Georgian Mathematical Journal Volume 8 (2001), Number 1, 1-11

POLYMERSIONS OF A DISK WITH CRITICAL POINTS ON THE BOUNDARY

R. ABDULAEV

Abstract. The existence of an interior extension with critical points on the boundary is proved for a given closed normal curve and for nonnormal curves of specific kind.

2000 Mathematics Subject Classification: 30G12.

Key words and phrases: Normal curve, interior extension, assemblage, critical point.

Introduction

The problem of the existence of an interior extension for a given closed normal curve was solved by Ch. Titus [1] in 1961. In 1967 S. Blanc [2] gave another solution of this problem. Subsequently, many authors investigated different aspects of this theory [3]–[9], but in al these papers the existence of interior extension was proved under the assumption that the images of critical points of this mapping do not lie on the given curve. On the other hand, M. Morse [10] introduced the concept of a partial branch point of the inverse function and extended his formula connecting the angular order of a curve with the sum of multiplicities of critical points for interior mappings possessing critical points on the boundary.

In this paper we investigate a problem of the existence of an interior extension with critical points on the boundary for a given closed normal curve (Section 3). In Section 4 an interior extension is constructed for a closed curve which is not normal.

1. Preliminaries

Two continuous mappings (paths) g : [a, b] → ℝ^{2} and h : [c, d] → ℝ^{2} are called equivalent if there exists a sense preserving homeomorphism χ : [a, b] → [c, d] such that g = h ∘ χ. The set of all equivalent paths is called a curve and each path belonging to the curve is called a representation of this curve. Let γ be a curve, and let γ(x) be some representation of γ defined on [a, b]. The point i(γ) = γ(a) is called the initial point of γ and t(γ) = γ(b) the terminal point of γ. By ⟨γ⟩ is denoted the set of images of γ(x). Let γ_{1} and γ_{2} be two curves such that i(γ_{2}) = t(γ_{1}), and let γ_{1}(x), x ∈ [a, b] and γ_{2}(x), x ∈ [c, d]

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

be the corresponding representations. Take some b′ > b and a homeomorphism χ : [b, b′] → [c, d], χ(b) = c and consider the mapping defined on [a, b′]

(γ_{1} · γ_{2})(x) = \begin{cases}γ_{1}(x), & when x ∈ [a, b], \\ γ_{2}(χ(x)), & when x ∈ [c, d]. \end{cases}

The curve containing the mapping (γ_{1} · γ_{2})(x) will be denoted by γ_{1} · γ_{2}.

Let γ(x) ∈ γ, x ∈ [a, b] and [c, d] ⊆ [a, b]. By γ[c, d] we denote the curve generated by the restriction of γ(x) to [c, d] and by γ[d, c] the curve generated by γ(ω(x)), where ω(x) is a sense reversing homeomorphism of [c, d] onto itself.

The point v ∈ ⟨γ⟩ is called a vertex (or node) if for some path ψ ∈ γ, card ψ^{−1}(v) > 1 [4], [6]. It is clear that the definition of a vertex does not depend on the choice of representation. A vertex v is called a double point if ψ^{−1} = {x_{1}, x_{2}} and ⟨ψ[x_{1} − ε, x_{1} + ε]⟩ separates ⟨ψ[x_{2} − ε, x_{2} + ε]⟩\ψ(x_{2}) for some ε > 0. The curve is called normal if it has a finite number of vertices and each of them is a double point [9].

Let f be an interior mapping [11], [12] of a domain D into ℝ^{2}. For every z ∈ D there exists a neighborhood U(z) such that the restriction of f to U(z)\{z} is a (μ(f, z) + 1)-sheeted covering of f(U(z))\{f(z)}. If μ(f, z) > 0, we say that the covering f|_{U(z)} is branched and the point z is called a critical point of the multiplicity μ(f, z). After G. Francis [6] we call an interior mapping having a finite number of critical points in D a polymersion.

A closed curve γ is called an interior boundary [1], [2] if there exists an interior mapping (polymersion) of the unit disk D^{2}, continuous in \overline{D}^{2} such that the restriction of f to the counterclockwise oriented unit circle S^{1} is a path belonging to γ. Another expression of this fact is: f is an interior extension of γ.

2. The existence of an interior extension for a normal closed curve

In this section a summary of the main result of the paper [5] is given for convenience.

Let w = γ(ζ), ζ ∈ S^{1}, be a representation of a closed normal curve and let α_{j}, j = \overline{0, N}, be the collection (raying) of paths (rays) with the following properties:

1. i(α_{j}) = a_{j} lies in some bounded component of ℝ^{2} − ⟨γ⟩, t(α_{j}) = b_{j} lies in the unbounded component of ℝ^{2}\⟨γ⟩. There is at least one a_{j} in each bounded component of γ.

2. Each α_{j} is transverse to γ.

3. α_{j}’s are pairwise disjoint and do not meet nodes of γ.

POLYMERSIONS OF A DISK WITH CRITICAL POINTS ON THE BOUNDARY 3

The crossing point of γ and α_{j} is called positive if γ crosses α_{j} from right to left, and negative otherwise. Index crossing points of α_{j} and γ by (j, k, ±1) (called hereafter letters), where the sign of the third coordinate coincides with the sign of the crossing and second coordinates k, are arranged in an increasing way along α_{j}. In what follows the expression “a point (a, b, c)” will mean the crossing point corresponding to the letter (a, b, c).

Choose i(γ) and trace along γ. We make a sequence α (called the word of γ) of letters as they are encountered in the tracing of γ. Denote by G(γ) (the grouping) and B(γ) (the branching) collections of letters of α satisfying following conditions

(a) The sets of A(γ) = G(γ) ∪ B(γ) are mutually disjoint.

(b) If A ∈ A(γ) and A′ ∈ A(γ), then γ^{−1}(A) is contained in one component of S^{1}\γ^{−1}(A′).

(c) Every set in G(γ) is of the form {(j, k, −1), (j, k′, 1)} with k < k′.

(d) Every negative letter (i.e., the letter with a negative third coordinate) occurs in some set of G(γ).

(e) All letters belonging to some B ∈ B(γ) have the same first coordinate. In other words, each B consists of letters corresponding to the crossing points of one path of the raying.

Such a set A(γ) is called an assemblage for γ. The assemblage is called maximal if ∑_{B∈B(γ)}(card B−1) = τ(γ), where τ(γ) is the tangent winding number

of γ.

Theorem ([5]). A normal closed curve has an interior extension iff it has a maximal assemblage. An assemblage determines multiplicities of critical points of corresponding interior extension f. Namely, for B ∈ B(γ) there exists x ∈ D^{2} such that f(x) = a_{j} and μ(f, x) = card B − 1.

Remark. The investigation of the problem of the existence of an interior extension was carried out in [5] under the condition that γ is smooth. But it is easy to show that the above-stated result remains true for a piece-wise smooth curve as well.

3. The Existence of an Interior Extension with Partial Critical Points

By U(z) we denote a neighborhood of z in ℝ^{2}. For a point ζ ∈ S^{1} denote U_{+}(ζ) = U(ζ) ∩ \overline{D}^{2}, U_{−}(ζ) = U(ζ) ∩ CD^{2}, U^{+}(ζ) = U(ζ) ∩ D^{2}, U^{−}(ζ) = U(ζ) ∩ \overline{D}^{2}. Let γ(ζ) be a representation of a normal curve and let γ[ζ_{1}, ζ_{2}] be a simple arc of this curve. Let ζ_{0} ∈ γ[ζ_{1}, ζ_{2}] and let U(γ(ζ_{0})) be a neighborhood of γ(ζ_{0}) such that U(γ(ζ_{0}))\⟨[ζ_{1}, ζ_{2}]⟩ consists of two components. Denote by U^{+}(γ(ζ_{0})) the component of U(γ(ζ_{0}))\⟨[ζ_{1}, ζ_{2}]⟩ lying to the left of γ[ζ_{1}, ζ_{2}] and by U^{−}(γ(ζ_{0})) the other component. By U_{+}(γ(ζ_{0})) and U_{−}(γ(ζ_{0})) denote U^{+}(γ(ζ_{0})) ∪ (⟨[ζ_{1}, ζ_{2}]⟩ ∩ U(γ(ζ_{0}))) and U^{−}(γ(ζ_{0})) ∪ (⟨[ζ_{1}, ζ_{2}]⟩ ∩ U(γ(ζ_{0}))), respectively.

Let f be an interior extension of γ. A point ζ̃ ∈ S^{1} is called a partial critical point of f (or a partial branch point of f^{−1}) of multiplicity μ(f, ζ̃) > 0 if there exists an extension F of f from U_{+}(ζ̃) to U(ζ̃) such that F : U(ζ̃)\{ζ̃} → F (U(f(ζ̃)))\{f(ζ̃)} is a μ(f, ζ̃) + 1-sheeted covering and F |_{U^{−}}(ζ̃) : U^{−}(ζ̃) → F (U^{−}(f(ζ̃)) is a homeomorphism.

The behavior of f(z) in U^{(}ζ̃) can be described in the following way [12]: ζ̃ ∈ S^{1} is a partial critical point of f of multiplicity μ(f, ζ̃) if there exist 2μ(f, ζ̃)

simple arcs ℓ_{j}, j = \overline{1, 2μ(f, ζ̃)}, ∩^{2μ(f,_{ζ})}_{j=1}⟨ℓ_{j}⟩ = \{ζ̃\} passing through some U^{+}(ζ̃)

such that the restriction of f to each component of U^{+}(ζ̃)\ ∩^{2μ(f,_{ζ})}_{j=1} ⟨ℓ_{j}⟩ is schlicht

and maps this component onto U^{+}(f(ζ̃)) or U^{−}(f(ζ̃)). If we number these components in a consecutive way then the components with odd indices will be taken onto U^{+}(f(ζ̃)), while the components with even indices onto U^{−}(f(ζ̃)).

Theorem 1. Let γ be a normal closed curve possessing a maximal assemblage A(γ) and let B = {(j, k_{0}, 1), (j, k_{1}, 1), …, (j, k_{μ}, 1)} ∈ B(γ), k_{i} < k_{j+1}, i = \overline{0, μ − 1}, μ > 0. Then there exists an interior extension f̃ of γ such that f̃^{−1} has a partial branch point of multiplicity μ at the point (j, k_{0}, 1).

Without loss of generality we can assume that the raying α_{0}, α_{1}, …, α_{N} is numbered so that the set B consists of letters belonging to α_{0}.

Let E be an open set such that it contains the ray α_{0} and does not contain other rays of the raying and nodes of the curve γ. Moreover, we can choose E in such a way that every component of E ∩ ⟨γ⟩ contains at most one crossing point belonging to α_{0}. Denote by ⟨γ_{k_{i}}⟩ the component of E ∩ ⟨γ⟩ containing the point (0, k_{i}, 1) ∈ B, i = \overline{0, μ − 1}.

Take a simple arc γ[ζ′, ζ′′] such that ⟨γ[ζ′, ζ′′]⟩ ⊂ ⟨γ_{k_{0}}⟩. Let (0, k_{0}, 1) = γ(ζ̃) =

γ(e^{i_{θ}}), ζ′ = e^{iθ′}, ζ′′ = e^{iθ′′}. Take 2μ + 2 points ζ_{k} = e^{iθ_{k}}, k = \overline{0, 2μ + 1}, such

that θ′ < θ_{0} < θ_{2} < ⋯ < θ_{2μ} < θ̃ < θ_{2μ−1} < θ_{2μ−3} < ⋯ θ_{1} < θ_{2μ+1} < θ′′. Let Ũ(γ(ζ̃)) be a neighborhood such that ⟨γ[ζ′, ζ′′]⟩ ⊂ Ũ(γ(ζ̃)) ⊂ E.

Denote by g_{1} a simple arc such that i(g_{1}) = γ(ζ_{0}), t(g_{1}) = γ(ζ_{1}), ⟨g_{1}⟩\{γ(ζ_{0}), γ(ζ_{1})} ⊂ Ũ^{+}(γ(ζ̃)), and by g_{2} a simple arc such that i(g_{2}) = γ(ζ_{1}), t(γ_{2}) = γ(ζ_{2}), ⟨g_{2}⟩\{γ(ζ_{1}), γ(ζ_{2})} ⊂ Ũ^{−}(γ(ζ̃)) etc. Without loss of generality we can assume that all curves g_{n}, n = \overline{1, 2μ}, have no common points except end points of the adjacent curves. Evidently, g_{2μ+1} is transversal to all curves g_{n} with odd indices. Denote g = g_{1} · g_{2} ⋯ g_{2μ+1} and parametrize this curve in some way by θ, θ_{0} ≤ θ ≤ θ_{2μ+1}. Denote

γ^{∗}(θ) = \begin{cases}γ(e^{iθ}), & 0 ≤ θ ≤ θ_{0}, \\ g(θ), & θ_{0} ≤ θ ≤ θ_{2μ+1}, \\ γ(e^{iθ}), & θ_{2μ+1} ≤ θ ≤ 2π. \end{cases}

Denote by D_{2k+1}, k = \overline{0, μ}, the domain bounded by the closed Jordan curve γ[ζ_{2k}, ζ_{2k+1}] · g_{2k+1}^{−1}, and by D_{2k}, k = \overline{1, μ}, the domain bounded by g_{2k}^{−1} · γ[ζ_{2k}, ζ_{2k−1}]. Take in each D_{2k}, k = \overline{1, μ}, a point a_{N+k} and draw simple disjoint paths α_{N+k} through E on the left of α_{0} such that i(α_{j}) = a_{j} and t(α_{j}) lie in the unbounded component of ℝ^{2}\⟨γ⟩, j = \overline{N + 1, N + μ}. Moreover, we can choose paths α_{j}, j = \overline{N + 1, N + μ}, in such a way that the crossing point of α_{j} and g_{2ℓ} preceeds, on g, the crossing point of α_{s} and g_{2ℓ} when N + 1 ≤ j < s ≤ ℓ + N, 1 ≤ ℓ ≤ μ. Number the crossing points along each α_{j}, j = \overline{N + 1, N + μ}, consecutively so that the second coordinate of ⟨α_{j}⟩ ∩ ⟨g_{2}⟩ is equal to μ. It is evident that all crossings ⟨α_{j}⟩ ∩ ⟨g_{2ℓ}⟩, j = \overline{N + 1, N + μ}, 1 ≤ ℓ ≤ μ, are negative. Choose two points p_{1} and p_{2} such that p_{1} ∈ ⟨α_{0}⟩ ∩ Ũ(γ(ζ̃)), p_{2} ∈ ⟨α_{0}⟩ ∩ Ũ(γ(ζ̃)), p_{1} preceeds p_{2} on α_{0} and (⟨γ^{∗}⟩\⟨g⟩) ∩ ⟨α_{0}[p_{1}, p_{2}]⟩ = ∅. Denote by q an arc such that i(q) = p_{1}, t(q) = p_{2}, ⟨q⟩∩(⟨γ⟩\⟨γ[ζ′, ζ_{0}]⟩) = ∅ and ⟨q⟩∩⟨γ[ζ′, ζ_{0}]⟩ consists of a unique point. Denote α̃_{0} = α_{0}[a_{0}, p_{1}] · q · α_{0}[p_{2}, b_{0}].

Consider the system of rays α̃_{0}, α_{1}, …, α_{N}, α_{N+1}, …, α_{N+μ}. As g_{2μ+1} is transversal to all g_{2k+1}, k = \overline{0, μ − 1}, we have at least one initial point of rays in each bounded component of ℝ^{2}\⟨γ^{∗}⟩. Thus we obtain the raying for γ^{∗}.

Now we are going to construct an assemblage for γ^{∗}. Since the existence of an assemblage does not depend on a choice of the initial point of the curve, we can assume that i(γ^{∗}) = γ(ζ_{0}). Then the word for γ^{∗} has a following form:

\begin{matrix}(N + 1, μ, −1)(N + 2, μ, −1)(N + 3, μ, −1) ⋯ (N + μ, μ, −1) \\ (N + 2, μ − 1, −1)(N + 3, μ − 1, −1)(N + 4, μ − 1, −1) ⋯ (N + μ, μ − 1, −1) \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ (N + μ − 1, 1, −1)(N + μ, 2, −1)(N + μ, 1, −1)S_{0} \\ (N + μ, μ + r_{1}, 1)(N + μ − 1, μ + r_{1}, 1) ⋯ (N + 1, μ + r_{1}, 1)S_{1} \\ (N + μ, μ + r_{2}, 1)(N + μ − 1, μ + r_{2}, 1) ⋯ (N + 1, μ + r_{2}, 1)S_{2} \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ (N + μ, μ + r_{n}, 1)(N + μ − 1, μ + r_{n}, 1) ⋯ (N + 1, μ + r_{n}, 1)S_{n}, \end{matrix} (1)

where n≥μ and S_{0}, S_{1}, …, S_{n} are the sets of letters belonging to rays α̃_{0}, α_{1}, …, α_{N}. Select from list (1) all letters belonging to ∪^{μ}_{i=1}⟨γ_{k_{i}}⟩∩⟨α̃_{0}⟩. Let r_{n_{1}}, r_{n_{2}}, …, r_{n_{μ}} be the second coordinates of these letters arranged in the order as they are encountered in word (1). Replace the numbering of arcs γ_{k_{1}}, …, γ_{k_{μ}} in the following way. The arc containing the letter with the second coordinate r_{n_{i}} will have a new number μ + i and be denoted by γ_{μ+i}^{∗}. Then we replace the second coordinates of letters belonging to γ_{μ+i}^{∗} by μ + i and mark the letters with changed second coordinates by the superscript ∗.

Consider the following list of sets:

{(N + 1, μ, −1), (N + 1, 2μ, 1)^{∗}}, {(N + 2, μ, −1), (N + 2, 2μ, 1)^{∗}}, …

{(N + μ, μ, −1), (N + μ, 2μ, 1)^{∗}},
{(N + 2, μ − 1, −1), (N + 2, 2μ − 1, −1)^{∗}}, …
{(N + μ, μ − 1, −1), (N + μ, 2μ − 1, 1)^{∗}}, …,

{(N + k, μ − (k − 1), −1), (N + k, 2μ − k + 1, 1)^{∗}},  (2)
{(N + k + 1, μ − (k − 1), −1), (N + k + 1, 2μ − k + 1, 1)^{∗}}, …,
{(N + μ, μ − (k − 1), −1), (N + μ, 2μ − k + 1, 1)^{∗}}, …
{(N + μ − 1, 2, −1), (N + μ − 1, μ + 2, 1)^{∗}},

{(N + μ, 2, −1), (N + μ, μ + 2, 1)^{∗}}, {(N + μ, 1, −1), (N + μ, μ + 1, 1)^{∗}}.

Include all sets of list (2) as well as all sets of G(γ) into G(γ^{∗}). B(γ^{∗}) consists of all sets of B(γ) except B_{0}. We claim that G(γ^{∗}) ∪ B(γ^{∗}) is an assemblage for γ^{∗}. It is evident that conditions (a), (c) (d), (e) are fulfilled.

Recall that B = {0, k_{0}, 1), (0, μ + 1, 1)^{∗}, (0, μ + 2, 1)^{∗}, …, (0, 2μ, 1)^{∗}} was included in A(γ) and hence the set [θ_{0}, θ_{2μ+1}]∪(γ^{∗})^{−1}( ∪^{μ}_{j=1}⟨γ_{μ+j}^{∗}⟩) does not separate,

on S^{1}, any (γ^{∗})^{−1}(A) for A ∈ A(γ), A ≠ B. Thus we have to check that for any two sets A_{1} and A_{2} from the (2) (γ^{∗})^{−1}(A_{1}) does not separate, on S^{1}, the set (γ^{∗})^{−1}(A_{2}).

Consider two sets A_{i} = {(N + k_{i}, m_{i}, −1), (N + k_{i}, μ + m_{i}, 1)}, i = 1, 2. Suppose that (γ^{∗})^{−1}(N + k_{1}, m_{1}, −1) preceeds (γ^{∗})^{−1}(N + k_{2}, m_{2}, −1). Then either m_{1} > m_{2}, or m_{1} = m_{2} and k_{1} < k_{2}. In the former case μ + m_{2} < μ + m_{1} and hence (N + k_{1}, μ + m_{1}, 1)^{∗} follows (N + k_{2}, μ + m_{2}, 1)^{∗}. In the latter case the crossing points (N + k_{1}, μ + m_{1}, 1)^{∗} and (N + k_{2}, μ + m_{1}, 1)^{∗} lie on the same γ_{μ+m_{1}}^{∗} and as k_{1} < k_{2}, (N + k_{2}, μ + m_{1}, 1)^{∗} preceeds (N + k_{1}, μ + m_{1}, 1)^{∗} on γ_{μ+m_{1}}^{∗}. In both cases (γ^{∗})^{−1}(A_{1}) does not separate (γ^{∗})^{−1}(A_{2}) on S^{1}. Thus A(γ^{∗}) is an assemblage for γ^{∗}. Moreover, A(γ^{∗}) is a maximal assemblage since

∑_{B∈B(γ^{∗})}(card B − 1) = ∑_{B∈B(γ)\B_{0}}(card B − 1) = τ(γ) − μ = τ(γ^{∗}).

Denote by f^{∗}(ζ), ζ ∈ \overline{D}^{2}, the interior extension of γ^{∗} and let ℓ be the arc of S^{1} corresponding to g under the mapping f^{∗}.

Now we are going to construct some bordered Riemann surface using the procedure of identification of boundary arcs of the domains D_{k}, k = \overline{1, 2μ + 1}. Write boundaries of D_{1} and D_{2} in the form

∂D_{1} = γ[ζ_{0}, ζ̃] · γ[ζ̃, ζ_{1}]g_{1}^{−1},
∂D_{2} = g_{2}^{−1} · γ[ζ_{1}, ζ̃] · γ[ζ̃, ζ_{2}]

and paste \overline{D}_{1} and \overline{D}_{2} along ⟨γ[ζ_{1}, ζ̃]⟩. Denote by D̃_{2} the obtained domain. We have

∂D̃_{2} = γ[ζ_{0}, ζ̃] · γ[ζ̃, ζ_{2}] · g_{2}^{−1} · g_{1}^{−1}

and

∂D_{3} = γ[ζ_{2}, ζ̃] · γ[ζ̃, ζ_{3}] · g_{3}^{−1}

After pasting \overline{D̃}_{2} and \overline{D}_{3} along ⟨γ[ζ_{2}, ζ̃]⟩, we obtain the domain D̃_{3} which is no longer one-sheeted and

∂D̃_{3} = γ[ζ_{0}, ζ̃] · γ[ζ̃, ζ_{3}] · g_{3}^{−1} · g_{2}^{−1} · g_{1}^{−1}.

After 2μ steps of pasting, we obtain the many-sheeted domain D_{2μ+1} and

∂D̃_{2μ+1} = γ[ζ_{0}, ζ̃] · γ[ζ̃, ζ_{2μ+1}] · g_{2μ+1}^{−1} · g_{2μ}^{−1} ⋯ g_{1}^{−1} = γ[ζ_{0}, ζ_{2μ+1}] · g^{−1}.

Denote by G the image of D_{2μ+1} under the mapping h(w) = \sqrt[μ+1]{w − γ(ζ̃)}. G is a simply-connected domain bounded by a simple closed curve.

Let D′ be a simply connected domain such that D′ ⊂ C\overline{D}^{2} and \overline{D}′∩\overline{D}^{2} = ⟨ℓ⟩. Consider the function h(f^{∗}(e^{iθ})), e^{iθ} ∈ ⟨ℓ⟩, where we mean that the points f^{∗}(e^{iθ}) belonging to ⟨g⟩ are located on the boundary of the many-sheeted domain D̃_{2μ+1}. Then e^{iθ} → h(f^{∗}(e^{iθ})) is a homeomorphism of ⟨ℓ⟩ onto some arc of the boundary of the domain G. Let H be a homeomorphism of \overline{D}′ onto \overline{G} such that

H|_{⟨ℓ⟩} = h ∘ f^{∗}.  (3)

The existence of such a homeomorphism H is evident (it follows from Lemma 6.1 of [11]). Consider the function

f̃(z) = \begin{cases}f^{∗}(z), & if z ∈ \overline{D}^{2}, \\ h^{−1} ∘ H, & if z ∈ \overline{D}′. \end{cases}

By (3) f̃(z) is continuous in \overline{D}^{2} ∪ \overline{D}′ ∪ ⟨ℓ⟩. As f^{∗}(z) and h^{−1} ∘ H are interior mappings in D^{2} and D′, respectively, and H and h are homeomorphisms on ⟨ℓ⟩ and ⟨g⟩, respectively, f̃(2) is an interior function in D^{2} ∪ D′ ∪ ⟨ℓ⟩. The behavior of f̃ in a vicinity of the point m ∈ ⟨∂(D^{2} ∪ D′ ∪ ⟨ℓ⟩⟩, f̃(m) = γ(ζ̃) is defined by the behavior of h at the point γ(ζ̃) and hence m is a partial critical point of f̃ of multiplicity μ.

Now we will prove the inverse statement.

Theorem 2. If a normal curve γ has an interior extension possessing partial critical points, then there exists an interior extension of γ without partial critical points.

Let f̃ be an interior extension of γ. Denote by ζ_{j}, j = \overline{1, m}, partial critical points of f̃, and by μ_{j} their multiplicities. Let U_{0}(ζ_{j}) be a neighborhood of ζ_{j} such that f̃|_{\overline{U}_{0}(ζ_{j})∩S^{1}} is a simple arc. Let ℓ_{j} = [ζ′_{j}, ζ′′_{j}] be an arc of S^{1} such that ζ_{j} ∈ ⟨ℓ_{j}⟩ ⊂ U_{0}(ζ_{j}) and let U(ζ_{j}) be a neighborhood of ζ_{j} such that ∂U^{−}(ζ_{j}) = ℓ′_{j} · ℓ_{j}^{−1}, where ℓ′ is a simple arc ⟨ℓ′_{j}⟩ ∩ \overline{D}^{2} = \{ζ′_{j}, ζ′′_{j}\}. Denote L_{j} = f̃ ∘ ℓ_{j}, and consider a neighborhood U(γ(ζ_{j})) such that ∂U(γ(ζ_{j})) is a simple arc and ∂U(γ(ζ_{j})) ∩ ⟨γ⟩ = {f̃(ζ′_{j}), f̃(ζ′′_{j})}. Denote ∂U^{−}(γ(ζ_{j})) = L′_{j} · L_{j}^{−1} and let φ :

⟨ℓ′_{j}⟩ → ⟨L′_{j}⟩ be an arbitrary homeomorphism φ(ζ′_{j}) = f̃(ζ′_{j}), φ(ζ′′_{j}) = f̃(ζ′′_{j}). Denote by Φ the homeomorphism

Φ : \overline{U}^{−}(ζ_{j}) → \overline{U}^{−}(γ(ζ_{j}))

coinciding with φ on ∂U^{−}(ζ_{j}).

The mapping

f_{j} = \begin{cases}f̃(z), & z ∈ \overline{D}^{2}, \\ Φ(z), & z ∈ \overline{U}^{−}(ζ_{j}) \end{cases}

is an interior mapping of the simply connected domain D^{2} ∪ U^{−}(ζ_{j}) ∪ ⟨ℓ_{j}⟩ and from the definition of a partial critical point it follows that ζ_{j} is a (interior) critical point of f_{j} of multiplicity μ_{j}.

Repeat the above described procedure for every partial critical point of f̃. Finally, we obtain a function f without partial critical points. The restriction of f to the boundary of the domain of definition is a representation of a normal curve Γ which differs from γ. Applying the theorem of Section 2 to Γ we conclude that there exists an assemblage A(Γ). As the partial critical points ζ_{j}, j = \overline{1, m}, of f̃ have became the interior critical points of f, each ζ_{j}, j = \overline{1, m}, must be the initial point for a ray α_{j}, j = \overline{1, m}, included into the raying R(Γ) generating A(Γ).

Take m points z_{j}, z_{j} ∈ U^{+}(γ(ζ_{j})), j = \overline{1, m}, and connect z_{j} with ζ_{j} by a simple arc λ_{j} in U^{+}(γ(ζ_{j})). Consider a new system R′ of rays obtained from R(Γ) by replacing of α_{j} by α′_{j} = λ_{j} ·α_{j}. It is clear that R′ satisfies all conditions of raying for γ.

Let B ∈ B(Γ), and let B consist of points belonging to α′_{j}. If the crossing point of α_{j} with L′_{j} does not belong to B we include B into B(γ). If the crossing point of α_{j} with L′_{j} belongs to B, we replace this point by the crossing point of α′_{j} with ℓ and include obtained set B′ into B(γ). All other sets of B(Γ) are included into A(γ) unchanged as well as all sets of G(Γ). It is evident that the set B(γ) ∪ G(Γ) is an assemblage for γ. As card B′ = card B the multiplicity of every z_{j} is equal to μ_{j}, j = \overline{1, m}. Hence by Theorem 24.2 of [11] A(γ) is a maximal assemblage for γ.

4. The existence of an interior exstension of a nonnormal curve

Let γ(ζ) ≡ γ(e^{iθ}) be a parametrization of a closed curve γ. The point ζ̂ = e^{i_{θ}} is called a reversing point if there exist ε′ > 0, ε′′ > 0 and a sense reversing

homeomorphism ψ_{ζ}: [e^{i(θ−ε′)}, e^{i_{θ}}] → [e^{i_{θ}}, e^{i(_{θ}+ε′′)}] such that

γ(ψ_{ζ}(e^{iθ})) = γ(e^{iθ}), θ ∈ [θ̂ − ε′, θ].

Let f be the interior extension of γ and let ζ̂ be a reversing point of γ. We say that ζ̂ is a reversing critical point of f(z) (or f(ζ̂) is a reversing branch point of f^{−1}) of multiplicity μ(f, ζ̂) ≥ 0 if there exist some U(ζ̂), U(f(ζ̂)) and μ(f, ζ̂) simple arcs ℓ_{j} passing through U^{+}(ζ̂) and such that ⟨ℓ_{j}⟩ have a unique common point at ζ̂, f(z) is schlicht in every component v_{j}, j = \overline{1, μ(f, ζ̂) + 1}, of U(ζ̂)\ ∪ ⟨ℓ_{j}⟩ and f(z) takes every V_{j} onto U(f(ζ̂))\⟨γ[e^{i(θ−ε′)}, e^{i(θ+ε′′)}]⟩.

Let Q = (q_{1}, q_{2}, …, q_{m}) be an ordered m-tuple of points of ⟨γ⟩ distinct from the chosen initial point of γ. It is allowed that there are coinciding points among q_{j} but we require that every pair of points with succesive indices consists of distinct points. Denote by γ_{1} the arc of γ such that i(γ_{1}) = i(γ) and t(γ_{1}) = q_{1}, by γ_{2} the arc of γ or γ^{−1}, i(γ_{2}) = q_{1}, t(γ_{2}) = q_{2} etc, i(γ_{m+1}) = q_{m}, t(γ_{m+1}) = t(γ). Consider the curve γ_{Q} = γ_{1} · γ_{2} ⋯ γ_{m+1}.

The curve coincides with γ iff every q_{j−1} preceeds q_{j} on γ, j = \overline{2, m}. Denote by Q′ the subset of Q consisting of points q_{k} not separating q_{k−1} and q_{k+1} on γ, k = \overline{2, m − 1}. Denote n = card Q′. It is clear that n is even. Let \overline{μ} = (μ_{1}, μ_{2}, …, μ_{m}) be a vector where μ_{j}, j = \overline{1, m}, are positive integers.

Denote |\overline{μ}| = ∑_{j=1}^{m} μ_{j} and ν = n/2.

Theorem 3. Let γ be a normal curve and let f be an interior extension of γ with a partial critical point ζ̃ of multiplicity μ̃. Then for any Q ⊂ U(f(ζ̃)) and \overline{μ} such that |\overline{μ}| + ν = μ̃ there exists an interior extension f̂ of γ_{Q} such that every q_{j} ∈ Q\Q′ is a partial branch point and q_{j} ∈ Q′ is a reversing branch point of f̂^{−1} of multiplicity μ_{j}, j = \overline{1, m}.

Denote n_{k} = card(Q′ ∩ \{q_{1}, q_{2}, …, q_{k}\}) and by ζ̂_{j}, j = \overline{1, m}, the points corresponding to q_{j}, j = \overline{1, m}, under the mapping f. Let the set E, points ζ_{k}, k = 0, 2μ + 1, and curves g_{k}, k = \overline{1, 2μ̃ + 1}, be chosen as in Section 3. We assume that U(f(ζ̃)) is chosen so that ⟨g_{k}⟩ ∩ U(f(ζ̃)) = ∅, k = \overline{1, 2μ̃ + 1}. Denote f|_{S^{1}} = γ(ζ). Then

∂D_{j} = γ[ζ_{j−1}, ζ̂_{1}] · γ[ζ̂, ζ_{j}]g_{j}^{−1}, j = 1, 2μ_{1} + 1 + n_{1}.

Paste \overline{D}_{1} and \overline{D}_{2} along ⟨γ[ζ̂_{1}, ζ_{1}]⟩ and denote the obtained domain by \overline{D̂}_{2}. We have

∂D̂_{2} = γ[ζ_{0}, ζ̂_{1}] · γ[ζ̂_{1}, ζ_{2}] · g_{2}^{−1} · g_{1}^{−1}.

After pasting \overline{D̂}_{2} and \overline{D}_{3} along ⟨γ[ζ_{2}, ζ̂_{1}]⟩, we obtain \overline{D̂}_{3} and

∂D̂_{3} = γ[ζ_{0}, ζ̂_{1}] · γ[ζ̂_{1}, ζ_{3}] · g_{3}^{−1} · g_{2}^{−1} · g_{1}^{−1}.

After 2μ_{1} + n_{1} steps of pasting we obtain the many-sheeted domain D̂_{2μ_{1}+1+n_{1}} with the boundary

∂D̂_{2μ_{1}+1+n_{1}} = γ[ζ_{0}, ζ̂_{1}] · γ[ζ̂_{1}, ζ_{2μ_{1}+1+n_{1}}] · g_{2μ_{1}+1+n_{1}}^{−1} ⋯ g_{1}^{−1}.

The next domain \overline{D}_{2μ_{1}+n_{1}+2} is pasted with \overline{D̂}_{2μ_{1}+n_{1}+1} along ⟨γ[ζ_{2μ_{1}+n_{1}+1}, ζ̂_{2}]⟩. The procedure is continued by pasting \overline{D}_{2μ_{1}+n_{1}+k} with \overline{D̂}_{2μ_{1}+n_{1}+k−1} along ⟨γ[ζ_{2μ_{1}+n_{1}+k}, ζ̂_{2}]⟩ for k ≤ 2μ_{1} + n_{2} + 1. The domain \overline{D}_{2μ_{1}+2μ_{2}+n_{2}+2} is pasted with \overline{D̂}_{2μ_{1}+2μ_{2}+n_{2}+1} along ⟨γ[ζ_{2μ_{1}+2μ_{2}+n_{2}+1}, ζ̂_{3}]⟩. The general rule of pasting

is the following: the domain \overline{D}_{2 ∑_{j=1}^{s−1} μ_{j}+n_{s−1}+k} is pasted with \overline{D̂}_{2 ∑_{j=1}^{s−1} μ_{j}+n_{s−1}+k−1}

along ⟨γ[ζ_{2 ∑_{j=1}^{s−1} μ_{j}+k}, ζ̂_{s}]⟩ for 2 ≤ k ≤ 2μ_{s} + n_{s} + 1. The domain \overline{D}_{2 ∑_{j=1}^{s} μ_{j}+n_{s−1}+2} is

pasted with \overline{D̂}_{2 ∑_{j=1}^{s} μ_{j}+n_{s−1}+1} along ⟨γ[ζ_{2 ∑_{j=1}^{s} μ_{j}+n_{s}+1}, ζ̂_{s+1}]⟩, 1 ≤ s ≤ m − 1, n_{0} = 0

and the domain \overline{D̂}_{2 ∑_{j=1}^{m} n_{m}} = \overline{D̂}_{2( ∑_{j=1}^{m} μ_{j}+ν)} = \overline{D̂}_{2_{μ}} is pasted with \overline{D}_{2_{μ}+1} along

⟨γ[ζ̂_{m}, ζ_{2_{μ}+1}]⟩. We have

∂D̂_{2 ∑_{j=1}^{s−1} μ_{j}+n_{s−1}+k} = γ[ζ_{0}, ζ̂_{1}] · γ[ζ̂_{1}, ζ_{2 ∑_{j=1}^{s−1} μ_{j}+n_{s−1}+k}] · g_{2 ∑_{j=1}^{s−1} μ_{j}+n_{s−1}+k}^{−1} ⋯ g_{1}^{−1}.

and

∂D̂_{2_{μ}+1} = γ[ζ_{0}, ζ̂_{1}] · γ[ζ̂_{1}, ζ̂_{2}] ⋯ γ[ζ̂_{m}, ζ_{2_{μ}+1}] · g_{2_{μ}+1}^{−1} · g_{2_{μ}}^{−1} · g_{1}^{−1}.  (4)

We claim that D̂_{2_{μ}+1} is a simply connected domain. Indeed, all domains involved in the pasting procedure are simply connected and each domain is pasted with only one other domain along only one boundary arc.

Let ĥ : D̂_{2_{μ}+1} → G be a homeomorphism onto a one-sheeted simply connected domain. Let the domain D′ be the same as in the proof of Theorem 1, and let Ĥ be a homeomorphism of D′ onto G such that

Ĥ|_{⟨ℓ⟩} = ĥ(f(e^{iθ})), e^{iθ} ∈ ⟨ℓ⟩.

From (4) it follows that the function

f̂ = \begin{cases}f(z), & z ∈ \overline{D}^{2}, \\ h^{−1} ∘ Ĥ, & z ∈ \overline{D}′ \end{cases}

is an interior extension of γ_{Q}.

To complete the proof, we have to show that the point q_{j} is a branch point (partial or reversing) of multiplicity μ_{j}, j = \overline{1, m}. It is clear that without loss of generality we can assume that f̂ is defined in \overline{D}^{2}, and the preimages of q_{j} under f̂ are the same points ζ̂_{j}, j = \overline{1, m}.

Take some ζ̂_{s} and calculate the number of domains D_{k} such that q_{s} belongs to \overline{D}_{k}. From the construction of D̂_{k} it follows that q_{s} belongs to \overline{D}_{2 ∑_{j=1}^{s−1} μ_{j}+n_{s−1}+k},

1 ≤ k ≤ 2μ_{s} + n_{s} + 1. Hence q_{s} belongs to 2μ_{s} + 1 domains D_{k} if n_{s} = n_{s−1} and to 2μ_{s} + 2 domains D_{k} if n_{s} = n_{s+1} + 1. Thus ζ̂_{s} is a partial critical point of multiplicity μ_{s} if n_{s} = n_{s−1} (q_{s} separates q_{s−1} and q_{s+1}) and ζ̂_{s} is a reversing critical point of multiplicity μ_{s} if n_{s} = n_{s−1} + 1 (q_{s} does not separate q_{s−1} and q_{s+1}).

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(Received 30.07.2000)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: ramaz@imath.acnet.ge