| The topology of Julia sets for rational functions is one of important problems. The connectivity of Julia sets for polynomials have a complete description because of the proof of the Branner-Hubbard conjecture. For the dynamics of rational functions, the connectivity of quadratic rational functions has also a complete description: In fact, from the estimate for the upper bound of the number of periodic stable domains by Shishikura [50], we known the quadratic rational functions have no Herman ring. Furthermore, Yin Yongcheng proved that the Julia set of a quadratic rational function is either connected or a Cantor set( [60], see also [40]). Shishikura also proved that the Julia sets of Newton's iterate of polynomials are always connected [51]. However, in general, the topology of the Julia sets of rational functions will be more complicate than polynomials, for example, Milnor and Tan Lei (see [40]) first proved that there is a quadratic rational function of the following formwhose Julia set is a Sierpinski curve (Sierpinski carpet); McMullen [38] studied the Julia sets of rational functionswhich is regarded as the rational perturbation of zn. He proved that when 1/n + 1/d < 1 and |a| sufficiently small, the Julia set of Fa(z) is a Cantor annulus, i.e. the homeomorphic image of the Cartesian product of the Cantor set and the unit circle. These topological structures of Julia sets can not appear for polynomials.For the dynamics of quadratic rational functions fa,b(z), it has been studied by Goldberg and Keen [33] (for the case |a| > 1), and Yin Yongcheng [61] (for the case a = 1).Recently, the family of rational functions Fa(z) first studied by McMullen (which is called McMullen family now) makes great interesting. For example, a lot of studies on this family were given by Blanchard, Devaney et. al, Roesch, and Steinmetz (see [11,12,19-21,23-26,49,53,54]). It was discovered that there are various topological structures for the Julia set of Fa(z). It can be a Cantor set, a Cantor annulus, or a Sierpinski curve. Furthermore, it was also studied the topology of hyperbolic components and the bifurcation set in the parameter plane.The McMullen family Fa(z) has two critical points∞and 0, where∞is a super-attracting fixed point and 0 is the unique pole of Fa(z). These critical points have very simple orbits. Besides, Fa(z) has also n + d critical points, which is called free critical points. They are symmetrically located on the critical circle {|z| = (|a|d/z)1/(n+d)}, and their iterate images are also symmetric with respect to the origin. The free critical orbits are tending to infinity or not simultaneously. This means that essentially there is only one free critical point or free critical value for Fa(z). In the case that the free critical orbits tend to infinity, Devaney et.al. [25] gave a classification theorem for the topology of the Julia sets of Fa(z). Let B be the Fatou component which contains∞, and T be the Fatou component which contains 0. ThenTheorem 1. (Devaney et.al.) Suppose that the free critical orbits of the McMullen rational function Fa(z) tend to infinity, then(1) if B contains a free critical value, then the Julia set of Fa(z) is a Cantor set, Fa(z) restricted on the Julia set is conjugated to the shift transformation on the symbol space of n + d symbols.(2) if T contains a free critical value, the the Julia set of Fa(z) is a Cantor annulus consist of quasi-circles.(3) if one of iterate inverse images of T contains a free critical value, the the Julia set is a Sierpinski curve.In the literature, we don't find the discussion for the topology of the Julia sets of Fa(z) when the free critical orbits are bounded, even for the connectivity of the Julia sets.In the first part of this dissertation, we studied the topology of the Julia sets of the family of rational functionswith two parameters. This family can be regarded as the rational perturbation of the unimodal polynomials Pb(z) = zn + b. It is well-known that there are a lot of important works on the dynamics of the unimodal polynomials Pb(z), see [5,22,35,36] etc. It is also interested in the dynamics of the rational perturbation Fa,b(z). For example, Blanchard et.al. [13] studied the dynamics of Fa,b(z) for n = 2 and some special values of parameter b. Here, we call Fa,b(z) the generalized McMullen family. In this paper, we gave some discussions on the topological properties of the Julia sets o Fa,b(z), in particular, the connectivity of the Julia sets.Similar to the McMullen family, the infinity∞is a super-attracting fixed point of the generalized McMullen rational function Fa,b(z), and it is a critical point with multiple n - 1. The origin 0 is the unique pole of Fa,b(z), and it is also a critical point with multiple n - 1. As before, we denote B the Fatou component containing∞and T the Fatou component containing the origin.Fa,b(z) has also 2n simply critical points a1/2n, which are called free critical points. They are located on the critical circle {|z| = |a|1/2n}. These free critical points have two images v,v+, which are called free critical values. Unlike the McMullen family, the orbits of these two free critical values have independent behaviors. Hence, the dynamical properties of Fa,b(z) are more complicate and the study on the topology of the Julia sets is more difficult.To discuss the connectivity of the Julia set of Fa,b(z), we first discuss the existence of the Herman ring. It is known that the quadratic rational functions have no Herman ring [50]. Bamon and Bobenrieth [6] also proved that the rational functions gλ(z) = 1 + 1/λzd,λ∈C\{0}, have no Herman ring. In this paper, we provedTheorem 2. The generalized McMullen rational functions Fa,b(z) have no Herman ring.Furthermore, according to the behaviors of the critical orbits, we divide our discussion into three cases to study the topology of the Julia sets of Fa,b(z). 1. Escape case: the orbits of both free critical values are tending to infinity; 2. Semi-escape case: the orbit of one free critical value tends to infinity, and the orbit of another free critical value is bounded; 3. Non-escape case: the orbits of both free critical values are bounded. Let Ja,b denote the Julia set of Fa,b(z). We have1. Escape case.Theorem 3. Suppose that the orbits of both free critical values of Fa,b(z) tend to infinity.(1) If B = T, then both free critical values are in B, and the Julia set Ja,b of Fa,b(z) is a Cantor set.(2) If B≠T, then(2.1) when the two free critical values are in different Fatou components, the Julia set Ja,b of Fa,b(z) is connected. In particular, if one free critical value is in T, then Ja,b is a Sierpinski curve;(2.2) when the two free critical values are in the same Fatou component, then the Julia set Ja,b of Fa,b(z) is disconnected. Ja,b has infinitely many components. Every Fatou component is either simply connected or doubly connected. In particular, if both free critical values are in T, then Ja,b is a Cantor annulus consist of quasi-circles.2. Semi-escape case.Let Ka,b = {z∈(?): Fa,bn(z) (?)∞} denote the filled Julia set of Fa,b(z), where, Fa,bn(z) is the n-th iterate of Fa,b(z). Then Ja,b = (?)Ka,b. A component of Ka,b which contains a free critical point is called a critical component of Ka,b.Theorem 4. Suppose that the orbit of the free critical value v tends to infinity, but the orbit of the free critical value v+ is bounded.(1) If the free critical value v is not in B, then the Julia set Ja,b of Fa,b(z) is connected.(2) If the free critical value v is in B, then B = T and the Julia set Ja,b of Fa,b(z) is disconnected. Moreover,(2.1) if every critical component of the filled Julia set Ka,b is not periodic, then Ja,b is a Cantor set;(2.2) if one of critical component of Ka,b is periodic, then this critical component is homeomorphic to the filled Julia set of a quadratic polynomial.3. Non-escape case.Theorem 5. Suppose that the orbits of both free critical values of Fa,b(z) are bounded.(1)If every Fatou component contains at most one free critical value, then the Julia set Ja,b of Fa,b(z) is connected.(2) If there exists a Fatou component D1 which contains both free critical values, then the Julia set Ja,b of Fa,b(z) is disconnected. In this case, the Fatou component D1 is periodic, and it has only one inverse image D0.(2.1) If D0 has period 1, then it is completely invariant. The Julia set Ja,b = (?)D0, which consists of infinitely many disjoint and non-nested topological circles and uncountably many points.(2.2) If D0 has period great than 1, then the boundary of D0 consists of infinitely many disjoint and non-nested topological circles and uncountably many points.Some examples were given to show the existence of the cases discussed in above theorems.The second part of this dissertation is to study the Chebyshev polynomials on the Julia sets of polynomials. The Chebyshev polynomials on a compact set have many applications in approximation theory [7], numerical computation [45] and potential theory [3], etc.. Generally, it is difficult to obtain the Chebyshev polynomials for a given compact set [32]. In 1983, Barnsley et.al. [7] obtained the 2n-th Chebyshev polynomials on the Julia sets of the quadratic polynomials Tλ(z) = (z-λ)2,λ∈C. In this paper, we obtained the dn-th Chebyshev polynomials on the Julia sets of arbitrary monic polynomials.Let K be a compact set in the plane (?). The center of the minimal disk which contains K is called the geometric center of K.Theorem 6. Let P be a monic polynomial with degree d≥2. Then the dn-th Chebyshev polynomial on the Julia set of P is Pn(z)- cP, where cP is the geometric center of the Julia set JP.For a given compact set K, the Chebyshev polynomials on the equipotential curves of K obtained by the Green function on the unbounded component of C \ K with singularity at∞are also interesting. One basic problem is if the Chebyshev polynomials on the compact set K and on its equipotential curves are always the same. For the unit circle, the interval, and the union of two intervals on the real line, the answer is yes (see [4,28,46]). In 1996, Stawiska [52] discussed the 2n-th Chebyshev polynomials on the equipotential curves of the Julia set of Tλ(z) = (z -λ)2. It was proved that whenλ∈[0,4], the 2n-th Chebyshev polynomial on the Julia set JT coincides with one on the equipotential curves of JT.We discussed the Chebyshev polynomials on the equipotential curves of the Julia sets for general polynomials. We obtainedTheorem 7. Let P be a monic polynomial with degree d≥2. Let Jp be its Julia set,ΓP(R) be the equipotential curve of level R > 0. Then the dn-th Chebyshev polynomial onΓP(R) is given by Pn(z)- cR,n, where cR,n is the geometric center of the equipotential curveΓP(Rdn)(= Pn(ΓP(R))) of level Rdn> 0.Obviously, if cR,n≠cP, then the Chebyshev polynomials on the Julia set JP and on the equipotential curvesΓP(R) are not equal. We gave an example so that they are different. We also gave a sufficient condition so that they are the same.Finally, as an application of the Chebyshev polynomials on the Julia sets, we gave a new proof of a result on the capacity of Julia set which is originally given by Brolin [17].
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