Generalized Rings Around the McMullen Domain

• Garijo, Antonio ^[2] ; Jang, HyeGyong ^[3] ; Marotta, Sebastian M ^[1]
  1. [1] Boston University

     Boston University

     City of Boston, Estados Unidos

     
  2. [2] Universitat Rovira i Virgil
  3. [3] University of Science, Pyongyang

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 1, 2019, págs. 233-264
• Idioma: inglés
• DOI: 10.1007/s12346-018-0287-y
• Enlaces
  • Texto completo
• Resumen

  • We consider the family of rational maps given by Fλ(z)=zn+λ/zd where n,d∈N with 1/n+1/d<1, the variable z∈C^ and the parameter λ∈C. It is known that when n=d≥3 there are infinitely many rings Sk with k∈N, around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter λ-plane where the Julia sets of Fλ are Cantor sets of simple closed curves. The rings Sk converge to the boundary of the McMullen domain as k→∞ and contain parameter values that lie at the center of Sierpiński holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of Fλ are Sierpiński curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1/n+1/d<1 where n is not necessarily equal to d. The number of Sierpiński holes and superstable parameters on S1 is τ1n,d=n-1, and on Sk for k>1 is given by τkn,d=dnk-2(n-1)-nk-1+1.

• Referencias bibliográficas
  • 1. McMullen, C.: Automorphisms of rational maps. In: Drasin, D., Kra, I., Earle, C.J., Marden, A., Gehring, F.W. (eds.) Holomorphic Functions...
  • 2. Devaney, R.L., Look, D., Uminsky, D.: The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54, 1621–1634...
  • 3. Devaney, R.L., Marotta, S.M.: The McMullen domain: rings around the boundary. Trans. Am. Math. Soc. 359, 3251–3273 (2007)
  • 4. Devaney, R.L.: Baby Mandelbrot sets adorned with Halos in families of rational maps. Contemp. Math. 396, 37–50 (2006)
  • 5. Roesch, P.: On Captures for the Family fλ(z) = z2 + λ/z2. Dynamics on the Riemann Sphere, pp. 121–130. European Mathematical Society,...
  • 6. Jang, H., So, Y., Marotta, S.M.: Generalized baby Mandelbrot sets adorned with halos in families of rational maps. J. Differ. Equ. Appl....
  • 7. Steinmetz, N.: Sierpi ´nski curve Julia sets of rational maps. Comput. Methods Funct. Theory 6, 317–327 (2006)
  • 8. Steinmetz, N.: On the dynamics of the McMullen family R(z) = zm + λ/zl . Conform. Geom. Dyn. 10, 159–183 (2006)
  • 9. Devaney, R.L., Josi´c, K., Shapiro, Y.: Singular perturbations of quadratic maps. Int. J. Bifurc. Chaos 14(1), 161–169 (2004)
  • 10. Blanchard, P., Devaney, R.L., Look, D.M., Seal, P., Shapiro, Y.: Sierpi ´nski curve Julia sets and singular perturbations of complex polynomials....
  • 11. Devaney, R.L.: Structure of the McMullen domain in the parameter planes for rational maps. Fundam. Math. 185, 267–285 (2005)
  • 12. Devaney, R.L., Holzer, M., Look, D., Moreno Rocha, M., Uminsky, D.: Singular perturbations of zn. In: Rippon, P., Stallard, G. (eds.)...
  • 13. Devaney, R.L.: The McMullen domain: satellite Mandelbrot sets and Sierpi ´nski holes. Conform. Geom. Dyn. 11, 164–190 (2007)
  • 14. Çilingir, F., Devaney, R.L., Russell, E.R.: Extending external rays throughout the Julia sets of rational maps. J. Fixed Point Theory...
  • 15. Devaney, R.L.: Dynamics of zn + λ/zn; Why the Case n = 2 is Crazy. In: Conformal Dynamics and Hyperbolic Geometry. Contemporary...
  • 16. Devaney, R.L., Russell, E.R.: Connectivity of Julia Sets for Singularly Perturbed Rational Maps. Chaos, CNN, Memristors and Beyond, pp....
  • 17. Devaney, R.L., Pilgrim, K.: Dynamic classification of escape time Sierpi ´nski curve Julia sets. Fundam. Math. 202, 181–198 (2009)
  • 18. Devaney, R.L.: Singular Perturbations of Complex Analytic Dynamical Systems. Nonlinear Dynamics and Chaos: Advances and Perspectives,...
  • 19. Devaney, R.L.: Singular perturbations of complex polynomials. Bull. Am. Math. Soc. 50, 391–429 (2013)
  • 20. Devaney, R.L.: A Mandelpinski maze for rational maps of the form zn + λ/zd . Indag. Math. 27, 1042–1058 (2016)
  • 21. Devaney, R.L.: Mandelpinski spokes in the parameter planes for rational maps. J. Differ. Equ. Appl. 22, 330–342 (2016)
  • 22. Devaney, R.L.: Mandelpinski structures in the parameter planes of rational maps. Proc. Oxtoby Centen. Conf. AMS Contemp. Math Ser. 678,...
  • 23. Milnor, J.: Dynamics in One Complex Variable, 2nd edn. Vieweg, Göttingen (2000)
  • 24. Beardon, A.F.: Iteration of Rational Functions. Springer, New York (1991)
  • 25. Carleson, L., Gamelin, T.W.: Complex Dynamics. Springer, New York (1993)
  • 26. Morosawa, S., Nishimura, Y., Taniguchi, M., Ueda, T.: Holomorphic Dynamics. Cambridge University Press, Cambridge (2000)
  • 27. Steinmetz, N.: Rational Iteration: Complex Analytic Dynamical Systems. De Gruyter Studies in Mathematics, vol. 16. Cambridge University...
  • 28. Whyburn, G.T.: Topological characterization of the Sierpi ´nski curve. Fundam. Math. 45, 320–324 (1958)
  • 29. Blanchard, P., Çilingir, F., Cuzzocreo, D., Devaney, R.L., Look, D., Russell, E.D.: Checkerboard Julia sets for rational maps. Int. J....
  • 30. Devaney, R.L.: A Myriad of Sierpinski Curve Julia Sets. Difference Equations, Special Functions, and Orthogonal Polynomials, pp. 131–148....
  • 31. Devaney, R.L., Look, D.M.: Buried Sierpi ´nski curve Julia sets. Discrete Contin. Dyn. Syst. 13, 1035– 1046 (2005)
  • 32. Devaney, R.L., Look, D.M.: A criterion for Sierpi ´nski curve Julia sets. Topol. Proc. 30, 163–179 (2006)
  • 33. Jang, H.G.: The rings around the McMullen domain in families of rational maps Fλ(z) = zn + λ/zm. Manuscript (2014)