Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II

We consider Walsh's conformal map from the exterior of a set $E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when $E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where $P$ is an algebraic polynomial of degree $n$. Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when $E$ is the union of $\ell$ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.


Introduction
In his 1956 paper [23], Walsh obtained a canonical generalization of the Riemann mapping theorem from simply connected domains to multiply connected domains.In his construction, an -connected domain in C := C ∪ {∞} is mapped onto the exterior of a generalized lemniscate, as indicated in the following theorem.
Theorem 1.1.Let E 1 , . . ., E ⊆ C be disjoint, simply connected, infinite compact sets and let that is, E c = C \ E is an -connected domain.Then there exists a unique compact set of the form where a 1 , . . ., a ∈ C are distinct and m 1 , . . ., m > 0 are real numbers with j=1 m j = 1, and a unique conformal map If E is bounded by Jordan curves, then Φ extends to a homeomorphism from E c to L c .
The compact set L in (1.2) consists of disjoint compact components L 1 , . . ., L , with a j ∈ L j for j = 1, . . ., .The components L 1 , . . ., L are labeled such that a Jordan curve surrounding E j is mapped by Φ onto a Jordan curve surrounding L j .The centers a 1 , . . ., a and the exponents m 1 , . . ., m in Theorem 1.1 are uniquely determined.The domain L c is the exterior of the generalized lemniscate {w ∈ C : |U (w)| = cap(E)} and is usually called a lemniscatic domain; see [5, p. 106].Here, cap(E) denotes the logarithmic capacity of E.
Walsh's conformal map onto a lemniscatic domain is a canonical generalization of the Riemann map.Indeed, if = 1 in Theorem 1.1, i.e., if E is simply connected, the exterior Riemann map R E : E c → D c , uniquely determined by the normalization R E (z) = d 1 z + d 0 + O(1/z) at infinity with d 1 > 0, satisfies R E (z) = d 1 Φ(z) + d 0 , which follows from [23,Thm. 4]; see also [18,Rem. 1.2].Thus, R E and Φ are related by a simple linear transformation, and L = {w ∈ C : |w − a 1 | ≤ cap(E)} is a disk, where a 1 = −d 0 /d 1 and cap(E) = 1/d 1 .
The reason why we are interested in Walsh's conformal map is that the lemniscatic domain L c in (1.2) has a very simple form and is in particular (the exterior of) a classical lemniscate if the exponents m 1 , . . ., m in (1.2) are rational.In addition, as in the case of the Riemann map where the Green's function for the complement of the unit disk is simply log|w|, also the Green's function for the complement of L in (1.2) has the simple form and g E (z) = g L (Φ(z)) holds with Φ from Theorem 1.1.Moreover, Walsh's conformal map allows the construction of Faber-Walsh polynomials on sets E with several components as in Theorem 1.1, generalizing the well-known Faber polynomials and the classical Chebyshev polynomials of the first kind; see Walsh's original article [24], the book of Suetin [21], and the article [20].After Walsh's seminal paper [23], further existence proofs of Theorem 1.1 were published by Grunsky [3,4,5], Jenkins [6], and Landau [10].The latter contains an iteration for computing Walsh's map, but it requires knowledge of the harmonic measure of the boundary.None of these papers contain any explicit example, which might be the reason that Walsh's map has not been widely used so far.However, in Walsh's Selected Papers [25, pp. 374-377], Gaier recognizes Walsh's conformal map onto lemniscatic domains as one of Walsh's major contributions.The first explicit examples of Walsh's map were derived in [19] and applied in [20] for polynomial approximation on disconnected compact sets.In [13], Nasser, Liesen and the second author obtained a numerical method for computing Walsh's conformal map for sets E bounded by smooth Jordan curves.The method relies on solving a boundary integral equation (BIE) with the generalized Neumann kernel.This numerical algorithm also yields a method for the numerical computation of the logarithmic capacity of compact sets [11].In [18], we obtained a characterization of the exponents m 1 , . . ., m in terms of Green's function and derived explicit formulas and examples of conformal maps onto lemniscatic domains for polynomial pre-images E.
One main objective is the actual computation of the lemniscatic domain L, that is, the computation of the exponents m j and the centers a j , and of the conformal map Φ.In this paper, we consider this question for sets E that are polynomial pre-images of [−1, 1], i.e., E = P −1 ([−1, 1]), where P is an algebraic polynomial of degree n.The set P −1 ([−1, 1]) consists of components, 1 ≤ ≤ n, where each component consists of a certain number of analytic Jordan arcs [17].In particular, the components E j of P −1 ([−1, 1]) are not bounded by Jordan curves and thus the BIE method for computing Φ and L from [13] does not apply here.
This paper is the second of a series of papers.In the first part [18], we characterized the exponents as m j = n j /n, where n j is the number of zeros of the polynomial P in the component E j and n is the degree of the polynomial P .Moreover, we derived some general characterizations for the centers a 1 , . . ., a and considered the cases = 1 and = 2 in more detail.
Building on these results, the main contributions in this paper are the following.
(1) In Section 3, we consider sets E consisting of = 2 real intervals, or, more generally, of two components that are symmetric with respect to the real line.For these sets, we have explicit formulas for the centers a 1 , a 2 (Theorem 3.1).If, in addition, the set E is symmetric with respect to the origin then the corresponding Walsh map Φ can be given explicitly; see Theorem 3.3 and the following two examples.
(2) In Section 4, we consider sets E consisting of = 3 real intervals.In this case, the centers a 1 , a 2 , a 3 are the solution of a nonlinear system of three equations, which can be solved numerically.This is illustrated in Example 4.1 considering a two-parameter family of three intervals.If, in addition, the set E is symmetric with respect to the origin then a 1 , a 2 , a 3 can be given by an explicit formula.In either case, the conformal map Φ can be computed numerically by solving a polynomial equation and using the mapping properties of Φ in Theorem 2.1.
(3) In Section 5, we propose an iterative method for the computation of a 1 , . . ., a for sets E consisting of an arbitrary number ≥ 1 of real intervals, or, more generally, of components symmetric with respect to the real line.The key idea is that a 1 , . . ., a are the zeros of a certain polynomial with prescribed critical values but unknown critical points, with an additional constraint on a 1 , . . ., a .Once the centers a j of L are computed, the conformal map Φ can be evaluated numerically.The iterative method for the computation of a 1 , . . ., a works for an arbitrary number of intervals.In all our numerical examples, the method converges in very few iteration steps (at most 7 steps for up to 20 intervals) and returns highly accurate approximations of a 1 , . . ., a .In all examples where a 1 , . . ., a are known explicitly, the computed values have an error of the order of the machine precision.In those examples, where the exact values of a 1 , . . ., a are not known explicitly, the values obtained with our new method agree to order 10 −14 with those obtained by a modification of the BIE method from [13].This suggests that both methods are very accurate.
In Section 2, we first collect results on Walsh's conformal map Φ and the lemniscatic domain for general compact sets E with certain symmetries.Second, we recall known important facts on sets E which are polynomial pre-images.In Appendix A, a rather general result concerning symmetric sets is proven, which is needed for the proofs in Sections 3 and 4.
2 General Compact Sets and Polynomial Pre-Images

Results for General Compact Sets
For a set K ⊆ C, we denote as usual K * := {z : z ∈ K} and −K := {−z : z ∈ K}.If K ⊆ C is compact then the Green's function (with pole at infinity) of K c is denoted by g K .If E = E 1 ∪ . . .∪ E and E * j = E j for all components of E then we label the components "from left to right": By [18,Lem. A.2], each E j ∩ R is a point or an interval, and we label E 1 , . . ., E such that x ∈ E j ∩ R and y ∈ E j+1 ∩ R implies x < y for all j = 1, . . ., − 1.As a first result, let us collect some mapping properties of Φ if the set E has certain symmetries.Theorem 2.1.Let the notation be as in Theorem 1.1.
and Φ(z) is in the same quadrant as z.
(iv) Assume that E * j = E j for j = 1, . . ., .Let R \ E = I 0 ∪ . . .∪ I with open intervals I j ordered from left to right, and let similarly R \ L = J 0 ∪ . . .∪ J with open intervals J j ordered from left to right.Then Φ maps I j onto J j , that is Φ(I j ) = J j for j = 0, . . ., , and Φ is strictly increasing on R \ E, and in particular on each interval I j .
(v) Assume that E * j = E j for j = 1, . . ., .Then the Green's function g E has − 1 critical points z 1 , . . ., z −1 ∈ C \ E. These are simple and satisfy z j ∈ I j for j = 1, . . ., − 1.Moreover, Φ(z j ) = w j ∈ J j are the critical points of g L .In particular, if z ∈ I j with z < z j (resp.z > z j ) then w = Φ(z) ∈ J j with w < w j (resp.w > w j ), and (2.1) In the next theorem, we consider sets E that are symmetric with respect to the origin and investigate the effect on the exponents m j and the centers a j of the lemniscatic domain.
Theorem 2.2.Let E = ∪ j=1 E j be as in Theorem 1.1 with centers a 1 , . . ., a and exponents m 1 , . . ., m of L. If E = −E and if j 1 , j 2 are such that E j 2 = −E j 1 , then m j 2 = m j 1 and a j 2 = −a j 1 .In particular, if E j = −E j , then a j = 0.Moreover, the set of critical points of g E is symmetric with respect to zero.
Proof.If E = −E then Φ(−z) = −Φ(z) by [19,Lem. 2.2] or [18, Lem.2.6], and g E (−z) = g E (z) since g E (z) and g E (−z) both satisfy the properties of the Green's function, hence, in particular, (∂ z g E )(−z) = −(∂ z g E )(z).This shows that the set of critical points of g E is symmetric with respect to zero.Next, let γ be a smooth closed curve in C \ E with wind(γ; z) = δ j 1 ,k for z ∈ E k , then −γ surrounds the component −E j 1 = E j 2 .We use [18,Thm. 2.3] and the substitution z = −u to obtain Similarly,

Results for Polynomial Pre-Images of an Interval
We consider polynomial pre-images of [−1, 1], that is, where P is a polynomial of degree n ≥ 1 with complex coefficients of the form Each component E j consists of a certain number of analytic Jordan arcs, see [17] for details, and E c is connected; see [18,Thm. A.4] or [17,Lem. 1 (viii)].All zeros of P are in E and each E j contains at least one zero of P , compare [18,Thm. 3.1].By [15, Proof of Thm.5.2.5], the Green's function of E c is given by where P (z) 2 − 1 has a branch cut along E and behaves as P (z) at ∞; compare also the beginning of [18,Sect. 3].Note that the critical points of g E are the critical points of P in C \ E. Again by [15, Thm.5.2.5], the logarithmic capacity of E is In the next theorem, we summarize results from [18] that are needed in the sequel.
Theorem 2.3.Let E be a polynomial pre-image as in (2.2).

(i)
The set E has connected components if and only if P has exactly − 1 critical points z 1 , . . ., z −1 (counting multiplicities) for which (ii) The exponents m j in Theorem 1.1 are given by where n j ≥ 1 is the number of zeros of P in E j .
Since the exponents m j are determined in Theorem 2.3 (ii), let us turn our attention to the determination of the centers a j .If all components E j of E are symmetric with respect to the real line, we know that a 1 , . . ., a are the solution of a certain nonlinear system of equations.
Equations (2.13)-(2.14)are a nonlinear system of equations for a 1 , . . ., a , provided that the critical points w 1 , . . ., w −1 of Q outside L can be expressed explicitly in terms of a 1 , . . ., a ; see (2.11).This is possible for = 2 and = 3, and we consider these cases in Sections 3 and 4. For larger , this is no longer practical.In Section 5, we develop a numerical method to compute a 1 , . . ., a for arbitrary ≥ 2.
Before we start this investigation, let us conclude this section with a theorem which gives a necessary and sufficient condition for a union of real intervals to be a polynomial pre-image of [−1, 1], and which follows from [14, Lem.2.1]; see also [16,Lem. 1].
Theorem 2.5.The set is the pre-image of [−1, 1] under a polynomial P of degree n, that is, if and only if c 1 , . . ., c 2n satisfy the following system of equations: Remark 2.6.(i) The set E in Theorem 2.5 is the union of disjoint intervals, where ∈ {1, 2, . . ., n}, that is, E has − 1 gaps.
(ii) With the help of c 1 , . . ., c 2n satisfying (2.17), the corresponding polynomial P of degree n (unique up to sign) can be given by Note that P (z) and −P (z) generate the same pre-image E.
(iii) Figure 1 shows the real graph of a polynomial of degree n = 7 whose pre-image consists of the = 4 disjoint intervals which has been computed with the help of (2.17) and (2.18).

Sets with Two Components
If the polynomial pre-image ) has two components that are symmetric with respect to the real line, the centers a 1 , a 2 are given explicitly in Theorem 3.1, which follows from the more general result in [18,Cor. 3.10].This covers the important case that E consists of two real intervals, that is Theorem 3.1.Let P be a polynomial of degree n ≥ 1 as in (2.3) with either real or purely imaginary coefficients such that Let n 1 , n 2 be the number of zeros of P in E 1 , E 2 , respectively, and let z 1 be the critical point of P in C \ E. Then the points a 1 , a 2 are real with a 1 < a 2 and are given by with the positive real n-th root in (3.1).
In the following example for a one-parameter family of two intervals, we apply the formulas of Theorem 3.1 in order to obtain the lemniscatic domain and compute the corresponding conformal map Φ.

and the critical point of
).Then, using the correct branch of the square root as indicated after (2.4), we obtain see Theorem 2.3 (iii).We use the mapping properties of Φ established in Theorem 2.1 to determine the correct value of w.The image of z √ 4) =: a 1 (0) and a 2 → 1/ 3 √ 4 =: a 2 (0), so that the corresponding set L "converges" to {w ∈ C : √ 4), compare Figure 2.However, the centers a 1 , a 2 of L do not converge to the center a 1 = 0 of the lemniscatic domain . This shows that a discontinuity in the connectivity (in our example from = 2 components to = 1 component) leads to a discontinuity in the centers.
If the polynomial P is real and even, then ) is symmetric with respect to the real line as well as symmetric with respect to the origin, and we obtain an explicit formula for the conformal map Φ.
Theorem 3.3.Let the polynomial P (z) = n j=0 p j z j with p n = 0 be real, even and assume that P has exactly one critical point z 1 with critical value ) while the other n − 2 critical points are in E. Then the following assertions hold: , and E = −E, E * = E, and 0 / ∈ E. In particular, only the following two cases can occur: , and and if with the positive real n-th root in both cases.For p 2 0 − 1, the positive branch is taken if p 0 > 1 and the negative branch if p 0 < −1.Moreover, in both cases, a 1 = −a 2 and with that branch of the n-th root and of the outer square root which yields positive real values of Φ(z) for sufficiently large z on the positive real line.
Proof.(i) The fact that E has = 2 components is an immediate consequence of Theorem 2.3 (i), since P has exactly one critical point outside E. Thus, E = E 1 ∪ E 2 with disjoint, simply connected, infinite compact sets E 1 , E 2 .Since P is real and even, we have E * = E and E = −E.By Theorem A.1 (ii) and Corollary A.2, assertion (i) follows.
(ii) and (iii): The number of zeros of P in E 1 and in E 2 is the same, i.e., n 1 = n 2 = n/2.Since P is even, z = 0 is a critical point of P .Since 0 / ∈ E, we have z 1 = 0 and thus We consider the two cases pointed out in (i).
Case 1: 2 ) n/2 and (iii) follows from Theorem 2.3 (iii).Note that P (z)/(2p n ) is a real polynomial with positive leading coefficient 1/2.Therefore, the complex roots have to be taken as indicated in the theorem.
Case 2: E * 1 = E 2 .We reduce this case to case 1 as follows.The polynomial P (z) := P (iz) also satisfies the assumptions of the theorem, and the set E = P −1 ([−1, 1])) = −iE falls under case 1, so that the corresponding set L and conformal map Φ : E c → L c are determined by the formulas in case 1.By [19, Lem.2.3], we have a 2 = i a 2 and Φ(z) = i Φ(−iz), which yields after a short calculation the formulas in case 2.
and n 1 = n 2 = 1.The critical point of P is z 1 = 0 / ∈ E, so that P satisfies the assumptions in Theorem 3.3.
which is in accordance with [19,Cor. 3.3].Figure 3 shows the sets E and L and the image of a grid under Φ.For the numerical evaluation of Φ(z) we use the modified formula with the main branch of the square root, and Φ(0) = 0; compare to [19,Thm. 3.1].

Sets with Three components
Let the polynomial P be given as in (2.3) and assume that E given in (2.2) has = 3 components, that is, with n 1 + n 2 + n 3 = n.Moreover, Q has two critical points w 1,2 in C \ L which are the solutions of see Theorem 2.3 (v).A short computation shows that If all three components E 1 , E 2 , E 3 of E are symmetric with respect to R, then a 1 , a 2 , a 3 are the solution of the non-linear system of equations (2.13)-(2.14) in Theorem 2.4, which can be solved numerically; see the following example.
is the polynomial pre-image of a polynomial P of degree n = 3 if and only if α, β, γ 1 , γ 2 satisfy the equations for k = 1 and k = 2. Simplifying this system gives Hence, for α, β > 0, α + β < 1, and γ 1 , γ 2 given by (4.6), the set (4.4) is a polynomial pre-image.By (2.18), the polynomial of degree n = 3 (unique up to sign) with ) is given by In particular, The critical points of P are given by Next, we compute a 1 , a 2 , a 3 for the set in (4.The Mathematica command NSolve returns six distinct triples (a 1 , a 2 , a 3 ), which are permutations of each other, where only one satisfies a 1 < a 2 < a 3 .Figure 5 illustrates the conformal map Φ.We compute w = Φ(z) by solving Q(w) = P (z) + P (z) 2 − 1 and determine the correct solution w (out of the n = 3 solutions) using the mapping properties of Φ in Theorem 2.1.
If E has "double symmetry", that is, E is symmetric with respect to zero and each component is symmetric with respect to R, then the centers a 1 , a 2 , a 3 can be given explicitly.
with the positive real n-th root.In (4.7), z 2 can also be replaced by z 1 .
Example 4.3.Let us construct a polynomial P of degree n = 3 with a pre-image consisting of three symmetric intervals By Theorem 2.5, E is a polynomial pre-image if and only (2.17) holds, which is equivalent to β = 1 − α.This implies 0 < α < 1/2.By Remark 2.6 (ii), The critical points of P are z 1,2 = ∓ (1 − α + α 2 )/3.We have a 2 = 0, a 1 = −a 3 and, by (4.7), where a short calculation shows This yields the explicit formula with positive real roots.Since a 2 = 0, a 1 = −a 3 , and n 1 = n 2 = n 3 = 1, we have explicitly determined the canonical domain L c .In the limiting case α → 0, the set E degenerates to {−1, 0, 1} and a 3 → 1.In the other limiting case α → 1/2, the set E tends to the interval [−1, 1] and lim α→1/2 .This shows again that a discontinuity in the connectivity (in this example from = 3 components to = 1 component) leads to a discontinuity of the centers.
Figure 6 illustrates the sets E and L, and the conformal map Φ for several values of α.For z ∈ C \ E, we determine w = Φ(z) by solving the polynomial equation Q(w) = P (z) + P (z) 2 − 1, where the correct value is determined with the help of the mapping properties of Φ in Theorem 2.1.

Several Intervals
In this section, we consider the case that E = P −1 ([−1, 1]) consists of components, each symmetric with respect to the real line.This includes in particular the important case when E consists of real intervals.
For a given polynomial P with critical points z 1 , . . ., z −1 ∈ C\E, we want to compute the centers a 1 < . . .< a of L with the help of (2.13) and (2.14) in Theorem 2.4.In other words, we are looking for the polynomial where n 1 , . . ., n are known (these are the number of zeros of P in E 1 , . . ., E , see Theorem 2.3 (ii)), such that the critical values of Q at the critical points w 1 , . . ., w −1 ∈ C \ L are given; see (2.13).Additionally, a 1 , . . ., a must satisfy (2.14).In his 1961 paper [7], Kammerer considered a similar problem and gave an algorithm for computing the (real) polynomial of degree n with n − 1 prescribed oscillating critical values (and additionally two interpolatory conditions).Existence and uniqueness of such a polynomial has been proved in [2]; see also [1,8,9,12,22] for further results on polynomials with prescribed critical values.In [14, Sect.6], Peherstorfer and the first author provided a modification of Kammerer's algorithm for computing a polynomial pre-image consisting of intervals.
In the following, we modify Kammerer's algorithm for the computation of Q, that is, for the computation of a 1 , . . ., a .For the sake of simplicity, let us concentrate first on the case of real intervals.
Let the polynomial P be as in (2. 3) and such that 1 , . . ., a [k] as initial values, compute a [k+1] 1 , . . ., a [k+1] such that where 2. Compute the solutions w Note that E j ∩ R is indeed a point or an interval; see [18,Lem. A.2].In particular, Then the algorithm works with the same initial values.
In our MATLAB implementation of the above algorithm, we solve the nonlinear system of equations for a  j − a j | in the final step is reported.We observe that the algorithm converges in very few iteration steps in all examples.Moreover, in the examples where a j are known explicitly, the algorithm terminates with an error close to machine precision, which suggests that the algorithm is highly accurate.In Example 3.5, the initial guess is already the exact solution and the iteration stops in its first step.Figure 7 shows the error curves |a In Example 4.1, the exact values of a 1 , a 2 , a 3 are not known.For a comparison, we first map E c onto a region bounded by Jordan curves (through the successive application of inverse Joukowski maps to the intervals) and then apply the numerical boundary integral equation (BIE) method from [13], which computes the centers a 1 , . . ., a , the exponents m 1 , . . ., m , the capacity cap(E) and the conformal map Φ.The difference between the values a j computed by the two methods is close to machine precision, suggesting that both methods are very accurate.For comparison, we compute a 1 , . . ., a 4 by adapting the BIE method from [13] as described in Example 5.3.The difference of the computed values is of order 10 −15 , suggesting that both methods are very accurate.With the obtained values a 1 , . . ., a 4 , the conformal map w = Φ(z) is evaluated as follows.We solve the polynomial equation Q(w) = P (z) + P (z) 2 − 1, see Theorem 2.3 (iii), and determine the correct value of w using the mapping properties of Φ given in Theorem 2.1.Figure 8 shows the sets E and L and the image of a grid under the conformal map Φ.
Example 5.5.Consider the polynomial for which For comparison, we compute a 1 , . . ., a 5 also by adapting the BIE method from [13] as described in Example 5.3.The difference of the computed results is of the order 10 −14 , i.e., both results agree up to almost machine precision.
The computed values a 1 , . . ., a 5 are used to compute numerically the conformal map Φ as described in Example 5.4. Figure 9 illustrates the conformal map.The difference of the values of Φ on the grid computed by this method and by the BIE method from [13] is of the order 10 −14 .We have again a very good agreement between both methods, which suggests that both methods are very accurate.

A Appendix
Given a set E = ∪ j=1 E j with certain symmetries, we characterize the possible relations between the components E 1 , . . ., E .In this section, unlike in the previous parts of the paper, the components E j are allowed to shrink to points, and we require that the E j are nonempty simply connected compact sets.Note that if E j has at least two points, then it is an infinite compact set.
(iii) If = 3 then there exists j 0 , j 1 , j 2 with {j 0 , j 1 , j Proof.First, let = 1, i.e., let E be a simply connected nonempty compact set with E = −E.If E consists of a single point then E = −E implies E = {0}.Otherwise, E is an infinite compact set (since E is simply connected with at least two distinct points), and we can apply Theorem 1.1 to E. The symmetry E = −E implies that L = {w ∈ C : |w| ≤ cap(E)} since a 1 = 0 by Theorem 2.2, and that Φ is odd.The assumption 0 ∈ E c leads to the contradiction 0 = Φ(0) ∈ L c .This shows that 0 ∈ E and concludes the proof of (i).
We write the permutation σ as a product of disjoint cycles.Since f is an involution, i.e., f (f (z)) = z, we have σ 2 = id, which shows that the cycles have length 1 or 2 (transpositions).A cycle of length 1, say (j), corresponds to a set with −E j = f (E j ) = E j , which satisfies 0 ∈ E j by (i).Since the sets E 1 , . . ., E are disjoint, there can be no other cycle of length 1.Thus, σ is the product of (disjoint) transpositions and possibly one cycle of length 1.
The method of proof of Theorem A.1 yields an analogous statement for sets that are symmetric with respect to the real line.
Corollary A.2. Let E = ∪ j=1 E j with E * = E and disjoint, simply connected nonempty compact sets E 1 , . . ., E .Then {1, 2, . . ., } is partitioned into sets {j 0 } with one element, for which E * j 0 = E j 0 , and sets {j 1 , j 2 } with two distinct elements, for which E * j 1 = E j 2 .In the special case = 2, i.e., E = E 1 ∪ E 2 with E * = E, then either E * 1 = E 1 and Proof.Let f (z) = z and f (E) = E * = E, for which f (f (z)) = z for all z ∈ C. Proceeding as in the proof of Theorem A.1, we obtain that f (E j ) = E σ(j) with a permutation σ, which decomposes into cycles of length 1 or 2. Hence the components of E are either symmetric (E * j = E j , corresponding to a cycle (j)) or come in pairs E * j 1 = E j 2 (corresponding to a cycle (j 1 , j 2 ) of length 2).
Remark A.3.The method of proof of Theorem A.1 can be generalized to E = ∪ j=1 E j with disjoint, simply connected nonempty compact sets E 1 , . . ., E , and a continuous function f : E → E with f (E) = E and f k = f • . . .• f = id for some integer k ≥ 2 (where k is minimal with this property).Then σ defined as in the proof of Theorem A.1 is a permutation that satisfies σ k = id.In particular, σ can be written as a product of disjoint cycles, where the length of each cycle divides k.Indeed, σ k = id implies c k = id for every cycle c of σ, hence a cycle c has length l(c) ≤ k.Write k = ql(c) + r with 0 ≤ r < l(c).Then id = c k = c r , which implies r = 0 since r < l(c) and l(c) is the smallest positive exponent with c l(c) = id.Hence, l(c) divides k.
and Φ maps the upper (lower) half-plane without E bijectively onto the upper (lower) half-plane without L. (ii) If E = −E * , i.e., E is symmetric with respect to the imaginary axis, then Φ(z) = −Φ(−z), Φ(iR \ E) = iR \ L, and Φ maps the left (right) half-plane without E bijectively onto the left (right) half-plane without L.

c 1 c 2  c 3 c 4 c 5 c 6 c 7 c 8  c 9 c 10 Figure 1 :
Figure 1: Real graph of a polynomial P of degree n = 7 whose pre-image of [−1, 1] consists of = 4 intervals; see Example 5.4 for the explicit formula of P .

Remark 3 . 4 .
(i) In Theorem 3.3, the following equivalence holds: E contains a real point if and only if E * 1 = E 1 and E * 2 = E 2 , which follows from [18, Lem.A.2].In more detail: If x ∈ E ∩ R, then without loss of generality, x ∈ E 1 and hence

3 . 3 . 5 (
the roles of R and iR in (ii) switch.Let us consider two illustrative examples for Theorem 3.Example Two symmetric real intervals).For 0 < α < β, consider the polynomial
and of a Jordan arc symmetric with respect to the real line with endpoints b 3 = √ α 2 + i and b 4 = b 3 = √ α 2 − i (where Re(b 3 ) > 0) intersecting the interval at the critical point α; see [17, Thm.1].By (3.2), the points a 1 , a 2 are given by

Figure 5 :
Figure 5: Left: The set E in Example 4.1 with α = 0.05 and β = 0.3 in black and a grid.Right: Corresponding lemniscatic domain (∂L in black), the points a 1 , a 2 , a 3 (black dots), and the image of the grid under Φ.

Theorem
whose boundary is a double figure eight self-intersecting at ±2 −4/3 .As observed in Example 3.2, the centers of L do not converge to the center 0 of the lemniscatic domain {w ∈ C : |w| ≤ 1/2} corresponding to [−1, 1]

2 .
The algorithm also works for more general sets E = ∪ n j=1 E j with E * j = E j for j = 1, . . ., .In this case, let us define b 1 , . . ., b 2 by

j
− a j | for Example 3.2 and Example 4.3.

Figure 7 :
Figure 7: Error curves |a [k] j − a j | for the sets E from Example 3.2 (left) and 4.3 (right).Missing dots mean that the error is exactly zero.

Table 1 :
Number of iteration steps until convergence and final maximal error between the computed and exact values of a 1 , . . ., a , where available; see Example 5.3.Example 5.3.We apply our algorithm to all previous examples.Table 1 lists the number of iteration steps in our algorithm until convergence.For those sets where a 1 , . . ., a are known explicitly, also the maximal error max j=1,..., |a