SciELO - Scientific Electronic Library Online

 
vol.84 issue1Publications in the field of Agrarian Sciences in the Anais da Academia Brasileira de Ciências: what next?Preparation and evaluation of a biomimetic scaffold with porosity gradients in vitro author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Article

Indicators

Related links

Share


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 Pc(z)=zn - czn-j

 

 

Carlos ArteagaI; Alexandre AlvesII

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 Mn of the family of polynomials Pc(z) = zn - czn-j, where c is a complex parameter. We prove that limn→∞ Mn = 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 Mn da família de polinômios Pc(z) = zn - czn-j, onde c é um parâmetro complexo. Provamos que limn→∞ Mn = 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 Pc(z) = zn - czn-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 Pc(z) = zn - czn-j, where c is a complex parameter. By definition, the connectedness locus Mn of this family of polynomials consists of all parameters c such that the Julia set of Pc(z) is connected or equivalentely if the orbit of every critical point of Pc(z) is bounded (see Carleson and Gamelin 1992). Since for all parameter c; z = 0 is a superattracting fixed point of Pc(z), we deduce that Mn consists of all parameter c such that the orbit of every non-zero critical point of Pc(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 Mn.

THEOREM A. Mn is a non-empty compact subset of the plane and lim (Mn) = 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 Mn.

PROOF. Let cD and let . Since n > 3 j, we have that . Let zc be a non-zero critical point of Pc(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 |zc| < 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 Mn, 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, Mn is bounded.

Let and let L be a positive integer such that Lj - J > 1. Suppose by contradiction that Mn is not closed. Then, there exists d in the boundary Mn of Mn 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 Mn, there exists c Mn 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 Mn. Therefore, Mn 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