SURGERY ON POSTCRITICALLY FINITE RATIONAL MAPS BY BLOWING UP AN ARC

KEVIN M. PILGRIM AND TAN LEI

Abstract. Using Thurston’s characterization of postcritically finite rational functions as branched coverings of the sphere to itself, we give a new method of constructing new conformal dynamical systems out of old ones. Let f(z) be a rational map and suppose that the postcritical set P (f) is finite. Let α be an embedded closed arc in the sphere and suppose that f|α is a homeomorphism. Define a branched covering g as follows. Cut the sphere open along α. Glue in a closed disc D. Map S^{2} − Int(D) via f and Int(D) by a homeomorphism to the complement of f(α). We prove theorems which give combinatorial conditions on f and α for g to be equivalent in the sense of Thurston to a rational map. The main idea in our proofs is a general theorem which forces a possible obstruction for g away from the disc D on which the new dynamics is defined.

1. Introduction

A rational map f(z) = p(z)/q(z) where p and q are relatively prime complex polynomials determines a holomorphic map of the Riemann sphere ℂ̂ to itself, and so defines a holomorphic dynamical system. The set of points C(f) where the derivative of f vanishes is the set of critical points of f; these are the points where the local degree of the map is larger than one. Counted with multiplicity, there are 2d − 2 critical points where d is the degree of f. The postcritical set P (f) of f is defined by P (f) = \overline{∪_{n>0}f^{n}(C(f))}. The map f is said to be postcritically finite if P (f) is finite. The Riemann sphere decomposes into the open Fatou set F (f) of points whose asymptotic behavior is stable under small perturbations, and the closed Julia set J(f) of points whose behavior under iteration is chaotic.

A branched covering f : S^{2} → S^{2} is an orientation-preserving continuous map such that for every x ∈ S^{2}, there exist local coordinates near x and f(x) such that, in these coordinates, x and f(x) are the point 0 ∈ ℂ and f is given by z ↦ z^{n}, n ≥ 1. The integer n is called the local degree of f near n. The sets C(f) and P (f) are defined analogously as for rational maps.

Convention: We assume throughout this paper that all rational maps and branched coverings are postcritically finite.

Research at MSRI is supported in part by NSF grant DMS-9022140.

Two branched coverings f and g of the sphere to itself are said to be equivalent if there are homeomorphisms ψ_{0} and ψ_{1} such that ψ_{0} ∘ f = g ∘ ψ_{1}, and such that ψ_{0} is isotopic to ψ_{1} through homeomorphisms ψ_{t}, t ∈ [0, 1] with ψ_{t}|P (f) = ψ_{0}|P (f) for all t.

Thurston has given necessary and sufficient combinatorial conditions for a postcritically finite branched covering g to be equivalent to a rational map, and has also shown that (with one understood exception) this map is determined up to conformal congugacy by g; see [DH2].

If f(z) is a rational map, suppose by cutting and pasting we obtain a branched covering g. When is g equivalent to a rational map? This question is, by Thurston’s theorem, a combinatorial one. In general, given d and N, there are infinitely many distinct equivalence classes of branched coverings f of degree d with |P (f)| = N. Only finitely many of these can contain rational maps, so it is useful to have constructions which, when applied to rational maps, yield new rational maps. In this paper we define a certain kind of cutting and pasting process on f(z) which is very robust: with fairly weak hypotheses which do not include restrictions on the degree of f or its global combinatorics, the result g will be equivalent to a rational map.

The rough idea of our construction is the following. Let f(z) be a postcritically finite rational map, let α be a closed, embedded topological arc in ℂ̂ and suppose that f|α is a homeomorphism. Define a new branched covering g by cutting open the sphere along α, gluing in a disc D, mapping the complement of D by f, and mapping D to the complement of f(α) in the sphere. A local model of g near D is the map z ↦ \frac{1}{2}(z + 1/z) on a neighborhood of the closed unit disc; this maps the unit circle onto the interval [−1, 1]. Then deg(g) = deg(f) + 1, and the local degree of g is one more than the local degree of f at the endpoints of α. More generally, one can map the disc D in a similar fashion n times around the sphere so that deg(g) = deg(f)+n. We call this construction blowing up the arc α n times. Note that by construction f = g away from D. The precise conditions on α for this construction to be welldefined, and the statements of our main theorems, are somewhat technical and are given precisely in Sections 2.3 and 2.4. We prove that under these conditions, the combinatorial class of g depends only on combinatorial data (Proposition 2).

For example, we derive the following as a corollary to our first main theorem, Theorem A. To set up the statement, we recall that in a periodic Fatou component of a postcritically finite rational map f, there is a canonical family of arcs called internal rays which are mapped homeomorphically under f to other internal rays (see Section 3.5 for the definition of internal ray).

Theorem: If α is a periodic or preperiodic internal ray of f, then the branched covering g which is f blown up n times along α is equivalent to a rational map.

We also apply the blowing up construction to obtain rational maps from Möbius transformations of finite order (see Theorem 4.2).

The central idea in our proofs is Theorem 3.2, Arcs intersecting obstructions. This theorem asserts that under suitable hypotheses, an obstruction Γ to the existence of a rational map equivalent to a branched covering g cannot intersect any family of arcs Λ which maps in a certain way. We use this theorem to “push” an obstruction away from the region D in the sphere where we add new dynamics. We then conclude that the obstruction Γ yields an obstruction for the original rational map f, which is impossible.

Our theorems may also be thought of as combination theorems between conformal dynamical systems. Other examples of such theorems include mating (see [Tan], [ST],

[Luo] ), where two polynomials of the same degree are glued together along their circles at infinity, and tuning (see [Ree], [Ahm]), where the dynamics of a polynomial is glued into the dynamics of a rational map near a superattracting cycle. In both of these examples the degree of the new map is the same as that of the old one. To date, all known combination theorems involving mating and tuning have hypotheses on the degree of the map and on the combinatorics of both the original map f(z) and the new dynamics which is being glued into f(z). Also, a definition of mating or tuning for branched coverings has not yet been given in such a manner as to be defined in terms of purely combinatorial data. Our techniques of proof also allow us to establish that certain blown-up quadratic polynomials may be mated (in a suitably generalized way) with an arbitrary quadratic polynomial to yield a rational map; see Section 4.4. Moreover, the technique generalizes easily to similar statements in higher degrees.

Organization of the paper.

• §2: we state needed definitions, give the precise definition of the blowing up construction, state our first two main theorems (Theorems A and B) precisely, and prove that the combinatorial class of g depends only on combinatorial data (Proposition 2).

• §3: we prove Theorems A and B.

• §4: we prove other related results.

• §5: we give examples illustrating our main theorems.

• §6: is an appendix containing a proof of Theorem 3.2, Arcs intersecting obstructions.

Acknowledgments: The pictures were drawn using programs written by Curt Mc-Mullen. The authors also thank MSRI, where some of this work was prepared.

Example: blowing up a preperiodic arc in the basilica. Let f(z) = z^{2} −1. Let α = [0, 1]. The filled-in Julia set of f(z) is the set of points whose orbits are bounded; this set is the black region in Figure 1. Let g be the branched covering which is f blown up once along α. The Julia set of the rational map equivalent to g is shown in Figure 2; see also [Pil1] for a discussion of this example and for a description of g as a branched covering.

Figure 1. The filled-in Julia set of f(z) = z^{2} − 1.

Figure 2. Julia set of the rational map corresponding to blowing up f(z) = z^{2} − 1 along [0, 1]. The white regions form the basin of infinity.

2. Blowing up an arc

§§2.1 and 2.2 give definitions needed to formulate our construction and theorems precisely. §2.3 states conditions under which blowing up will be a well-defined operation on equivalence classes. §2.4 gives the precise statements of Theorems A and B. §2.5 gives the construction of blowing up an arc, a proof that the equivalence class of the new map g depends only on combinatorial information, and a proposition (Orbits off α persist) which we will use in our proofs.

2.1. Marked branched covers.

Definition 1. A marked branched covering is a pair (f, X) where f : S^{2} → S^{2} is a branched covering and X is a finite set containing P (f) such that f(X) ⊂ X.

Two marked branched coverings (f, X), (g, Y) are said to be equivalent if they satisfy the definition of equivalence of branched coverings with the sets P (f) and P (g) replaced by X and Y respectively. There is a functor from the category of marked branched coverings (f, X) and equivalences to the category of branched coverings and equivalences. The image of a marked brached covering (f, X) under this functor is f; the image of an equivalence ψ_{0} : (S^{2}, X) → (S^{2}, Y) is the map of pairs ψ_{0} : (S^{2}, P (f)) → (S^{2}, P (g)). This functor “forgets” that ψ_{0} must send X − P (f) to Y − P (g). We call this the forgetful functor sending (f, X) to f.

Examples:

1. Let f be a rational map and X a finite forward-invariant set containing P (f). Then (f, X) is a branched cover marked by the set X. We will call (f, X) a marked rational map. Note that if (f, X) is a marked rational map with deg(f) = 1 and |X| ≥ 3, then f is necessarily a Möbius transformation of finite order.

2. Let p(z) = z^{2} − 1. Then p : [−1, 0] → [−1, 0] is an orientation-reversing homeomorphism. Postcompose p(z) with a homeomorphism h isotopic to the identity relative to P (p) such that for f(z) = h ∘ p(z), f([−1/4, 0]) = [−1, −3/4] and f([−1, −3/4]) = [−1/4, 0]. Let X = P (f) ∪ {−1/4, −3/4}. Then (f, X) is a branched cover marked by the set X.

In both examples, the image of (f, X) under the forgetful functor is combinatorially equivalent to a rational map. In the first case, however, the set X − P (f) of “new” marked points are already preperiodic under the dynamics of a rational map, whereas in the second case we have added new preperiodic points by our choice of representative of the combinatorial class of p(z).

2.2. Arcs and curves in (S^{2}, X). Let X be a finite set in the sphere. An arc α in (S^{2}, X) we define to be the image of the unit interval [0, 1] under a map j into the sphere such that the endpoints e(α) := j({0, 1}) ⊂ X, j|(0, 1) is an embedding, and j(0, 1) ∩ X = Ø. Thus we allow the endpoints e(α) of α to coincide (though we will not cut along such arcs in our construction). A simple closed curve in (S^{2}, X) is a simple closed curve in S^{2} − X; it is said to be peripheral if it is isotopic into every neighborhood of some point in X and essential if it is not contractible in S^{2} − X.

Two arcs (resp. curves) η_{0}, η_{1} in (S^{2}, X) are said to be isotopic relative to X, written η_{0} ≃_{X} η_{1}, if there is a continuous, one-parameter family η_{t}, t ∈ [0, 1] of such arcs (respectively curves) joining η_{0} to η_{1}. The isotopy class relative to X of an arc or simple closed curve η we will denote by [η]_{X}, or by [η] when the set X is understood.

A set of pairwise nonisotopic arcs (essential curves) in (S^{2}, X) will be called an arc (curve) system in (S^{2}, X). If Λ and Λ′ are arc or curve systems in (S^{2}, X) we write Λ ≃_{X} Λ′ if every element of Λ is isotopic to a unique element of Λ′ relative to X, and conversely.

The intersection number between a pair α, β consisting of either arcs or curves in (S^{2}, X) is defined by

α · β = min_{α≃_{X}α′,β≃_{X}β′} #{(α′ − e(α′)) ∩ (β′ − e(β′))}.

Invariance up to isotopy of arc systems.

Let (f, X) be a marked branched covering.

If λ is an arc in (S^{2}, X), we will call the closure λ̃ of a connected component of f^{−1}(λ − e(λ)) a lift of λ under f. Thus each arc λ in (S^{2}, X) has deg(f) distinct lifts λ̃. If the endpoints of λ are distinct, each lift maps homeomorphically under f onto λ. If Λ is an arc system in (S^{2}, X), a lift Λ̃ is defined to be an arc system in (S^{2}, X) each of whose elements λ̃ is a lift of some element λ ∈ Λ.

An arc system Λ in (S^{2}, X) is said to be forward invariant under f up to isotopy relative to X if there exists a subset Λ_{0} ⊂ Λ (possibly all of Λ) and a lift Λ̃_{0} of Λ_{0} such that Λ̃_{0} ≃_{X} Λ. If Λ ≃_{X} Λ′ and Λ is forward invariant up to isotopy relative to X, then so is Λ′. (An isotopy between an element λ ∈ Λ and an element λ′ ∈ Λ′ can be lifted to an isotopy between λ̃ ∈ Λ̃ and an arc λ̃′ which is a lift of λ̃′.)

The next proposition allows us to extract arc systems which are forward-invariant up to isotopy from a collection of arcs which are literally forward-invariant.

Proposition 1. Suppose L is a collection of arcs l in (S^{2}, X) such that for each l ∈ L, f|l is a homeomorphism and f(l) ∈ L. Then there is a subset Λ ⊂ L which is an arc system forward invariant under f up to isotopy, and such that each l ∈ L is isotopic relative to X to a unique element of Λ.

Proof. Let Λ_{0} ⊂ f(L) be an arc system chosen arbitrarily so that each arc f(l), l ∈ L is isotopic relative to X to a unique element of Λ_{0}. Then for each λ ∈ L, f(λ) ≃_{X} λ_{0} for a unique λ_{0} ∈ Λ_{0}. Hence by lifting isotopies, λ ≃_{X} λ̃_{0}, where λ̃_{0} is a lift of λ_{0}. Hence every element of L is isotopic to a lift of an element of Λ_{0}. We now enlarge our set Λ_{0} so that the last property holds. Let L_{1} denote the set of elements of L which are not isotopic to any element of Λ_{0}. Let Λ_{1} ⊂L_{1} be any arc system for which each element of L_{1} is isotopic to a unique element of Λ_{1}, and for which each element λ_{1} ∈ Λ_{1} is isotopic to some element of L_{1}. Then Λ = Λ_{0} ∪ Λ_{1} is an arc system with the desired properties.

2.3. Conditions under which blowing up is defined. Now suppose that (f, X) is a branched covering of degree ≥ 1 marked by the finite set X, and let α be an arc in S^{2}. Assume that the following conditions are satisfied:

Blowing Up Conditions:

1. f|α is a homeomorphism;

2. α is a union of arcs α_{j}, j = 1, 2, ..., L in (S^{2}, f^{−1}(X));

3. Int(α) ∩ (X ∪ C(f)) = Ø.

Note that the arcs f(α_{j}) are arcs in (S^{2}, X).

Remarks: Allowing α to consist of more than one arc in (S^{2}, f^{−1}(X)) is indeed useful: for example, consider the example of blowing up the “airplane” in Section 5. More generally, we may apply the blowing up construction to finite collections α^{1}, α^{2}, ... of arcs for which Int(α^{i}) ∩ α^{j} = Ø when i ≠ j and for which each arc satisfies the Blowing Up Conditions.

2.4. Statement of main results. We assume the following in this subsection. Let (f, X) be a marked rational map with deg(f) ≥ 2. Let α be an arc in ℂ̂ satisfying the Blowing Up Conditions. Let (g, X) be marked branched cover which is the map (f, X) blown up n times along α.

Theorem A If for all j = 1, 2, ..., L,

1. f(α_{j}) is isotopic relative to X to an arc which is contained in a finite arc system Λ_{j} which is forward invariant under f, and

2. there is a subset Λ_{j,0} ⊂ Λ_{j} with a lift Λ̃_{j,0} such that Λ̃_{j,0} ≃_{X} Λ_{j} and Λ̃_{j,0}∩Int(α) =

Ø, then g is combinatorially equivalent to a rational map.

Corollary 1. If α is a periodic or preperiodic internal ray of f, then g is equivalent to a rational map.

Corollary 2. Suppose α is a finite union of periodic or preperiodic internal rays of f. If for all n ≥ 0, the intersection number relative to X satisfies f^{n}(f(α))·f(α) ≠ 0 only when f^{n}(f(α)) = f(α), then g is equivalent to a rational map.

The definition of internal ray is given in Section 3.5.

Next, assume f(z) is a polynomial. According to work of Douady and Hubbard, there is a canonically defined finite topological tree T contained in the filled-in Julia set of f, depending only on f and e(α), such that f(T) ⊂ T with vertices mapped to vertices. In Section 3.4 we outline the construction of T.

Theorem B: (Blowing up an arc in a generalized Hubbard tree) If α ⊂ T, then g is combinatorially equivalent to a rational map.

Remark: The Blowing Up Conditions strongly restrict the possibilities for α. The novel feature of this theorem, relative to the first, is that we do not require that the α_{j} are eventually periodic up to isotopy relative to X.

2.5. Definition of blown-up map. Let (f, X) and α satisfy the Blowing Up Conditions of Section 2.3 . The construction is outlined in Figure 3 for the case n = 4.

We first cut the sphere along the interior of α. Let W be an open disc containing Int(α) bounded by a Jordan curve such that \overline{W} ∩ f^{−1}(X) = α ∩ f^{−1}(X) and ∂W ⊃ e(α). Since Int(α) ∩ (X ∪ C(f)) = Ø, we may choose W so that f|\overline{W} is a homeomorphism. Let h_{t} : ℂ̂ − Int(α) ↪ ℂ̂ be a continuous family of embeddings satisfying

1. h_{0} = id;

2. h_{t}|(ℂ̂ − W) = id for all t;

Figure 3. Construction of the blown-up map g from f.

3. h_{1}(ℂ̂ − Int(α)) = ℂ̂ − (D − e(α)), where D is a closed disc. The Jordan curve ∂D is a union of two closed arcs ∂D_{+} and ∂D_{−} with endpoints e(α) which we denote by {e_{−1}, e_{+1}}. The map h_{1}^{−1} extends uniquely to a continuous “collapsing” map c : ℂ̂ − Int(D) → ℂ̂ sending ℂ̂ − (D − e(α)) homeomorphically to ℂ̂ − Int(α) and each arc ∂D_{+} and ∂D_{−} homeomorphically to α.

We add the following new dynamics. Let Δ = \{z | |z| < 1 \}, and let ϕ_{n} : \overline{Δ} → ℂ̂ be the map given by

ϕ_{n}(z) = A_{2}(A_{1}(z))^{2n},

where A_{1}(z) = −i\frac{z+1}{z−1} and A_{2}(z) = \frac{z−1}{z+1}. The Möbius transformation A_{1} is a rigid rotation with respect to the spherical metric sending the ordered triple (−1, 0, 1) to (0, i, ∞) and sending \overline{Δ} to Im(z) ≥ 0; the map z ↦ z^{2n} maps the half-plane Im(z) ≥ 0 n times around the sphere sending the real axis to the nonnegative real axis; the Möbius transformation A_{2} is a rigid rotation sending (0, 1, ∞) to (−1, 0, 1). Thus the map ϕ_{n} wraps the closed unit disc \overline{Δ} around the sphere n times in such a way that ϕ_{n} is locally injective on \overline{Δ} − \{−1, 1\}, ϕ_{n}(−1) = −1, ϕ_{n}(1) = 1, and ϕ_{n}(∂Δ) = [−1, 1]. Let ∂Δ_{+}, ∂Δ_{−} denote respectively the intersection of \overline{Δ} with Im(z) ≥ 0 and Im(z) ≤ 0.

We now describe the gluing of f and ϕ_{n}. Choose a homeomorphism

h′ : (ℂ̂, [−1, 1]) → (ℂ̂, f(α))

which sends −1 to f(e_{−1}) and +1 to f(e_{+1}). The compositions (ϕ_{n}|_{∂Δ_{±}})^{−1} ∘ (h′)^{−1} ∘ f ∘ c|_{∂D_{±}} : ∂D_{±} → ∂Δ_{±} are homeomorphisms of closed arcs and together define a homeomorphism from ∂D to ∂Δ. Since D and \overline{Δ} are closed discs we can extend this to a homeomorphism h′′ between D and \overline{Δ}.

We now define a new branched covering g by

g(z) = \{ \begin{matrix}f ∘ c(z), & if z ∈ ℂ̂ − Int(D), \\ h′ ∘ ϕ_{n} ∘ h′′(z), & if z ∈ \overline{D}. \end{matrix}

The two definitions agree on the boundary, by construction, hence they give a welldefined branched covering of the sphere to itself such that deg(g) = deg(f) + n.

We may also apply the above construction simulaneously to finite collections of arcs α^{1}, α^{2}, ... which satisfy the Blowing Up Conditions and which satisfy Int(α^{i}) ∩ α^{j} = Ø, i ≠ j. We omit the details of this more general construction but emphasize that we may assume that for i ≠ j, \overline{W^{i}} ∩ \overline{W^{j}} ⊂ e(α^{i}) ∩ e(α^{j}) ⊂ X.

Proposition 2 (g depends only on classes). The combinatorial class of g depends only on the integer n, the combinatorial class of (f, X), and the isotopy classes of the arcs in (S^{2}, f^{−1}(X)) comprising α.

Proof. We prove in detail only the case of blowing up along a single arc; the case of blowing up along multiple arcs in similar, since we may choose the closed discs \overline{W^{i}} to intersect only in X.

If (f_{1}, X_{1}) is equivalent to (f_{2}, X_{2}), then (f_{1}, f_{1}^{−1}(X_{1})) is equivalent to (f_{2}, f_{2}^{−1}(X_{2})): an equivalence ψ_{0} : (S^{2}, X_{1}) → (S^{2}, X_{2}) may be lifted to an equivalence ψ_{1} : (S^{2}, f_{1}^{−1}(X_{1})) → (S^{2}, f_{2}^{−1}(X_{2})).

Let ϕ_{0}, ϕ_{1} : (S^{2}, f_{1}^{−1}(X_{1})) → (S^{2}, f_{2}^{−1}(X_{2})) be a pair of homeomorphisms giving an equivalence between (f_{1}, f_{1}^{−1}(X_{1})) and (f_{2}, f_{2}^{−1}(X_{2})). Then ϕ_{0} ∘ f_{1} = f_{2} ∘ ϕ_{1} and ϕ_{0} ≃ ϕ_{1} relative to f_{1}^{−1}(X_{1}). Let α^{2} = ϕ_{1}(α^{1}). Then α^{2} is an arc in S^{2} which is comprised of arcs in (S^{2}, f_{2}^{−1}(X_{2})). We must prove that if (g_{i}, X_{i}) is (f_{i}, X_{i}) blown up n times along α^{i}, i = 1, 2, then g_{1} and g_{2} are combinatorially equivalent. For convenience, if β^{i}, i = 1, 2 are arcs which are unions of finitely many arcs β_{j}^{i} in (S^{2}, Y), we will write β^{1} ≃_{Y} β^{2} if β_{j}^{1} ≃_{Y} β_{j}^{2} for all j.

Step 1: We first show the following. Suppose for i = 1, 2, g^{i} is the map produced from the map f by blowing up an arc α^{i}, where α^{1} ≃_{f^{−1}X} α^{2}, and W_{i}, D_{i}, h′_{i}, h′′_{i}, and h_{t}^{i} are the discs and maps used in the definition of g_{i}. Then g_{1} is equivalent to g_{2}.

Since f|\overline{W}_{i} is a homeomorphism, f(\overline{W}_{i}) is a closed disc. There exists a homeomorphism ψ_{0} : (S^{2}, X) → (S^{2}, X) isotopic to the identity relative to X such that ψ_{0}(f(\overline{W}_{1}), f(α^{1})) = (f(\overline{W}_{2}), f(α^{2})). For since α^{1} ≃_{f^{−1}X} α^{2}, f(α^{1}) ≃_{X} f(α^{2}). By results of Epstein [EP], there is an ambient isotopy of S^{2} carrying ∂f(\overline{W}_{1}) ∪ f(α^{1}) to ∂(f(\overline{W}_{2})) ∪ f(α^{2}) fixing X pointwise. Let Ψ : S^{2} × I → S^{2} be such an isotopy such that Ψ(·, 0) = id, Ψ(·, 1) = ψ_{0}, and Ψ(X, t) = id_{X} for all t.

Since X ⊃ P (f), f|S^{2} − f^{−1}X → S^{2} − X is a covering map. Hence there is a lift Ψ̃ of Ψ which is an isotopy fixing X pointwise between the identity and a map which we call ψ^{1}. Note that ψ^{1} : (\overline{W}_{1}, α^{1}) → (\overline{W}_{2}, α^{2}).

Recall that g_{i}|S^{2} − \overline{W}_{i} = f|S^{2} − \overline{W}_{i}, i = 1, 2. We will define a homeomorphism ψ_{1} : (S^{2}, X) → (S^{2}, X) such that g_{2}ψ_{1} = ψ_{0}g_{1} and show ψ_{0} is isotopic to ψ_{1} relative to X. The map ψ_{1} will be defined as four maps ψ^{(1)} = ψ^{1}|S^{2} − W_{1}, ψ^{(2)}, ψ^{(3)}, ψ^{(4)} which piece together.

We now define ψ^{(2)}. Let K_{i} = (g_{i}|D_{i})^{−1}(f(α^{i})), i = 1, 2. Then K_{i} is a graph in S^{2} with exactly two vertices which are the endpoints e(α^{i}) ⊂ f^{−1}(X) and with n + 2 edges, each of which joins one vertex to the other. There is a well-ordering on the set of edges of K_{i}: E < E′ if the edge E separates Int(E′) from Int(∂_{−}D_{i}) in D_{i}. Write the set of edges in increasing order as {E_{0}^{i}, ..., E_{n}^{i}}. Since we glue in the same new dynamics ϕ_{n} in the construction of g_{1} and g_{2}, there is a homeomorphism ψ^{(2)} : K_{1} → K_{2} such that g_{2}ψ^{(2)} = ψ_{0}g_{1} and ψ^{(2)}(E_{l}^{1}) = E_{l}^{2}, l = 0, 1, ..., n, i.e. ψ^{(2)} preserves the well-ordering of the edges.

We now define ψ^{(3)}. For l = 0, 1, ..., n − 1, the complementary component U_{l}^{i} of D^{i} − K_{i} bounded by E_{l}^{i} ∪ E_{l+1}^{i} is a Jordan domain mapping homeomorphically under g_{i} onto S^{2} − f(α^{i}), i = 1, 2. Hence for each l, there is a homeomorphism ψ^{(3),l} : U_{l}^{1} → U_{l}^{2} such that g_{2}ψ^{(3),l}|U_{l}^{1} = ψ_{0}g_{1}|U_{l}^{1}. Since ψ^{(2)} preserves the ordering of edges, the homeomorphisms ψ^{(3),l} and ψ^{(2)} agree on the common boundaries of their domains of definition.

A similar argument shows that there is a homeomorphism ψ^{(4)} : W_{1}−D_{1} → W_{2}−D_{2} such that g_{2}ψ^{(4)}|W_{1} − D_{1} = ψ_{0}g_{1}|W_{1} − D_{1} and which agrees with ψ^{(1)} on ∂W_{1} and with ψ^{(2)} on ∂D_{1}.

Hence the maps ψ^{(1)}, ψ^{(2)}, ψ^{(3)}, ψ^{(4)} define a homeomorphism of the sphere ψ_{1} such that ψ_{0}g_{1} = g_{2}ψ_{1}.

That ψ_{0} and ψ_{1} are isotopic relative to X follows from the fact that ψ_{1}|S^{2} − W_{1} is isotopic (via Ψ̃) to the identity relative to X and the fact that X∩\overline{W}_{1} ⊂ e(α^{1}) ⊂ ∂W_{1}. This completes the proof of Step 1.

Step 2: We now prove the general statement. Let (f_{1}, f_{1}^{−1}X_{1}), (f_{2}, f_{2}^{−1}X_{2}) be marked covers and ϕ_{0}, ϕ_{1} as above give an equivalence carrying the arc α^{1} to α^{2}. By replacing f_{2} with the conjugate ϕ_{0}f_{2}ϕ_{0}^{−1} we may assume that ϕ_{0} = id and that f_{1}(α^{1}) = f_{2}(α^{2}).

By Step 1, the combinatorial class of g_{i} is independent of the choice of discs W, D. Let W_{1}, D_{1} be any such choice for g_{1}. Hence we may assume that (W_{2}, D_{2}) = ϕ_{1}(W_{1}, D_{1}) and that f_{1}(\overline{W}_{1}, α^{1}) = f_{2}(\overline{W}_{2}, α^{2}).

Let ψ_{0} : (S^{2}, X)) → (S^{2}, X) be the identity map (after conjugation, we may assume X = X_{1} = X_{2} ). We will find a homeomorphism ψ_{1} such that ψ_{0}g_{1} = g_{2}ψ_{1} and ψ_{0} is isotopic to ψ_{1} relative to X. As in the proof of the previous step, we define ψ_{1} in four pieces ψ_{1}|S^{2} − W_{1} = ψ^{(1)}, ψ^{(2)}, ψ^{(3)}, ψ^{(4)}. The details are all similar to those in the proof of the previous step.

Remark: A similar proof shows that when blowing up along any finite collection of arcs α^{1}, α^{2}, ... of arcs for which Int(α^{i}) ∩ α^{j} = Ø, i ≠ j, the equivalence class of the resulting map is independent of the choices and representatives. Moreover, the resulting class is the same whether we blow up one arc at a time or all of them simultaneously.

The next proposition gives a condition under which arc systems in (S^{2}, X) forward invariant under f up to isotopy are also forward invariant under g up to isotopy.

Proposition 3 (Orbits off α persist). Suppose Λ is an arc system in (S^{2}, X) which is forward invariant under f up to isotopy relative to X. If, under f, there exists a lift Λ̃ ≃_{X} Λ such that Λ̃ ∩ Int(α) = Ø, then Λ is forward invariant under g up to isotopy relative to X.

Proof. Let Λ_{0} ⊂ Λ be an arc system and Λ̃_{0} a lift of Λ_{0} under f such that Λ̃_{0} ≃_{X} Λ. Since g|ℂ̂−D = f ∘c, if λ̃_{0} is a lift under f of λ_{0} ∈ Λ_{0} such that λ̃_{0} ∩Int(α) = Ø, then h_{1}(λ̃_{0}) = c^{−1}(λ̃_{0}) is a lift under g of λ_{0}. Since h_{1} is isotopic to the identity relative to X, λ̃_{0} ≃_{X} h_{1}(λ̃_{0}). Hence h_{1}(Λ̃_{0}) is a lift of Λ_{0} under g such that h_{1}(Λ̃_{0}) ≃_{X} Λ_{0}.

3. Proofs of Theorems A and B

In §3.1 we state the characterization of marked rational maps as marked branched coverings as an immediate corollary to the proof of Thurston’s characterization of rational maps as branched coverings. In §3.2 we state the main ingredient in our proof, Theorem 3.2, Arcs intersecting obstructions; its proof we relegate to an Appendix. In §3.3 and we prove Theorem A; §3.4 gives the proof of Theorem B, which depends on a construction given in §3.5.

3.1. Characterization of marked rational maps. Suppose (f, X) is a marked branched covering. To a curve system Γ in (S^{2}, X) we associate the Thurston linear transformation which is the linear map

f_{Γ} : ℝ^{Γ} → ℝ^{Γ}

given by

f_{Γ}(γ) = ∑_{γ′⊂f^{−1}(γ)} \frac{1}{deg(f : γ′ → γ)}[γ′]_{Γ},

where [γ′]_{Γ} denotes the element of Γ isotopic to γ′, if it exists. If there are no such elements, the sum is taken to be zero. Here, we regard elements of Γ as the basis vectors of ℝ^{Γ}. This transformation depends only on the isotopy classes of curves relative to X and is natural with respect to iteration of f. Since the entries of f_{Γ} are nonnegative, the Perron-Frobenius Theorem ([Gan], Chapter XIII) implies that there is a nonnegative real eigenvalue λ(f_{Γ}) equal to the spectral radius of f_{Γ}.

A nonnegative square matrix A_{ij} is said to be irreducible if for each (i, j), there is an n ≥ 0 such that A_{ij}^{n} > 0. We say that Γ is irreducible if the matrix for f_{Γ} is irreducible. The Perron-Frobenius theory easily gives that if Γ is any curve system with λ(f_{Γ}) > 0, then then Γ contains an irreducible curve system Γ′ for which λ(f_{Γ′}) = λ(f_{Γ}).

A curve system Γ is said to be a multicurve if its elements are nonperipheral and pairwise disjoint. The proof of Thurston’s characterization of rational maps as branched coverings of the sphere given in [DH2] extends immediately to the setting of branched covers marked by additional points.

Theorem 3.1 (Characterization of marked rational maps). Let (f, X) be a marked branched covering. Then (f, X) is combinatorially equivalent to a marked rational map (R, Y) if and only if for every irreducible multicurve Γ in (S^{2}, X), either

1. λ(f_{Γ}) = 1 and the orbifold associated to (f, P (f)) has signature (2, 2, 2, 2), in which case R is covered by an integral endomorphism of the torus, or

2. λ(f_{Γ}) < 1, in which case the rational map R is unique up to conjugation by elements of Aut(ℂ̂).

Definition 2 (Thurston obstruction.). If (f, X) is a marked branched covering for which the orbifold associated to (f, P (f)) is not the (2, 2, 2, 2) orbifold, an (irreducible) multicurve Γ for which λ(f_{Γ}) ≥ 1 is called an (irreducible) Thurston obstruction.

Example: Let Γ = {γ_{0}, ..., γ_{p−1}} be a multicurve such that the following holds: for each 0 ≤ j ≤ p − 1, γ_{j} has a (not necessarily unique) lift δ ≃_{X} γ_{j−1(modp)} and deg f : δ → γ_{j} = 1. Then Γ is called a Levy cycle and is an example of an irreducible Thurston obstruction.

Remarks:

1. See [DH2] for the proof and for the definition of the orbifold associated to a rational map. A rational map R is said to be covered by a torus endomorphism if there is a complex torus T, a holomorphic covering map R̃ : T → T, and a surjective holomorphic map p such that R ∘ p = p ∘ R̃. The endomorphism R̃ is said to be integral if it has a lift to an automorphism of ℂ of the form z ↦ nz, some n ≥ 2. The first case in the above theorem is rare and does not arise, for example, if there are periodic critical points of f or if |P (f)| ≥ 5.

2. In [DH2] the curve systems Γ in the theorem are required to be f-invariant, i.e. every lift δ of an element γ ∈ Γ is either inessential, peripheral, or isotopic to an element of Γ. However, by taking preimages we may always extend an irreducible multicurve Γ to an f-invariant multicurve without decreasing the leading eigenvalue of the corresponding Thurston transformation.

To apply Theorem 3.1 we will need to rule out the first rare case. Let (f, X) be a marked rational map f(z). If f is covered by an integral torus endomorphism, then f is not a polynomial. Moreover, there can be no arcs in (S^{2}, X) with distinct endpoints which are periodic under f up to isotopy relative to X. For the Euclidean metric on ℂ pushes down to an (orbifold) metric on ℂ̂ which is uniformly expanded under f. Hence all inverse branches of f are uniformly contracting with respect to this orbifold metric. This implies that if λ̃ is any lift under f of an arc λ in (S^{2}, X), then the diameter of λ̃ is strictly less than that of λ. Since the diameter of any arc in (S^{2}, X) with distinct endpoints is positive, this shows that there are no arcs with distinct endpoints which are periodic under f up to isotopy relative to X. Hence to prove our two theorems, we may assume that the orbifold O_{f} is not the (2, 2, 2, 2) orbifold.

If g is f blown up along α then the Euler characteristic χ(O_{g}) < χ(O_{f}) ≤ 0. The (2, 2, 2, 2) orbifold has Euler characteristic zero, hence O_{g} cannot be the (2, 2, 2, 2) orbifold. So g is combinatorially equivalent to a rational map if and only if there are no irreducible Thurston obstructions for g.

3.2. Arcs intersecting obstructions. Let (f, X) be a marked branched covering, Λ be an arc system in (S^{2}, X), and Γ a curve system in (S^{2}, X).

The unweighted Thurston linear transformation f_{#,Γ} : ℝ^{Γ} → ℝ^{Γ} is defined by

f_{#,Γ}(γ) = ∑_{γ′⊂f^{−1}(γ)}[γ′]_{Γ} for γ ∈ Γ,

where [γ′]_{Γ} denotes the element in Γ homotopic to γ′ and zero otherwise. As before, this map is independent of choice of representatives and natural with respect to iteration. Note that 0 ≤ (f_{Γ})_{i,j} ≤ (f_{#,Γ})_{i,j} for all i, j and that (f_{Γ})_{i,j} = 0 if and only if (f_{#,Γ})_{i,j} = 0. This implies that f_{Γ} is irreducible if and only if f_{#,Γ} is irreducible.

In analogy with the unweighted Thurston linear transformation, we define

f_{#,Λ} : ℝ^{Λ} → ℝ^{Λ}

by setting

f_{#,Λ}(λ) = ∑_{λ′⊂f^{−1}(λ)}[λ′]_{Λ} for λ ∈ Λ,

where [λ′]_{Λ} denotes the element of Λ homotopic to λ′ (if it exists). We say that Λ is irreducible if f_{#,Λ} is irreducible. It is straightforward to verify that the transformation f_{#,Λ} depends only on [Λ]_{X} and not on the choice of representatives and that f_{#,Λ} is natural with respect to iteration of f, i.e. (f^{n})_{#,Λ} = (f_{#,Λ})^{n}.

If α and β are arcs in (S^{2}, X) we define the multiplicity mult(f : α → β) to be the number of lifts of β which are isotopic to α relative to X. Thus (f_{#,Λ})_{ij} = mult(f : λ_{i} → λ_{j}).

The following proposition follows immediately from the decomposition theorem for nonnegative square matrices in [Gan], Chapter XIII.

Proposition 4 (Decomposition of invariant arc systems). Let Λ be a finite arc system forward invariant under f up to isotopy relative to X. Then there are disjoint (as subsets of Λ) irreducible arc subsystems Θ_{1}, Θ_{2}, ..., Θ_{s} of Λ and integers n_{1}, n_{2}, ..., n_{s} such that for each λ ∈ Λ, there is a (not necessarily unique) Θ_{k} such that λ is isotopic to a lift of an element of Θ_{k} under f^{n_{k}}.

For Λ a curve system or an arc system in (S^{2}, X), set Λ̃ (resp. Λ̃(f^{n})) to be the union of those components of f^{−1}(Λ) (resp. f^{−n}(Λ)) which are isotopic to elements of Λ. If Λ is irreducible, each component of Λ is isotopic to some (not necessarily unique) component of Λ̃(f) (resp. Λ̃(f^{n})).

The following theorem appears in a different form in [ST].

Theorem 3.2 (Arcs intersecting obstructions). Let (f, X) be a marked branched covering, Γ an irreducible Thurston obstruction in (S^{2}, X), and Λ an irreducible arc system in (S^{2}, X). Suppose furthermore that #(Γ ∩ Λ) = Γ · Λ. Then either

1. Γ · Λ = 0 and Γ · f^{−n}(Λ) = 0, f^{−n}(Γ) · Λ = 0 for all n ≥ 1; or

2. Γ · Λ ≠ 0 and

(a) Each component of Γ (resp. Λ) is isotopic to a unique component of Γ̃ (resp. Λ̃), the mapping f : Γ̃ → Γ (resp. f : Λ̃ → Λ) is a homeomorphism, Γ̃ ∩ (f^{−1}(Λ) − Λ̃)) = ∅, and Λ̃ ∩ (f^{−1}(Γ) − Γ̃) = ∅. More precisely,

(i) for each γ ∈ Γ, there is exactly one curve γ′ ⊂ f^{−1}(γ) such that γ′ ∩Λ̃ ≠ ∅. Moreover, the curve γ′ is the unique component of f^{−1}(γ) which is isotopic to an element of Γ;

(ii) for each λ ∈ Λ, there is exactly one connected component λ′ of f^{−1}(λ) such that λ′ ∩ Γ̃ ≠ ∅. Moreover, the arc λ′ is the unique component of f^{−1}(λ) which is isotopic to an element of Λ;

(b) the transformations f_{#,Γ} and f_{#,Λ} are transitive permutations of the basis vectors.

(c) The above results remain true if we replace f by f^{n}, for any n ≥ 1 (though transitivity may fail).

(d) For any arc λ in (S^{2}, X) which is isotopic to an arc in f^{−n}(Λ) for some n > 1 but which is not isotopic to an arc in Λ, Γ · λ = 0. Similarly, for any curve γ in (S^{2}, X) which is isotopic to a curve in f^{−n}(Γ) for some n > 1 but which is not isotopic to a curve in Γ, Λ · γ = 0.

We prove this theorem in the Appendix.

Corollary 3. If Λ contains more than one periodic cycle, or if the multiplicity of f from some element α ∈ Λ to β ∈ Λ is larger than one, then there are no irreducible Thurston obstructions which intersect Λ.

Proof. If so then f_{#,Λ} is not a transitive permutation of the basis vectors.

3.3. Proof of Theorem A. By Theorem 3.1, Characterization of marked rational maps, and our discussion at the end of Section 3.1, (g, X) is combinatorially equivalent to a marked rational map if there exist no irreducible Thurston obstructions Γ for g. Let Γ be such an obstruction and let Γ̃ be the union of those components of g^{−1}(Γ) isotopic to elements of Γ, relative to X.

By the second hypothesis of the theorem and Proposition 3, Orbits off α persist, for each j, Λ_{j} is forward invariant under g up to isotopy relative to X.

By Proposition 4, Decomposition of invariant arc systems, for each j, there is an irreducible arc system Θ_{j} ⊂ Λ_{j} and an integer n_{j} such that f(α_{j}) is isotopic relative to X to a lift of an element of Θ_{j} under g^{n_{j}}.

We now show that we may choose representatives of Γ so that Γ̃ ∩ D = Ø. This will complete the proof of the theorem: for since g|ℂ̂ − (D − e(α)) = f ∘ c and c : ℂ̂ − (D − e(α)) → ℂ̂ − Int(α) is isotopic to the identity relative to X, Γ is then an irreducible Thurston obstruction for the marked rational map (f, X). By Theorem 3.1, Characterization of marked rational maps, and the fact that f is not covered by an integral torus endomorphism, this gives a contradiction.

First, we may assume Γ minimizes the number of intersections with f(α_{j}) and Θ_{j} for all j. Write ∂D = ∪_{j}α_{0,j}^{±}, where α_{0,j}^{±} are the two unique components of (g|D)^{−1}(f(α_{j})) which are contained in ∂_{±}D respectively. The proof now breaks down into several cases:

Case 1: Γ · f(α_{j}) = 0. Then Γ ∩ f(α_{j}) = Ø, hence Γ̃ ∩ g^{−1}(f(α_{j})) = Ø, and so Γ̃ ∩ α_{0,j}^{±} = Ø.

Case 2: Γ · f(α_{j}) ≠ 0. We will again show that Γ̃ ∩ α_{j}^{±} = Ø.

We first show that in this case f(α_{j}) ∈ Θ_{j} up to isotopy. By our choice of Θ_{j}, f(α_{j}) is isotopic to a lift of an element θ ∈ Θ_{j} under f^{n_{j}}. First, if Γ · Θ_{j} = 0, then by an application of Theorem 3.2, Arcs intersecting obstructions, Part 1, we would have Γ · f(α_{j}) = 0, contradicting our assumption. Hence Γ · Θ_{j} ≠ 0. Next, if f(α_{j}) ∉ Θ_{j} up to isotopy, then by Theorem 3.2, Arcs intersecting obstructions, Part 2(d) ( with Λ = Θ_{j} and λ = f(α_{j})), Γ · f(α_{j}) = 0, contradicting our assumption. Hence f(α_{j}) ∈ Θ_{j} up to isotopy.

Neither of the arcs α_{0,j}^{+}, α_{0,j}^{−} can be isotopic to elements of Θ_{j}. For if so, then α must consist of a single arc in (S^{2}, X), i.e. L = 1 in the definition of the blown up map g. Then α itself is isotopic to an element of Θ_{j} since Θ_{j} is irreducible. It then follows that ∂D ⊂ g^{−1}(f(α)) consists of two arcs α_{0}^{+} and α_{0}^{−} in (S^{2}, X) which are isotopic relative to X, by construction of the blown-up map g. Hence the multiplicity of g : α_{0}^{±} → f(α) is at least n ≥ 2, where n is the number of times we blow up α. Hence the transformation g_{#,Θ} is not a transitive permutation of the basis vectors. By Theorem 3.2, Arcs intersecting obstructions, Γ · Θ = 0, hence Γ · f(α) = 0, contradicting our assumption.

By the previous paragraph α_{0,j}^{±} ∉ Θ_{j} up to isotopy. By Theorem 3.2, Part 2a(ii), there is exactly one lift λ′ of f(α_{j}) which intersects (as a subset of S^{2}) the set Γ̃, and this lift λ′ is isotopic to an element of Θ_{j}. Hence α_{0,j}^{±} ∩ Γ̃ = Ø.

In both cases we have shown that for all j, Γ̃ ∩ α_{0,j}^{±} = Ø. Thus Γ̃ ∩ ∂D = Ø, which implies that Γ̃ ∩ D = Ø: there can be no components of Γ contained entirely in D since Int(D) ∩ X = Ø.

Proof of Corollary 1 Suppose α is a periodic or preperiodic internal ray of f. Points in the interior of an internal ray lie in Fatou components and have infinite forward orbits under f. Hence no points in the interior of an internal ray can be preperiodic under f. We may therefore assume that α consists of a single arc in (S^{2}, f^{−1}X), and hence that f(α) is a periodic or preperiodic internal ray with endpoints in X.

Two internal rays either coincide, or intersect only in their endpoints. Hence if l_{i} = f^{i}(f(α)), i > 0, then l_{i} ∩ Int(f(α)) ≠ Ø if and only if α = l_{i} for some l_{i}, i.e. f(α) is a periodic internal ray.

Let L = {l_{i}}_{i>0}. Then L is a finite collection of arcs in (S^{2}, X) which, as a subset of the sphere, is forward invariant under f. By Proposition 1 there exists a subset Λ ⊂ L which is an arc system forward invariant under f such that each element of L is isotopic relative to X to a unique element of L. By the definition of forward invariance there is a subset Λ_{0} ⊂ Λ and a lift Λ̃_{0} ≃_{X} Λ.

If f(α) ∉ Λ, then Int(f(α)) ∩ Λ = Ø, hence Int(α) ∩ f^{−1}(Λ) = Ø. Since Λ̃_{0} ⊂ f^{−1}(Λ), Λ̃_{0} ∩ Int(α) = Ø. The Corollary then follows by Theorem A.

If f(α) ∈ Λ, let λ′ be any arc which is isotopic to f(α) relative to X and such that Int(f(α)) ∩ λ′ = Ø. Let Λ′ be the arc system (Λ − {f(α)}) ∪ {λ′}. Then Λ ≃_{X} Λ′ and so Λ′ is forward invariant under f up to isotopy relative to X. By construction Λ′ ∩ Int(f(α)) = Ø, hence f^{−1}(Λ′) ∩ Int(α) = Ø. If Λ′_{0} is the unique subset of Λ′ for which Λ′_{0} ≃_{X} Λ_{0}, and if Λ̃′_{0} is the lift of Λ′_{0} obtained by lifting isotopies between elements of Λ′_{0} and Λ_{0}, then Λ̃′_{0} ≃_{X} Λ̃_{0} ≃_{X} Λ. Since Λ̃′_{0} ⊂ f^{−1}(Λ′), Λ̃′_{0} ∩ Int(α) = Ø. The Corollary then follows by Theorem A.

Proof of Corollary 2 The argument is similar to the one given above. The hypothesis guarantees the existence of the arc λ′ used in the last paragraph.

3.4. Proof of Theorem B. Let f(z) be a postcritically finite monic polynomial and X a finite set such that f(X) ⊂ X and X ⊃ P (f) . Let K(f) and F (f) denote the filled-in Julia set and Fatou set of f, respectively.

In [Poi], §I.1 (originally in [DH1]), it is shown that X − {∞} determines uniquely a finite topological tree T with vertices V (T) satisfying the following properties.

1. T ⊂ K(f) and T ∩ F (f) only in internal rays;

2. f(T) ⊂ T and f(V (T)) ⊂ V (T);

3. V (T) ⊃ X − {∞}.

The definition of internal and external rays is given in the following section.

Our first step in the proof is to construct a “spider” S for T. The construction of S is given in the following section. S is a finite topological graph with vertices V (S) and edges E(S) satisfying the following properties:

1. V (S) = X ⊃ P (f);

2. Every edge of S is a topological arc joining a finite vertex to infinity;

3. f(S) ⊂ S;

4. f maps edges of S homeomorphically to edges of S;

5. S ∩ T = X − {∞};

Note that the edges of S are arcs in (S^{2}, X).

Given the existence of the spider S with the properties above, we now prove Theorem B.

The edges E(S) form a finite collection of arcs l in (S^{2}, X) such that f(l) ∈ E(S) for each l ∈ E(S). By Proposition 1, there is an arc system Λ ⊂ E(S) which is forward invariant under f up to isotopy relative to X, i.e. there is a subset Λ_{0} ⊂ Λ and a lift Λ̃_{0} ≃_{X} Λ. Note that since Λ̃_{0} ≃_{X} Λ ⊂ X, the endpoints of Λ̃ are points in X.

We now show that Λ is forward invariant under g up to isotopy relative to X. By Proposition 3, Orbits off α persist, it is enough to show that Λ̃_{0} ∩ Int(α) = Ø. Since α ⊂ T and f(T) ⊂ T, f(α) ⊂ T. Since T ∩ S ⊂ X, f(α) ∩ S ⊂ X. Hence α ∩ f^{−1}(S) ⊂ f^{−1}(X). Now, Λ̃_{0} ⊂ f^{−1}(S), and as remarked above, the endpoints of Λ̃_{0} are points in X, hence α ∩ Λ̃_{0} ⊂ X. But Int(α) ∩ X = Ø, hence Int(α) ∩ Λ̃_{0} = Ø.

Hence Λ is forward invariant under g up to isotopy relative to X. By Proposition 4, there are disjoint (as subsets of Λ) irreducible arc systems Θ_{k} ⊂ Λ and integers n_{k} such that each element of Λ is a lift of an element of Θ_{k} under n_{k} for some k.

Let Γ be an irreducible Thurston obstruction for g and let Γ̃ be the union of components of g^{−1}(Γ) which are isotopic relative to X to elements of Γ. We may assume that Γ minimizes the number of intersections with each f(α_{j}) and with each λ ∈ Λ. As in the previous proof it suffices to show that Γ̃ ∩ ∂D = Ø, where D is the disc used in the construction of g.

Let K = (g|D)^{−1}(Λ); this is a finite, connected graph in D with vertices equal to g^{−1}(X) ∩ D ⊂ K.

We first reduce to the case when Γ̃ ∩ K = Ø. As in the previous proof, there are two cases. Let λ ∈ Λ.

Case 1: Γ · λ = 0. Then Γ ∩ λ = Ø by our choice of representatives of Γ, so Γ̃ ∩ (g|D)^{−1}(λ) = Ø.

Case 2: Γ · λ ≠ 0.

Let λ̃ be any lift of λ under g which is contained in D (and hence in K). By Proposition 4, λ is isotopic to a lift of an element of some irreducible arc system Θ_{j} ⊂ Λ under f^{n_{k}}, some n_{k}.

Since λ ∈ Λ ⊂ E(S), λ is an edge of the spider S and hence joins a point of X−{∞} to the point ∞. By construction D ∩ ∞ = Ø. Since λ̃ ⊂ D is a lift of λ under g, one of the endpoints of λ̃ is contained in D and maps onto ∞. Hence this endpoint is strictly preperiodic under g. Hence λ̃ cannot be contained in the irreducible arc system Θ_{k}. By Theorem 3.2, Arcs intersecting obstructions, Γ̃ ∩ λ̃ = Ø.

Together, the two cases show that Γ̃ ∩ K = Ø.

So we may assume Γ̃ ∩ K = Ø. We conclude the proof by showing that if Γ̃ ∩ ∂D ≠ Ø, then Γ does not have minimal intersection number with f(α), contrary to our assumption.

Suppose γ̃ ⊂ Γ̃ intersects ∂D. Since Γ̃ ∩ K = Ø, γ̃ ∩ Int(D) is contained in a component V of D − K. Since V (S) = X ⊃ P (f), and since each edge E(S) is isotopic relative to X to an element of some Λ_{k}, each component of S^{2} −Λ is an open disc containing no critical values of g. Hence each component of g^{−1}(S^{2} − Λ) maps homeomorphically under g to its image. Since V is contained in such a component, g|V is a homeomorphism. Let W ⊂ V be any component of V −γ̃ which “outermost”, i.e. not separated from ∂D by any component of γ̃∩D. Then W is a disc in S^{2}−g^{−1}X bounded by exactly one subarc of γ̃ and one subarc of ∂_{+}D or ∂_{−}D. Since g|V is a homeomorphism and W ⊂ V, g(W) is a disc in S^{2} − X bounded by one subarc of f(α) and one subarc of γ ⊂ Γ, contradicting the assumption that Γ ∩ f(α) is minimized.

3.5. Construction of S. We first recall some facts about the dynamics of postcritically finite polynomials from [Mil].

Let f(z) be a postcritically finite polynomial. The filled-in Julia set K(f) = {z|f^{n}(z) ↛ ∞} is a connected, compact subset of ℂ whose complement is connected. The Julia set J(f) = ∂K(f) is locally connected. For each Fatou component Ω, there is exactly one point x ∈ Ω such that f^{n}(x) ∈ P (f) for some n. Let Ω_{x} denote the Fatou component containing such a point x. A classical theorem of Böttcher implies the following: there are holomorphic isomorphisms ϕ_{x} : (Δ, 0) → (Ω_{x}, x) such that for all w ∈ Δ, ϕ_{f(x)}(w^{d_{x}}) = f(ϕ_{x}(w)), where d_{x} is the local degree of f near x. Since J(f) is locally connected, each Fatou component has locally connected boundary, and a theorem of Carathéodory implies that the maps ϕ_{x} extend continuously to S^{1} = ∂Δ. Let R_{t} denote the radial line {r exp(2πit)|0 ≤ r ≤ 1}. The image R_{x,t} = ϕ_{x}(R_{t}) we call the ray of angle t in Ω_{x}; if x = ∞ R_{x,t} is called an external ray; otherwise we call it an internal ray. Each periodic point x ∈ J(f) is the landing point of at least one and at most finitely many external rays. The landing point of an external ray is periodic if and only if x is periodic, though the periods may differ. Internal and external rays map homeomorphically onto their images.

Construction of S. Our spider will be a union

S = ⋃_{x∈X−{∞}} E_{x},

where E_{x} denotes the set of edges of the graph S incident to x.

• If x ∈ J(f) ∩ (X − {∞}), each element e ∈ E_{x} will be an external ray landing at x, and E_{x} will be the union of all external rays landing at x.

• If x ∈ F (f) ∩ (X − {∞}), each element e ∈ E_{x} will be the union of exactly one internal ray in Ω_{x} and one external ray landing at a common point q ∈ ∂Ω_{x}. The set E_{x} will be a finite union of such pairs.

Since external and internal rays map homeomorphically onto their images, and the image of an internal (external) ray is again an internal (external) ray, edges will map homeomorphically to edges.

Each x ∈ J(f) ∩ (X − {∞}) is eventually periodic, so the set E_{x} of external rays landing at x is finite (see [Mil], §18), and f(E_{x}) = E_{f(x)}. Since T ⊂ K(f), E_{x} ∩ T ⊂ X − {∞}.

We now define E_{x} for x ∈ F (f) ∩ (X − {∞}). First, choose one element x from each periodic cycle in (X − {∞}) ∩ F (f). Let p be the period of x. Then T ∩ Ω_{x} consists of a finite collection of eventually periodic internal rays. Thus there exists a periodic internal ray R_{x}, of some period kp > 0, such that R_{x} ∩ T = {x}. Let q be the landing point of the internal ray R_{x}; then q is periodic of period k. There exists a periodic external ray R_{∞} landing at q. Since q is not a critical point, f^{k} is a local homeomorphism near q sending R_{x} to itself, hence the ray R_{∞} also has period k. Let e_{x} = R_{x} ∪ R_{∞}; this will be one edge in the collection E_{x}. Define

E_{x} = ⋃_{n=0}^{k−1} f^{np}e(x).

E_{x} is finite since q is periodic. For y ∈ f^{n}(x), 1 ≤ n < p, define

E_{y} = f^{n}(E_{x}).

Thus E_{y} is finite. Since R_{x} is periodic, R_{x} ∩ T = {x}, f(X − {∞}) ⊂ X − {∞}, and f(T) ⊂ T, we have that E_{y} ∩ T ⊂ X − {∞} for all y.

We now inductively define E_{x} for strictly preperiodic x ∈ X−{∞}∩F (f). Suppose the collection E_{y} has been defined, where f(x) = y. Choose any element e_{y} ∈ E_{y}. The edge e_{y} is a union of an internal ray R_{y} and external ray R_{∞}; let q be their common landing point. Choose any preimage R_{x} of R_{y} joining x to a point q′ in ∂Ω_{x}. Since a small disc near q pulled back to q′ is again a disc, there is a preimage R′_{∞} of R_{∞} joining ∞ to q′. Let e_{x} = R_{x} ∪ R′_{∞} and E_{x} = {e_{x}}. Then f : e_{x} → e_{y} and E_{x} ∩ T = {x}.

This completes the construction of S.

4. Complements

4.1. Blowing up multiple arcs. Theorems A and B generalize in the obvious way to blowing up along finite collections α^{1}, α^{2}, ... of arcs for which Int(α^{i})∩α^{j} = Ø, i ≠ j.

4.2. Blowing up Möbius transformations. Let G be a finite connected graph in S^{2} with vertex set X and edge set E. Suppose in addition that no edge joins a vertex to itself and that no two vertices are joined by more than one edge. Assign to each edge e ∈ E an integer m(e) ≥ 0. Let (M, X) be a marked Möbius transformation of finite order and suppose M(G) = G.

Theorem: (Blowing up Möbius transformations) Let (g, X) be the marked branched covering (M, X) blown up m(e) times along the edge e ∈ E. Let G′ denote the subgraph of G which is the union of edges e′ (and their vertices) for which m(M^{i}e′) > 0 for some i. Then (g, X) is equivalent to a marked rational map if and only if

1. G′ is connected, and

2. if G′ is nonempty, no component of S^{2} − G′ contains more than one point of X.

Proof. If G′ is empty there is nothing to prove. Otherwise, there is at least one edge which is blown up. The vertices of this blown up edge become periodic critical points for g, hence the orbifold of the (unmarked) branched covering g is not the (2, 2, 2, 2) orbifold. Hence by Theorem 3.1, Characterization of marked rational maps, (g, X) is equivalent to a marked rational map if and only if (g, X) has no Thurston obstructions.

Note that G′ is invariant under M. Let X′ denote the vertices of G′. Then for each connected component G′′ of G′, G′′ has at least two vertices.

We now prove the necessity of (1) and (2). First, assume that (g, X) is equivalent to a marked rational map. Then (g^{∘n}, X) is also equivalent to a marked rational map for every n ≥ 1.

If (2) fails, then there is a component U of S^{2} − G′ such that |U ∩ (X − X′)| ≥ 2. Let γ be any simple closed curve in U separating G′ from U ∩ (X − X′). Then γ is nonperipheral since |X′| ≥ 2 and |U ∩ (X − X′)| ≥ 2 by assumption. Choose n ≥ 1 so that M^{∘n}(γ) = γ. We may further assume that γ is disjoint from the discs W within which we alter the map (M, X) to obtain the map (g, X). It then follows that g^{∘n}(γ) = γ and that (g^{∘n})|γ is a homeomorphism. Then γ is a Levy cycle for (g^{∘n}, X), so (g^{∘n}, X) is not equivalent to a rational map.

If (1) fails, then there is a simple closed curve γ in S^{2} −(G′ ∪X) separating G′ into two subsets, each of which has at least two elements of X. Hence γ is nonperipheral. The same argument as in the previous paragraph then shows that (g^{∘n}, X) is not equivalent to a rational map.

We now prove the sufficiency of (1) and (2). We may assume |X| ≥ 4, since otherwise there are no nonperipheral simple closed curves in (S^{2}, X) and hence no Thurston obstructions. Next, we may assume that for every e ∈ E and for every i > 0, M^{i}e ≃_{X} e if and only if M^{i}e = e, by our requirement that there is at most one edge between any two vertices and the assumption M(G) = G.

For each orbit of an edge in E′ under M, choose one edge e′_{j}. Let ϵ_{j} ≃_{X} e′_{j} be an arc in (S^{2}, X) chosen so that ϵ_{j} ∩ G = e(ϵ_{j}). Then in particular, Int(ϵ_{j}) ∩ e′ = Ø for each e′ ∈ E′. Let Λ_{j} denote the arc system which is the orbit of ϵ_{j} under M. Then Λ_{j} is forward-invariant under M and the interior of every element of Λ_{j} is disjoint from the edges along which we blow up. Hence by Proposition 3, Orbits off α persist, each Λ_{j} is forward-invariant up to isotopy under (g, X).

The transformations g_{#,Λ_{j}} are irreducible but are not transitive permutations of the basis vectors. The transformation M_{#,Λ_{j}} is a transitive permutation of the basis vectors since Λ_{j} is the orbit of a single arc under M and is therefore irreducible. Hence as matrices g_{#,Λ_{j}} ≥ M_{#,Λ_{j}} ≥ 0, so g_{#,Λ_{j}} is irreducible since M_{#,Λ_{j}} is irreducible. Moreover, if λ ∈ Λ_{j} and λ ≃_{X} e′ where m(e′) ≥ 1, then mult(g : λ → g(λ)) = m(e′) + 1 ≥ 2, and so the entry of g_{#,Λ_{j}} corresponding to the ordered pair (λ, g(λ)) is at least two.

Let Γ be an irreducible Thurston obstruction for (g, X). We may assume that the elements of Γ are chosen so that Γ · Λ_{j} = |Γ ∩ Λ_{j}| for each j. By Theorem 3.2, Arcs intersecting obstructions, Γ ∩ Λ_{j} = Ø for each j. Since ∪_{j}Λ_{j} = G′, Γ ⊂ S^{2} − G′. Conditions (1) and (2) imply that every component of S^{2} − (G′ ∪ X) is either simplyconnected or is a once-punctured disc. Hence there are no nonperipheral simple closed curves in S^{2} − (G′ ∪ X), and so there are no Thurston obstructions for (g, X).

Example: Let f_{p/q}(z) = z^{2} + c_{p/q}, q ≥ 2, where p < q and p and q are relatively prime, where c_{p/q} is the center of the hyperbolic component tangent to the main cardioid in the p/q limb of the Mandelbrot set. We call such a polynomial “starlike”. (There is a topological tree T which is a union of q internal rays, each joining a point in the finite superattracting cycle containing 0 to a common repelling fixed point. Thus T is a “star”. The tree T is mapped homeomorphically under f into itself with rotation number p/q.) Let M_{p/q} = e^{2πip/q}z and X_{p/q} = {∞} ∪_{n∈ℤ} e^{2πinp/q}. Let G be the orbit of the arc [1, ∞] under M_{p/q}. Then f_{p/q} is equivalent to (M_{p/q}, X_{p/q}) blown up once along α = [1, ∞] ⊂ G.

Remark: Let M_{p/q} be as above. Let Y_{p/q} = X_{p/q} − {∞} and let G be the unit circle. Let α be the circular segment joining 1 and e^{2πi \frac{1}{q}}. Then blowing up (M, Y_{p/q}) once along α yields a quadratic non-polynomial rational map. Mary Rees (personal communication) has informed us that all quadratic non-polynomial maps with obstructed tunings arise in this fashion.

4.3. Killing obstructions by blowing up. The blowing up construction can be applied to marked branched covers which are not equivalent to rational maps in such a way that the result is equivalent to a rational map.

Let p(z) = z^{2} −1. Extend p to a degree two branched cover of the closed disc D by adjoining the map z → z^{2} along the circle at infinity S_{∞}^{1}. Now let f be the branched cover obtained by gluing the action of p on two copies D_{+}, D_{−} of D along S_{∞}^{1}. The map f is called the mating of f with itself and is not equivalent to a rational map. For let R_{t} denote the external ray of angle t for p. Then after adjoining on S_{∞}^{1}, R_{t} becomes a topological arc R_{t}^{∞} in D joining ∞ · e^{2πit} to the landing point of R_{t} in J(p). For the map f, the unions of arcs R_{1/3}^{∞}, R_{2/3}^{∞} in D_{+} and D_{−} are joined together to form a simple closed curve γ ⊂ S^{2} − P (f) which maps to itself homeomorphically by degree one, and hence is an obstruction. Note that the interval [−1, 0] ⊂ D_{−} is mapped homeomorphically to itself under f. Consider the map g = (f, P (f)) blown up once along α = [−1, 0] ⊂ D_{−}. Then by Theorem 3.2, Arcs Intersecting obstructions, no obstruction can intersect α. Hence any obstruction for g is isotopic relative to P (g) into D_{+}, where there are no such obstructions. So g is equivalent to a rational map while f is not.

4.4. Generalized matings. One may think of the map g in the previous example as the polynomial z^{2} − 1 mated with the rational map which is z^{2} − 1 blown up along α. (The resulting branched covering is well- defined since there are no critical points in the interior of the disc D in which we glue new dynamics.) Using techniques similar to the ones employed in the proofs of our theorems one may prove the following. To set up the statement, suppose f_{1}(z) is a starlike quadratic polynomial with tree T. Let α be the union of two consecutive edges in T. Then α satisfies the hypothesis of Corollary 2, so blowing up n times along α yields a branched covering equivalent to a rational map r_{n}(z). The map r_{n}(z) has a fixed critical point of local degree two which we may take to be at infinity.

Theorem 4.1 (Matings with blown-up starlike polynomials exist). Let f_{2}(z) be an arbitrary postcritically finite quadratic polynomial. Then the mating of f_{2}(z) with r_{n}(z) is equivalent to a rational map g(z).

Similar results can be formulated for higher-degree maps as well as for generalizations of tunings. The proofs are all similar to those given here: an obstruction is forced to lie outside the region on the sphere in which the new dynamics is glued by Theorem 3.2, Arcs intersecting obstructions.

4.5. Special maps. The following theorem is useful since there are interesting examples of maps produced by blowing up arcs to which Theorems A’ and B’ do not apply.

Theorem: (Special maps) If g is (f, X) blown up along α, and if |P (g)| = 3, then g is equivalent to a rational map.

Figure 4. The tree for the Douady Rabbit.

Figure 5. Douady’s Rabbit.

Proof. The orbifold of g is hyperbolic, so g is equivalent to a rational map if and only if there are no Thurston obstructions, which do not exist since there are no nonperipheral curves in (S^{2}, P (G)).

5. Examples

In this section we give some examples and applications.

5.1. Examples illustrating Theorems.

Blowing up a periodic arc in the “rabbit”. Let f(z) = z^{2} + c, where c ≈ −0.12256117+0.74486177i is chosen so that 0 is periodic of period three and Im(z)> 0. Let X = {0, c, f(c)}. Then f is “starlike”, so there is a tree T (shown in Figure 4) with vertices consisting of the set X and the alpha fixed point p of f and which is mapped homeomorphically to itself under f with rotation number 1/3. T is a union of the three internal rays for f which are fixed under f^{∘3}.

We let α be the union of the two internal rays joining 0 and c to p. By Corollary 2, blowing up once along α yields a branched covering equivalent to a rational map. The Julia set of f is shown in Figure 5, and that of f blown up along α is shown in Figure 6. Blowing up the “airplane”. Let f(z) = z^{2} + c, where c ≈ −1.75488 is chosen so that the critical point at zero is periodic of period three with orbit lying in the Figure 6. The Douady Rabbit blown up along α.

Figure 7. The airplane.

real line. Then c < 0 < f(c). Let X = {c, 0, f(c)}. The tree T_{X} is the interval [c, f(c)] with vertices X. The polynomial f sends the invervals [c, 0] and [0, f(c)] homeomorphically to the interval [c, f(c)]. We let α = [c, 0] ⊂ T_{X}. Then the Blowing Up conditions are satisfied, and by Theorem B, (f, X) blown up once along α is equivalent to a rational map.

The Julia set of f is shown in Figure 7. The Julia set of the result of blowing up f along α is shown in Figure 8; the map is given by g(z) = a\frac{(z−1)^{3}}{1−3z}, where a ≈ −2.37123, normalized so that 1 ↦ 0 by local degree three, 0 ↦ −a by local degree two, and −a ↦ 1 by local degree one.

A degree four map with S_{3} symmetry. Let G be a topological graph in S^{2} which is a triangle with vertices X and edges E. Let M be the identity map. By Theorem 4.2, Blowing up Möbius transformations, the marked covering (g, X) blown up once along each edge in E is equivalent to a degree four rational map g with three fixed critical points each of mulitiplicity two. The full mapping class group of orientation-preserving homeomorphisms h of S^{2} − X up to isotopy is isomorphic to S_{3} and commutes with g up to equivalence since the blowing up construction in this case is performed symmetrically. That is, h ∘ g is equivalent to g ∘ h. It follows that the conformal automorphism group of g is S_{3}. A formula for this map is given in [McM] and is used there for giving a generally convergent iterative algorithm for solving the cubic.

Figure 8. The airplane blown up along α.

Remark: More generally one can apply Theorem 4.2, Blowing up Möbius transformations to other symmetric graphs and the identity map to produce symmetric postcritically finite rational maps; cf. [DM] for a detailed discussion of rational maps with icosahedral symmetry.

6. Appendix: Proof of Theorem Arcs intersecting obstructions

We will use the following easy fact:

Lemma 1. Let f be a branched covering from S^{2} to S^{2}. Let B be a subset of S^{2} such that f|_{B} : B → f(B) is a bijection (resp. a k-to-1 mapping). Let A be any subset of S^{2}. Then #f^{−1}(A) ∩ B = #A ∩ f(B) (resp. #f^{−1}(A) ∩ B = k · #A ∩ f(B)).

For Λ a curve system or an arc system in (S^{2}, X)), set Λ̃ (resp. Λ̃(f^{n})) to be the union of those components of f^{−1}(Λ) (resp. f^{−n}(Λ)) which are isotopic to elements of Λ. If Λ is irreducible, each component of Λ is isotopic to some (possibly non-unique) component of Λ̃(f) (resp. Λ̃(f^{n})).

By the definition of irreducible arc system, there exists at least one subset Λ̃′ of Λ̃ such that Λ̃′ is isotopic to Λ and f : Λ̃′ − X → Λ − X is a homeomorphism. Then

f_{#,Γ}(γ_{j}) · Λ ≤ ∑_{γ′⊂f^{−1}(γ_{j})} #(γ′ ∩ Λ̃′) = #(f^{−1}(γ_{j}) ∩ Λ̃′) = #(γ_{j} ∩ Λ) = γ_{j} · Λ (1)

where the first inequality follows since f : Λ̃′ − X → Λ − X is a homeomorphism and the second to last equality follows by the lemma. Assume f_{#,Γ}(γ_{j}) = ∑_{i} b_{ij}γ_{i}. Then by (1), (∑_{j} ∑_{i} b_{ij}γ_{i}) · Λ ≤ Γ · Λ . But

(∑_{j} ∑_{i} b_{ij}γ_{i}) · Λ = (∑_{i}(∑_{j} b_{ij})γ_{i}) · Λ ≥ ∑_{i} γ_{i} · Λ = Γ · Λ ,  (2)

where the last inequality is obtained as follows. By irreducibility, f_{Γ} has no zero row no zero column, so the same is true for f_{#,Γ}. But each entry of f_{#,Γ} is a non negative integer, so b_{ij} ≥ 1 if b_{ij} ≠ 0. We have then either A) Γ · Λ = 0 or B) Γ · Λ ≠ 0 and all inequalities above become equalities.

In case that Γ · Λ = 0, we claim that Γ · f^{−1}(Λ) = 0. Let γ_{i} ∈ Γ. Then γ_{i} ∼ γ′ with γ′ a curve in f^{−1}(γ_{j}) for some γ_{j} ∈ Γ. Then γ_{i} · f^{−1}(Λ) ≤ #(γ′ ∩ f^{−1}(Λ)) = k#(γ_{j} ∩ Λ) = 0. The case for n > 1 is proved similarly. This prove part one of the theorem.

In case B), equalities in Equation 2 imply that for each i, there is a unique j such that b_{ij} ≠ 0 and that b_{ij} = 1. Thus for each γ_{i}, there is a unique γ_{j} such that some component of f^{−1}(γ_{j}) is isotopic to γ_{i}. Also, there is a unique such component γ′ of f^{−1}(γ_{j}) isotopic to γ_{i}. Since f_{#,Γ} is irreducible, this implies that f_{#,Γ} is a transitive permutation of the basis vectors. As a consequence, γ′ is also the unique curve in f^{−1}(γ_{j}) isotopic to an element of Γ.

Equalites in Equation 1 tells us that f_{#,Γ}(γ_{j}) · Λ = #(f^{−1}(γ_{j}) ∩ Λ̃′) = γ_{j} · Λ. Combining with the facts that f_{#,Γ}(γ_{j}) = [γ′]_{Γ} = γ_{i}, γ′ · Λ ≤ #(γ′ ∩ Λ̃′) and γ′ ∩ Λ̃′ ⊂ f^{−1}(γ_{j}) ∩ Λ̃′, we get γ_{i} · Λ = γ_{j} · Λ and γ′ ∩ Λ̃′ = f^{−1}(γ_{j}) ∩ Λ̃′. As a consequence, γ′ is also the unique curve in f^{−1}(γ_{j}) such that γ′ ∩Λ̃′ ≠ ∅. Moreover f^{−1}(Γ)∩Λ̃′ = Γ̃∩Λ̃′.

The matrix f_{Γ} is a permutation with entry a_{ij} = 1/k_{j}, where k_{j} is the degree of f : γ′ → γ_{j}. Since λ(f_{Γ}) ≥ 1, we have k_{j} = 1 for all j and f_{Γ} = f_{#,Γ}. Thus f : Γ̃ → Γ is a homeomorphism.

Note that in the above argument we only used the following properties of the triple (Γ, Λ, Λ̃′): the matrix f_{Γ} has neither zero row nor zero column, with leading eigenvalue at least 1, the set Λ̃′ is isotopic to Λ and f : Λ̃′ − X → Λ − X is a homeomorphism. By what we just proved, the triple (Λ, Γ, Γ̃) verifies also these properties. So we can redo the same argument for this new triple. We conclude then for each λ ∈ Λ, there is exactly one curve λ′ in f^{−1}(λ) with λ′ ∩ Γ̃ ≠ ∅. Moreover, λ′ is the unique curve in f^{−1}(λ) which is isotopic to an element of Λ. The matrix f_{#,Λ} is a transitive permutation. As a consequence, Λ̃′ = Λ̃.

Replace Λ̃′ by Λ̃ in the beginning of the proof, we conclude finally that Γ̃∩f^{−1}(Λ) = Γ̃∩Λ̃ = f^{−1}(Γ)∩Λ̃ and #Γ̃∩Λ̃ = #Γ∩Λ = Γ·Λ. As a consequence, Γ̃∩(f^{−1}(Λ)−Λ̃) = ∅ and (f^{−1}(Γ) − Γ̃) ∩ Λ̃ = ∅.

Now take any arc λ isotopic to some arc in f^{−1}(Λ) but to none of the arcs in Λ, it is isotopic to an arc λ′ ∈ f^{−1}(Λ) − Λ̃. So Γ · λ ≤ #(Γ̃ ∩ λ′) = 0.

This proves the part 2(a), 2(b) and (2(d), n = 1) of the theorem. To get 2(c) and (2(d), n > 1), we just need to note that if Γ, Λ are irreducible for f, they are also irreducible for f^{n}, for all n ≥ 1. This gives the theorem.

The following theorem can be proved in exactly the same manner as Theorem 3.2.

Theorem 6.1 (Arcs intersecting arcs). Let Λ_{1}, Λ_{2} be two irreducible systems of arcs. Then either

1. Λ_{1} · Λ_{2} = 0 and f^{−n}(Λ_{1}) · f^{−n}(Λ_{2}) = 0 for all n ≥ 1, or

2. Λ_{1} · Λ_{2} ≠ 0 and (f^{−n}(Λ_{1}) − Λ̃_{1}(f^{n})) · Λ_{2} = 0 = Λ_{1} · (f^{−n}(Λ_{2}) − Λ̃_{2}(f^{n})) for all n ≥ 1. In this case we also have (a) The transformations f_{♯,Λ_{i}} are transitive permutations of the basis vectors; (b) for each λ ∈ Λ_{i}, there is a unique component of f^{−1}(λ) intersecting Λ_{i′}, i ≠ i′ ∈ {1, 2}; (c) (a) and (b) remain true if we replace f by f^{n}, n ≥ 1.

Corollary 4. If Λ_{1} contains more than one periodic cycle, or if some element of Λ maps with multiplicity greater than one, then Λ_{1} · Λ_{2} = 0 for any irreducible arc system Λ_{2}.

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[Mil] John Milnor. Dynamics in One Complex Variable: Introductory Lectures. SUNY Stonybrook Institute for Mathematical Sciences Preprint (1990/5).

[Pil1] Kevin M. Pilgrim. Rational Maps Whose Fatou Components are Jordan Domains. Ergodic Theory and Dynamical Systems (to appear).

[Pil2] Kevin M. Pilgrim. Cylinders for Iterated Rational Maps. PhD thesis, University of California at Berkeley, May, 1994.

[Poi] Alfredo Poirier. On Post-Critically Finite Polynomials Part Two: Hubbard Trees. SUNY Stony Brook Institute for Mathematical Sciences Preprint (1993/7).

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Kevin M. Pilgrim,, Mathematics Department, Cornell University, White Hall, Ithaca, NY 14853, USA

  E-mail address: pilgrim@math.cornell.edu

Tan Lei, Department of Mathematics, University of Warwick, Coventry CV4 7AL, England

  E-mail address: tanlei@maths.warwick.ac.uk