The Study Of The Topological And Geometric Properties Of Julia Sets

The study of the topological and geometric properties of the Julia sets of holomorphic maps is an important problem in complex dynamics. It mainly includes the connected-ness, locally connectivity and the regularity of Julia sets (or their subsets), etc. For the topological properties of the Julia sets of polynomials, many authors contributed on this. For the quadratic polynomials fc(z)=z2+c, where c belongs to the Mandelbrot set, Yoccoz proved that if fc is not infinitely renormalizable, then the Julia set J(fc) of fc is locally connected [38]. Lyubich proved that if c is real, then J(fc) is locally connected [46]. For the case when fc has a Siegel disk, Petersen proved that if the rotation number on the Siegel disk is of bounded type, then J(fc) is also locally connected [57]. If the rotation number is of David type, then Petersen and Zakeri proved that the boundary of the Siegel disk is a Jordan curve and the Julia set is locally connected [58]. Moreover, Avila, Buff and Cheritat proved that there exists Siegel disk whose boundary is a smooth curve [4].For cubic polynomials, Branner and Hubbard proved that all but countably many components of the Julia set are single points. Recently, this result has been extended to all polynomials in [63]. For the rational maps, if it is geometrically finite, then, with the possible exception of countable components, every Julia component is either a point or a Jordan curve [59].The first example of rational map with disconnected Julia set whose components are all Jordan curves was discovered by McMullen [49]. He showed that if f(z)=z2+λ/z3and A is small enough, then the Julia set of f is homeomorphic to the product of the middle third Cantor set and the unit circle. These types of Julia sets are called Cantor circles. Later, many authors focus on the family, which is commonly referred as the McMullen maps:fλ(z)=zm+λ/zl, where l, m≥2and λ∈C\{0}. It can be shown that when1/l+1/m<1and A is small enough, the Julia set of fλ is a Cantor set of circles (see [49,§7] and [24,§3]).The first part of this thesis mainly focuses on the rational maps whose Julia sets are Cantor circles. Up to now, it is known that McMullen maps (or the rational maps after perturbing on their Fatou sets) are the only examples such the Julia sets of rational maps are Cantor circles. This motivates us to think about the question whether there exist other rational maps whose Julia sets are Cantor circles. Although Haissinsky and Pilgrim have constructed a class of rational maps whose Julia sets are Cantor circle which is different from McMullen maps "essentially" by quasiconformal surgery. However, they did not give the specific expressions of these maps [35]. Therefore, a natural question is whether one can give the specific expressions of these maps.In fact, in this thesis, we not only give the specific expressions of these types of rational maps, but also find out all the rational maps whose Julia sets are Cantor circles "essentially". The word "essentially" means we consider this problem in the sense of the topological conjugacy classes on the Julia sets. Specifically, we find the specific expressions of a family of rational maps (McMullen maps are the special cases) such that their Julia sets are Cantor circles after choosing appropriate parameters. On the other hand, for each given rational map whose Julia set is a Cantor set of circles, we can always find a map in the family which we have found such they are topologically conjugate on their corresponding Julia sets (Theorems3.1.1and3.1.2).Based on this, we calculate and give a lower and upper bound of the number of different topological conjugacy classes on the Julia sets of rational maps whose Julia sets are Cantor circles. Moreover, we give the specific numbers of the different topological conjugacy classes for5≤d≤36, where d is the degree of the rational maps.Hyperbolic rational maps possess simple dynamical properties. We. find out a series of non-hyperbolic rational maps whose Julia sets are Cantor circles. As far as we known, this is the first example of non-hyperbolic rational maps whose Julia sets are Cantor circles. The idea behind the construction is the attracting basin of the original rational maps has been replaced by a simply connected parabolic basin.The second part of this thesis mainly focuses on the study of the geometric properties of the Julia sets of McMullen maps. According to the Escape Trichotomy Theorem, if all the critical points of a McMullen map are attracted by∞, then the corresponding Julia set is either a Cantor set, a Cantor set of eircles or a Sierpiriski carpet [24]. We prove that in this case, the Julia set of a McMullen is quasisymmetrically equivalent either to a standard Cantor set, a standard Cantor set, of circles or a round Sierpinski carpet (which is also standard in some sense).For the case when the parameter of the McMulle family is real, a sufficient and necessary condition such the Julia set Jλ of McMullen map fλ is a Sierpinski carpet was gave in [62,76]. Based on this, we give a sufficient and necessary condition to guarantee that Jλ is quasisymmetrically equivalent to a round Sierpinski carpet. In particular, there exists non-hyperbolic rational map whose Julia set is quasisymmetrically equivalent to a round Sierpinski carpet. Moreover, for case when λ is complex, we give a sufficient condition to guarantee that Jλ is quasisymmetrically equivalent to a round Sierpinski carpet (but we need to set l=m≥3, where l, m is the integers in fλ).From the topological point of view, all Cantor sets of circles are the same since they are all topologically equivalent (homeomorphic) to the "stand" Cantor set of circles C×S1, where C is the middle third Cantor set and S1is the unit circle. Therefore, to obtain much richer structure of all Cantor sets of circles, we can look at the Cantor circles equipped with metric from the point of view of quasisymmetric geometry. In fact, a basic problem in the quasisymmetric geometry is to determine whether two given homeomorphic metric spaces are quasisymmetrically equivalent to each other. The conformal dimension of a metric space is the infimum of the Hausdorff dimensions of all metric spaces which are quasisymmetrically equivalent to the original metric space. According to the combina-torial analysis on the rational maps which we have found in the first part of this thesis, we show that this family gives a series of specific examples, such the Julia sets of them are homeomorphic, but not quasisymmetrically equivalent to the Julia sets of McMullen maps. In particular, these examples can be used to verify that there exist hyperbolic rational maps whose Julia sets are Cantor circles and whose conformal dimensions are arbitrarily close to2.The third part of this thesis studies the regularity of the boundaries of a family of entire functions. The study of the topological and geometric properties of the boundaries of Siegel disks (subsets of Julia sets) is an important problem. After assuming the rotation number satisfies some arithmetical condition, Douady conjectured that the boundary of the Siegel disk must be a Jordan curve. For this problem, Douady, Zaker, Shishikura and Zhang contributed to the rational case (see [26,77,66,80]). But this problem is far from being solved for transcendental case (see [32,41,78,79]). We consider the one-dimensional family fa(z)=e2πiθ sin(z)+αsin3(z):a∈C\{0}, and prove that the boundary of the Siegel disk centered at the origin of fa is a quasicircle which passes through2,4or6critical points of fa counted with multiplicity if θ is of bounded type.The fourth part of this thesis considers the realize problem of the critical tableau. As a powerful tool, the critical tableau was introduce by Branner and Hubbard when they studied the cubic polynomials [11]. It plays a central role in the study of locally connec-tivity of the Julia sets. Branner and Hubbard proved that:If an abstract critical marked tableau satisfies some rules, then this tableau must be realized by a cubic polynomial. We give a new proof of the realization theorem of tableaux by using tree dynamics induced by polynomials and the realization theorem of tree dynamics (see [30,18]).In the last part of this thesis, we study the boundary behavior of the immediate basin of infinity of McMullen maps, and give an asymptotic formula of its Hausdorff dimension.