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Geography of the cubic connectedness locus: Intertwining surgery. (English) Zbl 0959.37036

The prevalence of Mandelbrot sets in one-parameter complex analytic families is a well-studied phenomenon in conformal dynamics. Its explanation by A. Douady and J. H. Hubbard [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 287-343 (1985; Zbl 0587.30028)] has given rise to the theory of renormalization, and has inspired many efforts, starting with the seminal work of B. Branner and A. Douady [Lect. Notes Math. 1345, 11-72 (1987; Zbl 0668.58026)], to invert this procedure by means of surgery on quadratic polynomials. In this paper the authors exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. These products were observed by J. Milnor in computer experiments which inspired Lavaurs’ proof of non local-connectivity for the cubic connectedness locus. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction is an intertwining surgery on two quadratics. Using quasiconformal surgery techniques of Branner and Douady, the authors show that any two quadratics may be intertwined to obtain a cubic polynomial. The associated asymptotic geography of the cubic connectedness locus is discussed.

MSC:

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F25 Renormalization of holomorphic dynamical systems
28A80 Fractals
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References:

[1] B. BIELEFELD , Changing the order of critical points of polynomials using quasiconformal surgery , Thesis, Cornell, 1989 .
[2] B. BIELEFELD , Questions in quasiconformal surgery , pp. 2-8, in Conformal Dynamics Problem List, ed. B. Bielefeld, Stony Brook IMS Preprint 1990 / 1991 , and in part 2 of Linear and Complex Analysis Problem Book 3, ed. V. Havin and N. Nikolskii, Lecture Notes in Math., Vol. 1574, Springer-Verlag, 1994 .
[3] B. BRANNER and A. DOUADY , Surgery on complex polynomials , in Proceedings of the Symposium on Dynamical Systems, Mexico, 1986 , Lecture Notes in Math., Vol. 1345, Springer-Verlag, 1987 . Zbl 0668.58026 · Zbl 0668.58026
[4] B. BRANNER and N. FAGELLA , Homeomorphisms between limbs of the Mandelbrot set , MSRI Preprint 043-95. · Zbl 0961.37011
[5] 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. MR 90d:30073 | Zbl 0668.30008 · Zbl 0668.30008 · doi:10.1007/BF02392275
[6] X. BUFF , Extension d’homéomorphismes de compacts de \Bbb C , Manuscript, and personal communication.
[7] A. DOUADY , Does a Julia set depend continuously on a polynomial ? , Proc. of Symp. in Applied Math., Vol. 49, 1994 . MR 1315535 | Zbl 0934.30023 · Zbl 0934.30023
[8] A. DOUADY and J. H. HUBBARD , Étude dynamique des polynômes complexes, I & II , Publ. Math. Orsay, 1984 - 1985 . Article | Zbl 0552.30018 · Zbl 0552.30018
[9] A. DOUADY and J. H. HUBBARD , On the dynamics of polynomial-like mappings , Ann. scient. Éc. Norm. Sup., 4e série, Vol. 18, 1985 , pp. 287-343. Numdam | MR 87f:58083 | Zbl 0587.30028 · Zbl 0587.30028
[10] A. EPSTEIN , Counterexamples to the quadratic mating conjecture , Manuscript in preparation.
[11] D. FAUGHT , Local connectivity in a family of cubic polynomials , Thesis, Cornell 1992 .
[12] L. GOLDBERG and J. MILNOR , Fixed points of polynomial maps II , Ann. Scient. Éc. Norm. Sup., 4e série, Vol. 26, 1993 , pp. 51-98. Numdam | MR 95d:58107 | Zbl 0771.30028 · Zbl 0771.30028
[13] P. HAÏSSINSKY , Chirurgie croisée , Manuscript, 1996 .
[14] J. H. HUBBARD , Local connectivity of Julia sets and bifurcation loci : three theorems of J.-C. Yoccoz , in Topological methods in Modern Mathematics, Publish or Perish, 1992 , pp. 467-511 and 375-378. · Zbl 0797.58049
[15] J. KIWI , Non-accessible critical points of Cremer polynomials , IMS at Stony Brook Preprint 1995 /2. · Zbl 0966.37015
[16] P. LAVAURS , Systèmes dynamiques holomorphes : Explosion de points périodiques , Thèse, Université de Paris-Sud, 1989 .
[17] O. LEHTO and K. I. VIRTANEN , Quasiconformal Mappings in the Plane , Springer-Verlag, 1973 . MR 49 #9202 | Zbl 0267.30016 · Zbl 0267.30016
[18] M. LYUBICH , On typical behavior of the trajectories of a rational mapping of the sphere , Soviet. Math. Dokl., Vol. 27, 1983 , No. 1, pp. 22-25. MR 84f:30036 | Zbl 0595.30034 · Zbl 0595.30034
[19] M. LYUBICH , On the Lebesgue measure of a quadratic polynomial , IMS at Stony Brook Preprint 1991 / 2010 .
[20] M. LYUBICH , Dynamics of quadratic polynomials, I. Combinatorics and geometry of the Yoccoz puzzle , MSRI Preprint 026-95.
[21] R. MAÑ;É , P. SAD and D. SULLIVAN , On the dynamics of rational maps , Ann. Scient. Éc. Norm. Sup., 4e série, Vol. 16, 1983 , pp. 51-98. Numdam | Zbl 0524.58025 · Zbl 0524.58025
[22] C. MCMULLEN , Complex Dynamics and Renormalization , Annals of Math. Studies, Princeton Univ. Press, 1993 . · Zbl 0791.52014
[23] C. MCMULLEN , Renormalization and 3-Manifolds which Fiber over the Circle , Annals of Math. Studies, Princeton Univ. Press, 1996 . MR 97f:57022 | Zbl 0860.58002 · Zbl 0860.58002
[24] C. MCMULLEN and D. SULLIVAN , Quasiconformal homeomorphisms and dynamics III : The Teichmüller space of a rational map , Preprint, 1996 .
[25] J. MILNOR , Dynamics in one complex variable : Introductory lectures , IMS at Stony Brook Preprint 1990 / 1995 .
[26] J. MILNOR , Remarks on iterated cubic maps , Experimental Math., Vol. 1 1992 , pp. 5-24. Article | MR 94c:58096 | Zbl 0762.58018 · Zbl 0762.58018
[27] J. MILNOR , On cubic polynomials with periodic critical point , Manuscript, 1991 .
[28] J. MILNOR , Hyperbolic components in spaces of polynomial maps , with an appendix by A. Poirier, IMS at Stony Brook Preprint 1992 / 1993 .
[29] J. MILNOR , Periodic orbits, external rays and the Mandelbrot set ; An expository account, Preprint, 1995 .
[30] S. NAKANE and D. SCHLEICHER , Non-local connectivity of the tricorn and multicorns , in Proceedings of the International Conference on Dynamical Systems and Chaos, World Scientific, 1994 . Zbl 0989.37537 · Zbl 0989.37537
[31] M. SHISHIKURA , The parabolic bifurcation of rational maps , Colóquio Brasileiro de Matemática 19, IMPA, 1992 .
[32] R. WINTERS , Bifurcations in families of antiholomorphic and biquadratic maps , Thesis, Boston University, 1989 .
[33] B. YARRINGTON , Local connectivity and Lebesgue measure of polynomial Julia sets , Thesis, SUNY at Stony Brook, 1995 .
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