Fixed points and boundary behaviour of the Koenigs function

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 471–488 FIXED POINTS AND BOUNDARY BEHAVIOUR OF THE KOENIGS FUNCTION Manuel D. Contreras, Santiago D´ıaz-Madrigal, and Christian Pommerenke Universidad de Sevilla, Escuela Superior de Ingenieros, Departamento de Matem´atica Aplicada II Camino de los Descubrimientos, s/n, ES-41092 Sevilla, Spain; [email protected], [email protected] Technische Universit¨at, Institut f¨ur Mathematik DE-10623 Berlin, Germany; p[email protected]erlin.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 ϕbe a holomorphic map in the unit disc Dwith ϕ(D)⊂D. A point a∈∂Dis called a boundary contact point of ϕ, if the non-tangential or angular limit L:= ∠limz→aϕ(z) lies in ∂D. Moreover, boundary contact points have multipliers. That is, and also in the non-tangential sense, ϕalways has a derivative ϕ0(a)∈C∞\{0}at that point a. If L=a, the point ais called a boundary fixed point of ϕand, in this case, ϕ0(a)∈(0,+∞)∪ {∞}. In what follows, a fixed point of ϕwill mean a fixed point in the classical sense (ϕ(a) = awith a∈D) or a boundary fixed point. The famous Denjoy–Wolff point (DW-point) of every non-trivial ϕwill be denoted by τϕor, 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 ϕnof the function ϕ. Of course, many 2000 Mathematics Subject Classification: Primary 30D40; Secondary 30C45, 37C25. This research has been partially supported by the Ministerio de Ciencia y Tecnolog´ıa and the European Union (FEDER) project BFM2003-07294-C02-02 and by La Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa. 472 M. D. Contreras, S. D´ıaz-Madrigal, and Ch. Pommerenke things are known, so we are trying to determine clearly what our contributions are. Almost all of our results assume that ϕhas an inner DW-point (τ∈Dand |ϕ0(τ)| 6= 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 ϕis usually said to have an elliptic DW-point, whenever ϕis not trivial, τ∈Dand |ϕ0(τ)|= 1. So, suppose that ϕhas an inner DW-point. If a∈∂Dis a boundary fixed point of ϕwith ϕ0(a)6=∞, by [13, p. 80], the curve r∈[0,1) 7→ ϕ(ra)∈D tends non-tangentially to aand, according to the Lehto–Virtanen theorem [13, Chapter 4], we get that ais also a boundary fixed point of ϕn, for all n∈N. On the other hand, if ais a boundary fixed point of ϕnwith finite derivative, we can only claim that ais 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 ϕ0(a)6=∞is crucial there. In fact, in Section 3, we provide an example (Example 1) of a univalent holomorphic map ϕ:D→Dhaving a boundary fixed point a∈∂Dsuch that ais not even a boundary contact point of ϕ2. Necessarily, ϕ0(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 (ϕt) is a family (indexed by the non-negative real numbers) of holomorphic self-maps of the unit disk, satisfying the following three conditions: (a) ϕ0is the identity in D, (b) ϕt+s=ϕt◦ϕs, for all t, s ≥0, (c) for every z∈D, limt→0ϕt(z) = z. The fractional iterates ϕtare always univalent. Moreover, if one of the iterates ϕt(t > 0) has an inner (resp. boundary) DW-point, all the ϕt(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 (ϕt) 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 (ϕt)be a semigroup of analytic functions with inner DWpoint and a∈∂D. Then, the point ais a boundary fixed point of ϕtfor some t > 0if and only if it is a boundary fixed point for all the iterates ϕt. In particular, if a holomorphic mapping ϕ:D→Dwith inner DW-point can be embedded into a semigroup of analytic functions, then all the iterates of ϕhave 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→Dis also a boundary fixed point of ϕ2, which we know to be incorrect in general. Anyway, our approach is different and, perhaps, clearer. Fixed points and the Koenigs function 473 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→Csuch that σ◦ϕ=ϕ0(τ)σ with σ0(τ) = 1, for every holomorphic self-map ϕof Dwith inner DW-point and ϕ0(τ)6= 0. This map σis called the Koenigs function associated to ϕand σis univalent, whenever ϕis. It is possible to prove that if (ϕt) is a semigroup with inner DW-point, all the Koenigs functions associated to ϕt(t > 0) coincide, so we can talk about the Koenigs function of the semigroup (ϕt). Theorem 2. Let (ϕt)be a semigroup of analytic functions with inner DWpoint and let σbe the corresponding Koenigs function. Assume that a∈∂D. Then the following are equivalent: (1) The point ais a boundary fixed point of ϕtfor some (resp. all)t > 0. (2) The radial limit limr→1−σ(ra)exists and is ∞. (3) The unrestricted limit limz→aσ(z)exists and is ∞. That is, the function σadmits a continuous extension from D∪{a}into C∞, where we assign σ(a) := ∞. In terms of prime end theory and denoting the corresponding Carath´eodory map by ˆσ, we notice that the above statement two says that ˆσ(a) is an accessible prime end and the statement three that the impression of ˆσ(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 (ϕt)be a semigroup of analytic functions with inner DWpoint and a∈∂Da boundary fixed point of ϕtfor some (resp. all)t > 0. Then, each fractional iterate ϕtadmits a continuous extension from D∪{a}into D∪{a}, where ϕt(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–Carath´eodory theorem, if a∈∂Dis a boundary contact point of some ϕnwith ϕ0 n(a)6=∞, then ais also a boundary contact point of each iterate ϕk,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 474 M. D. Contreras, S. D´ıaz-Madrigal, and Ch. Pommerenke called an ω-limit point of a curve γ: [0,1) →Cif there exists a strictly increasing sequence (rn)⊂[0,1) convergent to 1 such that γ(rn)→ξ. The set of all ω-limit points of γis called its ω-limit and it is denoted by ω(γ). Theorem 4. Let (ϕt)be a semigroup of analytic functions with inner DWpoint and a∈∂D. Then the following are equivalent: (1) The point ais a boundary fixed point of ϕtfor some (resp. all)t > 0. (2) The point ais a boundary contact point of ϕtfor all t > 0. (3) There is t > 0such that the point ais a boundary contact point of ϕnt , for every n∈N. (4) For all t > 0, the ω-limit of the curve r∈[0,1) −→ ϕt(ra)∈D is completely included in ∂D. (5) For some t > 0and every n∈N, the ω-limit of the curves r∈[0,1) −→ ϕnt(ra)∈D are completely included in ∂D. In general, the above analysis cannot be quickly translated to the DW-boundary 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 = (ϕt)be a semigroup of analytic functions with boundary DW-point and a∈∂D. Then, the point ais a boundary fixed point of ϕtfor some t > 0if and only if it is a boundary fixed point for all ϕt. 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→Dbe a univalent function with inner DW-point and let σbe the corresponding Koenigs function. Likewise, let γ: [0,1) →Dbe a curve. Then, the following statements are equivalent. (1) For all n∈N, the limit limr→1ϕnγ(r)exists and lies in ∂D. (2) For all n∈N, the ω-limit of the curve ϕn◦γ: [0,1) →Dis completely contained in ∂D. (3) The ω-limit of the curve σ◦γ: [0,1) →Cis ∞. That is, limr→1σγ(r)=∞. Fixed points and the Koenigs function 475 Corollary 7. Let ϕ:D→Dbe a univalent function with inner DW-point and let σbe the corresponding Koenigs function. Assume that a∈∂D. Then, the following statements are equivalent. (1) The point ais a boundary contact point of ϕn, for every n∈N. (2) For all n∈N, the ω-limit of the curve r∈[0,1) →ϕn(ra)∈Dis completely contained in ∂D. (3) The radial limit limr→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(ϕn◦γ)⊂∂D, for all n” by “ω(ϕn◦γ)T∂D6=∅, for all n” (see Example 4). Theorem 8. Let ϕ:D→Dbe a function with inner DW-point τand ϕ0(τ)6= 0, and let σbe the corresponding Koenigs function. Assume that a∈∂D and let γ: [0,1) →Dbe a curve such that ω(γ) = {a}. If ais a boundary contact point for every iterate of ϕ, then the following two equivalent statements hold: (1) For every n∈N, the intersection ω(ϕn◦γ)T∂Dis not empty. (2) The ω-limit of the curve σ◦γ: [0,1) →Ccontains ∞. Even for radial curves, the behaviour of the just mentioned ω-limit of the curve σ◦γcan be very “pathological” in a certain sense. In fact, in Section 3, we present a non-univalent holomorphic function ϕ:D→Dhaving the point 1 as a boundary fixed point and 0 as the DW-point with 0 <|ϕ0(0)|<1, such that the ω-limit of the curve r∈[0,1) →σ(r)∈Cis 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 ϕn◦γ, where γis curve in Dtending to the corresponding boundary fixed point of ϕ. In a certain sense, this complements the former theorem. Theorem 9. Let ϕ:D→Dbe a function with inner DW-point τand ϕ0(τ)6= 0, and let σbe 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) →Cdoes not contain ∞and sup|ϕn◦γ(t)−τ|:t∈[0,1)n→∞ −→ 0. (ii) The ω-limit of the curve σ◦γ: [0,1) →Ccontains ∞and ω(ϕn◦γ)T∂D6=∅, for every n∈N. 476 M. D. Contreras, S. D´ıaz-Madrigal, and Ch. Pommerenke It is worth mentioning that there are functions ϕsuch 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=ϕ0(τ). Since σ is univalent we have that λ6= 0. If the limit limr→1ϕn(γ(r)) = ηexists, then we have that ω(ϕn◦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∈Dthere is a natural number nsuch that λnw∈Ω. Moreover, w∈ω(σ◦γ). That is, there is a sequence (rm) in the interval (0,1) converging to 1 such that σγ(rm)→w. Therefore, bearing in mind that σis univalent, we have ϕnγ(rm)=σ−1λnσγ(rm)m→∞ −→ σ−1(λnw)∈D. That is, ω(ϕn◦γ)∩D6=∅. A contradiction. Finally, we see that (3) implies (1). Fix a natural number nand consider the curve r∈[0,1) 7→ λnσγ(r). By hypothesis, limr→1λnσγ(r)=∞. Since σis univalent it follows from [14, p. 162] that ϕnγ(r)=σ−1λnσγ(r) tends to a limit as r→1, and this limit lies in ∂Dbecause σis 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 cwith Re c > 0 such that ϕt(z) = σ−1e−ctσ(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 ais a boundary fixed point of ϕtfor some t > 0. (ii) The point ais a boundary fixed point of ϕtfor all t > 0. (iii) The unrestricted limit limz→aσ(z) exists and is ∞. (iv) The radial limit limr→1−σ(ra) exists and is ∞. (v) The point ais a boundary contact point of ϕtfor all t > 0. (vi) There is t > 0 such that the point ais a boundary contact point of ϕnt , for every n∈N. (vii) For all t > 0, the ω-limit of the curve r∈[0,1) −→ ϕt(ra)∈D is completely included in ∂D. Fixed points and the Koenigs function 477 (viii) For some t > 0 and every n∈N, the ω-limits of the curves r∈[0,1) −→ ϕnt(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∈Cwith Re c > 0 and w∈C,w6= 0, we define the spiral spirc[w] = {e−csw:s∈R}S{0}S{∞}; given real numbers s < t, we define the spiral segment spirc[e−sw, e−tw] as the subarc of spirc[w] that goes from e−swto e−tw; finally, spirc[∞, w] = {e−csw:s≤0}S{∞}. Take a zero-chain (Cm) in Dconverging to asuch that σ(Cm)is a zerochain in Ω such that ˆσ(a) = σ(Cm). To get (iii) we have to prove that the impression of the prime end ˆσ(a) is the single point ∞. By [5, Lemma 3.3] the corresponding impression I(ˆσ(a)) ⊂∂∞Ω must be of one of the following types: (a) Iˆσ(a)is a single point. (b) There is λ∈∂Dand η1< η2such that Iˆσ(a)= spirc[e−η1cλ, e−η2cλ]. (c) There is λ∈∂Dand η∈Rsuch that Iˆσ(a)= spirc[∞, e−ηcλ]. Since limr→1σ(ra) = ∞, we have that ∞ ∈ Iˆσ(a). Therefore, the possibility (b) cannot occur. If (c) is satisfied, then limmwmalways exists and it is equal to λe−ηc whenever wm∈σ(Cm) for all m∈N(see the proof of [5, Theorem 1.2. (2) implies (3)]). Moreover, for each m, there is rm∈(0,1) such that rma∈Cm. So, limmσ(rma) = e−ηc . A contradiction because the sequence (rm) must tend to 1 and, by hypothesis, limmσ(rma) = ∞. Therefore, Iˆσ(a)is a single point and, again using that ∞ ∈ Iˆσ(a)we get that Iˆσ(a)={∞}. (iv) implies (ii). Let us fix t > 0. We have to show that lim r→1ϕt(ra) = lim r→1σ−1e−ctσ(ra)=a. First of all, notice that by [14, p. 162], the limit limr→1σ−1e−ctσ(ra)does exist. So we only have to prove that the limit is equal to a. Clearly, we have that limr→1ϕ0(ra) = a. Take an increasing sequence (rn) tending to 1 with r0= 0. For each n, define the following curves γn:r∈rn−1, rn7−→ γn(r) := σ(ra), ηn:s∈0, t7−→ ηn(s) := e−csσ(rna). 478 M. D. Contreras, S. D´ıaz-Madrigal, and Ch. Pommerenke Since Ω is c-spirallike, the curve ηnis in Ω. Now we build the curve Γ = ∞ M n=1 (γ2n−1+η2n−1+e−ctγ2n+η− 2n). Notice that Γ is a curve in Ω joining 0 to ∞. So, by [14, p. 162], we have that there is ω∈∂Dsuch that limw∈Γ,w→∞ σ−1(w) = ω. On the one hand, the points σ(r2n−1a)∈γ2n−1⊂Γ and the sequence σ(r2n−1a)tends to ∞. So, ω= limn→∞ σ−1σ(r2n−1a)= limn→∞ r2n−1a=a. On the other hand, the points e−ctσ(r2n−1a)∈η2n−1⊂Γ and the sequence e−ctσ(r2n−1a)tends to ∞. So, ω= lim n→∞ σ−1e−ctσ(r2n−1a)= lim n→∞ ϕt(r2n−1a). Therefore, limn→∞ ϕt(r2n−1a) = a. Since the limit limr→1ϕt(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 limr→1σ(rξ) does exist and belong to CS{∞} for all ξ∈∂D. In particular, we have that the limit limr→1e−cntσ(rξ) exists for all nand all ξ. On the one hand, if this limit belongs to Ω, then the continuity of σ−1in Ω implies that the limit limr→1ϕnt(rξ) = limr→1σ−1e−cntσ(rξ)exists. On the other hand, if this limit belongs to ∂∞Ω, by [14, p. 162], the limit limr→1ϕnt(rξ) = limr→1σ−1e−cntσ(rξ)exists. Summing up, we have that for all nand for all ξ, the limit limr→1ϕnt(rξ) exists. It is worth pointing out that we want to prove (vi) and, at this moment, we only know that the limit limr→1ϕnt(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→1ϕnt(ra) = afor all n. We argue by induction. By hypothesis, we have that limr→1ϕt(ra) = a. Now suppose that limr→1ϕnt(ra) = a. Then, by the Lehto–Virtanen theorem [13, Chapter 4], we have that lim r→1ϕ(n+1)t(ra) = lim r→1ϕtϕnt(ra)= lim z→a, z∈ϕnt ([0,1)a)ϕt(z) = lim r→1ϕt(ra) = a. Proof of Corollary 3. By Theorem 2, we have that the function σhas a continuous extension to the point a. What we are going to do is to pass this continuous extension property from σto ϕt. So, fix t > 0 and let Uk=z∈D:|z−a|<1/k. Fixed points and the Koenigs function 479 We have to show that diamϕt(Uk)→0 as k→ ∞. Suppose this is false. Then, for some subsequence, there are Jordan arcs Cknin ϕt(Ukn) with diamCkn>b>0. We have σ(Ckn)⊂σϕt(Ukn)=e−ctσ(Ukn)n→∞ −→ ∞ by Theorem 2. Hence (Ckn) 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∈∂Dthe DW-point of the semigroup. Then there is a unique univalent function σ:D→Cwith σ(0) = 0 verifying the property (∗∗∗) Ω + t⊂Ω,for each t > 0,where Ω := σ(D), and such that ϕt(z) = τσ−1σ(τz) + t, t ≥0, z ∈D (see [1] or [15]). Fix t0>0 such that ais a fixed point for ϕt0. We claim that the following properties hold: (a) For all ξ∈∂D, the limit limr→1σ(rξ) exists and belongs to ∂∞Ω; (b) For all ξ∈∂D, the limit limr→1ϕt0(rξ) exists; (c) ais a fixed point of ϕnt0for all n; (d) limr→1σ(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 σsatisfying (∗ ∗ ∗), there is α∈−1 2π, 1 2π, such that Re eiα(1 −z)2σ0(z)≥0 for all z∈D. In particular, denoting g(z) = z/(1 −z), we have that Re eiασ0(z) g0(z)≥0 for all z∈D. Since gis convex, we see that the function eiασis close-to-convex with associated function g. Notice that g(D) = w∈C: Re w > −1 2. So, ∂g(D) = w∈C: Re w=−1 2 and we parametrize this curve by w(t) = −1 2+sin t 2(1 −cos t)i 486 M. D. Contreras, S. D´ıaz-Madrigal, and Ch. Pommerenke Proof. For each natural number ntake H2n=z∈C: Re z=−1 + 1 2n,Im z≥ −2n, H2n+1 =z∈C: Re z=−1 + 1 2n+ 1,Im z≤2n+ 1 and consider the domain Ω = z∈C: Re z > −1S n≥1 Hn. Let σbe the normalized Riemann map from Donto Ω. The sequence of Jordan arcs (Cn) given by C2n=−1 + 1 2n+ 1 −i, −1 + 1 2n−i, C2n+1 =−1 + 1 2n+ 2 +i, −1 + 1 2n+ 1 +i, for all n, generates a prime end pin Ω. Take the point a∈∂Dsuch that ˆσ(a) = p. Then the set of principal points of the prime end pis {z∈C: Re z=−1}S{∞} and, by [13, Theorem 2.16], this set coincides with the ω-limit of the curve r∈[0,1) 7→ σ(ra). In particular, ∞belongs to this ω-limit and limr→1σ(ra)6=∞. Notice that 1 2Ω⊂Ω. Therefore, the univalent function ϕ(z) = σ−11 2σ(z) for all z∈Dis well-defined and it is clear that the Koenigs function of ϕis the function σ. Since ∞belongs to ω-limit of the curve r∈[0,1) 7→ σ(ra), by Theorem 9, the ω-limit of the curve r∈[0,1) 7→ ϕn(ra) touches ∂Dfor all n. Finally, since limr→1σ(ra)6=∞, by Theorem 6, ais not a boundary contact point of ϕnfor all n. Example 5. There are a function ϕ:D→Dwith DW-point 0, ϕ0(0) 6= 0, and associated Koenigs function σ, and a boundary fixed point a∈∂Dof ϕn, for all n, such that, the ω-limit of the curve r∈[0,1) 7→ σ(ra) contains the DW-point 0 and ∞. Proof. Take 0 <λ<1 and ϕ(z) = z(z−λ)/(1 −λz). Then ϕ0(0) = −λ, ϕ(1) = 1, and 1 < ϕ0(1) <∞. Consider x1=λand xn=ϕ(xn+1) for all n. Then λ < xn+1 <1, xn< xn+1 for all nand xn→1. Moreover, σ(xn) = σϕ(xn)=−λσ(xn+1). Therefore, bearing in mind that 0 = σ(0) = σϕ(λ)=−λσ(λ), we have that σ(xn) = 0 for all nand 0 belongs to the ω-limit of the curve r∈[0,1) 7→ σ(ra). Since 1 is a fixed point of ϕnfor 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→Dwith DW-point 0, ϕ0(0) 6= 0, and associated Koenigs function σsuch that ∞is in ω(σ◦Γ) for all curves Γ in Dwith ω(Γ) in ∂D. Proof. Consider the finite Blaschke product ϕ(z) = cz m−1 Y k=1 z−zk 1−zkz,|c|= 1,0<|zk|<1 for all k. Since ϕis continuous on Dand ϕ(∂D) = ∂D, the ω-limit of the curves r7→ ϕn(Γ(r)) are in ∂Dfor all curves Γ in Dwith ω(Γ) in ∂Dand 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−1 n(zk) for all nand k= 0,...,m−1 where z0= 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 pr`oblemes extr´emaux dans les classes des fonctions convexes vers l’axe r´eel n´egatif. - Ann. Polon. Math. 38, 1980, 311–317. [5] Contreras, M. D., and S. D´ ıaz-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, G¨ottingen, 1975. 488 M. D. Contreras, S. D´ıaz-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 Ces`aro operator on Hp(D). - Ph.D. Thesis, University of Illinois, 1985. [16] Valiron, G.: Fonctions analytiques. - Presses Universitaires de France, Paris, 1954. Received 31 May 2004