C. Penrose, On quotients of shifts associated with dendrite Julia sets of quadratic polynomials, Ph.D. Thesis, University of Coventry (1994). ===================================== Genadi Levin : On explicit connections between dynamical and parameter spaces Journal Journal d'Analyse Mathématique Publisher Hebrew University Magnes Press ISSN 0021-7670 (Print) 1565-8538 (Online) Issue Volume 91, Number 1 / December, 2003 DOI 10.1007/BF02788792 Pages 297-327 =================================== Toolpath generation for layer manufacturing of fractal objects Author(s): W.K. Chiu, Y.C. Yeung, K.M. Yu Journal:Rapid Prototyping Journal Year:2006 Volume:12 Issue:4 Page:214 - 221 ISSN:1355-2546 -------------------------------------------------- The divisor periodic point of escape-time N of the Mandelbrot set Auteur(s) / Author(s) WANG XINGYUAN (1) ; ZHANG XU (1) ; Abstract This paper offers a rendering method based on the escape-time method to draw the Mandelbrot set in different colors. The rendering method chooses different colors according to the distance and the escape-time N, so it can effectively render the Mandelbrot set, which can show us the 3-D view of the Mandelbrot set. According to amplify the part of the Mandelbrot set drawn by the rendering method, we will find some cirques with one heart like the equipotential lines. The center point c0of the cirques with one heart whose color will be rendered by black has a period, and only those center points c0, whose periods are the divisors of the escape-time N, will show us the cirques with one heart like the equipotential lines around them. We call those center points c0 the divisor periodic points of the escape-time N. Journal Title Applied mathematics and computation ISSN 0096-3003 CODEN AMHCBQ 2007, vol. 187, no2, pp. 1552-1556 [5 page(s) (article)] (8 ref.) ----------------------------------- Finke, L. P., Potentialfunktionen von Juliamengen, Diploma Thesis, Universität Bremen, 1994. -------------------- Symbolic dynamics for angle-doubling on the circle I. The topology of locally connected Julia sets Book Series Lecture Notes in Mathematics Publisher Springer Berlin / Heidelberg ISSN 0075-8434 (Print) 1617-9692 (Online) Volume Volume 1514/1992 Book Ergodic Theory and Related Topics III DOI 10.1007/BFb0097523 Copyright 1992 ISBN 978-3-540-55444-8 DOI 10.1007/BFb0097524 Pages 1-23 --------------------------------------------- A count of maximal small copies in Multibrot sets Andrew Bridy et al 2005 Nonlinearity 18 1945-1953 ------------------------------------------- Bifurcations and discriminants for polynomial maps P Morton et al 1995 Nonlinearity 8 571-584 ================================== Douady, Adrien; Hubbard, John Hamal Itération des polynômes quadratiques complexes. (French) [Iteration of complex quadratic polynomials] C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123--126. //------------------------------------------------------------------ A. DOUADY and J. H. HUBBARD, Étude dynamique des polynomes complexes I, II (Publication Mathématiques d'Orsay, no. 84-02, 1984, no. 85-04, 1985). Zbl 0552.30018 ---------------------------------------------------------- Kleinian groups and iteration of quadratic maps Journal Archiv der Mathematik Publisher Birkhäuser Basel ISSN 0003-889X (Print) 1420-8938 (Online) Issue Volume 55, Number 2 / August, 1990 DOI 10.1007/BF01189128 Pages 111-116 -------------------------------------------------- A. DOUADY and J. H. HUBBARD, Iteration of complex quadratic polynomials (C.R.A.S., Vol. 294, 1982, pp. 123-126). MR 83m:58046 | Zbl 0483.30014 ------------------------------------------------------ On the symmetries of the Julia sets for the process zRightarrowzp+c A Lakhtakia et al 1987 J. Phys. A: Math. Gen. 20 3533-3535 doi:10.1088/0305-4470/20/11/051 ------------------------------------------------------------------ Journal Title - Mathematische Zeitschrift Article Title - Hausdorff dimension 2 for Julia sets of quadratic polynomials Volume - Volume 237 Issue - 3 First Page - 571 Last Page - 583 Issue Cover Date - 2001-07-10 Author - Stefan-M. Heinemann Author - Bernd O. StratmannDOI - 10.1007/PL00004881 Link - http://www.springerlink.com/content/n4cqr7e18e81m284 ------------------------------------------------------------------------------ Bessis, D.; Geronimo, Formal Power Series and Some Theorems of J. F. Rittin Arbitrary Characteristic Journal Monatshefte für Mathematik Publisher Springer Wien ISSN 0026-9255 (Print) 1436-5081 (Online) Issue Volume 127, Number 4 / May, 1999 DOI 10.1007/s006050050040 Pages 277-293 Abstract.  We study composition of power series and polynomials over algebraically closed fields of arbitrary characteristic. The so-called Boettcher function of a power series is introduced and investigated. It is the principal aim of this paper to prove some results going back to J. F. Ritt in this general setting. In particular, we determine the pairs of permutable polynomials and characterize polynomials which satisfy a certain rational functional equation and polynomials which have a common iterate. --------------------------------------------- Meblin transforms associated with Julia sets and physical applications Journal Journal of Statistical Physics Publisher Springer Netherlands ISSN 0022-4715 (Print) 1572-9613 (Online) Issue Volume 34, Numbers 1-2 / January, 1984 Category Articles DOI 10.1007/BF01770350 Pages 75-110 Subject Collection Physics and Astronomy SpringerLink Date Monday, June 20, 2005 D. Bessis1, J. S. Geronimo1, 2 and P. Moussa1 (1) Service de Physique Théorique, CEN-SACLAY, F91191 Gif sur Yvette Cedex, France (2) Present address: School of Mathematics, Georgia Institute of Technology, 30332 Atlanta, Georgia Received: 30 August 1983 Abstract We introduce the Mellin transform of the balanced invariant measure associated to the Julia set generated by a rational transformation. We show that its analytic continuation is a meromorphic function, the poles of which are on a semi-infinite periodic lattice. This allows one to have an understanding of the behavior of the measure near a repulsive fixed point. Trace identities corresponding to the fact that the analytically continued Mellin transform vanishes at negative integers are derived for the polynomial case. The quadratic map is first analyzed in detail, and the analytic properties of the inverse of the Green's function are exhibited. Of interest is the appearance of a dense set of spikes at dyadic points when the Julia set is disconnected. These results are used to study the residues of the Mellin transform. A certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems, the excitation spectrum of which displays unusual oscillations. The appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion. ============================================= The complex potential generated by the maximal measure for a family of rational maps Journal Journal of Statistical Physics Publisher Springer Netherlands ISSN 0022-4715 (Print) 1572-9613 (Online) Issue Volume 52, Numbers 3-4 / August, 1988 Category Articles DOI 10.1007/BF01019717 Pages 571-575 Subject Collection Physics and Astronomy SpringerLink Date Friday, January 21, 2005 Add to marked items Add to shopping cart Add to saved items Permissions & Reprints Recommend this article Articles The complex potential generated by the maximal measure for a family of rational maps Artur Oscar Lopes1 (1) Institute for Physical Science and Technology, University of Maryland, 20742 College Park, Maryland Received: 26 January 1988 Abstract The exact complex potential generated by the maximal measure for a family of rational maps is given. The results are of analytical nature because the complex potential does not change nicely if the coordinates of a rational map are changed. There exist applications of this result to the theory of moments. ----------------------------------------------- @ARTICLE{1995Nonli...8..571M, author = {{Morton}, P. and {Vivaldi}, F.}, title = "{Bifurcations and discriminants for polynomial maps }", journal = {Nonlinearity}, year = 1995, month = jul, volume = 8, pages = {571-584}, adsurl = {http://adsabs.harvard.edu/abs/1995Nonli...8..571M}, adsnote = {Provided by the Smithsonian/NASA Astrophysics Data System} } ---------------------------------------------------- Ring Wolfgang : On the Local Behaviour of Discrete Dynamical Systems, Karl-Franzens-University Graz, 1991. master thesis : ----------------------------------- M. Rivi: Local behaviour of discrete dynamical systems. Ph.D. Thesis, Universit`a di Firenze, 1999. ------------------------------------------- Singer, D., Stable orbits and bifurcation of maps on the interval [J], SIAM J. Appl.Math., 35(1998), 260-267 --------------------- W. P. Thurston. On the combinatorics and dynamics of iterated rational maps. Preprint. ------------------ Zbl 0664.58015 Milnor, John; Thurston, William On iterated maps of the interval. (English) [A] Dynamical systems, Proc. Spec. Year, College Park/Maryland, Lect. Notes Math. 1342, 465-563 (1988). ------------------------- Jukka A. Ketoja1 and Juhani Kurkijärvi2 Binary tree approach to scaling in unimodal maps Journal of Statistical Physics v 75, Numbers 3-4 / May, 1994 --------------------------- Brolin, Hans Invariant sets under iteration of rational functions. Ark. Mat. 6 1965 103--144 (1965). ------------------- Another choice for orbit traps to generate artistic fractal images Author: Ye R. Source: Computers and Graphics, Volume 26, Number 4, August 2002 , pp. 629-633(5) Publisher: Elsevier --------------------------------------------- Parameter space of one-parameter complex mappings Author: Liaw S.-S. Source: Chaos, Solitons and Fractals, Volume 13, Number 4, March 2002 , pp. 761-766(6) Publisher: Elsevier ------------------------------------------- General Mandelbrot sets and Julia sets with color symmetry from equivariant mappings of the modular group Authors: Chung K.W.1; Chan H.S.Y.; Chen N. Source: Computers and Graphics, Volume 24, Number 6, December 2000 , pp. 911-918(8) Publisher: Elsevier ----------------------- Physical meaning for Mandelbrot and Julia sets Author: Beck C.1 Source: Physica D, Volume 125, Number 3, 15 January 1999 , pp. 171-182(12) Publisher: Elsevier ------------------------------ The dynamics of complex polynomials and automorphisms of the shift Journal Inventiones Mathematicae Issue Volume 104, Number 1 / December, 1991. Pages 545-580 The dynamics of complex polynomials and automorphisms of the shift Paul Blanchard1, Robert L. Devaney1 and Linda Keen2 ---------------------------------- Nonconformal perturbations of z map z2 + c: the 1 : 3 resonance H Bruin et al 2004 Nonlinearity 17 765-789 ====================== Philip, A. G. ., Robucci, A., and Frame, M., "A new scaling along the spike of the Mandelbrot set, '' Computers & Graphics 16 (1992), 223-234. ============================= Hurwitz, H., Frame, M., and Peak, D., "Scaling symmetries in nonlinear dynamics: a view from parameter space," Physica D 81 (1995), 23-31. ========================== =========================== Counting stable cycles in unimodal iterations Lutzky, M. Physics Letters A, Volume 131, Issue 4-5, p. 248-250. 08/1988 ========================== Galois theory of periodic orbits of rational maps F Vivaldi et al 1992 Nonlinearity 5 961-978 doi:10.1088/0951-7715/5/4/007 Abstract. The periodic points of a rational mapping are roots of a polynomial. If the coefficients of the mapping are algebraic numbers, then the periodic orbits are also algebraic numbers. A sequence of algebraic number fields is naturally associated with rational mappings, namely the fields containing all orbits of a given period. The authors study the corresponding Galois groups. They show that the latter have subgroups that permute the points of an orbit in the same way as the dynamics. The subgroup having all orbits as invariant sets identifies a field which contains the multipliers of the orbits. They construct their minimal polynomial, thereby computing the multiplier of a cycle without computing the cycle itself. They show that the periodic orbits of the quadratic family are soluble by radicals if their period is less or equal to 4, and they exhibit examples of unsoluble orbits of period 5. Dynamics over algebraic number fields is discrete, and all numerical experiments are reproducible. ------------------------------------------------------- Feigenvalues for Mandelsets K M Briggs et al 1991 J. Phys. A: Math. Gen. 24 3363-3368 doi:10.1088/0305-4470/24/14/023 ======================================= An Improved Computation of Component Centers in the Degree-$n$ Bifurcation Set Author : Young Hee Geum Young Ik Kim Journal of Applied Mathematics & Computing 2002 v.10 n.1 -------------------------------------------------- [2] Intersection of the Degree-$n$ Bifurcation Set with the Real Line Author : Young Hee Geum Young Ik Kim Journal of the Korea Society of Mathematical Education. Series B. The Pure and Applied Mathematics 2002 v.9 n.2 --------------------------------------- [3] An Epicycloidak Boundary of the main Component in the degree-$n$ Bifurcation Set Author : Young Hee Geum Young Ik Kim Journal of Applied Mathematics & Computing 2004 v.16 n.1 ------------------------------------- [4] Locating and counting bifurcation points of satellite components from the main component in the degree-$n$ bifurcation set Author : Young Hee Geum Young Ik Kim Journal of Applied Mathematics & Computing 2006 v.22 n.1 --------------------------------- Bifurcations and discriminants for polynomial maps P Morton et al 1995 Nonlinearity 8 571-584 ---------------------------- IRRATIONAL POINTS IN THE MANDELBROT SET Fractals, Vol. 13, No. 3 (2005) 233-236 Author(s): P. B. VINOD KUMAR K. BABU JOSEPH -------------------------- Guckenheimer, J., and McGehee, R., "A proof of the Mandelbrot n2 conjecture," Institute Mittag-Leffler, Report 15, 1984. ========================== Coulette, P. and Tresser, C., "Iteration d'endomorphismes et groupe de renormalisation," J. De Physique Colloque C 539 (1978), C5-25. ================================================ C. L. Petersen, When Schroder meets Bottcher - convergence of level sets, C.R.Acad.Sci. Paris, Ser. I 339 (2004) 219-22. ----------------------------------------------------------------------- Gray codes and the symbolic dynamics of quadratic maps Cusick, T.W. Electronics Letters Volume 35, Issue 6, 18 Mar 1999 Page(s):468 - 469 ----------------------------------------- Quadratic dynamics in binary number systems Author: Paul E. Fishback a DOI: 10.1080/10236190412331334482 Published in: journal Journal of Difference Equations and Applications, Volume 11, Issue 7 June 2005 , pages 597 - 603 =========================================== ================================ Remarks on the period three cycles of quadratic rational maps Selim Berker et al 2003 Nonlinearity 16 93-100 doi:10.1088/0951-7715/16/1/306 --------------------------------- ================================== Algebraic Dynamics and Transcendental Numbers Book Series Lecture Notes in Physics Publisher Springer Berlin / Heidelberg ISSN 1616-6361 Subject Physics and Astronomy Volume Volume 550/2000 Book [Without Title] Copyright 2000 Page 372 ============================= //------------ Ewing, John H.; Schober, Glenn The area of the Mandelbrot set. (English) Numer. Math. 61, No.1, 59-72 (1992). ============================= Leung, Y.J.; Schober, G. Some coefficient problems and applications. (English) Pac. J. Math. 145, No.1, 71-95 (1990). ============================================= Ewing, J. H. and Schober, G. "The Area of the Mandelbrot Set." Numer. Math. 61, 59-72, 1992. //------------------------------ The Geometry of Julia Sets Jan M. Aarts, Lex G. Oversteegen Transactions of the American Mathematical Society, Vol. 338, No. 2 (Aug., 1993), pp. 897-918 =================================== A New Partition Identity Coming from Complex Dynamics Journal Annals of Combinatorics Publisher Birkhäuser Basel ISSN 0218-0006 (Print) 0219-3094 (Online) Subject Mathematics and Statistics Issue Volume 9, Number 3 / October, 2005 Category Original Paper DOI 10.1007/s00026-005-0254-6 Pages 245-257 ----------- A count of maximal small copies in Multibrot sets Andrew Bridy et al 2005 Nonlinearity 18 1945-1953 // Wstep do teorii iteracji ... " byl wydany przez Instytut Matematyczny PAN, ul Sniadeckich 8, Warszawa i tam trzeba o niego pytac (na pewno jest dostepny w tamtejszej bibliotece). Krzysztof Baranski ------------------- Bud-Sequence conjecture on M fractal image and M–J conjecture between C and Z planes from z?zw+c(w=?+i?) Computers & Graphics, Volume 22, Issue 4, August 1998, Pages 537-546 Ning Chen and Weiyong Zhu ---------------------------- Publisher: Taylor & Francis Issue: Volume 9, Number 7 / July 2003 Pages: 687 - 691 Unbounded Orbits and Binary Digits Marc Chamberland A1 and Mario Martelli† A2 ----------------------------------------- -------------------- Computers & Graphics Volume 28, Issue 5 , October 2004, Pages 779-784 Chaotic bands in the Mandelbrot set M. Romera, G. Álvarez and F. Montoya --------------------------- Physica A: Statistical Mechanics and its Applications Volume 292, Issues 1-4 , 15 March 2001, Pages 207-230 Misiurewicz point patterns generation in one-dimensional quadratic maps ================================== Gray codes and 1D quadratic maps Electronics Letters -- 25 June 1998 -- Volume 34, Issue 13, p. 1304-1306 G. Álvarez,1 M. Romera,1 G. Pastor,1 and F. Montoya1 ----------------------- Chen Ning, Zhu Weiyong. Bud-sequence conjecture on M fractal image and M-J conjecture between c and z planes from z?zw+c(w=?+i?)[J]. Computer & Graphics,1998,22(4):537-546. //======================== Comparing sequential visualisation methods for the Mandelbrot set Source Computational Methods In Sciences And Engineering; Vol. 01 archive Proceedings of the international conference on Computational methods in sciences and engineering table of contents Kastoria, Greece Pages: 148 - 151 Year of Publication: 2003 ISBN:981-238-595-9 Author V. Drakopoulos ------------------------------- On the dimension of a part of the Mandelbrot set R Van Damme 1989 J. Phys. A: Math. Gen. 22 5249-5257 ------------------------------------------- [4] M. Henon,1969, "Numerical Study of Quadratic Area-Preserving Mappings," Quarterly of Applied Mathematics, 27, 291-312. ----------------------------------------- D. Schleicher. The structure of the Mandelbrot set. Preprint 1995. ------------------------ author = "K. Keller", title = "Symbolic dynamics for angle-doubling on the circle {III}. {Sturmian} sequences and the quadratic map", journal = ETDS, volume = 14, year = 1994, pages = "787-805"} ---------------------------------------------------------- Symbolic dynamics for angle-doubling on the circle. IV. Equivalence of abstract Julia sets. Keller, Karsten Atti Sem. Mat. Fis. Univ. Modena 42 (1994), no. 2, 547--567, MathSciNet. -------------------------------------------------------------- The geometry of Julia sets. Aarts, Jan M.; Oversteegen, Lex G. Trans. Amer. Math. Soc. 338 (1993), no. 2, 897--918, MathSciNet. ----------------------------------- The components of a Julia set. Beardon, A. F. Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 1, 173--177, MathSciNet. --------------------------------------------- Karsten Keller Julia equivalences and Abstract Siegel disks , In: Progress in Complex Dynamics, H.Kriete (editor), Pitman Research Notes in Mathematical Sciences 387 (1998), 86-101. ------------------------------------------ 2005 European Mathematical Society, FIZ Karlsruhe & Springer­Verlag 0603.30030 Douady, A. Algorithms for computing angles in the Mandelbrot set. (English) Chaotic dynamics and fractals, Proc. Conf., Atlanta/Ga. 1985, Notes Rep. Math. Sci. Eng. 2, 155­168 (1986). A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168). ----------------------------------- B. BRANNER, Cubic polynomials : turning around the connectedness locus, The Technical University of Denmark, Mat-report 1992-2005. ----------------------------- B. BRANNER and J. H. HUBBARD, The iteration of cubic polynomials, Part I : The global topology of parameter space (Acta Mathematica, Vol. 160, 1988, pp. 143-206). -------------------------- Title: An Abstract Mandelbrot Set Algorithm for zn + c DOI No: doi:10.1142/S0218348X9800002X Source: Fractals [Complex Geometry, Patterns, and Scaling in Nature and Society], Vol. 6, No. 1 (1998) 1-10 Copyright: World Scientific Publishing Company Author(s): Albert Kern Michael Frame -------------------------------- An abstract Mandelbrot set algorithm for z^n + c. Kern, Albert; Frame, Michael Fractals 6 (1998), no. 1, 1--10, MathSciNet. ---------------------------- On the cusp and the tip of a midget in the Mandelbrot set antenna. Romera, M.; Pastor, G.; Montoya, F. Phys. Lett. A 221 (1996), no. 3-4, 158--162, MathSciNet. --------------------------- MathSciNet. --------------------------- Spirals in the Mandelbrot set. I, II, III. Stephenson, John Phys. A 205 (1994), no. 4, 634--645, 646--655, 656--664, MathSciNet. ================== Nonanalytic dynamics for generating the Mandelbrot set: a tutorial. Metzler, W.; Brelle, A.; Schmidt, K.-D. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2 (1992), no. 2, 241--250, MathSciNet. ===================== The Mandelbrot set for binary numbers. Senn, Peter Amer. J. Phys. 58 (1990), no. 10, 1018, MathSciNet. -------------- Looking at the Mandelbrot Set (in Computer Corner) Mark Bridger The College Mathematics Journal, Vol. 19, No. 4. (Sep., 1988), pp. 353-363, Jstor. ------------------------ The Mandelbrot set in a model for phase transitions. Peitgen, Heinz-Otto; Richter, Peter H. Workshop Bonn 1984 (Bonn, 1984), 111--134, Lecture Notes in Math., 1111, Springer, Berlin, 1985, MathSciNet. --------------------- The Fractal Geometry of Mandelbrot Anthony Barcellos The College Mathematics Journal, Vol. 15, No. 2. (Mar., 1984), pp. 98-114, Jstor. ------------------- General Mandelbrot Sets Generated by the Complex Iteration zn+1=Zmn+cZHU Zhi-liang,YAN De-jun, ZHU Wei-yong JOURNAL OF NORTHEASTERN UNIVERSITY (NATURAL SCIENCE) 2000 Vol.21 No.5 -------------------- Construction of Internal Structures of the General Mandelbrot Sets Based on the Method of Examining the Inner Loop WANG Xing-yuanGU Shu-sheng ??????(?????) JOURNAL OF NORTHEASTERN UNIVERSITY (NATURAL SCIENCE) 2000 Vol.21 No.5 P.490-493 ------------------------ Hooper K J.A note on some internal structures of the mandelbrot set[J]. Computers & Graphics,1991,15(2):295-297 -------------- Building Blocks for Quadratic Julia Sets Joachim Grispolakis, John C. Mayer, Lex G. Oversteegen Transactions of the American Mathematical Society, Vol. 351, No. 3 (Mar., 1999), pp. 1171-1201 ============================= Szyszkowicz, Mieczyslaw (1991) Block Iterations in the Complex Plane. Computer Graphics Forum 10 (1), 67-70. doi: 10.1111/ 1467-8659.1010067 ------------------------- The Fractal Geometry of Mandelbrot Anthony Barcellos The College Mathematics Journal, Vol. 15, No. 2. (Mar., 1984), pp. 98-114 -------------------------------- --------------------------------------- Comparing sequential visualisation methods for the Mandelbrot set Source Computational Methods In Sciences And Engineering; Vol. 01 archive Proceedings of the international conference on Computational methods in sciences and engineering table of contents Kastoria, Greece Pages: 148 - 151 Year of Publication: 2003 ISBN:981-238-595-9 ------------------------------------------ Smooth decomposition of generalized Fatou set explains smooth structure in generalized Mandelbrot set. Peinke, J.; Parisi, J.; Röhricht, B.; Rössler, O. E.; Metzler, W. Z. Naturforsch. A 43 (1988), no. 1, 14--16, MathSciNet. ----------------------------------------- On crossing the boundary of the Mandelbrot set. Handler, Ivan; Kauffman, Louis H.; Sandin, Dan Computers in geometry and topology (Chicago, IL, 1986), 151--177, Lecture Notes in Pure and Appl. Math., 114, Dekker, New York, 1989, MathSciNet. ------------------------------ Douady, A., and Hubbard, J., "On the dynamics of polynomial-like mappings," Ann. Sci. Ecole Norm. Sup,, 18 (1985), 287-343. -------------------------------------------- Guckenheimer, J., and McGehee, R., "A proof of the Mandelbrot n2 conjecture," Institute Mittag-Leffler, Report 15, 1984. ----------------------------------------------------------------- On the Structure of the Mandelbar Set Crowe, W. D. and R. Hasson and P. J. Rippon and P. E. D. Strain-Clark Nonlinearity, (1989), V. 2, pp. 541-553 ====================== On the coefficients of the mapping to the exterior of the Mandelbrot set. Ewing, John H.; Schober, Glenn Michigan Math. J. 37 (1990), no. 2, 315--320, MathSciNet. ==================== High-order cycles in the logistic map or centers of cardioids in the Mandelbrot set. Stephenson, John J. Statist. Phys. 58 (1990), no. 3-4, 579--597, MathSciNet. ============================================= The abstract Mandelbrot set---an atlas of abstract Julia sets. Keller, Karsten Topology, measures, and fractals (Warnemünde, 1991), 76--81, Math. Res., 66, Akademie-Verlag, Berlin, 1992, MathSciNet. =============================== The Orbit Diagram and the Mandelbrot Set Robert L. Devaney The College Mathematics Journal, Vol. 22, No. 1. (Jan., 1991), pp. 23-38, Jstor. ========================== Nonanalytic dynamics for generating the Mandelbrot set: a tutorial. Metzler, W.; Brelle, A.; Schmidt, K.-D. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2 (1992), no. 2, 241--250, MathSciNet. =========================== # Global analytical structure of the Mandelbrot set and its generalization. Huang, Yong-nian Sci. China Ser. A 35 (1992), no. 2, 175--185, MathSciNet. ------------------------ Counting hyperbolic components of the Mandelbrot set. Lutzky, M. Phys. Lett. A 177 (1993), no. 4-5, 338--340, MathSciNet. ============================== A count of maximal small copies in Multibrot sets Andrew Bridy et al 2005 Nonlinearity 18 1945-1953 ---------------------------------- The index on the Mandelbrot set. Fujimoto, Yoshihisa Internat. J. Bifur. Chaos Appl. Sci. Engrg. 3 (1993), no. 5, 1225--1233, MathSciNet. ===================== The Mandelbrot Set and sigma-Automorphisms of Quotients of the Shift Pau Atela Transactions of the American Mathematical Society, Vol. 335, No. 2. (Feb., 1993), pp. 683-703, Jstor. ============= Spirals in the Mandelbrot set. I, II, III. Stephenson, John Phys. A 205 (1994), no. 4, 634--645, 646--655, 656--664, MathSciNet. =============== # On the complement of the Mandelbrot set. Levin, G. M. Israel J. Math. 88 (1994), no. 1-3, 189--212, MathSciNet. ====================== On the cusp and the tip of a midget in the Mandelbrot set antenna. Romera, M.; Pastor, G.; Montoya, F. Phys. Lett. A 221 (1996), no. 3-4, 158--162, MathSciNet. ==================== An abstract Mandelbrot set algorithm for z^n + c. Kern, Albert; Frame, Michael Fractals 6 (1998), no. 1, 1--10, MathSciNet. ------------------ Determination of Mandelbrot set's hyperbolic component centres. Álvarez, G.; Romera, M.; Pastor, G.; Montoya, F. Chaos Solitons Fractals 9 (1998), no. 12, 1997--2005, MathSciNet. ================ Composed accelerated escape time algorithm to construct the general Mandelbrot sets. Liu, Xiangdong; Zhu, Zhiliang; Wang, Guangxing; Zhu, Weiyong Fractals 9 (2001), no. 2, 149--153, MathSciNet. ============== Journal of Mathematical Sciences Publisher: Springer New York ISSN: 1072-3374 (Paper) 1573-8795 (Online) DOI: 10.1007/BF01095412 Issue: Volume 52, Number 6 Date: December 1990 Pages: 3512 - 3522 Theory of iterations of polynomial families in the complex plane G. M. Levin =============================== The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653 ================================= Can We See the Mandelbrot Set? John Ewing College Mathematics Journal, Vol. 26, No. 2 (Mar., 1995) , pp. 90-99 =============================== The Mandelbrot Set and $\sigma$-Automorphisms of Quotients of the Shift Pau Atela Transactions of the American Mathematical Society, Vol. 335, No. 2 (Feb., 1993) , pp. 683-703 --------------------------------------------------------- Chaos: An Interdisciplinary Journal of Nonlinear Science -- December 1998 -- Volume 8, Issue 4, pp. 739-740 Ordering of the Mandelbrot-like set of the exponential map M. Romera, G. Pastor, G. Álvarez, and F. Montoya ============================== Mathematical Notes Publisher: Springer New York ISSN: 0001-4346 (Paper) 1573-8876 (Online) DOI: 10.1007/BF01236299 Issue: Volume 48, Number 5 Date: November 1990 Pages: 1126 - 1131 Symmetries on the Julia set G. M. Levin1 ============================ Journal of Computational Methods in Science and Engineering Issue: Volume 4, Numbers 1-2 / 2004 Pages: 115 - 123 Symbolic mathematical computing of bifurcations in dynamical systems Jose C. Valverde A1, Fernando L. Pelayo A1, Juan A. Martinez A1, Juan J. Miralles A1 -------------------------------------------------------------------- Parry, William . Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 1966 368--378. -------------------------------------- Author: Barnsley, M.F.; Geronimo, J.S.; Harrington, A.N. Title: Geometrical and electrical properties of some Julia sets. Source: Classical and quantum models and arithmetic problems, Lect. Notes Pure Appl. Math. 92, 1-68 (1984). Publication Year: 1984 ======================== Use of potential functions in 3D rendering of fractal images from complex functions Journal The Visual Computer Publisher Springer Berlin / Heidelberg ISSN 0178-2789 (Print) 1432-2315 (Online) Subject Computer Science Issue Volume 12, Number 4 / April, 1996 =================== Field lines in the Mandelbrot set Kenelm W. Philip Journal Title: Computers & Graphics Date: 1992 Volume: 16 Issue: 4 p. 443 - 447 ================= The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence Robert L. Devaney The American Mathematical Monthly, Vol. 106, No. 4 (Apr., 1999), pp. 289-302 doi:10.2307/2589552