摘要 :
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 ...
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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.
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摘要 :
For McMullen maps f (lambda) (z) = z (p) + lambda/z (p) , where lambda is an element of C\{0} = 3 and lambda is small enough, then the Julia set J(f (lambda) ) of f (lambda) is a Cantor set of circles. In this paper we show that t...
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For McMullen maps f (lambda) (z) = z (p) + lambda/z (p) , where lambda is an element of C\{0} = 3 and lambda is small enough, then the Julia set J(f (lambda) ) of f (lambda) is a Cantor set of circles. In this paper we show that the Hausdorff dimension of J(f (lambda) ) has the following asymptotic behavior dim(H)J(f(lambda))=1+log2/logp+O(vertical bar lambda vertical bar(2-4/p)),as lambda -> 0. An explicit error estimation of the remainder is also obtained. We also observe a 'dimension paradox' for the Julia set of Cantor set of circles.
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摘要 :We consider the family of rational maps given by F(z)=zn+/zd where n,dN with 1/n+1/d<1, the variable zC<^> and the parameter C. It is known that when n=d3 there are infinitely many rings Sk with kN, around the McMullen domain. The McMullen domain is a ...
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We consider the family of rational maps given by F(z)=zn+/zd where n,dN with 1/n+1/d<1, the variable zC<^> and the parameter C. It is known that when n=d3 there are infinitely many rings Sk with kN, around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter -plane where the Julia sets of F are Cantor sets of simple closed curves. The rings Sk converge to the boundary of the McMullen domain as k and contain parameter values that lie at the center of Sierpiski holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of F are Sierpiski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1/n+1/d<1 where n is not necessarily equal to d. The number of Sierpiski holes and superstable parameters on S1 is 1n,d=n-1, and on Sk for k>1 is given by kn,d=dnk-2(n-1)-nk-1+1.
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摘要 :
In this paper we consider the dynamical behavior of the family of complex rational maps given by F-lambda (z) = z(n) + lambda/z(d) where n >= 2, d >= 1. Despite the high degree of these maps, there is only one free critical orbit ...
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In this paper we consider the dynamical behavior of the family of complex rational maps given by F-lambda (z) = z(n) + lambda/z(d) where n >= 2, d >= 1. Despite the high degree of these maps, there is only one free critical orbit up to symmetry. Also, the point at infinity is always a superattracting fixed point. Our goal is to consider what happens when the free critical orbit tends to infinity. We show that there are three very different types of Julia sets that occur in this case. Suppose the free critical orbit enters the immediate basin of attraction of infinity at iteration j. Then we show: (1) If j = 1, the Julia set is a Cantor set; (2) If j = 2, the Julia set is a Cantor set of simple closed curves; (3) If j > 2, the Julia set is a Sterpinski curve.
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摘要 :
We study the family of complex maps given by F-lambda(z) = z(n) + lambda/z(n) + c where n >= 3 is an integer lambda. is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corr...
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We study the family of complex maps given by F-lambda(z) = z(n) + lambda/z(n) + c where n >= 3 is an integer lambda. is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.
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