Parabolic perturbation in the family z ?1 + 1=wz?


  • Juan Bobenrieth Universidad del Bío-Bío.



Rational maps, roots of unity, primitives, periodic cycles, hyperbolic components, mapas racionales, raíces de unidad, primitivas, ciclos periódicos, componentes hiperbólicos.


Consider the family of rational mapsFd = {z? fw(z) =1+ : w ? C\{0}} (d ? N, d ? 2)and the hyperbolic component A? = {w : fw has an attracting fixed point}. We prove that if w? ? ?A? 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 A? at the point w?; which contains w-values for which fw has an attracting periodic cycle of period q.

Author Biography

Juan Bobenrieth, Universidad del Bío-Bío.

Facultad de Ciencias,Departamento de Matemáticas.


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[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).



How to Cite

J. Bobenrieth, “Parabolic perturbation in the family z ?1 + 1=wz?”, Proyecciones (Antofagasta, On line), vol. 21, no. 1, pp. 1-7, May 2017.