Loewner Theory in annulus I: evolution families and differential equations

Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex Analysis, which plays a crucial role in such important achievements as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. We construct an analogue of this theory for the annulus. In this paper, the first of two articles, we introduce a general notion of an evolution family over a system of annuli and prove that there is a 1-to-1 correspondence between such families and semicomplete weak holomorphic vector fields. Moreover, in the non-degenerate case, we establish a constructive characterization of these vector fields analogous to the non-autonomous Berkson - Porta representation of Herglotz vector fields in the unit disk [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594].

There is an essentially one-to-one correspondence between Herglotz vector fields, evolution families, and Loewner chains.
Classical Loewner Theory originated from the so-called Parametric Representation of univalent functions proposed in 1923 by C. Loewner [34] and developed further by a number of specialists in Geometric Function Theory, among which we would like to mention the fundamental contributions of Kufarev [27] and Pommerenke [36], [37,Ch. 6]. In the case which they considered, and which is usually referred in modern literature to as the (classical) radial case, a Loewner chain is a family of univalent functions f t (z) = e t z+a 2 (t)z 2 +. . ., z ∈ D, such that f s (D) ⊂ f t (D) whenever t ≥ s. A (classical radial) Herglotz vector field is of the form G(w, t) := −wp(w, t), where p is holomorphic in z ∈ D, measurable in t ≥ 0, and satisfies conditions p(0, t) = 1 and Re p(w, t) > 0 for all z ∈ D and a.e. t ≥ 0. For any holomorphic vector field of this form there exists a unique classical radial Loewner chain (f t ) such that [s, +∞) ∋ t → ϕ s,t (z) := f −1 t • f s (z) solves, for any z ∈ D and s ≥ 0, the Loewner -Kufarev ODEẇ = G(w, t) with the initial condition w| t=s = z. The converse is also true: any classical radial Loewner chain is a solution to the Loewner -Kufarev PDE ∂f t (z)/∂t = −f ′ (z)G(z, t) driven by some classical radial Herglotz vector field G, while the evolution family (ϕ s,t ) corresponding to (f t ) solves the Loewner -Kufarev ODE. Moreover, the Loewner chain (f t ) can be reconstructed via its evolution family (ϕ s,t ) by means of the formula f s = lim t→+∞ e t ϕ s,t .
This relation between G, (f t ), and (ϕ s,t ) provides a representation of the class S of all normalized holomorphic univalent functions in D, since any f 0 ∈ S can be embedded as an initial element into a classical radial Loewner chain [37, Th. 6.5 on p. 159], [21]. This representation was one of the main tools in proof of the famous Bieberbach conjecture, given by de Branges [13].
A similar representation, designated in modern terminology by the attribute chordal, was proposed by Kufarev and his students for holomorphic univalent self-maps of the upper half-plane with the hydrodynamic normalization at the point of infinity [28]; see also references cited in [17, p. 543-544].
Komatu [24] was the first who was able to apply Loewner's ideas for parametric representation of univalent functions in the annulus. His approach was developed by Goluzin [20], Li En Pir [33], Lebedev [32], and Gutljanskiȋ [22]. More general cases of Loewner Evolution in multiply connected context were studied in [26] and by an essentially different method in [30,41]. The monograph [4] contains a self-contained detailed account on the Parametric Representation both in simply and multiply connected cases.
Until the last decade the attention of specialists in Loewner Theory was mainly paid to the radial case in the unit disk, first of all because of its applications in the study of the class S and its subclasses. The significance of the chordal Loewner Evolution as well as the Loewner Evolution in multiply connected domains was apparently underestimated. However, nowadays the Parametric Method, invented by Loewner to study the Bieberbach conjecture, has gone far beyond the scope of the original problem, and the distribution of active interest in various aspects of the theory has been changed. The most spectacular evidence of this fact is the well-celebrated Stochastic Loewner Evolution (SLE) invented in 2000 by Schramm [39]. It appears that the chordal Loewner -Kufarev equation driven by the Brownian motion is intrinsically related to several important lattice models in Statistical Physics, such as critical Ising model. We refer interested readers to [31]. For a wider discussion on connection of deterministic and stochastic Loewner Evolutions to Conformal Field Theory and Integrable Systems see [35] and references cited there.
To conclude, we would like to mention the survey paper [2] that covers the basics of the classical and modern Loewner Theory, its history, recent development and applications. every evolution family (D t ), (ϕ s,t ) of order d ∈ [1, +∞] can be described as the nonautonomous semiflow of some semicomplete weak holomorphic vector field G of order d in D := {(z, t) : t ≥ 0, z ∈ D t }. Conversely, every such semiflow constitutes an evolution family. See Theorem 5.1 and Definitions 2.1 and 2.5 for the exact formulation.
Further, in Theorem 5.6 we establish, for the case when all the annuli D t are nondegenerate, a constructive characterization of semicomplete weak holomorphic vector fields in D analogous to the non-autonomous version of the Berkson -Porta representation [10,Theorem 4.8], which characterizes Herglotz vector fields in the unit disk.
A part of our proofs rely on lifting evolution families from a system of annuli to a simply connected domain D. This technique, together with some new results on evolution families in D, is developed in Section 3.
Moreover, we include auxiliary Section 2 on Carathéodory differential equations driven by weak holomorphic vector fields. The first part contains standard facts about solutions to such ODEs which we regard as known. However, we include a sketch of the proofs, because up to our best knowledge, no literature gives a direct proof of these results formulated exactly as we need. The second part is devoted to the case of the ODEs which are semicomplete to the right. It contains an exact characterization of the solutions, which is applied later to prove Theorem 5.1.
In Section 6 we consider a number of examples. In particular, we prove that, in contrast to the simply connected case, an evolution family over a system of annuli can share any finite number of fixed points.
Finally, in Section 7 we discuss shortly how our results are related to the achievements of Komatu, Goluzin, Li En Pir, and Lebedev.

Holomorphic Carathéodory differential equations
In this section we obtain a characterization of solutions to Carathéodory ordinary differential equations driven by holomorphic vector fields.
Let E ⊂ R be a non-empty interval, bounded or unbounded, and consider a connected relatively open subset D of the set C × E. We fix E and D throughout this section. In this paper we apply the results of this section for the case E := [0, +∞) and D := D × [0, +∞) or D := {(z, t) : t ≥ 0, 0 < |z| < r(t)}, where r : [0, +∞) → [0, 1) is non-increasing and continuous. However, we prefer to consider the general case, which might be useful for further studies and at the same time does not require any substantial modification of the argument.
By AC d (X, Y ), where X ⊂ R and d ∈ [1, +∞], we denote the class of all locally absolutely continuous functions f : X → Y such that the derivative f ′ belongs to L d loc (X). Further, denote by D(z 0 , r), where r > 0 and z 0 ∈ C, the disk {z : |z − z 0 | < r}, and let D := D(0, 1) and T := ∂D.  [38,§VIII.8]. The facts we state below can be regarded as well-known. However, these results do not seem to be proved earlier in the literature directly and explicitly in the form that our context requires. That is why we prefer to sketch the proofs.
We start by introducing the class of vector fields we deal with.
Remark 2.2. It is easy to see that condition WHVF3 in Definition 2.1 is equivalent to WHVF3' For any closed disk B ⊂ C and any compact interval Given a weak holomorphic vector field G in D and a point (z, s) ∈ D, let us consider the following initial value problem for a Carathéodory ODE (2.1)ẇ = G(w, t), w(s) = z.
Problem (2.1) is equivalent (see, e. g., [19, §I.1]) to the integral equation L s z w = w, where is well-defined for any continuous function w : J → C such that J ⊂ E is an interval, s ∈ J, and (w(t), t) ∈ D for all t ∈ J.
Theorem 2.3. Let G be a weak holomorphic vector field in D of order d ∈ [1, +∞]. Then the following statements hold: (i) For any (z, s) ∈ D, the initial value problem (2.1) has a unique non-extendable solution t → w * s (z, t). In other words, there exists an interval J * (z, s) ⊂ E, s ∈ J * (z, s), and a solution w * s (z, ·) to the problem (2.1) defined in J * (z, s) such that any other solution t → w 0 (t) to (2.1) is a restriction of w * s (z, ·). (ii) For any (z, s) ∈ D the non-extendable solution w * s (z, ·) belongs to AC d J * (z, s), C . (iii) For any (z, s) ∈ D there exists no compact set K ⊂ D such that w * s (z, t), t : t ∈ J * (z, s), t ≥ s ⊂ K unless sup J * (z, s) = sup E. (iv) For any (z 0 , s 0 ) ∈ D and any t 0 ∈ J * (z 0 , s 0 ) there exists ε > 0 such that t 0 ∈ J * (z, s 0 ) whenever |z − z 0 | < ε. Moreover, the function z → w * s 0 (z, t 0 ) is holomorphic and injective in D(z 0 , ε).
First we prove the following lemma.
Lemma 2.4. Let G : D → C be a weak holomorphic vector field of order d ∈ [1, +∞] and K ⊂ D a compact set. Suppose I ⊂ E is a compact interval and B := D(z 0 , 2R) is a closed disk such that B × I ⊂ K. Then for any s ∈ I the following assertions hold: (a) for each z ∈ D(z 0 , R) and each sufficiently small interval J ⊂ I, J ∋ s, there exists a unique solution J ∋ t → w s (z, t) to the initial value problem (2.1); (b) for each z ∈ D(z 0 , R) the solution to (2.1) can be continued all over the interval Proof. First of all let us note that, since K ⊂ D is compact, there exists δ > 0 such that Then the argument of the proof of [10,Lemma 4.2] can be easily adapted to show that there exists a non-negative functionk K ∈ L d pr R (K), R depending only on G and K such that Choose any α ∈ (0, 1). There exists h > 0 that fulfills the following two conditions: By C(X, Y ) let us denote the class of all continuous functions from X to Y . Fix any s ∈ I. Given an interval J ⊂ J h (s) with s ∈ J, from (2.3) -(2.5) it follows easily that for any z ∈ D(z 0 , R), the operator L s z , given by (2.2), is a contracting self-mapping of C(J, B) endowed with the Chebyshov metric ρ(w 2 , w 1 ) := sup t∈J |w 2 (t) − w 1 (t)|. The metric space M := C(J, B), ρ is complete. Hence the Banach fixed point theorem implies statements (a) and (b) of the lemma.
The proof of (d) is similar. The operator L, defined by the formula Lw (z, t; s) := L s z w(z, ·; s) (t), is a contractive self-mapping of C(A, B) endowed with the metric ρ(w 2 , w 1 ) := sup (z,s,t)∈A w 2 (z, t; s) − w 1 (z, t; s) . The metric spaceM = C(A, B),ρ is complete. Hence it contains a solution to the equation Lw = w, which, in virtue of (a), has to coincide with A ∋ (z, s, t) → w s (z, t). This proves (d).
We are left with statement (c). Clearly, it is sufficient to show that L maps into itself the closed subspaceM hol ofM consisting of all maps fromM which are holomorphic in z.
So consider any w ∈M hol . Fix an arbitrary (z, s, t) ∈ A. For all ξ ∈ J h (s) and all ω ∈ U : with some constant C = C(z) > 0 not depending on ω and ξ. Here we combined (2.3) with the fact that the family w(·, ξ; s) : ξ ∈ J h (s) is normal in D(z 0 , R) and hence the mapping (ω, ξ) → w(z + ω, ξ; s) − w(z, ξ; s) /ω is bounded on U × J h (s). By construction, From (2.6) it follows that we can apply Lebesgue's dominated convergence theorem to conclude that there exists a finite limit of the integral in (2.7) as ω → 0. Thus Lw)(·, t; s) is differentiable in the complex sense in D(z 0 , R). This completes the proof.
Proof of Theorem 2.3. Assertion (a) of Lemma 2.4 implies that for any (z, s) ∈ D there exists a unique local solution to (2.1). Thus statement (i) of the theorem follows in the same way as in the classical theory of ODEs.
Fix now any (z, s) ∈ D and take an arbitrary compact interval I ⊂ J * (z, s). By condition WHVF3 with K := w * s (z, ξ), ξ : ξ ∈ I , for any t 1 , t 2 ∈ I such that t 1 < t 2 we have where k K ∈ L d (I, R) and does not depend on t 1 and t 2 . This proves statement (ii). Let K ⊂ D be any compact set. Then there is another compact set K 1 ⊂ D and constants R > 0 and δ > 0 such that D(z, 2R) × [s − δ, s + δ] ∩ E ⊂ K 1 whenever (z, s) ∈ K. We claim that any solution J ∋ t → w 0 (t) to the equationẇ = G(w, t) such that w 0 (t), t ∈ K for all t ∈ J, is extendable to a neighborhood of t * := sup J provided t * < sup E. Indeed, by assertion (b) of Lemma 2.4 applied with K 1 substituted for K, there exists δ 1 ∈ (0, δ] such that for any s ∈ J the initial value problem (2.1) with z := w 0 (s) has a solution defined on the interval (s − δ 1 , s + δ 1 ) ∩ E. In view of the uniqueness of solution to (2.1), this proves our claim, which, in its turn, implies statement (iii) of the theorem.
Let us prove statement (iv). For simplicity we may assume that t 0 > s 0 . Let J := [s 0 , t 0 ]. Taking w 0 := w * s 0 (z 0 , ·)| J and K := w 0 (t), t : t ∈ J in the above argument and using assertion (c) of Lemma 2.4 we can conclude that there exists a finite increasing sequence (s j ) n j=0 starting with s 0 and finishing with s n := t 0 such that for each j = 1, . . . , n the function w * s j−1 (z, t) is well-defined and holomorphic in z for all t ∈ I j := [s j−1 , s j ] and all z ∈ D(w 0 (s j−1 ), R). Since w * s j−1 w 0 (s j−1 ), s j = w 0 (s j ), there exists ε > 0 such that the composition f (z) := w * s n−1 (·, s n ) • . . . • w * s 1 (·, s 2 ) • w * s 0 (·, s 1 ) is well-defined and holomorphic in D(z 0 , ε). It follows that w * s 0 (·, t 0 ) is also well-defined in D(z 0 , ε) and coincides there with f . The fact that w * s 0 (·, t 0 ) is injective follows from the uniqueness of the solution in the same way as in the classical theory of ODEs.
We are left with the proof of statement (v). By statement (iv) the map w * s 0 (·, t 0 ) is defined in D(z 0 , ε) and is continuous at the point z 0 . It follows from assertions (b) and (d) of Lemma 2.4 that there exists ε 1 > 0 such that the map (z, for some ε 2 > 0 and continuous at the point (z 0 , s 0 ). Denote ζ 0 := w * s 0 (z 0 , t 0 ). Again by assertions (b) and (d) of Lemma 2.4, the map (ζ, t) → w * t 0 (ζ, t) is well-defined in O ε 3 (ζ 0 , t 0 ) for some ε 3 > 0 and continuous at (ζ 0 , t 0 ). Thus the map (z, s, t) → g(z, s, t) := w * t 0 (f (z, s), t) is well-defined in U(ǫ) for some ǫ > 0 and continuous at the point (z 0 , s 0 , t 0 ). To finish now the proof it remains to notice that g is a restriction of the mapping (z, s, t) → w * s (z, t).

2.2.
Semicomplete weak holomorphic vector fields and families of holomorphic functions generated by them. In this section we consider weak holomorphic vector fields G for which the solution to the initial value problem (2.1) exists globally to the right. Our proofs take advantage of the methods used in [10,11,12,14]. Without loss of generality we adopt the following Assumption. For any t ∈ E the set D t := {z : (z, t) ∈ D} is not empty. For simplicity we will assume that all D t 's are domains, which is enough for our purpose. However, our arguments (with minimal changes) are also valid for the case when some of the D t 's are not necessarily connected.
Recall that by Theorem 2.3, for any (z, s) ∈ D all solutions to the initial value problem (2.1) can be obtained as restrictions of the unique non-extendable solution In the following proposition we establish some important properties of the nonautonomous semiflows generated by semicomplete weak holomorphic vector fields. Proposition 2.6. Let G : D → C be a semicomplete weak holomorphic vector field of order d ∈ [1, +∞]. Then the formula ϕ s,t (z) := w * s (z, t) defines a family (ϕ s,t ) s∈E, t∈E s of mappings ϕ s,t : D s → D t such that the following assertions hold: (i) ϕ s,t is holomorphic in D s for any s ∈ E and any t ∈ E s ; (ii) ϕ s,s = id Ds for any s ∈ E; (iii) ϕ s,t = ϕ u,t • ϕ s,u for any s, u, t ∈ E such that s ≤ u ≤ t; (iv) for any compact set K ⊂ D there exists a non-negative functionk for any u, t ∈ E and any (z, s) ∈ K satisfying s ≤ u ≤ t.
Proof. The proof of the first 3 assertions is straightforward from previous results. Indeed, assertion (i) follows from Theorem 2.3 -(iv), assertion (ii) holds by the very definition of w * s (z, t), and (iii) is a consequence of Theorem 2.3 -(i).
We are left with assertion (iv).
Denote s 0 := min pr R (K). Then, by condition WHVF3 of Definition 2.1, there exists a non-negative function k K(T ) ∈ L d [s 0 , T ], R such that |G(w, t)| ≤ k K(T ) (t) for any (w, t) ∈ K(T ). Extend k K(T ) to E by setting k K(T ) (t) = 0 for all t ∈ E \ [s 0 , T ]. Take any non-decreasing sequence (T n ) ⊂ E such that T 1 > s 0 and T n → sup E as n → +∞. Definẽ k K := χ [s 0 ,T 1 ] k K(T 1 ) + +∞ n=2 χ (T n−1 ,Tn] k K(Tn) , where χ A stands for the characteristic function of a set A. Obviously,k K ∈ L d loc (E, R) and G ϕ s,t (z), t ≤k K (t) for all (z, s) ∈ K and all t ∈ E s .
To complete the proof it remains to recall that ϕ s,t (z) − ϕ s,u (z) = t u G ϕ s,ξ (z), ξ dξ for any (z, s) ∈ D and any u, t ∈ E s . The converse of Proposition 2.6 is also true: the properties (i)-(iv) turn out to be characteristic. The exact formulation of this fact uses the following two notions from the analysis of infinitely dimensional vector-functions of a real variable.
Let U and W be some domains in the complex plane C and I ⊂ R an interval containing at least two different points. By Hol(U, W ) we denote the set of all holomorphic functions from U to W endowed with the topology of the uniform convergence.
Definition 2.7. A mapping ϕ : I → Hol(U, W ) will be called absolutely continuous on I if for any ε > 0 and any compact set K ⊂ U there exists δ = δ(ε, K) > 0 such that for any finite set of pairwise disjoint intervals (I j ) n j=1 , I j := (a j , b j ), a j , b j ∈ I, we have where · K stands for the Chebyshov norm on K, i.e., f K := sup z∈K |f (z)| for any bounded function f : K → C. A mapping ϕ : I → Hol(U, W ) will be called locally absolutely continuous if ϕ is absolutely continuous on each compact interval J ⊂ I.
The function g is the derivative of ϕ at the point t 0 and will be denoted by (dϕ/dt)(t 0 ) orφ(t 0 ).
Now we can state the following Theorem 2.10. Let d ∈ [1, +∞] and let (ϕ s,t ) s∈E, t∈E s be a family of maps ϕ s,t : D s → D t such that assertions (i)-(iv) of Proposition 2.6 hold. Then there exist a null-set N ⊂ E and a semicomplete weak holomorphic vector field G : D → C of order d such that the following statements are true: (a) for any s ∈ E the map E s ∋ t → ϕ s,t ∈ Hol(D s , C) is locally absolutely continuous; The proof of Theorem 2.10 is preceded by the following three lemmas.
Lemma 2.11. Under the conditions of Theorem 2.10, for each s ∈ E and each t ∈ E s , the function ϕ s,t is not constant in D s .
Proof. Assume on the contrary that ϕ s 0 ,t 0 is constant for some s 0 , t 0 ∈ E with s 0 ≤ t 0 . Take u 0 := (s 0 + t 0 )/2. Then ϕ s 0 ,t 0 = ϕ u 0 ,t 0 • ϕ s 0 ,u 0 by (iii). Hence either ϕ s 0 ,u 0 is constant in D s 0 or ϕ u 0 ,t 0 is constant in D u 0 . Repeating this procedure several times if necessary, we see that one could assume from the very beginning that there exist R > 0 and z 0 ∈ C such that K : Combining (ii) and (iv), we see that there exists non- The contradiction finishes the proof.
Then under the conditions of Theorem 2.10, the map Proof. Fix an arbitrary (z 0 , s 0 ) ∈ D and t 0 ∈ E s 0 . We have to prove that ϕ s,t 0 (z 0 ) → ϕ s 0 ,t 0 (z 0 ) as s → s 0 . The proof is in two steps.
Step 1. First we prove continuity from the left. So assume that s < s 0 and that s ∈ E z 0 . From (ii) and (iv) it follows that ϕ s, Step 2. If E ∋ sup E and s 0 = sup E, then the proof is finished. So we may suppose that s 0 < s < sup E. Choose any s 1 > s 0 and R > 0 such that Then it follows from (ii) and (iv) that ϕ s 0 ,s (z) → z uniformly in D(z 0 , R) as s → s 0 + 0, s < s 1 . Using Rouche's theorem one can easily show that for all s ∈ (s 0 , s 1 ) close to s enough there exists a function ψ s 0 ,s : The proof is now finished.
Then under the conditions of Theorem 2.10, the mapẼ z 0 ∋ s → ϕ s,s+h (z 0 ) is continuous.
Note that s 0 ∈ I. Moreover, for any s ∈ I we have s ≤ s 0 + h. Hence by (iv) with To complete the proof it only remains to apply Lemma 2.12 with t 0 := s 0 + h.
Proof of Theorem 2.10. First of all we notice that statement (a) follows directly from assertion (iv) applied with K := K ′ × {s}, where s ∈ E and K ′ is an arbitrary compact subset of D s .
Let us prove (b). Consider any countable expanding system (K n ) of compact sets K n ⊂ D such that ∪ n∈N int K n = D, where int A stands for the interior of a set A. (Such a system exists in any locally compact separable metric space.) To simplify the notation we will denote by k n the functionk Kn from assertion (iv).
Fix any t ∈ E and any n ∈ N. If k n (t) < +∞ and t is a Lebesgue point of k n , then Fix any s ∈ E ′ := E \ {sup E}. Let K ⊂ D s be a compact set. Then, by construction, there exists n ∈ N such that K × {s} ⊂ K n . Therefore, by (2.8), It follows from (a) that E s ∋ t → ϕ s,t (z) is locally absolutely continuous for any (z, s) ∈ D. Hence there exists a null-set N(z, s) ⊂ E s such that dϕ s,t (z)/dt exists for all t ∈ E s \ N(z, s). Fix an arbitrary s ∈ E ′ and take any sequence (z k ) ⊂ D s with at least one accumulation point in D s . Set N(s) := [∪ k∈N N(z k , s)] ∪ M. Then, owing to (2.9), Vitali's principle implies that for any t ∈ E s \ N(s), there exists a finite limit Now take any s 1 ≤ s, s 1 ∈ E. We claim that t → ϕ s,t ∈ Hol(D s , C) is also differentiable for all t ∈ E s \ N(s 1 ). Indeed, by the above argument, . By Lemma 2.11, U ⊂ D s is a domain. Hence using again (2.9) and Vitali's principle, we conclude that Finally, for the case E ∋ sup E we set N : We are left with statement (c). Define the function G : D → C in the following way: G(·, t) := dϕ s,t /dt| s:=t for all t ∈ E \ N, and G(·, t) ≡ 0 for all t ∈ N. Then, from assertion (iii), it immediately follows dϕ s,t /dt = G(·, t) • ϕ s,t for all t ∈ E \ N and all s ∈ E ∩ (−∞, t]. So it remains to show that G is a semicomplete weak holomorphic field of order d. First of all, G(·, t) ∈ Hol(D t , C) for all t ∈ E by construction. Further, for any fixed z ∈ C such that E z = ∅, we have n ϕ s,s+1/n (z) − z) → G(z, s) as n → +∞ for a.e. s ∈ E z . By Lemma 2.13, for each fixed n ∈ N the function s → ϕ s,s+1/n is continuous. It follows that G(z, ·) is measurable on E z . Therefore, G satisfies conditions WHVF1 and WHVF2 from Definition 2.1 Let us check condition WHVF3. Fix any compact set K ⊂ D. By construction, there exists n ∈ N such that K ⊂ K n . Let (z, t) ∈ K. If t ∈ N, then trivially we have 0 = |G(z, t)| ≤ k n (t). So assume that t ∈ N. By construction, t is a Lebesgue point for k n . Hence, on the one hand, Note that I(K) := pr R (K) is compact. Thus to finish the proof of WHVF3, it is now sufficient to set k K := k n | I(K) .
Finally, by the above arguments, for any (z, s) ∈ D, E s ∋ t → ϕ s,t (z) solves the initial value problem (2.1). Thus the vector field G is semicomplete. This finishes the proof.

Evolution families in simply connected domains
The notion of evolution family is one of the three central notions in modern Loewner Theory in simply connected domains. In this paper we will use the following definition of evolution family in a domain of the complex plane, which for the case of the unit disk D was formulated in [10].
. for any T > 0 and any z ∈ D there exists a non-negative function k z, One can show that all non-trivial cases of evolution families in a (fixed) domain can be reduced in one or another way to the case when D = D. We will not go into details, except for proving the following result, which will be applied further in the paper.
Let (ϕ s,t ) 0≤s≤t be a family of holomorphic self-maps of the unit disk D and F : D → C any holomorphic univalent function. Then the formula The proof is based on following The above lemma was proved for the case D := D in [10, Proposition 3.5]. The same argument with obvious modifications works for any hyperbolic domain D. Therefore we omit the proof.
We will also take advantage of the following well-known statement.
Lemma 3.4. Let U ⊂ C be a domain and F : U → C a holomorphic function. Then for any compact set K ⊂ U there exists a constant C(K) > 0 such that Proof. The lemma follows immediately from the fact that the function R(z 1 , Proof of Proposition 3.2. Let F be any of the conformal mappings of D onto D. Let us assume first that (ϕ s,t ) is an L d -evolution family in D. We have to prove that (Φ s,t ) is an L d -evolution family in D. It easy to see that (Φ s,t ) satisfies conditions EF1 and EF2. We have to prove only EF3.
By Lemma 3.3 the mapping (s, t) → ϕ s,t ∈ Hol(D, D) is continuous. Hence the set K z,T := {ϕ s,t (z) : 0 ≤ s ≤ t ≤ T } ⊂ D is compact for any fixed z ∈ D and T > 0. Therefore, by Lemma 3.4 with U := D, there exists C = C(K z,T ) > 0 such that for all s, u, and t satisfying 0 ≤ s ≤ u ≤ t ≤ T . Now the fact that (Φ s,t ) satisfies EF3 follows immediately.
The converse statement, i.e. the fact that (ϕ s,t ) is an L d -evolution family provided so is (Φ s,t ), can be proved in a symmetric way.
The authors of [10] established a deep relationship between evolution families in the unit disk D and Carathéodory non-autonomous differential equations, driven by the so-called Herglotz vector fields.
where G is an arbitrary infinitesimal generator in D. The non-autonomous version of this statement follows easily: Remark 3.7. One of the immediate consequences of the above remark is that an infinitesimal generator G D : D → C can have at most one zero in D unless it vanishes identically.
In [10] it was proved that every Herglotz vector field in D is semicomplete [10, Theorem 4.4]. Moreover, there is a one-to-one correspondence between these vector fields and evolution families. Namely, for every Herglotz vector field G : D × [0, +∞) → C of order d there exists a unique L d -evolution family (ϕ s,t ) generated by the vector field G in the following sense: for any z ∈ D and any s ≥ 0, the function [s, +∞) ∋ t → ϕ s,t (z) solves the initial value problemẇ = G(w, t), w(s) = z [10, Theorem 5.2]. Conversely, every L d -evolution family (ϕ s,t ) in D is generated by a unique (up to a null-set) Herglotz vector field G : D × [0, +∞) → C of order d [10, Theorem 6.2]. Using Proposition 3.2 and Remark 3.6 one can easily conclude that these results hold also with D replaced by any simply connected domain D C. Hence, taking into account Proposition 2.6 with D × [0, +∞) substituted for D, we can state Theorem 6.2 from [10] in the following, a bit stronger form: The following statements hold: (i) for any s ≥ 0 the mapping [s, +∞) ∋ t → ϕ s,t ∈ Hol(D, D) is locally absolutely continuous; (ii) moreover, the above assertion (i) holds locally uniformly w.r.t. s, i.e. for any T > 0 and any K ⊂⊂ D there exists a non-negative function for any s, u, t ∈ [0, T ] such that s ≤ u ≤ t.
The proof for the case D := D is complete. For arbitrary simply connected domains D C, the proposition follows now from the Riemann Mapping Theorem and Proposition 3.2.

Evolution families over systems of doubly connected domains
4.1. Definition of an evolution family in doubly connected case. As we mentioned in the introduction, the most important new property of Loewner Evolution in multiply connected case is that the canonical domain has to evolve in time, while in simply connected case the conformal type does not change. On the level of evolution families one can explain this phenomenon by the fact that all the families satisfying Definition 3.1 for D := A r with some fixed r ∈ (0, 1) are exhausted by rotations (see Example 6.1). So instead of one fixed reference domain we consider families of reference domains. A natural choice of doubly connected reference domains are the annuli A r , where r ∈ [0, 1). (Note we do not exclude the case r = 0.) With each annulus A r we can associate a one-generated torsion-free Fuchsian group Γ such that A r is conformally equivalent to D/Γ. This group is unique up to conjugation by a Möbius transformation and the conjugation classes are uniquely defined by the multiplier λ of the generator of Γ. It is not difficult to calculate that λ = e −2πω(r) , where Hence it is natural to consider families of annuli (A r(t) ) t≥0 assuming some regularity of the function t → ω(r(t)). Namely, we introduce the following . We will say that (D t ) is a (doubly connected) canonical domain system of order d (or in short, a canonical L d -system) if the function t → ω(r(t)) belongs to AC d [0, +∞), [0, +∞) and does not increase. If r(t) ≡ 0, then the canonical domain system (D t ) will be called degenerate. If on the contrary r(t) does not vanish, then (D t ) will be called non-degenerate.
Finally, if there exists T > 0 such that r(t) > 0 for all t ∈ [0, T ) and r(t) = 0 for all t ≥ T , then we will say that (D t ) is of mixed type.
Remark 4.2. The condition that t → ω(r(t)) is of class AC d implies that t → r(t) also belongs to AC d [0, +∞), [0, 1) . In the non-degenerate case, i.e. when r(t) > 0 for all t ≥ 0, or if d = 1, then the converse is also true and we can replace ω(r(t)) by r(t) in the above definition. However, in general this is not the case and for some auxiliary statements (e. g. for Suppressing the language we will refer also to the pair E := (D t ), (ϕ s,t ) as an evolution family of order d and apply terms degenerate, non-degenerate, of mixed type to E whenever they are applicable to the canonical domain system (D t ).
It is clear that the notion of an evolution family over a degenerate canonical domain system given by Definition 4.3 is the same as the notion of evolution family in the domain D := D * given by Definition 3.1. This case is known to be equivalent to that of evolution families in the unit disk fixing the origin. We will discuss it briefly in Section 5.2, while the main attention in this paper will be paid to the non-degenerate case.

4.2.
Lifting evolution families to a simply connected domain. Given a canonical domain system (D t ) and a family (ϕ s,t ) satisfying algebraic conditions EF1 and EF2 from Definition 4.3, there is a lifting of (ϕ s,t ) to the upper half-plane H := {z : Im z > 0} (or any other hyperbolic simply connected domain) satisfying conditions EF1 and EF2 from Definition 3.1. Under some additional conditions such lifting is unique. An important role in our arguments is played by the class M(r 1 , r 2 ) of all functions ψ ∈ Hol(A r 1 , A r 2 ), 1 > r 1 ≥ r 2 ≥ 0, such that I(ψ • γ) = I(γ) for any oriented closed curve γ ⊂ A r 1 , where I(γ) stands for the index of the point z = 0 w.r.t. γ.
The lifting technique allows us to apply the theory of evolution families in simply connected domains to establish some useful results in the doubly connected case. One of these results is the following analogue of Proposition 3.9, which gives a sufficient (and in fact necessary) condition for (ϕ s,t ) to be an L d -evolution family over (D t ).
Theorem 4.4. Let D t = A r(t) be a canonical domain system of order d ∈ [1, +∞] and let (ϕ s,t ) 0≤s≤t be a family of holomorphic functions ϕ s,t : D s → D t satisfying conditions EF1 and EF2 in Definition 4.3. Suppose that at least one the following conditions holds: (a) for each t > 0, each s 0 ∈ [0, t) and any z ∈ D s 0 the mapping [s 0 , t] ∋ s → ϕ s,t (z) ∈ D * is continuous; (b) for each s ≥ 0 and any z ∈ D s the mapping [s, +∞) ∋ t → ϕ s,t (z) ∈ D * is continuous; (c) for each s ≥ 0 and t ≥ s the function ϕ s,t belongs to the class M r(s), r(t) . If there exists a point z 0 ∈ D 0 such that the function [0, +∞) ∋ t → ϕ 0,t (z 0 ) ∈ D * belongs to AC d [0, +∞), D * , then (ϕ s,t ) is an L d -evolution family over (D t ).
Using the lifting technique, we can also extend assertion (ii) of Theorem 3.8 to the doubly connected setting. for all s, u, t ∈ I satisfying s ≤ u ≤ t.
Our study of the vector fields corresponding to non-degenerate L d -evolution families is also based on the lifting technique. A suitable geometry of the covering space for nondegenerate case is the one of the strip S := {z : 0 < Re z < 1}. This is a motivation for following Theorem 4.6. Let D t = A r(t) be a non-degenerate canonical domain system of order d ∈ [1, +∞]. Then for any L d -evolution family (ϕ s,t ) over (D t ) there exists a unique L d -evolution family (Ψ s,t ) in the strip S such that where W τ (ζ) := exp(ζ log r(τ )) for all τ ≥ 0 and all ζ ∈ S.
The proofs are given in Section 4.4 and based on some lemmas we are going to establish in the next section.  (iii) ϕ s,t ∈ M r(s), r(t) for any t ≥ s; (iv) ϕ s,t is univalent in D s for any t ≥ s; Proof. To prove (i) we need only to apply condition EF3 from Definition 4.3 for S := s. Fix t ≥ s. From (i) it now follows that ϕ s,u (z) → ϕ s,t (z) pointwise in D s as u → t. The functions ϕ s,t , t ≥ s, form (for fixed s ≥ 0) a normal family in D s . Therefore, the pointwise convergence implies convergence of ϕ s,u to ϕ s,t in Hol(D s , C). This proves (ii). Now let us take any closed curve γ : [0, 1] → D s . Fix t ≥ s. Recall that ϕ s,s = id Ds . Hence, (ii) implies that g(u, x) := ϕ s,u γ(x) , u ∈ [s, t], x ∈ [0, 1], provides us with a homotopical deformation of the curve γ into the curve ϕ s,t • γ within the domain D t . It follows that I(ϕ s,t • γ) = I(γ). Since γ is chosen arbitrarily, (iii) is now also proved.
To prove (iv) we argue as in [11,Proposition 3]. Assume that there exist s ≥ 0, t > s and z 1 , z 2 ∈ D s such that z 1 = z 2 but ϕ s,t (z 1 ) = ϕ s,t (z 2 ). Denote ζ j (u) := ϕ s,u (z j ), j = 1, 2. Let u 0 := inf{u ∈ [s, t] : ζ 1 (u) = ζ 2 (u)}. Clearly, u 0 ∈ [s, t]. From (i) we know that the functions ζ j are continuous. Therefore, ζ 1 (u 0 ) = ζ 2 (u 0 ) := ζ 0 . In particular, u 0 = s. At the same time, by construction Now we claim that ϕ u,u 0 | U → id U in Hol(U, C) as u → u 0 −0. The pointwise convergence ϕ u,u 0 (z) → z, z ∈ U, is a consequence of condition EF3 in Definition 4.3 applied with (s, u, u, u 0 , t) substituted for (S, s, u, t, T ). Since the functions ϕ u,u 0 , u ∈ [u 1 , u 0 ], form a normal family in U, the pointwise convergence implies convergence in Hol(U, C). In particular, it follows that given a sufficiently small open neighborhood W of the point ζ 0 , the function ϕ u,u 0 is univalent on W provided u is sufficiently close to u 0 . The fact that this statement contradicts relations (4.3) and (4.4) proves assertion (iv).    Proof of Lemma 4.9. Owing to Lemma 4.8 we can assume that assertion (c) from Theorem 4.6 takes place. Let us construct first the family (Φ s,t ) and then prove that it is an L d -evolution family.
Then t → w(t) ∈ H is continuous and B t (w(t)) = z(t) for all t ≥ 0. Fix any s ≥ 0 and any t ≥ s. Taking into account EF2, we have (ϕ s,t • B s )(w(s)) = z(t). Hence (ϕ s,t • B s )(w(s)) = B t (w(t)). Note that according to Remark 4.10, B t : H → A r(t) is a covering map. It follows that there exists a unique lifting F of ϕ s,t • B s : H → D * w.r.t. B t which takes w(s) to w(t). Now put Φ s,t := F .
The above argument defines a family (Φ s,t ) 0≤s≤t ⊂ Hol(H, H). Equality (4.5) takes place by construction, while conditions EF1 and EF2 for (Φ s,t ) follow from conditions EF1 and EF2 for ϕ s,t and the uniqueness of the lifting. Now according to Theorem 3.9 it remains to find two points w 1 , w 2 ∈ H such that the functions t → w 1 (t) := Φ 0,t (w 1 ) and t → w 2 (t) := Φ 0,t (w 2 ) are of class AC d and w 1 (t) = w 2 (t) for all t ≥ 0.
Recall that for each s ≥ 0 and t ≥ s, the function Φ s,t was constructed to be a lifting of ϕ s,t • B s : H → D * w.r.t. B t . Therefore, for each t ≥ 0, the function φ t is the lifting of ϕ 0,t • exp| S r(0) w.r.t. exp| S r(t) . It follows that for each fixed ζ ∈ S r(0) the curve φ t • γ, where γ : [0, 1] → S r(0) ; θ → ζ + 2πiθ, is a lifting of the curve Consequently, taking into account that ϕ 0,t ∈ M r(0), r(t) , we get where I(·) stands for the index of the origin w.r.t. a curve. Now it remains to show that t → w 1 (t) and t → w 2 (t) are of class AC d . Recall that t → ζ(t) is a lifting of t → ϕ 0,t (z 0 ), which is of class AC d . Hence t → ζ(t) is of class AC d as well. Finally, by definition t → ω t is also of class AC d . Therefore w 1 (t) = w(t) = R ζ(t), ω t is locally absolutely continuous and the derivative By a similar argument, w 2 (t) ∈ AC d [0, +∞), H . The proof is now complete.

Proofs.
Proof of Theorem 4.4. We have to show that (ϕ s,t ) satisfies EF3. To this end we take advantage of Lemma 4.9 stating existence of an L d -evolution family (Φ s,t ) in H such that (4.5) holds. Below we use the notation introduced in the statement and proof of this lemma.
Equality (4.5) can be written in the following form:  for any s, u, t ∈ I such that s ≤ u ≤ t.
To prove the above claim we fix I := [S, T ] ⊂ [0, +∞) and a compact set K ⊂ S r(S) and consider the set K 1 := ∪ s∈I R(K, ω s ). Since R(ζ, ω s ) is jointly continuous in ζ and s on S r(S) × I, the set K 1 ⊂ H is compact. Furthermore, it follows from Lemma 3.3 that K 2 := ∪ S≤s≤t≤T Φ s,t (K 1 ) is a compact set in H. Finally, the function Q(w, ω) is holomorphic in a neighborhood of K 2 × [0, +∞) and the function τ → ω τ is continuous on I. Hence there exists a constant C 1 = C 1 (K 2 , I) > 0 such that Q(·, ω t ) − Q(·, ω u ) K 2 ≤ C 1 |ω t − ω u | for any t, u ∈ I. By the same reason there exists a constant C 2 = C 2 (K 2 , I) > 0 such that Now we can estimate the left-hand side in (4.9) for any s, u, t ∈ I, s ≤ u ≤ t, as follows: Thus our claim follows from assertion (ii) of Theorem 3.8 and from the fact that by Definition 4.1, τ → ω τ = ω(r(t)) is of class AC d . The proof is finished.
Proof of Proposition 4.5. By condition, (ϕ s,t ) is an evolution family of order d over a canonical L d -system (D t ). Hence, obviously, it satisfies the conditions of Theorem 4.4. Therefore the statement of Claim 1 in the proof of Theorem 4.4 is true. Now Proposition 4.5 follows easily from (4.8), because the map z → exp z contracts the Euclidian metric in S r for any r ∈ [0, 1).
Proof of Theorem 4.6. Let us first construct an L d -evolution family (Ψ s,t ) satisfying (4.2). Obviously, according to Lemma 4.7 the family (ϕ s,t ) fulfills the conditions of Theorem 4.4. Hence by Lemma 4.9, there exists an L d -evolution family (Φ s,t ) in H such that (4.5) holds for all s ≥ 0 and all t ≥ s. Recall that for each τ ≥ 0, B τ = exp •Q(·, ω τ ), where Q(·, ω τ ) maps H conformally onto S r(τ ) . Therefore, bearing in mind r(τ ) > 0 for any τ ≥ 0, we have B τ (w) = W τ Q(w, ω τ )/ log r(τ ) for all τ ≥ 0 and all w ∈ H. It follows, that the family (Ψ s,t ), defined by for all w ∈ H and τ ≥ 0, satisfies equality (4.2). It is also easy to see that (Ψ s,t ) fulfills conditions EF1 and EF2. Hence, according to Proposition 3.9, to prove the existence statement of Theorem 4.6 it remains to check that for any ζ 0 ∈ S the function [0, +∞) ∋ t → Ψ 0,t (ζ) belongs to AC d [0, +∞), C . Fix any T > 0 and any ζ ∈ S. Denote w := P −1 0 (ζ). Since t → Φ 0,t (w) is continuous, the set K := Φ 0,t (w) : t ∈ [0, T ] is compact. Hence there exists a positive constant C = C(ζ, T ) > 0 such that P t (w 2 ) − P u (w 1 ) ≤ C |w 2 − w 1 | + |r(t) − r(u)| for all w 1 , w 2 ∈ K and all u, t ∈ [0, T ]. Therefore, is of class AC d by definition. Together with the above inequality this means that [0, T ] ∋ t → Ψ 0,t (ζ) is also of class AC d , which completes the proof of the fact that (Ψ s,t ) is an L d -evolution family in S.
It remains to show the uniqueness of (Ψ s,t ). To this end we fix an arbitrary ζ 0 ∈ S and take w 0 := W 0 (ζ 0 ) ∈ A r(0) . Now denote w(t) := ϕ 0,t (w 0 ) and ζ(t) := Ψ 0,t (ζ 0 ) for all t ≥ 0. Then W t ζ(t) = w(t) for any t ≥ 0. The mapping W t is a covering map of S onto A r(t) . Hence for any s ≥ 0 and any t ≥ s, the function Ψ s,t is the lifting of the mapping ϕ s,t • W s w.r.t. W t that takes ζ(s) to ζ(t). It follows that the uniqueness of the family (Ψ s,t ) is implied by the uniqueness of the continuous function [0, +∞) ∋ t → ζ(t) ∈ S such that ζ(0) = ζ 0 and W t ζ(t) = w(t) for all t ≥ 0. Such a function t → ζ(t) is unique because, according to the definition of W t , the function t → ζ(t) log r(t) is a lifting of t → w(t) w.r.t. exp : S 0 → D * . With the value ζ(0) log r(0) being fixed, this completes the proof.

Evolution families and differential equations
This section contains our main results. We will establish a one-to-one correspondence between evolution families over canonical domain systems and semicomplete weak holomorphic vector fields, analogous to the correspondence between evolution families and Herglotz vector fields in the unit disk [10] and in complex hyperbolic manifolds [11]. Moreover, we will give a precise constructive description of semicomplete weak holomorphic vector fields which appear in our setting for the non-degenerate case. At the end of the section we consider the degenerate case, which turns out to be reducible to the case of evolution families in the unit disk.
Throughout this section we fix arbitrary d ∈ (B) For any semicomplete weak holomorphic vector field G : D → C of order d the formula ϕ s,t (z) := w * s (z, t), t ≥ s ≥ 0, z ∈ D s , where w * s (z, ·) is the unique non-extendable solution to the initial value problem (5.1)ẇ = G(w, t), w(s) = z, defines an L d -evolution family over the canonical domain system (D t ).
In the situation of the above theorem we will say that G is the vector field corresponding to the evolution family (ϕ s,t ). The phrase essentially unique in this theorem means that for any two vector fields G 1 and G 2 corresponding to the same evolution family, G 1 (·, t) = G 2 (·, t) for a.e. t ≥ 0.
Proof. Assertion (B) of the theorem follows readily from Proposition 2.6. According to Theorem 2.10, in order to prove (A), we only have to check that assertions (i) -(iv) of Proposition 2.6 hold for any L d -evolution family (ϕ s,t ) over (D t ). The first three of them hold by the very definition of an evolution family over a canonical domain system. To proof assertion (iv) of Proposition 2.6 we fix any compact set K ⊂ D and any T > max pr R (K). Since K is compact, there exist finite sequences (S j ) n j=1 ⊂ [0, T ] and (K j ) n j=1 such that K j ⊂ D S j is a compact set for each j = 1, . . . , n and K ⊂ ∪ n j=1 K j × [S j , T ]. Apply now Proposition 4.5 with K j and I j := [S j , T ] substituted for K and I, respectively. Further, extend the function k K j ,I j by zero to [0, S j ) and define k T K := n j=1 k K j ,I j ∈ L d [0, T ], R . Then, for any (z, s) ∈ K and any u, t ∈ [s, T ] with u ≤ t, Take any increasing sequence (T n ) such that T n > max pr R (K) for all n ∈ N and T n → +∞ as n → +∞. Now we finish the proof of assertion (iv) of Proposition 2.6 by setting Tn) . Finally, the fact that G is essentially unique follows from statement (iii) of the theorem with s := t. The proof is finished.

5.1.
Semicomplete weak holomorphic vector fields in non-degenerate case. In this section we are going to give a precise constructive description of semicomplete weak holomorphic vector fields in D := {(z, t) : t ≥ 0, z ∈ D t } for the case when the canonical domain system (D t ) is non-degenerate, i. e., when r(t) > 0 for all t ≥ 0.
Moreover, K r (x) is increasing on [−1, −r] and on [r, 1) with K r (±r) = 1 and K r (−1) = 0. Definition 5.3. Let r ∈ (0, 1). By the class V r we will mean the collection of all functions p ∈ Hol(A r , C) having the following integral representation where µ 1 and µ 2 are positive Borel measures on the unit circle T subject to the condition µ 1 (T) + µ 2 (T) = 1.
Remark 5.4. ¿From the proof of [45,Theorem 1] it is evident that given p ∈ V r , the measures µ 1 and µ 2 in representation (5.4) are unique.
Let F ∈ Hol(A r , C) for some r ∈ (0, 1). Denote by N (F ) the free term in the Laurent expansion of F , , ρ ∈ (r, 1).
Remark 5.5. Since N (K r )=1, we have N (p) = µ 1 (T) ≥ 0 for any p ∈ V r . Now we can formulate the main result of this section. (ii) for each w ∈ D := ∪ t≥0 D t the function p(w, ·) is measurable in E w := {t ≥ 0 : (w, t) ∈ D}; (iii) for each t ≥ 0 the function p(· , t) belongs to the class V r(t) ; (iv) C ∈ L d loc [0, +∞), R . The following proposition forms a main block for the proof of sufficiency in Theorem 5.6.
Denote by C the Carathéodory class of all functions q ∈ Hol(D, C) normalized by the condition q(0) = 1 and satisfying for all z ∈ D the inequality Re q(z) ≥ 0. This class has many nice properties, one of which can be formulated as follows.
Remark 5.9. The class C is a compact convex subset of Hol(D, C). Moreover, for any continuous convex functional L : C → R, Indeed, to prove the above statement one only has to apply the Krein -Milman theorem (see, e. g., [37, p. 181]) and take into account that the set of all extremal points of C coincides with {K θ 0 : θ ∈ R} (see, e. g., [23]). Our proof of Proposition 5.7 takes advantage of the following lemmas. Recall the notation S := {w : 0 < Re w < 1}.
Lemma 5.10. Let p ∈ Hol(S, C). Suppose that Re p(w) ≥ 0 for all w ∈ S. Then for any a, b ∈ R, a < b, (i) there exists C 1 (a, b, p) > 0 such that a, b, p), for all u ∈ (0, 1).
(ii) there exists C 2 (a, b, p) > 0 such that Therefore, it is sufficient to prove the lemma only for functions p ∈ Hol(S, C) satisfying Re p(w) ≥ 0 for all w ∈ S and normalized by p(1/2) = 1. So, further on we suppose that p fulfills these conditions. Let F stand for the conformal mapping of the unit disk D onto the strip S normalized by F (0) = 1/2, iF ′ (0) > 0, namely Consider the function q := p • F . It belongs to the Carathéodory class C. For each fixed u ∈ (0, 1) the functionals q → L D 1 (q) := L 1 (q • F −1 , u) and q → L D 2 (q) := L 2 (q • F −1 , u) are convex on the class C. Therefore, according to Remark 5.9 we can restrict ourselves to the case q = K θ 0 , θ ∈ R. In this case p = p θ := K θ 0 • F −1 .
The function F extends holomorphically to the closed unit disk minus {±1}. For any u ∈ U ε the preimage of the straight line {u + iv : v ∈ R} under F is a circular arc joining points ±1, while the preimages of the segments [ia, 1 + ia] and [ib, 1 + ib] are the hyperbolic geodesics symmetric w.r.t. the real line. In particular, it follows that there exists A(a, b, ε) > 0 such that for any u ∈ U ε , |dF (z)|/|d arg z| < A(a, b, ε) when z moves along I(u). Hence, for all u ∈ U ε and all θ ∈ R.
Using again properties of the function F , we conclude that there are positive constants B 1 (a, b) and B 2 (a, b) not depending on u such that for all u ∈ U ε and all z ∈ I(u).
Hence, according to (5.11) the integrand in the right-hand side of (5.9) can be estimated as follows for all z ∈ I(u). It follows that the integral itself is not greater than This proves statement (i). Statement (ii) can be proved in a similar way if one notices that for each fixed α ∈ R the function ρ → | Im K 0 (ρe iα )| is non-decreasing on (0, 1) and that 2π 0 | Im K 0 (ρe iα )| dα = 4 log 1 + ρ 1 − ρ .
Lemma 5.11. Let r ∈ (0, 1) and F ∈ Hol(A r , C). Suppose that there exist constants α ≥ 0, w 0 ∈ S \ {∞} and a function p ∈ Hol(S, C) such that for all w ∈ S, where W (w) := exp(w log r). Then for some constant C ∈ R and positive Borel measures µ 1 and µ 2 on the unit circle T subject to the condition µ 1 (T) + µ 2 (T) = 1.
Proof. Let us prove first that for all ρ ∈ (r, 1) and some constant M > 0 not depending on ρ.
By Theorem 4.6 there exists an L d -evolution family (Ψ s,t ) in S such that ϕ s,t • W s = W t • Ψ s,t for all s ≥ 0 and all t ≥ s, where W τ (w) := exp(w log r(τ )) for all τ ≥ 0 and all w ∈ S. Further by Theorem 3.8, there exists a null-set N 2 ⊂ [0, +∞) such that for every s ≥ 0, dΨ s,t (w) dt =G t (Ψ s,t (w)) for all w ∈ S and all t ∈ [s, +∞) \ N 2 , whereG t : S → C is an infinitesimal generator for each t ∈ [0, +∞) \ N 2 . Finally, the mapping t → r(t) is of class AC d . Hence t → log r(t) is differentiable for all t ≥ 0 aside some null-set N 3 . Thus setting in the above equalities z := W s (w) and then letting s := t, we conclude that for all t ∈ [0, +∞) \ N, where N := N 1 ∪ N 2 ∪ N 3 , and all w ∈ S, Now to show that G t is of form (5.5), we fix any t ∈ [0, +∞) \ N and take advantage of Berkson -Porta representation for infinitesimal generators in the unit disk D. Namely, according to Remark 3.6 with F given by (5.8),G t (w) = F ′ F −1 (w) H F −1 (w) for all w ∈ S, where F −1 (w) = i tg π(w − 1/2)/2 and H(z) := (τ − z)(1 −τ z)p(z) for some point τ ∈ D and some function p ∈ Hol(D, C) satisfying Re p(z) ≥ 0 for all z ∈ D. Writing w 0 := F −1 (τ ), we finally get that either where in both casesp t ∈ Hol(S, C) and Rep t (w) ≥ 0 for all w ∈ S.
Assume now thatG t is given by (5.22). Then, on the one hand, by Lemma 5.12 the function J(v) :=G t (iv + 1/2), v ∈ R, should have a finite purely imaginary limit for v → +∞ or for v → −∞. On the other hand, from (5.20) it follows that J(v) can be written as a periodic function of v plus the linear term −ir ′ (t)v/ r(t) log(t) . Therefore, r ′ (t) = 0 and J(v) is an imaginary constant. It follows that G t (z) = iC t z for all z ∈ A r(t) and some constant C t ∈ R. Note that p ≡ 0 belongs to V r for any r ∈ (0, 1). So setting p t (w) = 0 for all w ∈ D t we again obtain formula (5.5). The proof is now complete. Now we are going to establish some lemmas which will be used to prove sufficiency in Theorem 5. 6. In what follows in this section we assume that d ∈ [1, +∞], (D t ) is a non-degenerate canonical domains system of order d, and D := {(z, t) : t ≥ 0, z ∈ D t }.
Lemma 5.13. Let r ∈ (0, 1). Suppose p ∈ V r is given by (5.4). Then for any z ∈ A r , Proof. Inequality 1 − r 2k ≥ 1 − r, where k ∈ N, and the Laurent expansion of K r in A r , allows us to estimate |K r (z)| and |1 − K r (r/z)|, which together with (5.4) leads to (5.23).
Lemma 5.14. Let G : D → C. Suppose that there exist functions p : D → C and C : [0, +∞) → R such that conditions (i) -(iv) are fulfilled. Then G is a weak holomorphic vector field of order d in D.
Proof. Conditions WHVF1 and WHVF2 from Definition 2.1 hold trivially.
ρ ∈ r(0), 1 . Since by definition G(·, z) is measurable for all z ∈ A r(0) and for each T > 0 there exists a non-negative k T ∈ L d [0, T ], R such that |G(z, t)| ≤ k T (t) whenever |z| = ρ and t ∈ [0, T ], it follows with the help of the Lebesgue dominated convergence theorem that t → C(t) belongs to L d loc [0, +∞), R . This proves (iv). We are left with the proof of (ii). Fix any s ≥ 0 and any w ∈ D s . On one hand, by the construction we made in the proof of Proposition 5.7, p(w, t) = 0 for a.e. t ≥ s such that r ′ (t) = 0. On the other hand, t → p(w, t) is measurable in E * := {t ≥ s : r ′ (t) = 0}, because t → r ′ (t)p(t)/r(t) = G(w, t)/w − iC(t) is measurable and t → r(t) is locally absolutely continuous in [0, +∞). Thus t → p(w, t) is measurable in [s, +∞). Statement (ii) follows now easily.

5.2.
Degenerate case. In this section we will show that if a canonical domain system (D t ) is degenerate, i.e. D t := D * := D \ {0} for all t ≥ 0, then any evolution family (ϕ s,t ) over (D t ) can be extended to an evolution family in D with a common Denjoy -Wolff point at the origin. Namely, we prove the following Proof. Fix any s ≥ 0 and any t ≥ s. Since ϕ s,t is bounded in D * , the origin is its removable singularity. By Lemma 4.7, ϕ s,t ∈ M 0, 0 . Hence lim z→0 ϕ s,t (z) = 0. The fact that (φ s,t ) satisfies conditions EF1, EF2, and EF3 of Definition 3.1 follows from the corresponding conditions in Definition 4.3 except for EF3 with z = 0, which holds by the mere fact that φ s,t (0) = 0 for all s ≥ 0 and all t ≥ s. The proof is complete.
The converse statement is obvious: if (φ s,t ) is an L d -evolution family in D and φ s,t (0) = 0 for all s ≥ 0 and all t ≥ s, then (ϕ s,t ) := (φ s,t | D * ) is an L d -evolution family over (D t ) with D t = D * for all t ≥ 0.
Taking into account Theorem 5.1 and the above Proposition 5.15, one can deduce from the results of [10] a constructive characterization of semicomplete weak holomorphic vector fields for the degenerate case. Indeed, on the one hand, [10, Theorem 1.1] establishes the 1-to-1 correspondence between Herglotz vector fields and evolution families in D, while [10, Theorem 4.8] characterizes Herglotz vector fields in terms of the Berkson -Porta representation, see Remarks 3.6 and 3.7. On the other hand, Theorem 5.1 establishes the analogous 1-to-1 correspondence between evolution families and semicomplete weak holomorphic vector fields in the doubly connected settings. Hence, in view of Proposition 5.15, semicomplete weak holomorphic vector fields of order d ∈ [1, +∞] over the degenerate canonical domain system are exactly the Herglotz vector fields, given by [10,Theorem 4.8] with τ (t) = 0 for a.e. t ∈ [0, +∞).
In this way we obtain the following analogue of Theorem 5.6 for the degenerate case. Let us recall that by C we denote the Carathéodory class of all functions p ∈ Hol(D, C) such that p(0) = 1 and Re p(w) > 0 for all w ∈ D. (i) G(w, t) = w iC(t) − α(t)p(w, t) for a.e. t ≥ 0 and all w ∈ D t ; (ii) for each w ∈ D := ∪ t≥0 D t the function p(w, ·) is measurable in E w := {t ≥ 0 : (w, t) ∈ D}; (iii) for each t ≥ 0 the function p(· , t) belongs to the Carathéodory class C; (iv) C ∈ L d loc [0, +∞), R and α ∈ L d loc [0, +∞), [0, +∞) .

Examples
Example 6.1. A set of trivial examples can be obtained by considering static nondegenerate canonical domain systems (D t ), i.e. canonical domain systems for which D t does not depend on t and does not coincide with the punctured disk D * , say D t := A r for all t ≥ 0 and some constant r ∈ (0, 1). In this case by Theorem 5.6, the semicomplete weak holomorphic vector fields of order d are exactly the functions of the form G(w, t) = iC(t)w, where C belongs to L d loc [0, +∞), R . Hence, according to Theorem 5.1, the L d -evolution families (ϕ s,t ) over static non-degenerate canonical domain systems are just families of rotations, ϕ s,t (z) = ze i(θ(t)−θ(s)) , where θ ∈ AC d [0, +∞), R . Example 6.2. According to the classical Denjoy -Wolff theorem, a self-mapping of the unit disk cannot have more than one fixed point unless it is the identity map. The infinitesimal version of this statement implies that a Herglotz vector field G(z, t) in the unit disk (see Definition 3.5) cannot have more than one zero for almost every t ≥ 0 such that G(·, t) does not vanish identically, see Remark 3.7.
For mappings of the class M(r 1 , r 2 ) the situation is different. One can have any finite number of fixed points in A r . Now we show an example of an evolution family over an L ∞ -canonical system of annuli sharing an arbitrary finite number of fixed points.

Comments on parametric representation of slit mappings
Let 0 < m < 1 < M < +∞ and A := {ζ : m < |ζ| < M}. Denote by U(A) the class of all univalent holomorphic functions f : A → C * such that f (1) = 1 and for any closed curve γ ∈ A the index I(f • γ) of the origin w.r.t. the curve f • γ coincides with I(γ).
variables z := ζ/M, w := r(t)f /m t , the initial value problem (7.1) is equivalent tȯ w = G(w, t), w| t=0 = z, where G(w, t) := w[iC(t) − p(w, t)], C(t) := 1 − λ(t) Im K r(t) κ 2 (t) −1 r(t)/m t − λ(t) Im K r(t) m t κ 1 (t) , p(·, t) := p r(t), 1 − λ(t) ν t 2 , λ(t)ν t 1 for all t ≥ 0 and all w ∈ D t , and ν t j , j = 1, 2, t ≥ 0, stands for the Dirac measure on T with the atom at κ j (t). It is easy to see that functions C and p satisfy conditions (ii) -(iv) from Theorem 5.6 with d = +∞. Hence G is a semicomplete weak holomorphic vector fields of order +∞ and f (ζ, t) = m t ϕ 0,t (ζ/M)/r(t) for all t ≥ 0, where (ϕ s,t ) is the L ∞ -evolution family over A r(t) generated by G in the sense of Theorem 5.1 -(iv). Therefore we conclude that dynamics of system (7.1) -(7.2) can be regarded as a very special case of dynamics of evolution families we consider in this paper and that statement (B) of Lebedev's theorem follows from our results. In particular, equation (5.1) with the vector field G admitting the representation given in Theorem 5.6 is a generalization of the Komatu equation [24,20] (known also as the Goluzin -Komatu equation), since (7.1) -(7.2) reduces to the latter equation when λ ≡ 0 and m = 1.