This is a preprint of: “Univalent wandering domains in the Eremenko-Lyubich class”, Nuria Fagella, Xavier Jarque, Kirill Lazebnik, J. Anal. Math., 2019. DOI: [10.1007/s11854-027-0079-x] UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK Abstract. We use the folding theorem of [Bis15] to construct an entire function f in class B and a wandering domain U of f such that f restricted to f n (U ) is univalent, for all n ≥ 0. The components of the wandering orbit are bounded and surrounded by the postcritical set. 1. Introduction We consider the dynamical system formed by the iterates of an entire map f : C → C. We will consider only transcendental f , namely those maps f with an essential singularity at ∞. Such dynamical systems appear naturally as complexifications of one-dimensional real-analytic systems (interval maps or circle maps for instance) or as restrictions of analytic maps of R2n to certain invariant one complex-dimensional manifolds. The dynamics of f splits the complex plane into two complementary and totally invariant sets: The Fatou set (or stable set), where the iterates form a normal family, and its closed complement, the Julia set, J(f ), often a fractal formed by chaotic orbits. The Fatou set is open and is generally composed of infinitely many connected components, known as Fatou components, which map among each other under the function f . It was already Fatou [Fat20] who gave a complete classification of periodic Fatou compo- nents in terms of the possible limit functions of the sequence of iterates. His classification theorem states that an invariant Fatou component is either an immediate basin of attraction of an attracting or parabolic fixed point; or a Siegel disk, i.e. a topological disk on which f is conformally conjugate to a rigid irrational rotation; or a Baker domain if the iterates converge uniformly to infinity. This classification extends to periodic Fatou components, since a component of period p > 1 is invariant under f p . It is well-known that each of the periodic cycles of Fatou components is in some sense associated to the orbit of a singular value, that is, a point around which not all branches of f −1 are well defined. Singular values may be critical values (images of zeroes of f 0 ) or also asymptotic values which, informally speaking, are points that have at least one preimage “at infinity”, such as 0 for z → ez . 2010 Mathematics Subject Classification. Primary 30D05, 37F10, 30D30. The first and second authors were partially supported by the Spanish grant MTM2014-52209-C2-2-P and the Maria de Maeztu Excellence Grant MDM-2014-0445. 1 2 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK The set of singular values, S(f ), plays a crucial role in holomorphic dynamics precisely because of its relation with the periodic Fatou components of f . We mention two exam- ples. First, basins of attraction must contain a singular value (and hence its forward orbit). Second, it is known that the postsingular set, P (f ), i.e. the singular values of f together with their forward orbits, must accumulate on the boundary of a Siegel disk [Fat20, Mil06]. The relation with Baker domains is weaker and not so easy to state, and we refer the reader to [Ber95]. This connection allows one to glean information about possible stable orbit behaviours (namely periodic Fatou components) of a given map by understanding the dy- namics of its set of singular values. For this reason, many classes of entire functions have been singled out in terms of properties of their singular set. Important examples are the Speiser class S of maps with a finite number of singular values (whose members are also called of finite type) or the Eremenko-Lyubich class B = {f : C → C entire | S(f ) is bounded}. In the presence of an essential singularity at infinity there may exist Fatou components which are neither periodic nor eventually periodic. These are called wandering domains and are the subject of this paper. More precisely, a Fatou component U is a wandering domain if f k (U ) ∩ f j (U ) = ∅ for all k, j ∈ N, k 6= j. Perhaps because wandering domains do not exist for rational maps [Sul85], nor for maps in the Speiser class S [EL92, GK86], these rare Fatou components have not been subject of attention until quite recently, when maps with infinitely many singular values (like Newton’s method applied to entire functions) have started to emerge as interesting objects. Neverthe- less, many recent breakthrough results about wandering domains have appeared in the last several years. After the classical result [EL92] which states that maps in class B cannot have wandering domains whose orbits converge to infinity uniformly (called escaping wandering domains), there was reasonable doubt of whether functions in class B could have wandering domains at all. This question was answered affirmatively by C. Bishop in [Bis15] who con- structed a function in class B with an oscillating wandering domain, that is, a wandering domain whose orbits accumulate both at infinity and on a compact set (the collection of points that oscillate as such under f is termed the Bungee set BU (f ) - see [OS16] for details and properties of this set). This construction depends on a technique, also developed in [Bis15], termed quasiconformal folding. An oscillating wandering domain was constructed previously in [EL87] using approximation theory techniques, however it is not known whether this example is in class B. Indeed, these techniques give insufficient control over the singular set for this purpose. The question of existence of dynamically bounded wandering domains, i.e., wandering domains whose orbits do not accumulate at infinity, is unknown (this problem was first posed in [EL87]). It is a wide open problem to find the sharp relationship between wandering domains and the postsingular set P (f ), or even with the singular set S(f ). Results up to now show that some relation exists: if the domain is oscillating (i.e. lies inside BU (f )), any finite limit UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 3 function must be a constant in J(f ) ∩ P (f ) [Bak02] (see also [BHK+ 93]) and, in any case, there must be postsingular points inside or nearby the wandering components (see [BFJK] and [MBRG13] for the precise statements). A very related and natural question is whether a wandering domain could exist such that the function were univalent on each of the orbit components. Outside the class B the answer is, not surprisingly, affirmative, as shown in [EL87] and [FH09, Example 1]. The example of [EL87] is obtained using approximation theory, whereas the escaping wandering domain of [FH09, Example 1] is a logarithmic lift of an appropriately chosen invariant Siegel disk. But it is inside class B where this question makes most sense to be asked. As we mention above, wandering domains in class B can only be oscillating, that is, they must return infinitely often to a compact set, and hence a large amount of contraction is necessary. Let us keep in mind that Bishop’s example contains a critical point of very high order inside infinitely many of its components, which allows for this large contraction. Our main result in this paper shows that, nevertheless, univalent wandering domains are also possible inside this class of maps. Main Theorem. There exists an entire transcendental function f ∈ B and a wandering Fatou component U of f such that f |f n (U ) is univalent for all n ≥ 0. Our example uses the Folding Theorem in [Bis15] (see Section 2) and is in fact a careful modification of Bishop’s original construction. Very roughly speaking Bishop’s function behaves like (z − zn )dn for some dn → ∞, on some subsequence of wandering components. We replace these maps by (z − zn )dn + δn · (z − zn ) on subsets of the same components, which are univalent near the origin, and show that the critical values can be kept outside (but very close to) the actual wandering components. This is achieved by trapping the boundary of the wandering components inside annuli of decreasing moduli which separate the domains from the critical values. This paper is organized as follows. In section 2 we recall Bishop’s construction of a wandering domain in class B, together with other preliminary results we use later on in the proof. Section 3 describes the new map on D-components that we shall work with in the construction. The general strategy is then explained in Section 4, followed in Sections 5 and 6 by the estimates that show that a wandering domain exists. Finally Section 7 is dedicated to showing that the critical values do not belong to the new wandering domain orbit. Acknowledgements. We are grateful to Chris Bishop for his comments on a preliminary version of the paper. We would also like to thank Chris Bishop, Mikhail Lyubich and Lasse Rempe-Gillen for helpful discussions. We finally thank the Universitat de Barcelona and the Institut de Matemátiques de la UB for their hospitality during the visit that led to this work. 4 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK 2. Preliminaries The main result of this paper is based on Bishop’s construction of transcendental entire maps via quasiconformal folding [Bis15, Theorem 1.7]. Roughly speaking, given an infinite bipartite tree T , Bishop’s Theorem provides an entire function in B (bounded singular set) with prescribed behaviour (up to pre-composition with a quasiconformal map close to the identity) off a small neighborhood of T . As opposed to what occurs with other existence theorems (for example those in approximation theory - see for instance [Gai87]), one has fine control of the singular set of the final map, which makes this tool effective when constructing examples in restricted classes of functions such as class B. Our goal in this section is to provide the reader with the essential background to state (a simplified version of) Bishop’s Theorem (Subsection 2.1), and its application to produce a transcendental entire function in B having an oscillating wandering domain (Subsection 2.2). For a deeper discussion we refer to the source [Bis15] or to [FGJ15, Laz17] where some details are more explicit. Additionally, at the end of this section we recall some additional tools that will be used throughout the paper. 2.1. On Bishop’s quasiconformal folding construction. Let T be an unbounded con- nected bipartite graph with vertex labels in {−1, +1}. Then the connected components of C \ T are simply connected domains in C. We denote by R-components (respectively D- components) the unbounded (respectively bounded) components of C \ T . We will assume that T has uniformly bounded geometry, i.e. that edges are (uniformly) C 2 and that the diameters of edges satisfy certain uniform bounds (see Theorem 1.1 of [Bis15]). We define a neighbourhood of the graph given by [ n o T (r) := z ∈ C | dist(z, e) < r diam(e) , e edge of T where dist and diam denote the Euclidean distance and diameter respectively. We denote by Hr = {z = x + iy ∈ C | x > 0} the right half plane and by D = {z ∈ C | |z| < 1} the unit disk. For each connected component Ωj of C \ T , let τj : Ωj → ∆j be the Riemann map where ∆j = Hr or ∆j = D depending on whether Ωj is an unbounded or bounded component. We call ∆ = ∆j the standard domain for Ωj . We shall denote by τ the global map defined on ∪j Ωj such that τ |Ωj = τj . Each edge e of T is the common boundary of at most two complementary domains but corresponds via τ to exactly two intervals on ∂Hr , or one interval on ∂Hr and one in ∂D (see condition (i) in Theorem 2.1). The τ -size of an edge e is defined to be the minimum among the lengths of the two images of e under τ . Moreover we also define a map σ from the standard domains ∆j to C depending on whether ∆j equals Hr or D. More precisely we define σ(z) := cosh(z) if ∆j = Hr . Otherwise, if ∆j = D, then σ(z) := z d , d ≥ 2 possibly followed by a quasiconformal map ρ : D 7→ D which sends 0 to some w ∈ int(D) and is the identity on ∂D. UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 5 Now we are ready to state the main result in [Bis15], in a simplified version which is sufficient for our purposes. Theorem 2.1. Let T be an unbounded connected graph and let τ be a conformal map defined on each complementary domain C \ T as above. Assume that: (i) No two D-components of C \ T share a common edge. (ii) T is bipartite with uniformly bounded geometry. (iii) The map τ on a D-component with 2n edges maps the vertices to the 2nth roots of unity. (iv) On R-components the τ -sizes of all edges are uniformly bounded from below. Then there is an r0 > 0, a transcendental entire f , and a K-quasiconformal map φ of the plane, with K depending only on the uniformly bounded geometry constants, so that f = σ ◦ τ ◦ φ off T (r0 ). Moreover the only critical values of f are ±1 - corresponding to the vertices of T , and those critical values assigned by the D-components. As mentioned, the proof of the theorem is based on some quasi-conformal deformations of the maps τ and σ inside Tr0 so that the modified global map g = σ ◦ τ is quasiregular. In this situation the Measurable Riemann Mapping Theorem provides the quasiconformal map φ in the statement. We sketch here some details needed for our construction but we refer again to [Bis15] for a more detailed approach. If ∆ = Hr we first divide ∂Hr into intervals I of length 2π and vertices in 2πiZ. These intervals will correspond, after Bishop’s folding construction, to the images of the edges of T 0 ⊃ T (where T 0 is T with some decorations added, all of which are contained in a sufficiently small neighborhood T (r0 ) of T ) by a suitable quasiconformal deformation η̂ of τ . Secondly we define σ on ∂Hr . There are two cases to consider: either I is identified with a common arc of two R-components in which case we define σ(iy) := cosh(iy) for every iy ∈ I, or I is identified with a common arc of one R-component and one D-component, in which case we define σ(iy) := exp(iy) for every iy ∈ I. Finally we extend σ to Hr as a quasiconformal map which is cosh(x + iy) for x > 2π. 2.2. The prototype map. In [Bis15, Section 17], the author gives an application of The- orem 2.1 in order to construct a family of entire functions in class B depending on infinitely many parameters. One defines an unbounded connected graph T and, on some relevant com- plementary domains, one defines the maps σ ◦ τ depending on the parameters. By choosing the parameters appropriately, one is able to ensure that the resulting function has oscillat- ing wandering domains. Since our Theorem A is a modification of this example, we briefly describe it here. Again we refer to [Bis15] or [FGJ15, Laz17] for a detailed discussion. Consider the open half strip n πo S + := x + iy ∈ C | x > 0 and |y| < . 2 6 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK Following the previous notation we denote by (σ ◦τ )|S + the composition z 7→ cosh(λ sinh(z)). We remark that this map extends continuously to the boundary, sending ∂S + onto the real segment [−1, +1]. On the upper horizontal boundary of S + , we select points (an ± iπ/2)n>1 which are sent to {−1, +1} by (σ ◦ τ )|S + , such that an is close to nπ for every n > 1 (see [FGJ15] for more details). The following open disks will belong to the graph T : ∀n > 1, Dn := {z ∈ C | |z − zn | < 1} where zn := an + iπ. We complete the construction of T by firstly adding segments connecting vertically the points (ai , π/2) and (ai , π − 1), and the point (ai , π + 1) with infinity and secondly copying the structure through the symmetries z → ±z. Again, following  above notation, we will denote by (σ ◦ τ )|Dn the composition z 7→ ρn z − zn )dn for every n > 1 where, for every n, dn is a parameter to be fixed and ρn is a quasiconformal map sending 0 to wn (near 1/2) and such that ρn |∂D = Id. Figure 1 summarizes the construction. For suitable choices of the parameters {wn , dn , λ}, Theorem 2.1 gives an example of a tran- scendental map in B with an oscillating (non-univalent) wandering domain. The contribution of the present work is to show that by modifying the quasiregular maps on D-components (see Section 3), one is able to construct a function in class B which is then proven to have a univalent wandering domain. i(π + 1) D1 D2 D3  ρn (z − zn )dn iπ z1 z2 z3 i π2 S+ wn 0 1 a1 π a22π a3 −1 0 2 1 cosh(λ sinh(z)) Figure 1. The domains S + and (Dn )n>1 are depicted on the left. The dark gray areas represent for the preimages of the unit disk D under the map σ ◦ τ . 2.3. Other tools. The following statement is Koebe’s one quarter Theorem and a part of his distortion theorem. Theorem 2.2 ([Pom75a, Section 1.3]). Let F be a univalent function on the disk D(a, r) for some a ∈ C and r > 0. Then UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 7  (a) f (D(a, r)) ⊃ D F (a), 14 |F 0 (a)|r . (b) For all z ∈ D(a, r). r2 |z − a||F 0 (a)| r2 |z − a||F 0 (a)| 6 |F (z) − F (a)| 6 . (r + |z − a|)2 (r − |z − a|)2 As explained above the key idea behind Theorem 2.1 is to obtain the desired entire function f as the composition of a quasiregular map σ ◦ η, and a quasiconformal map φ given by the Measurable Riemann Mapping Theorem, that is f := (σ ◦ η) ◦ φ. In particular, f and σ ◦ η are not conjugate to each other. As it turns out, we shall have an explicit expression for σ ◦ η, at least in the domains where the relevant dynamics occur. In order then to control the dynamics of f one needs control on the correction map φ. The result that follows will be used to show that, in the cases we are interested in, we may assume that φ is arbitrarily close to the identity uniformly on C. Roughly speaking the next result states that if Φ : R2 → R2 is a K- quasiconformal map with dilatation µ supported on a planar set whose area is exponentially small near ifininity (see Definition 2.4 in [FGJ15] for a precise definition of (ε, h)-thin small set) then we may expect Φ to be close to a conformal map in C, and so, close to the identity after normalization. Theorem 2.3 (Bishop, personal communication). Suppose Φ : R2 → R2 is K-quasiconformal and is normalized to fix 0 and 1. Let E be the support of the dilatation of Φ (possibly un- bounded) and assume that E is (ε, h)-thin. Then (2.1) ∀z, w ∈ R2 , (1 − Ceβ )|z − w| − Ceβ 6 |Φ(z) − Φ(w)| 6 (1 + Ceβ )|z − w| + eβ where C and β only depend on K and h. For instance, as a consequence of the above result, and under suitable conditions on the behaviour and smoothness of Φ in a neighborhood of the real line, one can deduce bounds on the derivative of the map. Again we refer to [FGJ15] for precise statements and discussion. 3. The map on D-components In this section we describe a quasiregular map g in each of the D-components Dn of our graph that will be used in accordance with Theorem 2.1. Recall Dn is a disc of radius 1 centered at a point zn . We first translate Dn to the unit disc D by the map z → z − zn . Next we post-compose with a quasiregular map ψ which will be defined as z → z m + δz in a subdisc of D, interpolated with z → z m on ∂D. More precisely, consider for now the map z → z m + δz defined on D. The m − 1 critical points (ck ) and m − 1 critical values (vk ) of this map are given by  ( m−1 1 )  ( m−11 ) m − 1 −δ −δ ck = , vk = δ . m m m 8 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK If we fix δ and let m → ∞, the critical points will tend to ∂D, while the critical values tend, in absolute value, to δ. Fix δ << 1/2. In order to use Theorem 2.1 we would like to consider a map ψ which is the above described map z → z m + δz on a subdisc of D, interpolated with z → z m on ∂D. More precisely, consider a Euclidean circle γ centered at the origin of radius 32 δ, and note that for sufficiently large m, the critical values vk belong to int(γ), where int(γ) denotes the bounded component of C \ γ. Taking the pullback of γ under z → z m + δz we get a simple closed curve in D, that we denote γ −1 , so that the critical points ck belong to int (γ −1 ). Indeed, notice that int (γ −1 ) is mapped to int (γ), while the unit circle is mapped outside the disk of radius 1 − δ >> δ, so γ −1 ⊂ D. See Figure 2. We define the quasiregular map ψ on D by ( z m + δz, ifz ∈ int (γ −1 ) (3.1) ψ= zm ifz ∈ ∂D, and by linearly interpolating between the two definitions on the annulus D \ int (γ −1 ). It will be important that we establish that the dilatation of the map ψ is uniformly bounded independently of m and δ. Consider Figure 2. By symmetry, it is enough to bound the dilatation needed to interpolate between z → z m on ∂D and z → z m + δz on γ −1 in the shaded grey sector, denoted by Q, where the black vertex and white vertex are two preimages of +1, −1 under z → z m , respectively. We label w (the white vertex) the preimage of −1. Observe that the required dilatation is the ratio of the moduli of the two quadrilaterals shown at the bottom of Figure 2, since the map z m is conformal in the region under consideration and hence preserves moduli. We now argue that this quantity is uniformly bounded as δ → 0 and m → ∞. Indeed consider ψ(Q) as pictured, where the 23 δ term is clear since we know γ −1 is sent by ψ to the circle centered at the origin of radius 23 δ, and z m + δz does not change the argument of a complex number z ∈ R. On the other hand, arg(wm ) = π, and so arg(wm + δw) = π + o(1) as m → ∞. From this we get the 32 δeiθ term, with θ = π + o(1). We now explain the terms in the figure for the image of Q under z m . Notice that for x > 0, if xm + δx = 32 δ and x = 1 + o(1), then xm = 12 δ + o(1) as m → ∞. And if arg(x) = arg(w) with xm + δx = 3 2 δei(π+o(1)) = − 32 δeio(1) , then xm = − 52 δ + o(1) as m → ∞. Observe that log (ψ(Q)) tends to a Euclidean rectangle while log (h (Q)), with h(z) = z m , tends to a quadrilateral as m → ∞. In particular we have 1 mod (ψ (Q)) = (log(δ) + log (3/2) + o(1)) , and (3.2) π 1 1 (log(δ) + log (5/2) + o(1)) ≤ mod(Q) ≤ (log(δ) + log (1/2) + o(1)) , π π which implies that the quotient is uniformly bounded with m → ∞ and δ → 0. UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 9 γ −1 w Q γ 3 −1 2δ 1 ck ’s ψ zm −1 3 1 −1 1 2δ − 23 δeio(1) − 52 δ + o(1) 1 2δ + o(1) log log 3 log 2 + log δ + (π + o(1)) i 5 log 2 + log δ + 0(1) + πi πi πi 3 0 1 0 log 2 + log δ log 2 + log δ + 0(1) Figure 2. This figure illustrates the argument showing that the dilatation of ψ is uniformly bounded independent of m and δ << 1/2. Furthermore it will be important to note that the dilatation of the map ψ is supported on a region whose area tends to zero as m → ∞. This, together with the fact that the dilatation of ψ is bounded independent of m, δ, will be vital to our construction. We have described the map ψ on D. We recall we are defining a quasiregular map g on Dn , and that so far we have g(z) = ψ(z−zn ). We will post-compose with one more quasiconformal map ρ that is nearly identical to the one in [Bis15]. Namely ρ is a quasiconformal map of D which equals z → z + w in 14 D, and is a linear interpolation between this definition on 1 4 D and the identity on ∂D. The value w is a parameter in a small neighborhood of 1/2 that we will adjust later in the course of our construction. Thus our quasiregular map in D-components is g(z) = ρ ◦ ψ(z − zn ). There are m − 1 critical points, and m − 1 critical values that are located on a symmetric Euclidean circle of modulus approximately δ which 10 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK is centered around w. Lastly, notice that the bounded dilatation of g(z) = ρ ◦ ψ(z − zn ) in Dn is still concentrated in the annular region attached to ∂Dn whose area tends to 0 as m → ∞. 4. A general stratey for the construction Our construction will closely resemble the prototype explained in Section 2.2, with some modifications that we now describe. We have a collection of parameters λ, (wn ), (δn ), (dn ) whose definitions we now recall. In the strip S + we have the holomorphic map g(z) = cosh(λ sinh(z)). In the disc components Dn centered at zn , we have g(z) = ρn ◦ ψn (z − zn ), as described in the previous section. Namely, ψn (z) = z dn on ∂D and ψn (z) = z dn + δn z on a subdisc of D. We colloquially refer to (δn ) as the dilation parameters, since ψn effectively acts as a dilation by a factor of δn on a subdisc of D. Finally ρn (0) = wn , with wn contained in a small neighborhood of 1/2. Note that this means the critical values of ρn ◦ ψn (z − zn ) are centered around wn . Theorem 2.1 yields a quasiregular extension of g to all of C without modifying the definition of g in the disc components Dn or the strip S + . The essential part of this theorem is that the quasiregular constant is independent of the choices of the parameters λ, (wn ), (δn ), (dn ) - indeed varying the parameters does not affect the bounded geometry constants as verified explicitly in [FGJ15]. The entire function we consider will be f = g ◦ φ, where φ is a quasiconformal map gotten from invoking the Measurable Riemann Mapping Theorem, so that g ◦φ is indeed holomorphic. Furthermore, we normalize φ to be the identity near infinity (see [Bis15]). Our strategy to construct a wandering domain will be similar to the original argument given in [Bis15]. There the desired map f = g ◦ φ was obtained without modifying g on S + (that is, fixing λ conveniently), and then modifying g on the discs Dn for increasing n. At each step k of the construction the integrating map φk will change the definition of f itself, but the key point of the argument is to show that, on the one hand this process converges to a holomorphic map f , and on the other hand that the modifications do not destroy the desired dynamics. By ensuring the constants λ, (dn ) are sufficiently large, it may be verified as in [Bis15] or [FGJ15] that the orbit of 12 under f escapes to infinity, and that φ is ε-close to the identity uniformly on C for some small ε. Henceforth we will not change the parameter λ. We select a subsequence  of discs Dpn by defining pn to be the integer so that Dpn is the disc closest to f 2 . We let D̃n := Dpn . Henceforth we use the notations (d˜n ), (w̃n ), (δ̃n ) to denote the n 1 parameters corresponding to the subsequence of discs (D̃n ). Define an annulus An := An,µn to be the annulus containing ∂Dn bounded by two concen- tric circles of a distance µn away from ∂Dn (see Figure 3). Since φ is normalized to be the identity near infinity, we may now fix (µn ) → 0, so that the bounded component of C \ An is mapped by φ inside of the bounded component of C \ An,µn /2 , and so that the unbounded UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 11 component of C \ An is mapped by φ into the unbounded component of C \ An,µn /2 , for all n > 0 . We denote the bounded component of C \ An as Dn,µn . We further use the notation µ̃n = µpn and Ãn to denote the annulus Apn = Apn ,µ̃n surrounding ∂ D̃n . Likewise, D̃n,µ˜n = Dpn ,µpn . The general strategy for the first step in the construction of f , shown in Figure 3, is to assign the critical values (arising from critical points in D̃1 ) to land outside (but close to) the second preimage under f of the outer boundary of Ã2 . With this, an argument will show that these critical values do not lie in the same Fatou component as the wandering domain. On the other hand, as long as µ̃1 > µ̃2 , we will be able to ensure that f (D̃1,µ̃1 ) ⊂ f −2 (D̃2,µ̃2 ). Note that the distance of the critical values (arising from critical points in D̃1 ) from the point w1 is determined by δ1 , and w1 determines where the center of the disc D̃1 is sent. In the subsequent sections we will perform this procedure iteratively over n to yield an oscillating bounded wandering domain containing the orbit of D̃1,µ̃1 . Then we will prove that indeed there are no critical values in the orbit of our wandering domain. This will imply that the function f acts univalently on the whole orbit. D̃1 D̃1,µ˜1 π 1 z˜1 z˜2 f µ˜1 π µ˜2 D̃2 2 f f 1   2 1 f 1 f2 1 2 2 w1 f −2 (z˜2 ) Figure 3. This picture illustrates the desired dynamics of f before adjusting w1 to be inside f −2 (D̃2,µ2 ). After this adjustment, we will have f (D̃1,µ1 ) ⊂ f −2 (D̃2,µ2 ). As noted later in this section, changing w1 results in an extra dilatation for the quasiregular map g. We need to ensure that even after making the corresponding correction, we still have f (D̃1,µ1 ) ⊂ f −2 (D̃2,µ2 ). 12 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK 5. Determining the exponent and dilation parameters As noted earlier, we have fixed the parameter λ sufficiently large so that f n (1/2) escapes to infinity independently of further adjustments to (wn ), (δn ), (dn ). Observe that as each of the exponents d1 , d2 , ... → ∞, the corresponding correction maps φ(dn ) converge 1 uniformly on compact subsets to some fixed quasiconformal map φ0 . Moreover in the region S + we know that our function is f (z) = cosh(λ sinh(φ(dn ) (z))). We can take a univalent preimage of D̃1 under f (z) = cosh(λ sinh(φ(dn ) (z))) in S + . Since φ(dn ) converges uniformly on compact subsets to some fixed quasiconformal map φ0 , we know that f −1 (D̃1 ) converges to some fixed region, namely (cosh ◦λ sinh ◦ φ0 )−1 (D̃1 ). Indeed by a similar argument, as each dn → ∞ over all n, f −n (D̃n ) converges to some fixed region near 1/2, namely ((cosh ◦λ sinh ◦φ0 )−1 )◦n (D̃n )). We now iteratively define the sequence of exponents (dn ) and the dilation parameters (δn ). We will determine how to fix d˜n only at step n, and we will not change the exponents d1 , d2 , ..., d˜n−1 at step n, once we have determined these exponents at previous steps. We start with step 1. Consider increasing each dn → ∞. For reasons noted earlier, we have f −2 (Ã2 ) converging to some fixed topological annulus (cosh ◦λ sinh ◦φ0 )−2 (Ã2 ). Consider, as pictured in Figure 4, the two concentric circles C2 , C20 minimizing the ratio radius(C20 ) radius(C2 ) so that int(C20 ) ⊃ (cosh ◦λ sinh ◦φ0 )−2 (Ã2 ), and int(C2 ) ∩ (cosh ◦λ sinh ◦φ0 )−2 (Ã2 ) = ∅. What we really want is that (5.1) δ̃1 > radius(C20 ) and δ̃1 (1 − µ̃1 ) < radius(C2 ). Indeed, roughly speaking, the first inequality will allow us to ensure that the critical values of f coming form the critical points in D̃1 lie sufficiently far from the outer boundary of f −2 (Ã2 ), while the second inequality will show that the size of f (D̃1,µ̃1 ) is smaller than a disk contained in f −2 (D̃2,µ̃2 ). See Figure 5. As mentioned in the caption of Figure 4, even if we know that (1 + µ̃2 )/(1 − µ̃2 ) < (1 + µ̃1 )/(1 − µ̃1 ), it may not be the case that radius(C20 )/radius(C2 ) < (1 + µ̃1 )/(1 − µ̃1 ) due to the distortion of f . But, as we presently justify, by Theorem 2.2 we have that radius(Cn0 )/radius(Cn ) (5.2) → 1 as n → ∞. (1 + µ̃n )/(1 − µ̃n ) 1Herewe are using the notation φ(dn ) to emphasize that the correction map depends on a choice of the parameters (dn ). Indeed the correction map also depends on a choice of (δn ), (wn ), but that dependence is inessential for the purposes of this argument. UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 13 Ãn 2µ̃n Cn0 Cn (cosh ◦λ sinh ◦φ0 )−n Figure 4. Here we see the definitions of the concentric circles C20 , C2 . Note that the ratio rad(C20 )/rad(C2 ) may be larger than the ratio (1 + µ̃2 )/(1 − µ̃2 ) pertaining to the annulus Ã2 . (If f were a Euclidean similarity, we would have radius(C20 )/radius(C2 ) = (1 + µ̃2 )/(1 − µ̃2 )). In particular even if (1 + µ̃2 )/(1 − µ̃2 ) < (1 + µ̃1 )/(1 − µ̃1 ), it may not be the case that radius(C20 )/radius(C2 ) < (1 + µ̃1 )/(1 − µ̃1 ). However, as we argue, the ratio (radius(Cn0 )/radius(Cn ))/((1+ µ̃n )/(1− µ̃n )) → 1 as n → ∞, so that if necessary we can just consider instead f −n (Ãn ) for n > 2. Indeed, consider disks Bn−1 of a fixed unit size containing λ sinh φ0 (f n−1 (1/2)). By the expanding properties of cosh, we have that cosh : Bn−1 → cosh(Bn−1 ) is conformal, Ãn ⊂ cosh(Bn−1 ), and that as n → ∞, distn := dist(Ãn , ∂(cosh(Bn−1 ))) → ∞. It follows from Theorem 2.2 (b), that the distortion of the branch F of f −n mapping cosh(Bn−1 ) conformally to a neighborhood of 1/2, tends to 0 as n increases. More precisely, for points z on the boundary of Ãn , and z̃n the center of Ãn , the theorem yields max |F (z) − F (z̃n )| radius(Cn0 ) |z−z̃n |=1+µ̃n 1 + µ̃n (distn + (1 − µ̃n ))2 1 + µ̃n = ≤ 2 −→ , radius(Cn ) min |F (z) − F (z̃n )| 1 − µ̃n (distn − (1 + µ̃n )) n→∞ 1 − µ̃n |z−z̃n |=1−µ̃n and also max |F (z) − F (z̃n )| |z−z̃n |=1+µ̃n 1 + µ̃n (distn − (1 − µ̃n ))2 1 + µ̃n ≥ 2 −→ , min |F (z) − F (z̃n )| 1 − µ̃n (distn + (1 + µ̃n )) n→∞ 1 − µ̃n |z−z̃n |=1−µ̃n giving (5.2). Thus we have shown that we can find some sufficiently large n2 so that radius(Cn0 2 )/radius(Cn2 ) < (1+ µ̃1 )/(1− µ̃1 ). We also require that radius(Cn0 2 )/radius(Cn2 ) < 1/(1 − µ̃1 ). Once we have found such an n2 , we fix δ̃1 = radius(Cn0 2 ) · const1 , 14 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK where radius(Cn2 ) 1 1 < const1 < . radius(Cn0 2 ) 1 − µ̃1 As a consequence (1 − µ̃1 ) · δ̃1 = (1 − µ̃1 ) · radius(Cn0 2 ) · const1 < radius(Cn2 ) and therefore the inequalities (5.1) are satisfied. Up to this point, we have arranged for the image of D̃1,µ̃1 to be smaller than the n2 - preimage of D̃n2 ,µ̃n2 . In the next section we will perform surgery to move the image of D̃1,µ̃1 inside the preimage of D̃n2 ,µ̃n2 . On the other hand, since const1 > 1, the critical values coming from D̃1 land at a distance greater than radius(Cn0 2 ) from the center of the image of D̃1 . We will need this estimate to prove that there are no critical values in the orbit of our wandering domain. Now fix d1 , d2 , ..., d˜1 sufficiently large so that we have φ(dn ) − φ0 < ε1 on some large com- pact disc centered at 0 and containing Ãn2 . The quantity ε1 is chosen sufficiently small so that for k = n2 , we have that (cosh ◦λ sinh ◦φ(dn ) )−k (Ãk ) is ε01 close to (cosh ◦λ sinh ◦φ0 )−k (Ãk ) in the Hausdorff metric, where ε01 << modulus((cosh ◦λ sinh ◦φ0 )−k (Ãk )). Moreover we want this condition to hold true regardless of how we choose dp1 +1 , dp1 +2 , ..., d˜2 = dp2 , ... at further steps. This is possible because of uniform convergence of φ(dn ) to φ0 on compact subsets. Let us insist further on the choice of ε1 . We have a set of inequalities in step 1, for example (5.1), that are true if we replace φ(dn ) with the limiting map φ0 . All of these inequalities have some “wiggle room”, and we are choosing ε1 to be much smaller than that wiggle room, so that the inequalities hold even if we replace φ0 with φ(dn ) . We will also put a further condition on d˜1 . Recall that the (uniformly bounded) dilatation of g(z) on the disc D̃1 lives on a topological annulus we call Ũ1 which tends in area to zero as d˜1 → ∞. We require d˜1 to be sufficiently large so that a quasiconformal map supported on Ũ1 is ε1 close to the identity uniformly on S + . This ends the first step of the construction where we have fixed δ̃1 and d˜1 , and have given lower bounds for further dn . At step 2, we will determine how to fix d˜n2 . Notice that as we have fixed d˜1 , if we increase dp1 +1 , dp1 +2 , ... → ∞, the map φ(dn ) no longer converges to φ0 (since d˜1 now no longer tends to infinity), however φ(dn ) converges uniformly on compact subsets to some φ1 , and φ1 is ε1 close to φ0 on the relevant compact disc containing 0, Ã1 , Ãn2 . We go through a similar procedure as in step 1, choosing some n3 > n2 so that radius(Cn0 3 ) 1 1 + µ̃n2 < < radius(Cn3 ) 1 − µ̃n2 1 − µ̃n2 UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 15 Now, however, we look at φ1 rather than φ0 . Namely we are defining Cn3 , Cn0 3 by considering (cosh ◦λ sinh ◦φ1 )−n3 (Ãn3 ). Then, as before, we fix δ̃n2 = radius(Cn0 3 ) · const2 , where radius(Cn0 3 ) 1 1 < const2 < . radius(Cn3 ) 1 − µ̃n2 Now we fix dp1 +1 , dp1 +2 , ..., d˜n2 sufficiently large so that φ(dn ) − φ1 < ε2 on some large com- pact disc centered at 0 and containing Ãn3 . The quantity ε2 is chosen so that, for example, for k = n2 , n3 , we have (cosh ◦λ sinh ◦φ(dn ) )−k (Ãk ) is ε02 close to (cosh ◦λ sinh ◦φ1 )−k (Ãk ) in the Hausdorff metric, where ε02 << modulus((cosh ◦λ sinh ◦φ1 )−k (Ãk )). We also specify d˜n2 sufficiently large so that a quasiconformal map supported on Ũn2 is ε2 close to the identity uniformly on S + . We continue similarly, hence defining dn over all n. We note that at each step, the correction map φ(dn ) is εn close to φn , where φn is the limit obtained by fixing exponents already determined in previous steps and letting subsequent exponents tend to infinity. Moreover φn is ε1 -close to φ0 , ε2 -close to φ1 , ... and εn -close to φn−1 , at least on the relevant compact subsets. Thus, if we denote φ as the correction map associated to the iteratively determined choice of (dn ), (δn ) as above, we know that φ is ε1 -close to φ0 , φ is ε2 -close to φ1 , . . . , on the relevant compact subsets of C. 6. Performing Surgery to Yield a Wandering Domain Having fixed the parameters λ, (dn ), (δn ), we will now choose the parameters (wn ) itera- tively so as to yield a wandering domain. The hard work was already done in the previous section. We start at what we call step 0, where we have the entire function f = g ◦ φ, where φ is the correction map associated to the choice of parameters (dn ), (δn ) determined in the previous section, and for now we have set each wn = 1/2. Now in step 1 we consider the region f −n2 (Ãn2 ) and the corresponding circles Cn2 , Cn0 2 pictured in Figure 4. We set w̃1 to be the common center of the circles Cn2 , Cn0 2 . (Recall that Cn2 , Cn0 2 were defined using φ0 not φ, but we know φ0 is ε1 close to φ on the relevant compact subset, which is good enough). In other words, we modify our quasiregular map 2 g only in D̃1 , by setting ρ̃1 (0) = w̃1 to be the center of the circles Cn2 , Cn0 2 . We denote this new quasiregular map by g1 , so that g1 = g except in D̃1 . In particular we know that now g1 ◦ φ(D̃1,µ̃1 ) ⊂ (g1 ◦ φ)−n2 (D̃n2 ,µ̃n2 ) by the estimates in the previous section, and also that the critical points under g1 ◦ φ coming from D̃1 are sent outside Cn0 2 . However notice that g1 ◦φ is no longer holomorphic, since there is a new dilatation concentrated on an annulus we called Ũ1 near ∂ D̃1 . We can find a correction map η1 so that f1 := g1 ◦ φ ◦ η1 is holomorphic. 2recall that in a D-component Dn , we had the definition g(z) = ρn ◦ ψn (z − zn ). 16 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK Moreover we have already arranged for d˜1 to be sufficiently large so as to ensure η1 is close enough to the identity in the relevant region, so as to imply that we still have (see Figure 5) f1 (D̃1,µ̃1 ) ⊂ (f1 )−n2 (D̃n2 ,µ̃n2 ). Ãn2 D̃2,µ̃2 D̃1,µ̃1 2µ̃2 2µ̃1 f1 f1n2 −1 w̃1 Figure 5. The first step in the construction of the univalent wandering domain, after adjusting the parameters so that f1 (D̃1,µ̃1 ) ⊂ (f1 )−n2 (D̃n2 ,µ̃n2 ), and so that the critical values are outside (f1 )−n2 (Ãn2 ). Continuing to step 2, we now consider f1−n3 (An3 ) and the corresponding circles Cn3 , Cn0 3 . (Again note that Cn3 , Cn0 3 were defined using φ1 , not the new correction map φ ◦ η1 , but we have already arranged for φ ◦ η1 to be ε2 close to φ1 on the relevant compact subset). We set w̃n2 to be the common center of the circles Cn3 , Cn0 3 , i.e. we adjust our quasiregular map g1 only on the disc D̃n2 and denote this new quasiregular map g2 . We know that g2 ◦ φ ◦ η1 (D̃n2 ,µ̃n2 ) ⊂ (g2 ◦ φ ◦ η1 )−n3 (D̃n3 ,µ̃n3 ) and that the critical points under g2 ◦ φ ◦ η1 coming from D̃n2 are sent outside Cn0 3 . However g2 ◦ φ ◦ η1 is no longer holomorphic. Thus we find a new quasiconformal correction map η2 so that f2 := g2 ◦ φ ◦ η1 ◦ η2 is holomorphic. Moreover we have already arranged for d˜n2 to be sufficiently large so that f2 (D̃n2 ,µ̃n2 ) ⊂ (f2 )−n3 (D̃n2 ,µ̃n2 ). UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 17 Now we also have to worry about whether it is still the case that f2 (D̃1,µ̃1 ) ⊂ (f2 )−n2 (D̃n2 ,µ̃n2 ), however this was also built in to the definition of d˜n2 . In the previous section we arranged for the errors coming from η1 , η2 , ... to sum up so that continuing as above we will still have fk (D̃1,µ̃1 ) ⊂ (fk )−n2 (D̃n2 ,µ̃n2 ) for all k. Note that by construction, the errors k decrease exponentially fast with k so that summability is ensured. We continue iteratively defining as above some sequence of entire maps fk by adjusting the quasiregular map gk−1 from the previous step and then precomposing with a new correction map ηk . The parameters (dn ), (δn ) were chosen in the previous section so as to ensure that at step k we have D̃1,µ̃1 iterating after some number of steps into D̃nk ,µ̃nk under fk . Taking the limit under this procedure we have a limiting map we call f which is our desired function in class B with a wandering domain. In the next section we verify that there are no critical values in the orbit of our wandering domain. 7. Verifying that the wandering domain is indeed univalent We will prove that there are no critical values in the orbit of the Fatou component con- taining D̃1,µ̃1 . Since each component of the wandering domain is simply connected, this will imply that f acts univalently on the constructed wandering domain. Here it will be crucial that we have arranged for the critical points of f in D̃nk to land outside Cn0 k+1 . This means that the critical points of f in D̃nk iterate into the unbounded component of C \ Ãnk+1 . Hence all we need to prove is that the unbounded component of C \ Ãnk+1 must have empty intersection with the Fatou component containing D̃nk+1 ,µ̃nk+1 . To keep notation simple we will illustrate the argument by proving that the unbounded component of C \ Ã1 has empty intersection with the Fatou component containing D̃1,µ̃1 . The argument is similar to that given in [Laz17] where it is proven that the Fatou compo- nent containing D̃1,µ̃1 is bounded. We call this Fatou component U . Let u ∈ D̃1,µ̃1 ⊂ U , and suppose by way of contradiction that there exists a point v ∈ U ∩ (C \ Ã1 ). We may assume further that v lies in an R-component neighboring D̃1,µ̃1 . We call this R-component V . We note first that R is in the escaping set of f (proven in [Laz17]) and hence, since f ∈ B, that R is in the Julia set [EL92]. That means that f n (U ) ⊂ H over all n ∈ N. Denote by dH , df n (U ) , dU the hyperbolic distance in H, f n (U ), U respectively. We have then that (7.1) dH (f n (u), f n (v)) ≤ df n (U ) (f n (u), f n (v)) ≤ dU (u, v) for all n ∈ N where both inequalities are consequences of Schwarz’s lemma. We argue that the left hand side of (7.1) must tend to infinity with n, at least in some subsequence. This will be our needed contradiction. Remember through the course of our construction that we have arranged for |f (u) − 1/2| < 1/4 since u ∈ D̃1,µ̃1 and f (D̃1,µ̃1 ) ⊂ 18 NÚRIA FAGELLA, XAVIER JARQUE, AND KIRILL LAZEBNIK f −2 (D̃n2 ,µ̃n2 ). What about f (v)? We will argue that in fact we must have |f (v)| > 1. Indeed, consider a partition of Hr into Wk := {z : πk < Im(z) < (k + 1)π, Re(z) > 0} over k ∈ Z, as illustrated to the left of Figure 6. In the course of the proof of the folding theorem, the map τ : V → Hr was replaced by a locally quasiconformal τ̂ , and the graph T was decorated with edges to form a graph T 0 , so that τ̂ : V \ T 0 → Hr was quasiconformal and edges of T 0 were sent to the edges {k < Im(z) < (k + 1)π, Re(z) = 0} (see [Bis15, Laz17, FGJ15] for details). Pulling the regions Wk := {k < Im(z) < (k + 1)π} back under τ̂ we have regions τ̂ −1 (Wk ), and pulling back further by φ ◦ η1 ◦ η2 ◦ ... we have regions as pictured in Figure 6. Now if τ̂ −1 (Wk ) neighbors a D-component, recall that for z ∈ τ̂ −1 (Wk ), we have g(z) = η (τ̂ (z)) where η = exp on ∂Wk interpolated with η = cosh on Re(Wk ) > π. In particular this means that as long as z ∈ τ̂ −1 (Wk ), where τ̂ −1 (Wk ) neighbors a D-component, then |g(τ̂ (z))| > 1. Hence, we just need to argue that φ ◦ η1 ◦ η2 ◦ ...(v) ∈ τ̂ −1 (Wk ) where τ̂ −1 (Wk ) neighbors a D-component (remember our entire function is just f = g ◦ τ̂ ◦ (φ ◦ η1 ◦ η2 ◦ ...)). Indeed U can not intersect a region (τ̂ ◦φ◦η1 ◦η2 ◦...)−1 (W` ) ⊂ (φ◦η1 ◦η2 ◦...)−1 (V ) which does not neighbor (φ◦η1 ◦η2 ◦...)−1 (D̃1 ), the reason being that U also contains points inside (φ◦η1 ◦η2 ◦...)−1 (D̃1 ) and therefore U would have to cross the boundary of some region τ̂ −1 (Wl ). Namely U would have to cross (τ̂ ◦ φ ◦ η1 ◦ η2 ◦ ...)−1 ({y = (l + 1)π}) ∪ (τ̂ ◦ φ ◦ η1 ◦ η2 ◦ ...)−1 ({y = lπ}), but (τ̂ ◦ φ ◦ η1 ◦ η2 ◦ ...)−1 ({y = (l + 1)π}) ∪ (τ̂ ◦ φ ◦ η1 ◦ η2 ◦ ...)−1 ({y = lπ}) ⊂ J (f ), since they are sent to R by f , and this is a contradiction. 3π 2π W1 π φ ◦ η1 ◦ η2 ◦ · · · W0 τ̂ 0 −π W−2 −2π V Figure 6. Rough sketch of the preimages of the semistrips Wk under τ̂ and φ ◦ η1 ◦ η2 ◦ ...., to show that the wandering domain needs to be contained in the D−components. For clarity’s sake, only a few preimages are shown. Now we have established that |f (u) − 1/2| < 1/4 and |f (v)| > 1. Note that by the expanding properties of the exponential, upon subsequent applications of f we have that the distance between f (u), f (v) increases upon each iterate. Indeed f k (u), f k (v) must both lie in S + for 1 ≤ k < n2 (otherwise the wandering domain would have to cross ∂S + ⊂ UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS 19 J(f )), and we have that f (z) = cosh(λ sinh(φ(z)) in S + , where φ is uniformly close to the identity. Note, for example, that |f n2 (u) − f n2 (v)| > | expn2 (3/4) − expn2 (1)|, and that | expn (3/4) − expn (1)| → ∞ as n → ∞. Next note that we know from the construction that f n2 (u) ∈ D̃n2 ,µ̃n2 . This means that |f n2 +1 (u) − 1/2| < 1/4. What about f n2 +1 (v)? Well as observed in [Laz17], by considering preimages of R one sees that there is a ray belonging to J (f ) connecting zn − i to ∞ over all n (remember zn is the center of the D-component Dn ). This means that f n2 (v) must lie in an R-component neighboring Dn2 . Hence, as before, we know that |f n2 +1 (v)| > 1. Now we start over, only that we have n3 > n2 more iterations of f (z) = cosh(λ sinh(φ(z)) in S + before the points f n2 +1 (v), f n2 +1 (u) leave S + . Hence again the expanding properties of exp, together with the fact that |f n2 +1 (v)| > 1, |f n2 +1 (u)−1/2| < 1/4, imply that |f n2 +n3 (v)− f n2 +n3 (u)| > | expn3 (3/4) − expn3 (1)|. Hence since nk → ∞ as k → ∞, we know that |f n2 +n3 +...+nk (v) − f n2 +n3 +...+nk (u)| → ∞ as k → ∞. Moreover since by construction we know that f n2 +n3 +...+nk (u) ∈ D̃nk , we have that dH (f n2 +n3 +...+nk (u), f n2 +n3 +...+nk (v)) → ∞, which is our needed contradiction. References [Bak76] I. N. Baker. An entire function which has wandering domains. J. Austral. Math. Soc. Ser. A, 22(2):173–176, 1976. [Bak02] I. N. Baker. Limit functions in wandering domains of meromorphic functions. Ann. Acad. Sci. Fenn. Math., 27(2):499–505, 2002. [Ber95] Walter Bergweiler. Invariant domains and singularities. Math. Proc. Cambridge Philos. 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Departament de Matemàtiques i Informàtica, Institut de Matemàtiques de la Universitat de Barcelona (IMUB) and Barcelona Graduate School of Mathematics (BGSMath). Gran Via 585, 08007 Barcelona, Catalonia E-mail address: nfagella@ub.edu Departament de Matemàtiques i Informàtica, Institut de Matemàtiques de la Universitat de Barcelona (IMUB), and Barcelona Graduate School of Mathematics (BGSMath). Gran Via 585, 08007 Barcelona, Catalonia E-mail address: xavier.jarque@ub.edu California Institute of Technology. 1200 E California Blvd, Pasadena, CA 91125. E-mail address: lazebnik@caltech.edu