Homeomorphisms between limbs of the Mandelbrot set
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- by Dzmitry Dudko and Dierk Schleicher PDF
- Proc. Amer. Math. Soc. 140 (2012), 1947-1956 Request permission
Abstract:
We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994.References
- Artur Avila, Mikhail Lyubich, and Weixiao Shen, Parapuzzle of the Multibrot set and typical dynamics of unimodal maps, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 1, 27–56. MR 2735075, DOI 10.4171/JEMS/243
- Bodil Branner and Adrien Douady, Surgery on complex polynomials, Holomorphic dynamics (Mexico, 1986) Lecture Notes in Math., vol. 1345, Springer, Berlin, 1988, pp. 11–72. MR 980952, DOI 10.1007/BFb0081395
- Bodil Branner and Núria Fagella, Homeomorphisms between limbs of the Mandelbrot set, J. Geom. Anal. 9 (1999), no. 3, 327–390. MR 1757453, DOI 10.1007/BF02921981
- Bodil Branner and Núria Fagella, Extensions of homeomorphisms between limbs of the Mandelbrot set, Conform. Geom. Dyn. 5 (2001), 100–139. MR 1872159, DOI 10.1090/S1088-4173-01-00069-8
- Henk Bruin and Dierk Schleicher, Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 502–522. MR 2439637, DOI 10.1112/jlms/jdn033
- Adrien Douady, Descriptions of compact sets in $\textbf {C}$, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429–465. MR 1215973
- Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367, DOI 10.24033/asens.1491
- D. Dudko, The decoration theorem for Mandelbrot and Multibrot sets. Preprint, arXiv:1004.0633v1 [math.DS], 2010. Submitted.
- J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467–511. MR 1215974
- Jeremy Kahn and Mikhail Lyubich, A priori bounds for some infinitely renormalizable quadratics. II. Decorations, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 57–84 (English, with English and French summaries). MR 2423310, DOI 10.24033/asens.2063
- Jeremy Kahn and Mikhail Lyubich, A priori bounds for some infinitely renormalizable quadratics. III. Molecules, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 229–254. MR 2508259, DOI 10.1201/b10617-7
- E. Lau, D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials. Preprint 19, Institute of Mathematical Sciences, Stony Brook (1994).
- Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
- Mikhail Lyubich, Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures, Astérisque 261 (2000), xii–xiii, 173–200 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755441
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- John Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Astérisque 261 (2000), xiii, 277–333 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755445
- C. L. Petersen, P. Roesch, Carrots for dessert. Preprint, arXiv:1003.3947v1 [math.DS], 2010. Submitted.
- J. Riedl, Arcs in Multibrot Sets, Locally Connected Julia Sets and Their Construction by Quasiconforman Surgery. Thesis, Technische Universität München, 2001, available at the Stony Brook IMS thesis server.
- D. Schleicher, Internal addresses of the Mandelbrot set and Galois groups of polynomials. Preprint, arXiv:math/9411238v2 [math.DS], 1994.
- Dierk Schleicher, On fibers and local connectivity of Mandelbrot and Multibrot sets, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 477–517. MR 2112117, DOI 10.1051/0004-6361:20040477
Additional Information
- Dzmitry Dudko
- Affiliation: Research I, Jacobs University, Postfach 750 561, D-28725 Bremen, Germany – and – G.-A.-Universität zu Göttingen, Bunsenstraße 3–5, D-37073 Göttingen, Germany
- MR Author ID: 969584
- Email: d.dudko@jacobs-university.de
- Dierk Schleicher
- Affiliation: Research I, Jacobs University, Postfach 750 561, D-28725 Bremen, Germany
- MR Author ID: 359328
- Email: dierk@jacobs-university.de
- Received by editor(s): September 7, 2010
- Received by editor(s) in revised form: January 28, 2011
- Published electronically: September 23, 2011
- Additional Notes: The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft to the first author in the context of the Research Training Group 1493
- Communicated by: Bryna Kra
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1947-1956
- MSC (2010): Primary 30D05, 37F10, 37F45; Secondary 37F25
- DOI: https://doi.org/10.1090/S0002-9939-2011-11047-5
- MathSciNet review: 2888182