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Cubic polynomials with a parabolic point

Published online by Cambridge University Press:  22 January 2010

P. ROESCH
P. ROESCH*
Affiliation:
IMT, Toulouse, France (email: [email protected])
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Abstract

We consider cubic polynomials with a simple parabolic fixed point of multiplier 1. For those maps, we prove that the boundary of the immediate basin of attraction of the parabolic point is a Jordan curve (except for the polynomial z+z3 where it consists in two Jordan curves). Moreover, we give a description of the dynamics and obtain the local connectivity of the Julia set under some assumptions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1843 - 1867
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Blanchard, P.. Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[2]Carleson, L. and Gamelin, T. W.. Complex dynamics. Springer, New York, 1993.CrossRefGoogle Scholar
[3]Douady, A. and Hubbard, J. H.. Étude dynamique des polynômes complexes, I & II. Publ. Math. d’Orsay, 1984–85.Google Scholar
[4]Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. École Norm. Sup. (4) 18 (1985), 287–343.CrossRefGoogle Scholar
[5]Goldberg, L. and Milnor, J.. Fixed points of polynomial maps. II. Fixed point portraits. Ann. Sci. École Norm. Sup. (4) 26(1) (1993), 51–98.CrossRefGoogle Scholar
[6]Goluzin, G. M.. Geometric Theory of Functions of a Complex Variable (Translations of Mathematical Monographs, 26). American Mathematical Society, Providence, RI, 1969.CrossRefGoogle Scholar
[7]Kiwi, J.. Real laminations and the topological dynamics of complex polynomials. Adv. Math. 184(2) (2004), 207–267.CrossRefGoogle Scholar
[8]Kozlovski, O., Shen, W. and van Strien, S.. Rigidity for real polynomials. Ann. of Math. (2) 165 (2007), 749–841.CrossRefGoogle Scholar
[9]Lei, T. and Yongcheng, Y.. Local connectivity of the Julia set for geometrically-finite rational maps. Sci. China Ser. A 39 (1996), 39–47.Google Scholar
[10]Lei, T. and Yongcheng, Y.. Unicritical Branner–Hubbard conjecture. Complex Dynamics, Family and Friends. Ed. Schleicher, D.. A. K. Peters, 2009.Google Scholar
[11]Lyubich, M. and Minsky, Y.. Laminations in holomorphic dynamics. J. Differential Geom. 47(1) (1997), 17–94.CrossRefGoogle Scholar
[12]Milnor, J.. Local connectivity of Julia sets: expository lectures. The Mandelbrot Set, Theme and Variations (LMS Lecture Note Series, 274). Ed. Lei, Tan. Cambridge University Press, Cambridge, 2000, pp. 67–116.CrossRefGoogle Scholar
[13]Milnor, J.. Dynamics in One Complex Variable, 3rd edn.(Annals of Mathematics Studies, 160). Princeton University Press, Princeton, NJ, 2006.Google Scholar
[14]Milnor, J.. Cubic polynomial maps with periodic critical orbit, Part I. Complex Dynamics, Family and Friends. Ed. Schleicher, D.. A. K. Peters Ltd., Wellesley, MA, 2009.Google Scholar
[15]Naĭshul’, V. A.. Topological invariants of analytic and area-preserving mappings and their application to analytic differential equations in C 2 and CP 2. Funktsional. Anal. i Prilozhen. 14(1) (1980), 73–74 (in Russian).CrossRefGoogle Scholar
[16]Petersen, C. L.. On the Pommerenke–Levin–Yoccoz inequality. Ergod. Th. & Dynam. Sys. 13(4) (1993), 785–806.Google Scholar
[17]Petersen, C. L. and Roesch, P.. Parabolic tools, J. Difference Equ. Appl. (2009) accepted for publication.Google Scholar
[18]Petersen, C. L. and Zakeri, S.. On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. of Math. (2) 159 (2004), 1–52.CrossRefGoogle Scholar
[19]Pommerenke, C.. Boundary Behaviour of Conformal Maps. Springer, Berlin, 1992.CrossRefGoogle Scholar
[20]Roesch, P.. Puzzles de Yoccoz pour les applications à allure rationnelle. Enseign. Math. (2) 45(1–2) (1999), 133–168.Google Scholar
[21]Roesch, P.. Hyperbolic components of polynomials with fixed critical point of maximal order. Ann. Sci. École Norm. Sup. (4) 40(6) (2007), 901–949.CrossRefGoogle Scholar
[22]Roesch, P.. Cubic parabolic slice. Manuscript.Google Scholar
[23]Roesch, P. and Yin, Y.. The boundary of bounded polynomial Fatou components. C. R. Math. Acad. Sci. Paris 346(15–16) (2008), 877–888.CrossRefGoogle Scholar
[24]Steinmetz, N.. Rational iteration. Complex Analytic Dynamical Systems (de Gruyter Studies in Mathematics, 16). Walter de Gruyter, Berlin, 1993.Google Scholar
[25]Weiyuan, Q. and Yongcheng, Y.. Proof of the Branner–Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A 52(1) (2009), 45–65.Google Scholar
[26]Yoccoz, J.-C.. Petits diviseurs en dimension 1. Astérisque 231 (1995).Google Scholar