Branched coverings and cubic Newton maps

• Autorzy: Lei Tan
• Języki publikacji: EN

Abstrakty

EN

We construct branched coverings such as matings and captures to describe the dynamics of every critically finite cubic Newton map. This gives a combinatorial model of the set of cubic Newton maps as the gluing of a subset of cubic polynomials with a part of the filled Julia set of a specific polynomial (Figure 1).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1997
• Tom: 154
• Numer: 3
• Strony: 207-260

Daty

wydano

1997

otrzymano

1995-12-05

poprawiono

1996-09-09

poprawiono

1997-04-01

poprawiono

1997-05-22

Twórcy

autor

Lei Tan

• Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom

Bibliografia

• [B] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), 85-141.
• [CGS] J. Curry, L. Garnett and D. Sullivan, On the iteration of rational functions: Computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267-277.
• [D1] A. Douady, Systèmes dynamiques holomorphes, Séminaire Bourbaki, 35e année, 1982-1983, exp. no. 599, 1982.
• [D2] A. Douady, Algorithm for computing angles in the Mandelbrot set, in: Chaotic Dynamics and Fractals, M. F. Barnsley and S. G. Demko (eds.), Academic Press, New York, 1986, 155-168.
• [D3] A. Douady, Chirurgie sur les applications holomorphes, in: Proc. Internat. Congress Math., Berkeley, Calif., 1986, 724-738 (English version: preprint MSRI, 1986).
• [DH1] A. Douady et J. H. Hubbard, Étude dynamique des polynômes complexes, I et II, avec la collaboration de P. Lavaurs, Tan Lei et P. Sentenac, Publication d'Orsay 84-02, 85-04, 1984/1985.
• [DH2] A. Douady et J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), 263-297.
• [DH3] A. Douady et J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287-343.
• [F] D. Faught, Local connectivity in a family of cubic polynomials, Ph.D. thesis, Cornell Univ., Ithaca, N.Y., 1992.
• [Ha] F. von Haeseler, Über Attraktionsgebiete superattraktiver Zykle, Ph.D. thesis, Bremen Univ., Bremen, 1985.
• [HP] F. von Haeseler and H.-O. Peitgen, Newton's method and complex dynamical systems, Acta Appl. Math. 13 (1988), 3-58.
• [He] J. Head, The combinatorics of Newton's method for cubic polynomials, Ph.D. thesis, Cornell Univ., Ithaca, N.Y., 1987.
• [Le] S. Levy, Critically finite rational maps, Ph.D. Thesis, Princeton Univ., Princeton, N.J., 1985.
• [Me] H.-G. Meier, On the connectedness of the Julia-set for rational functions, preprint, Aachen Univ., 1989.
• [M1] J. Milnor, On cubic polynomials with periodic critical point (very rough draft), 5-28-91.
• [M2] J. Milnor, Dynamics in one complex variable: Introductory lectures, preprint Stony Brook 1990-5.
• [Prz] F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, in: Dynamical Systems and Ergodic Theory, K. Krzyżewski (ed.), PWN-Polish Sci. Publ., 1989, 229-235.
• [R1] M. Rees, A partial description of parameter space of rational maps of degree two: Part I, Acta Math. 168 (1992), 11-87.
• [R2] M. Rees, A partial description of parameter space of rational maps of degree two: Part II, Proc. London Math. Soc. (3) 70 (1995), 644-690.
• [R3] M. Rees, Realization of matings of polynomials as rational maps of degree two, manuscript, 1986.
• [Sa] D. Saupe, Discrete versus continuous Newton's method: A case study, Acta Appl. Math. 13 (1988), 59-80.
• [Sh1] M. Shishikura, The connectivity of the Julia set of rational maps and Fixed points, preprint, I.H.E.S., Bures-sur-Yvette, 1990.
• [Sh2] M. Shishikura, On a theorem of M. Rees for matings of polynomials, preprint, I.H.E.S., Bures-sur-Yvette, 1990.
• [ST] M. Shishikura and L. Tan, A family of cubic rational maps and matings of cubic polynomials, preprint 88-50, Max-Planck-Institut für Mathematik, Bonn.
• [Ta] L. Tan, Matings of quadratic polynomials, Ergodic Theory Dynam. Systems 12 (1992), 589-620.
• [TY] L. Tan and Y. C. Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. in China (Ser. A) 39 (1) (1996), 39-47.
• [Th] W. Thurston, The combinatorics of iterated rational maps, preprint, Princeton Univ., Princeton, N.J., 1983.
• [W] B. Wittner, On the bifurcation loci of rational maps of degree two, Ph.D. thesis, Cornell Univ., Ithaca, N.Y., 1986.