Parabolic perturbation in the family z →1 + 1=wzᵈ

• Bobenrieth, Juan ^[1]
  1. [1] Universidad del Bío-Bío

     Universidad del Bío-Bío

     Comuna de Concepción, Chile

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 21, Nº. 1, 2002, págs. 1-7
• Idioma: inglés
• DOI: 10.4067/S0716-09172002000100001
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• Resumen

  • Consider the family of rational mapsFd = {z→ fw(z) =1+ : w ∈ C\{0}} (d ∈ N, d ≥ 2)and the hyperbolic component A₁ = {w : fw has an attracting fixed point}. We prove that if w₀ ∈ ∂A₁ is a parabolic parameter with corresponding multiplier a primitive q-th root of unity, q ≥ 2; then there exists a hyperbolic component Wq; attached to A₁ at the point w₀; which contains w-values for which fw has an attracting periodic cycle of period q.

• Referencias bibliográficas
  • Citas [1] R. Bamon and J.Bobenrieth, ‘The rational maps z→1+1=wzᵈ have no Herman rings’, Proceedings of the American Mathematical...
  • [2] H. Jellouli, ‘Indice holomorphe et multiplicateur’, The Mandelbrot set, theme and variations (ed Tan Lei), London Mathematical Society...
  • [3] M. Lyubich, ‘The dynamics of rational transforms : the topological picture’, Russian Math. Surveys, (4) 41, pp. 43-117, (1986).
  • [4] J. Milnor, ‘Geometry and dynamics of quadratic rational maps’, Experimental Mathematics, (1) 2 , pp. 37-83, (1993).