To appear in Revista Matemática Iberoamericana, 2010 ( arXiv:0902.3116v1 [math.CV]) LOEWNER CHAINS IN THE UNIT DISK MANUEL D. CONTRERAS † , SANTIAGO DÍAZ-MADRIGAL † , AND PAVEL GUMENYUK ‡ Abstract. In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [5], of the radial and chordal variant of the Loewner diﬀerential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [5]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial diﬀerential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE. Contents 1. Introduction 1.1. Classical Loewner theory 1.2. Chordal Loewner equation 1.3. Generalization of classical evolution families 1.4. Main results 2. Evolution families and Herglotz vector fields in the unit disk 3. Loewner chains and evolution families 4. Loewner chains and partial diﬀerential equations 5. Remarks about semigroups References 2 2 3 4 5 7 13 26 29 32 Date: November 10, 2009. 2000 Mathematics Subject Classification. Primary 30C80; Secondary 34M15, 30D05. Key words and phrases. Loewner chains, evolution families. † Partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM2006-14449-C02-01, by La Consejerı́a de Educación y Ciencia de la Junta de Andalucı́a and by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”. ‡ Partially supported by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”, the Research Council of Norway and the Russian Foundation for Basic Research (grant #07-01-00120). 1 2 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK 1. Introduction 1.1. Classical Loewner theory. In 1923 Loewner [25] introduced the so-called parametric method in geometric function theory, mainly in hope to solve the famous Bieberbach problem about obtaining sharp estimates of Taylor coeﬃcients of normalized holomorphic univalent functions in the unit disk. It is worth recalling that the solution of this problem, given in 1984 by de Branges, relied also on this method. The modern form of the parametric method is mainly due to contributions by Kufarev [18] and Pommerenke [27]. Let us briefly recall the main constructions (see, e.g., [28, Chapter 6]). Let f0 (z) = z + a2 z 2 + . . . be a holomorphic univalent function in the unit disk D := {z : |z| < 1}. One can always embed this function into a uniparametric family (ft )t≥0 of holomorphic univalent functions in D satisfying the following two properties: ft (z) = et z + a2 (t)z 2 + . . . for any t ≥ 0 and fs (D) ⊂ ft (D) whenever t ≥ s ≥ 0. These type of families are called (classical) Loewner chains. One of the keystones of the parametric method is the fact that every such a family is diﬀerentiable in t almost everywhere on [0, +∞) and independently on z. Moreover, they satisfy the following PDE (1.1) ∂ft (z) ∂ft (z) =z p(z, t), ∂t ∂z where the driving term p(z, t) is measurable with respect to t ∈ [0, +∞) for all z ∈ D and holomorphic in z ∈ D with p(0, t) = 1 and Re p(z, t) > 0 almost everywhere on t ∈ [0, +∞). This equation is called the Loewner – Kufarev PDE. For each t ≥ s ≥ 0, the function ϕs,t := ft−1 ◦ fs is clearly a holomorphic univalent self-mapping of D and the whole family (ϕs,t )t≥s≥0 is referred to as the associated evolution family (sometimes transition family or semigroup family) of the Loewner chain. The remarkable fact is that, fixing z ∈ D and s ≥ 0, the functions w(t) = ϕs,t (z) are integrals of the characteristic equation for (1.1) (1.2) dw = −wp(w, t) dt with the initial condition w(s) = z. This equation is called the Loewner – Kufarev ODE and the right member of the equation, the associated vector field. Note that the family (ϕs,t ) is continuous in t ∈ [s, +∞) in the compact-open topology of Hol(D, C) for each s ≥ 0, and satisfies the algebraic conditions (1.3) ϕs,s = idD , s ≥ 0, and ϕs,t = ϕu,t ◦ ϕs,u , 0 ≤ s ≤ u ≤ t < +∞. Another crucial point in the parametric method is that the function f0 can be reconstructed by means of the integrals of (1.2). Namely, lim et ϕ0,t = f0 . t→+∞ LOEWNER CHAINS 3 Equation (1.2) can be considered on its own, without any a priori connection to Loewner chains. However, taking any driving term p(z, t) satisfying the above conditions, this equation has a unique solution w(t) = ϕs,t (z), assuming the initial condition w(s) = z. Then, it is possible to define fs := limt→+∞ et ϕs,t and generate in this way a Loewner chain. Clearly, (ϕs,t ) is an evolution family associated to this chain (ft ). In other words, within the framework of the classical parametric method, there is a one-to-one correspondence between this concept of evolution families, the driving terms (or the vector fields) appearing in Loewner equations and the so-called classical Loewner chains. 1.2. Chordal Loewner equation. In his original work [25], Loewner paid special attention to what we call now Loewner chains of slit mappings of the unit disk D. This Loewner chain (ft ) starts with a conformal mapping onto the complex plane minus a Jordan curve going to infinity and, thereafter, the family is obtained by erasing gradually this curve. In this case, the ODE equation (1.2) assumes the following form (see, e. g., [13, chapter III §2] or [12, chapter 3]) (1.4) dw κ(t) + w = −w , dt κ(t) − w where κ : [0, +∞) → R is a continuous function. The corresponding functions ϕs,t = ft−1 ◦ fs map D onto D with a slit generated by a Jordan curve starting from the boundary (see [11, Chapter 17]). These self-mappings of the unit disk are normalized at the origin: ϕs,t (0) = 0, ϕ�s,t (0) > 0. However, in many applications, one find quite similar examples but where the natural normalization is at a boundary point of the unit disk. In this case, it is possible to consider a real analogue of (1.4), the chordal Loewner equation (see, e. g., [2, chapter IV §7]), which is traditionally written for the upper half-plane U := {z : Im z > 0} instead of the unit disk D because there the associated vector field assumes the simpler form dw 2 = , w(0) = z, (1.5) dt ξ(t) − w where ξ : [0, +∞) → R is a real-valued driving term. In this chordal context, we could also talk again about driving terms, Loewner chains and evolution families. In contrast to the chordal variant, classical Loewner theory is mentioned in the recent literature as the radial case. The above chordal variant has been extended to cover a wider variety of new situations. For instance, the relationship between some kind of what we can name chordal Loewner chains and what deserve be named chordal evolution families has been considered by Goryainov and Ba in [17] and by Bauer in [3]. A recent burst of interest in Loewner theory is due in part to the so-called Schramm – Loewner evolution (SLE, also known as stochastic Loewner evolution), introduced in 2000 by Schramm [31]. SLE is an evolution model similar to Loewner chains (namely, given by 4 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK equation (1.4) or (1.5)) but with the driving term defined via a Brownian motion. In other words, it is a probabilistic version of the previously known radial and chordal Loewner chains. Both radial and chordal cases of SLE have important applications. In fact, they turn out to be very useful tools for the study of conformally invariant scaling limits of some classical statistical 2-dimensional lattice models, see [20 – 24]. Some recent developments concerning the relationship between properties of the driving term and the geometry of solutions to (1.4) and (1.5) can be found in [19, 26, 30]. 1.3. Generalization of classical evolution families. Loewner – Kufarev ODE (1.2) defines a holomorphic evolution in the unit disk. That is, for any initial point z ∈ D and any starting instance s ≥ 0, the solution w = w(t) to the initial value problem w(s) = z for equation (1.2) is unique, exists for all t ≥ s, and the dependence of w(t) on z reveals a holomorphic self-mapping of D. The same is true for the chordal Loewner equation (1.5) and its generalizations when being rewritten for D. Some natural questions arise: are there other examples of ODE with the same property?, what is the most general form of such type of equations?, is it possible to unify these holomorphic evolutions, bearing in mind the many similarities between them? The answer for the autonomous case (the vector field is of the form dw/dt = G(w)) comes from the theory of one-parametric semigroups of holomorphic functions (see the definition in Section 5). They have important applications in the theory of operators acting on spaces of analytic functions (see, e.g., [32, 33]) as well as in the theory of stochastic processes (see, e.g., [15, 16]). Berkson and Porta [4] found the most general form of such a function G, namely G(z) = (τ − z)(1 − τ z)p(z), z ∈ D, where p is a holomorphic function in D with Re p(z) ≥ 0 and τ ∈ D (again see Section 5 for more details). However, in the non-autonomous case and as far as we know, there were no satisfactory answers to the above questions before [5]. Certainly, a large number of examples related to chordal and radial Loewner diﬀerential equations has been treated in the literature but, at the same time, one can also find several (similar but diﬀerent) notions playing the role of Loewner chains, vector fields or, specially, evolution families. For instance, in [14] some classes of holomorphic univalent self-mappings, closed with respect to composition, are considered and evolution families within these classes are defined as two-parametric families (ϕs,t )0≤s≤t continuous with respect to t ∈ [s, +∞) in the open-compact topology of Hol(D, C) for each s ≥ 0 and satisfying the algebraic conditions (1.3). Moreover, in order to describe evolution families by means of diﬀerential equations, an additional condition is also imposed: namely, a certain functional applied to ϕ0,t is required to be (locally) absolutely continuous with respect to t. It is worth comparing this approach with the very classical case, where one can regard the equality ϕ�0,t (0) = e−t as a kind of additional condition ensuring diﬀerentiability in t. LOEWNER CHAINS 5 As we have just partially said, answers for the above questions under very general assumptions follow from results of the recent paper [5] by Bracci and the first two authors of this paper. Taking the whole class of holomorphic self-maps of D, they introduced a general notion of evolution family in the unit disk which includes, as very particular cases, one-parametric semigroups as well as all of those evolution families arising in Loewner theory, both for the radial and chordal variants. Now we cite their definition. Note that the functions ϕs,t are not assumed a priori to be univalent in D. Definition 1.1. A family (ϕs,t )0≤s≤t<+∞ of holomorphic self-maps of the unit disc is an evolution family of order d with d ∈ [1, +∞] (in short, an Ld -evolution family) if EF1. ϕs,s = idD , EF2. ϕs,t = ϕu,t ◦ ϕs,u for all 0 ≤ s ≤ u ≤ t < +∞, EF3. for all z ∈ D and for all T > 0 there exists a non-negative function kz,T ∈ Ld ([0, T ], R) such that � t |ϕs,u (z) − ϕs,t (z)| ≤ kz,T (ξ)dξ u for all 0 ≤ s ≤ u ≤ t ≤ T. One of the main results of [5] is that any evolution family (ϕs,t ) can be obtained via solutions to an ODE of the form dw/dt = G(w, t). Moreover, they characterize all the functions (or, in other words, all the vector fields) G that generate evolution families. Indeed, these vector fields resemble to a non-autonomous (the variable t is present) version of the celebrated Berkson-Porta representation theorem (see Section 2 for further definitions and full statements of these results). Nevertheless, a one-to-one correspondence between evolution families and certain type of vector fields is established in that paper. There, it is also explained how to recover the semigroup, radial and chordal cases in this new framework. Indeed, the three authors were able to formulate a similar theory of generalized evolution families for arbitrary hyperbolic complex manifolds [6]. In [5], the following natural question was left opened implicitly: is there a generalized notion of Loewner chain which can be put in one-to-one correspondence with those generalized evolution families or, equivalently, with those generalized Berkson-Porta vector fields? In the next subsection, we deal with this question presenting our main results about it. 1.4. Main results. As we mentioned in Section 1.1, Loewner – Kufarev equation (1.2) generates a special type of evolution families and there is a one-to-one correspondence between such evolution families and classical Loewner chains. In this paper we consider the analogous question for arbitrary evolution families in the sense of Definition 1.1. First of all, we give a suitable definition of Loewner chain for our general setting. 6 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK Definition 1.2. A family (ft )0≤t<+∞ of holomorphic maps of the unit disc will be called a Loewner chain of order d with d ∈ [1, +∞] (in short, an Ld -Loewner chain) if LC1. each function ft : D → C is univalent, LC2. fs (D) ⊂ ft (D) for all 0 ≤ s < t < +∞, LC3. for any compact set K ⊂ D and all T > 0 there exists a non-negative function kK,T ∈ Ld ([0, T ], R) such that � t |fs (z) − ft (z)| ≤ kK,T (ξ)dξ s for all z ∈ K and all 0 ≤ s ≤ t ≤ T . A Loewner chain (ft ) will be said to be normalized if f0 (0) = 0 and f0� (0) = 1 (notice that we only normalize the function f0 ). Our main results concerning relations between Loewner chains and evolution families are stated in the following three theorems. Theorem 1.3. For any Loewner chain (ft ) of order d ∈ [1, +∞], if we define ϕs,t := ft−1 ◦ fs , 0 ≤ s ≤ t, then (ϕs,t ) is an evolution family of the same order d. Conversely, for any evolution family (ϕs,t ) of order d ∈ [1, +∞], there exists a Loewner chain (ft ) of the same order d such that the following equation holds (1.6) ft ◦ ϕs,t = fs , 0 ≤ s ≤ t. Definition 1.4. A Loewner chain (ft ) is said to be associated with an evolution family (ϕs,t ) if it satisfies (1.6). Remark 1.5. We will actually prove (see Lemma 3.2) that any Loewner chain (ft ) associated with an evolution family (ϕs,t ) of order d ∈ [1, +∞] must be of the same order d. In general, fixed the evolution family (ϕs,t ), the algebraic equation (1.6) does not defined a unique Loewner chain. In fact, in some case, a plenty of diﬀerent Loewner chains are associated with the same evolution family. The following theorem gives necessary and suﬃcient conditions for the uniqueness for a normalized Loewner chain associated with a given evolution family. Theorem 1.6. Let (ϕs,t ) be an evolution family. Then there exists a unique normalized Loewner chain (ft ) associated with (ϕs,t ) such that ∪t≥0 ft (D) is either an Euclidean disk or the whole complex plane C. Moreover, the following statements are equivalent: (i) the family (ft ) is the only normalized Loewner chain associated with the evolution family (ϕs,t ); (ii) for all z ∈ D, |ϕ�0,t (z)| = 0; β(z) := lim t→+∞ 1 − |ϕ0,t (z)|2 LOEWNER CHAINS 7 (iii) there � exist at least one point z ∈ D such that β(z) = 0; (iv) ft (D) = C. t≥0 The Loewner chain (ft ) in the above theorem will be called the standard Loewner chain associated with the evolution family (ϕs,t ). In case of non-uniqueness (when conditions (i) – (iv) in Theorem 1.6 fail to be satisfied), we provide an explicit formula expressing all the associated normalized Loewner chains by means of the standard Loewner chain plus some Riemman map. In some sense, this formula tell us that the evolution procedures described by our Loewner chains are essentially unique up to a choice of the simply connected domain they are located in. Denote by S the class of all univalent holomorphic functions h in the unit disk D, normalized by h(0) = h� (0) − 1 = 0. Theorem 1.7. Suppose that under conditions of Theorem 1.6, � Ω := ft (D) �= C. t≥0 Then Ω = {z : |z| < 1/β(0)} and the set L[(ϕs,t )] of all normalized Loewner chains (gt ) associated with the evolution family (ϕs,t ), is given by the formula � � � � L[(ϕs,t )] = (gt )t≥0 : gt (z) = h β(0)ft (z) /β(0), h ∈ S . In Section 2 we state some results from [5] along with the necessary definitions. Moreover, we prove new statements concerning evolution families (see Definition 1.1), which we later use to obtain the main results of the paper. In Section 3 we reformulate and prove the theorems stated above. Namely, Theorem 1.3 follows from Theorems 3.1 and 3.3, while Theorems 1.6 and 1.7 follow from Theorem 3.6 and Proposition 3.4. Besides that, and in some cases, we establish a necessary and suﬃcient condition (Theorem 3.8) for a uniparametric family (ft )t≥0 of holomorphic (but not a priori univalent) maps defined in D to be a normalized Loewner chain associated with a given evolution family. In Section 4 we find an analogue (Theorem 4.1) of the Loewner – Kufarev PDE in this abstract context. We also show that there is a one-to-one correspondence between our concept of generalized Loewner chain and the generalized Berkson-Porta vector fields shown in [5]. In Section 5 we consider the special case of evolution families induced by semigroups of holomorphic functions in D. In particular, we show that the uniqueness of the Kœnigs function is a consequence of Theorems 1.3 and 1.6. 2. Evolution families and Herglotz vector fields in the unit disk Here we collect some known and new statements on evolution families (see Definition 1.1). 8 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK Let us first of all note that by [5, Corollary 6.3], given an evolution family (ϕs,t ), every function ϕs,t is univalent. The following statement turns out to be also quite useful. Lemma 2.1. [5, Lemma 3.6] Let (ϕs,t ) be an evolution family in the unit disc D of order d ∈ [1, +∞]. Then for each 0 < T < +∞ and 0 < r < 1, there exists R = R(r, T ) < 1 such that |ϕs,t (z)| ≤ R for all 0 ≤ s ≤ t ≤ T and |z| ≤ r. Any evolution family (ϕs,t ) is diﬀerentiable almost everywhere with respect to t. Besides the proof of this fact, a characterization of all vector fields generating evolution families in the disk is established in [5]. In order to give a strict statement of this result we need the following Definition 2.2. Let d ∈ [1, +∞]. A weak holomorphic vector field of order d in the unit disc D is a function G : D × [0, +∞) → C with the following properties: WHVF1. For all z ∈ D, the function [0, +∞) � t �→ G(z, t) is measurable; WHVF2. For all t ∈ [0, +∞), the function D � z �→ G(z, t) is holomorphic; WHVF3. For any compact set K ⊂ D and all T > 0 there exists a non-negative function kK,T ∈ Ld ([0, T ], R) such that |G(z, t)| ≤ kK,T (t) for all z ∈ K and for almost every t ∈ [0, T ]. Moreover, we say that G is a (generalized) Herglotz vector field (of order d) if for almost every t ∈ [0, +∞) it follows G(·, t) is the infinitesimal generator of a semigroup of holomorphic functions (see Section 5 for further details about semigroups of analytic functions and their infinitesimal generators). Theorem 2.3. [5, Theorems 6.2, 4.8] For any evolution family (ϕs,t ) of order d ∈ [1, +∞] there exists a (essentially) unique Herglotz vector field G(z, t) of order d such that for all z ∈ D, ∂ϕs,t (z) (2.1) = G(ϕs,t (z), t), a.e. t ∈ [0, +∞). ∂t Conversely, for any Herglotz vector field G(z, t) of order d ∈ [1, +∞] there exists a unique evolution family (ϕs,t ) of order d such that (2.1) is satisfied. Here by essential uniqueness we mean that two Herglotz vector fields G1 (z, t) and G2 (z, t) corresponding to the same evolution family must coincide for a.e. t ≥ 0. Herglotz vector fields can be further characterized in similar terms of the Berkson – Porta representation of infinitesimal generators. Definition 2.4. Let d ∈ [1, +∞]. A Herglotz function of order d is a function p : D × [0, +∞) → C with the following properties: LOEWNER CHAINS 9 HF1. For all z ∈ D, the function [0, +∞) � t �→ p(z, t) ∈ C belongs to Ldloc ([0, +∞), C); HF2. For all t ∈ [0, +∞), the function D � z �→ p(z, t) ∈ C is holomorphic; HF3. For all z ∈ D and for all t ∈ [0, +∞), we have Re p(z, t) ≥ 0. Theorem 2.5. [5, Theorems 1.2] Let G(z, t) be a Herglotz vector field of order d ∈ [1, +∞] in the unit disc. Then there exist a (essentially) unique measurable function τ : [0, +∞) → D and a Herglotz function p(z, t) of order d such that for all z ∈ D (2.2) G(z, t) = (z − τ (t))(τ (t)z − 1)p(z, t), a.e. t ∈ [0, +∞). Conversely, given a measurable function τ : [0, +∞) → D and a Herglotz function p(z, t) of order d ∈ [1, +∞], equation (2.2) defines a Herglotz vector field of order d. There is thus an (essentially) one-to-one correspondence between evolution families (ϕs,t ) of order d ∈ [1, +∞], Herglotz vector fields G(z, t) of order d, and couples (p, τ ) of Herglotz functions p(z, t) of order d and measurable functions τ : [0, +∞) → D. In what follows we say that the couple (p, τ ) is the Berkson – Porta data for (ϕs,t ). Now we state and prove some new assertions concerning evolution families, which we use in the proof of the main results. Denote by AC d (X, Y ), X ⊂ R, d ∈ [1, +∞], the class of all locally absolutely continuous functions f : X → Y such that the derivative f � belongs to Ldloc (X). Proposition 2.6. Let (ϕs,t ) be an evolution family of order d ∈ [1, +∞]. Then the following statements hold: (1) For any compact set K ⊂ D and all T > 0 there exists a non-negative function kK,T ∈ Ld ([0, T ], R) such that � t |ϕs,u (z) − ϕs,t (z)| ≤ kK,T (ξ)dξ u for all 0 ≤ s ≤ u ≤ t ≤ T and all z ∈ K. (2) For every z ∈ D the maps a(t) := ϕ0,t (z) and b(t) := ϕ�0,t (z) belong to AC d ([0, +∞), C) and b(t) �= 0 for all t ∈ [0, +∞). Proof. By Theorem 2.3, there is a Herglotz vector field of order d such that for all z ∈ D ∂ϕs,t (z) = G(ϕs,t (z), t), a.e. t ∈ [0, +∞). ∂t Proof of (1). By the very definition of Herglotz vector field there exists a non-negative function kK,T ∈ Ld ([0, T ], R) such that |G(z, t)| ≤ kK,T (t) (2.3) for all z ∈ K and for almost every t ∈ [0, T ]. Therefore, statement (1) is an easily consequence of the following inequalities �� t � �� t � � t � � � � ∂ϕ (z) s,ξ |ϕs,u (z) − ϕs,t (z)| = �� dξ �� = �� G(ϕs,ξ (z), ξ)dξ �� ≤ kK,T (ξ)dξ. ∂ξ u u u 10 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK Proof of (2). From the very definition of Herglotz vector field, evolution family of order d, and inequality (2.3) it follows that the map a belongs to AC d ([0, +∞), C). Moreover, since the functions ϕs,t are univalent [5, Corollary 6.3], we have b(t) �= 0 for all t. Fix T ∈ (0, +∞) and z ∈ D. There is R < 1 such that |ϕ0,t (z)| < R for all t ∈ [0, T ]. Then there is kR,T ∈ Ld ([0, T ], R) such that |G(w, t)| ≤ kR,T (t) for all |w| ≤ R and for almost every t ∈ [0, T ]. Therefore, � �� �� � � � � � �1 ∂ � �1 � ϕ (w) ∂ ϕ (w) 0,t 0,t � �=� � |b (t)| = �� dw dw � � 2π � 2π ∂t w2 w2 C(0,R)+ C(0,R)+ ∂t � �� �� �1 � 1 = �� G(ϕ0,t (w), t)dw �� ≤ kR,T (t) 2π R C(0,R)+ for almost every t ∈ [0, T ], where C(0, R)+ stands for the positively oriented circle of radius R centered at the point z = 0. This implies that b belongs to AC d ([0, +∞), C) and therefore completes the proof. � It appears to be useful to consider evolution families that consists of automorphisms of D. The following example is the most general form of such evolution families. Example 2.7. Take two functions a ∈ AC d ([0, +∞), D) and b ∈ AC d ([0, +∞), ∂D) and write b(t)z + a(t) ht (z) := for all t ≥ 0 and all z ∈ D. 1 + b(t)a(t)z −1 Then (ht ◦ h−1 s ) and (ht ◦ hs ) are evolution families of order d. Indeed, it is clear that both families of functions satisfy EF1 and EF2. Moreover, for any T < +∞ and z ∈ D there exists R < 1 such that � � � � z − a(s) � � (z)| = b(s) |h−1 � � ≤ R, 0 ≤ s ≤ T. s � 1 − a(s)z � Denote w = h−1 s (z). Then we have � � � b(t)w + a(t) b(u)w + a(u) �� � −1 −1 |ht ◦ hs (z) − hu ◦ hs (z)| = |ht (w) − hu (w)| = � − � � 1 + b(t)a(t)w 1 + b(u)a(u)w � 2 ≤ (|b(t) − b(u)| + |a(t) − a(u)|) (1 − R)2 for all 0 ≤ s ≤ u ≤ t ≤ T . These inequalities and the hypothesis on a and b imply that −1 the family (ht ◦ h−1 s ) satisfies EF3. Similarly, the family (ht ◦ hs ) satisfies EF3 as well. The following lemma allows us to transform evolution families by means of timedependent changes of variable in the unit disk. LOEWNER CHAINS 11 Lemma 2.8. Let (ψs,t ) be an evolution family or order d ∈ [1, +∞] and take two functions a ∈ AC d ([0, +∞), D) and b ∈ AC d ([0, +∞), ∂D). Write ϕs,t = ht ◦ ψs,t ◦ h−1 s and ϕ̃s,t = h−1 ◦ ψ ◦ h , where s,t s t ht (z) := b(t)z + a(t) 1 + b(t)a(t)z for all t ≥ 0 and all z ∈ D. Then (ϕs,t ) and (ϕ̃s,t ) are evolution families of order d. Proof. We present the proof for the family (ϕs,t ) and leave to the reader the one for the family (ϕ̃s,t ) which is quite similar. It is clear that the functions (ϕs,t ) satisfy properties EF1 and EF2. So we just have to prove that this family of functions satisfy EF3. Notice that, by Example 2.7, (ht ◦h−1 s ) is an evolution family. Fix z ∈ D and T ∈ (0, ∞). By Lemma 2.1 and the continuity of the functions a and b, there exists a number R < 1 such that −1 |ψs,t ◦ h−1 s (z)| ≤ R and |ϕs,t (z)| = |ht ◦ ψs,t ◦ hs (z)| ≤ R for all 0 ≤ s ≤ t ≤ T . Therefore, by Proposition 2.6 applied to the evolution families d (ht ◦ h−1 s ) and (ψs,t ), there are two functions k1 , k2 ∈ L ([0, T ], R) such that � t � t −1 −1 (2.4) |ψs,u (w) − ψs,t (w)| ≤ k1 (ξ)dξ and |hu ◦ hs (w) − ht ◦ hs (w)| ≤ k2 (ξ)dξ u u for all 0 ≤ s ≤ u ≤ t ≤ T and whenever |w| ≤ R. Moreover, there is a positive number M such that |ht (w1 ) − ht (w2 )| ≤ M |w1 − w2 | (2.5) whenever t ∈ [0, T ] and |w1 |, |w2 | ≤ R. Now, let us fix 0 ≤ s ≤ u ≤ t ≤ T and write z1 = ψs,u (h−1 s (z)) and z2 = hu (z1 ). Note that |z1 |, |z2 | ≤ R. The following chain of inequalities (where we use (2.4) and (2.5)) allows us to complete the proof |ϕs,t (z) − ϕs,u (z)| = ≤ ≤ = |ϕu,t (ϕs,u (z)) − ϕs,u (z)| = |ht ◦ ψu,t (z1 ) − hu (z1 )| |ht ◦ ψu,t (z1 ) − ht (z1 )| + |ht (z1 ) − hu (z1 )| M |ψu,t (z1 ) − z1 | + |ht (z1 ) − hu (z1 )| −1 M |ψu,t (z1 ) − ψu,u (z1 )| + |ht ◦ h−1 u (z2 ) − hu ◦ hu (z2 )| � t ≤ (M k1 (ξ) + k2 (ξ))dξ. u � Now we use Lemma 2.8 in order to establish a kind of decomposition for a given evolution family. 12 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK Proposition 2.9. Let (ϕs,t ) be an evolution family of order d ∈ [1, +∞]. Then there exist unique a ∈ AC d ([0, +∞), D), b ∈ AC d ([0, +∞), ∂D), and ψs,t : D → D, 0 ≤ s ≤ t < +∞, such that the following assertions hold (1) a(0) = 0, b(0) = 1, � (2) (ψs,t ) is an evolution family of order d such that ψs,t (0) = 0 and ψs,t (0) > 0 for all 0 ≤ s ≤ t, (3) ϕs,t = ht ◦ ψs,t ◦ h−1 s for all 0 ≤ s ≤ t < +∞, where ht (z) := b(t)z + a(t) 1 + b(t)a(t)z Proof. Write a(t) = ϕ0,t (0) and b(t) = ϕ�0,t (0) . |ϕ�0,t (0)| , t ≥ 0, z ∈ D. By Proposition 2.6, a ∈ AC d ([0, +∞), D) and b ∈ AC d ([0, +∞), ∂D). Now define ht as in the statement of the proposition and take � 2 ψs,t = h−1 t ◦ϕs,t ◦hs . Notice that h0 is the identity, ht (0) = a(t) and ht (0) = b(t)(1−|a(t)| ). By Lemma 2.8, the family (ψs,t ) is an evolution family of order d. Moreover, from the very definition of a it follows that ψ0,t (0) = 0 for all t. Using EF2, we deduce that ψs,t (0) = 0 |ϕ�0,t (0)| � for all s ≤ t. In a similar way, we show that ψ0,t (0) = 1−|a(t)| 2 > 0 for all t and then � ψs,t (0) > 0 for all 0 ≤ s ≤ t. The uniqueness is clear because from the equality ϕs,t = ht ◦ ψs,t ◦ h−1 we deduce s ϕ�0,t (0) that a(t) = ht (0) = ht (ψ0,t (0)) = ϕ0,t (0), b(t) = |ϕ� (0)| (which defines the functions ht 0,t uniquely) and ψs,t = h−1 t ◦ ϕs,t ◦ hs . The proof is now complete. � The following result gives the converse of Proposition 2.6(2). Proposition 2.10. Let (ϕs,t ) be a family of holomorphic self-maps of D. Suppose that conditions EF1 and EF2 are fulfilled. Then condition EF3 is equivalent to the following condition: EF4. The maps a(t) := ϕ0,t (0) and b(t) := ϕ�0,t (0) belong to AC d ([0, +∞), C) and b(t) �= 0 for all t ∈ [0, +∞). Proof. By Proposition 2.6 any evolution family satisfies EF4. Let (ϕs,t ) be a family of holomorphic self-maps of the unit disk satisfying EF1, EF2, and EF4. Write b0 (t)z + a(t) ht (z) := for all t ≥ 0 and all z ∈ D, 1 + b0 (t)a(t)z where b0 (t) = b(t)/|b(t)|. Define ψs,t = h−1 t ◦ ϕs,t ◦ hs for all 0 ≤ s ≤ t < +∞. It is clear � that the family (ψs,t ) satisfies EF1, EF2, ψs,t (0) = 0, and ψ0,t (0) = |b(t)|/(1 − |a(t)|2 ) for all 0 ≤ s ≤ t. Using [5, Theorem 7.3] with τ = z0 = 0 in that statement, we deduce that (ψs,t ) is an evolution family of order d. Finally, we just have to apply Lemma 2.8 to deduce that (ϕs,t ) is also an evolution family of order d. � LOEWNER CHAINS 13 3. Loewner chains and evolution families In this section we reformulate and prove our main results connecting evolution families with Loewner chains in a way similar to the one given in classical Loewner theory. First of all we prove that any Loewner chain of order d ∈ [0, +∞] generates an evolution family of the same order. Theorem 3.1. Let (ft ) be a Loewner chain of order d ∈ [1, +∞]. Set ϕs,t (z) := ft−1 (fs (z)), z ∈ D, 0 ≤ s ≤ t. Then (ϕs,t ) is a well-defined evolution family of order d in the unit disk and (trivially) satisfies the equality ft (ϕs,t (z)) = fs (z), z ∈ D, 0 ≤ s ≤ t. Proof. The proof of this theorem is quite long so we have divided it into several steps of independent interest on their own. In what follows, Ωt := ft (D), t ≥ 0. We also comment that ins Γ will denote the interior of a Jordan curve Γ and N (g, Γ) stands for the number of zeros (counting multiplicity), inside a rectifiable Jordan curve Γ contained in D, of a holomorphic map g defined in the whole unit disk. Finally, by ind(Γ, ξ) we denote the index of a closed rectifiable curve Γ with respect to a point ξ, and D(ξ, r) := {z ∈ C : |z−ξ| < r}. [Step 1] For every t ≥ 0 and every ω ∈ Ωt , there exist ε > 0, δ > 0 and a rectifiable Jordan curve γ with γ ∪ ins γ ⊂ D such that the following “locally uniform formula for the inverses” holds: � 1 ξfu� (ξ) −1 fu (w) = dξ, 2πi γ fu (ξ) − w whenever u ∈ [t − δ, t + δ] ∩ [0, +∞) and w ∈ D(ω, ε). � � Fix t ≥ 0 and ω ∈ Ωt . Denote z0 := ft−1 (ω) ∈ D and choose any r ∈ |z0 |, 1 and R ∈ (r, 1). Consider the complex domain Dt := ft (D(0, r)) ⊂ Ωt and define γ as the positively oriented circle of radius R centered at the origin. Since ft is univalent, it follows from the Argument Principle that for each w ∈ Dt , � 1 ft� (ξ) dξ = N (ft − w, γ) = 1. 2πi γ ft (ξ) − w Note that inf{|w − ft (z)| : w ∈ Dt , |z| = R} > 0, because r < R and ft is continuous and univalent in D. Moreover, by property LC3, we know that fs → ft uniformly on D(0, R) as s → t. This implies the existence of a number δ0 > 0 such that inf{|w − fu (z)| : w ∈ Dt , |z| = R} > 0, 14 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK for all non-negative u ∈ [t − δ0 , t + δ0 ]. In particular, this allows to consider, for every w ∈ Dt and every non-negative u ∈ [t − δ0 , t + δ0 ], the Argument Principle formula � 1 fu� (ξ) dξ = N (fu − w, γ). 2πi γ fu (ξ) − w Again, using property LC3 and the Weierstrass Theorem, we conclude that lim sup {|N (fu − w, γ) − 1| : w ∈ Dt } = 0. u→t But N (fu − w, γ) can take only integer values, so there exists δ1 ∈ (0, δ0 ) such that sup {|N (fu − w, γ) − 1| : w ∈ Dt } = 0, whenever u ∈ [t − δ1 , t + δ1 ] ∩ [0, +∞). In other words, we have showed that N (fu − w, γ) = 1, when u ∈ [t − δ1 , t + δ1 ] ∩ [0, +∞) and w ∈ Dt . At this point, we fix u ∈ [t − δ1 , t + δ1 ] and w ∈ Dt . Our idea is to apply now the generalized Argument Principle for the couple (id, fu − w) and the rectifiable closed curve γ (see, e. g., [10, p. 124, chapter V, Theorem 3.6]). Namely, recalling that fu −w is analytic in the unit disk with a unique zero (denoted by fu−1 (w)) which is contained in ins γ, we deduce that � 1 fu� (ξ) id(ξ) dξ = id(fu−1 (w))N (fu − w, γ) = fu−1 (w). 2πi γ fu (ξ) − w In order to finish the proof of Step 1 it is enough to define ε as the distance between ω and the boundary of Dt , which is positive since Dt is open and ω ∈ Dt by construction. [Step 2] For any r ∈ (0, 1) and any T > 0, we have that � � sup |(ft−1 ◦ fs )(z)| : 0 ≤ s ≤ t ≤ T, |z| ≤ r < 1. Fix r ∈ (0, 1) and T > 0 and suppose that the above supremum is 1. Then, there exist sequences (sn ), (tn ) and (zn ) such that: (a) for all n ∈ N, 0 ≤ sn ≤ tn ≤ T, |zn | ≤ r, (b) the following limits exist s := limn sn , t := limn tn , z0 := limn zn , β := limn (ft−1 ◦ fsn )(zn ), and n (c) 0 ≤ s ≤ t ≤ T, |z0 | ≤ r, β ∈ ∂D. We note that fs (z0 ) ∈ Ωs ⊂ Ωt and limn fsn (zn ) = fs (z0 ). Therefore, by [Step 1], there exist ε > 0, δ > 0 and a Jordan curve γ with γ ∪ ins γ ⊂ D such that � 1 ξfu� (ξ) −1 dξ, fu (w) = 2πi γ fu (ξ) − w LOEWNER CHAINS 15 whenever u ∈ [t − δ, t + δ] ∩ [0, +∞) and w ∈ D(fs (z0 ), ε). In particular, for n large enough, we have that � ξft�n (ξ) 1 −1 ftn (fsn (zn )) = dξ. 2πi γ ftn (ξ) − fsn (zn ) Clearly, by property LC3 and the above formula � � ξft�n (ξ) 1 1 ξft� (ξ) −1 ftn (fsn (zn )) = dξ −→ dξ = ft−1 (fs (z0 )) 2πi γ ftn (ξ) − fsn (zn ) 2πi γ ft (ξ) − fs (z0 ) as n → +∞. Since ft−1 (fs (z0 )) ∈ D, we obtain a contradiction, which finishes the proof of Step 2. [Step 3] Let γ : [a, b] → C be a rectifiable curve in D and T > 0. Then, for all t ∈ [0, T ], the curve γt : [a, b] → C, ξ �→ ft (γ(ξ)) ∈ Ωt is a well-defined rectifiable curve in Ωt . Moreover, sup{len(γt ) : t ∈ [0, T ]} < +∞, where, as usual, len(γt ) denotes the length of γt . The fact that γt is a well-defined rectifiable curve is widely known. So, suppose that the above supremum is +∞. In this case, there exists a sequence (tn ) in the interval [0, T ] such that limn tn = t ∈ [0, T ] and limn len(γtn ) = +∞. However, the well-known estimate len(γtn ) ≤ len(γ) max{|ft�n (ξ)| : ξ ∈ γ}, shows (recall that γ is a compact set) that there exists a subsequence (znk ) in the curve γ converging to some z0 ∈ γ such that limk ft�n (znk ) = ∞. However, by property LC3 k and Weierstrass’ Theorem, we deduce limk ft�n (znk ) = ft� (z0 ), obtaining in this way a k contradiction. [Step 4] In this step we will finally prove the theorem. By properties LC1 and LC2, we see that the functions ϕs,t (z) := ft−1 (fs (z)), z ∈ D, 0 ≤ s ≤ t are well-defined and, indeed, ϕs,t ∈ Hol(D, D), for any 0 ≤ s ≤ t. Hence, (ϕs,t ) will be an evolution family of order d if we are able to prove properties EF1, EF2, and EF3. The first two properties follow easily from the way we have defined the family (ϕs,t ). The third property is more diﬃcult to prove. We fix z ∈ D and T > 0. By [Step 2], there exists R1 := R1 (z, T ) ∈ (0, 1) such that sup{|ϕa,b (z)| : 0 ≤ a ≤ b ≤ T } ≤ R1 . 16 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK Applying again [Step 2], we obtain another R2 := R2 (z, T ) ∈ (0, 1) such that R2 > R1 sup{|ϕa,b (z)| : 0 ≤ a ≤ b ≤ T, |ξ| ≤ R1 } < R2 . Additionally, we denote by γ the positively oriented circle of radius R2 centered at the origin. As in [Step 3], we also consider the rectifiable curves γt := ft ◦ γ, which are Jordan curves due to the univalence of ft . Now, assume that 0 ≤ s ≤ u ≤ t ≤ T. Then, using property EF2, we obtain |ϕs,u (z) − ϕs,t (z)| = |ϕs,u (z) − ϕu,t (ϕs,u (z))| ≤ sup{|ϕu,t (ξ) − ξ| : |ξ| ≤ R1 }. But, for any |ξ| ≤ R1 , we have that |ft−1 (fu (ξ))| < R2 , so fu (ξ) ∈ ft (ins γ). Applying [28, Lemma 1.1], we see that fu (ξ) ∈ ins γt . The same argument shows that ft (ξ) ∈ ins γt . Therefore, using the Cauchy Integral Formula, for all |ξ| ≤ R1 we get � −1 � � � �ft (fu (ξ)) − ξ � = �ft−1 (fu (ξ)) − ft−1 (ft (ξ))� � � � � −1 −1 � ind(γt , fu (ξ)) � f (η) ind(γ , f (ξ)) f (η) t t t t = �� dη − dη �� 2πi 2πi γt η − fu (ξ) γt η − ft (ξ) �� � � � 1 ft−1 (η) ≤ |fu (ξ) − ft (ξ)| �� dη �� . 2π γt (η − fu (ξ))(η − ft (ξ)) We claim that d = d(z, T ) := inf{|ft (a) − fu (b)| : 0 ≤ u ≤ t ≤ T, |a| = R2 , |b| ≤ R1 } > 0. Therefore, recalling that ft−1 (Ωt ) ⊂ D and using the above estimation, we have � −1 � �ft (fu (ξ)) − ξ � ≤ 1 |fu (ξ) − ft (ξ)| 1 len(γt ). 2π d2 Now, by [Step 3], there exists C = C(z, T ) > 0 such that so sup{len(γt ) : t ∈ [0, T ]} ≤ C, C sup{|fu (ξ) − ft (ξ)| : |ξ| ≤ R1 }. 2πd2 Finally, by property LC3 with K := D(0, R1 ), there exists a non-negative function kz,T ∈ Ld ([0, T ]; R) such that � t C |ϕs,u (z) − ϕs,t (z)| ≤ k(η)dη. 2πd2 u Now it remains to prove that d > 0. Suppose on the contrary that d = 0. Then, there exist sequences (an ), (bn ), (un ) and (tn ) such that: (a) for all n ∈ N, 0 ≤ un ≤ tn ≤ T, |an | = R2 , |bn | ≤ R1 , (b) there exist the following limits u := limn un , t := limn tn , a := limn an , b := limn bn , (c) 0 ≤ u ≤ t ≤ T, |a| = R2 , |b| ≤ R1 , and (d) ftn (an ) − fun (bn ) → 0 as n → +∞. |ϕs,u (z) − ϕs,t (z)| ≤ LOEWNER CHAINS 17 By property LC3, we know that (fun ) and (ftn ) tends to fu and ft , respectively, in the compact-open topology of Hol(D, C). Therefore, by (b) and (d), we conclude that fu (b) = ft (a). However, using (c) from the definition of the Jordan curves γ and γt it is clear that a ∈ γ and ft (a) ∈ ft ◦ γ = γt . At the same time, |b| ≤ R1 . So by the choice of R2 we find that |ft−1 (fu (b))| < R2 . Thus, fu (b) ∈ ft (ins γ) = ins γt by [28, Lemma 1.1]. Obviously γt ∩ ins γt = ∅, so we have a contradiction, which finishes the proof. � The following lemma shows that if an evolution family has order d ∈ [1, +∞], then any Loewner chain associated with it is also of order d. From another point of view, the next lemma shows that the algebraic equation (1.6) implies indirectly conditions LC2 and LC3. Lemma 3.2. Let (ϕs,t ) be an evolution family of order d ∈ [1, +∞]. Assume that for all t ≥ 0 the function ft : D → C is univalent and ft ◦ ϕs,t = fs , 0 ≤ s ≤ t < +∞. Then the family (ft ) is a Loewner chain of order d. Proof. Let K be a compact subset of D and T > 0. By Lemma 2.1, there exists R1 ∈ (0, 1) such that |ϕs,t (z)| ≤ R1 for all z ∈ K whenever 0 ≤ s ≤ t ≤ T . Write R2 = (1 + R)/2. Again by Lemma 2.1, there exists R3 ∈ (0, 1) such that |ϕs,t (z)| ≤ R3 for all |z| = R2 and all 0 ≤ s ≤ t ≤ T . Since the function fT is continuous, there is a positive constant M such that |ft (ξ)| = |fT (ϕt,T (ξ))| ≤ M for all t ≤ T and any complex number ξ with |ξ| = R2 . Fix z ∈ K and 0 ≤ s ≤ t ≤ T . We have fs (z) − ft (z) = ft (ϕs,t (z)) − ft (z) � � � 1 ft (ξ) ft (ξ) = − dξ 2πi C(0,R2 )+ ξ − ϕs,t (z) ξ − z � � � 1 ft (ξ)(ϕs,t (z) − z) = dξ 2πi C(0,R2 )+ (ξ − ϕs,t (z))(ξ − z) � � � ϕs,t (z) − z ft (ξ) = dξ. 2πi (ξ − ϕs,t (z))(ξ − z) C(0,R2 )+ Therefore, � � � � � � ϕs,t (z) − z � ft (ξ) � |fs (z) − ft (z)| = � dξ �� 2πi (ξ − ϕs,t (z))(ξ − z) C(0,R2 )+ M ≤ R2 |ϕs,t (z) − z| (R2 − R1 )2 18 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK for all z ∈ K and 0 ≤ s ≤ t ≤ T . Now the conclusion of the lemma easily follows from the last inequality. � Now we prove the existence of a Loewner chain associated with a given evolution family. Theorem 3.3. Let (ϕs,t ) be an evolution family of order d ∈ [1, +∞]. Then there exists a normalized Loewner chain (ft ) of order d associated with the evolution family (ϕs,t ) such that the set Ω := ∪t≥0 ft (D) coincides with the disk {z : |z| < 1/β} if β > 0 and with the |ϕ� (0)| whole complex plane C if β = 0, where β = limt→+∞ 1−|ϕ0,t0,t (0)|2 . Proof. By Proposition 2.9 we have ϕs,t = ht ◦ ψs,t ◦ h−1 s , where (ψs,t ) is an evolution family � such that ψs,t (0) = 0 and ψs,t (0) > 0 for all t ≥ s ≥ 0, and ht is a conformal automorphism of D for each t ≥ 0, with h0 being the identity map. Now we build the Loewner chain for the evolution family (ψs,t ) and then a simple argument will allow us to finish the proof. By Theorems 2.3 and 2.5, there exist a measurable function τ : [0, +∞) → D and a Herglotz function p(z, t) of order d such that for all z ∈ D and all s ≥ 0, ∂ψs,t (z) = (ψs,t (z) − τ (t))(τ (t)ψs,t (z) − 1)p(ψs,t (z), t) a.e. t ∈ [0, +∞). ∂t � Since ψs,t (0) = 0, ψs,t (0) > 0, t ≥ s ≥ 0, we conclude that τ (t) ≡ 0. In this case, one can rewrite equation (3.1) in the form (3.1) ∂ψs,t (z) = −ψs,t (z)p(ψs,t (z), t). ∂t We will show that the functions ψs,t (z) (3.3) gs (z) := lim � , t→+∞ ψ0,t (0) (3.2) where the limit is attained uniformly on compact subsets of the unit disk, form a Loewner chain associated with (ψs,t ). Our proof of the existence of that limit follows the approach given in [28, Chapter 6]. However, for the sake of clearness and completeness, we include the details. ψ � (0) Assume for a moment that such a limit does exist. Then gs� (0) := limt→+∞ ψs,t � (0) = 0,t 1 > 0. Moreover, since all the functions ψs,t are univalent [5, Corollary 6.3], we conclude that the function gs is univalent for all s ≥ 0. Moreover, by construction � (0) ψ0,s ψt,u (ψs,t (z)) ψs,u (z) = lim � = gs (z), � u→+∞ u→+∞ ψ0,u (0) ψ0,u (0) gt ◦ ψs,t (z) = lim 0 ≤ s ≤ t < +∞. Therefore, by Lemma 3.2, the family (gt ) is a Loewner chain of order d associated with (ψs,t ). Also, it is clear that it is a normalized Loewner chain. Therefore, we have only to prove the existence of (3.3). LOEWNER CHAINS 19 By [5, Proof of Theorem 7.1], for all z ∈ D and t > s ≥ 0, � � t � (3.4) ψs,t (z) = z exp − p(ψs,ξ (z), ξ)dξ . s Write Λs,t (z) := �t s (p(0, ξ) − p(ψs,ξ (z), ξ)) dξ. Notice that � � t � � ψs,t (0) = exp − p(0, ξ)dξ > 0. s Therefore, (3.5) ψs,t (z) = z exp � ψ0,t (0) �� 0 s � p(0, ξ)dξ exp (Λs,t (z)) . Now in order to prove the existence of the limit (3.3), it is suﬃcient to show that Λs,t has a limit as t → +∞ which is attained uniformly on compact subsets of the unit disk. � � � � By property EF2, we have that ψ0,t (0) = ψs,t (0)ψ0,s (0) ≤ ψ0,s (0), because ψs,t (0) = 0 and ψs,t (D) ⊆ D. That is � ∂ψ0,t (0) ≤ 0, ∂t Since a.e. t ∈ [0, +∞). � � t � � ∂ψ0,t (0) � = −p(0, t) exp − p(0, ξ)dξ = −p(0, t)ψs,t (0), a.e. t ∈ [0, +∞), ∂t s we conclude that p(0, t) ≥ 0 for a.e. t ∈ [0, ∞). When Re p(·, ξ) > 0 (otherwise, p(·, ξ) is constant), necessarily p(0, ξ) > 0 and the holomorphic map z �→ p(z,ξ)−p(0,ξ) sends the unit disc into itself and fixes the origin. Then p(z,ξ)+p(0,ξ) � � � � |p(z, ξ) − p(0, ξ)| ≤ |z| �p(z, ξ) + p(0, ξ)� ≤ |z||p(z, ξ)| + |z||p(0, ξ)| ≤ |z| 1 + |z| 2 |z| |p(0, ξ)| + |z||p(0, ξ)| = p(0, ξ), 1 − |z| 1 − |z| where we have used [28, pages 39-40]. Therefore, by [28, Theorem 1.6], we have 2 |ψs,ξ (z)| 2 |ψs,ξ (z)| p(0, ξ) ≤ p(0, ξ) 1 − |ψs,ξ (z)| 1 − |z| � � � � � � ξ 2 exp − p(0, u)du � � 2 ψs,ξ (0) s ≤ p(0, ξ) = p(0, ξ). (1 − |z|)3 (1 − |z|)3 |p(ψs,ξ (z), ξ) − p(0, ξ)| ≤ 20 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK Now, we can bound the function Λs,· (z): � t |Λs,t (z) − Λs,u (z)| ≤ |p(0, ξ) − p(ψs,ξ (z), ξ)| dξ u � � ξ � � t 2 ≤ exp − p(0, u)du p(0, ξ)dξ (1 − |z|)3 u 0 � � � ξ �� � t 2 ∂ = − exp − p(0, u)du dξ (1 − |z|)3 u ∂ξ 0 � � � u � � � t �� 2 = exp − p(0, ξ)dξ − exp − p(0, ξ)dξ . (1 − |z|)3 0 0 Finally, from these last inequalities and the fact that � � t � lim exp − p(0, ξ)dξ ∈ [0, 1] t→+∞ 0 (recall that p(0, ξ) ≥ 0 for a.e. ξ ∈ [0, +∞)), we conclude that the limit (3.3) does exist. Now we consider the family ft = gt ◦ h−1 t . It easy to see that (ft ) satisfies the hypothesis of Lemma 3.2 and hence it is a Loewner chain of order d associated with (ϕs,t ). Since f0 = g0 , the Loewner chain (ft ) is normalized. Now, let us describe the set Ω = ∪t≥0 ft (D) = ∪t≥0 gt (D). An easy computation shows |ϕ� (0)| � � ψ0,t (0) = 1−|ϕ0,t0,t (0)|2 . In particular, since the map t �→ ψ0,t (0) is monotone, the number |ϕ�0,t (0)| t→+∞ 1 − |ϕ0,t (0)|2 β = lim is well-defined. � In view of the equality gt� (0) = 1/ψ0,t (0), Koebe’s theorem shows that gt (D) contains a � disk of radius 1/(4ψ0,t (0)) centered at the origin. In particular, if β = 0, then ∪t≥0 gt (D) = C. Suppose now that β > 0. We have proved that in this case ψs,t has a limit ψs as � t → +∞. Note that ψs� (0) = β/ψ0,s (0) → 1 as s → +∞, while ψs (D) ⊂ D and ψs (0) = 0. It follows that ψs → idD as s → +∞. Then gs tends to the mapping z �→ z/β as s → +∞ locally uniformly in D. Since gs (D) forms an increasing family of domains, it follows that ∪s≥0 gs (D) = {z : |z| < 1/β}. The proof is now finished. � In the above proof we have obtained that the function β : [0, +∞) → (0, 1] given by β(t) := 1 |ϕ� (0)| 1 − |ϕ0,t (0)|2 0,t for all t ≥ 0, z ∈ D LOEWNER CHAINS 21 is non-increasing and, as a consequence, the following limit exist β := lim β(t) ∈ [0, 1]. t→+∞ This number will play a crucial role in the study of uniqueness of Loewner chains associated with the evolution family (ϕs,t ). For this reason, in the next proposition we analyze in full generality the above limit. Proposition 3.4. Let (ϕs,t ) be an evolution family of order d ∈ [1, +∞] and define βz (t) := 1 − |z|2 |ϕ� (z)| 1 − |ϕ0,t (z)|2 0,t for all t ≥ 0, z ∈ D. Then (1) For all z ∈ D, the map β(z) : [0, +∞) → (0, 1] is absolutely continuous and non-increasing. In particular, there exists the following limit β(z) := lim βz (t). t→+∞ (2) The following assertions are equivalent: (a) There exists z ∈ D such that β(z) = 0. (b) For all z ∈ D we have β(z) = 0. (3) The following assertions are equivalent: (a) There exists z ∈ D with β(z) = 1. (b) For all z ∈ D, we have β(z) = 1. (c) For all t ≥ 0, the map ϕ0,t is an automorphism. (d) For all 0 ≤ s ≤ t, the map ϕs,t is an automorphism. (4) If there is z ∈ D such that β(z) < 1, then there is T ∈ [0, +∞) such that ϕ0,t is an automorphism for all 0 ≤ t ≤ T and ϕ0,t is not an automorphism for all t > T. Proof. By Proposition 2.9 we have ϕs,t = ht ◦ ψs,t ◦ h−1 s , where (ψs,t ) is an evolution family � such that ψs,t (0) = 0 and ψs,t (0) > 0 for all t ≥ 0, and ht is a conformal automorphism of D for each t ≥ 0, with h0 being the identity map. One can check that 1 − |z|2 1 − |z|2 � � βz (t) := |ϕ0,t (z)| = ψ0,t (z) 2 2 1 − |ϕ0,t (z)| 1 − |ψ0,t (z)| for all t ≥ 0, z ∈ D. Proof of (1). The absolute continuity of the function βz is just an easy consequence of Proposition 2.6. Denote by ρ�D the pseudo-hyperbolic distance in the unit disk. Since any holomorphic self-map of the unit disk is a contraction for ρ�D , given s < t and z, w ∈ D, we have ρ�D (ϕ0,t (w), ϕ0,t (z)) = ρ�D (ϕs,t (ϕ0,s (w)), ϕs,t (ϕ0,s (z))) ≤ ρ�D (ϕ0,s (w), ϕ0,s (z)). 22 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK That is � � � � � ϕ (w) − ϕ (z) � � ϕ (w) − ϕ (z) � � 0,t � � 0,s � 0,t 0,s � �≤� �. � 1 − ϕ0,t (w)ϕ0,t (z) � � 1 − ϕ0,s (w)ϕ0,s (z) � Dividing by |w − z| (w �= z) and taking limits as w → z, we deduce that |ϕ�0,t (z)| |ϕ�0,s (z)| ≤ . 1 − |ϕ0,t (z)|2 1 − |ϕ0,s (z)|2 Thus βz (t) ≤ βz (s) for all 0 ≤ s < t < +∞. � Proof of (2). Notice that we know that the number β(0) = limt→+∞ ψs,t (0) is well defined. Moreover, the family of functions (ψ0,t )t≥0 is normal. So there is a sequence (tn ) → +∞ such that the limit f (z) = limn ψ0,tn (z) exists for all z ∈ D and it is attained uniformly on compact subsets of D. The function f is either constant or univalent in D, with f (0) = 0 and f � (0) = β(0). Therefore f vanishes identically if and only if β(0) = 0. Otherwise, f is univalent and f � (z) �= 0 for all z ∈ D. Now, observe that 1 − |z|2 1 − |z|2 � |ψ (z)| = |f � (z)|. 0,t n n→+∞ 1 − |ψ0,tn (z)|2 1 − |f (z)|2 β(z) = lim βz (t) = lim βz (tn ) = lim t→+∞ n→+∞ That is, β(z) = 0 for some z ∈ D if and only if f � (z) = 0 for some z ∈ D if and only if f is zero (recall that f (0) = 0). Assertions (3) and (4) are easy and we leave their proofs to the reader. � |ϕ� (0)| Definition 3.5. Let ϕs,t be an evolution family and take β = limt→+∞ 1−|ϕ0,t0,t (0)|2 . Let (ft ) be a normalized Loewner chain associated with ϕs,t . We say that (ft ) is a standard Loewner chain if ∪t≥0 ft (D) = {z : |z| < 1/β} (obviously, when β = 0, by {z : |z| < 1/β} we mean the complex plane C). Note that if (ft ) is a Loewner chain associated with a given evolution family (ϕs,t ) and h is any univalent holomorphic function in Ω := ∪t≥0 ft (D), then the formula gt = h ◦ ft , t ≥ 0, defines a Loewner chain which is also associated with (ϕs,t ). In view of this remark, the following theorem gives a necessary and suﬃcient condition for an evolution family to have a unique normalized Loewner chain associated with it. Moreover, in case of nonuniqueness, the set of all normalized Loewner chains associated with (ϕs,t ) is explicitly described. As usual, we denote by S the class of all univalent holomorphic functions h in the unit disk D, normalized by h(0) = h� (0)−1 = 0. As above, β = limt→+∞ |ϕ�0,t (0)|/(1−|ϕ0,t (0)|2 ). Theorem 3.6. Let (ϕs,t ) be an evolution family. (1) There is a unique standard Loewner chain (ft ) associated with (ϕs,t ). (2) If β = 0, then there is a unique normalized Loewner chain (ft ) associated with (ϕs,t ) (and obviously, it is the standard one.) LOEWNER CHAINS 23 (3) If β > 0 and (gt ) is a normalized Loewner chain associated with (ϕs,t ), then there is h ∈ S such that � � (3.6) gt (z) = h βft (z) /β, where (ft ) is the unique standard Loewner chain associated with (ϕs,t ). Proof. Let (ft ) be the standard Loewner chain built in Theorem 3.3 and (gt ) another normalized Loewner chain associated with the evolution family (ϕs,t ). For each t ≥ 0 denote by kt : ft (D) → gt (D) the function kt = gt ◦ ft−1 . Write Ω1 = ∪t≥0 ft (D) = {z : |z| < 1/β} and Ω2 = ∪t≥0 gt (D). If s < t and w ∈ fs (D) with w = fs (z), we have that kt (w) = gt ◦ ft−1 (fs (z)) = gt ◦ ft−1 (ft (ϕs,t (z))) = gt (ϕs,t (z)) = gs (z) = gs (fs−1 (w)) = ks (w). That is, kt|fs (D) = ks . This property says that the function k : Ω1 → Ω2 defined by k(w) := kt (w) for some (or any) t such that w ∈ ft (D) is well-defined, univalent and onto. Moreover k(0) = 0 and k � (0) = 1. Notice that k ◦ ft = gt for all t. Now suppose that β = 0. Then Ω1 = C. Since Ω2 is a simply connected domain biholomorphic to C, we also have that Ω2 = C. In this case, k is a univalent entire function such that k(0) = k � (0) − 1 = 0. Then k is the identity and ft = gt for all t. This implies statement (2) and statement (1) for the case β = 0. If β > 0, denote by h : D → Ω2 the function h(z) = βk(z/β). Obviously, h belongs to S and satisfies (3.6). This proves statement (3). Finally, if (gt ) is also a standard Loewner chain associated with (ϕs,t ), then Ω2 = {z : |z| < 1/β}. In this case k : {z : |z| < 1/β} → {z : |z| < 1/β} is biholomorphic and k(0) = k � (0) − 1 = 1. That is k is the identity and ft = gt for all t ≥ 0. This proves statement (1) for β > 0. The proof is now complete. � Remark 3.7. It is clear from the above proof that one can define the standard Loewner chain as the unique normalized Loewner chain (ft ), associated with the evolution family (ϕs,t ), such that ∪t≥0 ft (D) is either an Euclidean disk or the whole complex plane. Our next theorem says that, in some particular cases, the univalence of the functions which form a Loewner chain can be replaced by an appropriate bound of these functions on certain hyperbolic disks. Theorem 3.8. Let (ϕs,t ) be an evolution family in the unit disk having a unique normalized Loewner chain associated with it. Suppose (ft )t≥0 is a family in Hol(D, C). Then (ft ) is the unique normalized Loewner chain associated with (ϕs,t ) if and only if the following three conditions are satisfied: (1) The function f0 is normalized, that is, f0 (0) = f0� (0) − 1 = 0. (2) The equation ft ◦ ϕs,t = fs holds for any 0 ≤ s ≤ t. 24 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK (3) For each R > 0, there exists some C > 0 (independent on t) such that for all t ≥ 0 the following inequality C |ft (z)| ≤ , β(t) where β(t) = |ϕ�0,t (0)| , 1 − |ϕ0,t (0)|2 t ≥ 0, holds for any z in the hyperbolic disk of radius R centered at ϕ0,t (0). Proof. Before dealing with the proof of the theorem, we comment (really recall) some facts and notations which will be used later on and which have been shown in the course of proofs of previous results. In our case, according to Theorem 3.6, we know that limt→+∞ β(t) = 0. Write ϕ�0,t (0) b(t)z + a(t) z − a(t) a(t) = ϕ0,t (0), b(t) = � , ht (z) = , and h−1 , t (z) = b(t) |ϕ0,t (0)| 1 + b(t)a(t)z 1 − a(t)z for all z ∈ D and all t ≥ 0. Clearly, a(0) = 0 and b(0) = 1. Finally, define ψs,t = h−1 t ◦ϕs,t ◦hs � for all 0 ≤ s ≤ t. One can easily prove that ψs,t (0) = 0 and ψs,t (0) = β(t)/β(s) > 0 for all 0 ≤ s ≤ t. Hence, by Proposition 2.9, (ψs,t ) is an evolution family. (⇒) Assume that (ft ) is the (unique) normalized Loewner chain associated with (ϕs,t ). By the very definition of normalized Loewner chains, we see that only property (3) requires a proof. Note that � |ψ0,t (0)| � = |ψ0,t (0)| = β(t) → 0 as t → +∞. 1 − |ψ0,t (0)|2 Therefore, according to Theorem 3.6 (now applied to the evolution family (ψs,t )), we deduce that (ψs,t ) has also a unique normalized Loewner chain associated with it. More� over, such a Loewner chain (gt ) satisfies the equality gt� (0)ψs,t (0) = gs� (0). Consequently, gt� (0)β(t) = gs� (0)β(s) for all t, s ≥ 0. But g0� (0)β(0) = 1. Thus gs� (0) = 1/β(s) for all s ≥ 0. Using the Distortion Theorem, we conclude that |gs (z)| ≤ for all s ≥ 0 and for all z ∈ D. 1 |z| β(s) (1 − |z|)2 eR − 1 ∈ (0, 1). Take z in the hyperbolic eR + 1 disk of radius R centered at the point a(s) = ϕ0,s (0). We have that Now, fix R > 0 and s ≥ 0 and consider r = −1 −1 ρD (h−1 s (z), 0)) = ρD (hs (z), hs (a(s))) = ρD (z, a(s)) ≤ R. LOEWNER CHAINS 25 Thus, |h−1 s (z)| ≤ r and |fs (z)| = |gs (h−1 s (z))| ≤ 1 r e2R − 1 = . β(s) (1 − r)2 2β(s) (⇐) First of all, bearing in mind Lemma 3.2 and property (1) combined with Theorem 3.6, we see that we only have to prove the univalence of each function ft . We start by defining gt := ft ◦ ht ∈ Hol(D, C), t ≥ 0. By property (2), we observe that gt ◦ ψs,t = gs , 0 ≤ s ≤ t. We notice that the family (gt ) satisfies the following three properties: (a) gt (0) = 0, for all t ≥ 0. (b) gt� (0) = β(t)−1 , for all t ≥ 0. (c) For all R > 0, there exists some C > 0 such that, for all t ≥ 0 and all |z| ≤ R, we have |gt (z)| ≤ Cβ(t)−1 . Now, fix s ≥ 0 and r ∈ (0, 1) and suppose that |z| ≤ r. Take also some R ∈ (0, 1) with R > r. By Schwarz Lemma, |ψs,t (z)| ≤ |z| ≤ r, for all t ≥ s. Then by the Cauchy Integral Formula, for all t ≥ s we have |gs (z) − β(t)−1 ψs,t (z)| = |gt (ψs,t (z)) − β(t)−1 ψs,t (z)| = |gt (ψs,t (z)) − gt (0) − gt� (0)ψs,t (z)| � � � � 1 � (ψs,t (z))2 � = � gt (ξ) 2 dξ �� 2πi C + (0,R) ξ (ξ − ψs,t (z)) 2πR ≤ |ψs,t (z)|2 max{|gt (ξ)| : |ξ| ≤ R}. 2πR2 (R − r) Therefore, by property (3), we can find C = C(R) (independent on t) such that |gs (z) − β(t)−1 ψs,t (z)| ≤ C β(t)−1 |ψs,t (z)|2 . R(R − r) 26 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK In fact, since limt→+∞ β(t) = 0 and by the Distortion Theorem, we deduce that � �2 C r −1 −1 � 2 |gs (z) − β(t) ψs,t (z)| ≤ β(t) |ψs,t (0)| R(R − r) (1 − r)2 C 1 β 2 (t) ≤ R(R − r)(1 − r)4 β(t) β 2 (s) C = β(t) → 0 as t → +∞. R(R − r)(1 − r)4 β 2 (s) Therefore, we conclude that gs = lim β(t)−1 ψs,t , t→+∞ in the compact-open topology of Hol(D, C). By Hurwitz’s Theorem and property (b), we find that gs is univalent. Since ht is an automorphism of the disk, we finally conclude that fs is univalent as well. The proof is complete. � Remark 3.9. The above proof shows that statement (3) in this last theorem can be replaced by “for all z ∈ D and for all s ≥ 0, the following inequality holds |fs ◦ hs (z)| ≤ where, as usual, hs (z) = b(s)z + a(s) 1 + b(s)a(s)z 1 |z| , β(s) (1 − |z|)2 , a(t) = ϕ0,a (0), and b(s) = ϕ�0,s (0) .” |ϕ�0,s (0)| 4. Loewner chains and partial differential equations In classical Loewner theory any Loewner chain satisfies the Loewner – Kufarev PDE, while the corresponding evolution family satisfies the Loewner – Kufarev ODE with the same driving term. Now we prove an analogue of this statement in our general setting. Theorem 4.1. The following assertions hold. (1) Let (ft ) be a Loewner chain of order d ∈ [1, +∞]. Then (a) There exists a set N ⊂ [0, +∞) (not depending on z) of zero measure such that for every s ∈ (0, +∞) \ N the function ∂fs (z) fs+h (z) − fs (z) := lim ∈C h→0 ∂s h is a well-defined holomorphic function on D. (b) There exist a Herglotz vector field G of order d and a set N ⊂ [0, +∞) (not depending on z) of zero measure such that for every s ∈ (0, +∞) \ N and every z ∈ D, ∂fs (z) = −G(z, s)fs� (z). ∂s z ∈ D �→ LOEWNER CHAINS 27 (2) Let G be a Herglotz vector field of order d ∈ [1, +∞] associated with the evolution family (ϕs,t ). Suppose that (ft ) is a family of univalent holomorphic functions in the unit disk such that ∂fs (z) = −G(z, s)fs� (z) for every z ∈ D, a.e. s ∈ [0, +∞). ∂s Then (ft ) is a Loewner chain of order d associated with the evolution family (ϕs,t ). Proof of (1.a). By the very definition of Loewner chain, the map s ∈ [0, +∞) �→ fs (z) ∈ C is absolutely continuous, for all fixed z ∈ D. Thus there exists a set of zero measure N1 (z) ⊂ [0, +∞) such that for every s ∈ [0, +∞) \ N1 (z) the following limit exists Ds (z) = ∂fs (z) fs+h (z) − fs (z) = lim . h→0 ∂s h Let kn ∈ Ldloc ([0, +∞), R) be a non negative function such that � t |fs (z) − ft (z)| ≤ kn (ξ)dξ s whenever |z| ≤ 1 − 1/n and 0 ≤ s ≤ t. For each n, there exists a set N2 (n) ⊂ [0, +∞) of zero measure such that for every s ∈ [0, +∞) \ N2 (t) there exists the limit � 1 s+h kn (η)dη. kn (s) = lim h→0 h s Let us define N := � ∞ � n=1 N1 � 1 n+1 �� � � � ∞ n=1 � N2 (n) . Obviously, N is a subset of [0, +∞) of zero measure, independent of z. We are going to prove that for all s ∈ [0, +∞) \ N the following limit fs+h (z) − fs (z) h exists and attained uniformly on compact subsets of D. First of all we show that for every s ∈ (0, +∞) \ N the family lim h→0 1 (fs+h − fs ) : 0 < h < 1 or − s < h < 0} h is a relatively compact set in Hol(D, C). To this aim, we consider two cases: (a) 0 < h < 1; (b) −s < h < 0. Case (a): Fix r ∈ (0, 1). Let n ∈ N be such that r < 1 − 1/n. Then, for every |z| ≤ r, � � � �1 � 1 s+h |Fh (z)| = �� (fs+h (z) − fs (z))�� ≤ kn (ξ)dξ ≤ C̃ < +∞, h h s Fs := {Fh := 28 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK where the last inequality takes place since s ∈ / N2 (n). Hence, sup{|Fh (z)| : |z| ≤ r, 0 < h < 1} < +∞ and consequently, by the Montel criterion, the subfamily of Fs with 0 < h < 1 is a normal family in D, as wanted. Case (b): the proof is similar to that of case (a) and we omit it. Thus the family Fs is relatively compact in Hol(D, C). Let ψ, φ be any pair of limit functions of Fs as h → 0. By the very definition of N , � � � � � � 1 1 1 Ds =ψ =φ , m+1 m+1 m+1 1 for every m ∈ N. But { m+1 } is a sequence accumulating at 0, hence by the identity principle ψ = φ. This shows that fs+h (z) − fs (z) , h→0 h exists for all s ∈ (0, +∞) \ N and is attained uniformly on compact subsets of D, which finishes the proof of (1.a). Proof of (1.b). By Theorem 3.1, there is an evolution family (ϕs,t ) of order d associated with (ft ). Let G : D×[0, +∞) → C be the Herglotz vector field whose positive trajectories are (ϕs,t ) (such a vector field exists by Theorem 2.3). Let N1 ⊂ [0, +∞) be the set of zero measure given by [5, Theorem 6.6] such that ∂ϕ∂u0,u (z) = G(ϕ0,u (z), u) for all u ∈ (0, +∞) \ N1 and all z ∈ D. Let N2 ⊂ [0, +∞) stand for the set of zero measure which has been denoted by N in part (1.a) of this theorem. Let N := N1 ∪ N2 . Diﬀerentiating with respect to t the equality ft (ϕ0,t (z)) = f0 (z), for z ∈ D and t ∈ (0, +∞) \ N we obtain lim ∂ϕ0,t ∂ft (z) + (ϕ0,t (z)) ∂t ∂t ∂ft = ft� (ϕ0,t (z)))G(ϕ0,t (z), t) + (ϕ0,t (z)). ∂t t Therefore, ft� (w)G(w, t) = − ∂f (w) for all w ∈ ϕ0,t (D). Since the ϕ0,t ’s are univalent, the ∂t identity principle for holomorphic maps implies that this equality is valid for the whole unit disk D. Proof of (2). Fix a point z in the unit disk. Then, up to a set of measure zero, we have 0 = ft� (ϕ0,t (z)) ∂ ∂ϕs,t ∂ft (ft (ϕs,t (z))) = ft� (ϕs,t (z)) (z) + (ϕs,t (z)) ∂t ∂t ∂t ∂ϕs,t = ft� (ϕs,t (z)) (z) − G(ϕs,t (z), t)fs� (ϕs,t (z)) ∂t � � ∂ϕs,t � = ft (ϕs,t (z)) (z) − G(ϕs,t (z), t) = 0. ∂t LOEWNER CHAINS 29 Therefore, ft (ϕs,t (z)) does not depend on t. Hence, ft (ϕs,t (z)) = fs (ϕs,s (z)) = fs (z) and the proof finishes just by applying Lemma 3.2. � 5. Remarks about semigroups A (one-parameter) semigroup of holomorphic functions is a continuous homomorphism Φ : t �→ Φ(t) = φt from the additive semigroup of non-negative real numbers into the composition semigroup of holomorphic self-maps of D. Namely, Φ satisfies the following three conditions: S1. φ0 is the identity in D, S2. φt+s = φt ◦ φs , for all t, s ≥ 0, S3. φt (z) tends to z as t tends to 0, uniformly on compact subsets of D. Let (φt ) be a semigroup of holomorphic self-maps of D. Let ϕs,t := φt−s for 0 ≤ s ≤ t < +∞. Then, by [5, Example 3.4], (ϕs,t ) is an evolution family of order ∞. Given a semigroup Φ = (φt ), it is well-known (see [33], [4]) that there exists a unique holomorphic function G : D → C such that, ∂φt (z) ∂φt (z) = G (φt (z)) = G (z) for all z ∈ D and t ≥ 0. ∂t ∂z The function G is known as the infinitesimal generator of the semigroup and, obviously, G (that clearly does not depend on t) is the Herglotz vector field associated with the evolution family (ϕs,t ). Berkson and Porta [4] proved that there exist τ ∈ D and a holomorphic function p : D → C with Re p(z) ≥ 0 such that G(z) = (τ − z)(1 − τ z)p(z), z ∈ D, and moreover, any function G of this form is the infinitesimal generator of some semigroup. In this very particular case when the evolution family is generated by a semigroup, the point τ has a dynamical meaning. To explain this meaning, we have to recall some notions from iteration theory. It can be easily deduced from the Schwarz-Pick lemma that a non-identity self-map φ of the unit disc can have at most one fixed point in D. If such a unique fixed point in D exists, it is usually called the Denjoy-Wolﬀ point. The sequence of iterates {φn } of φ converges to it uniformly on the compact subsets of D whenever φ is not a disc automorphism. If φ has no fixed points in D, the Denjoy-Wolﬀ theorem (see, e. g., [1]) guarantees the existence of a unique point τ on the unit circle ∂D which is the attractive fixed point, that is, the sequence of iterates {φn } converges to τ uniformly on the compact subsets of D. Such a point τ is again called the Denjoy-Wolﬀ point of φ. When τ ∈ ∂D is the Denjoy-Wolﬀ point of φ, the angular derivative φ� (τ ) is actually real-valued and, moreover, 0 < φ� (τ ) ≤ 1 (see [29]). As it is often done in the literature, we classify the holomorphic self-maps of the disc into three categories according to their behavior near the DenjoyWolﬀ point: (a) elliptic: the ones with a fixed point inside the unit disc D; 30 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK (b) hyperbolic: the ones with the Denjoy-Wolﬀ point τ ∈ ∂D such that φ� (τ ) < 1; (c) parabolic: the ones with the Denjoy-Wolﬀ point τ ∈ ∂D such that φ� (τ ) = 1. Going back to semigroups, we have to say that the point τ that appears in the BerksonPorta representation formula for the infinitesimal generator of the semigroup (φt ) is the Denjoy-Wolﬀ point of all the functions φt . In particular, all the functions share the DenjoyWolﬀ point. But something more can be said. If there is t0 > 0 such that the function φt0 is elliptic (resp. hyperbolic, parabolic) then all the functions of the semigroup are elliptic (resp. hyperbolic, parabolic). Besides the above classification of self-maps of the unit disk, there are two quite diﬀerent types of parabolic functions. To distinguish such functions, we have to recall the notion of hyperbolic step. Given a holomorphic self-map φ of D and a point z0 in D, we define the forward orbit of z0 under φ as the sequence zn = φn (z0 ). It is customary to say that φ is of zero hyperbolic step if for some point z0 the orbit zn = φn (z0 ) satisfies the condition limn→∞ ρD (zn , zn+1 ) = 0. It is well-known that the word “some” here can be replaced by “all”. In other words, the definition does not depend on the choice of the initial point of the orbit (see, for example, [8]). Using the Schwarz-Pick Lemma, it is easy to see that the maps which are not of zero hyperbolic step are precisely those holomorphic self-maps φ of D for which lim ρD (zn , zn+1 ) > 0 , n→∞ for some forward orbit {zn }∞ n=1 of φ, and hence for all such orbits. This is the reason why they are called maps of positive hyperbolic step. For a survey of these properties, the reader may consult [8]. It is easy to show that if φ is elliptic and is not an automorphism, then it is of zero hyperbolic step. If φ is hyperbolic, then it is of positive hyperbolic step. For parabolic maps the situation is more complicated: there are parabolic functions of zero hyperbolic step and of positive hyperbolic step. For example, the following dichotomy holds for parabolic linear-fractional maps: every parabolic automorphism of D is of positive hyperbolic step, while all non-automorphic linear-fractional parabolic self-maps of D are of zero hyperbolic step. For semigroups of holomorphic functions we can state the following Lemma 5.1. Let (φt ) be a semigroup of parabolic functions in the unit disk. If there exists t0 > 0 such that the function φt0 is of zero hyperbolic step, then all the functions φt , with t > 0, of the semigroup are of zero hyperbolic step. Proof. In this proof we will use diﬀerent well-known properties of the hyperbolic distance on simply connected domains in the complex plane that can be seen in [32]. By [35], there exists a univalent function h : D → C, with h(0) = 0, such that h ◦ φt = h + t for all t > 0. Write Ω = h(D) and denote by δΩ (w) the Euclidean distance from w ∈ Ω to ∂Ω. Since Ω + t ⊆ Ω for all t > 0, we can easily obtain that the function δΩ : [0, +∞) → R is non-decreasing (we are considering here the restriction of δΩ to LOEWNER CHAINS 31 the half-line [0, +∞)). By hypothesis, the sequence ρD (φnt0 (0), φ(n+1)t0 (0)) goes to zero. Moreover, by the Distance Lemma, we have that ρD (φnt0 (0), φ(n+1)t0 (0)) = ρΩ (h(0) + nt0 , h(0) + (n + 1)t0 ) = ρΩ (nt0 , (n + 1)t0 ) � � 1 |t0 | ≥ log 1 + , 2 min{δΩ (nt0 ), δΩ ((n + 1)t0 )} where ρΩ denotes the hyperbolic distance on Ω. Thus δΩ ((n + 1)t0 ) goes to ∞ and we conclude that limt→+∞ δΩ (t) = ∞. Now fix t > 0. Write Γn = [nt, (n + 1)t] and denote by lΩ (Γn ) the hyperbolic length of Γn in Ω. We have � |dw| 1 ρD (φnt (0), φ(n+1)t (0)) = ρΩ (nt, (n + 1)t) ≤ lΩ (Γn ) ≤ 2 ≤2 , δΩ (nt) Γn δΩ (w) where again we have used the monotonicity of δΩ on [0, +∞). Since the sequence δΩ ((n + 1)t) goes to ∞, the above inequality implies that ρD (φnt (0), φ(n+1)t (0)) tends to zero as n goes to ∞. The arbitrariness of t concludes the proof. � What can we say about the Loewner chains associated with the evolution families (ϕs,t ) = (φt−s )? If the semigroup (φt ) is elliptic and its Denjoy-Wolﬀ point is zero, then (see [35]) there is a complex number c and a univalent function h such that Re c ≥ 0, h(0) = 0, h� (0) = 1, and (5.1) h ◦ φt = e−ct h. The function h is called the Kœnigs function of the semigroup (φt ). ¿From equation (5.1), it is clear that the functions ft = ect h form a normalized Loewner chain associated with the evolution family (ϕs,t ) = (φt−s ). If Re c > 0, then ∪t≥0 ft (D) = C and, by Theorem 3.6, this is the unique normalized Loewner chain associated with (ϕs,t ). In particular, this implies the uniqueness of the Kœnigs function, a fact which is very well-known. If Re c = 0, then h is the identity map. Now suppose that the Denjoy-Wolﬀ point of the semigroup is on the boundary of the unit disc. Without loss of generality, we assume that such a point is 1. Then there is a univalent function h such that h(0) = 0 and h ◦ φt = h + t [35]. As in the elliptic case, the function h is referred to as the Kœnigs function of the semigroup (φt ). Siskakis [34] (see also [35]) proved that the Kœnigs function is unique in this case as well. As an easy application of our results we will reprove the uniqueness of the Kœnigs function. Similarly to the elliptic case, we have that the functions ft = h − t form a Loewner chain associated with the evolution family (ϕs,t ) = (φt−s ). Notice that this Loewner chain is not necessarily normalized. To proceed let us distinguish the diﬀerent type of semigroups. If the functions φt are hyperbolic, then by [7, Theorem 2.1], there is a horizontal strip Ω such that the range of h is included in Ω and ∪t≥0 ft (D) = Ω. In this case, there are much more Loewner chains associated with (ϕs,t ) but there is no other Loewner chain 32 M. D. CONTRERAS, S. DÍAZ-MADRIGAL, AND P. GUMENYUK (gt ) of the form gt = k − t, where k, k(0) = 0, is a univalent holomorphic function in D. Indeed, if such another function k does exist, then by Theorem 3.6, there is a univalent holomorphic function a : Ω → C such that a(h(z) − t) = k(z) − t for all t ≥ 0 and for all z ∈ D. Derivating with respect to t, we have a� (h(z) − t) = 1 for all t ≥ 0 and for all z ∈ D. That is a(z) = z + c for some constant c. Therefore h(z) − t + c = k(z) − t for all t ≥ 0 and for all z ∈ D. Since h(0) = k(0) = 0, we deduce that c = 0 and h = k. Consider now the parabolic case. According to the above lemma we have to distinguish two subcases. On one hand, if for some (or for any) t0 > 0 the function φt0 is of zero hyperbolic step, then by [9, Theorem 3.1 and Proposition 3.3], the range of h is not included in any horizontal half-plane. In this case, we have that ∪t≥0 ft (D) = C. Therefore, up to normalization, this is a unique Loewner chain associated with (ϕs,t ). On the other hand, if one (and then all) of the mappings φt is of positive hyperbolic step, then the range of h is included in a horizontal half-plane Ω. In fact, we can choose the half-plane such that ∪t≥0 ft (D) = Ω. By the same reason as in the hyperbolic case, there are much more Loewner chains associated with (ϕs,t ) but there is no other Loewner chains (gt ) of the form gt = k − t, where k, k(0) = 0, is a univalent holomorphic function in D. That is, again, the Kœnigs function of the semigroup is unique. References [1] M. 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