Structurally finite meromorphic functions

Masahiko Taniguchi

1 Definitions and main theorems

In general, for a possibly incomplete and branched holomorphic covering π : R → ℂ̂ of ℂ̂ by a simply connected Riemann surface R, we say that a point α in ℂ̂ is a singular value of the projection π if, for every neighborhood U of α, there exists a connected component V of π^{−1}(U) such that π : V → U is not a biholomorphic surjection, which is called a singular component of π^{−1}(U). In other words, a point α is not a singular value of π if and only if α is evenly covered by π, i.e., we can find a neighborhood U of α such that π maps every connected component of π^{−1}(U) biholomorphically onto U. The projection π is called a Speiser function if it has only a finite number of singular values.

Next, we say that a possibly incomplete and branched holomorphic covering π : R → ℂ̂ covers a domain D in ℂ̂ almost evenly if there are only a finite number of points in D which are not evenly covered, and at every such point α, there is a finite number of singular components of π^{−1}(B) for every sufficiently small disk B with center α.

If such a holomorphic covering π : R → ℂ̂, which covers ℂ̂ almost evenly, has no algebraic singularities, i.e. branch points, then the projection π is a meromorphic function with a polynomial Schwarzian derivative ([3], and also see [1]). And similarly, we can see that, when the projection π has a finite number of branch points, π has a rational Schwarzian derivative (cf. [2]). In the case of a holomorphic covering of ℂ which covers ℂ almost evenly, the author showed in [4] that the projection π is a structurally finite entire function. And we know that f is a sturcturally finite entire function, possibly postcomposed by a Möbius transformation, if and only if the Schwarzian derivative S(f) has the form

(\frac{P′}{P} + Q)′ − \frac{1}{2} (\frac{P′}{P} + Q)^{2}

with suitable polynomials P and Q.

Here recall that, for an entire function f, we can also consider the nonlinearity or the pre-Schwarzian derivative N(f) = f′′/f′ of f, which determines the affine structure induced by f.

Example 1 g(z) = e^{z} and f(z) = tan z has the same Schwarzian derivative S(g) = S(f) = −1/2, but N(g) = 1 and N(f) = 2f.

Definition 1.1 We say that a possibly incomplete and branched holomorphic covering π : R → ℂ̂ is structurally finite if it covers ℂ almost evenly and has ∞ as a Picard’s exceptional value.

Theorem 1.1 (Structure) A possibly incomplete and branched holomorphic covering π : R → ℂ̂ is structurally finite if and only if it is obtained from a structurally finite entire function ℂ → ℂ by applying the surgeries attaching ℂ̂ finitely many times.

Here the surgery attaching ℂ̂ to the covering π_{0} : R_{0} → ℂ̂ (in the sense of Schiffer) is the one constructing a new possibly incomplete and branched holomorphic covering π : R → ℂ̂, from π_{0} : R_{0} → ℂ̂ by taking a simple closed arc ℓ in ℂ̂ whose interior is in ℂ and disjoint from the set of all singular values of π_{0}, and a lift ℓ′ of ℓ with respect to π_{0}, by connecting ℂ − ℓ′ and ℂ̂ − ℓ crosswise, and by filling end points if necessary, so that π_{0} can be extended to a holomorphic projection π from a newly obtained simply connected Riemann surface R to ℂ̂.

Proof. First we suppose that the convering π : R → ℂ̂ is structurally finite and show the ”only-if” part.

Deforming π : R → ℂ̂ by the surgeries relaxing the relation between the singurality data as in [4] §5, we may assume that the singularities of π : R → ℂ̂ are in general position, i.e. every singular value is in ℂ and corresponds to either a single branch point or a logarithmic singurality of π.

Actually, first we can approximate the given π by a structurally finite holomorphic covering π_{1} : R_{1} → ℂ̂ whose singularities except for poles are in general position with respect to the synthetic Teichmüller topology (which is defined in the next section). Next, by applying Whitehead surgeries as in [4], we can deform π_{1} : R_{1} → ℂ̂ to another structurally finite holomorphic covering π_{2} : R_{2} → ℂ̂ whose singularities are in general position. Here it is clear that, if we show the assertion for π_{2} : R_{2} → ℂ̂, then we can conclude that the assertion also holds for the given π : R → ℂ̂.

Now fix a spider at ∞ (cf. [4] §4), and consider the plates, i.e. lifts of the complement of all legs of this spider at ∞ in ℂ (cf. [6]). On every plate Π, a leg ℓ corresponds to two borders of Π. If they are the same arc on R, we say that ℓ has trivial lifts on Π. And we say that a leg ℓ is polar if there is a non-trivial lift of ℓ ending at a pole, which in turn, is called a polar lift of ℓ.

To show the assertion by induction, first suppose that π has a single simple pole. Fix a plate Π_{0} having a polar lift ℓ_{0} as a border. Then Π_{0} is connected with another plate Π_{1} along ℓ_{0}. Furthermore since the single pole of f is simple, there are a finite number of legs ℓ_{1}, ⋯, ℓ_{N} and plates Π_{2}, ⋯, Π_{N+1} such that ℓ_{0}, ℓ_{1}, ⋯, ℓ_{j+1} are located in this order, the plates Π_{0}, ⋯, Π_{N} are mutually different, Π_{0} = Π_{N+1}, and Π_{j} and Π_{j+1} are connected along ℓ_{j}. Then by applying the positive permutation of spider legs {ℓ_{1}, ⋯, ℓ_{N}} just N −1 times (cf. [4] Definition 7.11), we have a new leg ℓ_{N+1}^{∗} neighboring to ℓ_{0} which also connected Π_{0} and Π_{1}. Deforming ℓ_{0} ∪ ℓ_{N+1}^{∗} continuously, we have a compact simple arc C in ℂ, and R is obtained from a simply connected Riemann surface R_{1} by the surgery attaching ℂ̂ to the holomorphic covering π_{1} : R_{1} → ℂ̂ along C in the sense of Schiffer, where π_{1} is the projection of a structurally finite holomorphic covering of ℂ by R_{1} induced from π.

Next suppose that the assertion holds if the number of poles is not greater than n (≥ 1). And consider such a π with n + 1 simple poles. Again, fix a plate Π_{0} having a polar lift ℓ_{0} as a border. Then similarly as in the previous paragraph, we can have a compact simple arc C in ℂ along which suitably deformed Π_{0} and another such one are connected crosswise. Hence we can decompose π : R → ℂ̂ into two structurally finite holomorphic covering π_{j} : R_{j} → ℂ̂ with n_{j} simple poles (j = 1, 2) where n_{j} are non-negative integers satisfying n_{1} + n_{2} = n + 1. Thus if both of n_{j} are positive, then by the induction we have the assertion for π. If one of n_{j}, say n_{1}, is 0, then the number of the singularities of π_{2} is less than that of π. Hence repeating the above decomposition finitely many times, we can decompose π into a finite number of structurally finite holomorphic coverings, each of which has at most n simple poles, and we can complete the proof of the ”only-if” part by induction.

Since the ”if” part is clear, we have finished the proof.

Theorem 1.2 (Representation) The projection of every structurally finite holomorphic covering π : R → ℂ̂ can be reperesnted as

∫ ^{z} \frac{P (t)}{∏_{j=1}^{k} (t − b_{j})^{m_{j}+1}} e^{Q(t)}dt

with suitable polynomials P and Q satisfying the residue condition:

(\frac{P (z)}{∏_{j≠ℓ} (z − b_{j})^{m_{j}+1}} e^{Q(z)})^{(m_{ℓ})} (b_{ℓ}) = 0

for every ℓ. Here m_{j} is the order of the pole b_{j}.

Proof. First by Structure Theorem, we can find a compact set K in ℂ, a conformal map ϕ of R − K into ℂ such that ℂ − ϕ(R − K) is compact, and a structurally finite entire function g such that π = g ∘ ϕ on R − K. Then it is well-know that ϕ can be extended to a quasiconformal map of R onto ℂ. In particular, π is a meromorphic function on ℂ. Furthermore, since ϕ has a simple pole at ∞, it is easy to see that π and hence also π′ has a finite order. The assumption implies that π′ has only a finite number of zeros and poles with residue 0, we conclude that π has the form as asserted.

Corollary 1 The projection of a structurally finite holomorphic covering of ℂ̂ is a meromorphic function on ℂ having a rational non-linearity.

Definition 1.2 We call the projection of a strucutrally finite holomorphic covering f : ℂ → ℂ̂ a structurally finite meromorphic function.

If such an f is obtained from a structurally finite entire function of type (p, q) by the surgeries attaching ℂ̂ n times, then we say that f is of type (p, q, n).

Example 2 The function (z^{2} + 1)/z is of type (0, 0, 1), (z + 1)e^{z}/z and e^{z}/z are of type (0, 1, 1), and ∫^{z} e^{t^{2}}dt/t^{2} is of type (0, 2, 1).

Actually, Representation Theorem implies the following Corollary 2 A meromorphic function f is structurally finite if and only if the non-linearity N(f) of f has the form

∑_{i=1}^{m} \frac{1}{z − a_{i}} − ∑_{j=1}^{k} \frac{m_{j} + 1}{z − b_{j}} + Q′(z)

with a suitable polynomial Q, points {a_{i}}_{i=1}^{k} and {b_{j}}_{j=1}^{k}, and positive integers {m_{j}}_{j=1}^{k}, such that

\frac{∏_{i=1}^{m} (z − a_{i})}{∏_{j=1}^{k} (z − b_{j})^{m_{j}+1}} e^{Q(z)}

satisfies the residue condition.

Finally, as in [5], we can show the following

Corollary 3 The Hausdorff dimension of the Julia set of a transcendental structurally finite meromorphic function is two.

Proof. Let f be a transcendental structurally finite meromorphic function. Then exactly as in [5], we can construct a compact subset E_{∞} such that the Hausdorff dimension of E_{∞} is two, and the orbit of every point in E_{∞} is contained in the set where |f′| ≥ 2 (cf. (4-d) in [5]). Hence E_{∞} is contained in the Julia set of f, which implies the assertion.

2 Several deformation spaces

Let f be a meromorphic function. Then we can consider several deformation spaces of f.

Definition 2.1 Let f_{1} and f_{2} be meromorphic functions. We say that f_{1} and f_{2} determine the same covering structure if there are a similarity g and a Möbius transformation h such that f_{2} = h ∘ f_{1} ∘ g^{−1}. We denote by C_{f} the covering structure determined by f, and call the set of all covering structures determined by meromorpihc functions g which are quasiconformally equivalent to f the prime Hurwitz space of f, which is denoted by H^{#}(f). Here we say that f and g are quasiconformally equivalent if there are quasiconformal self-maps φ and ψ of ℂ and ℂ̂, respectively such that g = ψ ∘ f ∘ φ^{−1}.

The prime Hurwitz distance d_{H^{#}} on H^{#}(f) is defined by setting

d_{H^{#}}(C_{f_{1}}, C_{f_{2}}) = inf_{φ} K(φ),

where the infimum is taken over all quasiconformal maps φ of ℂ which satisfy ψ ∘ f_{2} ∘ φ^{−1} = f_{1} with suitable quasiconformal maps ψ of ℂ̂. d_{H^{#}} is actually a distance and complete on H^{#}(f).

Definition 2.2 We say that meromorphic functions f_{1} and f_{2} determine the same isomorphism class if there is a similarity φ such that f_{2} = f_{1} ∘ φ. We call the set of all isomorphism classes I_{g} of meromorphic functions g which are quasiconformally equivalent to f the Hurwitz space of f, and denote it by H(f). Here, we distinguish the value ∞ and assume that ψ(∞) = ∞ for every quasiconformal map ψ appeared in quasiconformal equivalence relation g = ψ ∘ f ∘ φ^{−1}

If f is a Speiser function, then we can define the normalized Hurwitz distance d_{H} on H(f) by setting

d_{H}(I_{f_{1}}, I_{f_{2}}) = inf_{φ} K(φ),

where the infimum is taken over all quasiconformal maps φ of ℂ̂ satisfying f_{1} = ψ ∘f_{2} ∘φ^{−1} with quasiconformal maps ψ of ℂ̂ normalized as follows: We prescribe N + 2 points in ℂ and assume that each ψ fixes two of them, where N is the number of finite singular values of f. d_{H} is actually a distance and complete on H(f).

Remark We say that f and g are topologically equivalent if there are selfhomeomorphisms φ and ψ of ℂ and ℂ̂, respectively, such that ψ(∞) = ∞ and g = ψ ∘ f ∘ φ^{−1}. Let Top(f) be the set of all isomorphism classes of meromorphic functions topologically equivalent to f. If f is a Speiser function, we have

Top(f) = H(f).

Definition 2.3 The full deformation set F D(f) of f is the set of all meromorphic functions g on ℂ such that there are quasiconformal maps φ of ℂ which fix 0 and 1, and satisfy the qc-L^{∞} condition:

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

For every pair of functions f_{1} and 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 quasiconformal maps φ_{1} and φ_{2} of ℂ which fix 0 and 1, and satisfy the qc-L^{∞} conditions D_{f}(f_{j}; φ_{j}) < ∞. d is actually a distance and complete on F D(f). We call the distance d defined above 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 is denoted as F SD(f). (Cf. [4].)

Theorem 2.1 For every structurally finite meromorphic functions f of type (p, q, n) and with the polar type [n], the space SF (f) consists of all strucutrally finite meromorphic functions of type (p′, q, n) with the polar type [n], where p′ = p when q = 0 and p′ ≤ p when q > 0, is complete, in the sense that if f_{n} ∈ SF (f) converge to g ∈ F SD(f) then g ∈ SF (f), with respect to the synthetic Teichmüller topology.

Here, for a meromorphic function f with n poles counting multiplicities, we say that [n] = [n_{1}, ⋯, n_{k}] is the polar type of f if f has k poles which have orders n_{1}, ⋯, n_{k}, where ∑_{j=1}^{k} n_{j} = n.

Proof. First, it is easy to see that every structurally finite meromorphic functions of type (p′, q, n) with the polar type [n], where p′ = p when q = 0 and p′ ≤ p when q > 0, in SF (f).

On the other hand, let g be a meromorphic function belonging to F SD(f). Take a quasiconformal map φ of ℂ such that D_{f}(g; φ) < +∞. Let {b_{j}}_{j=1}^{k} be the poles of f. And for a sufficiently small ϵ > 0, take a disc D_{j} = {|z −b_{j}| < ϵ} for every j. (Here, when q = 0, then we regard ∞ also as a pole, and take one more disc D_{0} = {|z| > 1/ϵ} ∩ {∞}.)

Now we may assume that f(D_{j}) is disjoint from {|z| ≤ 2D_{f}(g; φ)}, and that the winding number of the image f(C_{j}) around 0 is −n_{j}， for every j = 1, ⋯, k. Then the assumption implies that g(φ(D_{j})) is disjoint from {|z| ≤ D_{f}(g; φ)} and that the winding number of g(φ(C_{j})) arround 0 is −n_{j}， for every k = 1, ⋯, k. Thus φ(D_{k}) contains no zeros and a pole with order n_{j} for every j, which implies that the polar type of g is [n].

Thus we have shown the assertion when q = 0. When q > 0, by the same argument as in the proof of [4] Theorem 2.18, we can show the assertion.

References

[1] R. L. Devaney and L. Keen, Dynamics of meromorphic maps; maps with polynomial Schwartian derivatives, Ann. scient. Êc. Norm. Sup., 22, (1989), 55–79.

[2] G. Elfving, Über eine Klasse von Riemannschen Flächen und ihre Uniformisierung, Acta Soc. Sci. Fenn. Nova Ser. A, Tom. II 3, (1934), 60 pp..

[3] R. Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten Acta Math. 58, (1932).

[4] M. Taniguchi, Synthetic deformation spaces of an entire function, Contemporary Math. 303, (2002), 107–136.

[5] M. Taniguchi, Size of the Julia set of a structurally finite transcendental entire function Math. Proc. Camb. Phil. Soc. 135, (2003), 181–192.

[6] M. Taniguchi, Covering structure and dynamical structure of a structurally finite entire function, Kodai Math. J., to appear.

Masahiko TANIGUCHI Department of Mathematics Kyoto University Kyoto 606, JAPAN