Skip to main content
SpringerLink
Log in

Mating Non-Renormalizable Quadratic Polynomials

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

Abstract

In this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1/2-limb of \({\mathcal {M}}\) , is non- renormalizable, and does not have any non-repelling periodic orbits.

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. Ahlfors L., Bers L.: Riemann Mapping Theorem for variable metrics. Ann. Math. 72, 385–404 (1960)

    Article  MathSciNet  Google Scholar 

  2. Branner B., Hubbard J.H.: The iteration of cubic polynomials. ii. Patterns and parapatterns. Acta Math. 3-4, 229–325 (1992)

    Article  MathSciNet  Google Scholar 

  3. Bullett S., Sentenac P.: Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Camb. Phil. Soc. 115, 451–481 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carleson, L., Gamelin, T.: Complex Dynamics. Berlin-Heidelberg NewYork: Springer-Verlag, 1993

    MATH  Google Scholar 

  5. Douady, A.: Algorithms for computing angles in the Mandelbrot set, In: Barnsley, Demko (ed.) “Chaotic Dynamics and Fractals,” London-NewYork: Academic Press, 1986, pp. 155–168

    Google Scholar 

  6. Douady A.: Systéms dynamiques holomorphes. Seminar Bourbaki. Astérisque. 105-106, 39–64 (1983)

    MathSciNet  Google Scholar 

  7. Douady A.: Disques de Siegel at aneaux de Herman. Seminar Bourbaki. Astérisque. 152-153, 151–172 (1987)

    MathSciNet  Google Scholar 

  8. Douady A., Earle C.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157, 23–48 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes I & II. Preprint, Pub. Math. d’Orsay 84-02, 85-05, 1984/1985

  10. Douady A., Hubbard J.H.: A proof of Thurston’s topological characterization of rational functions. Acta Math. 171, 263–297 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Epstein, A.: Counterexamples to the quadratic mating conjecture. Manuscript

  12. Hubbard, J.H.: Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, In: “Topological methods in modern mathematics” (Stony Brook, NY, 1991), Houston, TX: Publish or Perish, pp. 467–511, 1993

  13. Luo, J.: Combinatorics and holomorphic dynamics: Captures, matings, Newton’s method. Thesis, Cornell University, 1995

  14. Lyubich M.Yu.: The dynamics of rational transforms: The topological picture. Russian Math. Surveys 41, 43–117 (1986)

    Article  Google Scholar 

  15. Lyubich M.Yu.: Dynamics of quadratic polynomials I-II. Acta Math. 178, 185–297 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lyubich M.Yu.: Parapuzzles and SRB measures. Asterisque. 261, 173–200 (2000)

    MathSciNet  Google Scholar 

  17. Mañé R., Sad P., Sullivan D.: On the dynamics of rational maps. Ann. Sci. École Norm. Sup. (4) 16(2), 193–217 (1983)

    MATH  Google Scholar 

  18. McMullen, C.: Complex Dynamics and Renormalization. Ann. Math Stud. Vol. 135, 1994

  19. Milnor, J.: Dynamics in One Complex Variable: Introductory Lectures, Wiesbaden: Vieweg, 1999

  20. Milnor J.: Geometry and dynamics of quadratic rational maps. Experim. Math. 2, 37–83 (1993)

    MATH  MathSciNet  Google Scholar 

  21. Milnor, J.: Periodic orbits, external rays, and the Mandelbrot set: An expository account. Asterisque 261 (1999)

  22. Milnor J.: Pasting together Julia sets - a worked out example of mating. Expe. Math. 13(1), 55–92 (2004)

    MATH  MathSciNet  Google Scholar 

  23. Milnor, J.: Local connectivity of Julia sets: expository lectures. In: The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., 274, Cambridge: Cambridge Univ. Press, pp. 67–116, 2000

  24. Moore R.L.: Concerning upper semi-continuous collection of continua. Trans. Amer. Math. Soc. 27, 416–428 (1925)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rees, M.: Realization of matings of polynomials as rational maps of degree two. Manuscript, 1986

  26. Rees M.: A partial description of parameter space of rational maps of degree two: part I. Acta Math. 168, 11–87 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Roesch, P.: Holomorphic motions and parapuzzles, In: The Mandelbrot set, Theme and variations, London Math. Soc. Lecture Note Ser., 274, Cambridge: Cambridge Univ. Press, 2000

  28. Roesch, P.: Topologie locale des méthodes de Newton cubiques. Thesis, E.N.S, Lyon, 1997

  29. Shishikura, M.: On a theorem of M. Rees for matings of polynomials. In: The Mandelbrot set, Theme and variations, London Math. Soc. Lecture Note Ser., 274, Cambridge: Cambridge Univ. Press, 2000

  30. Shishikura M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. 147, 25–267 (1998)

    Article  MathSciNet  Google Scholar 

  31. Shishikura M.: Bifurcation of parabolic fixed points. In: The Mandelbrot set, Theme and variations, London Math. Soc. 274, Cambridge: Cambridge Univ. Press, 2000

  32. Shishikura M., Tan L.: A family of cubic rational maps and matings of cubic polynomials. Exp. Math. 9(1), 29–53 (2000)

    MATH  MathSciNet  Google Scholar 

  33. Slodkowski Z.: Natural extensions of holomorphic motions. J. Geom. Anal. 7(4), 637–651 (1997)

    MATH  MathSciNet  Google Scholar 

  34. Tan L.: Matings of quadratic polynomials. Erg. Th. and Dyn. Sys. 12, 589–620 (1992)

    MATH  Google Scholar 

  35. Tan L., Yin Y.: Local connectivity of the Julia set for geometrically finite rational maps. Sci. China Ser. A 39(1), 39–47 (1996)

    MATH  MathSciNet  Google Scholar 

  36. Yampolsky M.: Complex bounds for renormalization of critical circle maps. Erg. Th. Dyn. Sys. 18, 1–31 (1998)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  38. Zakeri S.: Biaccessibility in quadratic Julia sets. Erg. Th. Dyn. Sys. 20(6), 1859–1883 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Division, RTH-Royal Institute of Technology, st-10044, Stockholm, Sweden

    Magnus Aspenberg

  2. Mathematics Department, University of Toronto, 40 St George Street, Toronto, ON, M5S 2E4, Canada

    Michael Yampolsky

Authors
  1. Magnus Aspenberg
    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 L. Takhtajan

The first author was partially supported by the Foundation Blanceflor Boncompagni-Ludovisi, née Bildt and by the Fields Institute.

The second author was partially supported by an NSERC Discovery Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aspenberg, M., Yampolsky, M. Mating Non-Renormalizable Quadratic Polynomials. Commun. Math. Phys. 287, 1–40 (2009). https://doi.org/10.1007/s00220-008-0598-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0598-y

Keywords

Search

Navigation