Synthetic deformation space of an entire function

Masahiko Taniguchi

Department of Mathematics, Kyoto University, Kyoto 606, Japan

abstract:

Finitely constructable entire functions are called structurally finite. Structurally finite entire functions have many nice properties. In this note, we define a new kind of the deformation space of a general entire function, and discuss about completeness and stability of such deformation spaces in case of structurally finite entire functions.

1 Deformation spaces

The dynamic Teichmüller space of a holomorphic endomorphism (see [7], [8]) is rather tight to discribe variations of singular affine structures on ℂ. So we relax the connection between the domain and the target, and introduce a different kind of topology to the space of singular affine structures.

Definition 1.1 Let f be a non-affine entire function. Then the full deformation set F D(f) of f is the set of all entire functions g such that there is a quasiconformal self-map ϕ of ℂ satisfiying the qc-L^{∞} condition:

∥f − g ∘ ϕ∥_{∞} = sup_{ℂ}|f − g ∘ ϕ| < ∞.

Here we may assume that such a ϕ as above is always normalized, that is, fixes 0 and 1. Here the following lemma is clear.

Lemma 1.2 If g ∈ F D(f), then f ∈ F D(g).

Definition 1.3 For any two function f_{1}, f_{2} in F D(f), we set

d(f_{1}, f_{2}) = inf (log K(ϕ_{1} ∘ ϕ_{2}^{−1}) + ∥f_{1} ∘ ϕ_{1} − f_{2} ∘ ϕ_{2}∥_{∞}) ,

where the infimum is taken over all normalized quasiconformal automorphisms ϕ_{1}, ϕ_{2} of ℂ satisfying the qc-L^{∞} conditions between f and f_{1}, f_{2}, respectively.

Lemma 1.4 The psuedo-distance d is a distance, and F D(f) with this distance is a complete metric space.

Proof. If d(f_{1}, f_{2}) = 0, then there are sequences {ϕ_{j,n}} (j = 1, 2) such that

log K(ϕ_{1,n} ∘ ϕ_{2,n}^{−1}) + ∥f_{1} ∘ ϕ_{1,n} − f_{2} ∘ ϕ_{2,n}∥_{∞} → 0

as n, m → ∞. Then since {ϕ_{1,n} ∘ ϕ_{2,n}^{−1}} is a normal family, they converge to the identity. Moreover, since

∥f_{1} ∘ ϕ_{1,n} ∘ ϕ_{2,n}^{−1} − f_{2}∥_{∞} → 0,

we conclude that f_{1} and f_{2} coincide with each other. Thus, d is a distance.

Next, to show completeness, let {f_{n}} be an arbitrary Cauchy sequence. Then, there is a sequence of normalized quasiconformal ϕ_{n} such that

log K(ϕ_{n} ∘ ϕ_{m}^{−1}) + ∥f_{n} ∘ ϕ_{n} − f_{m} ∘ ϕ_{m}∥_{∞} → 0

as n, m → ∞. Fix such a sequence {ϕ_{n}}. Then a standard argument shows that, taking a subsequence if necessary, we may assume the existence of a normalized quasiconformal automorphism ϕ_{∞} such that ϕ_{n} converge to ϕ_{∞} locally uniformly and satisfy

log K(ϕ_{∞} ∘ ϕ_{n}^{−1}) → 0.

In particular, f_{n} are locally uniformly bounded because of the qc-L^{∞} condition between f_{1} and f_{n}, and hence we may assume that f_{n} converge locally uniformly to some non-constant entire function f_{∞}. Moreover, the above condition implies that

∥f_{n} ∘ ϕ_{n} − f_{∞} ∘ ϕ_{∞}∥_{∞} → 0,

since f_{n} ∘ ϕ_{n}(z) − f_{m} ∘ ϕ_{m}(z) tends to f_{n} ∘ ϕ_{n}(z) − f_{∞} ∘ ϕ_{∞}(z) as m → ∞ for every z ∈ ℂ. Thus f_{n} converge to f_{∞} in F D(f).

Definition 1.5 We call this distance the synthetic Teichmüller distance on F D(f). The space F D(f) equipped with this synthetic Teichmüller distance is called the full synthetic deformation space of f and written as F SD(f).

In this paper, we discuss about the full synthetic deformation spaces. But, to discuss some kind of stability, we also consider several other deformation spaces.

In the sequel, we consider only such an f that the singular values of f have a finite number of accumulating points.

Also, we call a point not virtually evenly covered a singular value of the covering. Here for such a function f as above, we say that a point α is virtually evenly covered if α is not a critical value and if there is a neighborhood U and a simple path L from the boundary of U to α such that every component D of f^{−1}(U − L) is relatively compact and f is a biholomorphic map of D onto U − L.

Recall that, if f belongs to the Speiser class, then singular values of the covering are nothing but singular values, i.e. values not evenly covered by f.

Definition 1.6 The set S(f) consisting of all entire functions g such that there are bijections between critical points counted with their multiplicity, and between singular values of the covering counted with their coincidence, of g and f.

Note that, when singular values of the covering are countable in number, then S(f) has a set of the natural local parameters, which gives a local injection to ℂ^{∞}.

Definition 1.7 For an entire function f, let Top(f) be the set of all entire functions topologically equivalent to f, and QC(f) the set of all entire functions quasiconformally equivalent to f.

Here we say that g is topologically (resp., quasiconformally) equivalent to f if there are self-homeomorphisms (resp., quasiconformal maps) ϕ and ψ of ℂ such that g = ψ ∘ f ∘ ϕ.

Now clearly,

QC(f) ⊂ Top(f) ⊂ S(f).

2 The case of structurally finite entire functions

To construct entire functions or singular affine structures, we need two kinds of building blocks; ones are quadratic blocks

az^{2} + bz + c : ℂ → ℂ (a ≠ 0)

and the others are exponential blocks (exp-blocks)

a exp bz + c : ℂ → ℂ (ab ≠ 0).

Remark 2.1 The trivial covering structure is given by similarities:

az + b : ℂ → ℂ (a ≠ 0).

We call such one a ℂ-block.

Definition 2.2 (Maskit surgery by connecting functions)

Let f_{j} : ℂ → ℂ (j = 1, 2) be two entire functions, and A_{j} be the set of singular values of f_{j}. Assume that there is a cross-cut L in ℂ (i.e. the image L of a continuous proper injection of ℝ into ℂ) such that

1. L ∩ A_{1} is coincident with L ∩ A_{2}, and is either empty or consists of a single point z_{0}, which is an isolated point of each A_{j},

2. L separates A_{1} − {z_{0}} from A_{2} − {z_{0}}, and

3. if L ∩ A_{1} = L ∩ A_{2} = {z_{0}}, then {z_{0}} is a critical value of each f_{j}: for a small disk U with center z_{0} such that U ∩ A_{j} = {z_{0}}, f_{j}^{−1}(U) has a relatively compact component W_{j} which contains a critical point for each f_{j}.

Then we say that an entire function f : ℂ → ℂ is constructed from f_{1} and f_{2} by a Maskit surgery with respect to L, and to {W_{j}} when they exist, if the following assumptions are satisfied: Let D_{j} be the component of ℂ − L containing A_{j} − {z_{0}}. Then there exist

1. components D̃_{1} and D̃_{2} of f_{1}^{−1}(D_{2}) and f_{2}^{−1}(D_{1}), respectively, such that f_{j} : D̃_{j} → D_{3−j} is biholomorphic and D̃_{j} ∩ W_{j} ≠ ∅ if L ∩ A_{j} are nonempty, 2. a cross-cut L̃ in ℂ such that f gives a homeomorphism of L̃ onto L,

and

3. a conformal map ϕ_{j} of ℂ − D̃_{j} onto U_{j} such that f_{j} = f ∘ ϕ_{j} on ℂ − D̃_{j} for each j, where U_{1} and U_{2} are components of ℂ − L̃.

Definition 2.3 We say that an entire function is structurally finite if it is constructed from a finite number of building blocks by Maskit surgeries.

We say that a structurally finite function is of type (p, q) if it is constructed from p quadratic blocks and q exp-blocks.

Remark 2.4 1. Suppose that f is structurally finite, then every g ∈ S(f) is of the same type as f.

2. Suppose that f belongs to the Speiser class, then

QC(f) = Top(f).

Compare with the Maskit combinations in [6]. Also, note that structural finiteness is characterized by the following topological condition.

Definition 2.5 We call that a holomorphic endomorphism of ℂ covers ℂ almost evenly if there are only a finite number of points which are not evenly covered, and at every such point α, there are only a finite number of components of f^{−1}(B) where f is not a homeomorphism onto B for every small disk B with center α.

Proposition 2.6 (Topological Characterization) Every structurally finite entire function covers ℂ almost evenly. Conversely, every entire function which covers ℂ almost evenly is structually finite.

Proof. The first assertion is clear.

Next, let f be an almost evenly covered entire function. Then we take a cross-cut L which togather with ∞ is freely homotopic to a simple closed curve C surrounding a single singular value, with a homotopy keeping the set of singular values fixing pointwise, in the Riemann sphere ℂ̂. Then there are a finite number of components of f^{−1}(L) which divides the singular affine structures non-trivially.

Applying such decompositions (of Klein types) a finitely many times, we can decompose the given almost evenly covered f into a finite number of exp-blocks and simple critical blocks

z ↦ a(z − z_{0})^{k} + c, (a ≠ 0)

with degree k (≥ 2).

Now, let f be such a simple critical block, and L a cross-cut passing through the critical value c of f. Take a suitable pair of components of f^{−1}(L − {c}) whose images by f cover L − {c}. Then we can decompose f into two simple critical blocks with degrees k_{1} and k_{2} satisfying k_{1}+k_{2} = k+1. Applying such decompositions (of Maskit types) a finite number of times, we can decompose f into k − 1 quadratic blocks.

Theorem 2.7 (Inclusion Theorem) For a structurally finite entire function f, the full deformation set F D(f) contains all structurally finite entire functions of the same type as that of f.

Corollary 2.8 For a structurally finite f,

S(f) ⊂ F D(f).

The proof of Inclusion Theorem will be given in the final section. Actually, the proof also implies the following

Corollary 2.9 If a structurally finite f of type (p, q) has p + q distinct singular values, then

QC(f) = Top(f) = S(f).

Definition 2.10 We define the set SF_{p,q} (with p + q ≥ 1) by setting

SF_{p,q} = {∫_{0}^{z}(c_{p}t^{p} + ⋯ + c_{0})e^{a_{q}t^{q}+···+a_{1}t}dt + b}

with c_{p}a_{q} ≠ 0 if q > 0, and we regard that SF_{p,0} = P oly_{p+1}; the set of all polynomials of degree exactly p + 1.

Such primitive functions have already appeared as typical examples in various contexts. See for instance, [1], [2], [3], and [9].

Now the topological characterization shows the following Corollary 2.11 Every element of SF_{p,q} is structurally finite and of type (p, q).

Thus, Inclusion Theorem implies that F D(f) contains SF_{p,q} for every f ∈ SF_{p,q}. In particular, the synthetic Teichmüller distance is finite on SF_{p,q} × SF_{p,q}, and we may equip SF_{p,q} with the synthetic Teichmüller topology.

Definition 2.12 For f ∈ SF_{p,q}, we set SD(f) = SF_{p,q}, and equip it with the sythetic Teichmüller topology, which we call the synthetic deformation space of f.

Actually, we can show the following

Theorem 2.13 (Representation Theorem) An entire function is structurally finite and of type (p, q) if and only if it belongs to SF_{p,q}.

For a proof, see [14], where the proof relies on the proof of Inclusion Theorem given here.

Proposition 2.14 For every P ∈ SF_{p,0} = P oly_{p+1}, SD(P) = F SD(P).

Proof. Fix Q ∈ F SD(P). Then since P is proper, so is Q. Indeed, if there were a compact ball B such that Q^{−1}(B) is non-compact, then U = (Q ∘ ϕ)^{−1}(B) would be non-compact, where ϕ is a quasiconformal map satisfying the qc-L^{∞} condition between P and Q. Hence sup_{U} |P | should be +∞. But since sup_{U} |P | ≤ ∥P − Q ∘ ϕ∥ + sup_{B} |z| < +∞, this is impossible.

Hence Q is polynomial, and then the degree should be equal to p + 1.

On the other hand, we have

Example 2.15 (Melting of ℂ-decorations)

f_{j}(z) = (1 + \frac{z}{j} ) e^{z} ∈ SF_{1,1}

converge to g(z) = e^{z} ∈ SF_{0,1} with respect to the synthetic Teichmüller topology as j tend to +∞.

Similarly,

g_{j}(z) = e^{2z} + \frac{2}{j} e^{z}

are structurally infinite, but converge to g(2z) with respect to the synthetic Teichmüller topology as j tend to ±∞.

Thus we can see that, even for an f ∈ SF_{p,q} with q > 0, F D(f) always so large that it contains structurally infinite functions, and SD(f) is neither closed nor open in F SD(f), except for the case that p = 0. Hence we need to consider slightly large complete subspaces. The following kind of compactness gives such ones, and also shows that SD(f) can be considered as a stratum of F SD(f).

Theorem 2.16 (Compactness Theorem) Suppose that f ∈ SF_{p,q} with q > 0, and let {f_{j}} be a seqeunce in SD(f) converging to some g in F SD(f). Then g is structurally finite, and of type (p′, q) with p′ ≤ p.

In particular,

SF_{≤p,q} = ∪_{p′≤p} SF_{p′,q}

with the synthetic Teichmüller topology is a complete metric space, and hence a completion of SD(f).

Proof. Let S = {α_{1}, ⋯, α_{m}} be a finite set of distinct singular values of g, and let r be the minimum among all the distances between distinct points in S. Fix an ϵ > 0 smaller than r/8 and, by considering sufficiently large j only, we may assume

∥g − f_{j} ∘ ϕ_{j}∥_{∞} < ϵ

for every j with a suitable normalized quasiconformal map ϕ_{j}.

Now let D_{ℓ} be the disk with center α_{ℓ} and radius r/4, i.e. the r/4neighborhood of α_{ℓ}. Then g^{−1}(D_{ℓ}) has a component where g is not biholomorphic. Let V be such a component. If V is relatively compact, then ϕ_{j}(V) is also relatively compact. Set a_{V} = g^{−1}(α_{ℓ}) ∩ V. Then the winding number of f_{j}∘ϕ_{j}(∂V) around f_{j}∘ϕ_{j}(a_{V}) is equal to that of g(∂V) around α_{ℓ}. Hence f_{j} has critical points in ϕ_{j}(V) and the number counted with their multiplicities is the same as that of g at a_{V}.

Next, suppose that V is not relatively compact. Since f_{j} ∘ ϕ_{j}(V) are bounded and f_{j} are structurally finite, f_{j} should have an asymptotic value in ϕ_{j}(V). Thus the number d_{ℓ} of components of g^{−1}(D_{ℓ}) where g is not biholomorphic is finite, and

∑_{ℓ=1}^{m} d_{ℓ} ≤ p + q.

Since S is arbitrary, this implies that g is structurally finite. In particular, every asymptotic value of g corresponds to a logarithmic singularity. And it remains to show that the type of g is (p′, q′) with p′ ≤ p, q′ = q.

Here, we have shown that p′ ≤ p, q′ ≤ q. Also note that, if a point α satisfies that every component of g^{−1}(D) is relatively compact, where D is the r/4-neighborhood of α, there are asymptotic values of no f_{j} in the r/8neighborhood D′ of α. Indeed, if some f_{j} has an asymptotic value in D′, then f_{j}^{−1}(D′) has a non-compact component V′. But ϕ_{j}(V′) should be contained in g^{−1}(D), which is impossible. Thus, if q′ < q, then we would have a disk neighborhood D of an asymptotic value of g such that some component W of g^{−1}(V) is simply connected, ∂W is a cross-cut, and ϕ_{j}(W) contains two paths determining different asymptotic values of some f_{j}. But then f_{j} can not be bounded on ϕ_{j}(W). This contradiction shows that q′ = q.

Now SF_{≤p,q} has another natural topology induced from the coefficents space of representatives. For instance, we define the line element ds by

ds = \frac{∑_{m=0}^{p} |dc_{m}|}{∑_{m=0}^{p} |c_{m}|} + \frac{|da_{q}|}{|a_{q}|} + ∑_{n=1}^{q−1} |da_{n}| + |db|

at every

f(z) = ∫_{0}^{z}(c_{p}t^{p} + ⋯ + c_{0})e^{a_{q}t^{q}+···+a_{1}t}dt + b

in SF_{≤p,q}. This distance is complete, and we call the induced topology the coefficient topology on SF_{≤p,q}. Thus Compactness Theorem shows the following

Corollary 2.17 (Equivalence Theorem) The synthetic Teichmüller topology is equivalent to the coefficient topology on SF_{p,q} for every p and q.

Finally, we state the following theorem about the size of the Julia set. Proofs will be given in [15].

Theorem 2.18 For every transcendental structurally finite entire function f, the Hausdorff dimension of the Julia set of f is two.

Remark 2.19 Compare with a theorem of Stallard ([12] II): For every transcendental entire function with bounded singular values, the Hausdorff dimension of J(f) is greater than 1.

3 The case of structurally infinite entire functions

In the case of structurally infinite entire functions, the situation becomes more complicated. We will show this by several examples.

First, entire functions in the same full synthetic deformation space are not necessarily in the same dynamic Teichmüller space. In particular, we can relax singular orbit relations in a connected component of the full synthetic deformaiton space.

Proposition 3.1 Set

f_{1}(z) = sin z

and

f_{2}(z) = \frac{z − c}{z} sin z

with a sufficiently small positive c. Then

f_{2} ∈ F D(f_{1}) − S(f_{1})

Proof. Critical values of f_{2} accumulates to ±1, which are the critical values of f_{1}. Also the critical points of f_{1} and f_{2} are asymptotically the same. Hence we can construct a quasiconformal map ϕ of ℂ which sends critical points of f_{1} to those of f_{2} and satisfies that

f_{1} = f_{2} ∘ ϕ

outside the preimage of a compact set under f_{1}.

Here note that

g(z) = \frac{sin z}{z}

and f_{2} are topologically non-equivalent, for the critical values of g accumulate to a single point 0. On the other hand, set B = {|z| < 2}, then there is a conformal map ϕ such that f_{2} = g ∘ϕ outside f_{2}^{−1}(B), which can be extended to a homeomorphism of ℂ onto itself. Hence, we can find a homeomorphism ψ of ℂ such that ∥f_{2} − g ∘ ψ∥_{∞} is finite.

Also note that both of f_{2} and g are bounded on the positive real axis, but that g has an asymptotic value, while f_{2} does not.

Proposition 3.2

\frac{sin z}{z} ∈ S(f_{2}) − F D(f_{2}) ⋃ Top(f_{2}).

Proof. The preimages of a point, say z = R with a sufficiently large R, under f_{2} have eventually almost periodic real parts and have bounded imaginary parts. Thus the modulus of any 4 preimages neighboring to each other is bounded and away from 0. On the other hand, though the preimages of z = R under g again have eventually almost periodic real parts, they have the imaginary parts tending to ±∞, which implies that the moduli of 4 preimages neighboring to each other tends to 0 or +∞ as they tend to ∞. Thus we conclude that no quasiconformal maps satisfy the qc-L^{∞} condition between f_{2} and g, and hence that g ∉ F D(f_{2}).

Next, topologically equivalent entire functions are not necessarily in the same full synthetic deformation space.

Proposition 3.3 Set

h_{1}(z) = z sin z

and

h_{2}(z) = ∫_{0}^{z} t ∏_{n=1}^{∞} (1 − \frac{t}{r_{n}} ) dt,

where {r_{n}} tend to +∞ so rapidly that |h_{2}(r_{n})| are strictly increasing and tend to +∞, and h_{2} is of order 0.

Then

h_{2} ∈ Top(h_{1}) − F D(h_{1}) ⋃ QC(h_{1}).

Proof. First suppose that there were a normalized quasiconformal map ϕ such that

∥h_{1} − h_{2} ∘ ϕ∥_{∞} < +∞.

Let {a_{n}} and {b_{n}} be the zeros of h_{1} and h_{2}. Then by the Koebe distortion theorem, ϕ^{−1}(b_{n}) should be asymptotically equal to a_{n}, for h_{1} ∘ ϕ^{−1}(b_{n}) are bounded. Then, since the order of f_{1} is one, that of h_{2} should be positive by the Hölder continuity of quasiconformal maps. But this is a contradiction.

Next suppose that there were quasiconformal maps ϕ and ψ such that h_{2} = ψ ∘ h_{1} ∘ ϕ. Here we may assume that ψ is normalized. Then a_{n} should be equal to ϕ(b_{n}), which implies again that the order of h_{2} should be positive.

Proposition 3.4 Set h_{3}(z) = 2h_{1}(z). Then

h_{3} ∈ QC(h_{1}) − F D(h_{1})

Proof. Let {α_{n}} and {β_{n}} be the critical values of h_{1} and h_{3}. Then both of α_{n} and β_{n} tend to ∞. Also note that, for arbitraliry fixed R, α_{n} for every sufficiently large n has a neighborhood U_{n} = {|z − α_{n}| < R} such that the component of h_{1}^{−1}(U_{n}) containing the critical point over α_{n} contains no other critical points. The same assertion holds for h_{2} and {β_{n}}.

Suppose that there were a quasiconformal map ϕ of ℂ such that

M = ∥h_{1} − h_{3} ∘ ϕ∥_{∞} < +∞.

Then as in the proof of Compactness Theorem, we can see that, for every sufficiently large n, the (R − M)-neighborhoods of α_{n} contain some β_{m}. This is impossible.

4 Proof of Inclusion Theorem

To describe the structure of an entire function, we use the following kind of a configuration graph. (Compare with dessins d’enfents, for instance, in [11].)

Definition 4.1 (Configuration tree) A configuration tree is a planar tree with the initial vertex (and hence whose edges have an orientation towards the initial vertex) and colored as follows:

1. There are two kind of vertices; white ones and black ones.

2. There are three kind of edges; white ones, black ones, and red ones.

3. Every connected component of the set of all white vertices and white edges is a subtree ℝ with vertices ℤ, which we call a ℤ-unit.

4. Every edge outside ℤ-units is colored black or red, according as the edge starts from a black vertex or from a white vertex.

Also a configuration tree is associated with the configuration data.

1. The singularity data; the center locus attached to every ℤ-unit and the decoration locus attached to every black edge, and

2. a spider at ∞ ([4]), which assigns each distinct singularity datum a mutually disjoint path to the infinity.

We call a pair of a red edge and the black vertex pointed by it a reduction pair. And if the red edge in a reduction pair has the opposite orientation under a change of the initial vertex, we delete the pair, and attach a new pair to every white vertex from which a black edge starts.

Further, if a white vertex is the initial one, then we may attach a reduction pair and regard that the newly attached black vertex is the initial one. Thus we may always assume that the initial vertex is black.

We say that such a new configuration tree is obtained by a change of the initial vertex, and that two configuration trees are equivalent, if, after suitable changes of the initial vertices of both, they are identical including colors.

Definition 4.2 (Realizability) We say that a configuration tree T is realizable (with suitable configuration data) if there is an entire function f which gives a tree equivalent to T under the following injunctions; a black edge and its starting black vertex represent a Maskit surgery attaching a quadratic block, while a red edge and its starting ℤ-unit represent a Maskit surgery attaching an exp-block where corresponding cross-cuts cut no legs of the spider.

We call T a configuration tree of f (with respect to the given configuration data).

We sometimes say that a black edge and the starting black vertex represent a ℂ-decoration and the decorated ℂ-block, respectively.

Also note that f may have several non-equivalent configuration tree with the same configuration data, which represent diferent orders of attaching blocks.

Definition 4.3 (Core) The core of a configuration tree is the smallest connected closed subtree containing all black vertices and non-white edges. And we call a tree is virtually compact if the core is compact.

A virtually compact tree is locally finite, and has a finite number of ends. Moreover, we can easily conclude the following theorem.

Theorem 4.4 Every configuration tree of a structurally finite function is virtually compact. Conversely, every virtually compact configuration tree is realizable by a structurally finite entire function.

Definition 4.5 For every vertex v of a configuration tree T with the initial vertex v_{0}, we write by Age(v) the number of all non-white edges in the simple path from v to v_{0}, and call it the age of v.

Corollary 4.6 If T is a configuration tree of a structurally finite entire function, then Age is a bounded function on the set of all vertices in T.

Now suppose that f is a structurally finite entire function of type (p, q).

Definition 4.7 We say that a structurally finite entire function of type (p, q) is simple if it has p + q distinct singurality data.

Then by standard arguments, we can show the following

Proposition 4.8 Every structurally finite entire function f can be approximated in SD(f) by simple functions of the same type.

Hence we may assume without loss of generality, that the given f is simple. (For existence of simple functions, see [14].) Furthermore, considering a quasiconformal equivalent of f, we may also assume that all singularity data are real and positive.

Here and in the sequel, we call an entire function g a quasiconformal equivalent of f if g is quasiconformally equivalent to f.

Definition 4.9 The standard function f_{p,q} of type (p, q) is a ℂ-block attached p quadratic blocks with decoration loci {1, ⋯, p} and q exp-blocks with center loci {p + 1, ⋯, p + q}. Here and in general, if the singularity data are real, we always consider a canonical spider at ∞ which is a spider with all legs parallel to the imaginary axis and going down to the real axis.

Next, we consider to deform the given f in SD(f) without changing the topological type (i.e. to deform f in SD(f) ∩ Top(f)).

Definition 4.10 we say that g is SD(f)-admissible if g is a quasiconformal equivalent of f and d(g, f) < +∞.

A typical deformation to obtain an SD(f)-admissible function is rearrangement of the legs of the spider.

Definition 4.11 Let (α, E) be the pair of a singularity datum α and a set E of singularity data of f such that there are no other singularity data in the minimal interval I = I_{α,E} containing α and E, and α is an end point of I.

Then we say that an SD(f)-admissible g is obtained from f by a simple move of (α, E) if g is a quasiconformal equivalent of f with quasiconformal maps ϕ_{1} and ϕ_{2} (i.e. g = ϕ_{1} ∘ f ∘ ϕ_{2}) such that

1. ϕ_{1} induces a cyclic permutation of the set {α} ∪ E such that α moves to the other side of I, and fixes all the other singularity data pointwise, 2. every leg of the canonical spider to a singurarity datum β is mapped by

ϕ_{1} to the leg to ϕ_{1}(β), except for the leg ℓ_{α} to α, and

3. the image ϕ_{1}(ℓ_{α}) and the leg to ϕ_{1}(α) in the canonical spider form a cross-cut, which separates E from all singularity data other than {α} ∪ E.

Now the following proposition implies Inclusion Theorem.

Proposition 4.12 Under the same circumstances as above, the standard function f_{p,q} of type (p, q) is SD(f)-admissible.

In particular, f_{p,q} is entire, and

d(f, f_{p,q}) < +∞.

This proposition in turn follows from two lemmas below: Rearrangement Lemma and Reduction Lemma. Here we may assume that the initial vertex of the configuration tree of f is black.

Lemma 4.13 (Rearrangement Lemma) Suppose that f have a configuration tree T_{f} with the black initial vertex and with Age ≤ 1. Then the standard function f_{p,q} of type (p, q) is SD(f)-admissible.

Proof. We may assume, by considering a quasiconformal equivalent of f, that f has the configuration tree with the same singularity data as that of f_{p,q}.

If some k ∈ ℤ is a center locus and k + 1 is a decoration locus of f, then by applying the simple move of the pair (k + 1, {k}) we obtain an SD(f)- admissible g_{1}. But the configuration tree of g_{1} (with respect to the canonical spider) has k as a decoration locus and k +1 as the center locus, and Age = 2 for vertices in the corresponding ℤ-unit W, while Age = 1 for every other non-initial vertex.

Next, by applying the simple move of the pair (k + 1, {k}) to g_{1}, we obtain an SD(f)-admissible g_{2} with the same configuration tree as that of g_{1} with respect to the canonical spider. Here, if black edges and ℤ-units of the configuration tree are numbered so that the ordered singularity data of g_{1} is the identitical bijection from the numbers to the data, then that of g_{2} is the permutation

(k, k + 1).

Again, by applying the simple move of the pair (k + 1, {k}) to g_{2}, we obtain an SD(f)-admissible g with the same configuration tree as that of f with respect to the canonical spider, and hence Age ≤ 1. But k is now a decoration locus of g, while k + 1 is a center locus of g.

Repeating such rearrangements, we can conclude that the standard function f_{p,q} is SD(f)- admissible.

Lemma 4.14 (Reduction Lemma) There is an SD(f)-admissible simple g having a configuration tree T_{g} with Age ≤ 1.

To prove this lemma, we assume that

max Age = M > 1

on the configuration tree of f. Let V = {v_{1}, ⋯, v_{n}} be the set of all vertices with Age = M that are connected directly (i.e. by single non-white edges) either to the same black vertex v corresponding to a decoration locus α or to the same ℤ-unit W with center locus α.

Then v or every w ∈ W have Age = M − 1 > 0, and v or some v_{0} ∈ W, which we call the root of W, is connected directly to some vertex with age M − 2 (which may be the initial vertex).

Here the following fact is easily seen.

Lemma 4.15 (Commutability Lemma I) Suppose that between two singularity data α and β there are no other singularity data, and that α and β, respectively, correspond to a vertex v_{α} in V and to v_{β} not in V ∪ {v} or V ∪ W. Then we can find an SD(f)-admissible simple g with the same configuration tree (with respect to the canonical spider) and the same set of the singularity data as those of f, but data corresponding to v_{α} and v_{β} are permuted.

Proof. By by the simple move of the pair (α, {β}) , we obtain a desired g.

Thus all the singularity data for V can be gathered near α, and we conclude the following

Lemma 4.16 There is an SD(f)-admissible simple f_{1} with the same configuration tree (with respect to the canonical spider) and the same set of the singularity data as those of f such that there is an open interval I in ℝ in which the singularity data are exactly α and those, say {α_{j}}, of V

Here we may assume that α_{j} are increasing, and further that {α, α_{1}, ⋯, α_{n}} equals to {1, ⋯, n + 1} with α = n′ + 1 > 1. Then by the simple move of {1, {2, ⋯, n + 1}}, we have the following Lemma 4.17 There is an SD(f)-admissible f_{2} with the same configuration tree (with respect to the canonical spider) and the same set of the singularity data as those of f_{1} such that the ordered singularity data of f_{2} corresponding to V ∪ {v} or V ∪ W is the cyclic permutation

(1, ⋯, n + 1)^{−1}

of that of f_{1}.

Repeating such rearrangements, we have an SD(f)-admissible f_{3} with the same configuration tree (with respect to the canonical spider) and the same set of the singularity data as those of f such that the ordered singularity data of f_{3} corresponding to V ∪ {v} or V ∪ W is the cyclic permutation

(1, ⋯, n + 1)^{−n′}

of that of f_{1}.

Under these circumstances, we divide into two cases.

1) The case that α corresponds to a black vertex v: In this case, by applying the similar rearrangements which corresponds to the permutation

(1, ⋯, n + 1)^{−n′−1},

we can obtain an SD(f)-admissible f_{V} with the same set of the singularity data as that of f, but Age ≤ M − 1 on vertices in {v} ∪ V.

2) The case that α corresponds to the root w_{0}: In this case, we need another index for v_{j}.

First taking another configuration tree of f if necessary, we may assume that no non-white edges end at w_{0}. Indeed, if there is such a vertex v^{0}, we can replace the corresponding edge to one towards the vertex connected directly with w_{0}. In this new tree, Age(v^{0}) = M − 1.

Then, since the root w_{0} divides the belonging ℤ-unit into two connected components, which in turn divide the set {v_{j}} into two classes V^{±} = {v_{j}^{±}}_{j=1}^{m^{±}} with m^{+} + m^{−} = n. Here we consider to reduce V^{±} to the empty sets. We discuss V^{+} only, for V^{−} can be treated similarly.

Let H(v_{j}^{+}) be the number of white edges in the simple path from v_{j}^{+} to w_{0}, and call it the height of v_{j}^{+}. Also note the following

Lemma 4.18 (Commutability Lemma II) Suppose that between two singularity data α and β there are no other singularity data, and that α and β, respectively, correspond to vertices u^{+} in V^{+} and to w which is either in V^{+} with H(u^{+}) ≠ H(w^{+}) or not in V^{+} ∪ W. Then we can find an SD(f)- admissible simple g with the same configuration tree, the same heights, and the same set of the singularity data (with respect to the canonical spider) as those of f, but data corresponding to u^{+} and w are permuted.

Thus as before we can gather all the singularity data for V^{+} near α. Here we assume that, by applying the simple move of the pair (α_{m}^{+}, E ∪ {α}) and then another simple move of the pair (α, {α_{m}^{+}}), we can decrease the height of v_{m^{+}}^{+} by 1, where E is the set of the singularity data between α and α_{m}^{+}. Also applying suitable simple moves and renumbering V^{+} if necessary, we may assume that the singularity data {α_{j}} for V^{+} are greater than α and increasing with respect to j.

Then similarly as in the proof of Rearrangement Lemma, we can decrease {H(v_{j}^{+})} one by one. And finally, we can obtain an SD(f)-admissible g with the same set of the singularity data {α_{j}}, such that [α, α_{m^{+}}^{+}] contains no other singularity data and that H(v_{j}^{+}) = 1 for every j. Thus we may assume that α = 1, and {α_{j}^{+}} equals to {2, ⋯, m^{+} + 1}.

Now, by applying the simple move of the pair (1, {2, ⋯, m^{+} + 1}), we can obtain an SD(f)-admissible f_{V} with the same singularity data as that of f, but Age ≤ M − 1 on vertices in V^{+}.

Proof of Reduction Lemma. Applying such reductions as above to all vertices with age M, we can obtain an SD(f)-admissible simple function with the same set of the singularity data as that of f, but with Age ≤ M −1.

Repeating this process a finite number of times, we have the desired g.

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