A note on the connectedness locus of the families of polynomials Pc ( z ) = z n − cz n − j

Let j be a positive integer. For each integer n > j we consider the connectedness locus Mn of the family of polynomials Pc (z) = z n − c z n−j, where c is a complex parameter. We prove that lim n!1 Mn = D in the Hausdorff topology, where D is the unitary closed disk {c; | c | ≤ 1}.


INTRODUCTION
In (Milnor 2009), J. Milnor considers the complex 1-dimensional slice S 1 of the cubic polynomials that have a superatracting fixed point.He gives a detailed pictured of S 1 in dynamical terms.In (Roesch 2007), Roesch generalizes these results for families of polynomials of degree n ≥ 3 having a critical fixed point of maximal multiplicity.This set of polynomials is described -modulo affine conjugacy-by the polynomials P c (z) = z n − c z n−1 .Roesch proved that the global pictures of the connectedness locus of this family of polynomials is a closed topological disk together with "limbs" sprouting off it at the cusps of Mandelbrot copies.In this note, we consider a positive integer j, and for each integer n > j, we consider the family of polynomials P c (z) = z n − c z n−j , where c is a complex parameter.By definition, the connectedness locus M n of this family of polynomials consists of all parameters c such that the Julia set of P c (z) is connected or equivalentely if the orbit of every critical point of P c (z) is bounded (see Carleson and Gamelin 1992).Since for all parameter c; z = 0 is a superattracting fixed point of P c (z), we deduce that M n consists of all parameter c such that the orbit of every non-zero critical point of P c (z) is bounded.We also consider the space of non-empty compacts subsets of the plane eqquiped with the Hausdorff distance (see Douady 1994).We obtain the following result about the size of M n .

PROOF OF THEOREM A
The proof of the Theorem is based in the following results.On the other hand, since n > 3 j, < and q (n − j − 1) − 1 > 1.Thus, Combinated with the estimate above, this gives positive integer q.Since k < 1, we deduce that the orbit {P c (zc )} is bounded and Lemma 2.1 is proved.
. By definition of M n , we have that, in order to prove Lemma 2.2, it is sufficient to prove that, for each non-zero critical point z c of P c (z) = z − cz , the orbit {P c (z c )} is not bounded.
We claim that k > and hence k > 1.
In fact, since z c = c, k = and the claim is proved.Now, we have that where the last inequality follows from the Claim above.
On the other hand, let s = q (n − 1) − 1.Then, s > 1 and Combinated with the estimates above, this gives positive integer q.Since k > 1, we conclude that, for each critical point zc of P c (z), the orbit {P c (z c ) }is not bounded, and Lemma 2.2 is proved.Now, we prove Theorem A. By Lemma 2.2, M n is bounded.
Let J = and let L be a positive integer such that L j − J > 1. Suppose by contradiction that M n is not closed.Then, there exists d in the boundary ∂ M n of M n such that the orbit{P d (z d )} is not bounded for some non-zero critical point z d of P d (z).Hence, there exists a positive integer Hence, the orbit {P c (z c )} is not bounded.This is a contradiction because c ∈ M n .Therefore, M n is closed, so it is compact.Now, Lemmas 2.1 and 2.2 and the fact that lim n ! 1 = 1 imply that lim n ! 1 M n =D in the Hausdorff topology, and Theorem A is proved.
THEOREM A. M n is a non-empty compact subset of the plane and lim (M n ) = D, in the Hausdorff topology, where D is the unitary closed disk {c; | c | ≤ 1}.
LEMMA 2.1.For n > 3 j, the closed unitary disk D is contained in M n .PROOF.Let c ∈ D and let k = .Since n > 3 j, we have that < , so k < .Let z c be a non-zero critical point of P c (z).Then, z c = c, and this implies that This and the fact that imply that Hence, since | c | < 1, Pc (zc ) | ≤ k | zc |.By induction, suppose that | P c (zc ) | ≤ k q | zc |.Then, where the last inequality is true because | zc | < 1 and k < 1.