#### Abstract

We provide a new proof for the description of holomorphic and biholomorphic flows on multiply connected domains in the complex plane. In contrast to the original proof of Heins (1941) we do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the corresponding universal coverings.

#### 1. Introduction

Holomorphic flows on simply connected domains in the complex plane constitute a rich and well-studied class of maps (cf. [1]). The basic description of holomorphic flows on multiply connected domains is due to Heins [2]. In this paper we provide a new proof of Heins' results and show that, with the notable exception of the punctured disc, every holomorphic flow on a multiply connected domain in the plane is biholomorphic. We do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the universal coverings.

Let be an interval and let , be sets. As customary, a mapping can be interpreted as a one-parameter family, , of self-mappings , where runs in and vice versa.

*Definition 1. *Let be a topological space. A family of mappings , is called a (semigroup) flow on , if , , and the composition semigroup rule
holds for every , .

A flow is called* trivial* if all its mappings are equal to the identity on . Flows can be viewed as semigroups of composition operators on . The set is called the* underlying domain* of the flow. If are defined and property (1) holds for every real , then the flow is called a* group flow*. It is easy to see that if is a topological space, is a homeomorphism, and the family is a flow on , then the family is a flow on . A subset is called an* invariant space* of a flow on if all its functions map into itself. In this case the restrictions form a flow on . A singleton invariant space of a flow is called a* fixed point* of the flow. Namely, is a fixed point of a flow if for every . By* holomorphic* (resp.,* biholomorphic*)* flow* on a domain in this paper we will understand, as usual, a flow on such that all are holomorphic (resp., biholomorphic) functions from into .

#### 2. Holomorphic Flows on Multiply Connected Domains in

Holomorphic flows on simply connected domains in the complex plane are well studied. As shown in [1], any holomorphic flow on is biholomorphic.

Theorem 2 (see [1]). *If is a holomorphic flow on , then one of the following holds:*(i)*does not have fixed points in and for some .*(ii)* has one fixed point in and .*

Recall that any 2-connected domains in are biholomorphic to one of the three canonical 2-connected domains: , , and an annulus.

Let be a holomorphic flow on or . Since each of the functions is homotopic to the , it follows that every is bounded near and, therefore, its singularity at can be removed by the Riemann's singularity theorem. Thus any can be extended at as , and together the whole flow can be extended to a holomorphic flow; say , on or on correspondingly, which fixes 0.

As shown in [1] any biholomorphic flow on the unit disc consists of Möbius transformations, and if 0 is a fixed point of this flow, then for some . However, in there exist holomorphic flows that are not biholomorphic. Namely, as shown in [1], any holomorphic flow on with a fixed point, restricted to an invariant domain of the flow containing the fixed point, generates a holomorphic flow on , which is not necessarily biholomorphic. Here are two examples.

*Example 3. *The flow with , , is a holomorphic flow on . It is biholomorphic if and only if .

*Example 4. *, , is a holomorphic flow on , which is not biholomorphic.

As mentioned above, the holomorphic flows on are restrictions on of holomorphic flows on that fix 0. Theorem 2 implies that these are only the flows of type , , all of which are biholomorphic.

The next theorem describes holomorphic flows on the other multiply connected domains in .

Theorem 5 (see [2]). *(a) Every holomorphic (and biholomorphic) flow on an annulus is of type for some real . (b) Every holomorphic (and biholomorphic) flow on a -connected domain in , , is trivial.*

In contrast to Heins’ original approach we prove this theorem by utilizing the techniques of universal coverings and of liftings of holomorphic flows on the universal coverings of their underlying domains in in an operator theoretic setting. First we provide several preliminary results that are interesting on their own.

Lemma 6. *Consider the strip . Let be a fixed positive number, and let be the mapping . Any holomorphic map of into itself that commutes with has the form for some real .*

*Proof. *Let . Denote . Clearly, is a harmonic function. The commuting relation , that is, , implies

Thus, for every the function is periodic in , with period . Hence, the integral over any interval with length is constant; that is, . Now is a harmonic function, and so it is linear; say, . Observe that
Therefore, and ; that is, () for any . Note that for any , since . Hence, ; that is, for any . Denote , where the limit exists -almost everywhere on . Since , we have for every . Therefore, ; that is, -almost everywhere. Similarly, for we have , and for every ; thus -almost everywhere. Since is a bounded harmonic function, we obtain that , and therefore, ; that is, for some real number .

Recall that a Möbius transformation is* hyperbolic* if it has exactly two distinct fixed points in , the topological boundary of ; that is, , for some , . Denote by the set of all hyperbolic transformations with common fixed points , together with the identity map.

Lemma 7. *Let belong to . If is a holomorphic map commuting with , then . The set of all such maps, namely, , forms a one-parameter group.*

*Proof. *Let and be the fixed points of . Denote by the strip . Let denote the homeomorphic extension of the Riemann mapping of onto . maps onto the compactification . We may choose so that , and . Transfer the mappings and to , by setting and . We have that , , is conformal, and commutes with . Hence is a shift in ; that is, there is a such that . Lemma 6 implies that for some . Obviously, the set of all such is a one-parameter group of shifts on with with two common distinct fixed points and . Finally, , (including ), are all conformal transformations of , that is, Möbius transformations with two common distinct points . Clearly, is a one-parameter group on .

Recall some facts about universal covering spaces of domains in . Let be a domain in , or, an open Riemann surface. Let be the (simply connected) universal covering space of , the covering mapping (the projection) of which is a local homeomorphism. Note that since is simply connected, then is homeomorphic either to or to . It is clear that under the complex structure on pulled back by from , is a holomorphic map. There is a group of Möbius transformations of onto itself with the following properties:(i) acts discretely on ;(ii) for every fixed ;(iii) for every .

Recall the standard construction and properties of the universal covering of (cf. [3]) that we will need in the sequel. If and are topological spaces and is an interval in , then a jointly continuous mapping is called a* homotopy* on . As mentioned above, the homotopy will be denoted also by , . If , are topological spaces and , then a* homotopy connecting * and is any continuous function such that , and . Two mappings are called* homotopic* to each other if there exists a homotopy that connects them.

Consider the pointed space . The covering space of is defined as the set of homotopy classes (with both ends fixed) of the family of all paths with initial point , equipped with the standard topology. The covering mapping is defined by , where . The group is defined as the family of homotopy classes of closed loops with . Such acts on any by , where is the end-to-end path initiating at , terminating at , with (see, e.g., [3, Ch. 1, Sec. 7]) for definition of the path groupoid.

In the case of the next lemma is proven in (cf. [3, Ch. 2, Sec. 2, Theorems 2 and 3]). The proof of the general case goes along similar lines.

Lemma 8 (uniqueness of homotopy lifting). *Let be a connected space, a fixed continuous mapping, and , a homotopy on such that . Then there is a unique homotopy with such that *(i)*;*(ii)* for every .*

Clearly, every function of a flow on is homotopic to .

Corollary 9. *Let be a proper subset of with universal covering , and let be the corresponding group of Möbius transformations. Assume that is a homotopy on with . Then there is a unique homotopy , , such that *(1)*;*(2)* for all ;*(3)* for every and all .*

*Proof. *We apply Lemma 8 with , and the homotopy with . Clearly, . By Lemma 8 there is a homotopy with , such that , and . If we denote by , then, clearly, conditions (1) and (2) are satisfied. The homotopy is uniquely defined, for two homotopies on with and being two liftings of the homotopy , , in contradiction with Lemma 8.

To prove condition (3), fix a and define . Clearly, , and, by property (iii) of covering mappings, . Thus, and are two homotopies on that satisfy conditions (1) and (2). By the uniqueness of homotopy liftings (Lemma 8) we obtain . Hence, for any .

Corollary 10. *Under the notations and assumptions of Corollary 9, if the homotopy is a flow on a set , then the lifted homotopy is a flow on .*

*Proof. *Fix a and define a new homotopy , , by
where is the lifted by Corollary 9 homotopy. By the continuity argument, is well defined and satisfies conditions (1) and (2) from Corollary 9. Indeed, , and for all . The semigroup property (1) of implies that for any we have
By the uniqueness part of Corollary 9 it follows that . Therefore, for any we have . Hence is a (semigroup) flow on .

In the next corollary we summarize the above results.

Corollary 11. *Let be a -connected domain in , , with universal covering and corresponding group of Möbius transformations . If admits other than the identity holomorphic self-map , then either *(i)* does not contain any hyperbolic Möbius transformations, or*(ii)* is commutative, in fact an infinite cyclic group, and is homeomorphic to an annulus.*

*Proof. *Suppose that the group contains a hyperbolic transformation with fixed points ; that is , . Let , be a holomorphic function on . Since the lifting of commutes with , it commutes also with . Lemma 7 implies that . Again by Lemma 7, with , we see that every holomorphic self-map of commuting with belongs to . Therefore, for every , thus . Since is a commutative one-parameter group and is discrete, we conclude that is a cyclic group. Therefore, is homeomorphic to an annulus and not to the punctured disc.

Let be a holomorphic function on a -connected domain that is homotopic to the identity, be a homotopy on connecting with , and be the unique, by Corollary 9, lifting of the family on the universal covering of with .

Lemma 12. *If is a non-simply-connected domain in and , is a holomorphic function on a -connected domain that is homotopic to the identity, then both and its lifting on the covering of are fixed-point free.*

*Proof. *Since is multiply connected, then its universal covering is homeomorphic to the unit disc . Suppose that is a fixed point for ; that is, . Since on , we have that for any . Therefore, . We claim that . To proof this we show first that there is a such that .

First we describe in detail the lifting of the homotopy connecting and , so that . For any given path in with initial point and we define as , , . Clearly, the mapping satisfies properties (1), (2) and (3) of Corollary 9. In particular, on . The uniqueness of the homotopy lifting implies that , and, in particular, .

Consider now the constant loop , and let . By , we have that ; that is, is a fixed point of . Since and since commutes with all , we see that . Hence, every point of type is a fixed point of . Since is a disc, the holomorphic mapping with more than one fixed point must be the identity. Consequently, , in contradiction with the hypotheses.

Proposition 13. *The only holomorphic mapping on a -connected domain in , , that is homotopic to the identity is the identity itself.*

*Proof. *Let be a map as in Theorem 5, and let be the lifting of on the universal covering of that commutes with the group , that is, such that for any . By Lemma 12, has no fixed points in , since is not simply connected. By the Denjoy-Wolff theorem, there is an , such that for every we have , where is the th composition of with itself. We have also that for every . Since , it follows that . Therefore, for every . Thus all Möbius transformations in have a common fixed point . If we map onto the upper half-plane by a Möbius transformation, so that is mapped at the point at and transfer on , then consists of Möbius mappings preserving the half-plane with fixed points at , that is, of type for some real , .

According to Corollary 11, if contains a hyperbolic mapping , then it is homeomorphic to an annulus. Hence is 2-connected, in contradiction with the hypotheses. If is not homeomorphic to an annulus, then all elements of must be parabolic, elliptic, or, the identity. No transformation of type with , real, can be elliptic, and, therefore, is parabolic; hence . Consequently, the members of are translations , . Since is discrete, it must be an infinite cyclic group of the form , where with some . Hence, the domain is homeomorphic to the punctured disc, which implies that is 2-connected, contrary to its hypothesis. This completes the proof.

*Proof of Theorem 5. *(a) If is an annulus in , then its universal covering is the strip . Let be a holomorphic flow on , and is its lifted flow on . Each commutes with the group of corresponding Möbius transformations, which consists of translations , . Let with . Since , Lemma 6 implies that , . Therefore the semigroup flow consists of translations , whose projection on the annulus are rotations. Hence for every , for some real . The semigroup property implies that there is a real such that .

(b) Let be a holomorphic flow on a -connected domain in , . Since the mappings are holomorphic functions on , then their liftings on , the universal covering of are holomorphic mappings on , since the projection is holomorphic. The proposition from the above implies that the flow is trivial.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The research of T. Tonev was partially supported by Grant no. 209762 from the Simons Foundation.