MATHEMATICAL SCIENCES

 

A note on the connectedness locus of the families of polynomials P_c(z)=zⁿ - cz^n-j

 

 

Carlos Arteaga^I; Alexandre Alves^II

^IDepartamento de Matemática, ICEX, Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, 31270-970 Belo Horizonte, MG, Brasil
^IIDepartamento de Matemática, CCE, Universidade Federal de Viçosa, Av. Peter Henry Rolfs, s/n, 36570-000 Viçosa, MG, Brasil

Correspondence to

 

 

---

ABSTRACT

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}.

Key words: Julia set, connectedness locus, hyperbolic components, principal components.

---

RESUMO

Seja j um inteiro positivo. Para cada inteiro n > j, consideramos o locus conexo M_n da família de polinômios P_c(z) = zⁿ - cz^n-j, onde c é um parâmetro complexo. Provamos que lim_n→∞ M_n = D na topologia de Hausdorff; onde D é o disco unitário {c;|c|<1}.

Palavras-chave: Conjunto de Julia, locus conexo, componentes hiperbólicas, componente principal.

---

 

 

1 INTRODUCTION

In (Milnor 2009), J. Milnor considers the complex 1-dimensional slice S₁ of the cubic polynomials that have a superatracting fixed point. He gives a detailed pictured of S₁ 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) = zⁿ - cz^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) = zⁿ - cz^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.

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 | d" 1}.

 

THEOREM A. Mn is a non-empty compact subset of the plane and

in the Hausdorff topology, where D is the unitary closed disk {c;|c|<1}.

 

2 PROOF OF THEOREM A

The proof of the Theorem is based in the following results.

LEMMA 2.1. For n > 3 j, the closed unitary disk D is contained in M_n.

PROOF. Let c ∈ D and let . Since n > 3 j, we have that . Let z^c be a non-zero critical point of P_c(z). Then, , and this implies that

This and the fact that

imply that

Hence, since .

By induction, suppose that . Then,

where the last inequality is true because |z^c| < 1 and k < 1.

On the other hand, since . Thus,

.

Combinated with the estimate above, this gives . Hence, for all positive integer q. Since k < 1, we deduce that the orbit is bounded and Lemma 2.1 is proved.

LEMMA 2.2. If n > j, then Mn is a subset of the disk .

PROOF. Let . 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 , the orbit is not bounded.

Let . We claim that and hence k > 1.

In fact, since ,

and the claim is proved.

Now, we have that

By induction, suppose that . Then,

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 . Hence, for all positive integer q. Since k > 1, we conclude that, for each critical point , the orbit is not bounded, and Lemma 2.2 is proved.

Now, we prove Theorem A. By Lemma 2.2, M_n is bounded.

Let 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 is not bounded for some non-zero critical point zd of Pd (z). Hence, there exists a positive integer q such that . Since , we can choose a local branch of in a neighborhood V of d such that , for all c ∈ V. Since d ∈ ∂ M_n, there exists c ∈ M_n ∩ V such that . By Lemma 2.2, . Then,

thus,

By induction, suppose that . It follows that,

Hence, the orbit 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 imply that in the Hausdorff topology, and Theorem A is proved.

 

REFERENCES

CARLESON L AND GAMELIN T. 1992. Complex Dynamics. Springer-Verlag, New York Inc.         [ Links ]

DOUADY A. 1994. Does a Julia set depend continuosly on the polynomials? Proceedings of Symposia in Applied Mathematics vol. 49.         [ Links ]

MILNOR J. 2009. Cubic Polynomials with Periodic Critical orbit, Part I, "Complex Dynamics Families and Friends", ed., D. Scheleicher, A.K. Peters, p. 333-411.         [ Links ]

ROESCH P. 2007. Hyperbolic components of polynomials with a fixed critical point of maximal order. Ann Scientifiques de L'Ecole Normal Sup vol. 40.AMS Classification: Primary 37F45; Secondary 30C10.         [ Links ]

 

 

Correspondence to:
Carlos Arteaga
E-mail: dcam@mat.ufmg.br

Manuscript received on November 22, 2010
Accepted for publication on June 10, 2011