Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 205–222

UNIFORMLY QUASIREGULAR SEMIGROUPS IN TWO DIMENSIONS

A. Hinkkanen

University of Illinois at Urbana–Champaign, Department of Mathematics Urbana, IL 61801, U.S.A.; aimo@math.uiuc.edu

Abstract. Let G be a semigroup of K -quasiregular or K -quasimeromorphic functions mapping a given open set U in the Riemann sphere into itself, for a fixed K, the semigroup operation being the composition of functions. We prove that if G satisfies an algebraic condition, which is true for all abelian semigroups, then there exists a K -quasiconformal homeomorphism of U onto an open set V such that all the functions in f ∘G∘f^{−1} are meromorphic functions of V into itself. In particular, if U is the whole sphere then the elements of f ∘ G ∘ f^{−1} are rational functions. We give an example of a semigroup generated by two functions on the sphere, each quasiconformally conjugate to a quadratic polynomial, that cannot be quasiconformally conjugated to a semigroup of rational functions. We give another such example of a semigroup of K -quasiconformal homeomorphisms. These results extend and complement a similar positive conjugacy result of Tukia and of Sullivan for groups of K -quasiconformal homeomorphisms.

1. Introduction and results

Let U be a non-empty open set in the extended complex plane or Riemann sphere \overline{C} = C ∪ \{∞\} where C denotes the complex plane. In two dimensions we may define a K -quasimeromorphic map of U (into \overline{C}) by saying that f is K -quasimeromorphic if we can write f = φ ∘ h where h is a K -quasiconformal map on U (hence, by definition, a homeomorphism) and φ is a meromorphic function defined on h(U). If f never takes the value ∞, we say that f is K - quasiregular. The composition of two quasiregular or quasimeromorphic maps is also quasiregular or quasimeromorphic. Hence we may talk about a semigroup of quasiregular or quasimeromorphic maps of U into itself, the semigroup operation being the composition of functions. A special case would be a group of quasiconformal maps of U onto itself. If all the elements of such a semigroup G are K -quasiregular or K -quasimeromorphic for a fixed K ≥ 1, we say that G is a K - quasiregular or a K -quasimeromorphic semigroup. If this is true for some K, we say that G is a uniformly quasiregular semigroup or a uniformly quasimeromorphic semigroup. A special case would be a K -quasiconformal group or a uniformly quasiconformal group.

We shall assume throughout that each element of G is non-constant in each component of the open set U. If U is connected and if this restriction is not

1991 Mathematics Subject Classification: Primary 30C62, 20M20.

Research partially supported by the Alfred P. Sloan Foundation and by the U.S. National Science Foundation grant DMS 94-00999.

imposed, then the subset of G consisting of non-constant functions is a subsemigroup while any homeomorphism would conjugate the constant functions in G to constant functions. Thus, if U is connected, our results extend immediately to the case when G is allowed to contain constant functions.

Uniformly quasiconformal groups and uniformly quasiregular or quasimeromorphic semigroups can certainly arise as follows. Let U and V be open sets in \overline{C}, let f be a K -quasiconformal map of U onto V , and let G′ be a group of conformal self-mappings of V (for suitable V , this could be a group of Möbius transformations of V onto itself), or let G′ be a semigroup of meromorphic functions of V into itself. Then G = f^{−1} ∘ G′ ∘ f is a K^{2} -quasiconformal group, or a K^{2} -quasiregular or a K^{2} -quasimeromorphic semigroup of functions taking U onto or into itself. In particular, in the case of a semigroup rather than a group, we could have U = V = \overline{C}, and G′ could be a semigroup of rational functions (containing at least one function of degree at least 2, say). Tukia [4] and Sullivan [3] proved that any K -quasiconformal group arises in this way, and Tukia [4] showed furthermore that if G is a K -quasiconformal group then we may take f to be K_{1} -quasiconformal where K_{1} depends on K only and K_{1} ≤ K. In this note, we shall generalize Tukia’s method so as to extend his result to K -quasiregular and K -quasimeromorphic semigroups generated by a single function, and to certain other suitable semigroups. We provide a counterexample to show that in case of a semigroup generated by two functions, each of which is individually conjugate to a quadratic polynomial, there does not have to exist a quasiconformal conjugacy of the whole semigroup to a semigroup of meromorphic (in this case, rational) functions.

Theorem 1. Let U be a non-empty open set in \overline{C}, and let G be a semigroup of K -quasiregular or K -quasimeromorphic functions of U into itself, generated by a single function. Then there is a K_{1} -quasiconformal map f of \overline{C} onto itself, taking U onto an open set V such that every element in the semigroup f ∘G∘f^{−1} is a meromorphic function of V into itself. In particular, if U = \overline{C} then G is quasiconformally conjugate to a cyclic semigroup of rational functions. Here K_{1} depends on K only and K_{1} ≤ K^{2} .

We may take K_{1} = (\sqrt{K^{2} + 1/K^{2}} + K − 1/K)/\sqrt{2} ≤ min\{K^{\sqrt{2}}, K\sqrt{2} \} (compare [4, p. 77]). This bound is obtained from Tukia’s by replacing K by K^{2} in his formula.

Theorem 2. There exist quadratic polynomials F_{1} and F_{2} and quasiconformal mappings φ_{1} and φ_{2} of \overline{C} such that the functions f_{j} = φ_{j} ∘ F_{j} ∘ φ_{j}^{−1} , for j = 1, 2, generate a uniformly quasiregular semigroup G in \overline{C}, but there is no quasiconformal homeomorphism f of \overline{C} such that f ∘ G ∘ f^{−1} consists of meromorphic (hence, rational) functions.

As the proof of Theorem 2 will show, the reason why Tukia’s method of proof for groups fails for general semigroups, is, in a sense, a lack of relations between the elements of a semigroup. If G is a group and g, h ∈ G then there is u ∈ G satisfying some relation involving g and h, such as g = h ∘ u or g = u ∘ h. It appears that free (uniformly quasimeromorphic) semigroups are particularly poor candidates for the existence of a quasiconformal conjugacy to a semigroup of meromorphic functions. However, for a semigroup admitting sufficiently many relations, a proof can be made to work with adequate care. There are many results of that kind, and we provide the following example. We denote the identity mapping of any set by Id. For any function f we denote the iterates of f by f^{n} so that f^{0} = Id and f^{n+1} = f ∘ f^{n} for all n ≥ 0.

Theorem 3. Let U be a non-empty open set in \overline{C}, and let G be a K - quasimeromorphic semigroup of functions mapping U into itself. Suppose that G has the following property:

(1.1)  for all g, h ∈ G there are φ, ψ ∈ G ∪ {Id} such that g ∘ φ = h ∘ ψ.

Then there exists a K_{1} -quasiconformal homeomorphism f of \overline{C} onto itself, taking U onto an open set V , such that the semigroup f ∘G∘f^{−1} consists of meromorphic functions taking V into itself. Here K_{1} depends only on K.

In particular, this holds if G is abelian, for then we may take φ = h and ψ = g in (1.1).

In Theorem 3, we get the same estimate for K_{1} as given after Theorem 1. One can ask what happens if the condition g ∘ φ = h ∘ ψ in (1.1) is replaced

by φ ∘ g = ψ ∘ h. A simple modification of the proof of Theorem 3 does not yield the desired result, basically because of the following fact: if f ∈ G and if F is any branch of f^{−1} then f ∘ F = Id while we need not have F ∘ f = Id. It remains an open question whether the conclusion of Theorem 3 still holds for some other reason, after modifying (1.1) in this way.

Even though, in connection with semigroups, it may be of greater interest to consider non-homeomorphic mappings, it may be worth pointing out that an easy-to-verify counterexample can be found for semigroups of homeomorphisms.

Theorem 4. Let h be a K -quasiconformal mapping of the strip S = {x+iy : 0 ≤ x ≤ 1} onto itself that fixes every boundary point of S including each of the prime ends at +i∞ and −i∞. Suppose that h is not the identity mapping of S. Define f_{1}(z) = h(z) + 1 when z ∈ S and f_{1}(z) = z + 1 when z ∈ \overline{C} \ S. Define f_{2}(z) = z + 1 for all z ∈ \overline{C}. Then f_{1} and f_{2} generate a K -quasiconformal semigroup G, but G cannot be quasiconformally or even topologically conjugated to a semigroup of conformal mappings.

Roughly speaking, if the semigroup G of Theorem 4 could be conjugated to a semigroup of conformal mappings, then both f_{1} and f_{2} would be conjugated to the same mapping, which is impossible if f_{1} ≠ f_{2} , that is, if h ≠ Id. The actual proof of Theorem 4 is not phrased exactly in this way, but a consideration of the elements (necessarily translations, if the point at infinity is fixed as we may assume) of the hypothetical conjugated semigroup shows that this would be the case.

Obviously, there are many other similar examples. For example, the function f_{2}(z) = z + 1 in Theorem 4 could be replaced by λz for some λ > 1, and then S would be replaced by the annulus {z : 1 ≤ |z| ≤ λ}.

The positive results (Theorems 1 and 3) are based on the concept of a grand orbit. If z ∈ U, we say that a point w ∈ U is in the grand orbit of z under the semigroup G provided that w = g(z) for some g belonging to the formal group generated by G. The elements of this formal group are obtained by combining finitely many elements, each of which lies in G or corresponds to a branch of the inverse of some element of G. More precisely, we require that there are x_{1}, x_{2}, …, x_{n} ∈ U, z_{1}, …, z_{n−1} ∈ U, and h_{1}, …, h_{n−1}, h′_{1}, …, h′_{n−1} ∈ G ∪ {Id} such that n ≥ 2, w = x_{1} , z = x_{n}, and

h_{i}(z_{i}) = x_{i} and h′_{i}(z_{i}) = x_{i+1} for 1 ≤ i ≤ n − 1.

Theorem 2 shows that there are essentially more uniformly quasimeromorphic semigroups than there are semigroups of meromorphic functions. For semigroups of meromorphic functions, one can develop a Fatou–Julia theory analogously to the iteration theory due to Fatou and Julia based on the concept of a normal family (for semigroups of rational functions, see [1]). Thus, for a semigroup G defined on an open set U, the set of normality or the Fatou set of G consists of those points z in U that have a neighbourhood D such that the restrictions of the elements of G to D form a normal family. Standard results for sequences of K -quasimeromorphic mappings show that the limit function of any convergent subsequence is then a constant (possibly infinity) or a non-constant K -quasimeromorphic function. The Julia set would be J(G) = U \ N(G). (In unpublished lecture notes of the author from 1989, for a course at the University of Texas at Austin, it is shown that for the iteration of a single meromorphic function f defined in a plane domain U without the condition that f(U) ⊂ U (for z ∈ N(G) one requires, in addition, that all the iterates of f are defined in the neighbourhood D), a Fatou–Julia theory can be developed, and that the classification of the components of the set of normality follows the same lines as in the classical case, the new cases that arise corresponding merely to the fact that the prime ends of U need not coincide with single topological boundary points of U.) One may ask, of course, under what circumstances a uniformly quasimeromorphic semigroup is topologically (even if not quasiconformally) conjugate to a semigroup of meromorphic functions. A topological conjugacy would still preserve the structure of the dynamics.

I would like to thank the referee for his helpful remarks.

2. Proof of Theorem 1

Of course, Theorem 1 is a special case of Theorem 3, but we prefer to give a separate proof to fix ideas that can be expanded later. Let the assumptions of Theorem 1 be satisfied, except that G need not be cyclic until explicitly specified, and set k = (K − 1)/(K + 1) ∈ [0, 1). Write ∂g = ∂g/∂z and \overline{∂}g = ∂g/∂ \overline{z}. For each g ∈ G, the complex dilatation μ(z, g) = \overline{∂}g/∂g is defined a.e., is measurable, and satisfies |μ(z, g)| ≤ k for a.e. z ∈ U. We look for an essentially bounded complex-valued function μ defined in \overline{C} with ∥μ∥_{∞} < 1 such that if f is a quasiconformal homeomorphism of \overline{C} onto itself with μ(z, f) = μ(z) for a.e. z then for every g ∈ G, the function f ∘ g ∘ f^{−1} , defined on the open set V = f(U), is meromorphic in V . This will be the case if, and only if, for each g ∈ G, we have μ(z) = μ(z, f) = μ(z, f ∘ g) for a.e. z ∈ U. The standard formula for μ(z, f ∘ g) [2, (5.6), p. 183] gives

(2.1)  μ(z, f ∘ g) = \frac{μ(z, g) + μ(g(z), f)e^{−2i arg ∂g(z)}}{1 + μ(z, g) μ(g(z), f)e^{−2i arg \overline{∂}g(z)}} ,

so that with μ(z, f) = μ(z) we require that

(2.2)  μ(z) = \frac{μ(z, g) + μ(g(z))e^{−2i arg ∂g(z)}}{1 + μ(z, g) μ(g(z))e^{−2i arg \overline{∂}g(z)}} ≡ T_{g,z}(μ(g(z))).

Here T_{g,z} is the Möbius transformation of the unit disk D onto itself given by

T_{g,z}(w) = \frac{a + \overline{b}w}{b + \overline{a}w}

where a = \overline{∂}g(z) and b = ∂g(z). In particular, if g is the identity map Id then T_{g,z}(w) ≡ w, that is, T_{Id,z} = Id, for all z. By the definitions and by the chain rule T_{h∘g,z} = T_{g,z} ∘ T_{h,g(z)} so that T_{h∘g,z}^{−1} = T_{h,g(z)}^{−1} ∘ T_{g,z}^{−1} .

The formula (2.1) states that

(2.3)  μ(z, f ∘ g) = T_{g,z}(μ(g(z), f)).

We set μ ≡ 0 outside U. To define μ in U so as to satisfy (2.2), we recall the ideas of Tukia [4]. Let us first ignore any problems concerning measurability and the fact that complex dilatations are defined and formulas such as (2.2) hold only almost everywhere.

Write B(z, r) = {w : |w − z| ≤ r} and B(r) = B(0, r). Define for z ∈ U,

S(z) = {μ(z, g) : g ∈ G} ⊂ B(k).

If h ∈ G then by (2.3),

S(h(z)) = {μ(h(z), g) : g ∈ G} = {T_{h,z}^{−1}(μ(z, g ∘ h)) : g ∈ G} ⊂ T_{h,z}^{−1}(S(z)) and so

(2.4)  T_{h,z}(S(h(z))) ⊂ S(z).

The above considerations are valid in any semigroup. Let us now assume that G is generated by a function γ. Thus G = {γ^{n} : n ≥ 1} or possibly G = {γ^{n} : n ≥ 0}. We may assume that γ ≠ Id, for otherwise there is nothing to prove. It clearly suffices to conjugate γ to a meromorphic function.

If G were a group, then we would have the equality T_{h,z}(S(h(z))) = S(z) instead of (2.4). This would allow us to continue along the lines of Tukia and complete the proof. In general, equality need not hold here even if G is a cyclic semigroup. In this case, if G contains the identity mapping, we have S(z) = T_{h,z}(S(h(z))) ∪ {0}, and usually we need not have to have 0 ∈ T_{h,z}(S(h(z))). If G is cyclic and does not contain the identity mapping then again usually S(z) \ T_{h,z}(S(h(z))) consists of one point (and in any case contains at most one point). Therefore the set S(z) needs to be replaced by another set with the required invariance property. It is hard to see how this could be done except by replacing S(z) by a larger set. This then means that the larger set might no longer be a subset of B(k). We can only hope that the larger set will be contained in B(k_{1}) for some fixed k_{1} < 1. The general principle is that one needs to add to S(z) the complex dilatations of all the functions (“words”) in the “group” generated by the elements of the semigroup and by the locally defined branches of their inverses. To ensure that all such dilatations lie in some B(k_{1}), one should know, for example, that only words of a fixed length in this group are needed to cover all words. For a general semigroup (for example, a free semigroup with at least two generators), it is not true for any finite N that only words of length at most N will suffice to ensure the required complete invariance of the enlarged S(z). If there are relations in the group then some fixed N may suffice, and this is what will happen in the proof of Theorem 3. Even if all words are needed, it could happen by coincidence, in some special case, that the enlarged set still lies in some fixed B(k_{1}). For more discussion and examples, see the remarks after the statement of Lemma 1 in Section 4, and the proof of Theorem 3.

In the case of a cyclic group, we add to S(z) all complex dilatations that would be there if G were a group generated by γ. Thus we let S_{1}(z) consist of the complex dilatations at z of all mappings g of the form γ^{n} where n ≥ 0, or a branch of γ^{−n} for some n ≥ 1 whenever such a branch is defined in a neighbourhood of z, or of the form γ^{−m} ∘ γ^{n} where m, n ≥ 0. Clearly S(z) ⊂ S_{1}(z). We proceed for a while without worrying about the question of for which points z, the set S_{1}(z) is well defined, with all branches of all inverses being defined in some neighbourhood of z (depending on the branch, of course).

Pick z ∈ U and h ∈ G. As g goes through all maps considered in the definition of S_{1}(h(z)), the maps g ∘ h go through exactly all the maps considered in the definition of S_{1}(z). As the γ^{n} and the branches of their inverses are locally quasiconformal, the formula (2.3) still applies, and we see that (2.4) holds with equality.

We note that each inverse branch of any γ^{n} is K -quasiconformal since γ^{n} is K -quasimeromorphic. Hence any function γ^{−m} ∘ γ^{n} is K^{2} -quasimeromorphic. Thus S_{1}(z) ⊂ B(k_{1}) where k_{1} = (K^{2} − 1)/(K^{2} + 1) < 1. So in the cyclic case, S(z) can be enlarged without having to enlarge the hyperbolically bounded set B(k) that is guaranteed to cover every (S(z) and here also) S_{1}(z) to more than this B(k_{1}).

We now complete the proof of Theorem 1 following Tukia’s method. For each S_{1}(z), there is a unique point P (z) such that among all hyperbolic disks (in the hyperbolic metric of the unit disk) containing S_{1}(z), there is one centred at P (z) with the smallest possible hyperbolic radius [4, p. 75]. Since Möbius transformations such as T_{h,z} that map the unit disk onto itself, preserve hyperbolic distances and hyperbolic disks, it follows from (2.4) with equality that P (z) = T_{h,z}(P (h(z))). Thus defining μ(z) = P (z), we find a function μ satisfying (2.2). The estimate for K_{1} follows directly from Tukia’s paper according to our reference, since S_{1}(z) ⊂ B(k_{1}).

We finally have to address the question of definability and measurability of μ. The set that formally contains G and, for each element of G, its at most countably many inverse branches, is countable. Each γ^{n} can be written as γ^{n} = φ_{n} ∘ h_{n}, where h_{n} is a quasiconformal homeomorphism and φ_{n} is meromorphic in h_{n}(U). Hence, for each n, there are only countably many points z ∈ U at which some branch of the inverse of γ_{n} has a branch point (preventing us from defining a single-valued branch of the inverse in some neighbourhood of z). For a.e. z ∈ U it is therefore true that the sets S(z) and S_{1}(z) are completely well defined, that is, all conceivable inverse branches can be considered and all functions involved have a well defined complex dilatation at z. It follows as in Tukia’s paper that the function μ defined above is measurable. Thus we have the estimate on |μ(z)| for almost every z, giving rise to the cited upper bound for K_{1} . In view of the remarks made at the beginning of Section 2, this completes the proof of Theorem 1.

3. Proof of Theorem 3

We first dispose of some technicalities. We proceed to define a complex dilatation by following a variant of Tukia’s method. This leads to a measurable function provided that we are dealing with countably many elements, including the elements of the semigroup and branches of their inverses. This situation arises when the semigroup itself is countable. Suppose that the case of a countable semigroup has been dealt with.

Let then G be any semigroup satisfying the assumptions of Theorem 3. As quasimeromorphic mappings belong locally to certain Sobolev spaces that are separable, it follows that G has a countable dense subset L. So for any g ∈ G there is a sequence h_{n} ∈ L such that h_{n} → g locally uniformly on U with respect to the spherical metric. Now L generates a countable semigroup ⟨L⟩. If ⟨L⟩ does not satisfy the assumptions of Theorem 3, then set L_{0} = ⟨L⟩. Suppose that L_{n} is a countable subsemigroup of G, and let M_{n} consist of all elements of L_{n} and all the elements φ, ψ of G that would have to be added to L_{n} so as to satisfy (1.1) for any pair of functions g, h ∈ L_{n}. Define L_{n+1} to be the semigroup generated by M_{n}. Then L_{n+1} is a countable subsemigroup of G. Set L_{∞} = ⋃_{n=0}^{∞} L_{n}. Then L_{∞} is a countable subsemigroup of G that satisfies the assumptions of Theorem 3. Thus, by our assumption concerning the countable case, there is a quasiconformal homeomorphism f of \overline{C} mapping U onto an open set V such that f ∘ L_{∞} ∘ f^{−1} consists of meromorphic functions mapping V into itself. But then, if g ∈ G, h_{n} ∈ L ⊂ L_{∞}, and h_{n} → g locally uniformly on U, the functions f ∘ h_{n} ∘ f^{−1} are meromorphic and tend to f ∘ g ∘ f^{−1} locally uniformly in V . Hence f ∘ g ∘ f^{−1} is meromorphic in V , as required, and the proof of Theorem 3 will be complete after dealing with the countable case.

Suppose then that G is countable and satisfies the assumptions of Theorem 3. Consider finite words of the form g_{1}∘g_{2}∘⋯∘g_{n} where each g_{j} is either an element of G, or the identity mapping (in case Id ∉ G), or is a branch of the inverse of some element of G. For almost every z ∈ U, it is true that for every such word, with the exceptions to be described below, there is a neighbourhood of z in which we may define the function described by this word, no matter how we choose the branches of the inverse functions at the appropriate points, and furthermore, each component function g_{j} is locally homeomorphic and quasiconformal at the appropriate points and has, at these points, a well defined complex dilatation. Thus the complex dilatation of each such word can be defined at z. We take S(z) to be the set of all such complex dilatations at z.

There are exceptions that may occur when h(U) is a proper subset of U for some h ∈ G. For example, if the word is h^{−1} ∘ g where g, h ∈ G and if g(z) ∉ h(U), then (h^{−1} ∘ g)(z) is not defined. For each z, we simply ignore all such words, depending on z. Also, even if g(z) ∈ h(U), for certain z there may be more branches of h^{−1} defined and locally homeomorphic at g(z) than for other values of z, and for each z, we simply use all branches that are defined and locally homeomorphic at g(z). The same principles apply to all words in an obvious way. This completes the definition of S(z).

For a.e. z ∈ U it is the case that S(z) is well defined at every point on the grand orbit of z under G, where the grand orbit of z under G is the set of all points of the form (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(z) for all words described above. Using (2.3), it is easily checked that for a.e. z ∈ U it is true that S(z) = T_{h,z}(S(h(z))) for all h ∈ G. This construction can be performed for any semigroup G, and makes sense and has the above invariance property for a.e. z ∈ U whenever G is countable.

We shall now prove that under the special conditions of Theorem 3, there is a number k_{1} < 1 such that S(z) ⊂ B(k_{1}) for a.e. z. Then we can define P (z) and μ(z) as in the proof of Theorem 1, and the rest goes through in the same way as there. In particular, K_{1} will depend only on K provided that k_{1} depends only on K.

Suppose that (1.1) of Theorem 3 holds. The function represented by a word g_{1}∘g_{2}∘⋯∘g_{n} is not changed if the word is normalized in such a way that n is even with n ≥ 2, g_{j} ∈ G ∪ {Id} whenever j is even, and g_{j} is the branch of the inverse of an element of G∪{Id} denoted by h_{j} if j is odd. So we assume that we consider a normalized word, and claim that the function it represents can be written in the form u ∘ v where v ∈ G ∪ {Id} while u is the branch of the inverse of an element of G ∪ {Id}. We prove this claim by induction on the even integer n ≥ 2. The claim is clearly true for n = 2 by definition of the normalization. Suppose that it is true for a certain n ≥ 2. Consider a normalized word g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n+2} as above. By the induction assumption, in some neighbourhood of z, the function g_{3} ∘ g_{4} ∘ ⋯ ∘ g_{n+2} is equal to u ∘ v where v ∈ G ∪ {Id} while u is the branch of the inverse of an element b of G ∪ {Id}. Also there is h_{1} ∈ G ∪ {Id} such that g_{1} is a branch of h_{1}^{−1} in some neighbourhood of an appropriate point. By (1.1), there are φ, ψ ∈ G ∪ {Id} with φ ∘ g_{2} = ψ ∘ b. (Note that (1.1) can be trivially satisfied if g = Id or h = Id in (1.1).) It follows that φ ∘ g_{2} ∘ u = ψ ∘ b ∘ u = ψ. Thus ψ ∘ v = φ ∘ g_{2} ∘ u ∘ v = φ ∘ h_{1} ∘ g_{1} ∘ g_{2} ∘ u ∘ v. Now define Ψ = ψ ∘ v ∈ G ∪ {Id} and Φ = φ ∘ h_{1} ∈ G ∪ {Id}. We see that

Ψ = Φ ∘ (g_{1} ∘ g_{2} ∘ u ∘ v) = Φ ∘ (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n+2}).

This shows that for a suitably chosen branch of Φ^{−1} in some neighbourhood of an appropriate point, which branch we simply denote by Φ^{−1} , we have

g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n+2} = Φ^{−1} ∘ Ψ,

as was to be proved.

Now, since Φ^{−1} and Ψ above are both locally K -quasiconformal, so that Φ^{−1}∘Ψ is locally K^{2} -quasiconformal, it follows that the element of S(z) generated by the word considered lies in B(k_{1}), where k_{1} = (K^{2}−1)/(K^{2}+1). This together with what has been said before, completes the proof of Theorem 3.

4. Proof of Theorem 2

4.1. To keep the calculations manageable, we construct a simple counterexample. However, it is clear from the principles involved that there are many other similar examples. We first describe what the example is going to be. Choose positive numbers δ and ε smaller than, say, 10^{−2} . We also require that 6\sqrt{δ} < ε and that

(4.1)  (1 + ε)^{2^{N}} = 2 + ε + \frac{1}{2}(\sqrt{δ} − δ)

for some integer N ≥ 2. We can take N as large as we like; such a choice would merely require us to choose sufficiently small numbers ε and δ. Set ω = exp(πi/(2^{N} − 1)) and a = 2 + ε. Thus

ω^{1+2+···+2^{N−1}} = ω^{2^{N} −1} = −1

and so the N^{th} iterate of the function z ↦ ωz^{2} is z ↦ −z^{2^{N}} . By choosing N to be large enough, we can make | arg ω| as small as we like. Define F_{1}(z) = z^{2} and F_{2}(z) = a + ω(z − a)^{2} . Define a 2-quasiconformal mapping φ_{1} of \overline{C} by taking φ_{1}(z) = z|z|/δ if |z| < δ and φ_{1}(z) = z if |z| ≥ δ. Define a 2-quasiconformal mapping φ_{2} of \overline{C} by taking φ_{2}(z) = a + (z − a)|z − a|/δ if |z − a| < δ and φ_{2}(z) = z if |z − a| ≥ δ. Then φ_{1}^{−1}(z) = z\sqrt{δ/|z|} if |z| < δ and φ_{1}^{−1}(z) = z if |z| ≥ δ. We have φ_{2}(z) = a + φ_{1}(z − a) and φ_{2}^{−1} = a + φ_{1}^{−1}(z − a). Finally, set f_{j} = φ_{j}^{−1} ∘ F_{j} ∘ φ_{j} for j = 1, 2. Let G be the semigroup generated by f_{1} and f_{2} . We shall prove that G has the required properties.

A calculation shows that f_{1}(z) = z^{2} when |z| ≥ \sqrt{δ}, that f_{1}(z) = z^{2}/\sqrt{δ} when |z| ≤ δ, and that f_{1}(z) = (z^{2}\sqrt{δ})/|z| when δ < |z| < \sqrt{δ}. Similarly, we have f_{2}(z) = a + ω(z − a)^{2} when |z − a| ≥ \sqrt{δ}, f_{2}(z) = a + ω(z − a)^{2}/\sqrt{δ} when |z − a| ≤ δ, and f_{2}(z) = a + (ω(z − a)^{2}\sqrt{δ} )/|z − a| when δ < |z − a| < \sqrt{δ}. Thus f_{2}(z) = a+ωf_{1}(z −a) for all z ∈ C. The complex dilatation μ_{1}(z) of f_{1} vanishes when |z| < δ or |z| > \sqrt{δ}, and is given by μ_{1}(z) = −\frac{1}{3}(z/\overline{z}) when δ < |z| < \sqrt{δ}. For the complex dilatation μ_{2}(z) of f_{2} , we have μ_{2}(z) ≡ μ_{1}(z − a). Hence each of f_{1} and f_{2} is 2-quasimeromorphic in \overline{C}. We set A_{1} = \{z : δ < |z| < \sqrt{δ} \} and A_{2} = \{z : δ < |z − a| < \sqrt{δ} \}. Thus, apart from a set of zero area, μ_{m}(z) is non-zero only if z ∈ A_{m}, for m = 1, 2.

To see that G is a 2-quasiregular semigroup (in C, and 2-quasimeromorphic in \overline{C}), consider an element g = g_{n} ∘ g_{n−1} ∘ ⋯ ∘ g_{1} of G, where each g_{j} is equal to f_{1} or f_{2} , in a small neighbourhood D of some point z ∈ C. If the complex dilatation of g does not vanish identically in D then there is a smallest j such that (g_{j} ∘ ⋯ ∘ g_{1})(D) intersects A_{m} and g_{j+1} = f_{m} where m = 1 or m = 2 (this includes the case j = 0, in which case we interpret g_{j} ∘ ⋯ ∘ g_{1} = Id). By symmetry, we may assume that m = 1. As we are looking at the fixed function g and are only interested in what happens at z, we may take D so small that (g_{j} ∘ ⋯ ∘ g_{1})(D) ⊂ B(2\sqrt{δ} ). Suppose that g_{l} = f_{1} for j + 1 ≤ l ≤ j + q for some q ≥ 1 while either j +q = n or g_{j+q+1} = f_{2} . The maximal dilatation of g_{j} ∘⋯∘g_{1} restricted to D is 1, and the maximal dilatation of g_{j+q} ∘ ⋯ ∘ g_{j+1} = f_{1}^{q} does not exceed 2. (Note that if w ∈ A_{1} then |f_{1}^{p}(w)| < δ for all p ≥ 1.) If j + q = n, there is nothing more to prove. Suppose that j + q < n and that g_{j+q+1} = f_{2} . Now g_{j+q} ∘ ⋯ ∘ g_{j+1} maps B(2\sqrt{δ} ) into itself. If w ∈ (g_{j+q} ∘ ⋯ ∘ g_{1})(D) then |w − a| ≥ 2 + ε − 2\sqrt{δ} > 2 + 2ε/3. Thus |f_{2}(w) − a| > 4 + 8ε/3 and so |f_{2}(w)| > 2 + 5ε/3 > a + 2\sqrt{δ}. Now f_{2} is conformal at w, and each of f_{1} and f_{2} is conformal at f_{2}(w). Furthermore, applying either f_{1} or f_{2} at f_{2}(w) increases the modulus so much that after this, no matter which elements of G are then applied, we are dealing with only locally conformal mappings. This shows that the maximal dilatation of any g ∈ G cannot be more than 2 in any sufficiently small neighbourhood of z. Thus G is a 2-quasimeromorphic semigroup in \overline{C}.

4.2. The proof that G cannot be conjugated by a quasiconformal mapping to a semigroup of rational functions is based on the following lemma.

Lemma 1. Suppose that G is a uniformly quasimeromorphic semigroup acting on an open set U in \overline{C} such that for some K -quasiconformal homeomorphism f, the semigroup f ∘ G ∘ f^{−1} consists of meromorphic functions. If z ∈ U and if g = g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n} is a function defined in a neighbourhood of z, where each g_{j} is an element of G or a branch of the inverse of some element of G defined in a neighbourhood of an appropriate point, then the maximal dilatation of g does not exceed K^{2} .

Remark 1. Lemma 1 also shows that if G is a uniformly quasimeromorphic semigroup that can be quasiconformally conjugated to a semigroup of meromorphic functions, then the set S(z) considered in the proof of Theorem 3 is uniformly bounded for all z for which it is well defined (that is, for almost every z, we have S(z) ⊂ B(k_{1}) for some fixed k_{1} < 1). Hence, if it is possible to prove that a quasiconformal conjugacy exists at all, then, in a sense, it must be possible to obtain such a conjugacy by the extension of Tukia’s method given in the proof of Theorem 3. However, even if this were theoretically possible for a given G, it need not be the case that it is clear from the algebraic structure of G that the sets S(z) are relatively compact in the unit disk D. For example, if G′ is a free semigroup of rational functions and if f is a quasiconformal self-map of \overline{C} then G = f^{−1} ∘ G′ ∘ f is a free uniformly quasimeromorphic semigroup which can be quasiconformally conjugated to a semigroup of rational functions. However, the fact that the sets S(z) are relatively compact in D for this G, is by no means clear from the algebraic properties of G, and to see in some way that the S(z) are bounded away from the unit circle (which they are), it would be necessary to have some different type of information concerning the complex dilatations of the elements of G.

Remark 2. Theorem 3 gives a sufficient condition on the algebraic structure of G that forces Tukia’s method to work. From the proof of Theorem 3 we see that another sufficient condition can be given as follows: there exists a fixed positive integer N such that the function corresponding to any word g_{1} ∘ ⋯ ∘ g_{n}, for any n ≥ 1, where each g_{j} is an element of G∪{Id} or a branch of the inverse of such an element, can also be represented by a word of this type with n ≤ N (this obviously guarantees that we find a suitable k_{1} < 1 depending only on K and N). In the proof of Theorem 3 we saw that this is true with N = 2 when the assumptions of Theorem 3 are satisfied. However, this assumption, when formulated as above for an arbitrary N with no other connection to the algebraic structure of G, even if it is more general than the assumption of Theorem 3, seems to be less natural than the condition (1.1) of Theorem 3.

4.3. Proof of Lemma 1. Let the set-up be as in the assumptions of Lemma 1. For each j, write h_{j} = g_{j} if g_{j} ∈ G, and otherwise let g_{j} be a branch of the inverse of h_{j} ∈ G. There are meromorphic functions p_{j} ∈ f ∘G∘f^{−1} such that p_{j} = f∘h_{j}∘f^{−1} . Hence we may locally write g_{j} = f^{−1}∘p_{j}^{−1}∘f when g_{j} is an inverse of h_{j}. Note that under these circumstances we are considering a branch of p_{j}^{−1} which is meromorphic in the small neighbourhood where it is defined. If g_{j} ∈ G then g_{j} = h_{j} = f^{−1} ∘p_{j} ∘f. We see that g = g_{1} ∘g_{2} ∘⋯∘g_{n} = f^{−1} ∘p∘f for some meromorphic function p defined in a small neighbourhood of f(z). This shows that the maximal dilatation of g in a neighbourhood of z does not exceed K^{2} . Lemma 1 is proved.

4.4. We return to the proof of Theorem 2. To complete the proof, we only need to show that for any k_{1} < 1, no matter how close to 1, there exists a word g generated by the elements of our particular semigroup G and their locally defined inverses, and a point z ∈ C such that the complex dilatation of g has modulus at least k_{1} in some small neighbourhood of z. We choose η_{1} ∈ (0, 10^{−2}), say, and take z ∈ B(η_{1}, η_{1}^{2}). Then for all z ∈ B(η_{1}, η_{1}^{2}), we have | arg z| < η_{1}/(1 − η_{1}) < 2η_{1} . We consider words g = g_{n} ∘ ⋯ ∘ g_{1} where each g_{j} is of the form

g_{j} = f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}}

for some large positive integers q_{1} and q_{2} depending on and varying with j. The main idea is that each individual complex dilatation will be of the form −\frac{1}{3}(z/\overline{z}), or such a number multiplied by several factors of the form z/\overline{z}. Here all the numbers z occurring are almost real so that any factor z/\overline{z} is very close to 1. When we calculate complex dilatations for composite functions we are therefore applying the formula (2.1) to numbers that are very close to −\frac{1}{3} . Unlimited composition leads to a sequence of numbers tending to the unit circle (if all the individual complex dilatations involved were equal to −\frac{1}{3} , the sequence would tend to −1). This shows that the complex dilatations will not remain in a fixed disk B(k_{1}) for any k_{1} < 1.

4.5. The function f_{2} has the circle {z : |z − a| = 1} as its Julia set J(f_{2}), so that for some sufficiently large integer q, a suitable branch of f_{2}^{−q} maps any preassigned disk outside this circle, such as the disk B(η_{1}, η_{1}^{2}), into a prescribed neighbourhood of the point a − 1 = 1 + ε ∈ J(f_{2}). We choose q = q_{1} so large that for all z ∈ B(η_{1}, η_{1}^{2}), for such a branch of f_{2}^{−q}, we have | arg f_{2}^{−q}(z)| < η_{2} and |f_{2}^{−q}(z) − (a − 1)| < η_{2} for some η_{2} > 0 to be determined later. Since by (4.1),

f_{1}^{N}(a − 1) = 2 + ε + \frac{1}{2}(\sqrt{δ} − δ) = a + \frac{1}{2}(\sqrt{δ} − δ) ≡ b,

it follows that if η_{3} > 0 is given, we may choose η_{2} so that |f_{1}^{N}(f_{2}^{−q}(z)) − b| < η_{3} and |arg(f_{1}^{N}(f_{2}^{−q}(z)) − a)| < η_{3} for all z ∈ B(η_{1}, η_{1}^{2}). We certainly take η_{3} < \frac{1}{2}(\sqrt{δ} − 3δ), so that δ < |f_{1}^{N}(f_{2}^{−q}(z)) − a| < \sqrt{δ}, and possibly take η_{3} to be even smaller. Then for all z ∈ B(η_{1}, η_{1}^{2}), the point f_{1}^{N}(f_{2}^{−q}(z)) lies in the region where the complex dilatation of f_{2} has modulus \frac{1}{3} . Also

|arg(f_{2}(f_{1}^{N}(f_{2}^{−q}(z))) − a) − arg ω| < 2η_{3}

for all z ∈ B(η_{1}, η_{1}^{2}). We next take a suitable branch of f_{1}^{−q_{2}} to ensure that it maps the set (f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q})(B(η_{1}, η_{1}^{2})) into a small neighbourhood of the point 1 = a − (1 + ε), consequently with a very small spread in the argument. Then we apply f_{2}^{N} and map the resulting set into a small neighbourhood of a − (2 + ε + \frac{1}{2}(\sqrt{δ} − δ)) = −\frac{1}{2}(\sqrt{δ} − δ). Now δ < \frac{1}{2}(\sqrt{δ} − δ) < \sqrt{δ} since δ is small. Applying next f_{1} we operate in a set where the complex dilatation of f_{1} has modulus \frac{1}{3} . We end up in a set which is contained in a small neighbourhood of the point (\frac{1}{2}δ)(1−\sqrt{δ} ) ≈ \frac{1}{2}δ, with a very small spread in the argument, and hence also in a small neighbourhood of the origin (this neighbourhood of (\frac{1}{2}δ)(1 − \sqrt{δ} ) is, in a sense, comparable to the disk B(η_{1}, η_{1}^{2})), and we are ready to apply a similar word g_{j}. We have now described the operation of one word g_{j}. We next calculate the derivatives with respect to z and \overline{z} of such a word g_{j}, say g_{1} , at a point z ∈ B(η_{1}, η_{1}^{2}). A similar result is obtained for a subsequent word g_{j} at any point z in a suitable neighbourhood of (\frac{1}{2}δ)(1 − \sqrt{δ} ).

4.6. We introduce some notation. We write w_{0} = z ≈ η_{1} ≈ 0, w_{1} = f_{2}^{−q_{1}}(z) ≈ 1 + ε, w_{2} = f_{1}^{N}(w_{1}) ≈ a, w_{3} = f_{2}(w_{2}) ≈ a, w_{4} = f_{1}^{−q_{2}}(w_{3}) ≈ 1, w_{5} = f_{2}^{N}(w_{4}) ≈ 0, and w_{6} = g_{1}(z) = g_{1}(w_{0}) = f_{1}(w_{5}) ≈ 0. We have | arg(w_{3} − a) − arg ω| < 2η_{3} , and | arg w_{3} − arg c| < 3η_{3} , say, where

(4.2)  c = a + ω\sqrt{δ} \frac{1}{2}(\sqrt{δ} − δ) = a + \frac{1}{2}ωδ(1 − \sqrt{δ} ).

For 0 ≤ j ≤ 6 with j ≠ 3, both w_{j} and w_{j} −a have argument very close to either 0 or π, with π occurring exactly for w_{0} − a, w_{1} − a, w_{4} − a, w_{5} − a, w_{6} − a, and w_{5} . Next, note that

\begin{aligned}(f_{2}^{q_{1}})′(w_{1}) & = ω^{2^{q_{1}} −1}2^{q_{1}}(w_{1} − a)^{2^{q_{1}} −1} = 2^{q_{1}}(w_{0} − a)/(w_{1} − a), \\ (f_{1}^{N})′(w_{1}) & = 2^{N}(w_{1})^{2^{N} −1} = 2^{N}w_{2}/w_{1}, \\ (f_{1}^{q_{2}})′(w_{4}) & = 2^{q_{2}}w_{3}/w_{4}, and \\ (f_{2}^{N})′(w_{4}) & = −2^{N}(w_{4} − a)^{2^{N} −1} = 2^{N}(w_{5} − a)/(w_{4} − a). \end{aligned}

We have μ_{2}(w_{2}) = −\frac{1}{3}(w_{2} − a)/\overline{(w_{2} − a)} and μ_{1}(w_{5}) = −\frac{1}{3}(w_{5}/\overline{w_{5}}). For 0 ≤ j ≤ 6, we set b_{j} = w_{j}/\overline{w_{j}}.

To make the absolute value of the argument of every one of the points w_{1} , w_{2} , w_{4} , w_{5} , w_{1} − a, w_{2} − a, w_{4} − a, w_{5} − a small, we only need to choose q_{1} and q_{2} sufficiently large. This can be done for any initially given η_{1} , even though the precise choice of q_{1} and q_{2} will then depend on (an upper bound for) η_{1} . The magnitude of | arg z| = | arg w_{0}| and of | arg(w_{0} − a)| is controlled by the choice of η_{1} . If we perform the same estimate for a function g_{j} other than g_{1} then the role of w_{0} is taken by a point of the form w_{6} arising from a previous calculation, and we can get | arg w_{6}| and | arg(w_{6} − a)| to be as small as we like by taking the previous q_{1} and q_{2} sufficiently large. To handle this, we may choose for q_{1} and q_{2} values that increase to infinity with j. Thus we may consider all of the above arguments to be o(1) as j → ∞, and the rate of convergence to zero implied by the o(1)-notation can be as fast as we like. Similarly, arg w_{3} = arg c + o(1) so that with c_{3} = c/\overline{c}, we have b_{3} = c_{3}(1 + o(1)).

4.7. Define f = f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} and g = f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}} . Then f ∘ g = f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}} is the function we are looking at, one of the g_{j}. We have

∂f_{2}(w_{2}) = \frac{3}{2}ω\sqrt{δ} \frac{w_{2} − a}{|w_{2} − a|}, \overline{∂}f_{2}(w_{2}) = \frac{−1}{2}ω\sqrt{δ} \frac{(w_{2} − a)^{3}}{|w_{2} − a|^{3}} .

Thus

∂g(z) = (∂f_{2})(w_{2})\frac{(f_{1}^{N})′(w_{1})}{(f_{2}^{q_{1}})′(w_{1})} = \frac{3}{2}ω\sqrt{δ} \frac{w_{2} − a}{|w_{2} − a|}2^{N−q_{1}} \frac{w_{2}}{w_{1}} \frac{w_{1} − a}{w_{0} − a}

and

\overline{∂}g(z) = (\overline{∂}f_{2})(w_{2})\frac{\overline{(f_{1}^{N})′(w_{1})}}{\overline{(f_{2}^{q_{1}})′(w_{1})}} = \frac{−1}{2}ω\sqrt{δ} \frac{(w_{2} − a)^{3}}{|w_{2} − a|^{3}} 2^{N−q_{1}} \frac{\overline{w_{2}}}{\overline{w_{1}}} \frac{\overline{w_{1} − a}}{\overline{w_{0} − a}}.

Further, we have

\begin{aligned}∂f(g(z)) & = (∂f_{1})(w_{5})\frac{(f_{2}^{N})′(w_{4})}{(f_{1}^{q_{2}})′(w_{4})} = \frac{3}{2} \sqrt{δ} \frac{w_{5}}{|w_{5}|}2^{N−q_{2}} \frac{w_{5} − a}{w_{4} − a} \frac{w_{4}}{w_{3}} , \\ \overline{∂}f(g(z)) & = (\overline{∂}f_{1})(w_{5})\frac{\overline{(f_{2}^{N})′(w_{4})}}{\overline{(f_{1}^{q_{2}})′(w_{4})}} = \frac{−1}{2} \sqrt{δ} \frac{w_{5}^{3}}{|w_{5}|^{3}} 2^{N−q_{2}} \frac{\overline{w_{5} − a}}{\overline{w_{4} − a}} \frac{\overline{w_{4}}}{\overline{w_{3}}} . \end{aligned}

Hence

∂(f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}})(z) = ∂(f ∘ g)(z)

= ∂f(g(z))∂g(z) + \overline{∂}f(g(z))∂\overline{g}(z)

= \frac{3}{2} \sqrt{δ} \frac{w_{5}}{|w_{5}|}2^{N−q_{2}} \frac{w_{5} − a}{w_{4} − a} \frac{w_{4}}{w_{3}} \frac{3}{2}ω\sqrt{δ} \frac{w_{2} − a}{|w_{2} − a|}2^{N−q_{1}} \frac{w_{2}}{w_{1}} \frac{w_{1} − a}{w_{0} − a}

+ \frac{−1}{2} \sqrt{δ} \frac{w_{5}^{3}}{|w_{5}|^{3}} 2^{N−q_{2}} \frac{\overline{w_{5} − a}}{\overline{w_{4} − a}} \frac{\overline{w_{4}}}{\overline{w_{3}}} \frac{−1}{2}ω^{−1}\sqrt{δ} \frac{\overline{(w_{2} − a)^{3}}}{|w_{2} − a|^{3}} 2^{N−q_{1}} \frac{w_{2}}{w_{1}} \frac{w_{1} − a}{w_{0} − a}

= \frac{9}{4}ωδ2^{2N−q_{1}−q_{2}} \frac{w_{5}}{|w_{5}|} \frac{w_{2}}{w_{1}} \frac{w_{1} − a}{w_{0} − a} \frac{w_{5} − a}{w_{4} − a} \frac{w_{4}}{w_{3}} \frac{w_{2} − a}{|w_{2} − a|}

× (1 + \frac{1}{9}ω^{−2} \frac{w_{5}^{2}}{|w_{5}|^{2}} \frac{\overline{w_{5} − a}}{w_{5} − a} \frac{w_{4} − a}{\overline{w_{4} − a}} \frac{b_{3}}{b_{4}} \frac{\overline{(w_{2} − a)^{4}}}{|w_{2} − a|^{4}} )

= \frac{9}{4}ω \frac{|c|}{c}δ2^{2N−q_{1}−q_{2}} \frac{|w_{2}|}{|w_{1}|} \frac{|w_{1} − a|}{|w_{0} − a|} \frac{|w_{5} − a|}{|w_{4} − a|} \frac{|w_{4}|}{|w_{3}|}(1 + \frac{1}{9}ω^{−2}c_{3})(1 + o(1))

since ∂\overline{g}(z) = \overline{\overline{∂}g(z)}. Also

\overline{∂}(f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}})(z) = \overline{∂}(f ∘ g)(z)

= ∂f(g(z))\overline{∂}g(z) + \overline{∂}f(g(z))\overline{∂}\overline{g}(z)

= \frac{3}{2} \sqrt{δ} \frac{w_{5}}{|w_{5}|}2^{N−q_{2}} \frac{w_{5} − a}{w_{4} − a} \frac{w_{4}}{w_{3}} \frac{−1}{2}ω\sqrt{δ} \frac{(w_{2} − a)^{3}}{|w_{2} − a|^{3}} 2^{N−q_{1}} \frac{\overline{w_{2}}}{\overline{w_{1}}} \frac{\overline{w_{1} − a}}{\overline{w_{0} − a}}

+ \frac{−1}{2} \sqrt{δ} \frac{w_{5}^{3}}{|w_{5}|^{3}} 2^{N−q_{2}} \frac{\overline{w_{5} − a}}{\overline{w_{4} − a}} \frac{\overline{w_{4}}}{\overline{w_{3}}} \frac{3}{2}ω^{−1}\sqrt{δ} \frac{\overline{w_{2} − a}}{|w_{2} − a|}2^{N−q_{1}} \frac{\overline{w_{2}}}{\overline{w_{1}}} \frac{\overline{w_{1} − a}}{\overline{w_{0} − a}}

= \frac{−3}{4}ωδ2^{2N−q_{1}−q_{2}} \frac{w_{5}}{|w_{5}|} \frac{\overline{w_{2}}}{\overline{w_{1}}} \frac{\overline{w_{1} − a}}{\overline{w_{0} − a}} \frac{w_{5} − a}{w_{4} − a} \frac{w_{4}}{w_{3}} \frac{(w_{2} − a)^{3}}{|w_{2} − a|^{3}}

× (1 + ω^{−2} \frac{w_{5}^{2}}{|w_{5}|^{2}} \frac{\overline{w_{5} − a}}{w_{5} − a} \frac{w_{4} − a}{\overline{w_{4} − a}} \frac{b_{3}}{b_{4}} \frac{\overline{(w_{2} − a)^{4}}}{|w_{2} − a|^{4}} )

= \frac{−3}{4}(1 + ω^{−2}b_{3})ω \frac{|w_{3}|}{w_{3}}δ2^{2N−q_{1}−q_{2}} \frac{|w_{2}|}{|w_{1}|} \frac{|w_{1} − a|}{|w_{0} − a|} \frac{|w_{5} − a|}{|w_{4} − a|} \frac{|w_{4}|}{|w_{3}|}(1 + o(1)).

Note that (1 + ω^{−2}b_{3})ω|w_{3}|/w_{3} = 2 Re(ω|w_{3}|/w_{3}) is real and positive. We conclude that

arg ∂(f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}})(z) = arg(\frac{ω}{c} + \frac{\overline{ω}}{9\overline{c}}) + o(1)

and

arg \overline{∂}(f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}})(z) = π + o(1).

Further, we have

∂(f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}})(z)

= \frac{9}{4}ωδ2^{2N−q_{1}−q_{2}} \frac{b^{2}}{(1 + ε)^{2}cκ}(1 + \frac{1}{9}ω^{−2}c_{3})(1 + o(1))

= AC_{1}(1 + o(1)),

say, where A > 0 and |C_{1}| = 1 with Re C_{1} > 0, since a − 1 = 1 + ε, and since in subsequent applications the role of |w_{0} − a| is taken by |w_{6} − a| which stabilizes to

a − \frac{1}{2}δ(1 − \sqrt{δ}) ≡ κ.

Also

\overline{∂}(f_{1} ∘ f_{2}^{N} ∘ f_{1}^{−q_{2}} ∘ f_{2} ∘ f_{1}^{N} ∘ f_{2}^{−q_{1}})(z)

= \frac{−3}{4}(1 + ω^{−2}c_{3})ωδ2^{2N−q_{1}−q_{2}} \frac{b^{2}}{(1 + ε)^{2}κc}(1 + o(1))

= BD_{1}(1 + o(1)), say, where B > 0 and |D_{1}| = 1, in fact, D_{1} = −1. We have now obtained estimates applicable to ∂g_{j} and \overline{∂}g_{j}. The terms denoted by o(1) in these estimates can be made arbitrarily small as j → ∞. Hence we may assume in subsequent calculations that these terms are made so small as to make the resulting estimates for g_{j} ∘ ⋯ ∘ g_{1} sufficiently accurate for our purposes.

4.8. We use again the formulas ∂(f ∘ g)(z) = ∂f(g(z))∂g(z) + \overline{∂}f(g(z))∂\overline{g}(z) and \overline{∂}(f ∘ g)(z) = ∂f(g(z))\overline{∂}g(z) + \overline{∂}f(g(z))\overline{∂}\overline{g}(z), replacing f and g by g_{j} and g_{j−1} ∘⋯∘g_{1} . We write ∂(g_{j−1} ∘⋯∘g_{1})(z) = A_{j−1}C_{j−1}(1+o(1)) so that A_{1} = A, and \overline{∂}(g_{j−1} ∘ ⋯ ∘ g_{1})(z) = B_{j−1}D_{j−1}(1 + o(1)) so that B_{1} = B. We obtain

A_{j}C_{j}(1 + o(1)) = ∂(g_{j} ∘ ⋯ ∘ g_{1})(z) = (AC_{1}A_{j−1}C_{j−1} + BD_{1}B_{j−1}\overline{D_{j−1}} )(1 + o(1))

and

B_{j}D_{j}(1+o(1)) = \overline{∂}(g_{j} ∘⋯∘g_{1})(z) = (B_{j−1}D_{j−1}AC_{1} +BD_{1}A_{j−1}\overline{C_{j−1}} )(1+o(1)).

Thus we define A_{j}, B_{j}, C_{j}, and D_{j} by the formulas

\begin{aligned}A_{j}C_{j} & = AC_{1}A_{j−1}C_{j−1} + BD_{1}B_{j−1}\overline{D_{j−1}}, \\ B_{j}D_{j} & = B_{j−1}D_{j−1}AC_{1} + BD_{1}A_{j−1}\overline{C_{j−1}} \end{aligned}

and the requirement that |C_{j}| = |D_{j}| = 1 while A_{j} ≥ 0 and B_{j} ≥ 0. Next define A′_{j} = A_{j}/A^{j} and B′_{j} = B_{j}/A^{j}, and set ρ_{j} = B_{j}/A_{j} = B′_{j}/A′_{j}. We have ρ_{1} = B/A, which can be made as close to \frac{3}{5} as we like. Further, set α_{j} = 1 − ρ_{j}^{2} and β_{j} = (2 − α_{j})/α_{j}. We have

(4.3)  \begin{aligned}|μ(z, g_{j} ∘ ⋯ ∘ g_{1})| & = |\frac{\overline{∂}(g_{j} ∘ ⋯ ∘ g_{1})(z)}{∂(g_{j} ∘ ⋯ ∘ g_{1})(z)}| = \frac{B_{j}}{A_{j}} (1 + o(1)) \\ & = \frac{B′_{j}}{A′_{j}} (1 + o(1)) = ρ_{j}(1 + o(1)) < 1 \end{aligned}

so that we may assume that ρ_{j} < 1 for all j ≥ 1. Thus 0 < α_{j} ≤ 1 and β_{j} ≥ 1 for all j ≥ 1.

We set θ_{j} = C_{1}C_{j}D_{j} so that |θ_{j}| = 1, and write λ_{j} = − Re θ_{j} ∈ [−1, 1]. A calculation now shows that (since D_{1} = −1)

(4.4)  A′_{j}C_{j} = A′_{j−1}C_{1}C_{j−1} − ρ_{1}B′_{j−1}\overline{D_{j−1}},
(4.5)  B′_{j}D_{j} = B′_{j−1}C_{1}D_{j−1} − ρ_{1}A′_{j−1}\overline{C_{j−1}},

so that

\begin{aligned}A′_{j}^{2} & = |A′_{j}C_{j}|^{2} = (A′_{j−1})^{2} + ρ_{1}^{2}(B′_{j−1})^{2} + 2ρ_{1}A′_{j−1}B′_{j−1}λ_{j−1}, \\ B′_{j}^{2} & = (B′_{j−1})^{2} + ρ_{1}^{2}(A′_{j−1})^{2} + 2ρ_{1}A′_{j−1}B′_{j−1}λ_{j−1}. \end{aligned}

This gives, by induction,

\begin{aligned}A′_{j}^{2}α_{j} & = A′_{j}^{2}(1 − ρ_{j}^{2}) = A′_{j}^{2} − B′_{j}^{2} = (1 − ρ_{1}^{2})((A′_{j−1})^{2} − (B′_{j−1})^{2}) \\ & = (1 − ρ_{1}^{2})^{j−1}(A′_{1}^{2} − B′_{1}^{2}) = (1 − ρ_{1}^{2})^{j−1}(1 − ρ_{1}^{2}) = (1 − ρ_{1}^{2})^{j} = α_{1}^{j} \end{aligned}

so that A′_{j} > 0, and

(4.6)  A′_{j}^{2} + B′_{j}^{2} = (1 + ρ_{1}^{2})((A′_{j−1})^{2} + (B′_{j−1})^{2}) + 4ρ_{1}A′_{j−1}B′_{j−1}λ_{j−1}.

Noting that 1 + ρ_{j}^{2} = 2 − α_{j} and applying A′_{j}^{2}α_{j} = α_{1}^{j} in (4.6) we obtain, after dividing through by α_{1}^{j−1} and recalling the definition of β_{j}, that

(4.7)  β_{j} = β_{1}β_{j−1} + 4ρ_{1}ρ_{j−1}λ_{j−1}/(α_{1}α_{j−1}).

We shall show that λ_{j} ≥ 0 for all j ≥ 1. Then (4.7) yields β_{j} ≥ β_{1}β_{j−1} , so that by induction, β_{j} ≥ β_{1}^{j}. Since β_{1} ≈ 17/8 > 1, we then have β_{j} → ∞ as j → ∞. We deduce that as j → ∞, we also have α_{j} → 0 and thus ρ_{j} → 1, and hence by (4.3), |μ(z, g_{j} ∘ ⋯ ∘ g_{1})| → 1 for z ∈ B(η_{1}, η_{1}^{2}), as required. This then completes the proof of Theorem 2.

To prove that λ_{j} ≥ 0 for all j ≥ 1 by induction on j, first note that λ_{1} = Re(C_{1}^{2}) ≥ 1 − E/N for some positive absolute constant E, since

C_{1} = ω \frac{|c|}{c} \frac{1 + \{c/(9ω^{2}\overline{c})\}}{|1 + \{c/(9ω^{2}\overline{c})\}|}

where c is as in (4.2). Thus λ_{1} > 0 if N is large enough. We also have | Im(C_{1}^{2})| ≤ E/N if E is suitably chosen.

Suppose that j ≥ 2 and that λ_{j−1} ≥ 0. Multiplying the product of (4.4) and (4.5) by C_{1} we obtain

\begin{aligned}A′_{j}B′_{j} & (A′_{j−1})^{−2}θ_{j} = C_{1}^{2}\{ρ_{j−1}(θ_{j−1} + ρ_{1}^{2}\overline{θ_{j−1}} ) − ρ_{1}(1 + ρ_{j−1}^{2})\} \\ & = C_{1}^{2}\{−ρ_{j−1}(1 + ρ_{1}^{2})λ_{j−1} − ρ_{1}(1 + ρ_{j−1}^{2}) + iρ_{j−1}(1 − ρ_{1}^{2}) Im θ_{j−1}\} \end{aligned}

so that

(4.8)  \begin{aligned}A′_{j}B′_{j}(A′_{j−1})^{−2}λ_{j} & = (Re(C_{1}^{2}))\{ρ_{j−1}(1 + ρ_{1}^{2})λ_{j−1} + ρ_{1}(1 + ρ_{j−1}^{2})\} \\ & + (Im(C_{1}^{2}))(Im θ_{j−1})ρ_{j−1}(1 − ρ_{1}^{2}) \\ & ≥ (1 − E/N)ρ_{1}(1 + ρ_{j−1}^{2}) − (E/N)ρ_{j−1}(1 − ρ_{1}^{2}). \end{aligned}

Here we have used the assumption that λ_{j−1} ≥ 0 and the fact that Re(C_{1}^{2}) > 0 while Im θ_{j−1} ∈ [−1, 1].

The rightmost expression in (4.8) is positive provided that

(4.9)  \frac{1 + ρ_{j−1}^{2}}{ρ_{j−1}} > \frac{E}{N − E} \frac{1 − ρ_{1}^{2}}{ρ_{1}} .

Since (1 + ρ_{j−1}^{2})/ρ_{j−1} ≥ 2 and ρ_{1} ≈ \frac{3}{5} , it follows that (4.9) holds provided that N is large enough. Thus A′_{j}B′_{j}(A′_{j−1})^{−2}λ_{j} > 0. This also shows that B′_{j} ≠ 0 (we have already seen that A′_{j} ≠ 0). Since A′_{j} > 0 and B′_{j} > 0, we now deduce from (4.8) that λ_{j} > 0. This completes the induction proof that λ_{j} ≥ 0, and hence the proof of Theorem 2 is also complete.

5. Proof of Theorem 4

Let the assumptions of Theorem 4 be satisfied. Every element of G is a composition of finitely many functions, each being f_{1} or f_{2} . When applying such a function at some z ∈ C, we apply the translation z + 1 every time, except that at most once, we apply h. Thus every element of G is K -quasiconformal. Let G′ be the group generated by f_{1} and f_{2} . By Lemma 1, G is quasiconformally conjugate to a semigroup of conformal mappings if, and only if, G′ is quasiconformally conjugate to a group of conformal mappings. Suppose that there is a quasiconformal mapping f of \overline{C} such that f ∘G′∘f^{−1} is a Möbius group. (Of course, even without reference to Lemma 1, one can observe that the group generated by a semigroup of Möbius transformations, consists of Möbius transformations only.) Let φ be a conformal mapping of f(S) onto S. Since h is the restriction of f_{2}^{−1} ∘ f_{1} to S, it follows that f ∘ h ∘ f^{−1} is a conformal mapping of f(S) onto itself. Thus with F = φ ∘ f, the map F ∘ h ∘ F^{−1} is a conformal mapping of S onto itself. But F (∂S) = ∂S and h|∂S = Id. Thus F ∘ h ∘ F^{−1} = Id and so h = Id, which is a contradiction. Hence, when h ≠ Id, the semigroup G cannot be quasiconformally conjugate to a semigroup of conformal mappings.

The same argument works if f is just a homeomorphism of \overline{C} onto itself, including an obvious extension of Lemma 1 to this particular situation, where the fact that f is merely a homeomorphism, not necessarily quasiconformal, is compensated for by the greatly simplifying fact that all the elements of the semigroup f ∘ G ∘ f^{−1} are Möbius transformations in \overline{C}. This proves Theorem 4.

References

[1] Hinkkanen, A., and G.J. Martin: The dynamics of semigroups of rational functions I. - Preprint, Australian National University, 1993. To appear in Proc. London Math. Soc.

[2] Lehto, O., and K.I. Virtanen: Quasiconformal Mappings in the Plane. - Springer-Verlag, Berlin, 1973.

[3] Sullivan, D.: The ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. - In: Riemann Surfaces and Related Topics, Proceedings of the 1978 Stony Brook Conference. Ann. of Math. Stud. 97, 1981, 465–496.

[4] Tukia, P.: On two-dimensional quasiconformal groups. - Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 1980, 73–78.

Received 29 December 1994