On a question of T . Sheil-Small regarding valency of harmonic maps

The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form f(e) = e, 0 ≤ t ≤ 2π, where φ is a continuously non-decreasing function that satisfies φ(2π)−φ(0) = 2Nπ, assume every value finitely many times in the disc? Introduction. Let D and T be the open unit disc and the unit circle respectively, and let N be a positive integer. An N-valent quasi-homeomorphism from the unit circle into itself is a circle mapping f : T → T of the form f(eit) = eiφ(t), 0 ≤ t ≤ 2π, where φ is a non-decreasing function that satisfies φ(2π) − φ(0) = 2Nπ. It can be seen that every such quasi-homeomorphism is a pointwise limit of a sequence of circle mappings fn : T→ T of the form fn(e) = eiφn(t), 0 ≤ t ≤ 2π, where φn is a continuously strictly increasing function that satisfies φn(2π)− φn(0) = 2Nπ. A 1-valent quasi-homeomorphism is referred to as quasi-homeomorphism. The celebrated Radó–Kneser–Choquet Theorem can be stated as follows. 2000 Mathematics Subject Classification. Primary 26C10; Secondary 30C15.


Introduction.
Let D and T be the open unit disc and the unit circle respectively, and let N be a positive integer.An N -valent quasi-homeomorphism from the unit circle into itself is a circle mapping f : T → T of the form f (e it ) = e iϕ(t) , 0 ≤ t ≤ 2π, where ϕ is a non-decreasing function that satisfies ϕ(2π) − ϕ(0) = 2N π.It can be seen that every such quasi-homeomorphism is a pointwise limit of a sequence of circle mappings f n : T → T of the form f n (e it ) = e iϕn(t) , 0 ≤ t ≤ 2π, where ϕ n is a continuously strictly increasing function that satisfies ϕ n (2π) − ϕ n (0) = 2N π.
A 1-valent quasi-homeomorphism is referred to as quasi-homeomorphism.
The celebrated Radó-Kneser-Choquet Theorem can be stated as follows.
Theorem A (Radó-Kneser-Choquet Theorem [3, pp. 29-34]).Suppose that F is the harmonic extension in D of a quasi-homeomorphism f from the unit circle into itself.Then F is univalent.
In an attempt to generalize this theorem to 2-valent quasi-homeomorphisms f from the unit circle into itself, it was suggested that the respective functions F are at most 4-valent.However, examples presented in [2] have shown that some of these mappings could be 6-valent or 8-valent.Furthermore, the construction procedure used in the paper suggested the possibility of finding a 2-valent quasi-homeomorphism from the unit circle into itself whose harmonic extension in D assumes a predetermined finite valency.But this remains an open problem.
In a personal communication with the first author about a decade ago, T. Sheil-Small raised the following question: If F is the harmonic extension in D of a mapping f of the form f (e it ) = e iϕ(t) , 0 ≤ t ≤ 2π, where ϕ is a continuously non-decreasing function that satisfies ϕ(2π) − ϕ(0) = 2N π, then does F assume every value finitely many times in the disc?In this note, we show that the answer to this question is positive.In fact, a more general result is shown henceforth to be true.The result of the note can be stated as follows.
Theorem 1. Suppose that F is the harmonic extension in D of an N -valent quasi-homeomorphism f from the unit circle into itself that takes on three distinct values.Then F takes on every point in D \ C(F, T) finitely many times.
As a consequence we have: Then F takes on every point in D finitely many times.
Before embarking on the proof of Theorem 1, we define an algebraic curve as a connected component of the preimage of a straight line or circle under an analytic function.
Proof of Theorem 1. Write F = u + iv, where u and v are the real and imaginary parts of F .Suppose that there exist a point ω ∈ D \ C(F, T) and a set S of countably infinitely many distinct values Consider the level set u = u 0 ; note that this is a set-union of mutually disjoint algebraic curves.Suppose that each of these curves carries a finite subset of S of the points z n .Then these curves are countably infinite and may be denoted by Suppose that some arc γ accumulates on a non-degenerate subarc J ⊂ T; denote the interior of J by J • .Let η ∈ J • .Note that in every direction towards η from D there exists a sequence of points in γ converging to η on which u attains the value u 0 .This entails by a result of Schwarz [1, Theorem 23] that u is continuous and is identically u 0 on J • .
Let g be the analytic completion on u.By the reflection principle, g is analytic on J • .Fix η ∈ J • .It is immediate that g([0, η]) is an arc that meets the vertical line L : u = u 0 in the (u, v)-plane at countably infinitely many points that are away from infinity.Since both arcs g([0, η]) and L are analytic, g([0, η]) ⊂ L, see [4,Theorem 7.19,, and equivalently u = u 0 on [0, η].But η is an arbitrary point of J • ; hence u is identically u 0 on the open circular sector with vertex at the origin and subtending J • and consequently on D, which yields a contradiction.
Thus every Jordan arc γ terminates in every direction at a point in T. We contend that every γ is a crosscut of D. For suppose otherwise, then some γ is a loop with a unique point η ∈ γ ∩ T. If G ⊂ D is the bounded region enclosed by γ, then, because u is a bounded harmonic function, the limit We infer, by the maximum principle, that u is identically u 0 in G and consequently in D, which gives a contradiction.This proves our claim.
Suppose now that γ terminates at two distinct points α, β ∈ T, and let γ ⊂ C be a crosscut of D similar to γ.It is immediate that γ can not terminate at both α, β.In fact, γ can neither terminate at α nor at β.For suppose γ terminates at α; then, since C is connected, there exists a continuum that meets both γ and γ .But then K ∪ γ ∪ γ bounds a Jordan subdomain K of D, which gives a contradiction.
It follows at once that γ and γ are either disjoint or cross at a singleton in D; namely a critical point of u.Thus C is a tree whose vertices are the critical points of u and the terminal points of the arcs γ.We show that this tree is finite.Suppose otherwise, then the crosscuts γ comprising C are countably infinite, and consequently the same are the endpoints of C. The latter points subdivide T into countably infinitely many subarcs λ.Let λ 1 and λ 2 be two of these arcs that share a common terminal point ν, and let G 1 and G 2 be the Jordan domains bounded by C ∪ λ 1 and C ∪ λ 2 respectively.Note that G 1 and G 2 have a common boundary arc, denoted by δ ⊂ C, with an endpoint at ν. Evidently, g(δ) is a line segment of the vertical line L : u = u 0 .Note that g, like u, has no critical points in the interior of δ since g and u share these points, and that u(z) = u 0 and u(z It follows that g(G 1 ) and g(G 2 ) lie on different sides of L. But by the hypotheses on f, u − u 0 cannot change the sign more than N times.This implies at once that the number of arcs λ is at most 2N ; thus the number of crosscuts γ comprising C is at most N .
We conclude that some crosscut γ, denoted by Γ, contains infinitely countably many points ζ n .We may assume without loss of generality that ζ n ∈ Γ for all n = 1, 2, . . . .On the other hand, by undergoing the same discussion on v instead of u we conclude that there exists a crosscut Γ that is contained in the level set v = v(z 0 ) and contains infinitely countably many of the points ζ n ∈ Γ, n = 1, 2, . . . .Since every |ζ n | ≤ ρ < 1 n = 1, 2, . . .and the arcs Γ and Γ are analytic, Γ and Γ coincide.
Suppose now that ξ ∈ T is a terminal point of Γ (or Γ ).Then u(z) → u 0 for z ∈ Γ z → ξ; hence u 0 ∈ C(u, ξ).By the same token we conclude that v 0 ∈ C(v, ξ).Therefore, ω ∈ C(F, T) and we have a contradiction to our original assumption.This completes the proof of Theorem 1.
For a function F : D → C and a point ζ ∈ T, let C(F, ζ) and C(F, T) denote the cluster sets of F at ζ and on T respectively.
C n , n = 1, 2, . . . .Label one of the points of S ∩ C n by ζ n for every n.Since |ζ n | ≤ ρ < 1 for all n, there exists a subsequence (ζ n k ) of (ζ n ) that converges to a point ζ.Evidently, |ζ| ≤ ρ < 1, F (ζ) = ω and ζ belongs to some level curve C : u = u 0 .This yields a contradiction since near ζ the curve C fails to be isolated from the level curves C n .It follows that the level set u = u 0 is a disjoint union of finitely many algebraic curves of which one, say C, carries countably infinitely many points z n that we denote by ζ 1 , ζ 2 , . . . .Observe the following:(1) C never encloses a Jordan domain in D because of the maximum principle for harmonic functions; (2) C is a union of analytic Jordan arcs γ that are mutually disjoint except possibly for a common critical point of u; (3) Every γ clusters in T.