Skip to main content
SpringerLink
Log in

Constructing Locally Connected Non-Computable Julia Sets

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Binder I., Braverman M., Yampolsky M.: Filled Julia sets with empty interior are computable. J. FoCM 7, 405–416 (2007)

    MATH  MathSciNet  Google Scholar 

  2. Buff X., Chéritat A.: Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive. Comptes Rendus Math. 341(11), 669–674 (2005)

    Article  MATH  Google Scholar 

  3. Buff X., Chéritat A.: The Brjuno function continuously estimates the size of quadratic Siegel disks. Ann. Math. 164(1), 265–312 (2006)

    Article  MATH  Google Scholar 

  4. Braverman M., Yampolsky M.: Non-computable Julia sets. J. Amer. Math. Soc. 19(3), 551–578 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Braverman M., Yampolsky M.: Computability of Julia sets. Moscow Math. J. 8(2), 185–231 (2008)

    MATH  MathSciNet  Google Scholar 

  6. Braverman, M., Yampolsky, M.: Computability of Julia Sets. Algorithms and Computations in Math. Vol.23 Berlin-Heidelberg-New York: Springer, 2001

  7. de Faria E., de Melo W.: Rigidity of critical circle mappings I. J. Eur. Math. Soc. (JEMS) 1(4), 339–392 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Douady, A.: Disques de Siegel et anneaux de Herman. Séminaire Bourbaki, Vol. 1986/87, Astérisque 152-153(4), 151–172 (1988)

  9. Douady, A.: Descriptions of compact sets in c. In: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991) Houston, TX: Publish or Perish, 1992

  10. Herman M.: Are there critical points on the boundaries of singular domains? Commun. Math. Phys. 99(4), 593–612 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. 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. Inou, H., Shishikura, M.: Renormalization for parabolic fixed points and their perturbations. Preprint, 2007

  13. Lyubich M.: Dynamics of rational transformations: topological picture. Russ. Math. Surv. 41(4), 43–117 (1986)

    Article  MathSciNet  Google Scholar 

  14. Marmi S., Moussa P., Yoccoz J.-C.: The Brjuno functions and their regularity properties. Commun. Math. Phys. 186, 265–293 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Petersen C.: Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177, 163–224 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Petersen C., Zakeri S.: On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. Math. 159(1), 1–52 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sullivan, D.: Conformal dynamical systems. In: Geometric Dynamics (Palis, ed.), Lecture Notes Math., Vol. 1007, Berlin- Heidelberg-New York: Springer-Verlag, 1983, pp. 725–752

  18. Thurston, W.: On the combinatorics and dynamics of iterated rational maps. Preprint

  19. Weihrauch K.: Computable Analysis. Springer-Verlag, Berlin (2000)

    MATH  Google Scholar 

  20. Yampolsky M.: Complex bounds for renormalization of critical circle maps. Erg. Th. & Dyn. Syst. 19, 227–257 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yampolsky M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Études Sci. 96, 1–41 (2002)

    MATH  MathSciNet  Google Scholar 

  22. Yampolsky, M.: Siegel disks and renormalization fixed points. In: Holomorphic Dynamics and, Renormalization, Fields Institute Communications 53, Providence, RI: Amer. Math. Soc., 2008

  23. Yoccoz J.-C.: Petits diviseurs en dimension 1. S.M.F., Astérisque 231, 3–242 (1995)

    MathSciNet  Google Scholar 

  24. Yampolsky M., Zakeri S.: Mating Siegel quadratic polynomials. J. Amer. Math. Soc. 14(1), 25–78 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Microsoft Research New England, One Memorial Drive, Cambridge, MA, 02142, USA

    Mark Braverman

  2. Mathematics Department, University of Toronto, 100 St. George Street, Toronto, Ontario, M5S 3G3, Canada

    Michael Yampolsky

Authors
  1. Mark Braverman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Yampolsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Yampolsky.

Additional information

Communicated by G. Gallavotti

This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow.

The second author’s research is supported by NSERC operating grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Braverman, M., Yampolsky, M. Constructing Locally Connected Non-Computable Julia Sets. Commun. Math. Phys. 291, 513–532 (2009). https://doi.org/10.1007/s00220-009-0858-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0858-5

Keywords

Search

Navigation