Abstract

ARTEAGA, Carlos  and  ALVES, Alexandre. A note on the connectedness locus of the families of polynomials P_c(z)=zⁿ - cz^n-j. An. Acad. Bras. Ciênc. [online]. 2012, vol.84, n.1, pp. 5-8.  Epub Feb 09, 2012. ISSN 0001-3765.  http://dx.doi.org/10.1590/S0001-37652012000100002.

Let j be a positive integer. For each integer n > j we consider the connectedness locus M_n of the family of polynomials P_c(z) = zⁿ - cz^n-j, where c is a complex parameter. We prove that lim_n→∞ M_n = D in the Hausdorff topology, where D is the unitary closed disk {c;|c|<1}.

Keywords : Julia set; connectedness locus; hyperbolic components; principal components.