The connectedness locus of the family of polynomials P c (z) = z n –cz

Abstract

For the study of the two-dimensional space of cubic polynomials, Milnor [J. Milnor, Cubic polynomials with periodic critical orbit, Part I, in Complex Dy-namics Families and Friends, D. Scheleicher and A.K. Peters, eds., 2009, pp. 333–411] considers the complex one-dimensional slice 𝒮 n of the cubic polynomials which have a super-attracting orbit of period n. These polynomials are described – modulo affine conjugacy – by the family f c (z) = z 3 + cz 2. Milnor gives a detailed description of 𝒮 n and a number of questions were left open which were resolved recently by Roesch [P. Roesch Hyperbolic components of polynomials with a fixed critical point of maximal order, Ann. Scientifiques de L' Ecole Normal Sup. 40 (2007)] and Faught [D. Faught, Local connectivity in a family of cubic polynomials, Thesis Cornell University, 1992]. Roesch generalizes these results for the families of polynomials f c (z) = z d + cz d−1; d ≥ 3. In the present note, we consider for n ≥ 3, the family of polynomials P c (z) = z n − cz, where c is a complex parameter. It is proved that the connectedness locus ℳ n of the family of polynomials P c (z) contains the unitary disc {c; |c| ≤ 1}, and it is a subset of the disc . Thus, we obtain that lim n→∞ℳ n = {c; |c| ≤ 1}. Also, we prove that the sets of centred polynomials of the hyperbolic components of ℳ n and the Misiurewicz points are dense in the boundary of ℳ n . We observe that the family of polynomials z m (z n − c) has been considered by Rabii [M. Rabii, Local connectedness of the Julia set of the family z m (z n + b), Ergodic Theory Dyn. Syst. 18 (2) (1998), pp. 457–470]. She studied the local connectedness of the Julia set of these families.