G. Álvarez, M. Romera, G. Pastor, F. Montoya, "Determination of Mandelbrot set hyperbolic component centres", Chaos Solitons and Fractals, 9 (1998), 1997-2005. (PDF)
Pau Atela (1992). Bifurcations of dynamic rays in complex polynomials of degree two. Ergodic Theory and Dynamical Systems, 12 , pp 401-423
P. Collet and J. P. Eckmann. Iterated Maps on the Interval as Dynamical Systems. Birkhauser, Boston, 1980.
Cowen C C: Analytic solutions of Böttcher's functional equation in the unit disk. Journal Aequationes Mathematicae Issue Volume 23, Number 1 / December, 1981
Adrien Douady and John H. Hubbard : Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 and 1985)
Drakopoulos V., Comparing rendering methods for Julia sets , Journal of WSCG 10 (2002), 155–161.
V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ?
V Dolotin , A Morozow : Algebraic Geometry of Discrete Dynamics. The case of one variable
Ewing John H., Schober Glenn : Coefficients associated with the reciprocal of the Mandelbrot set. J. Math. Anal. Appl. 170, No.1, 104-114 (1992).
Lisa R. Goldberg: On the multiplier of a repelling fixed point Inventiones Mathematicae Volume 118, Number 1 / December, 1994, 85-108
E. N. Gilbert, J. Riordan : Symmetry Types of Periodic Sequences, Illinois J. Math., 5 (December 1961), pp. 657-665, Monograph 2072
S. Grossmann and S. Thomae: "Invariant distributions and stationary correlation functions of one-dimensional discrete processes", Z. Naturforsch. 32a, 1353 (1977)
Bailin HAO : Number of periodic orbits in continuous maps of the interval . Ann. Combin., 4 (2000), 339-346
On Fractal Coloring Techniques  by Jussi Härkönen
J. Hubbard and P. Papadopol,: Superattractive Fixed Points in C^n. Indiana Univ. Math. J. 43 (1994), 321 - 365
Irwin Jungreis : The uniformization of the complement of the Mandelbrot set. Duke Math. J. 52, no. 4 (1985), 935–938
Genadi Levin : Multipliers of periodic orbits of quadratic polynomials and the parameter plane
Genadi Levin : Multipliers of periodic orbits of quadratic polynomials and the parameter plane; February 2007) math/0702011
G. Levin, F. Przytycki : External rays to periodic points, Israel J. Math. 94 (1996), 29-57. MR 97d:58164
Levin, G. M. : On the complement of the Mandelbrot set. Israel J. Math. 88 (1994), no. 1-3, 189--212,
M. Lutzky : Counting hyperbolic components of the Mandelbrot set. Physics Letters A Volume 177, Issues 4-5, 21 June 1993, Pages 338-340
Abstract: Expressions are given for the number of hyperbolic components, cardioids, and discs which are associated with N-cycles for the Mandelbrot set. We also discuss the distribution of hyperbolic components along the real axis.
Manning A : Logarithmic Capacity and Renormalizability for Landing on the Mandelbrot Set Bull. London Math. Soc..1996; 28: 521-526
Mark McClure : Bifurcation sets and critical curves Mathematica in Education and Research. Volume 11, issue 1 (2006).
J. Milnor : "Periodic Orbits, External Rays and the Mandelbrot Set ..."; Asterisque 261 (2000) `Geometrie Complexe et Systemes Dynamiques', pp. 277-333; ims99-3 = [Stony Brook IMS Preprint 1999#3]
Misiurewicz M, Nitecki Z : Combinatorial patterns for maps of the interval , Mem. Amer. Math. Soc. 94 (1991), no. 456
Andrey Morozov : Universal Mandelbrot Set as a Model of Phase Transition Theory (to appear in JETP Lett.) ESI Preprint - (PS , PDF ) 5 pages
J. Milnor and W. Thurston, On Iterated Maps of the Interval, in Dynamical Systems: Proceedings of the special year held at the University of Maryland, College Park, 1986-87, (ed. J. Alexander) Lecture Notes in Math., vol. 1342, Springer-Verlag, Berlin (1988) 465-563.
Lei Tan Similarity between the Mandelbrot set and Julia sets Source: Comm. Math. Phys. 134, no. 3 (1990), 587–617
Tan Lei Voisinages connexes des points de Misiurewicz Annales de l'institut Fourier, 42 no. 4 (1992), p. 707-735
Young Hee Geum, Young Ik Kim : An improved computation of component centers in the degree-n bifurcation set. The Korean Journal of Computational & Applied Mathematics Volume 10 , Issue 1-2 (September 2002) Pages: 63 - 73 2002
Young Hee Geum, Young Ik Kim : A STUDY ON COMPUTATION OF COMPONENT CENTERS IN THE DEGREE- n BIFURCATION SET. International Journal of Computer Mathematics, Volume 80, Issue 2 February 2003 , pages 223 - 232
The governing equation locating component centers in the degree- n bifurcation set is a polynomial with a very high degree and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor (n - 1) . Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of computational precision digits. Many cases are studied for 2\leq n \leq 25 and show a remarkably improved computation. Our study extends the results given by Peitgen and Richter [11].
Keywords: Bifurcation; Component Centers; Degree- n Bifurcation Set; Mandelbrot Set; Newton's Method

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