Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc

Received: 28 May 2024 Accepted: 26 January 2025 DOI: 10.1112/jlms.70077 Journal of the London Mathematical Society RESEARCH ARTICLE Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc Manuel D. Contreras1Santiago Díaz-Madrigal1 Pavel Gumenyuk2 1Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II and IMUS, Universidad de Sevilla, Sevilla, Spain 2Department of Mathematics, Politecnico di Milano, Milan, Italy Correspondence Manuel D. Contreras, Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II and IMUS, Universidad de Sevilla, Sevilla, 41092 Spain. Email: contrer[email protected] Funding information Ministerio de Innovación y Ciencia, Grant/Award Number: PID2022-136320NB-I00; INdAM (GNASAGA) [Correction added on 12 May, 2025, after first online publication: The copyright line was changed.] Abstract Let 𝜑be a univalent non-elliptic self-map of the unit disc 𝔻and let (𝜓𝑡)be a continuous one-parameter semigroup of holomorphic functions in 𝔻such that 𝜓1≠𝗂𝖽𝔻 commutes with 𝜑. This assumption does not imply that all elements of the semigroup (𝜓𝑡)commute with 𝜑.In this paper, we provide a number of sufficient conditions that guarantee that 𝜓𝑡◦𝜑=𝜑◦𝜓𝑡for all 𝑡>0:This holds, for example, if 𝜑and 𝜓1have a common boundary (regular or irregular) fixed point different from their common Denjoy–Wolff point 𝜏, or when 𝜓1has a boundary regular fixed point 𝜎≠𝜏at which 𝜑is isogonal, or when (𝜑 − 𝗂𝖽𝔻)∕(𝜓1−𝗂𝖽 𝔻)has an unrestricted limit at 𝜏. In addition, we analyze how 𝜑behaves in the petals of the semigroup (𝜓𝑡). MSC 2020 30C55, 37F44, 30D05 (primary) Contents 1. INTRODUCTION AND MAIN RESULTS ......................... 2 2. NOTATION AND PRELIMINARIES............................ 7 2.1. Notation ....................................... 7 2.2. Holomorphic self-maps of the unit disc ....................... 7 © 2025 The Author(s). The Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. J. London Math. Soc. (2) 2025;111:e70077. wileyonlinelibrary.com/journal/jlms 1of31 https://doi.org/10.1112/jlms.70077 2of31 CONTRERAS et al. 2.3. Holomorphic models for univalent self-maps .................... 8 2.4. Commuting holomorphic self-maps ......................... 9 2.5. One-parameter semigroups in the unit disc ..................... 11 3. THE STARTING EXAMPLE AND A CHARACTERIZATION OF SELF-MAPS COMMUTING WITH A SEMIGROUP .......................... 12 4. PROOF OF THEOREM 1.3................................. 13 5. PETALS AND COMMUTATIVITY I. AUXILIARY RESULTS ............... 15 6. PROOF OF THEOREM 1.4................................. 18 7. PETALS AND COMMUTATIVITY II. PROOF OF THEOREMS 1.8 AND 1.9 ....... 20 8. PETALS AND ISOGONALITY............................... 22 9. PROOF OF THEOREM 1.2................................. 28 10. COMMUTING ONE-PARAMETER SEMIGROUPS ................... 28 APPENDIX A: THE ELLIPTIC CASE ............................. 29 ACKNOWLEDGEMENTS................................... 30 REFERENCES......................................... 30 1 INTRODUCTION AND MAIN RESULTS This paper is motivated by the following natural and quite old problem in discrete holomorphic iteration: Given a holomorphic self-map of the unit disc, that is, 𝜑∈𝖧𝗈𝗅( 𝔻), determine or at least analyze those 𝜓∈𝖧𝗈𝗅( 𝔻)that commute with 𝜑,see[1, section 4.10]. This kind of questions has also been treated in the framework of fractional iteration of holomorphic self-maps, with the aim to study all continuous one-parameter semigroups in the unit disc (𝜓𝑡)that commute with a given continuous one-parameter semigroup (𝜑𝑡)in the sense that 𝜓𝑡◦𝜑𝑠=𝜑 𝑠◦𝜓𝑡for all 𝑡,𝑠 ⩾0, see, for instance, [2]and[3]. In this paper, we tackle an intermediate situation, which has interesting implications in the discrete as well as in the fractional framework. Given 𝜑∈𝖴( 𝔻), a univalent self-map of the unit disc, the centralizer (𝜑) of 𝜑is defined as (𝜑) ∶= {𝜓 ∈ (𝔻)∶𝜑◦𝜓=𝜓◦𝜑}, (𝔻)∶={𝜓∶𝔻→𝔻holomorphic injective}. The problem we are interested in is the following: Problem 1.1. Fix 𝜑∈(𝔻)and a continuous one-parameter semigroup in the unit disc (𝜓𝑡). Suppose 𝜓1∈(𝜑),with𝜓1≠𝗂𝖽𝔻. Does it necessarily follow that 𝜓𝑡∈(𝜑) for all 𝑡>0? If not, provide conditions on 𝜑and 𝜓1— and maybe on some finite number of 𝜓𝑡s— under which the relation 𝜓1∈(𝜑) implies that the whole semigroup (𝜓𝑡)is contained in (𝜑). First of all, it is worth mentioning that the assumption of 𝜑being univalent is completely natural and not restrictive. As we will see (Propositions 3.3 and A.1), univalence is a necessary condition for 𝜑to commute with a non-trivial continuous one-parameter semigroup. We would also like to underline that the context of this problem is really different from the other two mentioned above because of its lack of symmetry. In particular, for two continuous one-parameter semigroups in the unit disc, (𝜑𝑡)and (𝜓𝑡),wecanhave𝜓1∈(𝜑𝑡)for all 𝑡>0and at the same time, 𝜑1∉(𝜓𝑡)for some 𝑡>0; see Example 3.1. The same example shows that, in general, the answer to the question in Problem 1.1 is negative. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 3of31 When 𝜑is elliptic, the above Problem 1.1 was implicitly solved by Cowen [4]. For the sake of completeness, at the end of the paper, we include a brief appendix with a very simple, and independent from Cowen’s work, solution for the elliptic case; see Proposition A.1. In the rest of the paper, we restrict ourselves to the non-elliptic case. In that non-elliptic case, it is not difficult to give a complete answer to Problem 1.1 in terms of the Koenigs function 𝐻of the semigroup (𝜓𝑡). Indeed, (𝜓𝑡)⊂(𝜑) if and only there exists 𝑐∈ℂsuch that 𝐻◦𝜑=𝐻+𝑐, see Proposition 3.3. The latter condition means exactly that 𝜑 is affine with respect to 𝜓1(see Definition 2.6 and, in general, Section 2.4 for further details). Moreover, as a quite direct consequence of our results in [5], we can answer the question in Problem 1.1 positively if the semigroup (𝜓𝑡)is hyperbolic or parabolic of zero hyperbolic step. However, when (𝜓𝑡)is parabolic of positive hyperbolic step, giving any significant answer to Problem 1.1 that does not involve 𝐻or the infinitesimal generator 𝐺=1∕𝐻 ′is really not that easy. In the following theorem, we summarize our positive answers to Problem 1.1 including the two ones described in the former paragraph (as items (a) and (b)). For the definition of isogonality at a boundary point involved in (e), see Section 8. Other useful definitions can be found in the preliminaries, see Section 2. Note that in our terminology every non-elliptic self-map is different from the identity map 𝗂𝖽𝔻. Furthermore, by a repelling fixed point, we mean a boundary regular fixed point different from the Denjoy–Wolff point. Theorem 1.2. Let 𝜑∈(𝔻)be non-elliptic and (𝜓𝑡)be a continuous one-parameter semigroup in 𝔻such that 𝜓1≠𝗂𝖽𝔻and 𝜑◦𝜓1=𝜓 1◦𝜑. Assume that one of the following conditions holds: (a) 𝜑is affine with respect to 𝜓1, (b) 𝜓1is either hyperbolic or parabolic of zero hyperbolic step, (c) there exist 𝑟∈(0,+∞)⧵ℚsuch that 𝜓𝑟∈(𝜑), (d) the limit lim 𝑧→𝜏 𝜑(𝑧) − 𝑧 𝜓1(𝑧)−𝑧, where 𝜏is the Denjoy–Wolff point of 𝜓1(and hence also of 𝜑), exists unrestrictedly in 𝔻, (e) 𝜓1has a repelling fixed point at which 𝜑is isogonal, or (f) 𝜑and 𝜓1have a common boundary fixed point different from the Denjoy–Wolff point. Then, 𝜓𝑡∈(𝜑) for all 𝑡>0. In general, conditions (b) and (d)–(f) in Theorem 1.2 are only sufficient ones. At the same time, it is worth remarking that these conditions involve 𝜑and 𝜓1, but do not depend on the knowledge of other elements of the semigroup (𝜓𝑡), its Koenigs map, or infinitesimal generator. A priori, this is not the case for the necessary and sufficient condition (a), see Definition 2.6. However, in many cases, our next result allows to bypass this difficulty. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4of31 CONTRERAS et al. Theorem 1.3. Let 𝜑∈(𝔻)be parabolic of positive hyperbolic step, and let 𝜓∈(𝜑) ⧵ {𝗂𝖽𝔻}. Denote by 𝜏the Denjoy–Wolff point of 𝜑and let 𝑅𝜑,𝜓(𝑧) ∶= 𝜓(𝑧) − 𝑧 𝜑(𝑧) − 𝑧,𝑧∈ 𝔻. Then, the following statements hold. (A) The sequence (𝑅𝜑,𝜓 ◦𝜑◦𝑛)converges locally uniformly in 𝔻to some function 𝑓𝜑,𝜓 ∈𝖧𝗈𝗅(𝔻,ℂ). (B) 𝜓is affine with respect to 𝜑if and only if the function 𝑓𝜑,𝜓 is constant in 𝔻. In fact, in such a case, 𝑓𝜑,𝜓(𝜁) = ∠ lim𝑧→𝜏 (𝜓(𝑧)−𝑧)∕(𝜑(𝑧)−𝑧)for all 𝜁∈𝔻. (C) Let (𝑆, ℎ𝜑,𝑧↦𝑧+1),where𝑆=ℍ∶= {𝑧∶ Im 𝑧 > 0} or 𝑆=−ℍ, stand for the canonical holomorphic model of 𝜑.Then𝑓𝜑,𝜓 ◦ℎ−1 𝜑extends holomorphically to a map 𝐺∶𝑆→𝑆∪ℝwith 𝐺(𝑤 + 1) = 𝐺(𝑤) for all 𝑤∈𝑆. (D) The self-map g(𝑤) ∶= 𝑤 + 𝐺(𝑤) is univalent in 𝑆,g(ℎ𝜑(𝔻)) ⊂ ℎ𝜑(𝔻),and 𝜓=ℎ −1 𝜑◦g◦ℎ𝜑=ℎ −1 𝜑◦(ℎ𝜑+𝑓 𝜑,𝜓). Coming back to Theorem 1.2, condition (f) is probably the most interesting and deepest result of the paper. In fact, according to the following theorem, this condition implies the stronger conclusion that 𝜑is an element of (𝜓𝑡). We exclude from the statement hyperbolic automorphisms, because in this case, the result is essentially known: If 𝜑is a hyperbolic automorphism, then (𝜓𝑡)is a hyperbolic one-parameter group and all the three conditions in Theorem 1.4 below hold, except that in condition (a) the words “for some 𝑡0>0” have to be replaced by “for some 𝑡0∈ℝ”; see, for example, [1, section 4.10] (see also [5]). Theorem 1.4. Let 𝜑∈(𝔻)be a non-elliptic self-map different from a hyperbolic automorphism, and suppose that it has a boundary fixed point 𝜎different from its Denjoy–Wolff point. Let (𝜓𝑡)be a continuous one-parameter semigroup such that 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. Then the following conditions are equivalent: (a) 𝜑=𝜓 𝑡0for some 𝑡0>0; (b) (𝜓𝑡)⊂(𝜑); (c) 𝜎is a boundary fixed point also for (𝜓𝑡). Remark 1.5. If at least one of the self-maps 𝜑or 𝜓1— and hence both of them†— are hyperbolic, then all the equivalent conditions (a), (b), and (c) in the above theorem are automatically satisfied. This follows from the fact that by [5, Propositions 6.6 and 6.9], in the hyperbolic case, we have (𝜑) = (𝜓1)={𝜓 𝑡∶𝑡⩾0}. Remark 1.6. Let 𝜑and 𝜓be two commuting holomorphic self-maps of 𝔻and suppose that 𝜑has a boundary fixed point 𝜎. It is known (see, e.g., [6]; see also Remark 3.2) that in general, 𝜎does not have to be a boundary fixed point for 𝜓, even if we additionally assume that 𝜎is regular, 𝜑is univalent, and 𝜓is embeddable in a continuous one-parameter semigroup. Therefore, the implications (b) ⇒(c) and (b) ⇒(a) in the theorem stated above can be regarded as an illustration †See [4, Corollary 4.1]. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 5of31 of a neat difference between commutativity with a self-map (and hence with all its natural iterates) and commutativity with all the fractional iterates. If a continuous one-parameter semigroup (𝜓𝑡)is contained in the centralizer of a non-elliptic self-map 𝜑, then by Theorem 1.4 every boundary fixed point of 𝜑has to be among boundary fixed points of (𝜓𝑡). The converse is not true: We may have a continuous one-parameter semigroup (𝜓𝑡)⊂(𝜑) with many boundary fixed points, while 𝜑has no boundary fixed point other than its Denjoy–Wolff point; see, for example, Example 3.1 with (𝜑𝑡)and (𝜓𝑡)interchanged. This leads us to the following natural question (in which, in fact, we impose some weaker assumptions). Problem 1.7. Let 𝜑∈(𝔻)be a non-elliptic self-map different from a hyperbolic automorphism and let (𝜓𝑡)be a continuous one-parameter semigroup such that 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. Further, suppose 𝜑∉{𝜓 𝑡∶𝑡⩾0}. Is there any relationship between 𝜑and the boundary fixed points of (𝜓𝑡) in this case? A natural way to attack this problem is through the concept of petal of a continuous oneparameter semigroup, concept intimately linked to the notion of boundary regular fixed point of such a semigroup. We refer the reader to Section 5for more details, or to the monograph [7, section 13] for a detailed exposition, which includes a number of related results. It is worth mentioning that this theory of petals plays also an important role in the proof of Theorem 1.4. We begin by showing that, as a general fact, 𝜑always maps petals into petals and in a rather specific way. The degenerate case of (𝜓𝑡)being a one-parameter group is excluded from the statement of the theorem below; it is covered separately, see Remark 7.1 in Section 7. Theorem 1.8. Let 𝜑∈(𝔻)be non-elliptic and let (𝜓𝑡) ⊄ 𝖠𝗎𝗍(𝔻)be a continuous one-parameter semigroup in 𝔻with Denjoy–Wolff point 𝜏∈𝜕𝔻such that 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. (A) Assume that Δis a hyperbolic petal of (𝜓𝑡)with associated boundary repelling fixed point 𝜎∈ 𝜕𝔻. Then, one and only one of the following three statements holds: (i) 𝜑(Δ) = Δ. In this case, there exists 𝑡0>0such that 𝜑=𝜓 𝑡0and, in particular, 𝜎is a repelling fixed point of 𝜑. (ii) Δ∩𝜑(Δ)=∅and there exists a hyperbolic petal Δ′of (𝜓𝑡)with associated repelling fixed point 𝜎′such that 𝜑(Δ) ⊂ Δ′and ∠lim 𝑧→𝜎 𝜑(𝑧)=𝜎 ′. (iii) Δ∩𝜑(Δ)=∅ and there exists a parabolic petal Δ′of (𝜓𝑡)such that 𝜑(Δ) ⊂ Δ′and ∠lim 𝑧→𝜎 𝜑(𝑧) = 𝜏.Inthiscase,𝜎is an irregular contact point for 𝜑. (B) Assume that Δis a parabolic petal of (𝜓𝑡). Then the following two statements hold: (a) 𝜑(Δ) ⊂ Δ; (b) 𝜑(Δ) = Δ if and only if 𝜑=𝜓 𝑡0for some 𝑡0>0. In the hyperbolic case, by the reason mentioned in Remark 1.5, only alternative (i) may occur, and moreover, (B) becomes trivial because a hyperbolic semigroup cannot have parabolic petals. In the parabolic case, based on Theorem 1.8, we are then able to give a quite complete answer to Problem 1.7 as follows. We again exclude from consideration the case (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻), in which the result, with some obvious modifications, holds trivially. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6of31 CONTRERAS et al. Theorem 1.9. Let 𝜑∈(𝔻)be a non-elliptic self-map with Denjoy–Wolff point 𝜏∈𝜕𝔻and let (𝜓𝑡) ⊄ 𝖠𝗎𝗍(𝔻)be a continuous one-parameter semigroup in 𝔻such that 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. Suppose that 𝜑is not an element of (𝜓𝑡). Then the following statements hold. (I) Both 𝜑and (𝜓𝑡)are parabolic. Moreover, (𝜓𝑡)has at most one parabolic petal. (II) Suppose (𝜓𝑡)has a parabolic petal Δ∗. Then for each hyperbolic petal Δof (𝜓𝑡),thereexista finite collection Δ1,Δ 2,…, Δ 𝑛of pairwise disjoint hyperbolic petals of (𝜓𝑡)such that Δ1=Δ, 𝜑(Δ 𝑘)⊂Δ 𝑘+1, 𝑘=1,…,𝑛−1, 𝜑(Δ 𝑛)⊂Δ ∗. Moreover, 𝜑(𝜎𝑘)=𝜎 𝑘+1, 𝑘=1,…,𝑛−1, 𝜑(𝜎 𝑛)=𝜏, where 𝜎𝑘(𝑘 =1,…,𝑛)stands for the repelling fixed point associated with Δ𝑘. (III) Suppose that (𝜓𝑡)has no parabolic petal. Then for each hyperbolic petal Δof (𝜓𝑡), there exists asequence(Δ𝑛)of pairwise disjoint hyperbolic petals of (𝜓𝑡)such that Δ1=Δ, 𝜑(Δ 𝑛)⊂Δ 𝑛+1,𝑛∈ℕ. Moreover, 𝜑(𝜎𝑛)=𝜎 𝑛+1,𝑛∈ℕ,and lim 𝑛→∞ 𝜎𝑛=𝜏, where 𝜎𝑛(𝑛 ∈ ℕ)stands for the repelling fixed point associated with Δ𝑛. Remark 1.10. The above two theorems imply that every repelling fixed point 𝜎of 𝜓∶=𝜓 1is a contact point for 𝜑and that 𝜑(𝜎) is also a boundary regular fixed point of 𝜓.Moreover,if𝜑(𝜎) ≠𝜎, then the (forward) orbit of 𝜎with respect to 𝜑(extended to a.e. point of 𝜕𝔻by angular limits) either hits the Denjoy–Wolff point within finite time, or it consists of pairwise distinct repelling fixed points of 𝜓, but tends to the Denjoy–Wolff point in the limit. In part, this resembles the situation with two general commuting holomorphic self-maps 𝜑,𝜓 ∈ 𝖧𝗈𝗅(𝔻)studiedbyBracci [6]. In this general case, 𝜑can have a repelling cycle consisting of repelling fixed points of 𝜓,but this possibility is ruled out if 𝜑is univalent and non-elliptic, see [6, Proposition 5.2]. At the same time, the results established in [6] do not seem to exclude some other situations not occurring in our case, such as a possibility that the orbit of 𝜎hits within finite time a common fixed point of 𝜑 and 𝜓different from their Denjoy–Wolff point. Moreover, in contrast to the context of Theorem 1.9, it is not clear in general whether any two orbits starting from different repelling fixed points of 𝜓 necessarily fall into the same category. Theorems 1.8 and 1.9 are proved in Section 7. Further, in Section 8, we consider the last two alternatives given in Theorem 1.8(A) and analyze in which case equality 𝜑(Δ) = Δ′occurs. This will be used, along with several other results, in the proof of Theorem 1.2 given in Section 9. Finally, in Section 10, based on our findings, we substantially improve a result by Elin et al. [2, section 5] on sufficient conditions for two parabolic continuous one-parameter semigroups (𝜑𝑡)and (𝜓𝑡)to commute, given that 𝜑1◦𝜓1=𝜓 1◦𝜑1. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 7of31 The part of the paper preceding Sections 7–10, the content of which we have already described, is organized as follows. In Section 2, we recall some preliminaries from holomorphic dynamics that will be needed to follow the paper. In Section 3, we present in detail an example related to Problem 1.1 and already mentioned above. In the same section, we establish a characterization for holomorphic self-maps 𝜑commuting with a non-elliptic semigroup (𝜓𝑡)in terms of the Koenigs function of (𝜓𝑡). In the next section, Section 4, we prove Theorem 1.3, which yields another characterization in terms of the boundary behaviour of 𝜓1and 𝜑along forward orbits under the discrete dynamics of 𝜓1. Section 5contains a brief survey on the theory of petals and some lemmata relating commutativity to petals, which we use in the proof of Theorem 1.4 given in Section 6,aswell as in the proof of Theorems 1.8 and 1.9. 2 NOTATION AND PRELIMINARIES Below we introduce some notation and basic theory used further in the paper. For more details and for the proofs of the results presented in this section, we refer the interested readers to the recent monographs [1, 7]. 2.1 Notation As usual, we denote the unit disc by 𝔻∶= {𝑧 ∈ ℂ∶|𝑧|<1},and𝔻∗∶= 𝔻⧵{0}.Wewriteℍ∶= {𝑤 ∈ ℂ∶Im𝑤>0}for the upper half-plane and ℍRe ∶= {𝑧 ∈ ℂ∶Re𝑤>0}for the right halfplane. Furthermore, denote by 𝖧𝗈𝗅(𝐷, 𝐸) the class of all holomorphic mappings of a domain 𝐷⊂ℂ into a set 𝐸⊂ℂ,andlet(𝐷, 𝐸) stand for the class of all univalent (i.e., injective holomorphic) mappings from 𝐷to 𝐸.Asusual,weendow𝖧𝗈𝗅(𝐷, 𝐸) and (𝐷, 𝐸) with the topology of locally uniform convergence. In case 𝐸=𝐷,wewillwrite𝖧𝗈𝗅(𝐷) and (𝐷) instead of 𝖧𝗈𝗅(𝐷,𝐷) and (𝐷, 𝐷), respectively. For a self-map 𝜑∶𝐷→𝐷of a domain 𝐷⊂ℂand 𝑛∈ℕ,we denote by 𝜑◦𝑛the 𝑛th iterate of 𝜑, and let 𝜑◦0∶= 𝗂𝖽𝐷.Moreover,if𝜑is an automorphism of 𝐷, then for every 𝑛∈ℕ, we denote by 𝜑◦−𝑛 the 𝑛th iterate of 𝜑−1. 2.2 Holomorphic self-maps of the unit disc The study of the dynamics of an arbitrary holomorphic self-map 𝜑of the unit disc 𝔻is a classical and well-established branch of Complex Analysis. An important role is played by the fixed points, all of which — except for at most one — lie on the boundary and hence should be understood in the sense of angular limits: 𝜎∈𝜕 𝔻is a boundary fixed point of 𝜑∈𝖧𝗈𝗅(𝔻)if 𝜑(𝜎) ∶= ∠lim𝑧→𝜎 𝜑(𝑧) exists and coincides with 𝜎. It is known that the angular derivative 𝜑′(𝜎) exists at every boundary fixed point 𝜎, but it can be infinite. In this latter case, 𝜎is referred to as a super-repelling fixed point; otherwise, that is, when 𝜑′(𝜎) is finite, it is in fact a positive real number and the boundary fixed point 𝜎is said to be regular (BRFP for short). The central result in the area is the Denjoy–Wolff Theorem, which states that if 𝜑is different from an elliptic automorphism (i.e., not an automorphism of 𝔻possessing a fixed point in 𝔻), then the sequence of the iterates (𝜑◦𝑛)converges locally uniformly in 𝔻to a certain point 𝜏∈𝔻.This 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8of31 CONTRERAS et al. point is called the Denjoy–Wolff point of 𝜑.Moreover,if𝜏∈𝜕𝔻, it is the unique boundary fixed point at which the angular derivative 𝜑′(𝜏) is finite and belongs to (0,1]. In particular, for every BRFP 𝜎different from the Denjoy–Wolff point 𝜏,wehave𝜑′(𝜎) ∈ (1, +∞). By this reason such points are referred to as repelling fixed points of 𝜑. According to the position of the Denjoy–Wolff point 𝜏and to the value of the multiplier 𝜑′(𝜏), holomorphic self-maps 𝜑∈𝖧𝗈𝗅( 𝔻)different from elliptic automorphisms are divided into three categories. Namely, 𝜑is called: (a) elliptic if 𝜏∈𝔻, (b) hyperbolic if 𝜏∈𝜕 𝔻and 𝜑′(𝜏) < 1,and (c) parabolic if 𝜏∈𝜕 𝔻such that 𝜑′(𝜏) = 1. The identity mapping 𝗂𝖽𝔻and all elliptic automorphisms of 𝔻are conventionally included in the category (a) of elliptic self-maps. Similarly, for an elliptic automorphism different from 𝗂𝖽𝔻,its Denjoy–Wolff point is defined to be its unique fixed point in 𝔻. Parabolic self-maps can have very different properties depending on the so-called hyperbolic step. Denote by 𝜌𝔻the hyperbolic distance in 𝔻,andlet𝜑∈𝖧𝗈𝗅(𝔻)be non-elliptic. Thanks to the Schwarz–Pick Lemma, for the orbit (𝑧𝑛)∶= (𝜑◦𝑛(𝑧0))of any point 𝑧0∈𝔻, there exists a finite limit 𝑞(𝑧0)∶=lim 𝑛→+∞ 𝜌𝔻(𝑧𝑛,𝑧 𝑛+1).Itisknown,see,forexample,[1, Corollary 4.6.9], that either 𝑞(𝑧0)>0for all 𝑧0∈𝔻,or𝑞≡0in 𝔻. The self-map 𝜑is said to be of positive or of zero hyperbolic step depending on whether the former or the latter alternative occurs. If 𝜑is hyperbolic, then it is always of positive hyperbolic step. However, there exist parabolic self-maps of zero as well as of positive hyperbolic step. 2.3 Holomorphic models for univalent self-maps An indispensable role in our study is played by the concept of a holomorphic model, which goes back to Pommerenke [8], Baker and Pommerenke [9], and Cowen [10] and which is discussed below for the special case of a univalent self-map. The terminology we use is mainly borrowed from [11]. Definition 2.1. Aholomorphic model of 𝜑∈(𝔻)is any triple ∶=(𝑆,ℎ,𝛼), where 𝑆is a Riemann surface, 𝛼is an automorphism of 𝑆,andℎis a univalent map from 𝔻into 𝑆satisfying the following two conditions: (HM1) ℎ◦𝜑=𝛼◦ℎ,and (HM2) 𝑆=⋃𝑛⩾0𝛼◦−𝑛(ℎ(𝔻)). The Riemann surface 𝑆is called the base space, and the map ℎis called the intertwining map of the holomorphic model . Every 𝜑∈(𝔻)⧵{𝗂𝖽 𝔻}admits an essentially unique holomorphic model. More precisely, the following fundamental theorem holds. Theorem 2.2 [11, Theorem 1.1]. Every 𝜑∈(𝔻)admits a holomorphic model. Moreover such a model is unique up to a model isomorphism; that is, if (𝑆1,ℎ 1,𝛼 1)and (𝑆2,ℎ 2,𝛼 2)are holomorphic models for 𝜑, then there exists a biholomorphic map 𝜂of 𝑆1onto 𝑆2such that 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 9of31 ℎ2=𝜂◦ℎ1,𝛼 2=𝜂◦𝛼1◦𝜂−1. The type of a univalent self-map (elliptic, hyperbolic, or parabolic) is reflected in, and actually can be fully determined from the kind of holomorphic model 𝜑admits. For an open interval 𝐼⊂ℝ, we define 𝑆𝐼∶= ℝ×𝐼={𝑥+𝑖𝑦∶𝑥∈ℝ,𝑦∈𝐼}. Theorem 2.3 ([10], see also [11]). Let 𝜑∈(𝔻)⧵{𝗂𝖽 𝔻}. The following statements hold. (1) 𝜑is an elliptic automorphism with multiplier 𝜆∈𝜕 𝔻⧵{1}if and only if 𝜑admits a holomorphic modeloftheform𝜑∶= (𝔻,ℎ,𝑧↦𝜆𝑧),whereℎ ∈ 𝖠𝗎𝗍(𝔻). (2) 𝜑is an elliptic self-map with multiplier 𝜆∈𝔻∗(and hence it is not an automorphism) if and only if 𝜑admits a holomorphic model of the form 𝜑∶= (ℂ,ℎ,𝑧↦𝜆𝑧). (3) 𝜑is a hyperbolic self-map with multiplier 𝜆∈(0,1)if and only if 𝜑admits a holomorphic model of the form 𝜑∶= (𝑆𝐼,ℎ,𝑧↦𝑧+1),where𝐼=(𝑎,𝑏)is a bounded open interval of length 𝑏−𝑎=𝜋∕|log 𝜆|. (4) 𝜑is a parabolic self-map of positive hyperbolic step if and only if 𝜑admits a holomorphic model of the form 𝜑∶= (𝑆𝐼,ℎ,𝑧↦𝑧+1),where𝐼is an open unbounded interval different from the whole ℝ. (5) 𝜑is a parabolic self-map of zero hyperbolic step if and only if 𝜑admits a holomorphic model of the form 𝜑∶= (ℂ,ℎ,𝑧↦𝑧+1). Remark 2.4. In the above theorem, we may assume the following: - incase(1),ℎ′(𝜏) > 0, where 𝜏is the Denjoy–Wolff point of 𝜑; - incase(2),ℎ′(𝜏)=1, where 𝜏is the Denjoy–Wolff point of 𝜑; - in cases (3) and (5), ℎ(0) = 0; - incase(4),Reℎ(0) = 0 and 𝑆𝐼=𝑆 (0,+∞) =ℍor 𝑆𝐼=𝑆 (−∞,0) =−ℍ. Using the uniqueness part of Theorem 2.2, one can show (see, e.g., [1, Corollary 4.6.12] for details) that the above assumptions play the role of a normalization under which the holomorphic model 𝜑for a given 𝜑∈(𝔻)⧵{𝗂𝖽 𝔻}is unique. Note that the normalization for cases (3) and (5) would also work in case (4), but we prefer to use another normalization, so that for parabolic self-maps of positive hyperbolic step, the base space 𝑆𝐼of 𝜑coincides with ℍor −ℍ. Moreover, replacing, if necessary, 𝜑with 𝑧↦𝜑(  𝑧) we may assume 𝑆𝐼=ℍ. Definition 2.5. The unique holomorphic model 𝜑of a self-map 𝜑∈(𝔻)⧵{𝗂𝖽 𝔻}defined in Theorem 2.3 and normalized as in Remark 2.4 is called the canonical (holomorphic) model for 𝜑. The intertwining map ℎof the canonical model 𝜑is called the Koenigs function,andΩ∶=ℎ(𝔻) is called the Koenigs domain of 𝜑. 2.4 Commuting holomorphic self-maps It is clear that if two holomorphic self-maps 𝜑,𝜓 ∈ 𝖧𝗈𝗅(𝔻)⧵{𝗂𝖽 𝔻}commute, that is, 𝜑◦𝜓=𝜓◦𝜑, and if one of them is elliptic, then the other is also elliptic and they share the Denjoy–Wolff point. The situation is not so evident when we consider non-elliptic self-maps. In 1973, Behan [12], see 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 16 of 31 CONTRERAS et al. Following the conventional terminology in dynamical systems, we call the above BRFP 𝜎the 𝛼point of the corresponding hyperbolic petal Δ. In case of a parabolic petal, by its 𝛼-point we mean the Denjoy–Wolff point 𝜏. In the rest of this section, we suppose 𝜑∈(𝔻)is a non-elliptic self-map with Denjoy–Wolff point 𝜏∈𝜕 𝔻, that 𝜑is different from a hyperbolic automorphism, and that (𝜓𝑡)is a continuous one-parameter semigroup of 𝔻satisfying 𝗂𝖽𝔻≠𝜓1∈(𝜑). Lemma 5.2. In the above assumptions, 𝜑()⊂,wherestands for the backward invariant set of (𝜓𝑡). In particular, the image 𝜑(Δ) of any petal Δof (𝜓𝑡)is contained again in some petal of (𝜓𝑡) (which may coincide with Δ). Proof. Denote by ℎthe Koenigs function of (𝜓𝑡). Using Abel’s equation (2.2), and taking into account that ℎis univalent, it is easy to see that ℎ()=⋂ 𝑡⩾0 ℎ(𝔻)+𝑡 = ⋂ 𝑛∈ℕ ℎ(𝔻)+𝑛. (5.1) The second equality holds because, again by (2.2), ℎ(𝔻)+𝑡 2⊂ℎ(𝔻)+𝑡 1whenever 𝑡2⩾𝑡1⩾0. By the hypothesis 𝜑∈(𝜓1). Therefore, by [5, Theorem 3.4], there exists a univalent holomorphic map g∶ℎ( 𝔻)→ℎ(𝔻)such that ℎ◦𝜑=g◦𝜑and g(𝑤 + 1) = g(𝑤) + 1 for any 𝑤∈ℎ(𝔻). Therefore, ℎ(𝜑())=g(ℎ())=g(⋂ 𝑛∈ℕ ℎ(𝔻)+𝑛)=⋂ 𝑛∈ℕ g(ℎ(𝔻)+𝑛 )=⋂ 𝑛∈ℕ g(ℎ(𝔻))+𝑛. (5.2) Combining the fact that g(ℎ(𝔻))⊂ℎ(𝔻)with the equalities (5.1)and(5.2), we see that ℎ(𝜑())⊂ℎ().Sinceℎis univalent, this proves the inclusion 𝜑()⊂. Finally, the statement regarding the petals follows immediately from this inclusion because the map 𝜑is continuous and open. □ Note that the petals of (𝜓𝑡)are pairwise disjoint. Therefore, in Lemma 5.2 proved above, the petal Δ′that contains 𝜑(Δ) is unique. Lemma 5.3. Let Δ′be the petal of (𝜓𝑡)defined as above, that is, 𝜑(Δ) ⊂ Δ′.Then 𝜑(𝜎) ∶= ∠ lim 𝑧→𝜎 𝜑(𝑧) = 𝜎′, where 𝜎and 𝜎′stand for the 𝛼-points of the petals Δand Δ′, respectively. Proof. Fix any point 𝑧0∈Δ. Then according [7, Proposition 13.4.2], there is a unique regular backward orbit (𝑧𝑡)⊂Δof (𝜓𝑡)starting from 𝑧0and converging to 𝜎as 𝑡→+∞. More precisely, for each 𝑡⩾0there is a unique point 𝑧𝑡∈Δsuch that 𝜓𝑡(𝑧𝑡)=𝑧 0and the map [0, +∞) ∋ 𝑡 ↦ 𝑧𝑡is continuous and injective, with lim𝑡→+∞ 𝑧𝑡=𝜎and sup 𝑡⩾0 𝜌𝔻(𝑧𝑡,𝑧 𝑡+1) < +∞. (5.3) 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 17 of 31 Since 𝜑commutes with 𝜓1and hence with 𝜓𝑛for any 𝑛∈ℕ,wehave 𝜓𝑛(𝜑(𝑧𝑛))=𝜑 (𝜓𝑛(𝑧𝑛))=𝜑(𝑧 0). Taking into account that 𝑧𝑛∈Δand hence 𝜑(𝑧𝑛)⊂Δ ′for all 𝑛∈ℕ∪{0}, we see that the points 𝑤𝑛lie on the backward orbit of 𝑤0∶= 𝜑(𝑧0)∈Δ ′. Therefore, 𝑤𝑛→𝜎 ′as 𝑛→+∞. Thus, we see that 𝜑(𝑧𝑛)→𝜎 ′for a sequence (𝑧𝑛)converging to 𝜎and such that the hyperbolic distance 𝜌𝔻(𝑧𝑛,𝑧 𝑛+1), according to (5.3), is bounded. This leads, see, for example, [11, Theorem 1.4], to the desired conclusion that 𝜑has angular limit at 𝜎equal to 𝜎′.□ Lemma 5.4. If Δis a parabolic petal of (𝜓𝑡),then𝜑(Δ) ⊂ Δ. Proof. By the definition of a parabolic petal, the 𝛼-point 𝜎of the petal Δcoincides with the Denjoy– Wolff point 𝜏of the semigroup (𝜓𝑡), which in turn coincides with the Denjoy–Wolff point of 𝜑. Therefore, 𝜑(𝜎) = 𝜎. According to Lemma 5.3, it follows that the petal Δ′containing 𝜑(Δ) must be also parabolic. By [7, Theorem 13.5.7], (𝜓𝑡)has at most two parabolic petals. If it has exactly one parabolic petal, then we are done. If there are two distinct parabolic petals, then (again according [7,Theorem 13.5.7]) the images 𝑃𝑘,𝑘=1,2, of the two parabolic petals with respect to the Koenigs map ℎof (𝜓𝑡)are of the form 𝑃1={𝑤∶Im𝑤<𝑎}and 𝑃2={𝑤∶Im𝑤>𝑏}for some 𝑏⩾𝑎.As a consequence, the semigroup (𝜓𝑡), and hence the self-map 𝜓1, are of zero hyperbolic step. Since 𝜑∈(𝜓1),by[5, Proposition 4.3], 𝜑is affine with respect to 𝜓1, that is, ℎ◦𝜑=ℎ+𝑐for a suitable 𝑐∈ℂ. It follows that if Δand Δ′were two different parabolic petals, then the translation by 𝑐or −𝑐 would map 𝑃1into 𝑃2, which is clearly impossible. Thus, 𝜑(Δ) ⊂ Δ′=Δ.□ Corollary 5.5. In the notation of Lemma 5.3,𝜑(Δ) ⊂ Δ if and only if 𝜑(𝜎)=𝜎. Proof. If Δis a hyperbolic petal, then the corollary follows from Lemma 5.2 and the fact that any two distinct petals have distinct 𝛼-points unless both petals are parabolic. So suppose Δis a parabolic petal. Then 𝜑(Δ) ⊂ Δ by Lemma 5.4. Moreover, as we have seen in the proof of Lemma 5.4,𝜑(𝜎) = 𝜎 because in this case, 𝜎=𝜏is the Denjoy–Wolff point for (𝜓𝑡) and hence also for 𝜑. These two observations complete the proof. □ Lemma 5.6. Suppose that𝜑is not an element of (𝜓𝑡).Then(𝜓𝑡)can have at most one parabolic petal. Proof. Suppose that (𝜓𝑡)has two distinct parabolic petals. Then, as in the proof of Lemma 5.4,we see that ℎ◦𝜑=ℎ+𝑐, where the constant 𝑐must be real because each of the half-planes 𝑃1and 𝑃2 are mapped by the translation 𝑤↦𝑤+𝑐into itself. Moreover, ℎ(𝔻)+𝑐=ℎ (𝜑(𝔻))⊂ℎ(𝔻)and hence, by [5, Theorem 3.1 (B)], 𝑐⩾0. It follows that 𝜑=𝜓 𝑐, which contradicts the hypothesis. □ We conclude this section with a lemma, which is definitely known for specialists and holds also for non-univalent self-maps, but lacking a reference we prefer to include a short proof making use of commutativity. Lemma 5.7. Let 𝜑1,𝜑 2∈(𝔻)be such that 𝜑◦𝑘 1=𝜑 ◦𝑘 2for some 𝑘∈ℕ.If𝜑1is non-elliptic, then 𝜑2=𝜑 1. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 18 of 31 CONTRERAS et al. Proof. Since 𝜑1is univalent and non-elliptic, so is 𝜑3∶= 𝜑◦𝑘 1.Let(𝑆,ℎ,𝑧↦𝑧+1) stand for the canonical holomorphic model of 𝜑3. Clearly, both 𝜑1and 𝜑2commute with 𝜑3. Therefore, see, for example, [5, Theorem 3.4], 𝜑𝑚=ℎ −1◦g𝑚◦ℎ,𝑚=1,2, with certain g𝑚∈(𝑆) satisfying g𝑚(𝑤 + 1) = g𝑚(𝑤)+1for all 𝑤∈𝑆. The equality 𝜑◦𝑘 2=𝜑 ◦𝑘 1=𝜑 3implies g◦𝑘 𝑚(𝑤) = 𝑤 + 1, 𝑚 = 1, 2, for all 𝑤∈𝑆. In particular, g1,g2∈ 𝖠𝗎𝗍(𝑆). Using the one-to-one correspondence between the automorphisms of 𝑆and those of 𝔻induced by a conformal mapping between 𝑆and 𝔻and taking into account that the iterates of any 𝑓 ∈ 𝖠𝗎𝗍(𝔻)have the same fixed points as 𝑓, it is not difficult to see that both g1 and g2are translations along ℝ.Sinceg◦𝑘 1=g◦𝑘 2, it follows that g1=g2.Thus,𝜑2=𝜑 1.□ 6PROOF OF THEOREM 1.4 Recall that by the hypothesis 𝜑∈(𝔻)is non-elliptic and different from a hyperbolic automorphism and that 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}. In particular, it follows that the semigroup (𝜓𝑡)is non-trivial and non-elliptic. Moreover, by Behan’s Theorem, 𝜑and (𝜓𝑡)have the same Denjoy–Wolff point 𝜏. Denote by (𝑆,𝐻,(𝑧↦𝑧+𝑡) 𝑡⩾0)the canonical holomorphic model of (𝜓𝑡). 𝑃𝑟𝑜𝑜𝑓 𝑜𝑓 (a) ⇒(b) is completely trivial. 𝑃𝑟𝑜𝑜𝑓 𝑜𝑓 (b) ⇒(c). Since (𝜓𝑡)⊂(𝜑),byProposition3.3 we have 𝐻◦𝜑=𝐻+𝑐for some 𝑐∈ℂ. The Koenigs function 𝐻has angular limits at all points on 𝜕𝔻, possibly except for the Denjoy– Wolff point, see, for example, [7, Corollary 11.1.7]. Let us check that 𝐻(𝜎) ∶= ∠lim𝑧→𝜎 𝐻(𝑧) = ∞. Suppose 𝐻(𝜎) ≠∞. Then 𝐻(𝜑(𝑟𝜎))=𝐻(𝑟𝜎)+𝑐→𝐻(𝜎)+𝑐as 𝑟→1 −. By the hypothesis, 𝜎 is a boundary fixed point of 𝜑, that is, 𝜑(𝑟𝜎) → 𝜎 as 𝑟→1 −. According to the Lehto–Virtanen’s Theorem (see, e.g., [7, Theorem 3.3.1]) it follows that 𝐻(𝜎) = lim 𝑟→1−𝐻(𝜑(𝑟𝜎))=𝐻(𝜎)+𝑐 and hence 𝑐=0, that is, 𝜑=𝗂𝖽 𝔻. The latter contradicts the hypothesis. Thus, 𝐻(𝜎) = ∞ and, as a consequence, see, for example, [7, Proposition 13.6.1], 𝜎is a boundary fixed point of (𝜓𝑡). 𝑃𝑟𝑜𝑜𝑓 𝑜𝑓 (c) ⇒(a). By an old result of Heins [20, Lemma 2.1], 𝜓1cannot be a hyperbolic automorphism because it commutes with 𝜑, which is not a hyperbolic automorphism. Moreover, 𝜓1 is not a parabolic automorphism because by (c), it has a boundary fixed point 𝜎different from its Denjoy–Wolff point. It follows that the non-elliptic semigroup (𝜓𝑡)cannot be extended to a group. According to the regularity of 𝜎, we distinguish two cases. Case I: 𝜎is a repelling fixed point of (𝜓𝑡).LetΔbe the hyperbolic petal of (𝜓𝑡)associated with 𝜎, as explained in Section 5.Since(𝜓𝑡)is not a group, according to [7, Theorems 13.2.7, 13.4.12], there exists a univalent map gfrom 𝔻onto Δsuch that the formula ˆ 𝜓𝑡∶= g−1◦𝜓𝑡◦g,𝑡⩾0, defines a hyperbolic group (ˆ 𝜓𝑡)in 𝔻with Denjoy–Wolff point at −𝜎. By Corollary 5.5,𝜑(Δ) ⊂ Δ. Therefore, ˆ 𝜑∶=g−1◦𝜑◦gis an element of (𝔻).Moreover,𝜓1∈(𝜑) implies ˆ 𝜑∈(ˆ 𝜓1).Sinceˆ 𝜓1is a hyperbolic automorphism and by [5, Remark 6.4], we conclude that there exists 𝑡0>0such that either ˆ 𝜑=ˆ 𝜓𝑡0or ˆ 𝜑=ˆ 𝜓−1 𝑡0. If the latter alternative would occur, then (𝜓𝑡0◦𝜑)|Δ=𝗂𝖽 Δand hence, by the identity principle, 𝜓𝑡0◦𝜑=𝗂𝖽 𝔻, which is impossible because 𝜓1∉ 𝖠𝗎𝗍(𝔻). Therefore, ˆ 𝜑=ˆ 𝜓𝑡0. It immediately follows that 𝜓𝑡0|Δ=𝜑|Δand hence, again by the identity principle, 𝜓𝑡0=𝜑as desired. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 19 of 31 Case II: 𝜎is a super-repelling fixed point of (𝜓𝑡). According to [7, Corollary 13.6.7], we know that there is 𝑎∈ℝsuch that lim 𝑧→𝜎 Im𝐻(𝑧)=𝑎 and lim 𝑧→𝜎 Re 𝐻(𝑧) = −∞. (6.1) Note that both limits are unrestricted. Taking into account that 𝜎is a boundary fixed point of 𝜑, we therefore have 𝑎= lim 𝑟→1−Im 𝐻(𝜑(𝑟𝜎)). (6.2) If 𝜓1is hyperbolic or parabolic of zero hyperbolic step, then by [5, Proposition 4.3], 𝜑∈(𝜓1) implies 𝐻◦𝜑=𝐻+𝑐for a suitable 𝑐∈ℂand, in view of (6.1)and(6.2), it follows that Im 𝑐 = lim 𝑟→1−(Im 𝐻(𝜑(𝑟𝜎)) − Im 𝐻(𝑟𝜎))=𝑎−𝑎=0. Therefore, 𝑐is a real number. Certainly 𝑐≠0because 𝜑≠𝗂𝖽𝔻.Moreover,if𝑐were negative, we would obtain that 𝜓−𝑐◦𝜑=𝗂𝖽 𝔻, which is impossible because 𝜓𝑡∉ 𝖠𝗎𝗍(𝔻)for any 𝑡>0. Hence, 𝑐>0anditfollowsthat𝜑=𝜓 𝑡0with 𝑡0∶= 𝑐. This completes the proof in the case when (𝜓𝑡)is hyperbolic or parabolic of zero hyperbolic step. Fromnowonwesupposethat𝜓1is parabolic of positive hyperbolic step. For the sake of clarity, we will also assume that the base space for (𝜓𝑡)is 𝑆=ℍ. The proof in the other case, that is, for 𝑆=− ℍ, is completely similar. Then the fact that 𝜑∈(𝜓1)implies, by [5, Proposition 7.2], that there exists 𝐹∈𝖧𝗈𝗅( 𝔻,ℍ∪ℝ) such that 𝐻◦𝜑(𝑧)=𝐻(𝑧)+𝐹 (𝑒2𝜋𝑖𝐻(𝑧))for all 𝑧∈𝔻. We notice that the function 𝑤↦𝑤+𝐹(𝑒 2𝜋𝑖𝑤)is univalent in ℍ, a fact that will be used later. By (6.1)and(6.2), we have lim 𝑟→1−Im 𝐹(𝑒2𝜋𝑖𝐻(𝑟𝜎))= lim 𝑟→1−(Im 𝐻(𝜑(𝑟𝜎)) − Im 𝐻(𝑟𝜎))=𝑎−𝑎=0. (6.3) By (6.1), for any 𝑝=𝑒 2𝜋𝑖𝜃 ∈𝜕𝔻there exist a natural 𝑁=𝑁(𝑝)and a sequence (𝑟𝑛)=(𝑟 𝑛(𝑝)) in (0,1) convergent to 1 such that Re𝐻(𝑟𝑛𝜎)=𝜃−𝑛for all 𝑛⩾𝑁. For this sequence (𝑟𝑛),wehave lim 𝑛→∞ 𝑒2𝜋𝑖𝐻(𝑟𝑛𝜎) =lim 𝑛→∞ 𝑒2𝜋𝑖(𝜃−𝑛)𝑒−2𝜋 Im 𝐻(𝑟𝑛𝜎) =𝑝𝑒 −2𝜋𝑎.(6.4) Since 𝐻(𝔻)⊂ℍ,wehave𝑎⩾0. Below we consider separately the cases 𝑎>0and 𝑎=0. Subcase II.a: 𝑎>0. In this case, 𝑒−2𝜋𝑎 ∈ (0, 1).Let𝐶𝑎be the circle of radius 𝑒−2𝜋𝑎 centred at the origin. Taking into account that 𝐶𝑎⊂𝔻and using (6.3)and(6.4), we deduce that Im 𝐹(𝑧) = 0 for all 𝑧∈𝐶 𝑎. By the maximum principle for harmonic functions and the identity principle for holomorphic functions, it follows that 𝐹≡𝑟for some constant 𝑟∈ℝ. Hence, 𝐻◦𝜑=𝐻+𝑟.By essentially the same argument as in the case of zero hyperbolic case, we see that 𝑟>0and therefore, 𝜑=𝜓 𝑡0with 𝑡0∶= 𝑟, as desired. Subcase II.b: 𝑎=0.Since𝐹∈𝖧𝗈𝗅( 𝔻,ℍ∪ℝ), we have Im 𝐹 is a non-negative harmonic function in the unit disc. By [21, Corollary on p. 38], there exists 𝐸⊂𝔻of measure zero such that 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 20 of 31 CONTRERAS et al. the angular limit ∠lim 𝑧→𝑝 Im 𝐹(𝑧) exists for every 𝑝∈𝜕𝔻⧵𝐸.By(6.4), for every 𝑝∈𝜕𝔻, the sequence (𝑒2𝜋𝑖𝐻(𝑟𝑛(𝑝)𝜎))converges non-tangentially to 𝑝. Therefore, by (6.3), we deduce that ∠lim 𝑧→𝑝 Im 𝐹(𝑧) = 0, for all 𝑝 ∈ 𝜕𝔻⧵𝐸. (6.5) Now, let us consider the function in 𝖧𝗈𝗅(ℍ,ℂ)given by 𝐺(𝑤) ∶= 𝐹(𝑒2𝜋𝑖𝑤), 𝑤 ∈ ℍ. Similarly to the case 𝑎>0, it suffices to show that 𝐺is a real constant. Consider g(𝑤) ∶= 𝑤 + 𝐺(𝑤). This function is a univalent holomorphic self-map of ℍwith g′(∞) ∶= ∠ lim𝑤→∞ g(𝑤)∕𝑤 = 1. Therefore, gadmits the following representation, see, for example, [22, Chapter V.4: (V.42), (V.44) (V.45)], g(𝑤) = 𝑟 + 𝑤 + ∫ℝ(1 𝜉−𝑤−𝜉 1+𝜉 2)d𝜇(𝜉), 𝑤 ∈ ℍ,with a constant 𝑟∈ℝand a non-negative Borel measure 𝜇on ℝgiven by 𝜇([𝑎, 𝑏]) = lim 𝜂→0+ 1 𝜋∫𝑏 𝑎 Im g(𝜉 + 𝑖𝜂) d𝜉 for any 𝑎,𝑏 ∈ ℝ,𝑎<𝑏, possibly except for a countably many points at which 𝜇has atoms. Taking into account that by (6.5), lim𝜂→0+Im g(𝜉 + 𝑖𝜂) = 0 for a.e. 𝜉∈ℝ, it remains to see that Im gis bounded in ℍ⧵ℍ1, where for 𝑏∈ℝwe denote by ℍ𝑏the half-plane {𝑤∶ Im 𝑤 > 𝑏}. Note that g′(∞) ≠0and hence gis conformal at ∞with g(∞) = ∞. Recall that gis univalent in ℍ. Then by a standard argument, see, for example, [23, pp. 303–304], we get that g(ℍ1)⊃𝑈 𝑅∶= {𝑤 ∶ arg 𝑤 ∈ (𝜋∕2, 3𝜋∕2), |𝑤|>𝑅} for some 𝑅>0.Let𝑅0∶= max{1, 𝑅}. Since g(𝑤 + 1) = g(𝑤)+1for all 𝑤∈ℍ, it actually follows that ℍ𝑅0⊂⋃𝑘∈ℤ𝑈𝑅0+𝑘⊂g(ℍ1). Therefore, g(ℍ⧵ℍ1)=g(ℍ)⧵g(ℍ1)⊂ℍ⧵ℍ𝑅0,andwearedone. □ 7 PETALS AND COMMUTATIVITY II. PROOF OF THEOREMS 1.8 AND 1.9 Before we start proving Theorem 1.8, let us analyze (in the remark below) the case (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻), which in the statement of the theorem is excluded from consideration. As it has been already mentioned in Section 2.5, in this case, we can (and do) extend (𝜓𝑡)to a group by setting 𝜓𝑡∶= (𝜓−𝑡)−1 for all 𝑡<0. Remark 7.1. Suppose (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻)is a continuous one-parameter group in 𝔻with 𝜓1≠𝗂𝖽𝔻, and let 𝜑∈(𝔻)be a non-elliptic self-map commuting with 𝜓1. Clearly, (𝜓𝑡)cannonbeelliptic. The whole unit disc 𝔻is the unique petal of (𝜓𝑡), which is hyperbolic or parabolic depending on whether (𝜓𝑡)is a hyperbolic or a parabolic group. Moreover, if (𝜓𝑡)is hyperbolic, then by [20, Lemma 2.1], 𝜑is a hyperbolic automorphism having the same fixed points as (𝜓𝑡), and hence 𝜑=𝜓 𝑡0for some 𝑡0∈ℝ.If(𝜓𝑡)is parabolic, then 𝜑is not necessarily an automorphism of 𝔻,but it has to be a parabolic self-map with the same Denjoy–Wolff point as 𝜓1; see, for example, [1, Proposition 2.6.11]. In this case, 𝜑 ∈ 𝖠𝗎𝗍(𝔻)ifandonlyif𝜑=𝜓 𝑡0for some 𝑡0∈ℝ. Proof of Theorem 1.8. First of all, since 𝜓1is not a hyperbolic automorphism (otherwise, we would have (𝜓𝑡) ⊂ 𝖠𝗎𝗍(𝔻)), by [20, Lemma 2.1], 𝜑cannot be a hyperbolic automorphism either. Therefore, by Behan’s Theorem, see [12]or[1, Theorem 4.10.3], the Denjoy–Wolff point of 𝜑 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 21 of 31 coincides with the Denjoy–Wolff point 𝜏of the semigroup (𝜓𝑡). In particular, it follows that (𝜓𝑡) is non-elliptic. Proof of (A). By Lemma 5.2, there exists a petal Δ′of (𝜓𝑡)such that 𝜑(Δ) ⊂ Δ′.Being connected components of ◦, petals are pairwise disjoint. Therefore, either 𝜑(Δ) ⊂ Δ,or 𝜑(Δ) ∩ Δ = ∅. In the former case, by Corollary 5.5,𝜑(𝜎) = 𝜎 in the sense of angular limits. Moreover, taking into account that 𝜎≠𝜏, by Theorem 1.4 we have that in this case, 𝜑=𝜓 𝑡0for some 𝑡0>0and as a consequence 𝜑(Δ) = 𝜓𝑡0(Δ) = Δ.Thus,if𝜑(Δ) ⊂ Δ, then alternative (i) in Theorem 1.8(A) holds. Now suppose 𝜑(Δ) ∩ Δ = ∅.IfΔ′is a hyperbolic petal of (𝜓𝑡), then according to Lemma 5.3, alternative (ii) in Theorem 1.8(A) holds. If Δ′is a parabolic petal of (𝜓𝑡), then again by Lemma 5.3, 𝜑(𝜎) = 𝜏 in the sense of angular limits. In this case, taking into account that 𝜎≠𝜏and that 𝜑′(𝜏) ≠ ∞,by[24, Lemma 8.2], the angular derivative 𝜑′(𝜎) is ∞, and thus, alternative (iii) holds. This completes the proof of Theorem 1.8(A). Proof of (B). By hypothesis, Δis a parabolic petal for (𝜓𝑡). Hence, statement (a) is just Lemma 5.4. Since one of the implications in statement (b) is clear, see Section 5, it remains to prove that if 𝜑(Δ) = Δ, then 𝜑is contained in (𝜓𝑡). To this end, consider the canonical holomorphic model (𝑆,𝐻,(𝑧↦𝑧+𝑡) 𝑡⩾0)for the semigroup (𝜓𝑡). It is known, see, for example, [7, Theorem 13.5.7], that Π∶=𝐻(Δ)is a half-plane of the form Π={𝑧∶Im𝑧>𝑎}or Π={𝑧∶Im𝑧<𝑎}. Note that (𝑆,𝐻,𝑧↦𝑧+1)is the canonical model for 𝜓1. Recall also that 𝜑∈(𝜓1). Therefore, according to [5, Theorem 3.4], 𝜑=𝐻 −1◦g◦𝐻for some g∈(𝑆) satisfying g(𝑧 + 1) = g(𝑧)+1 for all 𝑧∈𝑆.(7.1) If 𝜑(Δ) = Δ, then g(Π) = Π and hence, taking into account (7.1), we conclude that g(𝑧) = 𝑧 + 𝑐 for all 𝑧∈Πand some constant 𝑐∈ℝ. Clearly, this equality hold for all 𝑧∈𝑆. Note that g(𝐻(𝔻))= 𝐻(𝜑(𝔻))⊂𝐻(𝔻).Since𝜓1is not an automorphism of 𝔻,by[5, Theorem 3.1 (B)], we have 𝑐⩾0, and of course, 𝑐≠0because 𝜑≠𝗂𝖽𝔻. Therefore, 𝜑=𝜓 𝑡0with 𝑡0∶= 𝑐 > 0.□ Proof of Theorem 1.9. Let (𝑆,𝐻,(𝑧↦𝑧+𝑡) 𝑡⩾0)stand for the canonical holomorphic model of (𝜓𝑡). As in the proof of Theorem 1.8, we see that 𝜑is not a hyperbolic automorphism. Proof of (I). The semigroup (𝜓𝑡)cannot be elliptic, because 𝜓1≠𝗂𝖽𝔻commutes with the nonelliptic self-map 𝜑. Furthermore, (𝜓𝑡)cannot be hyperbolic because otherwise having 𝜑∈(𝜓1) and 𝜓1∉ 𝖠𝗎𝗍(𝔻)would imply 𝜑=𝜓 𝑡0for some 𝑡0⩾0, see, for example, [5, Remark 6.4]; but this directly contradicts the hypothesis. Therefore, (𝜓𝑡)is parabolic, and by [4, Corollary 4.1] so is 𝜑. Moreover, by Lemma 5.6,(𝜓𝑡)can have at most one parabolic petal. Proof of (II). Let Δ∗be the unique parabolic petal of (𝜓𝑡).Itisknown,see,forexample, [7, Theorem 13.5.7], that Π∶=𝐻(Δ ∗)is a half-plane of the form Π={𝑧∶Im𝑧>𝑎} or Π={𝑧∶Im𝑧<𝑎}. Below we give a proof for the former alternative. The other case can be treated in a similar way. If (𝜓𝑡)is parabolic of zero hyperbolic step, then by [5, Proposition 4.3], we have 𝐻◦𝜑◦𝐻−1 = (𝑤 ↦ 𝑤 + 𝑐), where 𝑐∉[0,+∞)because 𝜑isnotanelementof(𝜓𝑡).Notealsothatby[5, Theorem 3.1 (B)], 𝑐 ∉ (−∞, 0) because 𝜓1is not an automorphism. Moreover, 𝜑(Δ∗)⊂Δ ∗by Lemma 5.4 and hence, Π+𝑐⊂Π. As a consequence, Im 𝑐 > 0. Since for each hyperbolic petal Δ,itsimage 𝐻(Δ) is a horizontal strip, see, for example, [7, Theorem 13.5.5 (2)], it follows that there exists 𝑛∈ℕsuch that 𝐻(Δ)+𝑛𝑐⊂Πand, as a result, we have 𝜑◦𝑛(Δ)⊂Δ ∗. Choose the smallest of 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 22 of 31 CONTRERAS et al. such 𝑛s. To complete the proof of assertion (II) for the case of zero hyperbolic step, it now suffices to appeal to Lemmas 5.2 and 5.3. Suppose (𝜓𝑡)is of positive hyperbolic step. We may suppose 𝑆=ℍ. The argument for the case 𝑆=− ℍis similar. By [5, Theorem 3.4], there exists g∈(ℍ)satisfying 𝐻◦𝜑=g◦𝐻and such that g(𝑤 + 1) = g(𝑤)+1for any 𝑤∈ℍ. As in the previous case, gcannot be of the form 𝑤↦𝑤+𝑐 with 𝑐∈ℝ. Therefore, see, for example, [5, Lemma 9.3 (II)], exp (2𝜋𝑖g(𝜁))=𝑓(𝑒 2𝜋𝑖𝜁), where 𝑓∈ 𝖧𝗈𝗅(𝔻),𝑓(0) = 0,|𝑓′(0)|<1. Since the iterates of such a self-map 𝑓∶𝔻→𝔻converge to 0, see, for example, [1, Theorem 3.1.13], it follows that for any 𝑤∈ℍ,Im g◦𝑛(𝑤) → +∞ as 𝑛→+∞. Therefore, any point 𝑧∈𝔻, 𝜑◦𝑛(𝑧) = 𝐻−1(g◦𝑛(𝐻(𝑧)))∈𝐻 −1(Π) = Δ∗ for 𝑛∈ℕlarge enough. As a consequence, for any hyperbolic petal Δthere exists 𝑛∈ℕsuch that 𝜑◦𝑛(Δ) ∩ Δ∗≠∅. Relying on the fact that petals are pairwise disjoint and applying Lemma 5.2 with 𝜑◦𝑛in place of 𝜑, we therefore conclude that 𝜑◦𝑛(Δ) ⊂ Δ∗. Now we can complete the proof of (II) in the same manner as in the previous case: Pass to the smallest 𝑛∈ℕsatisfying 𝜑◦𝑛(Δ) ⊂ Δ∗and appeal to Lemmas 5.2 and 5.3. Proof of (III). Suppose that (𝜓𝑡)has no parabolic petal and let Δbe some hyperbolic petal of (𝜓𝑡). Applying inductively Lemmas 5.2 and 5.3, we conclude that there exists a sequence of hyperbolic petals (Δ𝑛)with 𝛼-points 𝜎𝑛such that Δ1=Δ,𝜑(Δ𝑛)⊂Δ 𝑛+1,and𝜑(𝜎𝑛)=𝜎 𝑛+1 for all 𝑛∈ℕ. In order to see that Δ𝑛sare pairwise disjoint,†we suppose on the contrary that Δ𝑚+𝑘 =Δ 𝑚 for some 𝑚,𝑘 ∈ ℕ. Then 𝜑◦𝑘(Δ𝑚)⊂Δ 𝑚, and hence by Theorem 1.8(A) applied to Δand 𝜑replaced by Δ𝑚and 𝜑◦𝑘we would have that 𝜑◦𝑘=𝜓 𝑡0for some 𝑡0>0. By Lemma 5.7, this would further imply that 𝜑itself is contained in (𝜓𝑡), which contradicts the hypothesis of the theorem. To complete the proof, it remains to notice that according to [6, Theorem 1.4], 𝜓′ 1(𝜎𝑛)⩽𝜓′ 1(𝜎1)for all 𝑛∈ℕand, as a consequence, the Cowen–Pommerenke inequality [24, Theorem 4.1 (iii)] +∞ ∑ 𝑛=1 |𝜏−𝜎 𝑛|2 𝜓′ 1(𝜎𝑛)−1 ⩽2Re(1 𝜓1(0) −1 )<+∞ implies |𝜏−𝜎 𝑛|→0as 𝑛→+∞.□ 8PETALS AND ISOGONALITY Given 𝜑∈(𝔻)non-elliptic and (𝜓𝑡) ⊄ 𝖠𝗎𝗍(𝔻)a continuous one-parameter semigroup in the disc such that 𝜓1∈(𝜑) ⧵ {𝗂𝖽𝔻}, in Theorem 1.8, it was shown that 𝜑maps petals of (𝜓𝑡)into petals of (𝜓𝑡). In fact, if such a petal Δis parabolic, we always have 𝜑(Δ) ⊂ Δ. However, this is no longer the case when Δis hyperbolic. Apart from the same possibility, that is, apart from the case 𝜑(Δ) ⊂ Δ, and in general, we can have the other two following situations: ∙Situation A: 𝜑(Δ) ⊂ Δ′and Δ′is a hyperbolic petal. ∙Situation B: 𝜑(Δ) ⊂ Δ′and Δ′is a parabolic petal. In the following two results, we analyze when 𝜑maps Δonto Δ′in both situations. †An alternative way to prove this fact is to combine [6, Proposition 5.2] with Lemma 5.3. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 23 of 31 Proposition 8.1. Assume we are in Situation A and let 𝜎and 𝜎′be the 𝛼-points of Δand Δ′, respectively. Then the following conditions are equivalent: (a) 𝜑(Δ) = Δ′; (b) 𝜑is isogonal at 𝜎; (c) the spectral values of (𝜓𝑡)at 𝜎and 𝜎′coincide. Moreover, if the above equivalent conditions hold, then 𝜑is affine with respect to 𝜓1and hence, (𝜓𝑡)⊂(𝜑). Proposition 8.2. In Situation B we always have 𝜑(Δ) ≠Δ′. The proof of these two propositions require some backgrounds which we expose thereafter. We begin by recalling the definition of isogonality, which we extend to the case of infinite angular limit. Definition 8.3. A univalent function 𝑓∶𝔻→ℂis said to be isogonal or semi-conformal at a point 𝜁∈𝜕 𝔻if the following two angular limits exist: 𝑓(𝜁) ∶= ∠ lim 𝑧→𝜁 𝑓(𝑧) ∈ ℂ∪{∞} and ∠lim 𝑧→𝜁 arg 𝑓0(𝑧) 𝑧−𝜁, where arg is to be understood as a continuous map from ℂ⧵{0}to ℝ∕(2𝜋ℤ)and 𝑓0(𝑧) ∶= {𝑓(𝑧)−𝑓(𝜁) if 𝑓(𝜁) ≠∞, 1∕𝑓(𝑧) if 𝑓(𝜁) = ∞. Further, a univalent function Ψ∶ℍRe →ℂis said to be isogonal at 𝜔∈𝜕ℍRe if the composition 𝑓∶=Ψ◦𝐻𝜔, where 𝐻𝜔is the Cayley map of 𝔻onto ℍRe with 𝐻𝜔(0) = 0,𝐻𝜔(1) = 𝜔,isisogonal at 𝜁=1. Remark 8.4. If 𝑓∈(𝔻)and 𝜁is a contact point of 𝑓, then an elementary argument shows that the isogonality condition requires that the image of the radial segment [0, 𝜁] under 𝑓approaches the point 𝑓(𝜁) ∈ 𝜕𝔻orthogonally to 𝜕𝔻. Therefore, in this case, the isogonality at 𝜁is equivalent to ∠lim 𝑧→𝜁 Arg 1−𝑓(𝜁)𝑓(𝑧) 1−𝜁𝑧 =0, (8.1) where Arg 𝑤 stands for the unique value of the argument of 𝑤≠0contained in (−𝜋,𝜋]. Passing this remark to the right half-plane with the help of the Cayley map, we can say that a univalent self-map 𝑇∈(ℍRe)with a boundary fixed point at 0 is isogonal at 0 if and only if ∠lim 𝑤→0 Arg 𝑇(𝑤) 𝑤=0. (8.2) 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 24 of 31 CONTRERAS et al. Definition 8.5. Let 𝜎∈𝜕𝔻be a repelling fixed point of a continuous one-parameter semigroup (𝜓𝑡)with associated infinitesimal generator 𝐺. The triple (ℍRe,Ψ,𝑄 𝑡)is called a pre-model for (𝜓𝑡) at 𝜎if the following conditions are met: (i) for each 𝑡⩾0,𝑄𝑡is the automorphism of ℍRe given by 𝑄𝑡(𝑧)∶=𝑒 𝜆𝑡𝑧, where 𝜆∶=𝐺 ′(𝜎); (ii) the map Ψ∶ℍRe →𝔻is holomorphic and injective, ∠lim 𝑤→0 Ψ(𝑤) = 𝜎,andΨis isogonal at 0, that is, ∠lim 𝑤→0 Arg 1−𝜎Ψ(𝑤) 𝑤=0; (8.3) (iii) Ψ◦𝑄𝑡=𝜓 𝑡◦Ψfor all 𝑡⩾0. Remark 8.6. It is known [25, Theorem 3.10] that every continuous one-parameter semigroup, at each repelling fixed point 𝜎, admits a pre-model unique up to the transformation Ψ(𝑤) ↦ Ψ(𝑐𝑤), where 𝑐is an arbitrary positive constant. Moreover, Ψ(ℍRe)coincides with the hyperbolic petal Δ(𝜎) associated with 𝜎.ThemapΨcan be expressed via the Koenigs function ℎof (𝜓𝑡).Namely,if the strip ℎ(Δ(𝜎)) is 𝕊(𝑎,𝑏)={𝑤∶𝑎<Im𝑤<𝑏}, then the intertwining map Ψin the pre-model for (𝜓𝑡)at 𝜎is given by Ψ(𝑤) ∶= ℎ−1(𝑏−𝑎 2𝜋 log 𝑤 + 𝑏+𝑎 2𝑖+𝑠 ),𝑤∈ℍRe, where 𝑠is an arbitrary real constant. As the following result shows, it is also possible to talk about pre-models associated with parabolic petals. Theorem 8.7 [7, Proposition 13.4.10 and its proof]. Let (𝜓𝑡)be a parabolic semigroup in the unit disc with Denjoy–Wolff point 𝜏∈𝜕 𝔻and let Δbe a parabolic petal of (𝜓𝑡).Then,thereexist Ψ∈(ℍRe,𝔻)with Ψ(ℍRe)=Δand ∠lim 𝑤→0 Ψ(𝑤) = 𝜏 and a parabolic group (𝑄𝑡)in ℍRe with Denjoy–Wolff point at 0 such that, for all 𝑡⩾0, Ψ◦𝑄𝑡=𝜓 𝑡◦Ψ. Definition 8.8. In the notation of the above theorem, the triple (ℍRe,Ψ,𝑄 𝑡)is called a pre-model for (𝜓𝑡)associated with the parabolic petal Δ. Lemma 8.9. Let 𝑇be a univalent self-map of ℍRe.If 𝑇(0) = 0 in the sense of angular limits and if 𝑇is isogonal at 0, then for any 𝛽>0, lim 𝑥→0+ 𝑇(𝛽𝑥) 𝑇(𝑥) =𝛽. (8.4) 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License EXTENSION OF COMMUTATIVITY TO FRACTIONAL ITERATES 25 of 31 Proof. Since 𝑇is isogonal at 0 with 𝑇(0) = 0, the univalent function 𝑇(1−𝑧 1+𝑧 ),𝑧∈𝔻, satisfies at 𝜁=1the Visser–Ostrowski condition; see, for example, [26, Proposition 4.11 on p. 81]. It follows† that lim 𝑥→0+ 𝑥𝑇′(𝑥) 𝑇(𝑥) =1. As a consequence, log 𝑇(𝛽𝑥) 𝑇(𝑥) = 𝛽 ∫ 1 𝑡𝑥 𝑇′(𝑡𝑥) 𝑇(𝑡𝑥) d𝑡 𝑡⟶ 𝛽 ∫ 1 d𝑡 𝑡= log 𝛽 as 𝑥→0 +, as desired. □ Recall that a sequence (𝑧𝑛)⊂𝔻converging to some point 𝜎∈𝜕𝔻is said to converge to 𝜎 non-tangentially if the limit set Slope[(𝑧𝑛), 𝑛 → +∞] of Arg(1 − 𝜎𝑧𝑛)as 𝑛→+∞is compactly contained in the open interval (−𝜋∕2,𝜋∕2). Further, we say that (𝑧𝑛)converges to 𝜎tangentially, if Slope[(𝑧𝑛), 𝑛 → +∞] ⊂ {−𝜋∕2,𝜋∕2}. Clearly, (𝑧𝑛)converges to 𝜎non-tangentially if and only if it has no subsequence converging tangentially. Lemma 8.10. Suppose 𝑓∈(𝔻)is isogonal at some point 𝜁∈𝜕𝔻and that 𝑓(𝜁) ∈ 𝜕𝔻.If(𝑧𝑛)⊂𝔻 converges to 𝜁tangentially, and if (𝑤𝑛)given by 𝑤𝑛∶= 𝑓(𝑧𝑛)for all 𝑛∈ℕconverges to 𝑓(𝜁),then the convergence of (𝑤𝑛)is also tangential. Proof. Composing 𝑓with suitable conformal mappings we may replace 𝔻with ℍRe and suppose 𝑓(𝜁)=𝜁=0.Fixsome𝜂∈(0,𝜋∕2)and set 𝜃∶=1 2(𝜂 + 𝜋∕2).Since(𝑧𝑛)⊂ℍRe converges to 0 tangentially, there exists 𝑟>0such that 𝐴𝜃,𝑟 ∶= {𝑧 ∶ |Arg 𝑧|<𝜃, |𝑧|<𝑟}does not contain any point of (𝑧𝑛). The isogonality of 𝑓at 0 implies, see, for example, [26, Proposition 4.10 on p. 81], that there exists 𝜌>0such that 𝐴𝜂,𝜌 ⊂𝑓 (𝐴𝜃,𝑟).Since𝑓is univalent by hypothesis, it follows that 𝐴𝜂,𝜌 does not contain any point of (𝑤𝑛). Taking into account that 𝜂∈(0,𝜋∕2)in this argument is arbitrary, we conclude that if (𝑤𝑛)converges to 0, then the convergence must be tangential. □ Lemma 8.11. Let 𝐷1,𝐷 2,𝐷 3∈{𝔻,ℍRe}and let 𝑓3∶= 𝑓2◦𝑓1,where𝑓1∶𝐷 1→𝐷 2and 𝑓2∶𝐷 2→𝐷 3are two (holomorphic) univalent functions. The following two statements hold. (A) If 𝑓1is isogonal at some 𝜁1∈𝜕𝐷 1with 𝜁2∶= 𝑓1(𝜁1)∈𝜕𝐷 2and if 𝑓2is isogonal at 𝜁2with 𝜁3∶= 𝑓2(𝜁2)∈𝜕𝐷 3,then𝑓3is isogonal at 𝜁1with 𝑓3(𝜁1)=𝜁 3. (B) Conversely, if 𝑓3is isogonal at some 𝜁1∈𝜕𝐷 1and if 𝜁3∶= 𝑓3(𝜁1)∈𝜕𝐷 3, then: (i) 𝑓1is isogonal at 𝜁1, (ii) 𝜁2∶= 𝑓1(𝜁1)belongs to 𝜕𝐷2, (iii) 𝑓2is isogonal at 𝜁2, and (iv) 𝑓2(𝜁2)=𝜁 3. Proof. First of all, without loss of generality, we may suppose 𝐷1=𝐷 2=𝐷 3=𝔻. Proof of (A). Taking into account Remark 8.4, we have to show that for any sequence (𝑧𝑛)⊂𝔻 converging to 𝜁1non-tangentially, 𝑓3(𝑧𝑛)→𝜁 3and Arg ((1 − 𝜁3𝑓3(𝑧𝑛))∕(1 − 𝜁1𝑧𝑛))→0.This †In [27] one can find another proof of this fact (not assuming global univalence of 𝑇), see [27, Lemma 8.1] applied to 𝑧 ↦ 1∕𝑇(1∕𝑧), as well as some closely related results connecting asymptotic behavior of the hyperbolic derivative to isogonality and to existence of finite angular derivative. 14697750, 2025, 2, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70077 by Spanish Cochrane National Provision (Ministerio de Sanidad), Wiley Online Library on [29/05/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License