Inventiones mathematicae

, Volume 204, Issue 3, pp 869–893 | Cite as

Non-landing parameter rays of the multicorns

Article

Abstract

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn \(\mathscr {M}_d^*\), the connectedness locus of unicritical antiholomorphic polynomials of degree d, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period 1) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of \(\mathscr {M}_d^*\), and the Misiurewicz parameters are not dense on the boundary of \(\mathscr {M}_d^*\).

Mathematics Subject Classification

37F45 37F10 37F20 30D05 

References

  1. 1.
    Bonifant, A., Buff, X., Milnor, J.: Antipode preserving cubic maps: the Fjord Theorem. http://www.math.univtoulouse.fr/~buff/Preprints/Antipodal/Antipodal.pdf (2015)
  2. 2.
    Ble, G., Douady, A., Henriksen, C.: Round annuli. Contemp. Math. (In the Tradition of Ahlfors and Bers, III) 355, 71–76 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Buff, X., Epstein, A.L.: A parabolic Pommerenke–Levin–Yoccoz inequality. Fund. Math. 172, 249–289 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Branner, B., Hubbard, J.H.: The iteration of cubic polynomials, part I: the global topology of parameter space. Acta Math. 160, 143–206 (1988)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Crowe, W.D., Hasson, R., Rippon, P.J., Strain Clark, P.E.D.: On the structure of the Mandelbar set. Nonlinearity 2 (1989)Google Scholar
  6. 6.
    Douady, A.: Does a Julia set depend continuously on the polynomial? Complex dynamical systems. Proc. Sympos. Appl. Math. 49, 91–138 (1994)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Goldberg, L.R., Milnor, J.: Fixed points of polynomial maps II: fixed point portraits. Ann. Sci. École Norm. Sup., \(4^{e}\) série 26, 51–98 (1993)Google Scholar
  8. 8.
    Hubbard, J., Schleicher, D.: Multicorns are not path connected. Frontiers in complex dynamics: in celebration of John Milnor’s 80th birthday, pp. 73–102 (2014)Google Scholar
  9. 9.
    Inou, H., Mukherjee, S.: Discontinuity of straightening in antiholomorphic dynamics (2015, manuscript in preparation)Google Scholar
  10. 10.
    Inou, H.: Self-similarity for the tricorn. http://arxiv.org/pdf/1411.3081.pdf (2014)
  11. 11.
    Komori, Y., Nakane, S.: Landing property of stretching rays for real cubic polynomials. Conform. Geom. Dyn. 8, 87–114 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lavaurs, P.: Systèmes dynamiques holomorphes: explosion de points périodiques paraboliques. Ph.D. thesis, Université de Paris-Sud Centre d’Orsay (1989)Google Scholar
  13. 13.
    Milnor, J.: Remarks on iterated cubic maps. Exp. Math. 1, 5–24 (1992)MathSciNetMATHGoogle Scholar
  14. 14.
    Milnor, J.: On rational maps with two critical points. Exp. Math. 9, 481–522 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Milnor, J.: Dynamics in One Complex Variable, 3rd edn. Princeton University Press, New Jersey (2006)MATHGoogle Scholar
  16. 16.
    Mukherjee, S., Nakane, S., Schleicher, D.: On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials. Ergodic Theory Dyn. Syst. http://arxiv.org/abs/1404.5031 (2014, to appear)
  17. 17.
    Mukherjee, S.: Orbit portraits of unicritical antiholomorphic polynomials. Conform. Geom. Dyn. AMS 19, 35–50 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nakane, S.: Connectedness of the tricorn. Ergodic Theory Dyn. Syst. 13, 349–356 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nakane, S., Schleicher, D.: On multicorns and unicorns I: antiholomorphic dynamics, hyperbolic components and real cubic polynomials. Int. J. Bifurc. Chaos 13, 2825–2844 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Schleicher, D.: Rational parameter rays of the Mandelbrot set. Astérisque 261, 405–443 (2000)MathSciNetMATHGoogle Scholar
  21. 21.
    Shishikura, M.: Bifurcation of parabolic fixed points. In: The Mandelbrot Set, Theme and Variations, London Mathematical Society Lecture Note Series, vol. 274, pp. 325–364. Cambridge University Press, Cambridge (2000)Google Scholar
  22. 22.
    Tan, L.: Stretching rays and their accumulations, following Pia Willumsen. In: Dynamics on the Riemann Sphere: A Bodil Branner Festschrift, pp. 183–208. European Mathematical Society (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.Jacobs University BremenBremenGermany
  3. 3.Institute for Mathematical SciencesStony Brook UniversityStony BrookUSA

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