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Carrots for dessert

Published online by Cambridge University Press:  17 November 2011

CARSTEN LUNDE PETERSEN  and
PASCALE ROESCH
CARSTEN LUNDE PETERSEN
Affiliation:
IMFUFA, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark (email: [email protected])
PASCALE ROESCH
Affiliation:
Institut de Mathématiques de Toulouse, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France (email: [email protected])
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Abstract

We formulate and prove a precise statement of asymptotic shrinking of ‘carrot-fields’ around the Mandelbrot set M. This phenomenon was suggested in a seminal text on polynomial-like mappings by Douady and Hubbard [On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. t.18 (1985), 287–343]. This is helpful for understanding how copies of M sit in the bifurcation loci of families of rational maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 2025 - 2055
Copyright
Copyright © Cambridge University Press 2011

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References

[DH1]Douady, A. and Hubbard, J. H.. Etude dynamique des polynômes complexes. Publications mathématiques d’Orsay, 1984.Google Scholar
[DH2]Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. t.18 (1985), 287–343.CrossRefGoogle Scholar
[Du]Dudko, D.. The Decoration Theorem for Mandelbrot and Multibrot sets, manuscript. arXiv:1004.0633v1.Google Scholar
[Ha]Haïssinsky, P.. Chirurgie parabolique. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 195–198.CrossRefGoogle Scholar
[Hu]Hubbard, J. H.. Local connectivity of Julia sets and bifurcation loci: three theorems of J. C. Yoccoz. Topological Methods in Modern Mathematics. Eds. Goldberg, and Phillips, . Publish or Perish, Houston, TX, 1993, pp. 467–511.Google Scholar
[L]Levin, G. M.. Disconnected Julia set and rotation sets. Ann. Sci. Éc. Norm. Supér. t.29 (1996), 1–22.Google Scholar
[Mi]Milnor, J. W.. Periodic orbits, externals rays and the Mandelbrot set: an expository account. Géométrie complexe et systèmes dynamiques (Orsay, 1995). Astérisque (261) (2000), 277–333 xiii.Google Scholar
[McM]McMullen, C. T.. The Mandelbrot set is universal. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Ed. Lei, Tan. Cambridge University Press, Cambridge, 2000.Google Scholar
[P]Petersen, C. L.. On the Pommerenke–Levin–Yoccoz inequality. Ergod. Th. & Dynam. Syst. 13 (1993), 785–806.Google Scholar
[PR1]Petersen, C. L. and Roesch, P.. Parabolic tools. J. Difference Equ. Appl. 16(05-06) (2010), 715–738.CrossRefGoogle Scholar
[PR2]Petersen, C. L. and Roesch, P.. The Yoccoz Combinatorial Analytic Invariant (Fields Institute Communications, 53). American Mathematical Society, 2008.CrossRefGoogle Scholar
[PR3]Petersen, C. L. and Roesch, P.. The Parabolic Mandelbrot set, manuscript.Google Scholar
[PR4]Petersen, C. L. and Roesch, P.. Carrots in Mandelbrot-like families, manuscript in preparation.Google Scholar
[Po]Pommerenke, C.. Boundary Behaviour of Conformal Maps. Springer, 1992.CrossRefGoogle Scholar
[R]Roesch, P.. Holomorphic motions and puzzles. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Ed. Lei, Tan. Cambridge University Press, 2000, pp. 117–131.CrossRefGoogle Scholar