When the parameter A is a complex number, the similar topological properties of Fatou components and the Julia set of Fλis discussed in [63] for the case m≥3. The approach in [63] is based on a Yoccoz partition, but no suitable partition can be made forλ> 0. So the case with positive parameter need to be discussed separately. In this paper, we will discuss the topology of Fatou components and the Julia set of rational map Fλin (0.0.5) when the parameterλis positive, and is not limited to the case m=l. Letλ> 0. It is shown that the immediately attracting basin Bλof oo is always a Jordan domain if the Julia set of Fλis not a Cantor set, which not only answers Devaney's problem about (0.0.5) but also generalizes the result to (0.0.4) for positive parameter case. Further regularity of Bλis also discussed. It is obtained that Bλis a quasidisk unless there is a parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλis connected then it is also locally connected and all Fatou components are Jordan domains. Furthermore, we can get more detailed description for the topology of the Julia set of Fa whenλ> 0. In fact, we gives an complete description to the problem when the Julia set is a Sierpinski curve.
The rational map Fλ(z) in (0.0.4) is semi-conjugate to fη(z)=ηzm(1+1/z)d,1/m+1/l<1. under z→(1/λ)zd, whereη=λm-1 and d= m+l We study the dynamics of fηinstead of Fλbecause it has only one'free'critical point which is the image of Fλ's d critical points under z→(1/λ)zd. It will be more convenient technically and Fλinherits the dynamic property of fηwith no important information lost.
When the map fηis defined on the positive real axis, 1/m+1/l< 1, we denote byηo the parameter corresponding to the case that fηis tangent to y= x, and byη* the parameter corresponding to the case that the image of the critical value is equal to the greater zero of fη(x) - x. Let Bηbe the immediately attracting basin of oo for fη. We have Theorem 1.Ifη>η0, then (?)Bη= J(fη) is a Cantor set.If0<η<η0, then (?)Bηis a Jordan curve.
Theorem 1 answers Devaney's question [24] in the real positive parameter case and deduces the conclusion with m not necessary equal to l. And the simply con-nected Bv has more wonderful property with rare exception. Theorem 2. If 0<η<η0, then Bv is a quasidisk. If Ifη=η0, then Bv is not a quasidisk.
For the whole Julia set, we know J(fη) is a Cantor set whenη>η0, a Cantor set of circles when 0<η<η* and connected whenη<η0. We prove that Theorem 3.Ifη*≤η≤η0, then J(fn) is locally connected, and every Fatou component is a Jordan domain.
It is shown in Chapter 4 that for everyη∈[η*,η0] we have a corresponding parameter c= c(η)∈[-2,1/4] such that J(fη) contains an embedded image of the Julia set of quadratic polynomial pc(z)= z2+c. And: Theorem 4. Letη*≤η≤η0.The Julia set J(fη) is a Sierpinski curve if and only if one of the following conditions holds
(i)η≠η* and c(η) is not in the closure of any hyperbolic component of the Man-delbrot set;
(ii) c(η) is in a primitive hyperbolic component of the Mandelbrot set of period great than 1;
(iii) c(η) is the root of a primitive hyperbolic component of period great than 1.
For definitions of primitive hyperbolic component of the Mandelbrot and its root, see Chapter 5.
On the basic of above theorems, when Bλis simply connected, it is a Jordan domain whose boundary is a closed Jordan curve, and with rare exception (η=η0) this simply connected Bλis also a quasidisk. If Bλis not simply connected then it must be infinitely connected which corresponds to a Cantor Julia set. In this case, as the unique Fatou component, what properties will Bλhave? Since quasidisk is a simply connected uniform domain, we guess that Bλis also a uniform domain.
In order to study the infinitely connected uniform domain, we consider a kind of typical set-self-similar set, which is an attractor of a finite family of contracting similarities. We have
Theorem 5. The complement of a strong open set condition self-similar set on Riemann sphere is a uniform domain.
After this we study a kind of more general sets-self-conformal set. The only difference of self-conformal set from self-similar set is that the iterated function system is made up of finite conformal maps. Similarly, strong open set condition creates uniform domain.
Theorem 6. The complement of a strong open set condition self-conformal set on Riemann sphere is a uniform domain.
On the basic of these two theorem, we consider the Fatou component of the McMullen family Fλin (0.0.4). The Julia set is a strong open set condition self-conformal set when it is a Cantor set. Furthermore the attractor is invariant.
Corollary 7. For the function Fλin (0.0.4), if J(Fλ) is a Cantor set, then Bλis a uniform domain.
In fact, we can prove a more general conclusion: Corollary 8.If the Julia set of a hyperbolic rational map is a Cantor set, then the only Fatou component is a uniform domain.
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