From Newton’s method to exotic basins Part I: The parameter space
Barański, Krzysztof
Fundamenta Mathematicae, Tome 158 (1998), p. 249-288 / Harvested from The Polish Digital Mathematics Library
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This is the first part of the work studying the family 𝔉 of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of 𝔉 and give a detailed study of the subfamily ℱ2 consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in ℱ2 from Newton maps to maps with so-called exotic basins.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:212315 
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     year = {1998},
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Barański, Krzysztof. From Newton’s method to exotic basins Part I: The parameter space. Fundamenta Mathematicae, Tome 158 (1998) pp. 249-288. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv158i3p249bwm/

[00000] [Ba] K. Barański, Connectedness of the basin of attraction for rational maps, Proc. Amer. Math. Soc. (6) 126 (1998), 1857-1866. | Zbl 0898.58043

[00001] [BH] B. Branner and J. Hubbard, The iteration of cubic polynomials, II: Patterns and parapatterns, Acta Math. 169 (1992), 229-325. | Zbl 0812.30008

[00002] [CGS] J. H. Curry, L. Garnett and D. Sullivan, On the iteration of a rational function: computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267-277. | Zbl 0524.65032

[00003] [DH1] A. Douady et J. H. Hubbard, Etude dynamique des polynômes complexes, I et II, avec la collaboration de P. Lavours, Tan Lei et P. Sentenac, Publication d'Orsay 84-02, 85-04, 1984-1985.

[00004] [DH2] A. Douady et J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287-343. | Zbl 0587.30028

[00005] [HP] F. von Haeseler and H.-O. Peitgen, Newton's method and complex dynamical systems, Acta Appl. Math. 13 (1988), 3-58. | Zbl 0671.30023

[00006] [He] J. Head, The combinatorics of Newton's method for cubic polynomials, Ph.D. thesis, Cornell University, Ithaca, 1987.

[00007] [MSS] R. Ma né, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), 193-217.

[00008] [Mi1] J. Milnor, Dynamics in one complex variable: introductory lectures, preprint, SUNY at Stony Brook, IMS # 1990/5.

[00009] [Mi2] J. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), 37-83. | Zbl 0922.58062

[00010] [P1] F. Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fund. Math. 149 (1996), 95-118. | Zbl 0852.58052

[00011] [P2] F. Przytycki, Remarks on simple-connectedness of basins of sinks for iterations of rational maps, in: Banach Center Publ. 23, PWN, 1989, 229-235.

[00012] [Re1] M. Rees, A partial description of parameter space of rational maps of degree two, I: Acta Math. 168 (1992), 11-87; II: Proc. London Math. Soc. (3) 70 (1995), 644-690. | Zbl 0774.58035

[00013] [Re2] M. Rees, Components of degree two hyperbolic rational maps, Invent. Math. 100 (1990), 357-382. | Zbl 0712.30022

[00014] [Ro] P. Roesch, Topologie locale des méthodes de Newton cubiques, Ph.D. thesis, École Norm. Sup. de Lyon, 1997.

[00015] [Se] G. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), 39-72. | Zbl 0427.55006

[00016] [Sh] M. Shishikura, The connectivity of the Julia set of rational maps and fixed points, preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1990.

[00017] [Ta] Tan, Branched coverings and cubic Newton maps, Fund. Math. 154 (1997), 207-260. | Zbl 0903.58029