Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

The dual nest for degenerate Yoccoz puzzles


Author: Magnus Aspenberg
Journal: Conform. Geom. Dyn. 13 (2009), 187-196
MSC (2000): Primary 37F20; Secondary 30D05
DOI: https://doi.org/10.1090/S1088-4173-09-00197-0
Published electronically: July 27, 2009
MathSciNet review: 2525102
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Yoccoz puzzle is a fundamental tool in Holomorphic Dynamics. The original combinatorial argument by Yoccoz, based on the Branner-Hubbard tableau, counts the preimages of a non-degenerate annulus in the puzzle. However, in some important new applications of the puzzle (notably, matings of quadratic polynomials) there is no non-degenerate annulus. We develop a general combinatorial argument to handle this situation. It allows us to derive corollaries, such as the local connectedness of the Julia set, for suitable families of rational maps.


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

References
  • Magnus Aspenberg and Michael Yampolsky. Mating non-renormalisable quadratic polynomials. Comm. Math. Phys. 287(1):1–40, 2009.
  • 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, DOI https://doi.org/10.1007/BF02392761
  • 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 Michael Lyubich. A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics. Preprint, math.DS/0609045.
  • Jeremy Kahn and Michael Lyubich. A priori bounds for some infinitely renormalizable quadratics: II. Decorations. Preprint, math.DS/0609046.
  • Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI https://doi.org/10.1007/BF02392694
  • John Milnor, Local connectivity of Julia sets: expository lectures, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 67–116. MR 1765085
  • John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
  • Pascale Roesch. Cubic polynomials with a parabolic point. Preprint. arXiv:0712.3372 .
  • Pascale Roesch, Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 6, 901–949 (English, with English and French summaries). MR 2419853, DOI https://doi.org/10.1016/j.ansens.2007.10.001
  • Pascale Roesch and Yongcheng Yin, The boundary of bounded polynomial Fatou components, C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 877–880 (English, with English and French summaries). MR 2441925, DOI https://doi.org/10.1016/j.crma.2008.06.004

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 37F20, 30D05

Retrieve articles in all journals with MSC (2000): 37F20, 30D05


Additional Information

Magnus Aspenberg
Affiliation: Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn Str.4, 24 098 Kiel, Germany
MR Author ID: 862979
Email: aspenberg@math.uni-kiel.de, maspenberg@gmail.com

Received by editor(s): April 8, 2009
Published electronically: July 27, 2009
Additional Notes: The author gratefully acknowledges funding from the Research Training Network CODY of the European Commission
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.