A priori bounds for some infinitely renormalizable quadratics: II. Decorations

Jeremy Kahn; Mikhail Lyubich

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 1, page 57-84
  • ISSN: 0012-9593

Abstract

top
A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we provea priori bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.

How to cite

top

Kahn, Jeremy, and Lyubich, Mikhail. "A priori bounds for some infinitely renormalizable quadratics: II. Decorations." Annales scientifiques de l'École Normale Supérieure 41.1 (2008): 57-84. <http://eudml.org/doc/272200>.

@article{Kahn2008,
abstract = {A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we provea priori bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.},
author = {Kahn, Jeremy, Lyubich, Mikhail},
journal = {Annales scientifiques de l'École Normale Supérieure},
language = {eng},
number = {1},
pages = {57-84},
publisher = {Société mathématique de France},
title = {A priori bounds for some infinitely renormalizable quadratics: II. Decorations},
url = {http://eudml.org/doc/272200},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Kahn, Jeremy
AU - Lyubich, Mikhail
TI - A priori bounds for some infinitely renormalizable quadratics: II. Decorations
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 1
SP - 57
EP - 84
AB - A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we provea priori bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.
LA - eng
UR - http://eudml.org/doc/272200
ER -

References

top
  1. [1] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., 1973, McGraw-Hill Series in Higher Mathematics. Zbl0272.30012MR357743
  2. [2] A. Avila, J. Kahn, M. Lyubich & W. Shen, Combinatorial rigidity for unicritical polynomials, preprint IMS at Stony Brook, #5, 2005. To appear in Annals of Math.. Zbl1204.37047
  3. [3] D. Cheraghi, Combinatorial rigidity for some infinitely renormalizable unicritical polynomials, preprint IMS at Stony Brook, #7, 2007. Zbl1218.37060MR2719786
  4. [4] A. Douady & J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup.18 (1985), 287–343. Zbl0587.30028
  5. [5] F. P. Gardiner & N. Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs 76, Amer. Math. Soc., 2000. Zbl0949.30002
  6. [6] J. Kahn, A priori bounds for some infinitely renormalizable quadratics: I. bounded primitive combinatorics, preprint IMS at Stony Brook, #5, 2006. 
  7. [7] J. Kahn & M. Lyubich, Local connectivity of Julia sets for unicritical polynomials, preprint IMS at Stony Brook, #3, 2005. To appear in Annals of Math.. Zbl1180.37072
  8. [8] J. Kahn & M. Lyubich, Quasi-additivity law in conformal geometry, preprint IMS at Stony Brook, #2, arXiv:math.DS/0505191v2, 2005. To appear in Annals of Math.. Zbl1203.30011
  9. [9] M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), 185–247, 247–297. Zbl0908.58053MR1459261
  10. [10] C. T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies 135, Princeton University Press, 1994. Zbl0822.30002MR1312365
  11. [11] J. Milnor, Local connectivity of Julia sets: expository lectures, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser. 274, Cambridge Univ. Press, 2000, 67–116. Zbl1107.37305MR1765085
  12. [12] J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account (Géométrie complexe et systèmes dynamiques, Orsay, 1995), Astérisque261 (2000), 277–333. Zbl0941.30016MR1755445

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.