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Dynamics and limiting behavior of Julia sets of König's method for multiple roots
Topology and its Applications 233 (2018) 16–32
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Topology and its Applications www.elsevier.com/locate/topol
Dynamics and limiting behavior of Julia sets of König’s method for multiple roots ✩ Gerardo Honorato CIMFAV–Facultad de Ingeniería, Universidad de Valparaíso, Chile
a r t i c l e
i n f o
Article history: Received 22 November 2016 Received in revised form 31 July 2017 Accepted 14 October 2017 Available online 26 October 2017 MSC: primary 65H04, 37F10, 30D05 secondary 37F50 Keywords: Connectedness Complex dynamics Root-finding algorithms
a b s t r a c t A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f , there exists at least one point in Sd converging under iterations of Nf to α. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton’s method. We show that for all order σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of König’s method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Newton’s iterative method Nf = Id −
f f
(1.1)
where f is a complex polynomial, as well as other higher-order methods have been extensively studied and used in order to deal with the equation f (z) = 0 (see for example [1], [17], [32], [33], [36], and [49]). The iterative function Nf defines a rational map on the Riemann sphere C = C ∪ {∞}. The simple roots of the equation f (z) = 0, that is the roots of the equation f (z) = 0 which are not roots of the derivative f (z), are super-attracting fixed points of Nf . In other words, let α be a simple root of f (z). Then Nf (α) = α and Nf (α) = 0. For a review of the dynamics of Newton’s method, see for example [11], [24]. In a more ✩
This work was supported by MATHAMSUD 16-MATH-06 PHYSECO and CNPq (The Brazilian National Research Council). E-mail address: [email protected].
https://doi.org/10.1016/j.topol.2017.10.013 0166-8641/© 2017 Elsevier B.V. All rights reserved.
G. Honorato / Topology and its Applications 233 (2018) 16–32
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general setting, we let P olyd and Ratk denote the space of polynomials of degree d and the space of rational functions of degree k, respectively. A root-finding algorithm is a rational map Tf : P olyd → Ratk such that the roots of the polynomial map f are attracting fixed points of Tf . We will say that a root-finding algorithm Tf has order of convergence σ if the local degree of Tf at each simple root of f is σ. E. Schröder ([44], [45], 1870) showed that Newton’s method applied to quadratic polynomials in the complex plane has two basins of attraction associated with the corresponding roots. His proof is obtained by conjugating Newton’s method for a quadratic polynomial f with the so-called Newton’s method for multiple roots associated to f defined by Mf (z) = z −
f (z)f (z) . (f (z))2 − f (z)f (z)
(1.2)
In fact, for instance conjugating Newton’s method with a Möbius transformation sending one root to infinity and the other one to 0, we obtain the map z 2 which proves immediately the existence of two connected basins of attraction. More precisely, the basins of attraction associated to Newton’s method applied to the polynomial f (z) = z 2 − 1 are found explicitly by noting that Nf =
1 . Mf
Thus, it suffices to concentrate on the convergence of the iterates under Mf . Consequently, the points on the complex plane with positive real part converge to 1, while those with negative real part converge to −1. For a different approach, see A. Cayley ([14], 1890). For a historical background and detailed explanation of these facts, see [3], [6]. More recently, Newton’s method for multiple roots has also been studied by T. Pomentale [38] and W. Gilbert [22]. A natural generalization of Newton’s method (1.1) is König’s root-finding algorithm applied to polynomials, denoted Kf, σ , also known as the Basic Family (in short, König’s method). Let f : C → C be a meromorphic map, and let σ ≥ 2 be an integer. König’s method of order σ associated to f is the meromorphic map Kf, σ : C → C defined by Kf,σ = Id + (σ − 1)
(1/f )[σ−2] (1/f )[σ−1]
(1.3)
where (1/f )[k] is the kth derivative of 1/f . In the special cases σ = 2 and σ = 3, König’s method is Newton’s method and Halley’s method, respectively. König’s method applied to polynomials has been studied from both the numerical and the dynamical points of view. (See for example, [4], [13], [28], [29], [37], [50].) Here we are interested in studying dynamical and geometric aspects of König’s method for multiple roots (see Definition 2.1), denoted Mf, σ , which is a generalization of Newton’s method for multiple roots. Given a polynomial f , König’s method for multiple roots Mf, σ is none other than König’s method Kf, σ applied to the rational map f /f . Note that the term f /f has the effect of converting the multiple roots of f (z) into simple ones. In other words, if f has a multiple zero of order n at α, then f /f has simple zero at α. Although Kf, σ and Mf, σ are similar, they are different from a dynamical viewpoint. For example, infinity is never a fixed point for Mf, σ , which is exactly the opposite for Kf, σ . In addition, the fixed points of Mf, σ associated to the roots of f are always super-attracting, regardless their multiplicity. We next state our two main results. In [13], X. Buff and C. Henriksen show that the sequence of Julia sets of König’s method applied to a polynomial f converges to the union of the bisecting locus of the set of roots of the polynomial f and infinity as σ → ∞. A Voronoi cell of a root α is the set of points in the complex plane that are strictly closer to α
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than to any of the other roots. A Voronoi diagram is the collection of all Voronoi cells. In [28], B. Kalantari shows that for generic polynomials, the basins of attraction of König’s method for polynomials approximate uniformly the Voronoi diagram of the roots. In this article, we show an analogous result for König’s method for multiple roots. More precisely, we show that the Fatou set of König’s method for multiple roots applied to polynomials converges in the Hausdorff complementary topology to the Voronoi diagram as σ → ∞. A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f , there exists at least one point in Sd converging under iterations of Nf to α. A question arising naturally from the result concerns whether it also holds for root-finding algorithms of order higher than Newton’s method. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton’s method. We show that for all order σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result of [27] to higher order root-finding methods. In [26] a theoretical proof of disconnectedness of the Julia set for König’s method applied to polynomials is given. As far as we know, there are no pictures showing that situation. Here we give a proof and also provide pictures that illustrate disconnectedness of the Julia set. The paper is organized as follows. In Section 2, we introduce basic notation and definitions, and we give results concerning the nature of fixed points. Section 3 presents a classification of the dynamics of König’s method for multiple roots applied to quadratic polynomials. Section 4 contains one of our main results, which involves Voronoi diagrams and the Fatou sets for König’s method for multiple roots applied to polynomials. In Section 5, we give a characterization of König’s method for multiple roots applied to polynomials. Section 6 is devoted to the conjugacy classes. Section 6 contains the main result: The Julia set of König’s method for multiple roots applied to polynomials is, in general, non-connected. 2. Basic notions and standard definitions We will first introduce basic notation and definitions concerning complex dynamics. P (z) Let R(z) = Q(z) be a rational map of the extended complex plane into itself, where P and Q are polynomials with no common factors. We say that α is a fixed point of R if R(α) = α. The multiplier of R at a fixed point α is the complex number λ(α) = R (α). In addition, we say that a fixed point α is attracting, repelling or indifferent if the modulus of its multiplier λ(α) is, respectively, less than, greater than or equal to 1. A super-attracting fixed point is a fixed point α of R for which R (α) = 0. For z ∈ C, we define its orbit as the set orb(z) = {z, R(z), R2 (z), . . . , Rk (z), . . .} where Rk means the k-fold iterate of R. The basin of attraction of a (super)attracting fixed point α of R is the set B(α) = {z ∈ C : Rn (z) → α as n → ∞}. In other words, the basin of attraction of a (super)attracting fixed point is the set consisting of the points whose iterates tend to the fixed point α as n tends to ∞. Let z0 be a fixed point of Rn which is not a fixed point of Rj , for any j with 0 < j < n. We say that orb(z0 ) = {z0 , R(z0 ), . . . , Rn−1 (z0 )} is a cycle of length n or simply an n-cycle. Note that orb(zj ) = orb(z0 ) for any zj ∈ orb(z0 ), and R acts as a permutation on orb(z0 ). The multiplier of an n-cycle is the complex n−1 n−1 number λ(orb(z0 )) = (Rn ) (z0 ). Note that by the Chain Rule, (Rn ) (zj ) = k=0 R (Rk (zj )) = k=0 R (zk ) where the second product is a re-arrangement of the first. Thus, at each point zj of the cycle, the derivative (Rn ) has the same value. An n-cycle {z0 , z1 , . . . , zn−1 } is said to be attracting, repelling, indifferent if the modulus of its multiplier λ(orb(z0 )) is, respectively, less than, greater than, equal to 1. An n-cycle Ω = {z0 , z1 , . . . , zn−1 } is said to be super-attracting if λ(orb(z0 )) = 0. If z0 is in an attracting n-cycle of R, the basin of attraction of the orbit orb(z0 ), denoted B(orb(z0 )), is j n the set ∪n−1 j=0 R (B(z0 )), where B(z0 ) is the basin of attraction of z0 as an attracting fixed point of R . The Julia set of a rational map R, denoted J(R), is the closure of the set consisting of the repelling periodic points. Its complement, denoted F (R), is called the Fatou set. For further information, see [10], [35].
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We next give the relevant definitions concerning the root-finding algorithms we are interested in studying and prove some preliminary results Definition 2.1. Let f : C → C be a meromorphic map, and let σ ≥ 2 be an integer. We define König’s method for multiple roots of order σ associated to f , denoted Mf, σ : C → C, as König’s method applied to the meromorphic map f /f . In other words, we have Mf,σ = Kf /f , σ = Id + (σ − 1)
(f /f )[σ−2] (f /f )[σ−1]
(2.1)
where (f /f )[k] is the kth derivative of f /f . When σ = 2, König’s method for multiple roots is known as Newton’s method for multiples roots given in formula (1.2). The following concerns the nature of fixed points of König’s method applied to a meromorphic map. For a proof see [13]. Proposition 2.1. Let f : C → C be a meromorphic map. Let αi denote a zero of f , and let ni ≥ 1 denote its multiplicity. Then for any integer σ ≥ 2, the fixed points of König’s method Kf, σ : C → C are either (super)attracting or repelling. Furthermore: (1) The (super)attracting fixed points are exactly the zeroes αi an their multiplicities are 1 − (σ − 1)/(ni + σ − 2). When ni = 1, the local degree of Kf, σ at αi is at least equal to σ. (2) The extraneous fixed points of Kf,σ are exactly the zeroes of (1/f )[σ−2] . If βj is a zero of (1/f )[σ−2] with multiplicity mj , then it is a repelling fixed point of Kf, σ with multiplier 1 + (σ − 1)/mj . The equivalent statement for Mf,σ is as follows. Proposition 2.2. Let f : C → C be a polynomial. Let αi denote a zero of f , and let ni ≥ 1 denote its multiplicity. Then for any integer σ ≥ 2, the fixed points of König’s method for multiple roots Mf, σ : C → C are either super-attracting or repelling. Furthermore: (1) The super-attracting fixed points are exactly the zeroes αi . (2) The extraneous fixed points of Mf,σ are exactly the zeroes of (f /f )[σ−2] . If βj is a zero of (f /f )[σ−2] with multiplicity mj , then it is a repelling fixed point of Mf, σ with multiplier 1 + (σ − 1)/mj . Proof. Let f be a complex polynomial. Applying Proposition 2.1 to the rational map f (z) ni = f (z) z − αi i=1 N
(2.2)
where ni are the multiplicities of αi , the result follows. 2 Proposition 2.3. Let f : C → C be a polynomial of degree d with N zeroes. Let αi denote a zero of f , and let ni ≥ 1 denote its multiplicity. Then: (1) Mf, σ has at most (σ − 1)(N − 1) repelling fixed points in C. (2) The degree of Mf, σ is at most σ(N − 1). (3) The local degree of Mf, σ at αi , is at least equal to σ.
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Proof. Let f be a complex polynomial. For statement (1), consider the formula (2.2). Note that the repelling fixed points of Mf, σ are the zeroes of g(z) =
f f
[σ−2] (z) = (−1)(σ−2) (σ − 2)!
N i=1
ni . (z − αi )σ−1
(2.3)
To compute the zeroes of g it suffices to compute the poles of g, since a rational map has the same number of zeroes and poles in C. It follows from formula (2.3) that the zeroes of f contribute with N (σ − 1) poles of g. On the other hand, g has a zero of order σ − 1 at ∞. In fact at ∞, we have that f takes the form λz d + ao z d−1 + a1 z d−2 + · · · . Thus g(z) =
f f
[σ−2] (z) =
λ z σ−1
1 + O(|z|−1 ) .
Therefore, Mf, σ has at most N (σ − 1) − (σ − 1) = (N − 1)(σ − 1) repelling fixed points in C. Concerning (2), recall that, in general, the degree of Mf, σ minus 1 is equal to the number of fixed points of Mf, σ . It follows from statement (1) that the degree of Mf, σ is at most [N + (N − 1)(σ − 1)] − 1 = σ(N − 1). It remains to prove (3). Suppose that α is a simple root of f . Again, using formula (2.3) we obtain that
f f
[σ−2] (z) =
λ + O(1) (z − α)σ−1
(2.4)
for some complex number λ = 0. Hence Mf, σ (z) = z + (σ − 1)
(f /f )
[σ−2]
(z)
[σ−1]
(z)
(f /f )
= α + (z − α) + (σ − 1)
(λ/(z − αi )σ−1 )(1 + O(z − α)σ−1 ) −(λ/(z − αi )σ )(1 + O(z − α)σ )
= α + O(z − α)σ . Consequently, the local degree of Mf, σ is at least equal to σ at the root α.
2
Remark 2.2. It follows from formula (2.4) that the degree of the denominator of Mf, σ is greater than or equal to the degree of the numerator. Thus, ∞ cannot be a fixed point of Mf, σ . Now we will classify König’s method for multiple roots applied to quadratic polynomials. The particular case of σ = 2 was studied by E. Schröder, noting that Kf,σ and Mf,σ are conjugated by 1/z. (See [45], [3].) Proposition 2.4. Let f : C → C be a quadratic polynomial. Then: (1) If f has a double root at a ∈ C, then König’s method for multiple roots is the constant map z → a. (2) If f has two simple roots, König’s method for multiple roots is conjugated to the map w → wσ . Proof. Statement (1) is a straightforward computation.
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Fig. 1. The Julia set of Mf, σ applied to the polynomial f (z) = z(z 2 − 1)(z − i) for σ = 2, 10, 30, respectively.
Concerning (2), we look at the solutions of the equations Mf, σ (z) = a and Mf, σ (z) = b, where a and b are the zeroes of f . The solutions of the first equation is a along with its multiplicities. Indeed, according to the formula (2.3) the equation Mf, σ (z) = a is equivalent to z−
(z − a)(σ−1) + (z − b)(σ−1) (z − a)σ + (z − b)σ
(z − a)(z − b) = a ,
which has the root a with multiplicity σ as a solution. We have a similar result if we consider the simple root b. Therefore, we have shown that a and b are exceptional points for Mf, σ , for every σ ≥ 2. If this is the case, then Mf,σ is conjugated to z σ for every σ ≥ 2. (See for example [10].) 2 3. Voronoi diagrams for König’s method for multiple roots This section will prove that the Fatou set of König’s method for multiple roots converges in the Hausdorff complementary topology to the Voronoi diagram as σ → ∞. (See Fig. 1.) Consider a given finite set of points in the complex plane, say S = {α1 , α2 , . . . , αd }. Definition 3.1. A Voronoi cell for a point αk or Voronoi face of the site αk is the set V (αk ) = {z ∈ C | d(z, αk ) < d(z, αj ), for all j = k}. The Voronoi diagram of S is defined as the set
V =
d
V (αi ).
i=1
We take the definition from [18]. Voronoi diagrams and their applications have been extensively studied. (See, for example, [7], [18], [19], [23], [39].) Let f be a degree d ≥ 2 complex polynomial with at least two different roots. Here we will consider S as the set of roots of f (some of them are multiple). In [13], X. Buff and C. Henriksen show that the sequence of Julia sets of König’s method applied to a polynomial f converges to the union of the bisecting locus of the set of roots of the polynomial f and infinity as σ → ∞. In [28], B. Kalantari shows that for generic polynomials, the basins of attraction of König’s method for polynomials approximate uniformly the Voronoi diagram of the roots. In addition, he shows that none of basins of attractions associated with a root-finding
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algorithm may be realized as a Voronoi cell. Unlike in Kalantari’s proof, our arguments depend on a result of E. Schröder, which is improved in [13], and we allow multiple roots for the polynomial. Definition 3.2. Let A and B be subsets of a metric space (X, d). The Hausdorff metric is defined as dH (A, B) = max{sup d(a, B), sup d(b, A)}. a∈A
b∈B
We may assume that we are working either with the euclidean metric or the spherical metric, since they are uniformly equivalent. Let K be the set of all non empty compact subsets (strictly contained) of the sphere C. Then it is known that equipped with the Hausdorff metric, K is a complete metric space. For a more detailed discussion of Hausdorff metric, see [9]. We next record a useful characterization of convergence in Hausdorff metric which we will use to prove Theorem 3.6 below. For a proof the reader can consult, for example, [9]. Lemma 3.3. The Hausdorff limit A of {Am } is also the set of all accumulation points of the sequences am such that am ∈ Am , for all m ≥ 1. As for the topology on the class of open sets, we introduce the Hausdorff complementary metric. We set Ω = {A ⊂ C, A is a nonempty open set}. Note that there exists a bijection between Ω and K. Thus, we can define the metric as follows. Definition 3.4. The Hausdorff complementary metric on the set Ω is given by dcH (A, B) := dH (C \ A, C \ B) where A and B belong to Ω. It follows immediately that (Ω, dcH ) is complete metric space. For a review of Hausdorff complementary metric, see for example [30] and references therein. The next Lemma provides a useful tool in order to prove convergence depending on the order σ. For a proof the reader can consult [13]. Lemma 3.5. Let D be a Euclidean disk centered at z0 and α be a point of D. Assume that g is meromorphic on D and has a unique pole, possibly multiple at α. Then, there exists a neighborhood of Uz0 of z0 on which the sequence of functions (n + 1)g [n] /g [n+1] converges uniformly to the function z → α − z. Considering g = f /f and recalling that a zero of f is a pole for the meromorphic map g [n] , we have the following immediate consequence of this Lemma, which is essential for our purpose. Proposition 3.1. Let f be a polynomial and D a disk centered at z0 . Assume that f has unique zero, possibly multiple α ∈ D. Then, Mf,σ converges uniformly on a neighborhood Uz0 of z0 contained in D, to the constant function α as σ → ∞. Theorem 3.6. Let f be a polynomial of degree d ≥ 2. Then lim F (Mf, σ ) = V
σ→∞
where F (Mf, σ ) is the Fatou set of Mf, σ and the convergence is in Hausdorff complementary metric.
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Proof. Let f be a degree d polynomial with roots αi , where i = 1, . . . , d. Note that for every σ ≥ 1, we have that F (Mf, σ ) is non-empty, since it contains the roots of f . Fix a root of f , say α = α1 . For simplicity we will denote as Bσ (α) the basin of attraction of the fixed point α of Mf, σ , whenever σ ≥ 1. We first prove the following Lemma 3.7. V c (α) = {z ∈ C : z is an accumulation point of a sequence zσ ∈ Bσc (α), σ ≥ 1}. Proof. Let z be in the complement of the Voronoi cell of α, V c (α). Since V (α) is an open set, V c (α) is compact subset of C. We first prove that z is an accumulation point of a sequence zσ ∈ Bσc (α) for σ ≥ 1. The discussion will be divided into two cases, according to the position of z. Case 1. If z ∈ ∂V (αi ) for some i = 1, . . . , d, then |z − αi | = |z − αk | for some k = 1, . . . , d, where k = i. Let D be a spherical disk of radius and centered at z. In this case we claim that the Julia set J(Mf,σ ) intersects D for σ sufficiently large. Indeed, let z1 and z2 be two points in D which belong to different components of the complement of ∂V (αi ). By interchanging the roles of z1 and z2 if necessary, we may assume that αi is the root of f closest to z1 and αk is the root of f closest to z2 . By Proposition 3.1 there exists a neighborhood Uz1 of z1 on which the sequence Mf, σ converges uniformly to αi , for σ sufficiently large. Thus, there exists a positive integer n1 such that Mf, σ (Uz1 ) ⊂ Bσ (αi ) for σ ≥ n1 . This shows that z1 is in the basin of attraction of αi of Mf, σ when σ ≥ n1 . Similarly, there exists a positive integer n2 such that z2 is in the basin of attraction of αk of Mf, σ , for σ ≥ n2 . Therefore the Julia set J(Mf,σ ) intersects D for σ ≥ max{n1 , n2 } = n3 , as claimed. Now by shrinking D if necessary, we have that z is an accumulation point of a sequence zσ ∈ J(Mf, σ ) for every σ ≥ n3 . For every σ = 1, . . . , n3 − 1 we can choose an arbitrary point zσ in J(Mf, σ ). Considering the union of σ and σ , (and calling it σ for notational simplicity) we have that z is an accumulation point of the sequence zσ in J(Mf, σ ), σ ≥ 1. Since J(Mf, σ ) ⊂ Bσc (α), in particular we have that z is an accumulation point of a sequence zσ in Bσc (α) for σ ≥ 1. Case 2. If z belongs to the interior of V (αi ) for some i = 2, . . . , d, then by definition, αi is the closest root of f to z. The argument now proceeds similarly as above. By Proposition 3.1 there exists a neighborhood Uz of z and a positive integer n such that Mf, σ (Uz ) ⊂ Bσ (αi ) for σ ≥ n. Shrinking Uz if necessary, we obtain a sequence zσ ∈ Bσ (αi ) converging to z, for σ ≥ n. For every σ = 1, . . . , n − 1 we can choose an arbitrary point zσ in Bσ (αi ). In particular z is an accumulation point of a sequence zσ ∈ Bσ (αi ) ⊂ Bσc (α) for every σ ≥ 1. This completes the proof of Case 2. Suppose now that z is an accumulation point of a sequence zσ ∈ Bσc (α) for σ ≥ 1. Then z cannot be in V (α) since otherwise, again by Proposition 3.1 there exists a neighborhood Uz of z and a positive integer n such that Mf, σ (Uz ) ⊂ Bσ (α) for σ ≥ n. Hence, zσ ∈ Bσ (α) for σ sufficiently large, which is a contradiction. Thus z is in V c (α). This completes the proof of Lemma 3.7. 2 We now complete the proof of Theorem 3.6. As a consequence of the Lemma above and Lemma 3.3 we have that Vαc = limσ→∞ Bσc (α) in Hausdorff metric, which implies that Vα = lim Bσ (α), σ→∞
in Hausdorff complementary metric. We claim that d i=1
V (αi ) = lim F (Mf, σ ), σ→∞
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in the Hausdorff complementary metric. In order to prove this, again we consider the characterization in d Lemma 3.3. If z belongs to the complement of i=1 V (αi ), then for some i, j = 1, . . . , d, we have that |z − αi | = |z − αj |. Now we apply the same argument of Case 1 in the preceding Lemma to obtain a sequence zσ in J(Mf, σ ), σ ≥ 1. In particular the sequence zσ belongs to F c (Mf, σ ), σ ≥ 1. This shows that z is accumulated by elements in the complement of F (Mf, σ ). On the other hand, suppose that z is an accumulation point of a sequence zσ ∈ Fσc (α) for σ ≥ 1. Then d z cannot be in i=1 V (αi ), since otherwise by Proposition 3.1 there exists a neighborhood Uz of z and a positive integer n such that Mf, σ (Uz ) ⊂ Bσ (α) for σ ≥ n. In particular, zσ ∈ Fσ (α) for σ sufficiently d large, which is a contradiction. It follows that z is in the complement of i=1 V (αi ). This proves that d V = i=1 V (αi ) = limσ→∞ F (Mf, σ ) in Hausdorff complementary metric, as required. 2 We recall the definition of Bisecting locus. Definition 3.8. The bisecting locus B of the set S ⊂ C, is the set of points z ∈ C where the distance function δ(z) = d(z, S) is not differentiable. Remark 3.9. Note that z0 ∈ B if and only if |z0 − α1 | = |z0 − α2 | for some α1 , α2 ∈ S. In fact, let z0 be a point in the plane so that |z0 − α1 | = |z0 − α2 | for some α1 , α2 ∈ S. Let r = |z0 − α1 |. Without loss of generality, we may assume that z0 = (0, 0), and α1 , α2 are real numbers. Then ∂δ (0, 0) = lim h→0 ∂x
inf w∈S d((h, 0), w) − inf w∈S d((0, 0), w)
h r − |h| − r |h| = lim − . = lim h→0 h→0 h h
As a consequence δ is not differentiable in z0 , and so z0 ∈ B. Conversely, suppose that for z0 = α1 we have |z0 − α1 | < |z0 − αi |, for every i = 2, . . . , d. Then α1 is the element of S closest to z0 , that is, z0 ∈ V (α1 ). Since V (α1 ) is an open set, there exists > 0 such that B(z0 , ) ⊂ V (α1 ). So, for every z ∈ B(z0 , ) we have that δ(z) = |z − α1 |, which is differentiable in B(z0 , ), as a function of R2 . Thus z0 ∈ / B. As an immediate Corollary to this Remark and Theorem 3.6 we have the following: Corollary 3.1. Let f be a polynomial. The Voronoi diagram V coincides with the complement of the bisecting locus. Remark 3.10. If Mf, σ has an attracting periodic orbit not associated with the roots, it will disappear when σ goes to infinity: if z is in the basin of attraction of this periodic orbit, according to the position of z, we can use a similar argument as in the proof of Lemma 3.7. Remark 3.11. Note that even for a polynomial g it is possible that a sequence of polynomials gm approximate g such that the respective Julia sets do not converge. See [31]. 4. Characterization of König’s method for multiple roots applied to polynomials This section will study the image of the map Mf,σ : P olyd → Ratk . More precisely, we are interested in finding conditions under which a rational map is König’s method for multiple roots applied to polynomials. This kind of result about Newton’s method applied to polynomials has been studied, to my knowledge, by
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G. Saunders [47], J. Head [25], K. Nishizawa and M. Fujimura [21], and more recently, J. Ruckert and D. Schleicher [42], E. Crane [15], P. Roesch [41]. König’s method applied to generic polynomials is studied by X. Buff and C. Henriksen [13]. We will need the following that treats the case of Newton’s method applied to rational maps. (See [15].) Proposition 4.1. A rational map F is Newton’s method applied to some rational map g if and only if all of the fixed points of F are simple and each fixed point α of F satisfies 1−F1 (α) ∈ Z. Moreover, g is a polynomial if and only if these integers are all positive. Proposition 4.2. Let R be rational map of degree d ≥ 2, with only super-attracting and repelling fixed points in C. Denote its super-attracting fixed points by αi , with i ≥ 1. Assume that any repelling fixed point βi d
has a multiplier of the form 1 + 1/mi , with mi ∈ N. Then R = Mf , where f (z) = (z − αi )ni , for some i=1
ni ∈ N. Proof. Given the rational map R, define the map F =
1 [σId − R]. (σ − 1)
We let χ denote the fixed points of R (whether they be attracting or repelling). A brief computation shows that, for every χ, F satisfies 1 ∈ Z. 1 − F (χ) According to Proposition 4.1, there exists a rational map g(z) = c that F = Id −
i (z − χ)
li
, where c ∈ C and li ∈ Z, such
g . g
Therefore, R = Id + (σ − 1)
g . g
Since R is a rational map with only super-attracting points αi and repelling fixed points βi in C, whose multipliers are 1 + 1/mi , with mi ∈ N, we have that (z − βi )mi g(z) = c i σ−1 i (z − αi )
(4.1)
where c ∈ C, c = 0. We write g := gσ to emphasize the dependence on σ. In order to prove Proposition 4.2, we need only show that gσ = (f /f )[σ−2] for some complex polynomial f . This is the content of the following Lemma 4.1. For every σ ≥ 2, gσ = (f /f )[σ−2] for some complex polynomial f . Proof. The proof will proceed by induction on σ. Define the map (z − αi ) 1 =z− i h(z) = z − . mi g2 (z) i (z − βi )
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The superattracting fixed point of h are αi and ∞, which is a repelling fixed point with multiplier d/(d − 1). Hence by Proposition 4.1, there exists a polynomial f such that h(z) = Nf . This implies that g2 =
f . f
As a consequence the assertion is true for σ = 2. According to the induction hypothesis, for σ = k there exists a complex polynomial f so that gk = ln(f )[k−1] . Here ln(f ) denotes the logarithmic derivative of f . It is not difficult to see that ln(f )[k] = (ln(f )[k−1] ) = (gk ) = gk+1 , which is precisely the formula for k + 1. This completes the induction. 2 This completes the proof of Proposition 4.2.
2
5. Conjugacy classes In this section we will prove a Scaling Theorem for König’s method for multiple roots. It is well known that if a root-finding algorithm satisfies a Scaling Theorem, then the dimension of the parameter space can be reduced. For instance in the case of Newton’s method applied to cubic polynomials, the dimension of the parameter space can be reduced to dimension 1. For a review of this kind of result, see [1]. The situation is different for polynomials of degree 4. (See [5], [16], [2].) In the best case the parameter space can be reduced to dimension 2. However under stronger assumptions one can restrict the study of the parameter space to dimension 1. More precisely, to the one parameter family fλ (z) = (z 2 + λ)(z 2 − 1).
(5.1)
Lemma 5.1. (Scaling Theorem) Let f be an analytic function on the Riemann sphere, and let A(z) = αz +β, with α = 0, be an affine map. If g(z) = f ◦ A(z), then for any integer σ ≥ 2, we have Mf, σ ◦ A = A ◦ Mg, σ . That is, König’s method for multiple roots of f is analytically conjugated to König’s method for multiple roots of g. Proof. Since g(z) = f ◦ A(z), we have
g g
[σ] =α
σ+1
f f
[σ] ◦ A.
Thus, A ◦ Mg,σ (z) = αz + (σ − 1)
(f /f )[σ−2] ◦ A(z) + β = Mf, σ ◦ A. (f /f )[σ−1]
This completes the proof. 2 Lemma 5.2. Let f and g be complex polynomials and A(z) = αz + β an affine map, where α = 0. Then A conjugates Mf, σ ◦ A = A ◦ Mg, σ if and only if there exists a complex number λ = 0 so that g = λf ◦ A.
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Proof. Assume that there exists a complex number λ = 0 such that g = λf ◦ A. It follows from Lemma 5.1 that Mf, σ ◦ A = A ◦ Mg, σ . Conversely, suppose that Mf, σ ◦ A = A ◦ Mg, σ . Define h = f ◦ A. Then, by Lemma 5.1 we have that Mg, σ = Mh, σ . Hence the polynomials g and h have the same roots with same multiplicities. Indeed, if α is a root of g which is not a root of h, by Proposition 2.2 α is a superattracting fixed point of Mg, σ . In particular α is a superattracting fixed point of Mh, σ , which by Proposition 2.2 corresponds to a zero of h, yielding a contradiction. Now suppose that α is a root of g with multiplicity n ≥ 1, and also a root of h with multiplicity m ≥ 1, where n = m. Without loss of generality, we may suppose that g(z) = (z − α)n f (z) and h(z) = (z − α)m f (z), for some polynomial f such that f (α) = 0. By formula (2.3) we have that n g (z) = + fˆ(z), g(z) (z − α) where fˆ is analytical on a neighborhood of α. Similarly for h we have m h (z) = + fˆ(z). h(z) (z − α) Since Mg, σ = Mh, σ for every σ ≥ 2, in particular for σ = 2 we have that (h /h) (g /g) = . (g /g) (h /h) Therefore, (m + (z − α)fˆ(z))(z − α) (n + (z − α)fˆ(z))(z − α) = , −n + fˆ (z)(z − α)2 −m + fˆ (z)(z − α)2 for all z ∈ C. It follows from this formula that n = m, a contradiction. As a consequence, there exists a complex number λ = 0 so that g = λf ◦ A. 2 6. Topology of the Julia set for König’s method for multiple roots A well known result of J. Hubbard, D. Schleicher and S. Sutherland shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f , there exists at least one point in Sd converging under iterations of Nf to α. (See [27].) For an improvement of this result, see for example [12], [43]. In order to construct the set Sd , it is essential that the immediate basins of attraction of Newton’s method, denoted Bα , be simply connected. This fact provides a source of geometric control. Indeed, there exists a conformal isomorphism ϕ : D → B(α) whose composition with Newton’s method yields a map acting on the open unit disk, that is, h = ϕ−1 ◦ Nf ◦ ϕ. The map h extends to the complex plane as a Blaschke product. A channel of a root α is an unbounded connected component W of B(α) \ D with the additional property that there is a w ∈ W that can be connected to Nf (w) by a curve in W . Now, there exists a neighborhood of infinity where Newton’s method is conjugated to a linear map. This implies that infinity is accessible from all of the basins of attraction of the roots. It follows that every access to ∞ of
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B(α) corresponds to a unique channel of α. Thus, when basins of attraction are simply connected, accesses and channels are in bijection. We may summarize the usefulness of the simply connectedness of basins of attraction in the following two key points:
Each immediate basin B(α) has exactly mα distinct accesses to infinity, where mα is the number of critical points of Nf in B(α).
Measuring the width of channels. The measure is given in terms of the conformal modulus of the annulus formed by the channel modulo the dynamics. Simple connectedness of basins of attraction for Newton’s method for polynomials is shown by F. Przytycki in [40]. He proves that if α is an attracting fixed point for Newton’s method Nf , then B(α) is simply connected. A stronger result due to M. Shishikura establishes that a rational function of degree greater than one has exactly one repelling (or weakly repelling) fixed point, then its Julia set is connected. As a corollary the Julia set of Newton’s method of a complex polynomial is connected. (See [46].) The case of polynomials of degree three is studied by H. Meier in [34] and L. Tan in [48]. In [20] is proved that the immediate basins of attraction associated to the Halley’s method applied to generic polynomials with real roots and real coefficients, are simply connected. Recently, Shishikura’s result is extended to entire maps by K. Baranski, N. Fagella, X. Jarque and B. Karpinska [8]. In this section, we show that for all orders σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to it is disconnected. For an illustration, see Fig. 2. Therefore, Przytycki’s and Shishikura’s results does not apply for König’s method for multiple roots when σ ≥ 2. Consequently, our result establishes restrictions for extending the proof of the main result of J. Hubbard, D. Schleicher and S. Sutherland [27] to higher order root-finding algorithms. In fact, a disconnected Julia set implies that several accesses may correspond to the same channel. In [26] we prove an analogous result for Kf,σ . The proof depends on the fact that Kf,σ has a unique repelling fixed point at ∞, which is no longer true for Mf,σ . Thus, we cannot repeat the proof in our context. In this article we give a proof and provide pictures that illustrate disconnectedness of the Julia set. Additionally, our proof is simpler and more direct. In fact, we give an explicit polynomial f such that the imaginary axis is completely contained in the Fatou set of Mf,σ , when σ ≥ 2. Now, the imaginary axis splits the Riemann sphere into two components, each of which contains components of the Julia set. Consequently, the Julia set is disconnected. The following is the main result of this section. Theorem 6.1. For all order σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to this polynomial is not connected. Proof. Let σ ≥ 2. Consider the family (5.1) fλ (z) = (z 2 + λ)(z 2 − 1). Set λ = 0. König’s method for multiple roots applied to the polynomial f0 (z) = z 2 (z 2 − 1) is given by
Mf, σ (z) =
z σ [(z + 1)σ − (z − 1)σ ] . z σ (z + 1)σ + 2(z + 1)σ (z − 1)σ + z σ (z − 1)σ
(6.1)
According to Lemma 2.3, the repelling fixed points are the zeroes of the equation
f f
[σ−2] (z) = 0.
(6.2)
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Fig. 2. In counterclockwise direction: the Julia set of Mf, σ applied to the polynomial f (z) = z 4 − z 2 for σ = 3, 4, 20 and σ = 100, respectively.
Since Mf, σ is an odd function, there are at least two repelling fixed points which are symmetric with respect to the imaginary axis. We will show that iR ∪ {∞} is invariant under the action of Mf, σ (see Fig. 2). In fact for every y ∈ R, we have Mf, σ (iy) = i Mf, σ (y) where Mf, σ is the map defined by Mf, σ (y) =
i(yi)σ ((yi − 1)σ − (yi + 1)σ ) . (yi)σ (yi + 1)σ + 2(yi + 1)σ (yi − 1)σ + (yi)σ (yi − 1)σ
Since Mf, σ (y) = Mf, σ (y), it follows that Mf, σ is a real map. From Mf,σ (1/z) |z=0 = 0 it follows that infinity belongs to the immediate basin of attraction of 0. Thus, Mf is the restriction to the imaginary axis of the map Mf, σ . Consequently, the imaginary axis is invariant under Mf, σ . Next, we will show that iR ∪ {∞} is contained in the immediate basin of attraction of 0. Lemma 6.2. For all y ∈ R \ {0}, we have |Mf, σ (y)| < |y|. Proof. Clearly, this assertion is equivalent to showing that for every y ∈ R, we have (yi)σ−1 ((yi + 1)σ − (yi − 1)σ ) (yi)σ (yi + 1)σ + 2(yi + 1)σ (yi − 1)σ + (yi)σ (yi − 1)σ < 1. It is not difficult to see that applying the triangle inequality repeatedly to the inequality
(6.3)
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1 |yi|σ−1 |yi + 1|σ < σ σ σ σ σ σ |yi| |yi + 1| + 2|yi + 1| |yi − 1| − |yi| |yi − 1| 2
(6.4)
and taking into account the equality |yi + 1|σ = |yi − 1|σ ,
(6.5)
we have that inequality (6.4) yields inequality (6.3). Therefore, it suffices to prove inequality (6.4). Note that |yi| < |yi + 1| and |yi + 1| ≥ 1. Hence, for all σ ≥ 2, we have |yi|σ−1 < |yi + 1|σ−1 ≤ |yi + 1|σ . Therefore, |yi|σ−1 <
|yi|σ |yi|σ + |yi + 1|σ − . 2 2
Using (6.5), we see that |yi|σ−1 |yi + 1|σ <
=
1 1 |yi|σ |yi + 1|σ + |yi − 1|σ |yi + 1|σ − |yi|σ |yi − 1|σ 2 2 1 (|yi|σ |yi + 1|σ + 2|yi + 1|σ |yi − 1|σ − |yi|σ |yi − 1|σ ) , 2
which is exactly inequality (6.4). 2 We now complete the proof of Theorem 6.1. The inequality of the Lemma 6.2 implies that all points on the imaginary axis tend to zero under iteration of Mf, σ . Thus the imaginary axis is in the immediate basin of attraction of 0 and, consequently, in the Fatou set. Therefore, the imaginary axis splits the Riemann sphere into two components, each of which contains components of the Julia set, namely the boundaries of the immediate basins of attraction of z = 1 and z = −1. It follows that J(Mf,σ ) is disconnected. The proof is now complete. 2 Remark 6.3. Consider the one parameter family obtained by applying Mf, σ to fλ (z) = (z 2 + λ)(z 2 − 1). The previous result shows that the parameter λ = 0 is in the complement of the connectedness locus. The parameter λ = 0 is not structurally stable for Mfλ , σ , since the super-attracting fixed point 0 splits in two super-attracting fixed point (this fact is unrelated with the connectedness of the Julia set: in fact, in the quadratic family occurs the same situation). However when λ ∼ 0, the Julia set of Mfλ , σ seems to be connected, at least for σ = 2. Hence non-connectedness apparently does not persists under perturbations of λ. Up to now, we are unable to prove such phenomena. Remark 6.4. Note that in particular our result implies that Newton’s method and König’s method applied to rational maps is not always connected. Acknowledgements The author would like to thank Juan Rivera-Letelier and Pancho Valenzuela-Henriquez for helpful discussions and Mark McClure for the help with graphics.
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Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Dynamics and limiting behavior of Julia sets of König’s method for multiple roots ✩ Gerardo Honorato CIMFAV–Facultad de Ingeniería, Universidad de Valparaíso, Chile
a r t i c l e
i n f o
Article history: Received 22 November 2016 Received in revised form 31 July 2017 Accepted 14 October 2017 Available online 26 October 2017 MSC: primary 65H04, 37F10, 30D05 secondary 37F50 Keywords: Connectedness Complex dynamics Root-finding algorithms
a b s t r a c t A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f , there exists at least one point in Sd converging under iterations of Nf to α. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton’s method. We show that for all order σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of König’s method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Newton’s iterative method Nf = Id −
f f
(1.1)
where f is a complex polynomial, as well as other higher-order methods have been extensively studied and used in order to deal with the equation f (z) = 0 (see for example [1], [17], [32], [33], [36], and [49]). The iterative function Nf defines a rational map on the Riemann sphere C = C ∪ {∞}. The simple roots of the equation f (z) = 0, that is the roots of the equation f (z) = 0 which are not roots of the derivative f (z), are super-attracting fixed points of Nf . In other words, let α be a simple root of f (z). Then Nf (α) = α and Nf (α) = 0. For a review of the dynamics of Newton’s method, see for example [11], [24]. In a more ✩
This work was supported by MATHAMSUD 16-MATH-06 PHYSECO and CNPq (The Brazilian National Research Council). E-mail address: [email protected].
https://doi.org/10.1016/j.topol.2017.10.013 0166-8641/© 2017 Elsevier B.V. All rights reserved.
G. Honorato / Topology and its Applications 233 (2018) 16–32
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general setting, we let P olyd and Ratk denote the space of polynomials of degree d and the space of rational functions of degree k, respectively. A root-finding algorithm is a rational map Tf : P olyd → Ratk such that the roots of the polynomial map f are attracting fixed points of Tf . We will say that a root-finding algorithm Tf has order of convergence σ if the local degree of Tf at each simple root of f is σ. E. Schröder ([44], [45], 1870) showed that Newton’s method applied to quadratic polynomials in the complex plane has two basins of attraction associated with the corresponding roots. His proof is obtained by conjugating Newton’s method for a quadratic polynomial f with the so-called Newton’s method for multiple roots associated to f defined by Mf (z) = z −
f (z)f (z) . (f (z))2 − f (z)f (z)
(1.2)
In fact, for instance conjugating Newton’s method with a Möbius transformation sending one root to infinity and the other one to 0, we obtain the map z 2 which proves immediately the existence of two connected basins of attraction. More precisely, the basins of attraction associated to Newton’s method applied to the polynomial f (z) = z 2 − 1 are found explicitly by noting that Nf =
1 . Mf
Thus, it suffices to concentrate on the convergence of the iterates under Mf . Consequently, the points on the complex plane with positive real part converge to 1, while those with negative real part converge to −1. For a different approach, see A. Cayley ([14], 1890). For a historical background and detailed explanation of these facts, see [3], [6]. More recently, Newton’s method for multiple roots has also been studied by T. Pomentale [38] and W. Gilbert [22]. A natural generalization of Newton’s method (1.1) is König’s root-finding algorithm applied to polynomials, denoted Kf, σ , also known as the Basic Family (in short, König’s method). Let f : C → C be a meromorphic map, and let σ ≥ 2 be an integer. König’s method of order σ associated to f is the meromorphic map Kf, σ : C → C defined by Kf,σ = Id + (σ − 1)
(1/f )[σ−2] (1/f )[σ−1]
(1.3)
where (1/f )[k] is the kth derivative of 1/f . In the special cases σ = 2 and σ = 3, König’s method is Newton’s method and Halley’s method, respectively. König’s method applied to polynomials has been studied from both the numerical and the dynamical points of view. (See for example, [4], [13], [28], [29], [37], [50].) Here we are interested in studying dynamical and geometric aspects of König’s method for multiple roots (see Definition 2.1), denoted Mf, σ , which is a generalization of Newton’s method for multiple roots. Given a polynomial f , König’s method for multiple roots Mf, σ is none other than König’s method Kf, σ applied to the rational map f /f . Note that the term f /f has the effect of converting the multiple roots of f (z) into simple ones. In other words, if f has a multiple zero of order n at α, then f /f has simple zero at α. Although Kf, σ and Mf, σ are similar, they are different from a dynamical viewpoint. For example, infinity is never a fixed point for Mf, σ , which is exactly the opposite for Kf, σ . In addition, the fixed points of Mf, σ associated to the roots of f are always super-attracting, regardless their multiplicity. We next state our two main results. In [13], X. Buff and C. Henriksen show that the sequence of Julia sets of König’s method applied to a polynomial f converges to the union of the bisecting locus of the set of roots of the polynomial f and infinity as σ → ∞. A Voronoi cell of a root α is the set of points in the complex plane that are strictly closer to α
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than to any of the other roots. A Voronoi diagram is the collection of all Voronoi cells. In [28], B. Kalantari shows that for generic polynomials, the basins of attraction of König’s method for polynomials approximate uniformly the Voronoi diagram of the roots. In this article, we show an analogous result for König’s method for multiple roots. More precisely, we show that the Fatou set of König’s method for multiple roots applied to polynomials converges in the Hausdorff complementary topology to the Voronoi diagram as σ → ∞. A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f , there exists at least one point in Sd converging under iterations of Nf to α. A question arising naturally from the result concerns whether it also holds for root-finding algorithms of order higher than Newton’s method. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton’s method. We show that for all order σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result of [27] to higher order root-finding methods. In [26] a theoretical proof of disconnectedness of the Julia set for König’s method applied to polynomials is given. As far as we know, there are no pictures showing that situation. Here we give a proof and also provide pictures that illustrate disconnectedness of the Julia set. The paper is organized as follows. In Section 2, we introduce basic notation and definitions, and we give results concerning the nature of fixed points. Section 3 presents a classification of the dynamics of König’s method for multiple roots applied to quadratic polynomials. Section 4 contains one of our main results, which involves Voronoi diagrams and the Fatou sets for König’s method for multiple roots applied to polynomials. In Section 5, we give a characterization of König’s method for multiple roots applied to polynomials. Section 6 is devoted to the conjugacy classes. Section 6 contains the main result: The Julia set of König’s method for multiple roots applied to polynomials is, in general, non-connected. 2. Basic notions and standard definitions We will first introduce basic notation and definitions concerning complex dynamics. P (z) Let R(z) = Q(z) be a rational map of the extended complex plane into itself, where P and Q are polynomials with no common factors. We say that α is a fixed point of R if R(α) = α. The multiplier of R at a fixed point α is the complex number λ(α) = R (α). In addition, we say that a fixed point α is attracting, repelling or indifferent if the modulus of its multiplier λ(α) is, respectively, less than, greater than or equal to 1. A super-attracting fixed point is a fixed point α of R for which R (α) = 0. For z ∈ C, we define its orbit as the set orb(z) = {z, R(z), R2 (z), . . . , Rk (z), . . .} where Rk means the k-fold iterate of R. The basin of attraction of a (super)attracting fixed point α of R is the set B(α) = {z ∈ C : Rn (z) → α as n → ∞}. In other words, the basin of attraction of a (super)attracting fixed point is the set consisting of the points whose iterates tend to the fixed point α as n tends to ∞. Let z0 be a fixed point of Rn which is not a fixed point of Rj , for any j with 0 < j < n. We say that orb(z0 ) = {z0 , R(z0 ), . . . , Rn−1 (z0 )} is a cycle of length n or simply an n-cycle. Note that orb(zj ) = orb(z0 ) for any zj ∈ orb(z0 ), and R acts as a permutation on orb(z0 ). The multiplier of an n-cycle is the complex n−1 n−1 number λ(orb(z0 )) = (Rn ) (z0 ). Note that by the Chain Rule, (Rn ) (zj ) = k=0 R (Rk (zj )) = k=0 R (zk ) where the second product is a re-arrangement of the first. Thus, at each point zj of the cycle, the derivative (Rn ) has the same value. An n-cycle {z0 , z1 , . . . , zn−1 } is said to be attracting, repelling, indifferent if the modulus of its multiplier λ(orb(z0 )) is, respectively, less than, greater than, equal to 1. An n-cycle Ω = {z0 , z1 , . . . , zn−1 } is said to be super-attracting if λ(orb(z0 )) = 0. If z0 is in an attracting n-cycle of R, the basin of attraction of the orbit orb(z0 ), denoted B(orb(z0 )), is j n the set ∪n−1 j=0 R (B(z0 )), where B(z0 ) is the basin of attraction of z0 as an attracting fixed point of R . The Julia set of a rational map R, denoted J(R), is the closure of the set consisting of the repelling periodic points. Its complement, denoted F (R), is called the Fatou set. For further information, see [10], [35].
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We next give the relevant definitions concerning the root-finding algorithms we are interested in studying and prove some preliminary results Definition 2.1. Let f : C → C be a meromorphic map, and let σ ≥ 2 be an integer. We define König’s method for multiple roots of order σ associated to f , denoted Mf, σ : C → C, as König’s method applied to the meromorphic map f /f . In other words, we have Mf,σ = Kf /f , σ = Id + (σ − 1)
(f /f )[σ−2] (f /f )[σ−1]
(2.1)
where (f /f )[k] is the kth derivative of f /f . When σ = 2, König’s method for multiple roots is known as Newton’s method for multiples roots given in formula (1.2). The following concerns the nature of fixed points of König’s method applied to a meromorphic map. For a proof see [13]. Proposition 2.1. Let f : C → C be a meromorphic map. Let αi denote a zero of f , and let ni ≥ 1 denote its multiplicity. Then for any integer σ ≥ 2, the fixed points of König’s method Kf, σ : C → C are either (super)attracting or repelling. Furthermore: (1) The (super)attracting fixed points are exactly the zeroes αi an their multiplicities are 1 − (σ − 1)/(ni + σ − 2). When ni = 1, the local degree of Kf, σ at αi is at least equal to σ. (2) The extraneous fixed points of Kf,σ are exactly the zeroes of (1/f )[σ−2] . If βj is a zero of (1/f )[σ−2] with multiplicity mj , then it is a repelling fixed point of Kf, σ with multiplier 1 + (σ − 1)/mj . The equivalent statement for Mf,σ is as follows. Proposition 2.2. Let f : C → C be a polynomial. Let αi denote a zero of f , and let ni ≥ 1 denote its multiplicity. Then for any integer σ ≥ 2, the fixed points of König’s method for multiple roots Mf, σ : C → C are either super-attracting or repelling. Furthermore: (1) The super-attracting fixed points are exactly the zeroes αi . (2) The extraneous fixed points of Mf,σ are exactly the zeroes of (f /f )[σ−2] . If βj is a zero of (f /f )[σ−2] with multiplicity mj , then it is a repelling fixed point of Mf, σ with multiplier 1 + (σ − 1)/mj . Proof. Let f be a complex polynomial. Applying Proposition 2.1 to the rational map f (z) ni = f (z) z − αi i=1 N
(2.2)
where ni are the multiplicities of αi , the result follows. 2 Proposition 2.3. Let f : C → C be a polynomial of degree d with N zeroes. Let αi denote a zero of f , and let ni ≥ 1 denote its multiplicity. Then: (1) Mf, σ has at most (σ − 1)(N − 1) repelling fixed points in C. (2) The degree of Mf, σ is at most σ(N − 1). (3) The local degree of Mf, σ at αi , is at least equal to σ.
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Proof. Let f be a complex polynomial. For statement (1), consider the formula (2.2). Note that the repelling fixed points of Mf, σ are the zeroes of g(z) =
f f
[σ−2] (z) = (−1)(σ−2) (σ − 2)!
N i=1
ni . (z − αi )σ−1
(2.3)
To compute the zeroes of g it suffices to compute the poles of g, since a rational map has the same number of zeroes and poles in C. It follows from formula (2.3) that the zeroes of f contribute with N (σ − 1) poles of g. On the other hand, g has a zero of order σ − 1 at ∞. In fact at ∞, we have that f takes the form λz d + ao z d−1 + a1 z d−2 + · · · . Thus g(z) =
f f
[σ−2] (z) =
λ z σ−1
1 + O(|z|−1 ) .
Therefore, Mf, σ has at most N (σ − 1) − (σ − 1) = (N − 1)(σ − 1) repelling fixed points in C. Concerning (2), recall that, in general, the degree of Mf, σ minus 1 is equal to the number of fixed points of Mf, σ . It follows from statement (1) that the degree of Mf, σ is at most [N + (N − 1)(σ − 1)] − 1 = σ(N − 1). It remains to prove (3). Suppose that α is a simple root of f . Again, using formula (2.3) we obtain that
f f
[σ−2] (z) =
λ + O(1) (z − α)σ−1
(2.4)
for some complex number λ = 0. Hence Mf, σ (z) = z + (σ − 1)
(f /f )
[σ−2]
(z)
[σ−1]
(z)
(f /f )
= α + (z − α) + (σ − 1)
(λ/(z − αi )σ−1 )(1 + O(z − α)σ−1 ) −(λ/(z − αi )σ )(1 + O(z − α)σ )
= α + O(z − α)σ . Consequently, the local degree of Mf, σ is at least equal to σ at the root α.
2
Remark 2.2. It follows from formula (2.4) that the degree of the denominator of Mf, σ is greater than or equal to the degree of the numerator. Thus, ∞ cannot be a fixed point of Mf, σ . Now we will classify König’s method for multiple roots applied to quadratic polynomials. The particular case of σ = 2 was studied by E. Schröder, noting that Kf,σ and Mf,σ are conjugated by 1/z. (See [45], [3].) Proposition 2.4. Let f : C → C be a quadratic polynomial. Then: (1) If f has a double root at a ∈ C, then König’s method for multiple roots is the constant map z → a. (2) If f has two simple roots, König’s method for multiple roots is conjugated to the map w → wσ . Proof. Statement (1) is a straightforward computation.
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Fig. 1. The Julia set of Mf, σ applied to the polynomial f (z) = z(z 2 − 1)(z − i) for σ = 2, 10, 30, respectively.
Concerning (2), we look at the solutions of the equations Mf, σ (z) = a and Mf, σ (z) = b, where a and b are the zeroes of f . The solutions of the first equation is a along with its multiplicities. Indeed, according to the formula (2.3) the equation Mf, σ (z) = a is equivalent to z−
(z − a)(σ−1) + (z − b)(σ−1) (z − a)σ + (z − b)σ
(z − a)(z − b) = a ,
which has the root a with multiplicity σ as a solution. We have a similar result if we consider the simple root b. Therefore, we have shown that a and b are exceptional points for Mf, σ , for every σ ≥ 2. If this is the case, then Mf,σ is conjugated to z σ for every σ ≥ 2. (See for example [10].) 2 3. Voronoi diagrams for König’s method for multiple roots This section will prove that the Fatou set of König’s method for multiple roots converges in the Hausdorff complementary topology to the Voronoi diagram as σ → ∞. (See Fig. 1.) Consider a given finite set of points in the complex plane, say S = {α1 , α2 , . . . , αd }. Definition 3.1. A Voronoi cell for a point αk or Voronoi face of the site αk is the set V (αk ) = {z ∈ C | d(z, αk ) < d(z, αj ), for all j = k}. The Voronoi diagram of S is defined as the set
V =
d
V (αi ).
i=1
We take the definition from [18]. Voronoi diagrams and their applications have been extensively studied. (See, for example, [7], [18], [19], [23], [39].) Let f be a degree d ≥ 2 complex polynomial with at least two different roots. Here we will consider S as the set of roots of f (some of them are multiple). In [13], X. Buff and C. Henriksen show that the sequence of Julia sets of König’s method applied to a polynomial f converges to the union of the bisecting locus of the set of roots of the polynomial f and infinity as σ → ∞. In [28], B. Kalantari shows that for generic polynomials, the basins of attraction of König’s method for polynomials approximate uniformly the Voronoi diagram of the roots. In addition, he shows that none of basins of attractions associated with a root-finding
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algorithm may be realized as a Voronoi cell. Unlike in Kalantari’s proof, our arguments depend on a result of E. Schröder, which is improved in [13], and we allow multiple roots for the polynomial. Definition 3.2. Let A and B be subsets of a metric space (X, d). The Hausdorff metric is defined as dH (A, B) = max{sup d(a, B), sup d(b, A)}. a∈A
b∈B
We may assume that we are working either with the euclidean metric or the spherical metric, since they are uniformly equivalent. Let K be the set of all non empty compact subsets (strictly contained) of the sphere C. Then it is known that equipped with the Hausdorff metric, K is a complete metric space. For a more detailed discussion of Hausdorff metric, see [9]. We next record a useful characterization of convergence in Hausdorff metric which we will use to prove Theorem 3.6 below. For a proof the reader can consult, for example, [9]. Lemma 3.3. The Hausdorff limit A of {Am } is also the set of all accumulation points of the sequences am such that am ∈ Am , for all m ≥ 1. As for the topology on the class of open sets, we introduce the Hausdorff complementary metric. We set Ω = {A ⊂ C, A is a nonempty open set}. Note that there exists a bijection between Ω and K. Thus, we can define the metric as follows. Definition 3.4. The Hausdorff complementary metric on the set Ω is given by dcH (A, B) := dH (C \ A, C \ B) where A and B belong to Ω. It follows immediately that (Ω, dcH ) is complete metric space. For a review of Hausdorff complementary metric, see for example [30] and references therein. The next Lemma provides a useful tool in order to prove convergence depending on the order σ. For a proof the reader can consult [13]. Lemma 3.5. Let D be a Euclidean disk centered at z0 and α be a point of D. Assume that g is meromorphic on D and has a unique pole, possibly multiple at α. Then, there exists a neighborhood of Uz0 of z0 on which the sequence of functions (n + 1)g [n] /g [n+1] converges uniformly to the function z → α − z. Considering g = f /f and recalling that a zero of f is a pole for the meromorphic map g [n] , we have the following immediate consequence of this Lemma, which is essential for our purpose. Proposition 3.1. Let f be a polynomial and D a disk centered at z0 . Assume that f has unique zero, possibly multiple α ∈ D. Then, Mf,σ converges uniformly on a neighborhood Uz0 of z0 contained in D, to the constant function α as σ → ∞. Theorem 3.6. Let f be a polynomial of degree d ≥ 2. Then lim F (Mf, σ ) = V
σ→∞
where F (Mf, σ ) is the Fatou set of Mf, σ and the convergence is in Hausdorff complementary metric.
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Proof. Let f be a degree d polynomial with roots αi , where i = 1, . . . , d. Note that for every σ ≥ 1, we have that F (Mf, σ ) is non-empty, since it contains the roots of f . Fix a root of f , say α = α1 . For simplicity we will denote as Bσ (α) the basin of attraction of the fixed point α of Mf, σ , whenever σ ≥ 1. We first prove the following Lemma 3.7. V c (α) = {z ∈ C : z is an accumulation point of a sequence zσ ∈ Bσc (α), σ ≥ 1}. Proof. Let z be in the complement of the Voronoi cell of α, V c (α). Since V (α) is an open set, V c (α) is compact subset of C. We first prove that z is an accumulation point of a sequence zσ ∈ Bσc (α) for σ ≥ 1. The discussion will be divided into two cases, according to the position of z. Case 1. If z ∈ ∂V (αi ) for some i = 1, . . . , d, then |z − αi | = |z − αk | for some k = 1, . . . , d, where k = i. Let D be a spherical disk of radius and centered at z. In this case we claim that the Julia set J(Mf,σ ) intersects D for σ sufficiently large. Indeed, let z1 and z2 be two points in D which belong to different components of the complement of ∂V (αi ). By interchanging the roles of z1 and z2 if necessary, we may assume that αi is the root of f closest to z1 and αk is the root of f closest to z2 . By Proposition 3.1 there exists a neighborhood Uz1 of z1 on which the sequence Mf, σ converges uniformly to αi , for σ sufficiently large. Thus, there exists a positive integer n1 such that Mf, σ (Uz1 ) ⊂ Bσ (αi ) for σ ≥ n1 . This shows that z1 is in the basin of attraction of αi of Mf, σ when σ ≥ n1 . Similarly, there exists a positive integer n2 such that z2 is in the basin of attraction of αk of Mf, σ , for σ ≥ n2 . Therefore the Julia set J(Mf,σ ) intersects D for σ ≥ max{n1 , n2 } = n3 , as claimed. Now by shrinking D if necessary, we have that z is an accumulation point of a sequence zσ ∈ J(Mf, σ ) for every σ ≥ n3 . For every σ = 1, . . . , n3 − 1 we can choose an arbitrary point zσ in J(Mf, σ ). Considering the union of σ and σ , (and calling it σ for notational simplicity) we have that z is an accumulation point of the sequence zσ in J(Mf, σ ), σ ≥ 1. Since J(Mf, σ ) ⊂ Bσc (α), in particular we have that z is an accumulation point of a sequence zσ in Bσc (α) for σ ≥ 1. Case 2. If z belongs to the interior of V (αi ) for some i = 2, . . . , d, then by definition, αi is the closest root of f to z. The argument now proceeds similarly as above. By Proposition 3.1 there exists a neighborhood Uz of z and a positive integer n such that Mf, σ (Uz ) ⊂ Bσ (αi ) for σ ≥ n. Shrinking Uz if necessary, we obtain a sequence zσ ∈ Bσ (αi ) converging to z, for σ ≥ n. For every σ = 1, . . . , n − 1 we can choose an arbitrary point zσ in Bσ (αi ). In particular z is an accumulation point of a sequence zσ ∈ Bσ (αi ) ⊂ Bσc (α) for every σ ≥ 1. This completes the proof of Case 2. Suppose now that z is an accumulation point of a sequence zσ ∈ Bσc (α) for σ ≥ 1. Then z cannot be in V (α) since otherwise, again by Proposition 3.1 there exists a neighborhood Uz of z and a positive integer n such that Mf, σ (Uz ) ⊂ Bσ (α) for σ ≥ n. Hence, zσ ∈ Bσ (α) for σ sufficiently large, which is a contradiction. Thus z is in V c (α). This completes the proof of Lemma 3.7. 2 We now complete the proof of Theorem 3.6. As a consequence of the Lemma above and Lemma 3.3 we have that Vαc = limσ→∞ Bσc (α) in Hausdorff metric, which implies that Vα = lim Bσ (α), σ→∞
in Hausdorff complementary metric. We claim that d i=1
V (αi ) = lim F (Mf, σ ), σ→∞
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in the Hausdorff complementary metric. In order to prove this, again we consider the characterization in d Lemma 3.3. If z belongs to the complement of i=1 V (αi ), then for some i, j = 1, . . . , d, we have that |z − αi | = |z − αj |. Now we apply the same argument of Case 1 in the preceding Lemma to obtain a sequence zσ in J(Mf, σ ), σ ≥ 1. In particular the sequence zσ belongs to F c (Mf, σ ), σ ≥ 1. This shows that z is accumulated by elements in the complement of F (Mf, σ ). On the other hand, suppose that z is an accumulation point of a sequence zσ ∈ Fσc (α) for σ ≥ 1. Then d z cannot be in i=1 V (αi ), since otherwise by Proposition 3.1 there exists a neighborhood Uz of z and a positive integer n such that Mf, σ (Uz ) ⊂ Bσ (α) for σ ≥ n. In particular, zσ ∈ Fσ (α) for σ sufficiently d large, which is a contradiction. It follows that z is in the complement of i=1 V (αi ). This proves that d V = i=1 V (αi ) = limσ→∞ F (Mf, σ ) in Hausdorff complementary metric, as required. 2 We recall the definition of Bisecting locus. Definition 3.8. The bisecting locus B of the set S ⊂ C, is the set of points z ∈ C where the distance function δ(z) = d(z, S) is not differentiable. Remark 3.9. Note that z0 ∈ B if and only if |z0 − α1 | = |z0 − α2 | for some α1 , α2 ∈ S. In fact, let z0 be a point in the plane so that |z0 − α1 | = |z0 − α2 | for some α1 , α2 ∈ S. Let r = |z0 − α1 |. Without loss of generality, we may assume that z0 = (0, 0), and α1 , α2 are real numbers. Then ∂δ (0, 0) = lim h→0 ∂x
inf w∈S d((h, 0), w) − inf w∈S d((0, 0), w)
h r − |h| − r |h| = lim − . = lim h→0 h→0 h h
As a consequence δ is not differentiable in z0 , and so z0 ∈ B. Conversely, suppose that for z0 = α1 we have |z0 − α1 | < |z0 − αi |, for every i = 2, . . . , d. Then α1 is the element of S closest to z0 , that is, z0 ∈ V (α1 ). Since V (α1 ) is an open set, there exists > 0 such that B(z0 , ) ⊂ V (α1 ). So, for every z ∈ B(z0 , ) we have that δ(z) = |z − α1 |, which is differentiable in B(z0 , ), as a function of R2 . Thus z0 ∈ / B. As an immediate Corollary to this Remark and Theorem 3.6 we have the following: Corollary 3.1. Let f be a polynomial. The Voronoi diagram V coincides with the complement of the bisecting locus. Remark 3.10. If Mf, σ has an attracting periodic orbit not associated with the roots, it will disappear when σ goes to infinity: if z is in the basin of attraction of this periodic orbit, according to the position of z, we can use a similar argument as in the proof of Lemma 3.7. Remark 3.11. Note that even for a polynomial g it is possible that a sequence of polynomials gm approximate g such that the respective Julia sets do not converge. See [31]. 4. Characterization of König’s method for multiple roots applied to polynomials This section will study the image of the map Mf,σ : P olyd → Ratk . More precisely, we are interested in finding conditions under which a rational map is König’s method for multiple roots applied to polynomials. This kind of result about Newton’s method applied to polynomials has been studied, to my knowledge, by
G. Honorato / Topology and its Applications 233 (2018) 16–32
25
G. Saunders [47], J. Head [25], K. Nishizawa and M. Fujimura [21], and more recently, J. Ruckert and D. Schleicher [42], E. Crane [15], P. Roesch [41]. König’s method applied to generic polynomials is studied by X. Buff and C. Henriksen [13]. We will need the following that treats the case of Newton’s method applied to rational maps. (See [15].) Proposition 4.1. A rational map F is Newton’s method applied to some rational map g if and only if all of the fixed points of F are simple and each fixed point α of F satisfies 1−F1 (α) ∈ Z. Moreover, g is a polynomial if and only if these integers are all positive. Proposition 4.2. Let R be rational map of degree d ≥ 2, with only super-attracting and repelling fixed points in C. Denote its super-attracting fixed points by αi , with i ≥ 1. Assume that any repelling fixed point βi d
has a multiplier of the form 1 + 1/mi , with mi ∈ N. Then R = Mf , where f (z) = (z − αi )ni , for some i=1
ni ∈ N. Proof. Given the rational map R, define the map F =
1 [σId − R]. (σ − 1)
We let χ denote the fixed points of R (whether they be attracting or repelling). A brief computation shows that, for every χ, F satisfies 1 ∈ Z. 1 − F (χ) According to Proposition 4.1, there exists a rational map g(z) = c that F = Id −
i (z − χ)
li
, where c ∈ C and li ∈ Z, such
g . g
Therefore, R = Id + (σ − 1)
g . g
Since R is a rational map with only super-attracting points αi and repelling fixed points βi in C, whose multipliers are 1 + 1/mi , with mi ∈ N, we have that (z − βi )mi g(z) = c i σ−1 i (z − αi )
(4.1)
where c ∈ C, c = 0. We write g := gσ to emphasize the dependence on σ. In order to prove Proposition 4.2, we need only show that gσ = (f /f )[σ−2] for some complex polynomial f . This is the content of the following Lemma 4.1. For every σ ≥ 2, gσ = (f /f )[σ−2] for some complex polynomial f . Proof. The proof will proceed by induction on σ. Define the map (z − αi ) 1 =z− i h(z) = z − . mi g2 (z) i (z − βi )
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The superattracting fixed point of h are αi and ∞, which is a repelling fixed point with multiplier d/(d − 1). Hence by Proposition 4.1, there exists a polynomial f such that h(z) = Nf . This implies that g2 =
f . f
As a consequence the assertion is true for σ = 2. According to the induction hypothesis, for σ = k there exists a complex polynomial f so that gk = ln(f )[k−1] . Here ln(f ) denotes the logarithmic derivative of f . It is not difficult to see that ln(f )[k] = (ln(f )[k−1] ) = (gk ) = gk+1 , which is precisely the formula for k + 1. This completes the induction. 2 This completes the proof of Proposition 4.2.
2
5. Conjugacy classes In this section we will prove a Scaling Theorem for König’s method for multiple roots. It is well known that if a root-finding algorithm satisfies a Scaling Theorem, then the dimension of the parameter space can be reduced. For instance in the case of Newton’s method applied to cubic polynomials, the dimension of the parameter space can be reduced to dimension 1. For a review of this kind of result, see [1]. The situation is different for polynomials of degree 4. (See [5], [16], [2].) In the best case the parameter space can be reduced to dimension 2. However under stronger assumptions one can restrict the study of the parameter space to dimension 1. More precisely, to the one parameter family fλ (z) = (z 2 + λ)(z 2 − 1).
(5.1)
Lemma 5.1. (Scaling Theorem) Let f be an analytic function on the Riemann sphere, and let A(z) = αz +β, with α = 0, be an affine map. If g(z) = f ◦ A(z), then for any integer σ ≥ 2, we have Mf, σ ◦ A = A ◦ Mg, σ . That is, König’s method for multiple roots of f is analytically conjugated to König’s method for multiple roots of g. Proof. Since g(z) = f ◦ A(z), we have
g g
[σ] =α
σ+1
f f
[σ] ◦ A.
Thus, A ◦ Mg,σ (z) = αz + (σ − 1)
(f /f )[σ−2] ◦ A(z) + β = Mf, σ ◦ A. (f /f )[σ−1]
This completes the proof. 2 Lemma 5.2. Let f and g be complex polynomials and A(z) = αz + β an affine map, where α = 0. Then A conjugates Mf, σ ◦ A = A ◦ Mg, σ if and only if there exists a complex number λ = 0 so that g = λf ◦ A.
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Proof. Assume that there exists a complex number λ = 0 such that g = λf ◦ A. It follows from Lemma 5.1 that Mf, σ ◦ A = A ◦ Mg, σ . Conversely, suppose that Mf, σ ◦ A = A ◦ Mg, σ . Define h = f ◦ A. Then, by Lemma 5.1 we have that Mg, σ = Mh, σ . Hence the polynomials g and h have the same roots with same multiplicities. Indeed, if α is a root of g which is not a root of h, by Proposition 2.2 α is a superattracting fixed point of Mg, σ . In particular α is a superattracting fixed point of Mh, σ , which by Proposition 2.2 corresponds to a zero of h, yielding a contradiction. Now suppose that α is a root of g with multiplicity n ≥ 1, and also a root of h with multiplicity m ≥ 1, where n = m. Without loss of generality, we may suppose that g(z) = (z − α)n f (z) and h(z) = (z − α)m f (z), for some polynomial f such that f (α) = 0. By formula (2.3) we have that n g (z) = + fˆ(z), g(z) (z − α) where fˆ is analytical on a neighborhood of α. Similarly for h we have m h (z) = + fˆ(z). h(z) (z − α) Since Mg, σ = Mh, σ for every σ ≥ 2, in particular for σ = 2 we have that (h /h) (g /g) = . (g /g) (h /h) Therefore, (m + (z − α)fˆ(z))(z − α) (n + (z − α)fˆ(z))(z − α) = , −n + fˆ (z)(z − α)2 −m + fˆ (z)(z − α)2 for all z ∈ C. It follows from this formula that n = m, a contradiction. As a consequence, there exists a complex number λ = 0 so that g = λf ◦ A. 2 6. Topology of the Julia set for König’s method for multiple roots A well known result of J. Hubbard, D. Schleicher and S. Sutherland shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f , there exists at least one point in Sd converging under iterations of Nf to α. (See [27].) For an improvement of this result, see for example [12], [43]. In order to construct the set Sd , it is essential that the immediate basins of attraction of Newton’s method, denoted Bα , be simply connected. This fact provides a source of geometric control. Indeed, there exists a conformal isomorphism ϕ : D → B(α) whose composition with Newton’s method yields a map acting on the open unit disk, that is, h = ϕ−1 ◦ Nf ◦ ϕ. The map h extends to the complex plane as a Blaschke product. A channel of a root α is an unbounded connected component W of B(α) \ D with the additional property that there is a w ∈ W that can be connected to Nf (w) by a curve in W . Now, there exists a neighborhood of infinity where Newton’s method is conjugated to a linear map. This implies that infinity is accessible from all of the basins of attraction of the roots. It follows that every access to ∞ of
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B(α) corresponds to a unique channel of α. Thus, when basins of attraction are simply connected, accesses and channels are in bijection. We may summarize the usefulness of the simply connectedness of basins of attraction in the following two key points:
Each immediate basin B(α) has exactly mα distinct accesses to infinity, where mα is the number of critical points of Nf in B(α).
Measuring the width of channels. The measure is given in terms of the conformal modulus of the annulus formed by the channel modulo the dynamics. Simple connectedness of basins of attraction for Newton’s method for polynomials is shown by F. Przytycki in [40]. He proves that if α is an attracting fixed point for Newton’s method Nf , then B(α) is simply connected. A stronger result due to M. Shishikura establishes that a rational function of degree greater than one has exactly one repelling (or weakly repelling) fixed point, then its Julia set is connected. As a corollary the Julia set of Newton’s method of a complex polynomial is connected. (See [46].) The case of polynomials of degree three is studied by H. Meier in [34] and L. Tan in [48]. In [20] is proved that the immediate basins of attraction associated to the Halley’s method applied to generic polynomials with real roots and real coefficients, are simply connected. Recently, Shishikura’s result is extended to entire maps by K. Baranski, N. Fagella, X. Jarque and B. Karpinska [8]. In this section, we show that for all orders σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to it is disconnected. For an illustration, see Fig. 2. Therefore, Przytycki’s and Shishikura’s results does not apply for König’s method for multiple roots when σ ≥ 2. Consequently, our result establishes restrictions for extending the proof of the main result of J. Hubbard, D. Schleicher and S. Sutherland [27] to higher order root-finding algorithms. In fact, a disconnected Julia set implies that several accesses may correspond to the same channel. In [26] we prove an analogous result for Kf,σ . The proof depends on the fact that Kf,σ has a unique repelling fixed point at ∞, which is no longer true for Mf,σ . Thus, we cannot repeat the proof in our context. In this article we give a proof and provide pictures that illustrate disconnectedness of the Julia set. Additionally, our proof is simpler and more direct. In fact, we give an explicit polynomial f such that the imaginary axis is completely contained in the Fatou set of Mf,σ , when σ ≥ 2. Now, the imaginary axis splits the Riemann sphere into two components, each of which contains components of the Julia set. Consequently, the Julia set is disconnected. The following is the main result of this section. Theorem 6.1. For all order σ ≥ 2, there exists a complex polynomial f such that the Julia set of König’s method for multiple roots applied to this polynomial is not connected. Proof. Let σ ≥ 2. Consider the family (5.1) fλ (z) = (z 2 + λ)(z 2 − 1). Set λ = 0. König’s method for multiple roots applied to the polynomial f0 (z) = z 2 (z 2 − 1) is given by
Mf, σ (z) =
z σ [(z + 1)σ − (z − 1)σ ] . z σ (z + 1)σ + 2(z + 1)σ (z − 1)σ + z σ (z − 1)σ
(6.1)
According to Lemma 2.3, the repelling fixed points are the zeroes of the equation
f f
[σ−2] (z) = 0.
(6.2)
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Fig. 2. In counterclockwise direction: the Julia set of Mf, σ applied to the polynomial f (z) = z 4 − z 2 for σ = 3, 4, 20 and σ = 100, respectively.
Since Mf, σ is an odd function, there are at least two repelling fixed points which are symmetric with respect to the imaginary axis. We will show that iR ∪ {∞} is invariant under the action of Mf, σ (see Fig. 2). In fact for every y ∈ R, we have Mf, σ (iy) = i Mf, σ (y) where Mf, σ is the map defined by Mf, σ (y) =
i(yi)σ ((yi − 1)σ − (yi + 1)σ ) . (yi)σ (yi + 1)σ + 2(yi + 1)σ (yi − 1)σ + (yi)σ (yi − 1)σ
Since Mf, σ (y) = Mf, σ (y), it follows that Mf, σ is a real map. From Mf,σ (1/z) |z=0 = 0 it follows that infinity belongs to the immediate basin of attraction of 0. Thus, Mf is the restriction to the imaginary axis of the map Mf, σ . Consequently, the imaginary axis is invariant under Mf, σ . Next, we will show that iR ∪ {∞} is contained in the immediate basin of attraction of 0. Lemma 6.2. For all y ∈ R \ {0}, we have |Mf, σ (y)| < |y|. Proof. Clearly, this assertion is equivalent to showing that for every y ∈ R, we have (yi)σ−1 ((yi + 1)σ − (yi − 1)σ ) (yi)σ (yi + 1)σ + 2(yi + 1)σ (yi − 1)σ + (yi)σ (yi − 1)σ < 1. It is not difficult to see that applying the triangle inequality repeatedly to the inequality
(6.3)
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1 |yi|σ−1 |yi + 1|σ < σ σ σ σ σ σ |yi| |yi + 1| + 2|yi + 1| |yi − 1| − |yi| |yi − 1| 2
(6.4)
and taking into account the equality |yi + 1|σ = |yi − 1|σ ,
(6.5)
we have that inequality (6.4) yields inequality (6.3). Therefore, it suffices to prove inequality (6.4). Note that |yi| < |yi + 1| and |yi + 1| ≥ 1. Hence, for all σ ≥ 2, we have |yi|σ−1 < |yi + 1|σ−1 ≤ |yi + 1|σ . Therefore, |yi|σ−1 <
|yi|σ |yi|σ + |yi + 1|σ − . 2 2
Using (6.5), we see that |yi|σ−1 |yi + 1|σ <
=
1 1 |yi|σ |yi + 1|σ + |yi − 1|σ |yi + 1|σ − |yi|σ |yi − 1|σ 2 2 1 (|yi|σ |yi + 1|σ + 2|yi + 1|σ |yi − 1|σ − |yi|σ |yi − 1|σ ) , 2
which is exactly inequality (6.4). 2 We now complete the proof of Theorem 6.1. The inequality of the Lemma 6.2 implies that all points on the imaginary axis tend to zero under iteration of Mf, σ . Thus the imaginary axis is in the immediate basin of attraction of 0 and, consequently, in the Fatou set. Therefore, the imaginary axis splits the Riemann sphere into two components, each of which contains components of the Julia set, namely the boundaries of the immediate basins of attraction of z = 1 and z = −1. It follows that J(Mf,σ ) is disconnected. The proof is now complete. 2 Remark 6.3. Consider the one parameter family obtained by applying Mf, σ to fλ (z) = (z 2 + λ)(z 2 − 1). The previous result shows that the parameter λ = 0 is in the complement of the connectedness locus. The parameter λ = 0 is not structurally stable for Mfλ , σ , since the super-attracting fixed point 0 splits in two super-attracting fixed point (this fact is unrelated with the connectedness of the Julia set: in fact, in the quadratic family occurs the same situation). However when λ ∼ 0, the Julia set of Mfλ , σ seems to be connected, at least for σ = 2. Hence non-connectedness apparently does not persists under perturbations of λ. Up to now, we are unable to prove such phenomena. Remark 6.4. Note that in particular our result implies that Newton’s method and König’s method applied to rational maps is not always connected. Acknowledgements The author would like to thank Juan Rivera-Letelier and Pancho Valenzuela-Henriquez for helpful discussions and Mark McClure for the help with graphics.
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