| 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 F?(z)= zm+?/zl, A?C*= C\{0},1/m+1/l.(0.0.4) It was shown in [52] that the Julia set of F?is a Cantor set of circles if the parameter?is 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 F?(z)=zm+?/zm,??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 F?in (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 F?is 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 F?is 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 for?> 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 F?in (0.0.5) when the parameter?is positive, and is not limited to the case m=l. Let?> 0. It is shown that the immediately attracting basin B?of oo is always a Jordan domain if the Julia set of F?is 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 B?is also discussed. It is obtained that B?is a quasidisk unless there is a parabolic fixed point on the boundary of B?. It is also shown that if the Julia set of F?is 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 when?> 0. In fact, we gives an complete description to the problem when the Julia set is a Sierpinski curve.The rational map F?(z) in (0.0.4) is semi-conjugate to f?(z)=?zm(1+1/z)d,1/m+1/l<1. under z?(1/?)zd, where?=?m-1 and d= m+l We study the dynamics of f?instead of F?because it has only one'free'critical point which is the image of F?'s d critical points under z?(1/?)zd. It will be more convenient technically and F?inherits the dynamic property of f?with no important information lost.When the map f?is defined on the positive real axis, 1/m+1/l< 1, we denote by?o the parameter corresponding to the case that f?is tangent to y= x, and by?* the parameter corresponding to the case that the image of the critical value is equal to the greater zero of f?(x) - x. Let B?be the immediately attracting basin of oo for f?. We have Theorem 1.If?>?0, then (?)B?= J(f?) is a Cantor set.If0<?<?0, then (?)B?is 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<?<?0, then Bv is a quasidisk. If If?=?0, then Bv is not a quasidisk. For the whole Julia set, we know J(f?) is a Cantor set when?>?0, a Cantor set of circles when 0<?<?* and connected when?<?0. We prove that Theorem 3.If?*????0, then J(fn) is locally connected, and every Fatou component is a Jordan domain.It is shown in Chapter 4 that for every??[?*,?0] we have a corresponding parameter c= c(?)?[-2,1/4] such that J(f?) contains an embedded image of the Julia set of quadratic polynomial pc(z)= z2+c. And: Theorem 4. Let?*????0.The Julia set J(f?) is a Sierpinski curve if and only if one of the following conditions holds(i)???* and c(?) is not in the closure of any hyperbolic component of the Man-delbrot set;(ii) c(?) is in a primitive hyperbolic component of the Mandelbrot set of period great than 1;(iii) c(?) 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 B?is simply connected, it is a Jordan domain whose boundary is a closed Jordan curve, and with rare exception (?=?0) this simply connected B?is also a quasidisk. If B?is 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 B?have? Since quasidisk is a simply connected uniform domain, we guess that B?is 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 haveTheorem 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 F?in (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 F?in (0.0.4), if J(F?) is a Cantor set, then B?is 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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