Communications in Mathematical Physics

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

Constructing Locally Connected Non-Computable Julia Sets

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.

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References

  1. BBY07.
    Binder I., Braverman M., Yampolsky M.: Filled Julia sets with empty interior are computable. J. FoCM 7, 405–416 (2007)MATHMathSciNetGoogle 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)MATHCrossRefGoogle 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)MATHCrossRefGoogle Scholar
  4. BY06.
    Braverman M., Yampolsky M.: Non-computable Julia sets. J. Amer. Math. Soc. 19(3), 551–578 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. BY08a.
    Braverman M., Yampolsky M.: Computability of Julia sets. Moscow Math. J. 8(2), 185–231 (2008)MATHMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefADSMathSciNetGoogle 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)MATHCrossRefADSMathSciNetGoogle Scholar
  15. Pet96.
    Petersen C.: Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177, 163–224 (1996)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHGoogle Scholar
  20. Yam99.
    Yampolsky M.: Complex bounds for renormalization of critical circle maps. Erg. Th. & Dyn. Syst. 19, 227–257 (1999)MATHCrossRefMathSciNetGoogle Scholar
  21. Yam02.
    Yampolsky M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Études Sci. 96, 1–41 (2002)MATHMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

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

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