On The Julia Set Of Polynomial f(z) = z^m + P, With p Complex
by
P. Bhattacharyya
Dr.M.G.R.University, Chennai, India

Let f(z) be a rational function of a complex variable z. The iterates of f(z) are defined as

f0(z) = z, fn+1(z) = f(fn(z)), n=0, 1, 2, ...

In their monumental papers Fatou [, ] and Julia [] introduced the Julia set J(f) for f(z) defined as the set of those posints z ∈ [`C] = C ∪{ ∞}, where { f_n(z)} is not normal in the sense of Montel. Fatou and Julia proved the following important general properties of J(f) where f is a rational funtion of degree ≥ 2.

1. J(f) is a nonempty perfect set and is the whole plane if it has an interior point.

2. J(f_n) = J(f) for every integer n ≥ 1.

3. J(f) is completely invariant under the mapping in the sense that f(J(f)) = J(f) = f^-1(J(f)) for all the branches of the inverse function.

It should be mentioned that a similar theory exists for the transcendental entire and transcendental meromorphic functions.

Determination of the structure of J(f) even for a simple polynomial is a problem of considerable difficulty. For a polynomial J(f) depends on the coefficients of the polynomial in a complicated manner.

Jiu-Yi Cheng [] considered the case of f(z) = pz+z^m where p is real. Myrberg []-[], Brolin[] and Bhattacharyya and Arumaraj [, ] have considered the cases where f(z) is a polynomial of degree(f) = 2, 3 and 4 with real coefficients.

Except for a brief mention by Myrberg ([], p10) not much work seems to have been done on the structure of the Julia set of polynomials with complex coefficients. In this paper we investigate the structure of J(f) where f(z) = z^m +p, p complex.

We prove:

Theorem 1: Let f(z) = z² +p, where p is complex. Then J(f) is a Jordan curve if p ∈ S where S = {p: [1/4] -p ∈ S₁} and S₁ = { (r, q) : r < [1/2](1+cosq)}.

Theorem 2: Let f(z) = z² +p, where p is complex. Then J(f) is totally disconnected and its Lebesgue measure is zero if |p| > 2.

Let g⁰(z) = z, g¹(z) = [g⁰(z)]²+z = z²+z, and gⁿ⁺¹(z) = [gⁿ(z)]²+z , n=0, 1, 2, ... Let D_n = { z: |gⁿ(z)| ≤ 2} and D = ∩_n=0^∞D_n. Then,

Theorem 3: Let f(z) = z² +p, where p is complex. Then J(f) is connected if p ∈ D and is totally disconnected and is of Lebesgue measure zero if p ∉ D.

We make some obseravations on when the set D may be connected.

Theorem 4: The Tchebycheff's constant of the set D is one.

Theorem 5: Let f(z) = z² +ip, where p is real.

(a)Then J(f) is a Jordan curve if |p| < \dfrac√{3+2√3}4.
(b) If p > 1.16, then J(f) is totally disconnected and is of Lebesgue measure zero.

Theorem 6: Let f(z) = z^m +p, where m ≥ 2 integer and p is complex. Then J(f) is totally disconnected and its Lebesgue measure is zero if |p| > 2^[1/(m-1)].

Theorem 7: Let f(z) = z^m +p, where m ≥ 2 integer and p is complex. Let D be defined as in Theorem 3. If p ∈ D, then J(f) is connected, otherwise J(f) is totally disconnected and is of Lebesgue measure zero.

References

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Bhattacharyya, P and Arumaraj, Y.E, On the structure of the Fatou set for the polynomial z³+pz, p < -3, Math. RE. Toyama Univ., Vol 3, 1980, 123-141.

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Bhattacharyya, P and Arumaraj, Y.E, On the iteration of the polynomials of degree 4 with real coefficients, Annales Academicae Scientiarum Fennicae, Seris AI, Mathematica, Vol 7, 1982, 157-163.

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Brolin H, Invariant sets under iteration of rational functions, Ark. Mat., 6, 1965, 103-144.

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Fatou P, Sur les equations funtionells, Bull. Soc. Math. France, 47, 1919, 161-271.

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Jiu-Yi Cheng, On the Julia set of the polynomial f(z) = pz +z^m with p real, Annales Academiae Scientarum Fennicae, Sens A.I. Mathematica, Vol 14, 1989, 169-175.

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Date received: April 30, 2007