In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analy...In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.展开更多
Denote by HD（J（f）） the Hausdorff dimension of the Julia set J（f） of a rational function f. Our first result asserts that if f is an NCP map, and fn → f horocyclically,preserving sub-critical relations, then fn ...Denote by HD（J（f）） the Hausdorff dimension of the Julia set J（f） of a rational function f. Our first result asserts that if f is an NCP map, and fn → f horocyclically,preserving sub-critical relations, then fn is an NCP map for all n ≥≥ 0 and J（fn） →J（f） in the Hausdorff topology. We also prove that if f is a parabolic map and fn is an NCP map for all n ≥≥ 0 such that fn→4 f horocyclically, then J（fn） → J（f） in the Hausdorff topology, and HD（J（fn）） →4 HD（J（f））.展开更多
In this article, we investigate the dynamical properties of fλ（z） =λzke2, for λ（≠0） ∈Cand k≥2. We will show that the boundaries of some （or all ） Fatou components are Jordan curves and the Julia sets are S...In this article, we investigate the dynamical properties of fλ（z） =λzke2, for λ（≠0） ∈Cand k≥2. We will show that the boundaries of some （or all ） Fatou components are Jordan curves and the Julia sets are Sierpinski carpet, and they are locally connected for some certain λ.展开更多
Assume that f is a transcendental entire function.The ray arg z=θ∈[0,2π]is said to be a limiting direction of the Julia set J(f)of f if there exists an unbounded sequence{z_(n)}■J(f)such that lim rn→∞ arg z_(n)=...Assume that f is a transcendental entire function.The ray arg z=θ∈[0,2π]is said to be a limiting direction of the Julia set J(f)of f if there exists an unbounded sequence{z_(n)}■J(f)such that lim rn→∞ arg z_(n)=θ.In this paper,we mainly investigate the dynamical properties of Julia sets of entire solutions of the complex differential equations F(z)f^(n)(z)+P(z,f)=0,and f^(n)+A(z)P(z,f)=h(z),where P(z,f)is a differential polynomial in f and its derivatives,F(z),A(z)and h(z)are entire functions.We demonstrate the existence of close relationships Petrenko's deviations of the coefficients and the measures of limiting directions of entire solutions of the above two equations.展开更多
基金Project supported by the National Basic Research Program (973) of China (Nos. 2004CB719402 and 2002CB312106), the National Natural Science Foundation of China (Nos. 60375020 and 50305033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020335112)
文摘In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.
文摘Denote by HD（J（f）） the Hausdorff dimension of the Julia set J（f） of a rational function f. Our first result asserts that if f is an NCP map, and fn → f horocyclically,preserving sub-critical relations, then fn is an NCP map for all n ≥≥ 0 and J（fn） →J（f） in the Hausdorff topology. We also prove that if f is a parabolic map and fn is an NCP map for all n ≥≥ 0 such that fn→4 f horocyclically, then J（fn） → J（f） in the Hausdorff topology, and HD（J（fn）） →4 HD（J（f））.
基金National Natural Science Foundation of China(No.10871089)
文摘In this article, we investigate the dynamical properties of fλ（z） =λzke2, for λ（≠0） ∈Cand k≥2. We will show that the boundaries of some （or all ） Fatou components are Jordan curves and the Julia sets are Sierpinski carpet, and they are locally connected for some certain λ.
基金Supported by the National Natural Science Foundation of China(11971344)。
文摘Assume that f is a transcendental entire function.The ray arg z=θ∈[0,2π]is said to be a limiting direction of the Julia set J(f)of f if there exists an unbounded sequence{z_(n)}■J(f)such that lim rn→∞ arg z_(n)=θ.In this paper,we mainly investigate the dynamical properties of Julia sets of entire solutions of the complex differential equations F(z)f^(n)(z)+P(z,f)=0,and f^(n)+A(z)P(z,f)=h(z),where P(z,f)is a differential polynomial in f and its derivatives,F(z),A(z)and h(z)are entire functions.We demonstrate the existence of close relationships Petrenko's deviations of the coefficients and the measures of limiting directions of entire solutions of the above two equations.
