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Generalized Rings Around the McMullen Domain

  • Dades identificatives

    Identificador: imarina:5873994
    Autors:
    Garijo AJang HMarotta S
    Resum:
    © 2018, Springer Nature Switzerland AG. We consider the family of rational maps given by F λ (z) = z n + λ/ z d 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 S k 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 F λ are Cantor sets of simple closed curves. The rings S k 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 F λ 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 S 1 is τ1n,d=n-1, and on S k for k> 1 is given by τkn,d=dnk-2(n-1)-nk-1+1.
  • Altres:

    Autor segons l'article: Garijo A; Jang H; Marotta S
    Departament: Enginyeria Informàtica i Matemàtiques
    Autor/s de la URV: Garijo Real, Antonio
    Paraules clau: Singularly perturbed rational maps Sierpinski holes Mcmullen domain Baby mandelbrot sets
    Resum: © 2018, Springer Nature Switzerland AG. We consider the family of rational maps given by F λ (z) = z n + λ/ z d 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 S k 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 F λ are Cantor sets of simple closed curves. The rings S k 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 F λ 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 S 1 is τ1n,d=n-1, and on S k for k> 1 is given by τkn,d=dnk-2(n-1)-nk-1+1.
    Àrees temàtiques: Mathematics, applied Mathematics Matemática / probabilidade e estatística Interdisciplinar Ensino Engenharias iv Discrete mathematics and combinatorics Astronomia / física Applied mathematics
    Accès a la llicència d'ús: https://creativecommons.org/licenses/by/3.0/es/
    ISSN: 15755460
    Adreça de correu electrònic de l'autor: antonio.garijo@urv.cat
    Identificador de l'autor: 0000-0002-1503-7514
    Data d'alta del registre: 2023-02-18
    Versió de l'article dipositat: info:eu-repo/semantics/submittedVersion
    Enllaç font original: https://link.springer.com/article/10.1007/s12346-018-0287-y
    Referència a l'article segons font original: Qualitative Theory Of Dynamical Systems. 18 (1): 233-264
    Referència de l'ítem segons les normes APA: Garijo A; Jang H; Marotta S (2019). Generalized Rings Around the McMullen Domain. Qualitative Theory Of Dynamical Systems, 18(1), 233-264. DOI: 10.1007/s12346-018-0287-y
    URL Document de llicència: https://repositori.urv.cat/ca/proteccio-de-dades/
    DOI de l'article: 10.1007/s12346-018-0287-y
    Entitat: Universitat Rovira i Virgili
    Any de publicació de la revista: 2019
    Tipus de publicació: Journal Publications
  • Paraules clau:

    Applied Mathematics,Discrete Mathematics and Combinatorics,Mathematics,Mathematics, Applied
    Singularly perturbed rational maps
    Sierpinski holes
    Mcmullen domain
    Baby mandelbrot sets
    Mathematics, applied
    Mathematics
    Matemática / probabilidade e estatística
    Interdisciplinar
    Ensino
    Engenharias iv
    Discrete mathematics and combinatorics
    Astronomia / física
    Applied mathematics
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