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
The dynamics of complex quadratic polynomials P_c(z)=z^2+c has been studiedextensively. In this note, we get one upper bound of the radius of the filled-in Juliaset of P(z), by Bieberbach conjecture(Theorem of de Bran...The dynamics of complex quadratic polynomials P_c(z)=z^2+c has been studiedextensively. In this note, we get one upper bound of the radius of the filled-in Juliaset of P(z), by Bieberbach conjecture(Theorem of de Branges) in univalent functionsand gives an answer to the question of Douady. As its application, a lower bound ofthe Hausdorff dimension for the Julia set of P(z) is obtained.展开更多
The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorith...The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorithm, and followed from the conclusion, but also offered two estimations of some special Julia set's Hausdorff's dimension by approximate linearization method.展开更多
We study Misiurewicz points on the parameter space about a family of rational maps Tλ concerning renormalization transformation in statistical mechanic. We determine the intersection points of the Julia set J(Tλ) an...We study Misiurewicz points on the parameter space about a family of rational maps Tλ concerning renormalization transformation in statistical mechanic. We determine the intersection points of the Julia set J(Tλ) and the positive real axis R+and discuss the continuity of the Hausdorff dimension HD(J(f)) about real parameter λ.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The dynamics of complex quadratic polynomials P_c(z)=z^2+c has been studiedextensively. In this note, we get one upper bound of the radius of the filled-in Juliaset of P(z), by Bieberbach conjecture(Theorem of de Branges) in univalent functionsand gives an answer to the question of Douady. As its application, a lower bound ofthe Hausdorff dimension for the Julia set of P(z) is obtained.
文摘The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorithm, and followed from the conclusion, but also offered two estimations of some special Julia set's Hausdorff's dimension by approximate linearization method.
基金supported by National Natural Science Foundation of China(GrantNos.11371363,11231009,11261002 and 11201474)the Special Basic Scientific Research Funds of Central Universities in China
文摘We study Misiurewicz points on the parameter space about a family of rational maps Tλ concerning renormalization transformation in statistical mechanic. We determine the intersection points of the Julia set J(Tλ) and the positive real axis R+and discuss the continuity of the Hausdorff dimension HD(J(f)) about real parameter λ.
