Skip to main content
Log in

Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. By using a more general strategy, we prove that all of these Julia sets fail to be arc-wise connected, and that their critical point is non-accessible.

Using the first strategy we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists spiral.”

A weaker result is obtained using the second strategy: the existence of polynomials having an explicitly given external ray accumulating at a particular point, but having in its impression the symmetric point as well. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists sine.”

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

Access this article

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. Ahlfors, L.V.Complex Analysis. Third Ed., McGraw-Hill, New York, 1979.

    MATH  Google Scholar 

  2. Branner, B. The Mandelbrot set,Proc. Symp. Appl. Math., AMS,39, 75–105, (1989).

    MathSciNet  Google Scholar 

  3. Douady, A. and Hubbard, J.H. Étude dynamique des polynômes complexes I-II (also referred to asThe Orsay Notes)Publ. Math., Orsay, (1984).

  4. Douady, A. and Hubbard, J.H. On the dynamics of polynomial-like mappings,Ann. Sci. Éc. Norm. Super.,18, 287–343, 1985.

    MathSciNet  MATH  Google Scholar 

  5. Douady, A. Algorithms for computing angles in the Mandelbrot set,Chaotic Dynamics and Fractals, Barnsley, M.F. and Demko, S.G., Eds., Atlanta, Academic Press, 155–168, 1986.

    Google Scholar 

  6. Douady, A. Does a Julia set depend continuously on the polynomial,Proc. Symp. Appl. Math. AMS,49, 91–138, (1994).

    MathSciNet  Google Scholar 

  7. Goldberg, L.R. and Milnor, J. Fixed points of polynomial maps. II: Fixed point portraits,Ann. Sci. Éc. Norm. Super.,26, 51–98,(1993).

    MathSciNet  MATH  Google Scholar 

  8. Grispolakis, J., Mayer, J.C., and Oversteegen, L.G. Building blocks for Julia sets,Trans. AMS, to appear.

  9. Hubbard, J.H. Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz,Topological Methods in Modern Mathematics, Publish or Perrish, 467–511/375–378, 1993.

  10. Jiang, Y. The renormalization method and quadratic-like mapsMSRI, preprint No. 081-95, (revised), Berkely, 1995.

  11. Jiang, Y. Local connectivity of the Mandelbrot set at certain infinitely renormalizable points,MSRI, preprint No. 063-95, (revised), Berkely, 1995.

  12. Kiwi, J. Non-accessible critical points of Cremer polynomials,IMS at StonyBrook, preprint, 1995.

  13. Levin, G. Disconnected Julia sets and rotation sets,Institute of Math., The Hebrew University of Jerusalem, revised version of preprint No. 15, 1991/1992, May 1993.

  14. Lyubich, M. Geometry of quadratic polynomials: Moduli, rigidity and local connectivity,IMS at StonyBrook, preprint, 1993.

  15. Mayer, J. Complex dynamics and continuum theory,Continua with the Houston Problem Book, Cook, H., et al., Eds., Marcel-Dekker, 1995.

  16. McMullen, C.T. Complex dynamics and renormalization,Ann. Math. Studies, Study 135, Princeton University Press, Princeton, NJ, (1994).

    Google Scholar 

  17. Milnor, J. Dynamics in one complex variable: Introductory lectures,IMS at StonyBrook, preprint, #1990/5, 1990.

  18. Milnor, J. Local connectivity of Julia sets: Expository lectures,IMS at StonyBrook, preprint, #1992/11, 1992.

  19. Perez-Marco, R. Topology of Julia sets and hedgehogs,Ergod. Th. and Dynam. Sys., 1998.

  20. Pommerenke, Ch.Boundary Behavior of Conformai Maps. GMV,299, Springer-Verlag, 1992.

  21. Sørensen, D.E.K. Complex dynamical systems: Rays and non-local connectivity, Ph.D. thesis, MAT-DTU, 1994.

  22. Sørensen, D.E.K. Accumulation theorems for quadratic polynomials,Ergod. Th. Dynam. Sys.,16, 555–590, (1996).

    Article  Google Scholar 

  23. Sørensen, D.E.K. Describing quadratic Cremer point polynomials by parabolic perturbations,Ergod. Th. Dynam. Sys.,18, 739–758, (1998).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sørensen, D.E.K. Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set. J Geom Anal 10, 169–206 (2000). https://doi.org/10.1007/BF02921810

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02921810

Math Subject Classification

Key Words and Phrases

Navigation