© Hindawi Publishing Corp. THE SPECTRUM OF A CLASS OF ALMOST PERIODIC OPERATORS

For almost Mathieu operators, it is shown that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized 
eigenfunction which decays exponentially are inconsistent properties.


Introduction.
In a series of papers [14,15,16,17,18,19] the author has developed an approach to study the spectrum of the simplest kind of nontrivial almost periodic operators, which is heavily based on C * -algebraic methods. This approach originated in the belief that the involvement of irrational rotation C * -algebras in the investigation of almost Mathieu operators would yield an interdependence between the occurrence of localized eigenfunctions and the topological nature of the spectrum of these operators. In the sequel, we are going to establish such a connection. For almost Mathieu operators which are defined by H(α, β, θ)ξ n = ξ n+1 + ξ n−1 + 2β cos(2παn+ θ)ξ n , ξ ∈ 2 (Z), (1.1) where α, β, and θ are real parameters, the following version of localization has been established by Fröhlich et al. (see [7, page 6 and Section 3]). Consider an irrational number α which satisfies the following Diophantine condition: there exists a constant c > 0 such that |nα − m| ≥ c/n 2 for all m, n ∈ Z and n ≠ 0. Then there exists a constant β 0 > 0 such that for any β ≥ β 0 , the following condition holds: (L ) there exists a subset N 1 ⊂ R which has Lebesgue measure zero, and a constant 0 < r < 1, such that the following condition holds true: if for ξ = {ξ n } n∈Z there are numbers a > 0, χ ∈ R, and θ ∈ R\N 1 such that |ξ n | ≤ an 2 and ξ n+1 + ξ n−1 + 2β cos(2παn+ θ)ξ n = χξ n , for n ∈ Z, (1.2) then ξ decays exponentially of order r as |n| → ∞, that is, there exists a constant b > 0, such that |ξ n | ≤ br |n| for n ∈ Z.
The property (L ) entails that there exists a subset N 2 ⊂ R containing N 1 , which also has Lebesgue measure zero, such that for every θ ∈ R\N 2 , the operator H(α, β, θ) has pure point spectrum with eigenfunctions decaying exponentially of order r .
The work of Fröhlich et al. is paralleled to some extent by the work of Sinaȋ [21] (see also [4,Section 5.5]). It is well known that for irrational α, the spectrum of H(α, β, θ) does not depend on θ. We will denote this spectrum by Sp(α, β). In this paper, we are concerned with the following condition for the localization of eigenfunctions: (L) for every χ ∈ Sp(α, β), there exists a θ such that the difference equation ξ n+1 + ξ n−1 + 2β cos(2παn+ θ)ξ n = χξ n (1.3) has a nontrivial solution which decays exponentially as |n| → ∞. While condition (L ) has been established for the parameters stipulated above, no set of parameters has been found yet for which condition (L) holds true. The objective of this paper is to prove the following theorem. In [1,3,9,10,11], the occurrence of Cantor spectrum has been established under various conditions where property (L ) does not hold. In a number of papers (cf. [12,20]), it has been conjectured that Sp(α, β) should always be a Cantor set.
We are going to list several properties that condition (L) implies. These properties will be crucial in the proof of the theorem. The first important observation is that if (L) holds and if µ denotes the integrated density of states for H(α, β, θ), then the logarithmic potential associated with µ takes the constant value log |β| on Sp(α, β). This means that Sp(α, β) is a regular compactum, µ is its equilibrium distribution, and |β| is the logarithmic capacity of Sp(α, β). (The basic material from classical potential theory which will be used in this paper has been assembled in Appendix A.) This shows among other things that the integrated density of states as well as the (averaged) Lyapunov index (as defined in [5]) are uniquely determined by Sp(α, β). However, considerably more can be shown. The following assertion gives a characterization of the level curves of the conductor potential associated with Sp(α, β) in terms of the spectra of perturbed operators, which are bounded but not selfadjoint. Assertion 1.2. If (L) holds, then a complex number z is contained in the spectrum of the operator H δ (α, β)ξ n = ξ n+1 + ξ n−1 + β δe 2παni + δ −1 e −2παni ξ n , ξ ∈ 2 (Z), (1.4) if and only if log |z − s|dµ(s) = log |β|+|log |δ||.
In order to prove this assertion, we will consider the C * -algebra generated by the family of operators {H δ (α, β)/δ ∈ R\{0}}, which is an irrational rotation C * -algebra with rotation number α. This will put us in a position to study the resolvent of these operators in terms of certain series expansions which arise naturally with the irrational rotation C * -algebra. These series expansions can be looked upon as noncommutative versions of Fourier series in two variables. The exponential behavior of these series expansions at infinity is then expressed in terms of subharmonic functions. Finally, potential theoretic arguments can be invoked to accomplish the proof of Assertion 1.2.
Our second assertion, whose proof relies heavily on the first one, establishes the claimed connection between condition (L) and the topological nature of the spectrum of almost Mathieu operators. In order to render this paper accessible to a wider audience, we will include the exposition material which has been published by the author in [14,15,16,17,18,19]. The organization of this paper is as follows. In Section 2, we briefly discuss the irrational rotation C * -algebra in the context of our approach. In Section 3, we present a notion of multiplicity for elements in Sp(α, β) which was developed in [15]. In Section 4, we study the resolvent of the operators H δ (α, β) (according to [17]). In Section 5, we give the proofs of Assertions 1.2 and 1.3. In Appendix A, we present some material from classical potential theory. In Appendix B, we state and prove a result about conductor potentials of regular compact subsets of the real line, which is vital for the proof of Assertion 1.3.

2.
The irrational rotation C * -algebra. Throughout the paper, α denotes an irrational number. An irrational rotation C * -algebra Ꮽ = Ꮽ α is a C * -algebra which is generated by two unitaries u and v satisfying the relation uv = e 2παi vu. Such an algebra is uniquely determined, up to isomorphisms, by the number α. We let h(α, β) = u + u * + β(v + v * ). The operator H(α, β, θ) is the image of h(α, β) under a specific representation of Ꮽ on the Hilbert space 2 (Z). If π θ is the representation of Ꮽ which is determined on the generators u and v by π θ (u)ξ n = ξ n+1 , then π θ (h(α, β)) = H(α, β, θ). The symmetries of the operator h(α, β) can be expressed in terms of certain symmetries on Ꮽ. These are (uniquely determined) involutive conjugate linear automorphisms σ u and σ v of Ꮽ and anti-automorphisms σ u and σ (v) of Ꮽ such that Furthermore, there is a (uniquely determined) automorphism ρ with period four such that The operator h(α, β) is always a fixed point for σ u , σ v , σ u , and σ v , and h(α, β) is a fixed point for ρ if and only if β = 1. Since σ u and σ v commute, likewise There is a unique tracial state τ on Ꮽ, that is, τ is a state which has the trace property τ(ab) = τ(ba) for all a, b ∈ Ꮽ. Furthermore, if µ denotes the integrated density of states for H(α, β, θ), then we have for any continuous function f on Sp(α, β) the identity and ρ(w pq ) = w q,−p . For any element a ∈ Ꮽ, let a pq = τ(w −p,−q a). We call this number the Fourier coefficient of a at the position (p, q). The series p,q∈Z a pq w pq converges to the element a in the Hilbert space norm associated with τ. We will call this series the Fourier series of a. Proposition 2.1. Suppose that a ∈ Ꮽ has a finite Fourier series. Then the Fourier series p,q∈Z c pq (z)w pq of the resolvent (a − z) −1 has the following property: for every compact subset K of the resolvent set of a, the double sequence {sup |c pq (z)| : z ∈ K} p,q∈Z decays exponentially as |p| and |q| approach infinity.
Proof. Suppose that the Fourier coefficients of a vanish for |p|, |q| ≥ n. Then for complex numbers x and y with modules close to one, the spectrum of the operator a(x, y) = |p|,|q|≤n a pq x p y q w pq (2.5) is contained in C\K, and we have The series on the right-hand side of this identity is absolutely convergent. Thus, for z ∈ K. Suitable choices for |x| and |y| conclude the argument.
3. Point spectrum and a certain multiplicity for points in the spectrum. In the sequel, we assume throughout that β ≠ 0. We call a state ϕ on the C *algebra Ꮽ an eigenstate of h(α, β) for χ ∈ Sp(α, β) if the identity holds. The general theory of C * -algebras yields that for every χ ∈ Sp(α, β), there exists at least one eigenstate of h(α, β) for χ. Since h(α, β) is a selfadjoint operator and a state is a selfadjoint functional, any eigenstate ϕ also satisfies the following identity: Suppose that ϕ is a state on Ꮽ, and for any p, q ∈ Z, let x pq = ϕ(w pq ). Then ϕ satisfies condition (3.1) if and only if cos(π αq) x p−1,q + x p+1,q + β cos(π αp) x p,q−1 + x p,q+1 = χx pq for any p, q ∈ Z. We are now going to explain how the solutions of the combined system (3.3) and (3.4) can be generated by certain recursions (see [15,18]). To this end, we consider a modified system where certain phase angles have been introduced in the coefficients cos παq + θ 2 x p−1,q + x p+1,q + β cos παp + θ 1 x p,q−1 + x p,q+1 = χx pq , where θ 1 and θ 2 satisfy the condition For any p, q ∈ Z, we define 4×4 matrices A pq and B pq having the property that a double sequence {x pq } solves system (3.5) if and only if and a k = 0 for the remaining entries of the matrix. Furthermore, let Condition (3.6) ensures that the denominators in these formulae do not vanish. Apart from being invertible, the matrices A pq and B pq satisfy the following identity: There are exactly four linearly independent solutions of (3.5), which can be generated in the following manner: given any numbers x 00 , x 10 , x 01 , and x 11 , one can use the formulae in (3.7), as recursions on the two-dimensional lattice, to compute the values x pq : The four circles to the left represent the four input parameters, while the stars to the right represent the last two output parameters.) Since there are infinitely many ways to reach a position (p, q) by a finite succession of those four basic recursions, departing at the positions (0, 0), (1, 0), (0, 1), and (1, 1), we face the question whether this procedure produces consistent results. Identity (3.10) ensures that the outcome is independent indeed from the specific path we chose to reach the position (p, q).
2 )} n∈N as a sequence of pairs of nonvanishing phase angles which converges to (0, 0). Moreover, we assume that θ Moreover, exploiting (3.3) for p = q = 0 shows that x 0,−1 is uniquely determined by x 00 , x 10 , x 01 , x 11 , and x −1,0 . Observe that the matrices A pq and B pq are well defined even for θ 1 = θ 2 = 0 whenever p ≠ q and p ≠ −q − 1. So, anything that has been said earlier regarding the recursions on the twodimensional lattice remains intact for θ 1 = θ 2 = 0 as long as we do not appeal to any formulae involving A pq and B pq , when p = q or p = −q − 1, or to any formulae involving A −1 pq and B −1 pq , when p = −q or p = q + 1. (Observe that for θ 1 = θ 2 = 0, the matrices A p,−p and A p+1,p are singular.) This entails that any point in the sector {(p, q) ∈ Z 2 /p ≥ |q|} can be reached by a finite succession of recursions of the four types described above, departing at the positions has exactly five linearly independent solutions, which are determined at the positions (0, 0), (1, 0), (0, 1), (1, 1), and (−1, 0). Moreover, the values at any position (p, q) for |p| > 1 or |q| > 1 can be determined by iterative recursions. Our next objective is to give a more detailed description of the solutions {x pq } of the combined system (3.3) and (3.4) for which x 00 = x 11 = 0 (according to [15,16]). The following characterizations can be established with the aid of the recursions described above: 10 , for p ≥ 0. A more specific characterization of the solutions described above can be given if we express them in terms of the parameter χ: x pq = ω pq (χ)x 10 for p > |q|, (3.12) In order to establish this last property, one can use two-dimensional recursions. For instance, to cover the case where |p| > |q|, one considers the recursion which is, of course, also redundant. The initial values in this case are Remark 3.1. Without entering the details, we would like to mention that there is an alternative approach to obtaining the polynomials ω pq (χ) by considering the Fourier expansions (i.e., the expansions in e nθi ) of the polynomials of the second kind for the difference equations ξ n+1 + ξ n−1 + 2β cos(2παn+ θ)ξ n = χξ n , ξ n+1 + ξ n−1 + 2β −1 cos(2παn+ θ)ξ n = χξ n , (3.15) for n ≥ 0 as well as n ≤ 0. At this point, we interrupt the discussion of the combined system (3.3) and (3.4) to give an application of what has already been established (see [14]). Proof. Suppose that the opposite were true. So, there exist θ ∈ R, χ ∈ Sp(α, 1), and ξ ∈ 2 (Z), ξ = 1, such that H(α, 1,θ)ξ = χξ. Then is an eigenstate of h(α, 1) for χ. We have the following properties which are true because ϕ is a vector state: is a solution of the combined system (3.3) and (3.4). Since ϕ(w 11 ) = ϕ(w 1,−1 ) and ρ(w 11 ) = w 1,−1 , we also have x 00 = x 11 = 0. Furthermore, {x pq } is not the trivial solution for if it were, ϕ would be ρ-invariant, which is impossible in the light of (i) and (iii). We conclude that {x pq } must be a linear combination of the solutions described in (1), (2), or (3). This means, however, that x pq takes a constant nonvanishing value for infinitely many (p, q) with ||p| − |q|| = 1; thus, contradicting (ii).
We resume our general discussion. In [15, pages 297-298], we have defined a three-dimensional recursion along the positive diagonal p = q in order to establish the following properties. Notice that if ϕ is an eigenstate of h(α, β) for χ ∈ Sp(α, β), then the double sequence {ϕ(w pq )} is uniformly bounded. Also in [15], the following sufficient condition for the occurrence of two pure eigenstates was given. H(α, β, θ), then h(α, β) has two distinct pure eigenstates for χ.
For future references, we point out that the set Ω(α, β) is at most countable. The set παZ ∪ π(α + 1)Z is trivially countable, and for every θ in this set, there can be no more than countably many eigenvalues of H(α, β, θ) because 2 (Z) is a separable Hilbert space. As in [15], we call the total number of pure eigenstates of h(α, β) for an element χ ∈ Sp(α, β) the multiplicity of χ. To see this, we observe that by Scholium 3.3, there is only one σ -invariant eigenstate for χ. Since h(α, β) is a fixed point of σ , ϕ • σ is also an eigenstate for χ. If ϕ • σ = ϕ, then ψ • σ = ψ, otherwise there would be at least three pure eigenstates for χ. Therefore, ϕ • σ = ψ. Let x pq = ϕ(w pq ) − ψ(w pq ).
Since {x pq } is a solution of (3.3) and (3.4), we have, on the one hand, On the other hand, we have  (2). We now give another application. It was shown in [6] that the operator H(α, β, θ) has no eigenvalues for |β| < 1. The proof of this fact was based on Oseledec's theorem. Independently, by the methods developed so far, the following weaker statement was shown to be true in [16,Theorem 3.1].
It may seem that the exclusion of the exceptional set Ω(α, β) from consideration in the last theorem is a deficiency that could be overcome by a more powerful argument. However, as the reasoning leading up to the proof of Assertion 1.3 will show, this is not likely to be the case. Putting it informally, the set Ω(α, β) is the "blind spot" of the theory. In a sense, the very existence of such an exceptional set is necessary in order for this approach to work.

The resolvent of perturbed operators.
Suppose that α and β are fixed.
Our next objective is to study the Fourier expansion of the resolvent of these operators (according to [17]). Recall from Proposition 2.1 that the Fourier series of (h (γ,δ) − z) −1 decays exponentially as the lattice parameters p and q approach infinity, at any point in the resolvent set of h (γ,δ) . We will see that there are two types of series expansions for the resolvent of h (γ,δ) ; namely, those which represent the resolvent on the unbounded component of the resolvent set (we will refer to those series as being of type I) and those which represent the resolvent on the bounded components of the resolvent set (we will refer to those series as being of type II).
We are going to recast the resolvent problem for the operators h (γ,δ) slightly, so that it parallels the induction of eigenstates in Section 3. An element a ∈ Ꮽ is an inverse of h (γ,δ) − χ if and only if the following two conditions hold: where I denotes the unit in Ꮽ; Considering the Fourier series p,q∈Z x pq w pq of a, condition (4.2) is equivalent with cos(π αq) γ −1 x p−1,q + γx p+1,q + β cos(π αp) δ −1 x p,q−1 + δx p,q+1 = χx pq + ε pq for p, q ∈ Z,    The connection between these solutions and those in Section 3 is as follows: pq is of type (3). (4.10) The following test which indicates the presence of solutions of Scholium 4.1 of type (4.7) through (4.9) can be derived with the aid of the recursions discussed in Section 3.
We denote by R(γ, δ) the resolvent set of h (γ,δ) . For some χ ∈ R(γ, δ), consider the Fourier expansion x pq w pq . (4.11) Let y pq = γ p δ q x pq . We say that χ is of type I if χ ∉ Sp(α, β) and To see this, suppose that the claim were not true. By assumption, in each of the four sectors of the two-dimensional lattice Z 2 , which are separated by the lines p = q and p = −q, {y pq } is a scalar multiple of exactly one of the four double sequences in (4.7) through (4.9). It follows that in any of those four sectors S, where {x pq } does not vanish identically, we can define a solution {s pq } of (4.4) and (4.6) by carrying out the following two steps. First, let s pq = x pq in S and s pq = 0 elsewhere. Then scale { s pq } with a suitable number c to obtain {s pq }, that is, s pq = c s pq . Since {x pq } decays exponentially as |p|, |q| → ∞, the same is true for {s pq }. So, if χ were not of type II, then we could construct such exponentially decaying solutions of (3.3) and (3.4) for at least two distinct sectors. This would yield at least two distinct inverses of h (γ,δ) − χ in the C *algebra Ꮽ, thus contradicting the uniqueness of such an inverse.
With a little more effort, one can show the following refined statement. If χ is of type II and (h (γ,δ) − χ) −1 = p,q∈Z x pq w pq , then  Since for no values of β, γ, and δ any two distinct conditions among those four stated in (4.13) are valid, it follows that for any operator h (γ,δ) which has points of type II in its resolvent set, the resolvent at any two of those points always has the same form.   .3) for p = q = 0. If it does, then the absolutely convergent Fourier series p,q∈Z z pq w pq defines an element a in the C * -algebra Ꮽ with the property (h(α, β)−χ)a = a(h(α, β)−χ) = 0. In particular, if ϕ is any state on Ꮽ and we define a functional ϕ a by ϕ a (x) = ϕ(a * xa), x ∈ Ꮽ, then ϕ a = cψ for some eigenstate ψ of h(α, β) for χ and some constant c ≥ 0. This gives rise to an infinite-dimensional space of uniformly bounded solutions of (3.3) and (3.4), which clearly contradicts Scholium 3.3. So, {z pq } does not solve (3.3) for p = q = 0. Thus, we can scale {z pq } by a suitable constant c such that {cz pq } solves (4.4) and (4.6) for γ = δ = 1. It follows that χ ∉ Sp(α, β) and the absolutely convergent Fourier series p,q∈Z cz pq w pq is the inverse of h(α, β) − χ. Since |γ|, |δ| ≤ 1, we have for all p, q ∈ Z, γ −p δ −q z pq ≤ γ −|p| δ −|q| z |p|,|q| = γ −|p| δ −|q| y |p|,|q| = x |p|,|q| . (4.14) It follows that {cz pq γ −p δ −q } decays exponentially as |p|, |q| → ∞, and hence the limit of the absolutely convergent Fourier series p,q∈Z cz pq γ −p δ −q w pq is an inverse of h(γ, δ) − χ. The uniqueness of the inverse entails that (4. 15) We conclude that χ is of type I. The cases where |γ| ≥ 1, |δ| ≤ 1 or |γ| ≤ 1, |δ| ≥ 1 or |γ| ≥ 1, |δ| ≥ 1 are treated in a similar fashion. Suppose that Ω is a component of R(γ, δ). Let Ω I be the set of those points in Ω which are of type I, and let Ω II be the set of those points in Ω which are of type II. By Scholium 4.6, we have Ω = Ω I ∪ Ω II . In order to prove that either Ω I = φ or Ω II = φ, it suffices to show that both sets are relatively closed. Suppose that χ 1 ,χ 2 ,... is a sequence in Ω I converging to χ ∈ Ω. Then that is, the Fourier coefficient of (h (γ,δ) −χ n ) −1 at the position (p, q) converges to the Fourier coefficient of (h (γ,δ) − χ) −1 at the position (p, q). Since χ n is of type I, we have for all p, q ∈ Z. It now follows from Scholium 4.5 that χ is in Ω I . Next, suppose that χ 1 ,χ 2 ,... ∈ Ω II converge to χ ∈ Ω. Then at the positions in all but one of the four sectors separated by the lines p = q and p = −q, the Fourier coefficients of (h (γ,δ) − χ n ) −1 vanish. Since this property is preserved under limits, it follows from Scholia 4.3 and 4.4 that χ is of type II. A component containing points of type I only will be called of type I, too. Otherwise, it will be called of type II. The Fourier coefficients of (h (γ,δ) −χ) −1 approach zero as |χ| → ∞. However, on components of type II, the Fourier coefficients of (h (γ,δ) − χ) −1 are polynomials (see (4.10) and (4)), and thus they do not approach zero as |χ| → ∞ unless they vanish identically.
x pq w pq , (4.19) then {x pq γ p δ q } cannot be the Fourier coefficients of an inverse of h(α, β)−χ.
Hence, χ must be of type II. for sufficiently large |δ|, the spectrum of δ −1 h (γ,δ) is close to the spectrum of βv * . So, for large |δ|, the set K ∪ Sp(α, β) is contained in a single component of R(γ, δ). The claim now follows from Scholium 4.9.
Proof. (i) An application of Fatou's lemma shows that ρ I is submean, that is, for any z ∈ C\ Sp(α, β), we have whenever r is sufficiently small. Since {c pq (z)} decays exponentially as |p|, |q| → ∞, we have ρ(z) < 0 in C\ Sp(α, β). The set {z ∈ C\ Sp(α, β)/ρ I (z) < − log δ} consists of all points which are of type I for h (1,δ) , for δ ≥ 1. By Scholia 4.6 and 4.7, this set is open. In conclusion, ρ I is subharmonic. The function ρ II is submean for the same reason ρ I is. Also, since all points in C\ Sp(α, β) are clearly of type I for h (1,1) , we have ρ II (z) ≥ 0 in C. It follows from Scholium 4.10 that ρ II (z) < ∞ in C. The set {z ∈ C/ρ II (z) < log δ} consists of all points which are of type II for h (1,δ) , for δ ≥ 1. Again by Scholia 4.6 and 4.7, this set is open. We conclude that ρ II is subharmonic.
Whence, by Lemma 5.1, In particular, ρ II vanishes everywhere on Sp(α, β). By the definition of the functions ρ I and ρ II , it is clear that z is in the spectrum of h (1,δ) if and only if ρ II (z) = log |δ|, which settles the proof.
Before we move on to the proof of Assertion 1.3, we would like to point out some consequence of Assertion 1.2 for the spectra of the operators h (1,δ) . If (L) holds, then [24, Section 4.1, Theorem 1] says that the spectrum of h (1,δ) for |δ| ≠ 1 either consists of a finite number of mutually exterior analytic Jordan curves or consists of a finite number of Jordan curves composed of a finite number of analytic Jordan arcs, which are mutually exterior except that each of a finite number of points may belong to several Jordan curves.
This shows that the series p,q∈Z δ q d (+) pq (χ)w pq converges absolutely and since {δ q d (+) pq (χ)} solves (4.4) and (4.6) for γ = 1, its limit is an inverse of h (1,δ) ) is an open and closed subset which is a Cantor set. For every t ∈ R\K and n ∈ N, we define M(t, n) = χ ∈ K/∆ χ (t) > n , (5.30) which is an open subset of K. Whence, is open in K. Assertion 1.2 and Appendix B.2 in conjunction with the mean value property entail that for any n, the set M n contains the (finite) boundary points of maximal intervals of R\K. Since, by assumption, K is a Cantor set, this entails that M n is dense in K for every n. Whence, ∞ n=1 M n = χ ∈ K/m χ = ∞ is a dense G δ -subset of K, which is a perfect compactum. On account of the Baire category theorem, we conclude that m χ = ∞ for uncountably many χ ∈ K. Assuming that (L) holds, this contradicts Lemma 5.4 combined with the fact that the set Ω(α, β) is at most countable.
Remark 5.5. In [19], it is shown that the conclusion of Assertion 1.2 always holds for irrational numbers α which are sufficiently well approximable by rationals in terms of a Diophantine condition and for |β| ≥ 1. By virtue of duality, this translates into a similar statement for 0 < |β| < 1.

Appendices
A. Subharmonic functions and potential theory. In the sequel, we present without proofs the material from classical potential theory which has been used in the paper. is called the conductor potential of K.
for every θ ∈ (0,π). For convenience, we assume that a = 0. Then, |z − s| ≥ |s|| sin θ| for z ∈ A θ and s ≤ 0, and hence Since 0 −∞ (1/|s|)dµ(s) is finite by assumption, then it follows that the integral |z −s| −1 dµ(s) is uniformly bounded for z ∈ A θ , but close to a. By Lebesgue's dominated convergence theorem, we infer that the limit in question exists and it is equal to b. Since the function x (x − s) −1 dµ(s) is decreasing in Ᏽ and it takes the value 0 exactly once in case Ᏽ is a finite interval (see [23, Section 7.2, Corollary 3]) and it vanishes at infinity only in case Ᏽ is an infinite interval, we conclude that b ≠ 0. Moreover, where f µ (z) → w as z → a in Γ . Let η k : [0, 1] → C be differentiable injective curves (k = 1, 2) such that η k ([0, 1)) ⊂ Γ ∩ A θ for some θ ∈ (0,π) and η k (1) = a. Let ϕ be the angle between η 1 and η 2 at a. Then (B.5) entails that the images η 1 and η 2 of η 1 and η 2 with respect to f µ form the same angle ϕ at the point w ∈ ∂f µ (Γ ) = B (See Appendix B.1). Since the curves η 1 and η 2 eventually evolve exclusively on only one side of the line through 0 and w, as they approach the point w, the angle between η 1 and η 2 can never exceed the value π/2. However, since we can arrange ϕ to be any number in (0,π), we have reached a contradiction.

Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities. This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics. It is possible to make a division in two main categories: astronomy and astrodynamics. By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth. Many important topics of research nowadays are related to those subjects. By astrodynamics, we mean topics related to spaceflight dynamics. It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects. Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts. Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines. The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research. All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/mpe/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/ according to the following timetable: