Advertisement

Communications in Mathematical Physics

, Volume 291, Issue 2, pp 513–532 | Cite as

Constructing Locally Connected Non-Computable Julia Sets

  • Mark Braverman
  • Michael Yampolsky
Article

Abstract

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected.

Keywords

Turing Machine Rotation Number Blaschke Product Continue Fraction Expansion Critical Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BBY07.
    Binder I., Braverman M., Yampolsky M.: Filled Julia sets with empty interior are computable. J. FoCM 7, 405–416 (2007)zbMATHMathSciNetGoogle Scholar
  2. BC05.
    Buff X., Chéritat A.: Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive. Comptes Rendus Math. 341(11), 669–674 (2005)zbMATHCrossRefGoogle Scholar
  3. BC06.
    Buff X., Chéritat A.: The Brjuno function continuously estimates the size of quadratic Siegel disks. Ann. Math. 164(1), 265–312 (2006)zbMATHCrossRefGoogle Scholar
  4. BY06.
    Braverman M., Yampolsky M.: Non-computable Julia sets. J. Amer. Math. Soc. 19(3), 551–578 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. BY08a.
    Braverman M., Yampolsky M.: Computability of Julia sets. Moscow Math. J. 8(2), 185–231 (2008)zbMATHMathSciNetGoogle Scholar
  6. BY08b.
    Braverman, M., Yampolsky, M.: Computability of Julia Sets. Algorithms and Computations in Math. Vol.23 Berlin-Heidelberg-New York: Springer, 2001Google Scholar
  7. dFdM99.
    de Faria E., de Melo W.: Rigidity of critical circle mappings I. J. Eur. Math. Soc. (JEMS) 1(4), 339–392 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. Dou88.
    Douady, A.: Disques de Siegel et anneaux de Herman. Séminaire Bourbaki, Vol. 1986/87, Astérisque 152-153(4), 151–172 (1988)Google Scholar
  9. Dou93.
    Douady, A.: Descriptions of compact sets in c. In: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991) Houston, TX: Publish or Perish, 1992Google Scholar
  10. Her85.
    Herman M.: Are there critical points on the boundaries of singular domains? Commun. Math. Phys. 99(4), 593–612 (1985)zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. Her86.
    Herman, M.: Conjugaison quasi symétrique des homéomorphismes du cercle à des rotations and Conjugaison quasi symétrique des difféomorphismes du cercle à des rotations et applications aux disques singuliers de Siegel. Available from http://www.math.kyoto-u.ac.jp/~mitsu/Herman/index.html, 1986
  12. IS07.
    Inou, H., Shishikura, M.: Renormalization for parabolic fixed points and their perturbations. Preprint, 2007Google Scholar
  13. Lyu86.
    Lyubich M.: Dynamics of rational transformations: topological picture. Russ. Math. Surv. 41(4), 43–117 (1986)CrossRefMathSciNetGoogle Scholar
  14. MMY97.
    Marmi S., Moussa P., Yoccoz J.-C.: The Brjuno functions and their regularity properties. Commun. Math. Phys. 186, 265–293 (1997)zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. Pet96.
    Petersen C.: Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177, 163–224 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. PZ04.
    Petersen C., Zakeri S.: On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. Math. 159(1), 1–52 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  17. Sul83.
    Sullivan, D.: Conformal dynamical systems. In: Geometric Dynamics (Palis, ed.), Lecture Notes Math., Vol. 1007, Berlin- Heidelberg-New York: Springer-Verlag, 1983, pp. 725–752Google Scholar
  18. Thu.
    Thurston, W.: On the combinatorics and dynamics of iterated rational maps. PreprintGoogle Scholar
  19. Wei00.
    Weihrauch K.: Computable Analysis. Springer-Verlag, Berlin (2000)zbMATHGoogle Scholar
  20. Yam99.
    Yampolsky M.: Complex bounds for renormalization of critical circle maps. Erg. Th. & Dyn. Syst. 19, 227–257 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Yam02.
    Yampolsky M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Études Sci. 96, 1–41 (2002)zbMATHMathSciNetGoogle Scholar
  22. Yam08.
    Yampolsky, M.: Siegel disks and renormalization fixed points. In: Holomorphic Dynamics and, Renormalization, Fields Institute Communications 53, Providence, RI: Amer. Math. Soc., 2008Google Scholar
  23. Yoc95.
    Yoccoz J.-C.: Petits diviseurs en dimension 1. S.M.F., Astérisque 231, 3–242 (1995)MathSciNetGoogle Scholar
  24. YZ01.
    Yampolsky M., Zakeri S.: Mating Siegel quadratic polynomials. J. Amer. Math. Soc. 14(1), 25–78 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Microsoft Research New EnglandCambridgeUSA
  2. 2.Mathematics DepartmentUniversity of TorontoTorontoCanada

Personalised recommendations