Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T04:55:23.376Z Has data issue: false hasContentIssue false
  • English
  • Français

The solar Julia sets of basic quadratic Cremer polynomials

Published online by Cambridge University Press:  17 March 2009

A. BLOKH,
X. BUFF,
A. CHÉRITAT and
L. OVERSTEEGEN
A. BLOKH
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA (email: ablokh@math.uab.edu, overstee@math.uab.edu)
X. BUFF
Affiliation:
Université de Toulouse; UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, FranceCNRS; Institut de Mathématiques de Toulouse UMR 5219; F-31062, Toulouse, France (email: xavier.buff@math.univ-toulouse.fr, arnaud.cheritat@math.univ-toulouse.fr)
A. CHÉRITAT
Affiliation:
Université de Toulouse; UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, FranceCNRS; Institut de Mathématiques de Toulouse UMR 5219; F-31062, Toulouse, France (email: xavier.buff@math.univ-toulouse.fr, arnaud.cheritat@math.univ-toulouse.fr)
L. OVERSTEEGEN
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA (email: ablokh@math.uab.edu, overstee@math.uab.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010, pp. 51 - 65
Copyright
Copyright © Cambridge University Press 2009

References

[1]Blokh, A. and Oversteegen, L.. The Julia sets of quadratic Cremer polynomials. Topol. Appl. 153 (2006), 3038–3050.CrossRefGoogle Scholar
[2]Blokh, A. and Oversteegen, L.. Monotone images of Cremer Julia sets. Preprint, arXiv:0809.1193 2008 Houston J. Math. to appear.Google Scholar
[3]Buff, X. and Chéritat, A.. Quadratic Julia sets with positive area. Preprint, arXiv:org/abs/math/0605514v2, 2008.Google Scholar
[4]Bullett, S. and Sentenac, P.. Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Camb. Phil. Soc. 115 (1994), 451–481.CrossRefGoogle Scholar
[5]Chéritat, A.. Recherche d’ensembles de Julia de mesure de Lebesgue positive. Thesis, Orsay, 2001.Google Scholar
[6]Childers, D., Mayer, J. and Rogers, J. Jr. Indecomposable continua and the Julia sets of polynomials, II. Topol. Appl. 153(10) (2006), 1593–1602.CrossRefGoogle Scholar
[7]Childers, D.. Holomorphic Dynamics and Renormalization: a Volume in Honor of John Milnor’s 75th Birthday, Fields Institute for Research in Mathematical Sciences. Are there critical points on the boundaries of mother hedgehogs? Fields Inst. Commun. 53 (2008), 75–87.Google Scholar
[8]Douady, A. and Hubbard, J. H.. Étude dynamique des polynômes complexes I, II. Publ. Math. d’Orsay 84–02 (1984), 85–04 (1985)Google Scholar
[9]Grispolakis, J., Mayer, J. and Oversteegen, L.. Building Blocks for Julia sets. Trans. Amer. Math. Soc. 351(3) (1999), 1171–1201.CrossRefGoogle Scholar
[10]Heath, J.. Each locally one-to-one map from a continuum onto a tree-like continuum is a homeomorphism. Proc. Amer. Math. Soc. 124 (1996), 2571–2573.CrossRefGoogle Scholar
[11]Inou, H. and Shishikura, M.. The renormalization for parabolic fixed points and their perturbation, in preparation.Google Scholar
[12]Kiwi, J.. Non-accessible critical points of Cremer polynomials. Ergod. Th. & Dynam. Sys. 20 (2000), 1391–1403.CrossRefGoogle Scholar
[13]Kiwi, J.. Real laminations and the topological dynamics of complex polynomials. Adv. Math. 184(2) (2004), 207–267.CrossRefGoogle Scholar
[14]Lyubich, M. and Minsky, Y.. Laminations in holomorphic dynamics. J. Differential Geom. 47(1) (1997), 17–94.CrossRefGoogle Scholar
[15]Mayer, J. and Rogers, J. Jr. Indecomposable continua and the Julia sets of polynomials. Proc. Amer. Math. Soc. 117(3) (1993), 795–802.CrossRefGoogle Scholar
[16]Mañé, R.. On a theorem of Fatou. Bol. Soc. Bras. Mat. 24 (1993), 1–11.CrossRefGoogle Scholar
[17]McMullen, C. T.. Complex Dynamics and Renormalization (Annals of Mathematical Studies, 135). Princeton University Press, Princeton, NJ, 1994.Google Scholar
[18]McMullen, C. T.. Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 180 (1998), 247–292.CrossRefGoogle Scholar
[19]Milnor, J.. Dynamics in One Complex Variable, 2nd edn. Vieweg, Wiesbaden, 2000.CrossRefGoogle Scholar
[20]Munkres, J. R.. Topology, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ, 2000.Google Scholar
[21]Perez-Marco, R.. Topology of Julia sets and hedgehogs. Publ. Math. d’Orsay 94–48 (1994).Google Scholar
[22]Perez-Marco, R.. Fixed points and circle maps. Acta Math. 179 (1997), 243–294.CrossRefGoogle Scholar
[23]Petersen, C. L.. Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177(2) (1996), 163–224.CrossRefGoogle Scholar
[24]Pommerenke, Ch.. Boundary Behavior of Conformal Maps. Springer, Berlin, 1992.CrossRefGoogle Scholar
[25]Schleicher, D. and Zakeri, S.. On biaccessible points in the Julia set of a Cremer quadratic polynomial. Proc. Amer. Math. Soc. 128 (1999), 933–937.CrossRefGoogle Scholar
[26]Sørensen, D.. Describing quadratic Cremer polynomials by parabolic perturbations. Ergod. Th. & Dynam. Sys. 18 (1998), 739–758.CrossRefGoogle Scholar
[27]Sullivan, D.. Conformal Dynamical Systems (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 725–752.Google Scholar
[28]Yoccoz, J. C.. Petits diviseurs en dimension 1. Astérisque 231 (1995).Google Scholar
[29]Zakeri, S.. Biaccessibility in quadratic Julia sets. Ergod. Th. & Dynam. Sys. 20 (2000), 1859–1883.CrossRefGoogle Scholar