Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 471–488

FIXED POINTS AND BOUNDARY BEHAVIOUR OF THE KOENIGS FUNCTION

Manuel D. Contreras, Santiago Díaz-Madrigal, and Christian Pommerenke

Universidad de Sevilla, Escuela Superior de Ingenieros, Departamento de Matemática Aplicada II Camino de los Descubrimientos, s/n, ES-41092 Sevilla, Spain; contreras@esi.us.es, madrigal@us.es

Technische Universität, Institut für Mathematik DE-10623 Berlin, Germany; pommeren@math.tu-berlin.de

Abstract. We analyze the relationship between the fixed points of different iterates of an analytic self-map of the unit disk. We show that, in general, a boundary fixed point of such a function is not a fixed point of its iterates. However, in the context of fractional iteration, all the iterates have the same fixed points. We also present results, in terms of the Koenigs function, of self-maps whose behaviour are not so extreme as above.

1. Introduction and statement of the results

1.1. Let φ be a holomorphic map in the unit disc D with φ(D) ⊂ D. A point a ∈ ∂D is called a boundary contact point of φ, if the non-tangential or angular limit L := ∠ lim_{z→a} φ(z) lies in ∂D. Moreover, boundary contact points have multipliers. That is, and also in the non-tangential sense, φ always has a derivative φ′(a) ∈ C_{∞}\{0} at that point a. If L = a, the point a is called a boundary fixed point of φ and, in this case, φ′(a) ∈ (0, +∞) ∪ {∞}.

In what follows, a fixed point of φ will mean a fixed point in the classical sense (φ(a) = a with a ∈ D) or a boundary fixed point. The famous Denjoy–Wolff point (DW-point) of every non-trivial φ will be denoted by τ_{φ} or, when there is no confusion, simply by τ. The study of fixed points and boundary contact points is one of the central topics in iteration theory in the unit disk (see [8], [10], [11]) as well as in those related mathematical branches like composition operators. It is worth mentioning that boundary contact points are playing a more and more important role in many situations (see [2], [3] and the references therein).

This paper is about the collection of the boundary fixed points and the boundary contact points of the different iterates φ_{n} of the function φ. Of course, many things are known, so we are trying to determine clearly what our contributions are. Almost all of our results assume that φ has an inner DW-point (τ ∈ D and |φ′(τ)| ≠ 1). We consider the case of a boundary DW-point (τ ∈ ∂D) only in Theorem 5. To be precise with the terminology, we recall that φ is usually said to have an elliptic DW-point, whenever φ is not trivial, τ ∈ D and |φ′(τ)| = 1.

2000 Mathematics Subject Classification: Primary 30D40; Secondary 30C45, 37C25.

This research has been partially supported by the Ministerio de Ciencia y Tecnología and the European Union (FEDER) project BFM2003-07294-C02-02 and by La Consejería de Educación y Ciencia de la Junta de Andalucía.

So, suppose that φ has an inner DW-point. If a ∈ ∂D is a boundary fixed point of φ with φ′(a) ≠ ∞, by [13, p. 80], the curve r ∈ [0, 1) ↦ φ(ra) ∈ D tends non-tangentially to a and, according to the Lehto–Virtanen theorem [13, Chapter 4], we get that a is also a boundary fixed point of φ_{n}, for all n ∈ N. On the other hand, if a is a boundary fixed point of φ_{n} with finite derivative, we can only claim that a is a boundary contact point of φ. In other words, and under certain regularity assumptions, boundary fixed points “grow” in the forward direction. In spite of several ambiguous comments elsewhere, we have to say that the hypothesis φ′(a) ≠ ∞ is crucial there. In fact, in Section 3, we provide an example (Example 1) of a univalent holomorphic map φ: D → D having a boundary fixed point a ∈ ∂D such that a is not even a boundary contact point of φ_{2} . Necessarily, φ′(a) = ∞.

1.2. As one may expect, we can go further in the framework of fractional iteration. Some words are in order. We recall that a semigroup of analytic functions (φ_{t}) is a family (indexed by the non-negative real numbers) of holomorphic self-maps of the unit disk, satisfying the following three conditions:

(a) φ_{0} is the identity in D,

(b) φ_{t+s} = φ_{t} ∘ φ_{s}, for all t, s ≥ 0,

(c) for every z ∈ D, lim_{t→0} φ_{t}(z) = z. The fractional iterates φ_{t} are always univalent. Moreover, if one of the iterates

φ_{t} (t > 0) has an inner (resp. boundary) DW-point, all the φ_{t} (t > 0) have an inner (resp. boundary) DW-point and, indeed, all of the DW-points are the same [15]. In that case, we say that (φ_{t}) is a semigroup with inner (resp. boundary) DW-point. There are other two types of semigroups (trivial and elliptic) but we remit to the literature for more information.

Theorem 1. Let (φ_{t}) be a semigroup of analytic functions with inner DWpoint and a ∈ ∂D. Then, the point a is a boundary fixed point of φ_{t} for some t > 0 if and only if it is a boundary fixed point for all the iterates φ_{t}.

In particular, if a holomorphic mapping φ: D → D with inner DW-point can be embedded into a semigroup of analytic functions, then all the iterates of φ have the same collection of boundary fixed points. This theorem was also enunciated by Cowen in the relevant paper [7]. Unfortunately, Cowen’s proof uses that a boundary fixed point of a holomorphic map φ: D → D is also a boundary fixed point of φ_{2} , which we know to be incorrect in general. Anyway, our approach is different and, perhaps, clearer.

Boundary fixed points can be characterized by some properties of the classical Koenigs function, also in the context of fractional iteration. The famous result of Koenigs [14, Section 6.1] asserts the existence of a unique holomorphic map σ: D→C such that

σ ∘ φ = φ′(τ)σ

with σ′(τ) = 1, for every holomorphic self-map φ of D with inner DW-point and φ′(τ) ≠ 0. This map σ is called the Koenigs function associated to φ and σ is univalent, whenever φ is. It is possible to prove that if (φ_{t}) is a semigroup with inner DW-point, all the Koenigs functions associated to φ_{t} (t > 0) coincide, so we can talk about the Koenigs function of the semigroup (φ_{t}).

Theorem 2. Let (φ_{t}) be a semigroup of analytic functions with inner DWpoint and let σ be the corresponding Koenigs function. Assume that a ∈ ∂D. Then the following are equivalent:

(1) The point a is a boundary fixed point of φ_{t} for some (resp. all) t > 0.

(2) The radial limit lim_{r→1^{−}} σ(ra) exists and is ∞.

(3) The unrestricted limit lim_{z→a} σ(z) exists and is ∞. That is, the function σ admits a continuous extension from D∪{a} into C_{∞}, where we assign σ(a) := ∞.

In terms of prime end theory and denoting the corresponding Carathéodory map by σ̂, we notice that the above statement two says that σ̂(a) is an accessible prime end and the statement three that the impression of σ̂(a) is singleton. This theorem is false outside of the fractional iteration world. In fact, we can show, see Example 2, a univalent function verifying statement two but failing the first one and another univalent function, see Example 3, satisfying the second but not the third condition.

Corollary 3. Let (φ_{t}) be a semigroup of analytic functions with inner DWpoint and a ∈ ∂D a boundary fixed point of φ_{t} for some (resp. all) t > 0. Then, each fractional iterate φ_{t} admits a continuous extension from D∪{a} into D∪{a}, where φ_{t}(a) := a.

Apparently, if we want a more general (univalent) version of the last theorem we have to weaken the hypothesis of being a boundary fixed point. As we will see, the concept of boundary contact point will be really useful for this task.

We point out that, as a consequence of the Julia–Carathéodory theorem, if a ∈ ∂D is a boundary contact point of some φ_{n} with φ′_{n}(a) ≠ ∞, then a is also a boundary contact point of each iterate φ_{k}, k < n. Thus, once more under certain regularity assumptions, boundary contact points “grow” in the backward direction. For fractional iteration, there is also a strong relation between boundary fixed points and boundary contact points. Here, the dynamical concept of ω -limit appears in a very natural way. We want to point out that a point ξ ∈ C_{∞} is called an ω -limit point of a curve γ: [0, 1) → C if there exists a strictly increasing sequence (r_{n}) ⊂ [0, 1) convergent to 1 such that γ(r_{n}) → ξ. The set of all ω -limit points of γ is called its ω -limit and it is denoted by ω(γ).

Theorem 4. Let (φ_{t}) be a semigroup of analytic functions with inner DWpoint and a ∈ ∂D. Then the following are equivalent:

(1) The point a is a boundary fixed point of φ_{t} for some (resp. all) t > 0.

(2) The point a is a boundary contact point of φ_{t} for all t > 0.

(3) There is t > 0 such that the point a is a boundary contact point of φ_{nt}, for every n ∈ N.

(4) For all t > 0, the ω -limit of the curve

r ∈ [0, 1) ⟶ φ_{t}(ra) ∈ D

is completely included in ∂D.

(5) For some t > 0 and every n ∈ N, the ω -limit of the curves

r ∈ [0, 1) ⟶ φ_{nt}(ra) ∈ D

are completely included in ∂D.

In general, the above analysis cannot be quickly translated to the DW-bound-ary context. Clearly, new phenomena appear and it deserves a proper and further study. Anyway, we want to point out that our initial theorem also holds in this situation.

Theorem 5. Let Φ = (φ_{t}) be a semigroup of analytic functions with boundary DW-point and a ∈ ∂D. Then, the point a is a boundary fixed point of φ_{t} for some t > 0 if and only if it is a boundary fixed point for all φ_{t}.

Obviously, we can drop “boundary” above since fixed points of fractional iterates of semigroups with boundary DW-point are all of them contained in ∂D.

This theorem was also stated by Cowen in [7]. However, the same remarks given for the inner DW-point case are still valid here.

1.3. Now we leave the fractional iteration and give the univalent version of Theorems 2 and 3. Due to the undoubtedly dynamical aspect of the equivalences, we have decided to give an initial version with “general” curves and, after that, to show the corresponding corollary with “radial” curves.

Theorem 6. Let φ: D → D be a univalent function with inner DW-point and let σ be the corresponding Koenigs function. Likewise, let γ: [0, 1) → D be a curve. Then, the following statements are equivalent.

(1) For all n ∈ N, the limit lim_{r→1} φ_{n}(γ(r)) exists and lies in ∂D.

(2) For all n ∈ N, the ω -limit of the curve φ_{n} ∘ γ: [0, 1) → D is completely contained in ∂D.

(3) The ω -limit of the curve σ∘γ: [0, 1) → C is ∞. That is, lim_{r→1} σ(γ(r)) = ∞.

Corollary 7. Let φ: D → D be a univalent function with inner DW-point and let σ be the corresponding Koenigs function. Assume that a ∈ ∂D. Then, the following statements are equivalent.

(1) The point a is a boundary contact point of φ_{n}, for every n ∈ N.

(2) For all n ∈ N, the ω -limit of the curve r ∈ [0, 1) → φ_{n}(ra) ∈ D is completely contained in ∂D.

(3) The radial limit lim_{r→1^{−}} σ(ra) exists and is ∞.

1.4. It is quite natural to ask about what can happen in the non-univalent situation. Different examples unequivocally tell us that, in this context, everything is much more complicated and, roughly speaking, they suggest replacing the verb “to contain” by “to touch” in the corresponding assertions.

Anyway, it is worth mentioning that it is not possible to replace in the above corollary “ω(φ_{n} ∘ γ) ⊂ ∂D, for all n” by “ω(φ_{n} ∘ γ) ⋂ ∂D ≠ ∅, for all n” (see Example 4).

Theorem 8. Let φ: D → D be a function with inner DW-point τ and φ′(τ) ≠ 0, and let σ be the corresponding Koenigs function. Assume that a ∈ ∂D and let γ: [0, 1) → D be a curve such that ω(γ) = {a}. If a is a boundary contact point for every iterate of φ, then the following two equivalent statements hold:

(1) For every n ∈ N, the intersection ω(φ_{n} ∘ γ) ⋂ ∂D is not empty.

(2) The ω -limit of the curve σ ∘ γ: [0, 1) → C contains ∞.

Even for radial curves, the behaviour of the just mentioned ω -limit of the curve σ ∘ γ can be very “pathological” in a certain sense. In fact, in Section 3, we present a non-univalent holomorphic function φ: D → D having the point 1 as a boundary fixed point and 0 as the DW-point with 0 < |φ′(0)| < 1, such that the ω -limit of the curve r ∈ [0, 1) → σ(r) ∈ C is a compact connected subset of C_{∞} including ∞ and the DW-point 0 (see Example 5).

Thinking about the meaning, in this non-univalent framework, of having a non-common boundary contact point for the iterates of φ, we have arrived at an extreme dichotomy for the behaviour of the ω -limits of φ_{n} ∘ γ, where γ is curve in D tending to the corresponding boundary fixed point of φ. In a certain sense, this complements the former theorem.

Theorem 9. Let φ: D → D be a function with inner DW-point τ and φ′(τ) ≠ 0, and let σ be the corresponding Koenigs function. Let γ: [0, 1) → D be a curve. Then, only one of the two following conditions is satisfied:

(i) The ω -limit of the curve σ ∘ γ: [0, 1) → C does not contain ∞ and

sup{|φ_{n} ∘ γ(t) − τ| : t ∈ [0, 1)} ⟶^{n→∞} 0.

(ii) The ω -limit of the curve σ∘γ: [0, 1) → C contains ∞ and ω(φ_{n}∘γ) ⋂ ∂D ≠ ∅, for every n ∈ N.

It is worth mentioning that there are functions φ such that the associated Koenigs function touches the point ∞ for every radial limit (see Example 6).

2. Proofs of the results

In order not to repeat arguments, we will give the proofs in a different order.

Proof of Theorem 6. Let Ω be equal to σ(D). Denote λ = φ′(τ). Since σ is univalent we have that λ ≠ 0. If the limit lim_{r→1} φ_{n}(γ(r)) = η exists, then we have that ω(φ_{n} ∘ γ) is only the single point η. So, it is clear that (1) implies (2).

Let us see that (2) implies (3). We know that ω (σ ∘ γ) ⊂ C_{∞}. Suppose that there is a point w ∈ ω (σ ∘ γ) ∩ C. Since w ∈ C, 0 ∈ Ω (which is a open set), and λ ∈ D there is a natural number n such that λ^{n}w ∈ Ω. Moreover, w ∈ ω (σ ∘ γ). That is, there is a sequence (r_{m}) in the interval (0, 1) converging to 1 such that σ(γ(r_{m})) → w. Therefore, bearing in mind that σ is univalent, we have

φ_{n}(γ(r_{m})) = σ^{−1}(λ^{n}σ(γ(r_{m}))) ⟶^{m→∞} σ^{−1}(λ^{n}w) ∈ D.

That is, ω(φ_{n} ∘ γ) ∩ D ≠ ∅. A contradiction.

Finally, we see that (3) implies (1). Fix a natural number n and consider the curve r ∈ [0, 1) ↦ λ^{n}σ(γ(r)). By hypothesis, lim_{r→1} λ^{n}σ(γ(r)) = ∞. Since σ is univalent it follows from [14, p. 162] that

φ_{n}(γ(r)) = σ^{−1}(λ^{n}σ(γ(r)))

tends to a limit as r → 1, and this limit lies in ∂D because σ is finite in D.

Proof of Theorems 1, 2 and 4. Let Ω be equal to σ(D). To fix the notation, we have that there is c with Re c > 0 such that φ_{t}(z) = σ^{−1}(e^{−ct}σ(z)) for all t ≥ 0 and z ∈ D.

To prove these three theorems, we have to get that the following eight assertions are equivalent:

(i) The point a is a boundary fixed point of φ_{t} for some t > 0. (ii) The point a is a boundary fixed point of φ_{t} for all t > 0. (iii) The unrestricted limit lim_{z→a} σ(z) exists and is ∞. (iv) The radial limit lim_{r→1^{−}} σ(ra) exists and is ∞.

(v) The point a is a boundary contact point of φ_{t} for all t > 0. (vi) There is t > 0 such that the point a is a boundary contact point of φ_{nt}, for

every n ∈ N. (vii) For all t > 0, the ω -limit of the curve

r ∈ [0, 1) ⟶ φ_{t}(ra) ∈ D

is completely included in ∂D.

(viii) For some t > 0 and every n ∈ N, the ω -limits of the curves

r ∈ [0, 1) ⟶ φ_{nt}(ra) ∈ D

are completely included in ∂D.

First of all, bearing in mind that all functions of the semigroup have the same Koenigs function, by Theorem 6, we have that (iv), (v), (vi), (vii), and (viii) are equivalent. Moreover, it is obvious that (ii) implies (v), (ii) implies (i), and that (iii) implies (iv). To close the cycle we are going to prove that

(iv) implies (iii), (iv) implies (ii), and (i) implies (vi).

(iv) implies (iii). To simplify this implication we introduce some notation. Given c ∈ C with Re c > 0 and w ∈ C, w ≠ 0, we define the spiral spir_{c}[w] = {e^{−cs}w : s ∈ R} ⋃{0} ⋃{∞}; given real numbers s < t, we define the spiral segment spir_{c}[e^{−s}w, e^{−t}w] as the subarc of spir_{c}[w] that goes from e^{−s}w to e^{−t}w; finally, spir_{c}[∞, w] = {e^{−cs}w : s ≤ 0} ⋃{∞}.

Take a zero-chain (C_{m}) in D converging to a such that (σ(C_{m})) is a zerochain in Ω such that σ̂(a) = [σ(C_{m})]. To get (iii) we have to prove that the impression of the prime end σ̂(a) is the single point ∞. By [5, Lemma 3.3] the corresponding impression I(σ̂(a)) ⊂ ∂_{∞}Ω must be of one of the following types:

(a) I(σ̂(a)) is a single point.

(b) There is λ ∈ ∂D and η_{1} < η_{2} such that I(σ̂(a)) = spir_{c}[e^{−η_{1}c}λ, e^{−η_{2}c}λ].

(c) There is λ ∈ ∂D and η ∈ R such that I(σ̂(a)) = spir_{c}[∞, e^{−ηc}λ].

Since lim_{r→1} σ(ra) = ∞, we have that ∞ ∈ I(σ̂(a)). Therefore, the possibility (b) cannot occur. If (c) is satisfied, then lim_{m} w_{m} always exists and it is equal to λe^{−ηc} whenever w_{m} ∈ σ(C_{m}) for all m ∈ N (see the proof of [5, Theorem 1.2.

(2) implies (3)]). Moreover, for each m, there is r_{m} ∈ (0, 1) such that r_{m}a ∈ C_{m}. So, lim_{m} σ(r_{m}a) = e^{−ηc}. A contradiction because the sequence (r_{m}) must tend to 1 and, by hypothesis, lim_{m} σ(r_{m}a) = ∞. Therefore, I(σ̂(a)) is a single point and, again using that ∞ ∈ I(σ̂(a)) we get that I(σ̂(a)) = {∞}.

(iv) implies (ii). Let us fix t > 0. We have to show that

lim_{r→1} φ_{t}(ra) = lim_{r→1} σ^{−1}(e^{−ct}σ(ra)) = a.

First of all, notice that by [14, p. 162], the limit lim_{r→1} σ^{−1}(e^{−ct}σ(ra)) does exist. So we only have to prove that the limit is equal to a.

Clearly, we have that lim_{r→1} φ_{0}(ra) = a. Take an increasing sequence (r_{n}) tending to 1 with r_{0} = 0. For each n, define the following curves

\begin{aligned}γ_{n} & : r ∈ [r_{n−1}, r_{n}] ⟼ γ_{n}(r) := σ(ra), \\ η_{n} & : s ∈ [0, t] ⟼ η_{n}(s) := e^{−cs}σ(r_{n}a). \end{aligned}

Since Ω is c-spirallike, the curve η_{n} is in Ω. Now we build the curve

Γ = ⨁_{n=1}^{∞}(γ_{2n−1} + η_{2n−1} + e^{−ct}γ_{2n} + η2n^{−}).

Notice that Γ is a curve in Ω joining 0 to ∞. So, by [14, p. 162], we have that there is ω ∈ ∂D such that lim_{w∈Γ,w→∞} σ^{−1}(w) = ω. On the one hand, the points σ(r_{2n−1}a) ∈ γ_{2n−1} ⊂ Γ and the sequence (σ(r_{2n−1}a)) tends to ∞. So, ω = lim_{n→∞} σ^{−1}(σ(r_{2n−1}a)) = lim_{n→∞} r_{2n−1}a = a. On the other hand, the points e^{−ct}σ(r_{2n−1}a) ∈ η_{2n−1} ⊂ Γ and the sequence (e^{−ct}σ(r_{2n−1}a)) tends to ∞. So,

ω = lim_{n→∞} σ^{−1}(e^{−ct}σ(r_{2n−1}a)) = lim_{n→∞} φ_{t}(r_{2n−1}a).

Therefore, lim_{n→∞} φ_{t}(r_{2n−1}a) = a. Since the limit lim_{r→1} φ_{t}(ra) does exist, we have that it must be a.

(i) implies (vi). First of all, notice that, since Ω is c-spirallike, by [9, p. 431], the limit lim_{r→1} σ(rξ) does exist and belong to C ⋃{∞} for all ξ ∈ ∂D. In particular, we have that the limit lim_{r→1} e^{−cnt}σ(rξ) exists for all n and all ξ. On the one hand, if this limit belongs to Ω, then the continuity of σ^{−1} in Ω implies that the limit lim_{r→1} φ_{nt}(rξ) = lim_{r→1} σ^{−1}(e^{−cnt}σ(rξ)) exists. On the other hand, if this limit belongs to ∂_{∞}Ω, by [14, p. 162], the limit lim_{r→1} φ_{nt}(rξ) = lim_{r→1} σ^{−1}(e^{−cnt}σ(rξ)) exists.

Summing up, we have that for all n and for all ξ, the limit lim_{r→1} φ_{nt}(rξ) exists. It is worth pointing out that we want to prove (vi) and, at this moment, we only know that the limit lim_{r→1} φ_{nt}(ra) exists for all n (notice that we have not used the hypothesis (i) to get this preliminary fact) but we do not know if these limits belong to the boundary of the unit disc.

Now, we are going to get this last assertion. Namely, we show that

lim_{r→1} φ_{nt}(ra) = a for all n.

We argue by induction. By hypothesis, we have that lim_{r→1} φ_{t}(ra) = a. Now suppose that lim_{r→1} φ_{nt}(ra) = a. Then, by the Lehto–Virtanen theorem [13, Chapter 4], we have that

lim_{r→1} φ_{(n+1)t}(ra) = lim_{r→1} φ_{t}(φ_{nt}(ra)) = lim_{z→a, z∈φ_{nt}([0,1)a)} φ_{t}(z) = lim_{r→1} φ_{t}(ra) = a.

Proof of Corollary 3. By Theorem 2, we have that the function σ has a continuous extension to the point a. What we are going to do is to pass this continuous extension property from σ to φ_{t}.

So, fix t > 0 and let

U_{k} = {z ∈ D : |z − a| < 1/k}.

We have to show that diamφ_{t}(U_{k}) → 0 as k → ∞. Suppose this is false. Then, for some subsequence, there are Jordan arcs C_{k_{n}} in φ_{t}(U_{k_{n}}) with diamC_{k_{n}} > b > 0. We have

σ(C_{k_{n}}) ⊂ σ(φ_{t}(U_{k_{n}})) = e^{−ct}σ(U_{k_{n}}) ⟶^{n→∞} ∞

by Theorem 2. Hence (C_{k_{n}}) is a sequence of Koebe arcs [12, p. 267] for the function σ. This is a contradiction because a univalent function has no Koebe arcs.

Proof of Theorem 5. Denote τ ∈ ∂D the DW-point of the semigroup. Then there is a unique univalent function σ: D → C with σ(0) = 0 verifying the property

(∗ ∗ ∗)  Ω + t ⊂ Ω, for each t > 0, where Ω := σ(D),

and such that

φ_{t}(z) = τσ^{−1}(σ(\overline{τ}z) + t), t ≥ 0, z ∈ D

(see [1] or [15]). Fix t_{0} > 0 such that a is a fixed point for φ_{t_{0}} . We claim that the following properties hold:

(a) For all ξ ∈ ∂D, the limit lim_{r→1} σ(rξ) exists and belongs to ∂_{∞}Ω;

(b) For all ξ ∈ ∂D, the limit lim_{r→1} φ_{t_{0}}(rξ) exists;

(c) a is a fixed point of φ_{nt_{0}} for all n;

(d) lim_{r→1} σ(ra) = ∞.

Once we know that (d) is satisfied we conclude the proof arguing as in the proof of (iv) implies (ii) above.

Let us check (a). In [4], K. Ciozda obtained that, given a univalent function σ satisfying (∗ ∗ ∗), there is α ∈ [−\frac{1}{2}π, \frac{1}{2}π], such that

Re e^{iα}(1 − z)^{2}σ′(z) ≥ 0 for all z ∈ D.

In particular, denoting g(z) = z/(1 − z), we have that

Re \frac{e^{iα}σ′(z)}{g′(z)} ≥ 0 for all z ∈ D.

Since g is convex, we see that the function e^{iα}σ is close-to-convex with associated function g. Notice that g(D) = \{w ∈ C : Re w > −\frac{1}{2}\}. So,

∂g(D) = \{w ∈ C : Re w = −\frac{1}{2}\}

and we parametrize this curve by

w(t) = −\frac{1}{2} + \frac{sin t}{2(1 − cos t)}i

for all 0 ≤ t ≤ 2π. Take

β(t) = \{ \begin{matrix}lim_{τ→t^{+}} arg[w(τ) − w(t)] & if w(t) ≠ ∞, \\ lim_{τ→t^{+}} arg w(τ) + π & if w(t) = ∞. \end{matrix}

On the one hand, since the function t ∈ (0, 2π) ↦ sin t/2(1 − cos t) is increasing, we get that β(t) = \frac{3}{2}π for all t ∈ (0, 2π). On the other hand,

β(0) = lim_{τ→0^{+}} arg w(τ) + π = lim_{τ→0^{+}} arg[−\frac{1}{2} + \frac{sin τ}{2(1 − cos τ)}i] + π = \frac{1}{2}π + π = \frac{3}{2}π.

Therefore, by [13, Theorem 3.21], the limit lim_{r→1} σ(rξ) does exist and belongs to C ⋃{∞} for all ξ ∈ ∂D. The univalence of σ implies that this limit belongs to ∂_{∞}Ω.

Now we obtain (b). On the one hand, if this limit belongs to Ω, then the continuity of σ^{−1} in Ω implies that the limit lim_{r→1} φ_{nt_{0}}(rξ) = lim_{r→1} σ^{−1}(σ(rξ)+nt_{0}) exists. On the other hand, if this limit belongs to ∂_{∞}Ω, by [14, p. 162], the limit lim_{r→1} φ_{nt_{0}}(rξ) = lim_{r→1} σ^{−1}(σ(rξ) + nt_{0}) exists. Summing up, we have that for all n the limit lim_{r→1} φ_{nt_{0}}(ra) exists.

To get (c), we argue by induction. We have to see that lim_{r→1} φ_{nt_{0}}(ra) = a for all n. By hypothesis, we have that lim_{r→1} φ_{t_{0}}(ra) = a. Now suppose that lim_{r→1} φ_{nt_{0}}(ra) = a. Then, by the Lehto–Virtanen theorem [13, Chapter 4], we have that

lim_{r→1} φ_{(n+1)t_{0}}(ra) = lim_{r→1} φ_{t_{0}}(φ_{nt_{0}}(ra))

= lim_{z→a, z∈φ_{nt_{0}} ([0,1)a)} φ_{t_{0}}(z) = lim_{r→1} φ_{t_{0}}(ra) = a.

Finally, we deduce (d). Denote ξ = lim_{r→1} σ(ra) (this limit does exist by (a)). Let us see that ξ = ∞. Suppose on the contrary that ξ ≠ ∞. Then there are two possibilities: either there is n such that ξ + nt_{0} ∈ Ω or ξ + st_{0} ∈ ∂Ω for all s > 0. In the first case, a = lim_{r→1} φ_{nt_{0}}(ra) = lim_{r→1} σ^{−1}(σ(ra) + nt_{0}) ∈ Ω. A contradiction. In the second case, we have that the half-line {ξ + s : s > 0} is included in the boundary of Ω. Moreover, we have one of the following two situations:

Im σ(rb) < Im ξ for all r or Im σ(rb) > Im ξ for all r.

Suppose we have that Im σ(rb) < Im ξ for all r. Then, there is c ∈ R, such that the half-strip

{w ∈ C : 0 < Im w < Im ξ, Re w > c}

is included in Ω^{#} .This implies that the points of the half-line {ξ + s : s > 0} are simple boundary points and, by [6, Theorem 14.5.12(b)], the function σ^{−1} has a continuous one-to-one extension to Ω ⋃{ξ + s : s > 0}. But we know that σ^{−1}(ξ + nt_{0}) = a for all n. A contradiction. So, ξ cannot be different from ∞.

Proof of Theorem 9. Without loss of generality, we may assume that τ = 0. Denote λ = φ′(0). First, assume that r := sup_{z∈γ} |φ_{m}(z)| < 1 for some natural number m. If z ∈ γ and n > m, then

|φ_{n}(z)| = |φ_{n−m}(φ_{m}(z))| ≤ sup_{|ζ|≤r} |φ_{n−m}(ζ)|.

Since 0 is the Denjoy–Wolff point, we get that sup_{|ζ|≤r} |φ_{n−m}(ζ)| ⟶_{n→∞}0. The

Koenigs function satisfies σ(z) = σ(φ_{n}(z))/λ^{n} for all n and for all z. So, given z ∈ γ, we have

|σ(z)| = \frac{1}{|λ|^{m}} |σ(φ_{m}(z))| ≤ \frac{1}{|λ|^{m}} sup_{|ζ|≤r} |σ(ζ)| < ∞.

Therefore, if sup_{z∈γ} |φ_{m}(z)| < 1 for some natural number m, we have that (1) is satisfied.

Now, assume that

(∗)  sup_{z∈γ} |φ_{n}(z)| = 1 for all natural n.

In this case, it is trivial that ω(φ_{n}∘γ) ⋂ ∂D ≠ ∅, for every n ∈ N. Take 0 < ϱ < 1 such that σ has no zeros on |ζ| = ϱ. Then there is c > 0 such that

(∗∗)  |σ(ζ)| ≥ c for |ζ| = ϱ.

Since φ_{n}(0) = 0, the curve γ_{n} = {φ_{n}(z) : z ∈ γ} has 0 as initial point and, by (∗), we have that \overline{γ_{n}} ⋂ ∂D ≠ ∅. So there is z_{n} ∈ γ such that |φ_{n}(z_{n})| = ϱ. Now, using again that σ(z) = σ(φ_{n}(z))/λ^{n} and (∗∗), we have that

|σ(z_{n})| = \frac{1}{|λ|^{n}} |σ(φ_{n}(z))| ≥ \frac{c}{|λ|^{n}} ⟶_{n→∞} ∞

and this implies that |z_{n}| → 1.

Proof of Theorem 8. Denote a_{n} := lim_{r→1} φ_{n}(ra) ∈ ∂D for all n. Since φ_{n}(D) ⊂ D, the function

ψ_{n}(z) = log(1 − \overline{a_{n}}φ_{n}(z)) (z ∈ D)

satisfies that | Im ψ_{n}(z)| < \frac{1}{2}π. Therefore, ψ_{n} is a Bloch function.

Suppose that sup_{z∈γ} |φ_{n}(z)| < 1 for some natural number n. Then |ψ_{n}| is bounded on γ. On the one hand, by the Anderson–Clunie–Pommerenke theorem [13, Proposition 4.4], we get that |ψ_{n}| is bounded on the radial segment [0, a). On the other hand, since 1 − \overline{a_{n}}φ_{n}(ra)⟶_{r→1}0, we have that |ψ_{n}| is not bounded on

the radial segment [0, a). A contradiction.

So, sup_{z∈γ} |φ_{n}(z)| = 1 for all n and, by Theorem 9, we obtain that

lim sup_{z→a, z∈γ} |σ(z)| = +∞.

3. The examples

Example 1. There is a univalent function φ: D → D with φ(0) = 0 and boundary fixed point 1 such that φ_{2} does not have a radial limit at 1.

Proof. (a) We define R = [0, 1) and

(1)  A = {(1 − e^{−1/t})e^{it} : 0 < t ≤ π},

furthermore, for k = 1, 2, …,

(2)  r_{k} = 1 − e^{−k}, B_{k} = \{r_{k}e^{it} : \frac{1}{k} + exp(−e^{2k}) ≤ t ≤ π\}.

We consider the simply connected domain

(3)  Ω = D⧹(A ⋃[−1, −1 + e^{−1}] ⋃( ⋃_{k=1}^{∞} B_{k}))

which contains 0. Let φ be the Riemann map of D onto Ω that satisfies φ(0) = 0 and φ(1) = 1 in the sense that γ = φ(R) lies between the arc A and the arcs B_{k}; see Figure 1. This function φ has a continuous extension to \overline{D}⧹\{1\}. There are two important aspects: the arc A is very tangential at 1 and the gaps between A and the B_{k} are exceedingly small.

Figure 1 (not to scale).

Since γ = φ(R) lies to the left of R, the image φ ∘ φ(R) lies to the left of φ(R). Furthermore φ ∘ φ(R) goes from 0 to ∂D. Hence there exists z_{k} with

(4)  z_{k} ∈ φ(R), |φ(z_{k})| = \frac{1}{2}(r_{k} + r_{k+1}), arg φ(z_{k}) < π.

We claim that arg φ(z_{k}) > \frac{1}{2}π for large k.

(b) Suppose that arg φ(z_{k}) ≤ \frac{1}{2}π. Let c_{1}, c_{2}, …, denote suitable positive constants. Since φ(R) goes from 0 to 1, there exists b_{k} ∈ R with |φ(b_{k})| = \frac{1}{2}(r_{k} + r_{k+1}); see Figure 2. The hyperbolic distance λ in D satisfies [13, p. 92]

(5)  λ(b_{k}, z_{k}) ≤ inf_{C} ∫_{C} \frac{|dw|}{dist (w, ∂Ω)},

where C runs through all curves in Ω from φ(b_{k}) to φ(z_{k}). It follows from the geometry of Ω (see (1) and (2)) that this infimum is essentially attained if C is the arc C_{k} of \{|w| = \frac{1}{2}(r_{k} + r_{k+1})\} from φ(b_{k}) to φ(z_{k}). Now φ^{−1}(A) lies on ∂D above R and the φ^{−1}(B_{k}) lie on ∂D below R. Hence φ(b_{k}) has about the same distance [12, p. 318] from A and ⋃ B_{k}.

⟶^{φ}

Figure 2 (not to scale). The dotted lines are φ^{−1}(C_{k}) and C_{k}. Note that the arc appears in both parts of the figure. We have denoted v_{k} = φ(z_{k}) and u_{k} = φ(a_{k}).

Since arg φ(z_{k}) ≤ \frac{1}{2}π we therefore obtain that dist (w, ∂Ω) ≥ c_{1}e^{−k} for w ∈ C_{k}. Hence we see from (5) that λ(b_{k}, z_{k}) ≤ c_{2}e^{k}. If S_{k} is the hyperbolic segment from z_{k} to a_{k} ∈ R, that is orthogonal to R, then it follows that

(6)  λ(a_{k}, z_{k}) ≤ λ(b_{k}, z_{k}) ≤ c_{2}e^{k};

see Figure 2. The curve A is tangential to ∂D at 1 and φ(R) lies between A and ∂D. Hence, by (1), there exists t_{k} such that

(7)  (1 − e^{−1/t_{k}})e^{it_{k}} ∈ S_{k}, |z_{k}| > 1 − e^{−1/t_{k}}.

Since the euclidean radius of S_{k} is ∼ 1 − a_{k} as k → ∞, we have t_{k} ∼ 1 − a_{k}. Hence, by (7),

λ(0, z_{k}) > c_{3} log \frac{1}{1 − |z_{k}|} > \frac{c_{3}}{t_{k}} ≥ \frac{c_{4}}{1 − a_{k}} ≥ exp(c_{5}λ(0, a_{k}))

and we obtain from (6) that, for large k,

(8)  c_{2}e^{k} > exp(c_{5}λ(0, a_{k})) − λ(0, a_{k}) > exp(c_{6}λ(0, a_{k})).
There are x_{k} ∈ R such that |φ(x_{k})| = r_{k}. We have [13, Corollary 1.5]
(9)  λ(ς_{0}, ς_{1}) ≥ \frac{1}{4} log \frac{|φ(ς_{0}) − φ(ς_{1})|}{4 dist (φ(ς_{0}), ∂Ω)} for ς_{0}, ς_{1} ∈ D.

Now suppose that a_{k} ≤ x_{k}. Then there exists z′_{k} ∈ S_{k} with |φ(z′_{k})| = r_{k}. Since

dist (φ(z′_{k}), ∂Ω) ≤ exp(−e^{2k}),

by (1) and (2), we obtain from (9) that

λ(a_{k}, z_{k}) ≥ λ(z′_{k}, z_{k}) ≥ c_{7}(e^{2k} − k) for large k

which contradicts (6). Hence x_{k} < a_{k} and it follows as above from (9) that

λ(0, a_{k}) ≥ λ(x_{k−1}, x_{k}) ≥ c_{8}e^{2k}.

Thus we conclude from (8) that c_{2}e^{k} > exp(c_{9}e^{2k}) for large k and we have arrived at a contradiction.

(c) Hence we have π > arg φ(z_{k}) > \frac{1}{2}π for large k. Since φ(z_{k}) ∈ φ_{2}(R) by (4), we conclude the ω -limit of φ_{2}(R) contains points different from 1.

On the other hand, since φ is univalent and φ(1) = 1, we obtain from a Theorem of Lindelöf [13, Theorem 2.16] that this ω -limit contains 1. Hence φ_{2}(R) does not have a limit as x → 1, with x ∈ R.

Example 2. There are a univalent function φ: D → D with an inner DW-point and associated Koenigs function σ and a point a ∈ ∂D such that lim_{r→1} σ(ra) = ∞ and a is not a boundary fixed point of φ.

Proof. Take σ(z) = z/(1 − z^{2}) for all z ∈ D. We have that σ′(z) = (1 + z^{2})/(1 − z^{2})^{2} , σ(0) = 0, and σ′(0) = 1. The function p(z) = zσ′(z)/σ(z) = (1 + z^{2})/(1 − z^{2}) satisfies that p(0) = 1 and Re(p(z)) > 0 for all z ∈ D. So, by [12, Theorem 2.5], σ is starlike. Moreover, σ(−z) = −σ(z) for all z ∈ D. Therefore, −\frac{1}{2}σ(D) ⊂ σ(D).

Consider the univalent function φ(z) = σ^{−1}(−\frac{1}{2}σ(z)) for all z ∈ D. It is clear that the Koenigs function of φ is the function σ. We have that lim_{r→1} σ(r) = ∞ but

lim_{r→1} φ(r) = lim_{r→1} σ^{−1}(−\frac{1}{2}σ(r))

= lim_{s→+∞, s∈R} σ^{−1}(−\frac{1}{2}s) = lim_{s→−∞, s∈R} σ^{−1}(\frac{1}{2}s) = −1.

So 1 is not a fixed point of φ.

Example 3. There is a univalent function φ: D → D with an inner DWpoint and associated Koenigs function σ and a point a ∈ ∂D such that

lim_{r→1} σ(ra) = ∞

(so, a is a boundary contact point of φ_{n} for all n) and lim_{z→1} σ(z) ≠ ∞. Proof. For each natural number n take

H_{n} = \{z ∈ C : Re z = −1 + \frac{1}{2n}, Im z ≥ −n\}

and consider the domain

Ω = {z ∈ C : Re z > −1}⧹( ⋃_{n≥1} H_{n}).

Let σ be the normalized Riemann map from D onto Ω. Let C_{n} be the arc in Ω of the circumference centered in 0 that goes from −1 + (1/2n) − ni to H_{1} . The sequence of Jordan arcs (C_{n}) generates a prime end p in Ω. Take the point a ∈ ∂D such that σ̂(a) = p. Then the impression of the prime end is I(p) = {z ∈ C : Re z = −1} ⋃{∞}. So the limit lim_{z→1} σ(z) does not exist. Moreover, by [13, Theorem 2.16], lim_{r→1} σ(ra) = ∞.

Notice that \frac{1}{2}Ω ⊂ Ω. Therefore, the univalent function φ(z) = σ^{−1}(\frac{1}{2}σ(z)) for all z ∈ D is well-defined and it is clear that the Koenigs function of φ is the function σ. So, by Theorem 6, a is a boundary contact point of φ_{n} for all n.

Example 4. There are a univalent function φ: D → D with an inner DWpoint and a point a ∈ ∂D such that, for all n, the ω -limit of the curve r ∈ [0, 1) ↦ φ_{n}(ra) touches ∂D but it is not contained in ∂D.

Proof. For each natural number n take

H_{2n} = \{z ∈ C : Re z = −1 + \frac{1}{2n}, Im z ≥ −2n\},
H_{2n+1} = \{z ∈ C : Re z = −1 + \frac{1}{2n + 1}, Im z ≤ 2n + 1\}

and consider the domain

Ω = {z ∈ C : Re z > −1}⧹( ⋃_{n≥1} H_{n}).

Let σ be the normalized Riemann map from D onto Ω. The sequence of Jordan arcs (C_{n}) given by

\begin{aligned}C_{2n} & = [−1 + \frac{1}{2n + 1} − i, −1 + \frac{1}{2n} − i], \\ C_{2n+1} & = [−1 + \frac{1}{2n + 2} + i, −1 + \frac{1}{2n + 1} + i], \end{aligned}

for all n, generates a prime end p in Ω. Take the point a ∈ ∂D such that σ̂(a) = p. Then the set of principal points of the prime end p is {z ∈ C : Re z = −1} ⋃{∞} and, by [13, Theorem 2.16], this set coincides with the ω -limit of the curve r ∈ [0, 1) ↦ σ(ra). In particular, ∞ belongs to this ω -limit and lim_{r→1} σ(ra) ≠ ∞.

Notice that \frac{1}{2}Ω ⊂ Ω. Therefore, the univalent function φ(z) = σ^{−1}(\frac{1}{2}σ(z)) for all z ∈ D is well-defined and it is clear that the Koenigs function of φ is the function σ. Since ∞ belongs to ω -limit of the curve r ∈ [0, 1) ↦ σ(ra), by Theorem 9, the ω -limit of the curve r ∈ [0, 1) ↦ φ_{n}(ra) touches ∂D for all n. Finally, since lim_{r→1} σ(ra) ≠ ∞, by Theorem 6, a is not a boundary contact point of φ_{n} for all n.

Example 5. There are a function φ: D → D with DW-point 0, φ′(0) ≠ 0, and associated Koenigs function σ, and a boundary fixed point a ∈ ∂D of φ_{n}, for all n, such that, the ω -limit of the curve r ∈ [0, 1) ↦ σ(ra) contains the DW-point 0 and ∞.

Proof. Take 0 < λ < 1 and φ(z) = z(z − λ)/(1 − λz). Then φ′(0) = −λ, φ(1) = 1, and 1 < φ′(1) < ∞. Consider x_{1} = λ and x_{n} = φ(x_{n+1}) for all n. Then λ < x_{n+1} < 1, x_{n} < x_{n+1} for all n and x_{n} → 1. Moreover,

σ(x_{n}) = σ(φ(x_{n})) = −λσ(x_{n+1}).

Therefore, bearing in mind that 0 = σ(0) = σ(φ(λ)) = −λσ(λ), we have that σ(x_{n}) = 0 for all n and 0 belongs to the ω -limit of the curve r ∈ [0, 1) ↦ σ(ra). Since 1 is a fixed point of φ_{n} for all n, by Theorem 8, we have that ∞ belongs to this ω -limit, too.

Fixed points and the Koenigs function 487

Example 6. There is a function φ: D → D with DW-point 0, φ′(0) ≠ 0, and associated Koenigs function σ such that ∞ is in ω(σ ∘ Γ) for all curves Γ in D with ω(Γ) in ∂D.

Proof. Consider the finite Blaschke product

φ(z) = cz ∏_{k=1}^{m−1} \frac{z − z_{k}}{1 − \overline{z_{k}}z} , |c| = 1, 0 < |z_{k}| < 1 for all k.

Since φ is continuous on \overline{D} and φ(∂D) = ∂D, the ω -limit of the curves r ↦ φ_{n}(Γ(r)) are in ∂D for all curves Γ in D with ω(Γ) in ∂D and for all n. Hence we obtain from Theorem 9 that

lim sup_{r→1}|σ(Γ(r))| = +∞.

Valiron [16, p. 123] has given a more precise description of the behaviour of the Koenigs function for finite Blaschke products. His results show that |σ(z)| → +∞ as |z| → 1 except in smaller and smaller neighbourhoods of the points φ_{n}^{−1}(z_{k}) for all n and k = 0, …, m − 1 where z_{0} = 0.

References

[1] Berkson, E., and H. Porta: Semigroups of analytic functions and composition operators. - Michigan Math. J. 25, 1978, 101–115.

[2] Bourdon, P. S.: Essential angular derivates and maximum growth of Koenigs eigenfunctions. - J. Funct. Anal. 160, 1998, 561–580.

[3] Bourdon, P. S., and J. H. Shapiro: Mean growth of Koenigs eigenfunctions. - J. Amer. Math. Soc. 10, 1997, 299–325.

[4] Ciozda, K.: Sur quelques pròblemes extrémaux dans les classes des fonctions convexes vers l’axe réel négatif. - Ann. Polon. Math. 38, 1980, 311–317.

[5] Contreras, M. D., and S. Díaz-Madrigal: Fractional iteration in the disk algebra: prime ends and composition operators. - Rev. Mat. Iberoamericana (to appear).

[6] Conway, J. B.: Functions of One Complex Variable II. - Graduate Texts in Math. 159, Springer-Verlag, New York, 1995.

[7] Cowen, C. C.: Iteration and solution of functional equations for functions analytic in the unit disc. - Trans. Amer. Math. Soc. 256, 1981, 69–95.

[8] Cowen, C. C., and Ch. Pommerenke: Inequalities for the angular derivative of an analytic function in the unit disk. - J. London Math. Soc. (2) 26, 1982, 271–289.

[9] Eenigenburg, P. J.: Boundary behavior of starlike functions. - Proc. Amer. Math. Soc. 33, 1972, 428–432.

[10] Poggi-Corradini, P.: Angular derivatives at boundary fixed points for self-maps of the disk. - Proc. Amer. Math. Soc. 126, 1998, 1697–1708.

[11] Poggi-Corradini, P.: Canonical conjugations at fixed points other than the Denjoy– Wolff point. - Ann. Acad. Sci. Fenn. Math. 25, 2000, 487–499.

[12] Pommerenke, Ch.: Univalent Functions. - Vandenhoeck & Ruprecht, Göttingen, 1975.

488 M. D. Contreras, S. Díaz-Madrigal, and Ch. Pommerenke

[13] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. - Springer-Verlag, Berlin, 1992.

[14] Shapiro, J. H.: Composition Operators and Classical Function Theory. - Springer-Verlag, New York, 1993.

[15] Siskakis, A. G.: Semigroups of composition operators and the Cesàro operator on H^{p}(D). - Ph.D. Thesis, University of Illinois, 1985.

[16] Valiron, G.: Fonctions analytiques. - Presses Universitaires de France, Paris, 1954.

Received 31 May 2004