Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Infinitely Renormalizable Quadratic Polynomials


Author: Yunping Jiang
Journal: Trans. Amer. Math. Soc. 352 (2000), 5077-5091
MSC (2000): Primary 37Fxx; Secondary 37E20
Published electronically: July 12, 2000
MathSciNet review: 1675198
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles.


References [Enhancements On Off] (What's this?)

  • [AL] Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. MR 0200442
  • [BH] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math. 160 (1988), no. 3-4, 143–206. MR 945011, 10.1007/BF02392275
    Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229–325. MR 1194004, 10.1007/BF02392761
  • [CG] Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383
  • [CT] Charles Tresser and Pierre Coullet, Itérations d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A577–A580 (French, with English summary). MR 512110
  • [DHM] A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
    A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie II, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (French). With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac. MR 812271
  • [DH] 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
  • [FE1] Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 0501179
  • [FE2] Louis Nirenberg, Remarks on nonlinear problems, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 189–197. MR 609561
  • [HU] 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
  • [JI] Yunping Jiang, Renormalization and geometry in one-dimensional and complex dynamics, Advanced Series in Nonlinear Dynamics, vol. 10, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1442953
  • [JIM] Y. Jiang, On Mandelbrot set at infinitely renormalizable points, MSRI Preprint 95-63 (1995).
  • [MA] Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe für den Referenten. [Series for the Referee]. MR 665254
  • [MC1] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
  • [MC2] Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347
  • [MI1] J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, IMS preprint 1990/5, Stony Brook.
  • [MI2] J. Milnor, Local connectivity of Julia sets: expository lectures, IMS preprint 1992/11, Stony Brook.
  • [MV] Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
  • [PM] R. Perez-Marco, Topology of Julia sets and hedgehogs, Preprint, Université Paris-Sud, 94-48 (1994).
  • [SU] Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37Fxx, 37E20

Retrieve articles in all journals with MSC (2000): 37Fxx, 37E20


Additional Information

Yunping Jiang
Affiliation: Department of Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016 and Department of Mathematics, Queens College of CUNY, Flushing, New York 11367
Email: yunqc@jiang.math.qc.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02514-9
Keywords: Julia set, local connectivity, two-dimensional puzzle, three-dimensional puzzle, infinitely renormalizable quadratic polynomial, complex bounds, unbranched
Received by editor(s): September 25, 1997
Received by editor(s) in revised form: January 14, 1999
Published electronically: July 12, 2000
Additional Notes: The author is supported in part by grants from the NSF and from the PSC-CUNY
Article copyright: © Copyright 2000 American Mathematical Society