臺灣博碩士論文加值系統

　　這篇論文研究碎形中的一大主題—曼德博集合。在了解曼德博集合之前我們需要從了解朱利亞集合開始，最終的目標則是利用程式來生成碎形圖像並希望能應用至藝術領域。
　　於是在這篇論文中首先介紹了朱利亞集合的定義及性質，並利用理論整理出可行的演算法來生成朱利亞集合的圖像。在對朱利亞集合有一定程度的理解之後便能開始研究曼德博集合，其原因來自曼德博集合的定義是蒐集所有另朱利亞集合連通的點。然而，在生成曼德博集合是會受到其定義的阻礙，如何有效的檢測朱利亞集合是否連通？這個問題的答案就是—曼德博集合基本定理，有了這個定理後便能生成曼德博集合。
　　最後也給了一些曼德博集合與朱利亞集合的例子，並且介紹了三維中的曼德博集合與朱利亞集合。

In this thesis, we survey the big theme of fractals － Mandelbrot sets. We start to study Julia sets before study Mandelbrot sets, and the goal is generating figures of fractals and applying to arts.
Hence, we introduce the definition and properties of Julia sets firstly, and use this theory to arrange some useful algorithms for generating the figures of Julia sets. After we survey Julia sets, we can study Mandelbrot sets, since the definition of Mandelbrot sets is all of the points such that the Julia set is onnected. However, we obtain the obstacle when generating andelbrot sets, that is, how to check the Julia set is connected or not? The answer of this question is － the fundamental theorem of Mandelbrot sets, we can generate the figures of Mandelbrot sets by this theorem.
Finally, we give some examples of Mandelbrot sets and Julia sets, and introduce 3-dimensional Mandelbrot sets and Julia sets.

摘要i
Abstract ii
Acknowledgement iii
1 Introduction to Fractals on a Complex Plane 1
2 Birth of Complex Fractals - Julia and Fatou Sets 3
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Coloring Fractal Sets on a Complex Plane . . . . . . . . . . . . . . . . . . 8
2.3 Julia and Fatou Sets within the Polynomials of Various Orders . . . . . . . 16
2.4 Other Variations of Julia and Fatou Sets on a Complex Plane . . . . . . . 27
2.5 Theoroms of Julia Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Fractal Revolution - The Mandelbrot Set 34
3.1 History and Concept of the Mandelbrot Set . . . . . . . . . . . . . . . . . 34
3.2 Geometrical Structure and Atoms of the Mandelbrot Set . . . . . . . . . . 37
3.3 Mandelbrot Set within the Polynomials of Various Orders and Other Variations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Fundamental Theorem of Mandelbrot Set . . . . . . . . . . . . . . . . . . 44
4 Searching for the Holy Grail: 3D Mandelbrot Set 52
5 What’s Next? Further Extensions of The Mandelbrot Set 56
6 Examples for the Julia Sets of Other Functions 58
References 66
iv

[1] B. B. Mandelbrot. The Fractal Geometry of Nature. W. H. Freeman and Co., New York, NY, 1982.
[2] S. Datta. Infinite sequences in the constructive geometry of tenth-century Hindu temple superstructures. Nexus Network Journal, 12(3):471–483, 2010.
[3] I. M. Rian, J.H. Park, H. U. Ahn, and D. Chang. Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho. Building and
Environment, 42(12):4093–4107, 2007.
[4] K. Trivedi. Hindu temples: Models of a fractal universe. The Visual Computer, 5(4):243–258, 1989.
[5] R. Eglash. African Fractals: Modern Computing and Indigenous Design. Rutgers University Press, New Brunswick, NJ, 1999.
[6] M. Bader. Space-Filling Curves: An Introduction With Applications in Scientific Computing. Springer Science & Business Media, Berlin Heidelberg, 2012.
[7] H.O. Peitgen, H. Jurgens, and D. Saupe. Chaos and Fractals: New Frontiers of Science. Springer Science & Business Media, New York, NY, 2013.
[8] B. Rosenfeld. Geometry of Lie Groups, volume 393 of Mathematics and its applications. Springer Science & Business Media,Dordrecht, Holland, 1997.
[9] D. S. Alexander, F. Iavernaro, and A. Rosa. Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906–1942, volume 38 of
History of Mathematics. American Mathematical Society, Providence, 2012.
[10] P. Blanchard. Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1):85–141, 1984.
[11] J. Barrallo and D. M. Jones. Coloring algorithms for dynamical systems in the complex plane. In ISAMA 99 Proceedings, pages 31–38, San Sebastián, 1999. The University of the Basque Country.
[12] P.W. Carlson. Two artistic orbit trap
enderingmethods for newton M-set fractals. Computers & Graphics, 23(6):925–931, 1999.
[13] R. Ye. Another choice for orbit traps to generate artistic fractal images. Computers & Graphics, 26(4):629–633, 2002.
[14] J. Barrallo and S. Sanchez. Fractals and multi layer coloring algorithms. Visual Mathematics, 3(10):0–0, 2001.
[15] U. G. Gujar and V. C. Bhavsar. Fractals from z zα + c in the complex c-plane. Computers & Graphics, 15(3):441–449, 1991.
[16] U. G. Gujar, V. C. Bhavsar, and N. Vangala. Fractals from z zα+c in the complex z-plane. Computers & Graphics, 16(1):45–49, 1992.
[17] J. Kudrewicz. Fractals and Chaos. Wydawnictwa Naukowo-Techniczne, Warsaw, 4th edition, 2007.
[18] S. H. Boyd and M. J. Schultz. Geometric limits of Mandelbrot and Julia sets under degree growth. International Journal of Bifurcation and Chaos, 22(12):1250301, 2012.
[19] X. Wang and T. Jin. Hyperdimensional generalized M–J sets in hypercomplex number space. Nonlinear Dynamics, 73(1):843–852, 2013.
[20] S. V. Dhurandhar, V. C. Bhavsar, and U. G. Gujar. Analysis of z-plane fractal images from z zα + c for α < 0. Computers & Graphics, 17(1):89–94, 1993.
[21] E. F. Glynn. The evolution of the Gingerbread Man. Computers & Graphics, 15(4):579–582, 1991.
[22] C. A. Pickover, editor. The Pattern Book: Fractals, Art, and Nature. World Scientific,
Singapore, 1995.
[23] K.W. Shirriff. An investigation of fractals generated by z 1/zn + c. Computers &
Graphics, 17(5):603–607, 1993.
[24] T. Bedford, F. A. M., and M. Urbanski. The scenery flow for hyperbolic Julia sets.
Proceedings of the London Mathematical Society, 85(2):467–492, 2002.
[25] N. Fagella. Dynamics of the complex standard family. Journal of Mathematical Analysis and Applications, 229(1):1–31, 1999.
[26] S.M. Heinemann and B. O. Stratmann. Geometric exponents for hyperbolic Julia sets. Illinois Journal of Mathematics, 45(3):775–785, 2001.
[27] C. McMullen. Area and Hausdorff dimension of Julia sets of entire functions. Transactions of the American Mathematical Society, 300(1):329–342, 1987.
[28] A. Negi, M. Rani, and P. K. Mahanti. Computer simulation of the behaviour of Julia sets using switching processes. Chaos Solitons and Fractals, 37(4):1187–1192, 2008.
[29] G. Rottenfusser and D. Schleicher. Escaping points of the cosine family. In P. J. Rippon and G. M. Stallard, editors, Transcendental Dynamics and Complex Analysis,
volume 348 of London Mathematical Society Lecture Note Series, pages 396–424. Cambridge University Press, Cambridge, MA, 2008.
[30] D. Schleicher and J. Zimmer. Escaping points of exponential maps. Journal of the London Mathematical Society, 67(2):380–400, 2003.
[31] M. Rani and A. Negi. New Julia sets for complex Carotid-Kundalini function. Chaos Solitons and Fractals, 36(2):226–236, 2008.
[32] R. L. Devaney and F. Tangerman. Dynamics of entire functions near the essential singularity. Ergodic Theory and Dynamical Systems, 6(4):489–503, 1986.
[33] W. R. Mann. Mean value methods in iteration. Proceedings of the American Mathematical Society, 4:506–510, 1953.
[34] M. A. Krasnoselski. Two observations about the method of succesive approximations. Uspekhi Matematicheskikh Nauk, 10:123–127, 1955.
[35] M. Rani and V. Kumar. Superior Julia set. Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education, 8(4):261–277,
2004.
[36] Y. S. Chauhan, R. Rana, and A. Negi. New Julia sets of Ishikawa iterates. International Journal of Computer Applications, 7(13):34–42, 2010.
[37] V. Berinde. Iterative Approximation of Fixed Points, volume 1912 of Lecture Notes in Mathematics. Springer, Berlin Heidelberg, 2nd edition, 2007.
[38] A. Negi, M. Rani, and R. Chugh. Julia and Mandelbrot sets in Noor orbit. Applied Mathematics and Computation, 228:615–631, 2014.
[39] W. Nazeer, S. M. Kang, M. Tanveer, and A. A. Shahid. Fixed point results in the generation of Julia and Mandelbrot sets. Journal of Inequalities and Applications,2015:298, 2015.
[40] S. M. Kang, W. Nazeer, M. Tanveer, and A. A. Shahid. New fixed point results for fractal generation in Jungck Noor orbit with sconvexity. Journal of Function Spaces,
2015:963016, 2015.
[41] C. A. Pickover. Biomorphs: Computer displays of biological forms generated from mathematical feedback loops. Computer Graphics Forum, 5:313–316, 1986.
[42] V. G. Ivancevic and T. T. Ivancevic. Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals. Springer, Berlin, 2008.
[43] J. Leys. Biomorphic art: An artist’s statement. Computers & Graphics, 26:977–979, 2002.
[44] N. S. Mojica, J. Navarro, P. C. Marijuan, and R. LahozBeltra. Cellular ‘bauplants’: Evolving unicellular forms by means of Julia sets and Pickover biomorphs. Biosystems, 98:19–30, 2009.
[45] M. Levin. Morphogenetic fields in embryogenesis, regeneration, and cancer: nonlocal control of complex patterning. Biosystems, 109(3):243–261, 2012.
[46] K. Gdawiec, W. Kotarski, and A. Lisowska. Biomophs via modified iterations. Journal of Nonlinear Science and Applications, 9:2305–2315, 2016.
[47] J. Fraser, An Introduction to Julia Sets, 2009.
[48] R. L. Devaney. The fractal geometry of the Mandelbrot set 2: How to count and how
to add. Fractals, 3(4):629–640, 1995.
[49] A. Katunin and K. Fedio. On a visualization of the convergence of the boundary of generalized Mandelbrot set to (n−1)sphere. Journal of Applied Mathematics and
Computational Mechanics, 14(1):63–69, 2015.
[50] T. V. Papathomas and B. Julesz. Animation with fractals from variations on the Mandelbrot set. The Visual Computer, 3(1):23–26, 1987.
[51] W. D. Crowe, R. Hasson, P. Rippon, and P. E. D. StrainClark. On the structure of the Mandelbar set. Nonlinearity, 2(4):541–553, 1989.
[52] S. Nakane. Connectedness of the tricorn. Ergodic Theory and Dynamical Systems, 13(2):349–356, 1993.
[53] S. Nakane and D. Schleicher. On Multicorns and Unicorns I: Antiholomorphic dynamics, hyperbolic components and real cubic polynomials. International Journal of Bifurcation and Chaos, 13(10):2825–2844, 2003.
[54] S. M. Kang, A. Rafiq, A. Latif, A. A. Shahid, and Y. C. Kwun. Tricorns and Multicorns of siteration scheme. Journal of Function Spaces, 2015:417167, 2015.
[55] J. Barrallo. Expanding the Mandelbrot set into higher dimensions. In G. W. Hart and R. Sarhangi, editors, Bridges 2010: Mathematics, Music, Art, Architecture, Culture, pages 247–254, Pecs, 2010.
[56] J. Cheng and J. Tan. Generalization of 3D Mandelbrot and Julia sets. Journal of Zhejiang University SCIENCE A, 8(1):134–141, 2007.
[57] K. Nagashima and H. Morimatsu. 3D representation of the Mandelbrot set. The Visual Computer, 10(6):356–359, 1994.
[58] A. V. Norton. Generation and display of geometric fractals in 3-D. Computer Graphics, 16(3):61–67, 1982.
[59] A. V. Norton. Julia sets in the quaternions. Computers & Graphics, 13(2):267–278, 1989.
[60] X.Y. Wang and Y.Y. Sun. The general quaternionic M–J sets on the mapping z zα+c (α 2 N). Computers & Mathematics with Applications, 53(11):1718–1732,
2007.
[61] C. J. Griffin and G. C. Joshi. Octonionic Julia sets. Chaos Solitons and Fractals, 2(1):11–24, 1992.
[62] C. J. Griffin and G. C. Joshi. Associators in generalized octonionic maps. Chaos Solitons and Fractals, 3(3):307–319, 1993.
[63] C. J. Griffin and G. C. Joshi. Transition points in octonionic Julia sets. Chaos Solitons and Fractals, 3(1):67–88, 1993.
[64] X. Wang and T. Jin. Hyperdimensional generalized M–J sets in hypercomplex number space. Nonlinear Dynamics, 73(1):843–852, 2013.
[65] D. Rochon. A generalized Mandelbrot set for bicomplex numbers. Fractals, 8(4):355–368, 2000.
[66] C. Matteau and D. Rochon. The inverse iteration method for Julia sets in the 3-dimensional space. Chaos Solitons and Fractals, 75:272–280, 2015.
[67] X.Y. Wang and W. J. Song. The generalized M-J sets for bicomplex numbers. Nonlinear Dynamics, 72(1):17–26, 2013.
[68] A. Zireh. A generalized-Mandelbrot set of polynomials of type ed for bicomplex numbers. Georgian Mathematical Journal, 15(1):189–194, 2008.
[69] A. A. Bogush, A. Z. Gazizov, Y. A. Kurochkin, and V. T. Stosui. Symmetry properties of quaternionic and biquaterionic analogs of Julia sets. Ukrainian Journal of
Physics, 48(4):295–299, 2003.
[70] Y. A. Kurochkin and S. Y. Zhukovich. Set symmetry, generated by octonion analog of Julia–Fatou algorithm. Vestnik Brestskaga Universiteta—Serya 4. Fizika Matematyka, 2:74–49, 2010.
[71] A. Katunin. On the symmetry of bioctonionic Julia sets. Journal of Applied Mathematics and Computational Mechanics, 12(2):23–28, 2013.
[72] V. Garant-Pelletier and D. Rochon. On a generalized Fatou-Julia theorem in multicomplex spaces. Fractals, 17(3):241–255, 2009.
[73] I.H. Lin. Classical Complex Analysis: A Geometric Approach, volume 2. World Scientific, Singapore, 2011.
[74] S. Ishikawa. Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44:147–150, 1974.
[75] Andrzej Katunin. A Concise Introduction to Hypercomplex Fractals, CRC Press, 2017.
[76] Kenneth Falconer, Fractal geometry-mathematical foundations and applications, Wiley, 2003.