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## Anais da Academia Brasileira de Ciências

*Print version* ISSN 0001-3765

### An. Acad. Bras. Ciênc. vol.84 no.1 Rio de Janeiro Mar. 2012 Epub Feb 09, 2012

#### http://dx.doi.org/10.1590/S0001-37652012000100002

**MATHEMATICAL SCIENCES**

**A note on the connectedness locus of the families of polynomials P_{c}(z)=z^{n} - cz^{n-j}**

**Carlos Arteaga ^{I}; Alexandre Alves^{II}**

^{I}Departamento de Matemática, ICEX, Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, 31270-970 Belo Horizonte, MG, Brasil

^{II}Departamento de Matemática, CCE, Universidade Federal de Viçosa, Av. Peter Henry Rolfs, s/n, 36570-000 Viçosa, MG, Brasil

**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*) =

*z*-

^{n}*cz*, where

^{n-j}*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*) =

*z*-

^{n}*cz*, onde

^{n-j}*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 *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}*cz*. 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

^{n-1}*j*, and for each integer

*n*>

*j*, we consider the family of polynomials

*P*(

_{c}*z*) =

*z*-

^{n}*cz*, where

^{n-j}*c*is a complex parameter. By definition, the

**connectedness locus**

*M*of this family of polynomials consists of all parameters

_{n}*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*consists of all parameter

_{n}*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* M*n 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*is not closed. Then, there exists

_{n}*d*in the boundary ∂

*M*of

_{n}*M*such that the orbit is not bounded for some non-zero critical point

_{n}*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*, there exists

_{n}*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*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.

_{n}

**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