Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 263–274

MAPPING PROPERTIES OF FATOU COMPONENTS

M.E. Herring

Imperial College of Science, Technology and Medicine, Department of Mathematics London SW7 2BZ, United Kingdom

Abstract. If f(z) is meromorphic in C and N_{1} and N_{2} are components of the Fatou set of f(z) such that f(z): N_{1} → N_{2} , it is shown that D = N_{2} \ f(N_{1}) is a set which contains at most two points. If f(z) is entire then D contains at most one point. Examples show that these results are sharp and also that the points of D are in general neither Picard exceptional nor Nevanlinna deficient.

1. Introduction

Let f(z): C (or Ĉ) → Ĉ be a non-constant meromorphic function which is not a Möbius transformation. We define the the n^{th} iterate of f(z) at z as f^{n}(z), where f^{1} = f(z) and f^{n+1}(z) = f(f^{n}(z)). The subset of the sphere such that {f^{n}}, n ∈ N, is defined meromorphically on some open neighbourhood, U of z and forms a normal family in U is termed the Fatou set of f(z), F (f), and consists of a countable union of connected open components. The complement of F (f) in Ĉ, J(f) is termed the Julia set of f(z) and is non-empty and perfect.

Non-rational functions are not defined continuously at transcendental singularities and so these points must lie in J(f).

The Fatou set has the property of complete invariance, that is z ∈ F (f) if and only if f(z) ∈ F (f); for rational f(z), J(f) has this property whilst for non-rational f(z), z ∈ J(f) \ {∞} if and only if f(z) ∈ J(f).

Let N_{1} be any component of F (f). Then f(N_{1}) is a connected subset of F (f) and so lies inside some component of F (f), say N_{2} .

If f(z) is rational then there are at worst finitely many algebraic singularities of f^{−1} in N_{2} , and so f(z): N_{1} → N_{2} is a finite branched cover. However this need not be the case if f(z) is non-rational. For instance, consider the function f(z) = e^{z} − 1. Let H = {z : Re(z) < 0}. Then f(H) ⊂ H and so by Montel’s theorem {f^{n}}, n ∈ N, forms a normal family in H which is then contained in some (forward invariant) component of F (f), say N. Now −1 ∈ H but −1 is a Picard omitted value of f(z) and so F (N) ⊂ N but −1 is not in f(N).

It is already known (e.g. [4]) that if f(z): N_{1} → N_{2} , then f(N_{1}) is open and dense in N_{2} , but nothing further seems to be known about the set N_{2} \ f(N_{1}). Our main result is

1991 Mathematics Subject Classification: Primary 30D05, 58F08.

Theorem 1. If f(z) is meromorphic in C and f(z): N_{1} → N_{2} , where N_{i} are components of F (f), then |N_{2} \ f(N_{1})| ≤ 2.

Theorem 2. If a ∈ N_{2} \ f(N_{1}), then a is admitted as an asymptotic value of f(z) along some path γ ⊂ N_{1} which runs to ∞.

We then have two obvious corollaries.

Corollary 1. If N_{1} is bounded, then f(z): N_{1} → N_{2} is a finite branched cover.

Corollary 2. If f(z) does not admit asymptotic values, then for all components N_{1} of F (f), f(z): N_{1} → N_{2} is surjective.

In [7, pp. 7–11], a family of meromorphic functions which do not admit asymptotic values is introduced.

Theorem 3. If f(z) is transcendental and F (f) is connected, then each point of F (f) \ f(F (f)) is a Picard exceptional value of f(z), and so the latter set contains at most two points.

We are able to prove more under assumptions on the structure of N_{1} :

Theorem 4. Let f(z) be transcendental meromorphic in C such that F (f) contains at least two components, with N_{1} unbounded and let Γ be the component of ∂N_{1} which contains ∞. Then if ∂N_{1} \ Γ is bounded in C, |N_{2} \ f(N_{1})| ≤ 1.

Corollary 3. If f(z) is transcendental entire and N_{1} is unbounded, then N_{1} and N_{2} are simply-connected and |N_{2} \ f(N_{1})| ≤ 1.

Corollary 4. If f(z) is a non-entire transcendental meromorphic function, and N_{1} is of finite connectivity ≥ 2, then |N_{2} \ f(N_{1})| ≤ 1.

It has been shown in [5] that for non-entire meromorphic functions F (f) may have unbounded components of arbitrary connectivity.

Theorem 5. Let f(z) be a non-entire transcendental meromorphic function, with N_{1} of connectivity at most two. Then if N_{2} is simply-connected, |N_{2} \ f(N_{1})| ≤ 1 whilst if N_{2} is multiply-connected then f(N_{1}) = N_{2} .

We may also investigate this problem for holomorphic self maps of the punctured plane. We need only consider transcendental non-entire functions which as in [3] are conjugate to

f(z) = z^{k} exp(g(z) + h(\frac{1}{z}))

where g, h are entire, k ∈ Z (if k is non-negative, then h is non-constant). It is shown in the same paper that then the connectivity of any component of F (f) is at most two, there being at most one multiply-connected component.

We then have:

Theorem 6. Let f(z) be a transcendental non-entire self map of C^{∗} = C \ {0} and f(z): N_{1} → N_{2} where N_{1} is a component of F (f). Then if N_{2} is simply-connected |N_{2} \ f(N_{1})| ≤ 1 whilst if N_{2} is multiply-connected (doubly) then f(N_{1}) = N_{2} .

Note that if f(z) is as above and F (f) consists of a single component, then by Theorem 3, f(z): F (f) → F (f) must be surjective as the two Picard exceptional values of f(z) lie in J(f).

In Section 4 we shall give various examples to show the sharpness of Theorems 1, 3, and 4 (with its corollaries). The obvious way to construct examples is by using functions, such as tan z, which have Picard exceptional values. We shall show, however, that a value in N_{2} \ f(N_{1}), although asymptotic need not be a Picard exceptional value, nor even Nevanlinna deficient. In particular, in Section 4, Example 4 for the entire function

f(z) = ∫_{0}^{z} exp(−e^{t}) dt,

F (f) contains a sequence of different components N_{i}, i = 1, 2, …, such that f(N_{i}) ⊂ N_{i}, and each N_{i} \ f(N_{i}) is a singleton {a_{i}}, where the value a_{i} has Nevanlinna deficiency δ(a_{i}, f) = 0.

2. Results needed for the proofs

Consider a domain D and let f(z) be a function single valued and meromorphic in D, where we assume that the complement of D in Ĉ has non-empty interior. By a spherical rotation, this is equivalent to supposing that D is bounded. When we refer to the capacity of a given set this should always be taken to mean the logarithmic capacity. For a non-isolated point z_{0} of ∂D = Γ we define

C_{D}(f, z_{0}) = {α : ∃z_{n} ∈ D, z_{n} → z_{0}, f(z_{n}) → α}.

Suppose further that E is a closed subset of Γ of capacity zero and z_{0} ∈ E is a limit point of Γ \ E. We define

C_{Γ\E}(f, z_{0}) = ⋂_{r>0} \overline{⋃_{z∈Γ∩D_{r}, z∉E} C_{D}(f, z)},

where D_{r} is a disc of spherical radius r centred at z_{0} . We denote by Ω_{D,E}(f, z_{0}) the open set

C_{D}(f, z_{0}) \ C_{Γ\E}(f, z_{0}).

With these assumptions we have:

Lemma 1 [14, Theorem VIII.39, p. 332]. If z_{0} ∈ E, then ∂C_{D}(f, z_{0}) ⊂ C_{Γ\E}(f, z_{0}).

Lemma 2 [14, Theorem VIII.41, p. 335]. Let Ω be a connected component of Ω_{D,E}(f, z_{0}), where z_{0} ∈ E. Then in any neighbourhood of z_{0} in D, f(z) takes any value of Ω infinitely often with the possible exception of at most a set of capacity zero. If the set E = {z_{0}}, then the exceptional set above contains at most two points.

Lemma 3 [12, Theorem 10, p. 25]. Let D be a simply-connected domain which is of hyperbolic type and z_{0} ∈ E as above. If C_{D}(f, z_{0}) is not the entire sphere then for any connected component Ω of Ω_{D,E}(f, z_{0}), f(z) takes any value of Ω infinitely often in a neighbourhood of z_{0} with at most one exception.

Lemma 4 [8, Theorem 3.5, p. 56]. Let w = f(z) be analytic and univalent in the disc D. Then the radial limits f(e^{iθ}) = lim_{r→1−} f(re^{iθ}) exist on ∂D outside a set of capacity zero, and can be constant only on sets of capacity zero.

Lemma 5 [8, Theorem 5.14, p. 109]. Let w = f(z) be analytic and bounded, |w| < 1 in |z| < 1 and suppose that

lim_{r→1−} |f(re^{iθ})| = 1

outside a set of capacity zero. Then either f(z) reduces to a finite Blaschke product or f(z) assumes every value in |w| < 1 infinitely often with at most one exception.

Lemma 6. If N is a simply or doubly-connected domain, D is the unit disc and Φ: D → N is the uniformizing map then Φ(e^{iθ}) = lim_{r→1−} Φ(re^{iθ}) exists outside a set of capacity zero on ∂D. Given a value α, then Φ(e^{iθ}) takes the value α at most on a set of capacity zero.

Proof. Because of Lemma 4 we may assume that N is doubly-connected. Let α, β be finite values which lie in different components of ∂N and put

Ψ = log \frac{Φ − α}{Φ − β}: D → C,

where we begin by choosing some z ∈ D and some branch of log{(Φ(z) − α)/(Φ(z)−β)}. By the monodromy theorem the continuation of this branch gives a single-valued analytic function Ψ in D. In fact, Φ gives a conformal map of D to the universal cover of N and the log maps this universal cover univalently to a simply-connected region B. Thus Ψ: D → B is univalent. Hence by Lemma 4 the radial limit Ψ(e^{iθ}) exists except for a set E of capacity zero, and for any c ∈ Ĉ, E_{c} = {e^{iθ} : Ψ(e^{iθ}) exists and = c} is a set of capacity zero for e^{iθ} ∉ E ∪ E_{∞}, e^{iθ} lies in E_{c} for some finite c which implies that Φ(e^{iθ}) = T (e^{c}) ≠ α, β, where T (w) = (α − βw)/(1 − w), and thus e^{c} ≠ 0, ∞.

For a given u the set {e^{iθ} : Φ(e^{iθ}) = u} is a subset of ∪E_{c_{n}} ∪ E ∪ E_{∞}, where e^{c_{n}} = T^{−1}(u), which is a union of countably many sets of capacity zero and hence has capacity zero.

3. Proofs

Proof of Theorem 2. (See [4, p. 242]). A brief sketch may be useful. This follows from the Star theorem of Gross (see e.g. [11, p. 287]). If z_{0} ∈ N_{1} , w_{0} = f(z_{0}) ∈ N_{2} and g is the branch of f^{−1} such that g(w_{0}) = z_{0} , the Star theorem ensures that we can continue g analytically along some polygonal path δ in N_{2} up to a point β ∈ D(α, ϵ) ⊂ N_{2} and then along some circular arc δ′ in D(α, ϵ) which goes from β up to α. Then g(δ ∪ δ′) = γ is in N_{1} by the complete invariance of F (f) and γ goes to ∞ since α ∉ f(N_{1}). i.e. the continuation is not possible over α.

Proof of Theorem 3. Suppose that F (f) is a single connected component. Then if U_{r} = {z : r < |z| < ∞}, by Picard’s theorem f(U_{r}) covers each point of the complex sphere infinitely often with the exception of at most two points. By the complete invariance of F (f), f(U_{r} ∩ F (f)) then covers each point of F (f) infinitely often with at most two exceptions and the result follows.

Proof of Theorem 1. Since Theorem 3 already covers the case when F (f) has a single component we assume that there are at least two components. Thus the complement of N_{1} has a non-empty interior and we may apply Lemmas 1 and 2 with D = N_{1} .

Suppose that α ∈ N_{2} \ f(N_{1}). Then α is admitted as an asymptotic value of f(z) along a path γ to ∞ in N_{1} . Thus α ∈ C_{N_{1}}(f, ∞). Now

\begin{aligned}C_{∂N_{1}\\{∞\}}(f, ∞) & = ⋂_{r>0} \overline{⋃_{υ∈∂N_{1}∩D_{r}, υ≠∞} C_{N_{1}}(f, υ)}, D_{r} = \{z : |z| > r\} \\ & = ⋂_{r>0} \overline{⋃_{υ∈∂N_{1}∩D_{r}, υ≠∞} f(υ)}, \end{aligned}

which is contained in ∂N_{2} . Thus

Ω_{N_{1},∞}(f, ∞) = C_{N_{1}}(f, ∞) \ C_{∂N_{1}\{∞}}(f, ∞)

is non-empty since it contains α ∈ N_{2} . From Lemma 1

∂C_{N_{1}}(f, ∞) ⊂ C_{∂N_{1}\{∞}}(f, ∞) ⊂ J(f)

and so there must be a connected component Ω of Ω_{N_{1},{∞}}(f, ∞) which contains N_{2} . From the second part of Lemma 2 we see that each point of Ω is assumed infinitely often by f(z) in N_{1} with at most two exceptions. Thus |N_{2} \f(N_{1})| ≤ 2 as claimed.

Proof of Theorem 4. Let Γ be the component of ∂N_{1} which contains ∞. If Γ were a point then it would have to be a limit point of bounded components of ∂N_{1} , which contradicts the hypothesis, and so Γ must be a non-degenerate continuum. If we let M be the component of Ĉ \ Γ which contains N_{1} , then M must be simply-connected, and as J(f) is infinite M must be of hyperbolic type.

Suppose that α ∈ N_{2} \ f(N_{1}), which is then admitted as an asymptotic value on some path to ∞ in N_{1} , say γ. Then α ∈ C_{M}(f, ∞). As in the previous proof we see that C_{Γ\{∞}}(f, ∞) ⊂ ∂N_{2} , and ∂C_{M}(f, ∞) ⊂ C_{Γ\{∞}}(f, ∞) so that Ω_{M,{∞}}(f, ∞) contains α and has a connected component Ω which must contain N_{2} .

Now C_{M}(f, ∞) ⊂ \overline{N_{2}} ≠ Ĉ. By using Lemma 3 we see that with the possible exception of a singleton, each value in Ω is taken infinitely often by f(z) in any neighbourhood of ∞ in M. Under the hypothesis we may choose such a neighbourhood which lies entirely within N_{1} and so |N_{2} \ f(N_{1})| ≤ 1.

Proof of Corollary 3. If f(z) is transcendental entire and F (f) is connected, then this follows from Theorem 3. Otherwise F (f) has at least two components, N_{1} is unbounded and so is simply-connected (cf. [1]), in which case ∂N_{1}\Γ is empty and the result follows from Theorem 4. Note that if N_{2} were multiply-connected then by [2, Theorem 3.1] N_{1} would be bounded which is a contradiction.

Proof of Corollary 4. If N_{1} is of finite connectivity ≥ 2 then by [6, Theorem 3.1] N_{1} is not completely invariant and so there must be other components of F (f). The result then follows directly from Theorem 4.

Proof of Theorem 5. Let ψ_{i}: D → N_{i} be the uniformizing maps of N_{1} , N_{2} ; then by Lemma 6, ψ_{1}(e^{iθ}) exists and is finite outside a set E of capacity zero. Pick e^{iθ} ∉ E, then ψ_{1}(e^{iθ}) = α ∈ ∂N_{1} \ {∞} and fψ_{1}(e^{iθ}) = f(α) ∈ ∂N_{2} . If λ is any branch of ψ_{2}^{−1} then we may continue F = λfψ_{1} to an analytic function in D, and by the properties of the map ψ_{2} we see that |F (re^{iθ})| → 1 as r → 1^{−}. Then by Lemma 5 we see that F is either a finite Blaschke product, in which case F (D) = D and f(N_{1}) = N_{2} , or F (z) takes every value D infinitely often with at most one exceptional point, so that N_{2} \ f(N_{1}) is at most a singleton. If however N_{2} is multiply-connected then given any w ∈ N_{2} , there are infinitely many points p_{n} such that ψ_{2}(p_{n}) = w with at most one of the p_{n} not taken by F (z), so that f(N_{1}) = N_{2} .

Proof of Theorem 6. If f(z) is a self map of the punctured plane then the components of F (f) are of connectivity at most two and so we may use the proof of Theorem 5 to show the claim.

This concludes the proofs of the theorems. There now follows a series of examples which demonstrate the sharpness of Theorem 3 and Corollaries 3 and 4.

4. Examples

Example 1. Consider the meromorphic function f(z) = tan z. Then from [9, p. 61] we see that J(f) = R ∪ {∞} and F (f) consists of two completely invariant components H^{+} and H^{−}, the upper and the lower planes respectively. Now f(z) ≠ ±i in C and so H^{+} \ f(H^{+}) = {i} and so the bound in Corollary 4 is sharp.

Example 2. Consider f(z) = \frac{1}{2} tan z. From [9, pp. 62–63] we see that J(f) is an unbounded totally disconnected subset of R, so F (f) is a single component. Also f(z) ≠ ±\frac{1}{2}i in C, so that F (f) \ f(F (f)) = \{±\frac{1}{2}i\} and the bounds in Theorems 1 and 3 are sharp.

The question of whether the case |N_{2} \ f(N_{1})| = 2 can occur when F (f) is not connected remains open.

Example 3. Consider

(1)  f_{p}(z) = ∫_{0}^{z} e^{−t^{p}} dt = z − \frac{z^{p+1}}{p + 1} + ⋯,  p ≥ 2.

From (1), zero is a parabolic fixed point of f with multiplicity p + 1 ≥ 3. It is then easy to see that for x ∈ R^{+} , z ∈ C the following properties of f_{p} hold:

(2)  f_{p}(e^{2πυiz/p}) = e^{2πυi/p}f_{p}(z), υ = 1, 2, …, p,
(3)  f_{p}(x) ≤ x,
(4)  f_{p}(R^{+}) ⊆ R^{+}.

Note also that f′(z) = e^{−z^{p}} which is non-zero in the finite plane, so that f^{−1} has no finite branch points.

Define sets A_{υ}, υ = 1, 2, …, p, by

A_{υ} = \{z : |arg z − \frac{2υπ}{p} | ≤ \frac{π}{2p}\},

and non-zero constants

α_{υ} = e^{2πυi/p} ∫_{0}^{∞} exp(−r^{p}) dr.

Then by [11, p. 267] we have that as z → ∞, z ∈ A_{υ}, f_{p}(z) → α_{υ} whilst as z → ∞ in the rest of the plane |f(z)| is unbounded, so the only finite transcendental singularities of f_{p}^{−1} are α_{1}, …, α_{p}.

The points ∞ and α_{υ} have Nevanlinna deficiencies δ(∞) = 1 and δ(α_{υ}) = 1/p, so that each α_{υ} is non-Picard exceptional.

We now claim that there are p disjoint, simply-connected components of F (f_{p}), say N_{υ}, υ = 1, 2, …, p, such that:

f(N_{υ}) ⊆ N_{υ}, e^{2πυi/p}R^{+} ⊆ N_{υ}.

For, by using the Leau–Fatou Flower Theorem (cf. [10, Theorem 7.2, p. 45] we see that there are p disjoint forward invariant components N_{υ} of F (f) such that zero lies in each ∂N_{υ} and that a segment [0, ϵ], ϵ > 0 of R^{+} lies in one N_{υ}, say N_{p}. From (3) and (4) we then see that f_{p}^{n}|_{R^{+}} → 0 as n → ∞ and R^{+} is contained in N_{p}. Using this fact and (2) we may label each N_{υ} so that N_{υ} contains e^{2πυi/p}R^{+} , which is then unbounded and so simply-connected (cf. [1]).

Now each α_{υ} ∈ e^{2πυi/p}R^{+} ⊂ N_{υ}. Suppose that α_{υ} ∈ f_{p}(N_{υ}). Then there exists some β ∈ N_{υ} with f_{p}(β) = α_{υ} and some branch g of f_{p}^{−1} defined analytically on some open neighbourhood of β with g(α_{υ}) = β.

Continuation of g along any path γ in N_{υ} which does not return to α is clearly possible. If U = D(α_{υ}, r) ⊂ N_{υ} so that g is analytic in U, suppose γ leaves \overline{U} at z_{1} and returns a first time to \overline{U} at z_{2} . If σ is the circumference of U and τ is the arc from z_{1} to z_{2} on ∂U then γ is homotopic in N_{υ} \ {α_{υ}} with fixed ends z_{1} , z_{2} to τσ^{n} for some integer n. Thus on analytic continuation on γ, g returns to the same branch and remains analytic at α_{υ}. But this is impossible, since in a parabolic basin such as N_{υ} the function f_{p} cannot be univalent.

So α_{υ} ∉ f(N_{υ}), and we have a function f_{p} demonstrating the sharpness of Corollary 3. Further the exceptional points of α_{υ} are not Picard exceptional.

We now show that exceptional points do not have to be even Nevanlinna exceptional.

Example 4. Consider the transcendental entire function f(z) defined as

(6) f(z) = ∫_{0}^{z} exp(−e^{t}) dt = ∫_{0}^{z} ∑_{n=0}^{∞} \frac{(−1)^{n}}{n!} e^{nt} dt = z + ∑_{n=1}^{∞} \frac{(−1)^{n}}{n!n} (e^{nz} − 1).

From (6), we see that for all z ∈ C

f(z + 2πi) = f(z) + 2πi,

and so in particular, f(2kπi) = 2kπi + f(0) = 2kπi, k ∈ Z. Thus each point 2kπi is a fixed point of f(z), and as f′(z) = exp(−e^{z}), f′(2kπi) = e^{−1} < 1, it is actually an attracting fixed point of f(z). As such, each 2kπi lies in F (f).

Suppose that z = x + iy, x > 0, |y| ≤ \frac{1}{2}π. Evaluating f(z) along the paths γ_{1} : Im z = 0, Re z = t ∈ [0, x]; γ_{2} : Re z = x, Im z = t ∈ [0, y] we get

f(z) = ∫_{0}^{x} exp(−e^{t}) dt + ∫_{0}^{y} i exp(−e^{x+it}) dt.

As x > 0 we have that

α(x) = ∫_{0}^{x} exp(−e^{t}) dt < ∫_{0}^{x} e^{−t} dt,

so that α(x) → α ∈ [0, 1] as x → ∞. As |y| ≤ \frac{1}{2}π, if 0 ≤ t ≤ |y|, then sin ≥ 2t/π, so that

|∫_{0}^{y} i exp(−e^{x+it}) dt| ≤ |∫_{0}^{y} exp(−e^{x cos t}) |dt|| = ∫_{0}^{|y|} exp(−e^{x cos t}) dt

≤ ∫_{0}^{π/2} exp(−e^{x cos t}) dt = ∫_{0}^{π/2} exp(−e^{x sin s}) ds

≤ ∫_{0}^{π/2} exp(\frac{−2t}{π}e^{x}) dt

= \frac{1}{2}π(e^{−x} − e^{−x}e^{−e^{x}}) → 0 as x → ∞.

Thus if x → ∞, |y| ≤ \frac{1}{2}π then

f(z) = f(x + iy) → ∫_{0}^{∞} exp(−e^{t}) dt = α (≠ 0), say.

Now for k ∈ Z, set

A_{k} = \{z : Re z > 0, | Im z − 2kπ| ≤ \frac{1}{2}π\}.

We have from (7) that as z → ∞, with z lying in some A_{k}, that

f(z) → α + 2kπi = α_{k}, say.

We claim that |f(z)| is unbounded as z → ∞ on the following paths:

(8) γ_{k} : R^{+} + (2k + 1)πi, k ∈ Z.

(9) Any path γ: [0, 1] → Ĉ such that Im γ(t) is unbounded.

(10) Any path γ: [0, 1] → Ĉ such that Re γ(t) < x_{0} for sufficiently large t. To show (8), if z = x + (2k + 1)πi , k ∈ Z, x > 0, then

f(z) = ∫_{0}^{(2k+1)π} i exp(−e^{it}) dt + ∫_{0}^{x} exp(−e^{t+(2k+1)πi}) dt
= O(1) + ∫_{0}^{x} exp(e^{t}) dt

so that |f(z)| → ∞ as |x| → ∞, and (8) is shown.

To show (9), suppose that γ is any such path to ∞. We may find values z_{k} = x_{k} + 2πki ∈ γ for an unbounded sequence of integers k and x_{k} ∈ R. Then f(z_{k}) = f(x_{k}) + 2πki. As f(z) is real on the real line, |f(z)| ≥ 2π|k| → ∞ as |k| → ∞.

In proving (10) we may now assume because of (9) that Im γ(s) is bounded with Re γ(s) → −∞, and | Im γ(s)| < y_{0} , say. Then if z = x + iy lies on γ,

f(z) = ∫_{0}^{x} exp(−e^{t}) dt + ∫_{0}^{y} i exp(−e^{x+it}) dt.

Now t < 0, so e^{t} < 1, e^{−e^{t}} > e^{−1} and then

|∫_{0}^{x} exp(−e^{t}) dt| > |x|e^{−1} → ∞

and

|∫_{0}^{y} exp(−e^{x+it}) dt| ≤ |∫_{0}^{y} exp(−e^{x cos t}) |dt||
≤ |∫_{0}^{y} exp(e^{x}) |dt|| ≤ |y_{0}| exp(e^{x}) → |y_{0}| as x → −∞.

Thus as z → ∞ on γ, |f(z)| → ∞, and the claims (8)–(10) above are shown. It is clear that f′ is non-zero in the finite plane, and so f^{−1} has no finite algebraic singularities. We now claim that the finite asymptotic values of f(z) are the points {a_{k}}. Suppose not, and so there is a path γ → ∞ so that as z → ∞ on γ, f(z) → a ∈ C \ {α_{k}}, k ∈ Z. By claims (8)–(10) we see that γ must eventually lie in the strip S in the right half-plane between an A_{k} and one of the lines l_{k}, l_{k−1} ; without loss of generality we may assume that it lies between A_{k} and l_{k−1} , so S = Re(z) > 0, (2k − 1)π ≤ Im(z) ≤ (2k − \frac{1}{2})π. We may then take the line δ : Im(z) = (2k − \frac{1}{2}π) contained in A_{k} and join γ and δ by a vertical line l, so that these paths together with ∞ enclose a region R contained inside the horizontal strip S.

Note that there exists A > 0 such that |f(z)| ≤ exp(e^{|z|})|z| < exp(Ae^{|z|}) for all large enough |z|.

Clearly |f| is bounded on ∂R, |f| < M, say. Consider the function g, where

g(z) = exp(ϵ exp(m\{z − (2k − \frac{3}{4}πi)\})), ϵ > 0, \sqrt{2} < m < 2.

Then for z ∈ R,

|g(z)| = exp(ϵ(exp mx) cos(my − m(2k − \frac{3}{4})))
> exp(ϵ(exp(mx) cos(m\frac{1}{4}π))) > 1,

since \sqrt{2} < m < 2. Thus |f/g| < M on ∂R, and as x → ∞ in R,

|f/g| < exp\{Ae^{x(1+v(x))} − ϵ cos(\frac{1}{4}mπ)e^{mx}\}

where v(x) = (|z|/x) − 1 → 0 as x → ∞. Hence we see that f/g → 0 as z goes to ∞ in R. We now choose a z ∈ R and take a curve Γ ⊆ R which connects γ and δ so that z lies in the finite part R_{0} of R cut off by Γ. Using the above and the maximum principle we see that, so long as Γ is chosen far enough to the right then |f/g| < M on Γ, and ∂R_{0} , so |f/g| < M in R_{0} .

So |f(z)| < M|g(z)|, and as ϵ is arbitrary and z fixed, we see that |f(z)| < M. However the choice of z was arbitrary and so this is true of all z ∈ R. We then use [13, Theorem 340, p. 172] to see that the asymptotic values a, α_{k} are equal, and so the only finite asymptotic values of f are the points α_{k} which are then exactly the finite transcendental singularities of f^{−1} .

The positive real axis, R^{+} , is mapped into itself by f(z) which has an attracting fixed point at 0, and on R, f′ < 1, so by Rolle’s theorem, f(x) = x has no real solutions other than 0. Also, for x > 0 (< 0), f(x) < 0 (> 0), and so we may then see that f^{n} → 0 on R as n → ∞, so R lies in an unbounded (and so simply-connected) forward invariant component N_{0} of F (f).

Using (7) we see that each R + 2kπi lies in some simply connected, forward invariant component N_{k} of F (f) which has an attracting fixed point of f(z) at 2kπi, so that the N_{k} must be disjoint. We then use exactly the same reasoning as in Example 3, to see that each α_{k} ∈ N_{k} \ f(N_{k}).

It is shown by [11, p. 290] that the points α_{k} are non-Nevanlinna exceptional points of f(z), and so the claim at the end of the introduction is shown.

The author wishes to thank Professor I.N. Baker for many helpful suggestions.

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274 M.E. Herring

[9] Devaney, R.L., and L. Keen: Dynamics of meromorphic maps with polynomial Schwarzian derivative . - Ann. Sci. École Norm. Sup. (4) 22, 1989, 55–79.

[10] Milnor, J.: Dynamics in one complex variable. Introductory lectures. - Preprint S.U.N.Y. Stonybrook, 1990/5.

[11] Nevanlinna, R.: Analytic Functions. - Springer-Verlag, Berlin, 1970.

[12] Noshiro, K.: Cluster Sets. - Springer-Verlag, Berlin, 1960.

[13] Pólya, G., and G. Szegö: Problems and Theorems in Analysis, Vol. 1. - Springer-Verlag, New York, 1972.

[14] Tsuji, M.: Potential Theory in Modern Function Theory. - Maruzen Co. Ltd., Tokyo, 1959.

Received 23 August 1993