1 Introduction
Let
$\widehat {\mathbb C}$
denote the Riemann sphere and let
$f:\widehat {\mathbb C}\to \widehat {\mathbb C}$
be a rational map of degree
$d\geq 2$
. By the Riemann–Hurwitz formula, f has
$2d-2$
critical points, counted with multiplicity; the critical set of f is denoted
$C_f$
. The postcritical set of f, denoted
$P_f$
, is the smallest forward invariant subset of the Riemann sphere that contains the critical values of f. If the postcritical set is finite, the rational map is said to be postcritically finite. Associated to a postcritically finite rational map is a ramification portrait; that is, a finite directed graph
$\Gamma _f$
that encodes the action of f restricted to
$C_f\cup P_f$
. As an example, consider the polynomial
$f:z\mapsto z^2-2$
. The critical set is
$C_f=\{0,\infty \}$
, the postcritical set is
$P_f=\{-2,2,\infty \}$
, and the ramification portrait is
In the ramification portrait above, there is an edge from vertex x to vertex y if and only if
$y=f(x)$
. This edge is weighted with the positive integer
$\mathrm {deg}_f(x)$
, the local degree of f at x. To lighten notation, we record the weight of the edge from x to
$f(x)$
if and only if
$\mathrm {deg}_f(x)>1$
; that is, if and only if
$x\in C_f$
. The ramification portrait above is a polynomial ramification portrait; that is, there is a fixed vertex mapping to itself with full degree (the vertex
$\infty $
).
In this article, we study which graphs are isomorphic to ramification portraits from postcritically finite polynomials. There are immediate necessary conditions that arise from local degree restrictions and from Riemann–Hurwitz restrictions (see §2). We prove that in the polynomial setting, these conditions are also sufficient. A weighted finite directed graph as above which satisfies these conditions is called an abstract polynomial portrait.
Theorem 1.1. Let
$\Gamma $
be an abstract polynomial portrait. Then, there exists a polynomial
$f:\widehat {\mathbb C}\to \widehat {\mathbb C}$
so that
$\Gamma _f\simeq \Gamma $
.
To prove Theorem 1.1, we construct an explicit topological polynomial
$g:S^2 \to S^2$
so that
$\Gamma _g\simeq \Gamma $
. We build g so that it has no obstructing multicurves. It then follows from Thurston’s topological characterization of rational maps that g is combinatorially equivalent to a polynomial f, so
$\Gamma _f\simeq \Gamma _g$
.
We cannot strengthen Theorem 1.1 by removing the hypothesis that
$\Gamma $
is a polynomial ramification portrait because of the following two phenomena. The first is dynamical and related to Thurston’s theorem. The second is non-dynamical and related to the Hurwitz problem.
1.1 Ramification portraits that can only be realized topologically
Consider the following abstract ramification portrait
$\Gamma $
:
Suppose there is a rational map
$f:\widehat {\mathbb C}\to \widehat {\mathbb C}$
so that
$\Gamma \simeq \Gamma _f$
. Then, f is a quadratic rational map with two periodic cycles of period 2. However, a quick computation reveals that a quadratic rational map can have at most one periodic cycle of period
$2$
, so no such f exists.
Even though no rational map f exists so that
$\Gamma _f\simeq \Gamma $
, it is possible to construct a topological branched cover
$g:S^2\to S^2$
so that
$\Gamma _g\simeq \Gamma $
. For example, after identifying
$S^2$
with
$\widehat {\mathbb C}$
, we could take the squaring map
$s: z\mapsto z^2$
and postcompose with an orientation-preserving homeomorphism
$h:S^2\to S^2$
so that
$h(\infty )=1, h(1)=\infty ,$
$h(0)=2$
, and
$h(4)=0$
. Then,
$g:=h\circ s$
is a branched cover with
$\Gamma _g\simeq \Gamma $
. By Thurston’s theorem, the map g will necessarily admit an obstructing multicurve (see §2).
We are aware of a few methods to construct ramification portraits that can only be realized topologically and that are similar in spirit to the example above. It would be interesting to put these examples into a more general context.
Question 1.2. Which abstract ramification portraits
$\Gamma $
can only be realized topologically?
1.2 Ramification portraits that cannot even be realized topologically
The Hurwitz problem is to characterize which branch data arise from branched covering maps
$S^2\to S^2$
. See [Reference Barański1, Reference Edmonds, Kulkarni and Stong5, Reference Khovanskii and Zdravkovska8, Reference Thom10]. For example, it is known that there is no branched cover with the branch data
$(2,2),(2,2),(3,1)$
. That is, there is no branched cover
$f:S^2\to S^2$
of degree
$4$
with exactly three critical values
$\{v_1,v_2,v_3\}\subseteq S^2$
, so that:
-
•
$f^{-1}(\{v_1\})$
contains exactly two points, each mapping forward with local degree two; -
•
$f^{-1}(\{v_2\})$
contains exactly two points, each mapping forward with local degree two; and -
•
$f^{-1}(\{v_3\})$
contains exactly two points, one mapping forward with local degree 3 and the other mapping forward with local degree 1.
This fact has dynamical consequences. Indeed, any abstract ramification portrait
$\Gamma $
with this branch data cannot be the ramification portrait of a branched covering map
$S^2\to S^2$
and, therefore,
$\Gamma $
cannot be the ramification portrait of a rational map
$\widehat {\mathbb C}\to \widehat {\mathbb C}$
. For example, the following ramification portrait has branch data
$(2,2),(2,2),(3,1)$
.
While the general Hurwitz problem is unsolved, we note that all polynomial branch data are realizable (see [Reference Edmonds, Kulkarni and Stong5, Preposition 5.2]). We will not use this fact to construct the branched cover
$g:S^2\to S^2$
in the proof of Theorem 1.1.
1.3 Thurston’s theorem
Let
$S^2$
denote an oriented topological 2-sphere and let
$f\colon \, S^2 \to S^2$
be an orientation-preserving branched cover of degree
$d\ge 2$
so that the postcritical set
$P_f$
is finite. We call such a map f a Thurston map. For convenience in stating the theorem, we assume that the orbifold of f is hyperbolic. (This condition essentially excludes power maps
$z\mapsto z^n$
, Chebyshev maps, and Lattès maps; see [Reference Douady and Hubbard3].) Two Thurston maps
$f:(S^2,P_f)\to (S^2,P_f)$
and
$g:(S^2,P_g)\to (S^2,P_g)$
are combinatorially equivalent provided that there are orientation-preserving homeomorphisms
$\phi _0:(S^2,P_f)\to (S^2,P_g)$
and
$\phi _1:(S^2,P_f)\to (S^2,P_g)$
so that:
-
•
$\phi _0\circ f = g\circ \phi _1$
; and -
• the homeomorphisms
$\phi _0$
and
$\phi _1$
are isotopic relative to
$P_f$
.
In the 1980s, Thurston proved that every Thurston map f is combinatorially equivalent to a rational map or it is obstructed. In the latter case, f admits an invariant curve system called an obstructing multicurve.
A multicurve
$\Delta $
is a finite collection of simple disjoint curves in
$S^2\setminus P_f$
, no two of which are homotopic. All components
$\delta \in \Delta $
are also required to be essential (
$\delta $
does not bound a disk) and non-peripheral (
$\delta $
does not bound a disk with exactly one puncture). The multicurve
$\Delta $
is said to be invariant for f provided that for all
$\delta \in \Delta $
, every component of
$f^{-1}(\{\delta \})$
is either:
-
• homotopic to some
$\delta '\in \Delta $
in
$S^2\setminus P_f$
; or -
• ‘erased’, that is, it is peripheral or inessential.
Given an invariant multicurve
$\Delta $
for f, Thurston defined an associated linear transformation
${\mathbb R}^\Delta \to {\mathbb R}^\Delta $
that encodes how different components of
$f^{-1}(\Delta )$
map to
$\Delta $
. The matrix for this transformation has non-negative real entries, so there is a leading eigenvalue
$\unicode{x3bb} $
which is real and non-negative. The multicurve
$\Delta $
is an obstruction provided that
$\unicode{x3bb} \geq 1$
. If the Thurston map f admits an obstruction, f is said to be obstructed. If not, f is said to be unobstructed.
Theorem 1.3. (Thurston’s topological characterization of rational maps [Reference Douady and Hubbard3])
Let
${f:(S^2,P_f)\to (S^2,P_f)}$
be a Thurston map and suppose that f has a hyperbolic orbifold. Then, f is combinatorially equivalent to a rational map F if and only if f is unobstructed. In this case, F is unique up to conjugation by Möbius transformations.
1.4 Levy cycles
For a given Thurston map
$f:(S^2,P_f)\to (S^2,P_f)$
, verifying the criterion in Thurston’s theorem is difficult as it involves an infinite search in general. In this article, we will work with Thurston maps that are topological polynomials; that is, there is some
$\omega \in S^2$
that is a fully ramified fixed point of f. More is known about Thurston’s criterion in the case of topological polynomials.
A Levy cycle for the Thurston map
$f:(S^2,P_f)\to (S^2,P_f)$
is a circularly ordered collection of simple closed curves
$\{\delta _0,\ldots , \delta _{n-1},\delta _{n}=\delta _0\}$
on
$S^2\setminus P_f$
such that:
-
• no two curves are homotopic relative to
$P_f$
; -
• the curves are pairwise disjoint;
-
• each curve is essential and non-peripheral; and
-
• for all
$1\leq i\leq n$
, at least one component of
$f^{-1}(\delta _i)$
is homotopic to
$\delta _{i-1}$
and maps to
$\delta _i$
by degree
$1$
.
Levy proved the following results in his thesis [Reference Levy9].
Theorem 1.4. (Levy)
Let
$f:(S^2,P_f)\to (S^2,P_f)$
be a Thurston map that is a topological polynomial. Then, f is obstructed if and only if f admits a Levy cycle.
Theorem 1.5. (Levy)
Let
$\Gamma $
be an abstract polynomial ramification portrait such that every critical vertex is periodic. Then, every Thurston map realizing
$\Gamma $
is unobstructed.
The proof of the latter can be strengthened to give the following result. See, for example, Hubbard [Reference Hubbard6, Theorem 10.3.9].
Theorem 1.6. (Levy–Berstein)
Suppose
$\Gamma $
is an abstract polynomial ramification portrait such that each cycle contains a critical vertex. Then, every Thurston map realizing
$\Gamma $
is unobstructed.
We will use Levy’s first theorem in an essential way in our proof of Theorem 1.1. Indeed, given an abstract ramification portrait
$\Gamma $
, we will construct a topological polynomial
$g:S^2\to S^2$
so that
$\Gamma _g\simeq \Gamma $
and so that g cannot possibly admit a Levy cycle. Theorem 1.1 immediately follows.
In [Reference Kelsey7, Theorem 1.1], Kelsey uses self-similar groups to give a partial converse to the Levy–Berstein theorem. In the discussion that follows, an attractor of an abstract ramification portrait is a cycle that contains a critical vertex and a non-attractor is a cycle that does not contain a critical vertex.
Theorem 1.7. (Kelsey)
Suppose
$\Gamma $
is an abstract polynomial ramification portrait and that
$\Gamma $
satisfies at least one of the following properties:
-
(1)
$\Gamma $
contains a cycle (which is necessarily a non-attractor) of length at least two that does not contain any critical values; -
(2)
$\Gamma $
contains at least two cycles (which are necessarily non-attractors) that do not contain any critical values; -
(3)
$\Gamma $
contains at least two non-attractor cycles that have length at least two; -
(4)
$\Gamma $
contains at least four non-attractor cycles.
Then, there is an obstructed Thurston map that realizes
$\Gamma $
.
In Theorem 1.8, we show that certain abstract polynomial ramification portraits have only unobstructed representatives and, in Theorem 1.9, we show that certain abstract polynomial ramification portraits have obstructed representatives. We need a definition to state Theorems 1.8 and 1.9. Let
$\Gamma $
be an abstract polynomial ramification portrait and let v be a vertex of
$\Gamma $
. Then, v is the source vertex of exactly one edge of
$\Gamma $
. We let
$\tau (v)$
denote the target vertex of this edge.
Theorem 1.8. Suppose
$\Gamma $
is an abstract polynomial ramification portrait that has at least four postcritical vertices and satisfies one of the following properties:
-
(i)
$\Gamma $
has a single non-attractor cycle and it has length one; -
(ii) every finite postcritical vertex of
$\Gamma $
is in a single non-attractor cycle, this cycle has length
$p^k$
for some prime number p and some positive integer k, and the finite postcritical vertices can be enumerated as
$\{v_i: 0\leq i < p^k\}$
such that
$\tau (v_i) = v_{i+1}$
(mod
$p^k$
) for every
$i\in \{0,\dots ,p^k -1\}$
and if
$v_j$
is a critical value, then j is a multiple of
$p^{k-1}$
.
Then, every Thurston map with ramification portrait isomorphic to
$\Gamma $
is unobstructed.
The hypothesis that there are at least four postcritical vertices is not restrictive, since by Thurston’s characterization theorem, a Thurston map with fewer than four postcritical points is unobstructed. The proof is along the lines of the argument for the Levy–Berstein theorem. If the abstract ramification portrait can be realized by an obstructed Thurston map, then by Levy [Reference Levy9], there must be a Levy cycle. This implies that, in the teminology of Hubbard [Reference Hubbard6], there must be a degenerate Levy cycle. One then shows that this is impossible if the ramification portrait satisfies property (i) or (ii). The proof is given in §4. Part (if not all) of case (i) of Theorem 1.8 was previously known. The case of a single non-attractor cycle of length one and no other finite cycles was observed by Kelsey [Reference Kelsey7, p. 52].
Theorem 1.9. Suppose
$\Gamma $
is an abstract polynomial ramification portrait that has at least four postcritical vertices and satisfies one of the following properties:
-
(i) every finite postcritical vertex of
$\Gamma $
is in a single non-attractor cycle, this cycle has length
$p^k$
for some prime number p and some positive integer k, the finite vertices can be enumerated as
$\{v_i: 0\le i < p^k\}$
such that
$v_0$
is a critical value,
$\tau (v_i) = v_{i+1}$
mod
$p^k$
for every
$i\in \{0,\dots ,p^k -1\}$
, and there is a critical value
$v_j$
such that j is not a multiple of
$p^{k-1}$
; -
(ii) every finite postcritical vertex of
$\Gamma $
is in a single non-attractor cycle of length at least two and this cycle does not have prime-power length; -
(iii)
$ \Gamma $
contains a non-attractor cycle of length at least two that does not contain all of the finite critical values; -
(iv)
$\Gamma $
has at least two non-attractor cycles of length one.
Then, there exists an obstructed Thurston map whose ramification portrait is isomorphic to
$\Gamma $
.
The proof of Theorem 1.9 is constructive and relies on a combinatorial lemma, Lemma 5.1. Given an abstract polynomial ramification portrait
$\Gamma $
that satisfies any of conditions (i)–(iv) of the theorem, we describe a construction of an obstructed Thurston map with ramification portrait isomorphic to
$\Gamma $
. We introduce rose maps and prove the lemma in §5. We then prove the theorem in §6.
Combining Theorems 1.8, 1.9, and the Levy–Berstein theorem, we classify the abstract polynomial ramification portraits that are completely unobstructed. We summarize this result in the following theorem, which we prove in §7.
Theorem 1.10. Suppose
$\Gamma $
is an abstract polynomial ramification portrait. Then, every Thurston map with ramification portrait isomorphic to
$\Gamma $
is unobstructed if and only if
$\Gamma $
satisfies at least one of the following conditions:
-
(i)
$\Gamma $
has at most three postcritical vertices; -
(ii) every cycle of
$\Gamma $
is an attractor; -
(iii)
$\Gamma $
has a single non-attractor cycle and it has length one; -
(iv) every finite postcritical vertex of
$\Gamma $
is in a single non-attractor cycle, this cycle has length
$p^k$
for some prime number p and some positive integer k, the finite postcritical vertices can be enumerated as and if
$$ \begin{align*} \{v_i: 0\le i < p^k\} \text{ such that }\tau(v_i) = v_{i+1}\;\mathrm{mod}\; p^k \text{ for every }i\in \{0,\dots,p^k -1\}, \end{align*} $$
$v_j$
is a critical value, then j is a multiple of
$p^{k-1}$
.
1.5 Notes and references
In addition to the references we have already highlighted, we would like to mention two somewhat related works: in [Reference DeMarco, Koch and McMullen2], the authors prove that given a finite set X with
$|X|\geq 2$
, a map
$F:X\to X$
, and prescribed multiplicities at points of X, there is some rational map
$f:\widehat {\mathbb C}\to \widehat {\mathbb C}$
so that the restriction of f to its postcritical set coincides with the map F in such a way that the multiplicity of each point
$x\in X$
agrees with the local degree of f at the corresponding postcritical point. We note that the map
$F:X\to X$
, together with the multiplicities at points of X, is analogous to our abstract ramification portrait
$\Gamma $
; however, our abstract ramification portraits satisfy a Riemann–Hurwitz condition, so they have a natural degree,
$d\geq 2$
. In our article, we are interested in the question of whether there is a rational map of degree d that realizes a given abstract ramification portrait. In [Reference DeMarco, Koch and McMullen2], the degree of the rational maps constructed can be arbitrarily large.
In [Reference Doyle and Silverman4], the authors begin with a similar sort of abstract combinatorial object called a portrait and they construct a moduli space of endomorphisms
$\mathbb P^N\to \mathbb P^N$
(using GIT) that consists of points realizing the given combinatorial data. While we prove the existence of maps with a given abstract ramification portrait, in [Reference Doyle and Silverman4], it is not proven that the moduli spaces are non-empty.
2 Preliminaries
2.1 Ramification portraits associated to Thurston maps
Let
$f:(S^2,P_f)\to (S^2,P_f)$
be a Thurston map of degree d. The ramification portrait of f is the weighted directed graph
$\Gamma $
such that the vertex set
$V(\Gamma )$
is the union of the set
$C_f$
of critical points and the set
$P_f$
of postcritical points, and for each vertex v, there is an edge from v to
$f(v)$
with weight the local degree
$\deg _f(v)$
of f at v. By the Riemann–Hurwitz formula,
$$ \begin{align*} \sum_{v\in C_f} (\deg_f(v)-1) = 2d-2. \end{align*} $$
Since f has degree d, at each vertex v, the sum of the weights of the incoming edges is at most d. Note that f is a topological polynomial if and only if there is a vertex v such that
$f(v) = v$
and
$\deg _f(v)=d$
.
2.2 Abstract ramification portraits
Suppose
$\Gamma $
is a finite weighted directed graph (with the weights positive integers) such that each vertex of
$\Gamma $
is the source of exactly one edge. Let
$\tau \colon \, V(\Gamma )\to V(\Gamma )$
be the function which takes a vertex v to the target of the edge with source v. We call the weight of the edge from v to
$\tau (v)$
the degree of
$\tau $
at v and denote it by
$\deg (v)$
. A vertex v is critical if
$\deg (v)> 1$
, and is postcritical if there are a critical vertex w and a positive integer k such that
$\tau ^{\circ k}(w) = v$
. If v is a critical vertex, then
$\tau (v)$
is called a critical value. We denote the set of critical vertices by
$C_{\Gamma }$
and we denote the set of postcritical vertices by
$P_{\Gamma }$
. We say that
$\Gamma $
is an abstract ramification portrait if it satisfies the following:
-
• every vertex of
$\Gamma $
is either critical or postcritical; -
• there is an integer
$d\ge 2$
such that
$\sum _{v\in C_{\Gamma }} (\deg (v)-1) = 2d-2$
; and -
• for each vertex v, the sum of the weights of the edges with target v is at most d.
We call d the degree of the abstract ramification portrait. We say that an abstract ramification portrait
$\Gamma $
is realized by a Thurston map f if
$\Gamma $
is isomorphic to the ramification portrait of f (as weighted directed graphs). An abstract ramification portrait
$\Gamma $
is realizable if it is realized by some Thurston map.
An abstract ramification portrait of degree d is an abstract polynomial ramification portrait if there is a vertex v such that
$\tau (v)=v$
and
$\deg (v)=d$
. In this case, we choose such a vertex and call it
$\infty $
; the other vertices are called finite. We call a cycle (of the action of
$\tau $
on
$V(\Gamma )$
) finite if all of its vertices are finite; that is, a cycle is finite if it does not consist of the singleton
$\infty $
.
2.3 Finite subdivision rules
We define finite subdivision rules in the present context of Thurston maps. A finite subdivision rule
$\mathcal {R}$
consists of the structure of a finite CW complex
$S_{\mathcal {R}}$
on the 2-sphere (called the model subdivision complex), a subdivision
$\mathcal {R}(S_{\mathcal {R}})$
of
$S_{\mathcal {R}}$
, and an orientation-preserving, continuous cellular map
$\sigma _{\mathcal {R}}\colon \, \mathcal {R}(S_{\mathcal {R}})\to S_{\mathcal {R}}$
(called the subdivision map) whose restriction to each open cell is a homeomorphism onto an open cell. Furthermore, for each closed 2-cell
$\widetilde {t}$
of
$S_{\mathcal {R}}$
, there are: (i) a cell structure t (called the tile type of
$\widetilde {t}$
) on the 2-disk
$D^2$
such that the 1-skeleton of t is
$\partial D^2$
and (ii) a continuous surjection
$\psi _t\colon \, t\to \widetilde {t}$
(called the characteristic map of
$\widetilde {t}$
) whose restriction to each open cell is a homeomorphism onto an open cell.
Since, by definition, a homeomorphism cannot be a Thurston map, a subdivision map
$\sigma _{\mathcal {R}}$
is not a Thurston map if it has degree one. Since the critical values of
$\sigma _{\mathcal {R}}$
are vertices of
$S_{\mathcal {R}}$
and
$\sigma _{\mathcal {R}}$
takes vertices of
$S_{\mathcal {R}}$
to vertices of
$S_{\mathcal {R}}$
, a subdivision map
$\sigma _{\mathcal {R}}$
is a Thurston map if it has degree at least 2. Conversely, a Thurston map f is the subdivision map of a finite subdivision rule if and only if there exists a connected finite f-invariant graph G that contains the postcritical set of f. Such a graph G serves as the 1-skeleton of a model subdivision complex.
3 Realizing a ramification portrait by an unobstructed map
In this section, we prove Theorem 1.1. We begin with an example to illustrate the construction. Consider the abstract ramification portrait
$\Gamma $
that is shown as follows.
The proof defines an ordering of the finite postcritical vertices of
$\Gamma $
. In this case, we use the ordering given by
$a_1 < a_2 < a_3 < a_4 < a_5 < a_6$
. Following the terminology that will be defined in the proof, the ordered sets
$(a_1,a_2,a_3)$
and
$(a_4)$
are called type-
$1$
chains and the ordered set
$(a_5,a_6)$
is called a type-
$2$
chain. (The first element of a type-
$1$
chain is the image of a critical vertex that is not postcritical, and the first element of a type-
$2$
chain is a periodic critical vertex.) The model subdivision complex
$S_{\mathcal {R}}$
is shown in Figure 1 as a stereographic projection of
$S^2$
to the plane. The 1-skeleton will always be a star graph with central vertex
$\infty $
. The vertices of
$\Gamma $
are identified with the vertices of
$S_{\mathcal {R}}$
. The ordering of the finite postcritical vertices chosen above determines the counterclockwise ordering of the labels of the vertices in Figure 1. The tile type t is shown in Figure 2;
$S_{\mathcal {R}}$
is the image of t under the characteristic map
$\psi \colon \, t\to S_{\mathcal {R}}$
. The label of a vertex v of t is
$\psi (v)$
; if
$\psi (v)\ne \infty $
, then v is called a finite vertex.
Figure 1 The model subdivision complex
$S_{\mathcal {R}}$
.
Figure 2 The tile type t, which
$\psi $
maps to
$S_{\mathcal {R}}$
by identifying edges in pairwise fashion.
The following is an outline of the construction of the subdivision
$\mathcal {R}(S_{\mathcal {R}})$
of
$S_{\mathcal {R}}$
. We add edges to
$S_{\mathcal {R}}$
in three stages.
In stage
$1$
, we add edges to
$S_{\mathcal {R}}$
that will ensure that the subdivision map cannot have any Levy cycles. Figure 3 shows the construction after the first stage for our example from the point of view of the tile type t. The tile t is a 12-gon. The label of every vertex v of t appears adjacent to v and outside of t. Every other vertex is labeled
$\infty $
and the labels of the finite vertices are in the proper cyclic order. Every finite vertex v has an image label, which is
$\sigma _{\mathcal {R}}(\psi (v))$
. It is drawn inside t. (Of course, this is abuse of notation, since we have not finished the construction yet and, hence, have not defined the subdivision map
$\sigma _{\mathcal {R}}$
yet.)
Figure 3 The construction after stage
$1$
.
In stage
$2$
, we add edges to ensure that
$\mathcal {R}(S_{\mathcal {R}})$
will have the correct number of subtiles and, hence,
$\sigma _{\mathcal {R}}$
will have the correct degree. This is done in a way that ensures that the ramification portrait of
$\sigma _{\mathcal {R}}$
will be isomorphic to
$\Gamma $
. No further changes are made in the second stage for our example since there is already the correct number of subtiles.
In stage
$3$
, we simply add stickers as needed in each subtile so that each subtile has the proper number of vertices and edges. (A sticker is an edge with a vertex of valence one, resembling a stick pin with a spherical head.) Figure 4 shows the result for our example.
Figure 4 The subdivision of the tile type after stage
$3$
.
All of this is done while ensuring that in every subtile, every other vertex is a vertex of t with label
$\infty $
and that the image labels are in the proper cyclic order.
It is now straightforward to define the subdivision map
$\sigma _{\mathcal {R}}$
so that its restriction to each open cell is a homeomorphism to an open cell and it takes each finite vertex to its image label. Figure 5 shows the subdivision
$\mathcal {R}(S_{\mathcal {R}})$
.
Figure 5 The subdivision of the model subdivision complex.
The five new edges in Figure 3 ensure that if we extend the subtiling so that it combinatorially describes a finite subdivision rule, then the subdivision map cannot have a Levy cycle. This can be proven by contradiction as follows. Let
$\gamma $
be an element of a Levy cycle. In the model subdivision complex, the new arc whose barycenter has label
$a_4$
bounds an open disk D such that
$D \cap P_f = \{a_3\}$
and its boundary contains
$\infty $
. Lemma 3.1 implies, for every positive integer n, that
$\gamma $
can be isotoped rel the postcritical set to be disjoint from all new edges of the nth subdivision
$\mathcal {R}^{n}(S_{\mathcal {R}})$
. Since the disk D contains the single postcritical point
$a_3$
and its boundary contains
$\infty $
, the vertex
$a_3$
cannot be in the connected component
$D_{\gamma }$
of
$S_{\mathcal {R}}\setminus \gamma $
which does not contain
$\infty $
. In the next two subdivisions, there will be new edges enclosing the postcritical points
$a_4$
,
$a_2$
, and
$a_1$
, so none of them can be in
$D_{\gamma }$
. There is a new edge joining the vertex
$a_5$
to
$\infty $
, so
$a_5$
cannot be in
$D_{\gamma }$
. In the next subdivision, there will be a new edge joining
$a_6$
to an
$\infty $
, so
$a_6$
cannot be in
$D_{\gamma }$
. Hence, no finite postcritical point can be in
$D_\gamma $
, so there are no Levy cycles and, hence, the subdivision map is equivalent to a rational map. This concludes our example.
We call an edge of a subdivision
$\mathcal {R}^n(S_{\mathcal {R}})$
(of a finite subdivision rule
$\mathcal {R}$
) a new edge if it is not contained in an edge of
$S_{\mathcal {R}}$
. The following lemma plays a crucial role in the proof of Theorem 1.1.
Lemma 3.1. Suppose f is a Thurston map which is also the subdivision map
$\sigma _{\mathcal {R}}$
of a finite subdivision rule
$\mathcal {R}$
. Suppose
$\{\delta _0,\dots ,\delta _{k-1}, \delta _{k}=\delta _0\}$
is a Levy cycle for f and let n be a positive integer. Then, for each
$i\in \{1,\dots ,k\}$
,
$\delta _i$
can be isotoped rel
$P_f$
so that it is disjoint from each new edge of the subdivision
$\mathcal {R}^n(S_{\mathcal {R}})$
.
Proof. We first assume
$n=1$
. Let E be the
$1$
-skeleton of
$S_{\mathcal {R}}$
and let
$E_1$
be the
$1$
-skeleton of
$\mathcal {R}(S_{\mathcal {R}})$
. Each
$\delta _i$
can be isotoped so that
$\delta _i\cap E_1$
is finite. For each
$i\in \{0,\dots ,k\}$
, let
$a_i$
be the minimum of
$\#(\delta \cap E)$
, where
$\delta $
is a curve that is isotopic rel
$P_f$
to
$\delta _i$
, and let
$b_i$
be the minimum of
$\#(\delta \cap E_1)$
, where
$\delta $
is a curve that is isotopic rel
$P_f$
to
$\delta _i$
. Let
$i\in \{1,\dots ,k\}$
and let
$\delta $
be a curve that is isotopic rel
$P_f$
to
$\delta _i$
such that
$a_i = \#(\delta \cap E)$
. Let
$\gamma $
be a component of
$f^{-1}(\delta )$
which maps to
$\delta $
by degree
$1$
and is isotopic rel
$P_f$
to
$\delta _{i-1}$
. Then,
$a_{i-1} \le b_{i-1} \le \#(\gamma \cap E_1) = a_i$
. Since this is true for every i and i varies cyclically, each of these inequalities is an equality. So,
$a_{i-1} = a_i$
and
$a_{i-1} = b_{i-1}$
. This implies that
$\gamma $
does not intersect
$E_1 \setminus E$
. This establishes the result for
$n=1$
.
Now, suppose that
$n>1$
. Let p be a positive integer with
$p\ge n$
and
$p\equiv 1 \mod k$
. Then,
$\{\delta _0,\dots ,\delta _{k-1}, \delta _{k}=\delta _0\}$
is a Levy cycle for
$f^{\circ p}$
. By the previous paragraph applied to
$f^{\circ p}$
, each
$\delta _i$
can be isotoped so that it does not intersect any new edge of
$\mathcal {R}^p(S_{\mathcal {R}})$
. Since each new edge of
$\mathcal {R}^n(S_{\mathcal {R}})$
is a union of new edges of
$\mathcal {R}^p(S_{\mathcal {R}})$
, then each
$\delta _i$
can be isotoped so that it is disjoint from each new edge of
$\mathcal {R}^n(S_{\mathcal {R}})$
.
Proof of Theorem 1.1
Let
$\Gamma $
be an abstract polynomial ramification portrait. Let
${C' = C_{\Gamma }\setminus \{\infty \}}$
(the set of finite critical vertices) and let
$P' = P_{\Gamma } \setminus \{\infty \}$
(the set of finite postcritical vertices). Let
$V_{\Gamma } = \{\tau (x): x\in C_{\Gamma }\}$
(the set of critical values) and let
${V' = V_{\Gamma } \setminus \{\infty \}}$
(the set of finite critical values). Let
$A = \{v\in V': v = \tau (c)\ \textrm {for some} c\in C_{\Gamma }\setminus P_{\Gamma }\}$
. For each
$v\in A$
, we choose an element
$c_v \in C_{\Gamma }\setminus P_{\Gamma }$
with
$\tau (c_v) = v$
.
Let n be the cardinality of
$P'$
. A key step is to appropriately order the elements of
$P'$
by naming them
$a_1,\dots ,a_n$
. To do this, we partition
$P'$
into chains. We define the chains recursively. We will put postcritical vertices that have already been placed in chains in a set
$\widetilde {A}$
such that
$\tau (\widetilde {A})\subseteq \widetilde {A}$
. To begin the construction, let
$\widetilde {A} = \emptyset $
and let
$i=1$
.
The ordering. Suppose for the recursive step that
$A\not \subset \widetilde {A}$
,
$i\in \{1,\dots ,n\}$
, and that we have already defined
$a_j$
for
$j\in \{1,\dots ,i-1\}$
. Suppose that there is a vertex in
$P'\setminus \widetilde {A}$
that is not periodic under
$\tau $
. Because
$\tau (\widetilde {A})\subseteq \widetilde {A}$
, we can choose an element
$v\in A\setminus \widetilde {A}$
such that v is not the image under
$\tau $
of a postcritical vertex. Let
$a_i = v$
and add v to
$\widetilde {A}$
. If
$\tau (v) \notin \widetilde {A}$
, we let
$a_{i+1} = \tau (v)$
and add
$\tau (v)$
to
$\widetilde {A}$
. We continue until we reach an index j such that
$\tau (a_j)$
is in
$\widetilde {A}$
. At this point, we stop this iteration of the recursion. We define the ordered set
$(a_i,\dots ,a_{j})$
to be the chain of each of its elements. We call it a type-1 chain. It begins with an element of A. The first element of the chain is
$a_i$
and the last element of the chain is
$a_{j}$
. The length of the chain is
$j+1-i$
. After redefining i to be
$j+1$
, we continue this recursive step as long as possible.
Once we can no longer continue this recursion, the elements of
$P'$
which remain are exactly the elements of finite (attractor) cycles which are connected components of
$\Gamma $
. To start the next recursion, we choose a critical vertex v in a remaining attractor cycle and let
$a_i = v$
. Let k be the number of elements in the attractor cycle. For
$1\le j < k$
, let
$a_{i+j} = \tau ^{\circ j}(a_i)$
. As before,
$a_i$
is the first element of the chain,
$a_{i+k-1}$
is the last element of the chain, and the length of the chain is k. We call it a type-2 chain. We continue recursively to choose all of the points in the other attractor cycles. After doing this, the elements of
$P'$
are
$a_1,\dots ,a_n$
in order.
Figure 6 One or two chains of type
$1$
.
Construction of
$S_{\mathcal {R}}$
. We next construct the associated finite subdivision rule
$\mathcal {R}$
. The
$1$
-skeleton of the model subdivision complex
$S_{\mathcal {R}}$
is a tree as in Figure 1. There is one central vertex. We identify
$\infty \in \Gamma $
with this central vertex. There are n ‘stickers’ (a sticker is an edge of the graph with a vertex of valence one, like a stick pin with a spherical head) from
$\infty $
to valence
$1$
vertices
$a_1,\dots ,a_n$
, in counterclockwise order. We identify
$a_1,\dots ,a_n\in \Gamma $
with these valence
$1$
vertices. The tile type t is a
$(2n)$
-gon, which we think of as an n-gon with each edge bisected. The characteristic map
$\psi :t\to S_{\mathcal {R}}$
maps the edge barycenters to the sticker heads and the other vertices to
$\infty $
. The edge barycenters are called finite vertices and the others are called
$\infty $
-vertices. More generally, a vertex of some subdivision of t which is not a vertex of t is called a finite vertex. Every vertex v of t is labeled by
$\psi (v)$
. These vertex labels are placed outside t in Figures 6–10. We use clockwise order on
$\partial t$
.
If k is an integer with
$k\ge 2$
, a k-doodle is a graph with three vertices and k edges (none of them loops) such that one vertex (the central vertex) has valence k, one vertex (the head) has valence
$1$
, and the third vertex (the foot) has valence
$k-1$
. Note that a
$2$
-doodle is a bisected arc.
We define a subdivision
$\mathcal {R}(t)$
of t. We do this in three stages. We first define a subtiling of t into subtiles such that the finite vertices in each subtile are in the proper cyclic order. This means that there might be fewer than n of them, but they will have image labels, which are distinct elements of
$\{a_1,\ldots ,a_n\}$
, and, when taken in clockwise order, their image labels have the same cyclic order as in
$(a_1,\ldots ,a_n)$
. We then add arcs and k-doodles as determined by the critical vertices of
$\Gamma $
so that we have d subtiles. Finally, we add stickers as necessary to get the subdivision
$\mathcal {R}(t)$
. The tiles of the first stage will be defined so that the resulting subdivision map does not have Levy cycles. We do this by ensuring that there is an iterated subdivison
$\mathcal {R}^n(t)$
of t such that for each finite vertex v of t except possibly one, either there is a new edge from v to an
$\infty $
-vertex or there is an arc (made out of two or four new edges) from the
$\infty $
-vertex before v to the
$\infty $
-vertex after v.
As we construct
$\mathcal {R}(t)$
, we will give image labels to its vertices. The image label of a vertex is the vertex it will map to under the analog of
$\psi $
from
$\mathcal {R}(t)$
to
$S_{\mathcal {R}}$
. So, for a vertex v in t, the image label of v is defined to be
$\tau (\psi (v))$
. We will keep track of the critical vertices that have already been accounted for during the construction of
$\mathcal {R}(t)$
in a set
$\widetilde {C}$
. For the beginning of the construction, we define
$\widetilde {C} = \emptyset $
.
Figure 7 A chain of type
$1$
followed by a chain of type
$2$
.
Figure 8 A chain of type
$2$
followed by a chain of type
$1$
. Here,
$i=n$
.
Figure 9 Two chains of type
$2$
.
Figure 10 A single chain and it has type
$1$
.
Stage 1. Suppose
$a_i$
is the last element of a chain and
$a_j$
is the first element of the next chain (in cyclic order). So, either
$1\le i < n$
and
$j=i+1$
or
$i=n$
and
$j=1$
. The construction in stage 1 depends on the types of the chains which contain
$a_i$
and
$a_j$
. We consider various cases.
If
$a_i$
and
$a_{j}$
are in distinct chains of type
$1$
or if they are in the same chain of type
$1$
(there is only one chain) and
$\tau (a_i)\ne a_j$
, then we add a k-doodle, with k being the degree of the critical vertex
$c_{a_{j}}$
, with head the
$\infty $
-vertex after
$a_i$
and with foot the
$\infty $
-vertex before
$a_i$
. We give the central vertex of the k-doodle image label
$a_{j}$
and add
$c_{a_{j}}$
to
$\widetilde {C}$
. See Figure 6, which, like Figures 7–10, is drawn with
$k=2$
.
If
$a_i$
is in a chain of type
$1$
and
$a_{j}$
is in a chain of type
$2$
, then we add
$k-1$
edges joining
$a_j$
to the
$\infty $
-vertex before
$a_i$
(where
$k = \deg (a_j)$
) and add
$a_{j}$
to
$\widetilde {C}$
. See Figure 7.
Suppose
$a_i$
is in a chain of type
$2$
and
$a_j$
is in a chain of type
$1$
(this can only occur if
$i=n$
and
$j=1$
). If
$a_i$
is in a chain of length
$1$
, then we do not do anything at this stage. If
$a_i$
is in a chain of length greater than
$1$
, then we add a k-doodle (with
$k = \deg (c_{a_{j}})$
) with head the
$\infty $
-vertex after
$a_i$
and with foot the
$\infty $
-vertex before
$a_i$
. We give the central vertex of the k-doodle image label
$a_{j}$
and add
$c_{a_{j}}$
to
$\widetilde {C}$
. Figure 8 shows both possibilities.
If
$a_i$
and
$a_{j}$
are both in chains of type
$2$
, and
$a_i$
is in a chain of length
$1$
, then to
$a_{j}$
we add
$k-1$
edges joining it to the
$\infty $
-vertex before
$a_{j}$
(where
$k= \deg (a_j)$
) and add
$a_{j}$
to
$\widetilde {C}$
. If
$a_i$
and
$a_{j}$
are both in chains of type
$2$
, and
$a_i$
is in a chain of length greater than
$1$
, then to
$a_{j}$
we add
$k-1$
edges joining it to the
$\infty $
-vertex before
$a_{i}$
(where
$k=\deg (a_j)$
) and add
$a_{j}$
to
$\widetilde {C}$
. The two possibilities are shown in Figure 9.
Now, suppose that there is a single chain, it has type
$1$
, and
$\tau (a_n) = a_1$
. This is the only remaining case. The Riemann–Hurwitz condition implies that if there is just one finite critical value, then
$\Gamma $
has only two critical vertices and their degrees both equal the degree of
$\Gamma $
. Hence, the finite critical vertex is the only vertex of
$\Gamma $
which
$\tau $
maps to the finite critical value. This is impossible in the present case because
$a_1\in A$
and
$\tau (a_n)=a_1$
. So, either one of the
$a_i$
elements is a critical vertex or one of the
$a_i$
elements with
$i>1$
is a critical value.
First, suppose that
$r\in \{1,\dots ,n\}$
and
$a_r$
is a critical vertex with degree k. We add
$k-1$
arcs in t from
$a_r$
to the
$\infty $
-vertex before
$a_r$
, and we add
$a_r$
to
$\widetilde {C}$
. See the left side of Figure 10.
If none of the
$a_i$
elements is a critical vertex, then some
$a_i$
with
$i>1$
is a critical value. In this case, suppose
$r\in \{2,\dots ,n\}$
such that
$a_r$
is a critical value. Let
$k_1 = \deg (c_{a_1})$
and let
$k_r = \deg (c_{a_r})$
. We add a
$k_r$
-doodle with head the
$\infty $
-vertex after
$a_n$
and with foot the
$\infty $
-vertex before
$a_n$
. We give its central vertex image label
$a_r$
and we add
$c_{a_r}$
to
$\widetilde {C}$
. We then add a
$k_1$
-doodle with head the
$\infty $
-vertex after
$a_n$
and with foot the
$\infty $
-vertex before
$a_n$
as indicated in Figure 10. We give its central vertex image label
$a_1$
and we add
$c_{a_1}$
to
$\widetilde {C}$
. See the right side of Figure 10. This completes stage
$1$
of the construction.
Verification of stage 1 consistency. We first verify that the stage 1 edge constructions can be made so that interiors of distinct edges are disjoint. This is clear unless one of the edges is as in Figure 7 or the right side of Figure 9. In both cases, the second chain
$(a_j,\ldots )$
has type 2. Figures 7–9 show that a stage 1 edge can join the
$\infty $
-vertex immediately preceding vertex
$v_j$
to a vertex with image label in the chain
$(a_j,\ldots )$
or an adjacent
$\infty $
-vertex only in the situation of the left side of Figure 9. So, edges as in Figure 7 or the right side of Figure 9 can be constructed to meet other edges only in endpoints. Thus, the stage 1 edges can be constructed so that their interiors are disjoint.
We now look at what we have after stage
$1$
. Every subtile except for the central one is either a
$2$
-gon or a
$4$
-gon, and so there are only one or two finite vertices. For a
$2$
-gon, there is only one finite vertex and so its image label is in proper cyclic order.
For a
$4$
-gon, there are two finite vertices, so their image labels are in proper cyclic order if they are distinct. Chains of type 2 consist of the vertices of connected components of
$\Gamma $
which are cycles. Using this and the facts that elements of chains are distinct and distinct chains are disjoint, one verifies that the image labels of the 4-gons in Figures 7–10 are distinct. The only potential problem is if, in the notation of Figure 6,
$\tau (a_i) = a_j$
. Suppose that this happens. Then,
$a_i$
and
$a_j$
are in different chains. Because
$\tau (a_i) = a_j$
, and
$a_i$
and
$a_j$
are in different chains,
$a_j$
is not periodic under
$\tau $
. However, if there exists such a vertex when a chain is defined, then the first vertex of that chain must not be the image of a postcritical vertex. So, it is not possible that
$\tau (a_i) = a_j$
. Hence, the two finite vertices of every 4-gon have different image labels.
Now, we verify that the same is true for the central subtile s. We first observe that the vertices of s alternate between finite postcritical vertices and
$\infty $
-vertices, as required.
Note that if x is a vertex of t with label a, then the image label of x is
$\tau (a)$
. So, considering only the vertices of s which are vertices of t, if they do not have image labels in the proper cyclic order, then the label of one of them is the last member of a chain. In this case, the left side of either Figures 8, 9, or 10 occurs. However, in these cases, it is not hard to see that our image labels are in proper cyclic order. So, the image labels of the vertices of s which are vertices of t are in the proper cyclic order.
It remains to interpolate the vertices of s which are not vertices of t. These occur in Figure 6, and the right sides of Figures 8 and 10. In the first two cases, the image label of the vertex in question is
$a_j$
, and in the last case, this image label is
$a_1$
. A short case analysis shows that proper cyclic order also occurs in these cases. We conclude that the image labels of the vertices of s are in proper cyclic order after stage 1.
Stage 2. For the second stage, we add subtiles (much as in stage 1) corresponding to the critical vertices in
$C'$
that are not in
$\widetilde {C}$
. We do this recursively. Each time we add subtiles because of an element c of
$C' \setminus \widetilde {C}$
, we add this element to
$\widetilde {C}$
. This guarantees that we visit every element of
$C'$
exactly once, including the visits made in stage 1. Since
$C'$
is finite, this process will terminate.
Suppose
$c\in C' \setminus \widetilde {C}$
. Let
$k = \deg (c)$
.
If
$c\in P'$
, then we add
$k-1$
edges from c to the
$\infty $
-vertex of t before c and we add c to
$\widetilde {C}$
. Image labels of finite vertices of all tiles remain in proper cyclic order.
Now, suppose
$c\not \in P'$
, but that there is another element
$c'\in C' \setminus \widetilde {C}$
with
$c'\not \in P'$
and
$\tau (c')\ne \tau (c)$
. Let
$k' = \deg (c')$
. Let
$i,j\in \{1,\dots ,n\}$
such that
$\tau (c) = a_i$
and
$\tau (c') = a_j$
. Choose a subtile s of t. If s does not contain a vertex with image label
$a_i$
, then there is a unique
$\infty $
-vertex in s such that we can add a k-doodle with head and foot this
$\infty $
-vertex, and with image label
$a_i$
, and still have the image labels be in cyclic order. We do this and we add c to
$\widetilde {C}$
. Image labels of finite vertices of all tiles remain in proper cyclic order. Suppose s does contain a vertex v with image label
$a_i$
. Then, we add a
$k'$
-doodle to s with head the
$\infty $
-vertex after v, with central vertex with image label
$a_j$
, and with foot the
$\infty $
-vertex before v. We then add a k-doodle to s with head the
$\infty $
-vertex after v, with central vertex with image label
$a_i$
, and with foot the
$\infty $
-vertex before v. There are a new subtile in s with the same image labels (and in the same cyclic order) as for s, some
$2$
-gons if
$k>2$
or
$k'> 2$
, and two
$4$
-gons with a vertex labeled
$a_i$
and a vertex labeled
$a_j$
. So, we still have the image labels of the finite vertices of all of the subtiles in cyclic order. Finally, we add c and
$c'$
to
$\widetilde {C}$
.
To complete stage
$2$
, we need to consider the case that all elements of
$C' \setminus \widetilde {C}$
have the same image
$a_i$
under
$\tau $
. Let
$r = \#( C' \setminus \widetilde {C} )$
and let m be the sum of the degrees of the elements of
$C' \setminus \widetilde {C}$
. At every step of the construction thus far, the number of subtiles of t increases by
$\deg (v)-1$
, where v is the vertex added to
$\widetilde {C}$
. So, the number of subtiles of t created thus far is
$$ \begin{align*} 1+\Sigma_{v\in\widetilde{C}}(\mathrm{deg}(v)-1) & = 1+\Sigma_{v\in C'}(\mathrm{deg}(v)-1)-m+r\\ & = 1+d-1-m+r\\ & = (d-m)+r. \end{align*} $$
Because there remain m tiles to add with a vertex labeled
$a_i$
and the total number of tiles at the end of this stage with a vertex labeled
$a_i$
is at most d (the sum of the weights of the edges of
$\Gamma $
with target vertex
$a_i$
is at most d), there is currently a subtile that does not have a vertex with image label
$a_i$
. There is a unique
$\infty $
-vertex in this subtile such that we can add a k-doodle with head and foot this
$\infty $
-vertex, and with image label
$a_i$
, and still have the image labels be in cyclic order. We do this and we add c to
$\widetilde {C}$
. This completes the recursive step, so we can continue the recursion until
$C' = \widetilde {C}$
. This completes the second stage. At this point, there are d subtiles of t and in each subtile, the image labels of the finite vertices are in proper cyclic order.
Stage 3. Suppose s is a subtile of the construction after stage two and
${i\in \{1,\dots ,n\}}$
. If s does not have a vertex with image label
$a_i$
, then there is a unique
$\infty $
-vertex of s to which we can add a sticker whose other vertex has image label
$a_i$
and still have the image labels of the finite vertices in cyclic order. We do this for every such s and i. This completes stage
$3$
.
Completion of the construction of
$S_{\mathcal {R}}$
. At this point, every subtile has n finite vertices and their image labels are in proper cyclic order. We define this to be the subdivision
$\mathcal {R}(t)$
and we define its image under the characteristic map
$t\to S_{\mathcal {R}}$
to be
$\mathcal {R}(S_{\mathcal {R}})$
. It is straightforward to define a subdivision map that takes
$\mathcal {R}(S_{\mathcal {R}})$
to
$S_{\mathcal {R}}$
, which takes each open cell homeomorphically to an open cell, takes each
$\infty $
-vertex to
$\infty $
, and takes a vertex with image label
$a_i$
to the vertex
$a_i$
. This completes the definition of the finite subdivision rule
$\mathcal {R}$
. It is clear from this construction that the ramification portrait of the subdivision map
$\sigma _{\mathcal {R}}$
is isomorphic to
$\Gamma $
.
Verification that
$\sigma _{\mathcal {R}}$
has no Levy cycle. We prove by contradiction that
$\sigma _{\mathcal {R}}$
cannot have a Levy cycle. Suppose
$\{\delta _0,\dots ,\delta _{k-1},\delta _k=\delta _0\}$
is a Levy cycle for
$\sigma _{\mathcal {R}}$
. Choose any
$i\in \{1,\dots ,k\}$
and consider the component
$D_i$
of
$S^2 \setminus \delta _i$
that does not contain
$\infty $
. Since
$\delta _i$
is essential and is not peripheral,
$D_i$
must contain at least two points of P. We will obtain a contradiction by showing that
$D_i$
can contain at most one point of P. For this, consider a finite postcritical point
$a_j$
of
$\sigma _{\mathcal {R}}$
in
$S_{\mathcal {R}}$
.
Suppose that there exists a positive integer m such that
$\tau ^{\circ m}(a_j)$
is a critical vertex. Then,
$a_j$
and
$\infty $
are joined by a new edge in
$\mathcal {R}^{m+1}(S_{\mathcal {R}})$
. By Lemma 3.1, we can isotope
$\delta _i$
in
$S^2\setminus P_f$
to be disjoint from this new edge, so
$a_j$
and
$\infty $
are in the same component of
$S^2 \setminus \delta _i$
. Hence,
$a_j\notin D_i$
. So,
$a_j\notin D_i$
if either
$a_j$
is in a type-
$2$
chain or we are in the situation of the left half of Figure 10.
Now, suppose that
$a_j$
is in a type-
$1$
chain that is not followed by a type-
$2$
chain. Let
$a_r$
be the last element of the type-
$1$
chain that contains
$a_j$
. We are in the situation of either Figure 6 or the right half of Figure 10. So, there is a pair of new edges of
$\mathcal {R}(S_{\mathcal {R}})$
that bounds an open disk D that contains
$a_r$
and no other postcritical points. Hence,
$a_r$
cannot be in the open disk
$D_i$
since if so, Lemma 3.1 implies that we can isotope
$D_i$
rel
$P_f$
into D. Similarly, in
$\mathcal {R}^{r-j+1}(S_{\mathcal {R}})$
, there is a pair of new edges that bounds an open disk that contains
$a_j$
and no other postcritical points.
Finally, suppose
$a_j$
is in a type-
$1$
chain which is followed by a type-
$2$
chain. This is the only remaining possibility. Let
$a_r$
be the last element of the type-
$1$
chain containing
$a_j$
. Suppose that
$j\ne r$
. Let u be the vertex of t with label
$a_j$
. Let
$t'$
be the tile of
$\mathcal {R}^{r-j}(t)$
which contains u. Then, the label of u relative to
$t'$
is
$a_r$
, that is, the structure map of
$t'$
from
$t'$
to
$S_{\mathcal {R}}$
maps u to
$a_r$
. Let v be the finite vertex of
$t'$
following u. The label of v relative to
$t'$
is
$a_{r+1}$
. Because (i)
$j\ne r$
, (ii)
$a_r$
is in a type-1 chain, and (iii)
$a_{r+1}$
is in a type-2 chain, the definition of chains implies that v is not the finite vertex of t following u. So, the edge
$e_1$
of
$t'$
joining v and the
$\infty $
-vertex of
$t'$
following u must be a new edge. However, as in Figure 7, there is a new edge
$e_2$
in
$\mathcal {R}(t')$
joining v and the
$\infty $
-vertex of
$t'$
preceding u. As before, the two new edges in
$\mathcal {R}^{r-j+1}(S_{\mathcal {R}})$
corresponding to
$e_1$
and
$e_2$
bound an open disk which contains
$a_j$
and no other postcritical point. We have reduced to the case in which
$j=r$
. Chains are defined so that at most one postcritical point has this property. So, the open disk
$D_i$
can contain at most one postcritical point, which contradicts the assumption that
$\delta _i$
is an element of a Levy cycle.
Since
$\sigma _{\mathcal {R}}$
is a Thurston map whose ramification portrait is isomorphic to
$\Gamma $
and
$\sigma _{\mathcal {R}}$
has no Levy cycle, the proof of Theorem 1.1 is complete.
4 Completely unobstructed ramification portraits
Proof of Theorem 1.8
Suppose
$\Gamma $
is an abstract polynomial ramification portrait of degree d that has at least four vertices and satisfies condition (i) or (ii) of the statement of the theorem. Suppose f is a Thurston map with ramification portrait isomorphic to
$\Gamma $
and with
$\infty $
a fixed critical point such that
$\deg _f(\infty ) = d$
. We prove by contradiction that f is unobstructed.
Suppose f is obstructed. Then, f has a Levy cycle and (in the terminology of [Reference Hubbard6, §10.3]) f has a degenerate Levy cycle
$\{\delta _0,\dots ,\delta _{n-1},\delta _n = \delta _0\}$
. This means the following. For each
$i\in \{0,\dots ,n\}$
, let
$D_i$
be the disk bounded by
$\delta _i$
in the 2-sphere such that
$\infty \not \in D_i$
. For each
$i\in \{1,\dots ,n\}$
, one component of
$f^{-1}(D_i)$
is a disk
$D_{i-1}'$
such that:
-
(a)
$D_i\cap D_j=\emptyset $
if
$i\ne j\in \{1,\dotsc ,n\}$
; -
(b) the boundary of
$D_{i-1}'$
is isotopic to
$\delta _{i-1}$
rel
$P_f$
; -
(c)
$D_{i-1}' \cap P_f = D_{i-1}\cap P_f$
; and -
(d)
$f|\colon \, D_{i-1}' \to D_i$
is a homeomorphism.
A key point for the Levy–Berstein theorem is that a postcritical point in one of the
$D_i$
disks cannot be a critical point, because that would violate part (d). However, it also cannot have an iterate that is a critical point, because that would imply that some
$D_j$
contains a critical point. Since each
$D_i$
must contain at least two postcritical points, there must be at least two postcritical points in non-attractor cycles. This gives the contradiction for case (i).
Now, suppose condition (ii) holds. Then, for some prime number p and positive integer k, we can enumerate the finite postcritical points of f as
$\{v_i: 0\leq i < p^k\}$
such that
$f(v_i) = v_{i+1}$
(mod
$p^k$
) for every
$i\in \{0,\dots ,p^k -1\}$
, and if
$v_j$
is a critical value, then j is a multiple of
$p^{k-1}$
. Since the sets
$P_f\cap D_i$
partition the set of finite postcritical points and they all have the same cardinality, there is a positive integer m such that
$\#(P_f \cap D_i) = p^m$
for all i. Then,
$np^m = p^k$
and
$n = p^r$
, where
$r = k-m$
. For some
$i\in \{1,\dots ,n\}$
,
$v_0\in D_i$
. Then,
$\{v_{jp^r}\colon \, 0\le j < p^m\} \subset D_i$
and so
$\{v_{jp^{k-1}}\colon \, 0\le j < p\} \subset D_i$
. Thus,
$D_i$
contains every finite critical value of f.
Let D be the disk bounded by
$\delta _i$
that contains
$\infty $
and let
$\widetilde {D} = f^{-1}(D)$
. Then, D does not contain any finite critical values of f. It follows that the restriction of f to
$\widetilde {D}\setminus \{\infty \}$
is a covering map onto
$D\setminus \{\infty \}$
. However, every connected covering space of a once-punctured disk is a once-punctured disk. Since f is d-to-1 near
$\infty $
, the space
$\widetilde {D}\setminus \{\infty \}$
is a once-punctured disk which maps by f to
$D\setminus \{\infty \}$
with degree d. Hence,
$\partial \widetilde {D}=f^{-1}(\partial D)=f^{-1}(\delta _i)$
. This contradicts the assumption that
$f^{-1}(\delta _i)$
has a connected component which maps to
$\delta _i$
with degree 1. Thus, f is unobstructed.
5 Rose maps
To prove Theorem 1.9, we need a topological description for topological polynomials which may not be subdivision maps. We begin this section by discussing our approach to this. We define a rose to be the boundary of the union of finitely many closed topological disks in the 2-sphere which are disjoint except for having exactly one point in common. We view a rose as a graph with exactly one vertex. Its edges are called petals.
Let
$S_1^2$
and
$S_2^2$
be two copies of
$S^2$
, and suppose that we have a finite branched covering map
$g\colon \, S_1^2\to S_2^2$
whose critical values lie in a finite set
$P\subseteq S_2^2$
. The restriction of g to
$S_1^2\setminus g^{-1}(P)$
is a covering map from
$S_1^2\setminus g^{-1}(P)$
onto
$S_2^2\setminus P$
. In the context of covering maps, a straightforward thing to do in this situation is to use the fact that
$S_2^2\setminus P$
is homotopic to a rose with
$\#(P)-1$
petals—the fundamental group of
$S_2^2\setminus P$
is a free group on
$\#(P)-1$
generators. Let
$R_2$
be a rose in
$S_2^2\setminus P$
which is a spine and let
$R_1=g^{-1}(R_2)$
. Because every connected component of
$S_2^2\setminus R_2$
contains at most one branch value of g, every connected component of
$S_1^2\setminus R_1$
is a disk, equivalently,
$R_1$
is connected. The restriction of g to
$R_1$
is a covering map onto
$R_2$
and this restriction determines g up to homotopy.
With this in mind, we construct finite branched covering maps
$g\colon \, S_1^2\to S_2^2$
as follows. Let
$R_2\subseteq S_2^2$
be a rose with
$n\ge 1$
petals. We orient
$S_2^2$
and label n connected components of
$S_2^2\setminus R_2$
each bounded by one petal with
$1,\dotsc ,n$
in counterclockwise order. We label the remaining connected component of
$S_2^2\setminus R_2$
with
$\infty $
. Suppose that we have a finite connected graph
$R_1\subseteq S_1^2$
whose vertices have small neighborhoods which look like small neighborhoods of the vertex of
$R_2$
. Here is what this means. The connected components of
$S_1^2\setminus R_1$
are labeled with
$1,\dotsc ,n$
(duplications allowed) and
$\infty $
. We choose a barycenter for every edge of
$R_1$
and call the resulting edges half edges. These barycenters are not vertices of
$R_1$
. Let v be a vertex of
$R_1$
. Then,
$2n$
half edges contain v. After orienting
$S_1^2$
, they can be written as
$\epsilon _1,\dotsc ,\epsilon _{2n}$
in counterclockwise order around v so that
$\epsilon _{2i-1}$
and
$\epsilon _{2i}$
are in the boundary of a connected component of
$S^2_1\setminus R_1$
with label i for every
$i\in \{1,\ldots ,n\}$
. Furthermore,
$\epsilon _{2i}$
and
$\epsilon _{2i+1}$
are in the boundary of a connected component of
$S_1^2\setminus R_1$
with label
$\infty $
for every
$i\in \{1,\ldots ,n\}$
, where
$\epsilon _{2n+1}=\epsilon _1$
. It is a straightforward matter, by mapping vertices, then half edges, and then disks, to construct a finite branched covering map
$g\colon \, S_1^2\to S_2^2$
such that
$g|_{R_1}$
is a covering map onto
$R_2$
which maps vertices to vertices and edges to edges. This can be done so that g has at most one critical point in every connected component of
$S_1^2\setminus R_1$
.
We define a rose map to be a map of pairs
$g\colon \, (S_1^2,R_1)\to (S_2^2,R_2)$
, where
$S_1^2$
and
$S_2^2$
are two oriented copies of
$S^2$
,
$g\colon \, S_1^2\to S_2^2$
is an orientation-preserving finite branched covering map,
$R_2\subseteq S_2^2$
is a rose,
$R_1= g^{-1}(R_2)$
is a graph with pullback graph structure, and every connected component of
$S_2^2\setminus R_2$
contains at most one critical value of g. The next lemma guarantees the existence of the rose maps that we will use for the proof of Theorem 1.9. We will precompose a rose map
$g\colon \, S_1^2\to S_2^2$
with a homeomorphism
$h\colon \, S_2^2\to S_1^2$
to obtain a desired topological polynomial
$f=g\circ h$
. In the lemma, the connected components of
$S_2^2\setminus R_2$
are labeled, which induces a labeling of the connected components of
$S_1^2\setminus R_1$
, which induces a labeling of the vertices of the graph dual to
$R_1$
.
Lemma 5.1. Suppose that
$\Gamma $
is an abstract polynomial ramification portrait whose finite postcritical vertices are
$v_1,\dots ,v_n$
. Let u be a finite critical value of
$\Gamma $
with the maximum number of incoming edges from critical vertices. Let v be any finite critical value of
$\Gamma $
. Then, there exists a rose map
$g\colon \, (S_1^2,R_1)\to (S_2^2,R_2)$
realizing the branch data of
$\Gamma $
such that n connected components of
$S_2^2\setminus R_2$
each bounded by a petal of
$R_2$
are labeled
$v_1,\dots ,v_n$
in counterclockwise order, the remaining connected component is labeled
$\infty $
, and
$R_1$
has a dual graph
$R_1^*$
for which the following statements hold.
-
(1) The boundary of one connected component of
$S_1^2\setminus R_1^*$
contains exactly two critical points: one vertex with label u and one vertex with label
$\infty $
. -
(2) If u has an incoming edge from a non-critical vertex, then the boundary of one connected component of
$S_1^2\setminus R_1^*$
contains exactly two critical points: one vertex with label v and one vertex with label
$\infty $
.
Proof. We will construct
$R_1$
rather explicitly.
To prepare for the construction of
$R_1$
, let U be the set of critical vertices which
$\tau $
maps to u and let W be the set of remaining finite critical vertices. So,
$C_\Gamma =U\amalg W\amalg \{\infty \}$
. The Riemann–Hurwitz condition gives that
$$ \begin{align*} \sum_{w\in U}^{}(\deg_\tau(w)-1)+\sum_{w\in W}^{}(\deg_\tau(w)-1)=d-1, \end{align*} $$
where d is the degree of
$\Gamma $
. So,
$$ \begin{align} \sum_{w\in W}^{}(\deg_\tau(w)-1)=d-1+\#(U)-\sum_{w\in U}^{}\deg_\tau(w)\ge \#(U)-1. \end{align} $$
In particular, if W is empty, then
$\#(U)=1$
. In this case, there is exactly one choice for
$R_1$
up to isomorphism. Only two connected components of
$S_1^2\setminus R_1$
are not monogons and one of these has label
$\infty $
. Statement (1) is true in this case and statement (2) is true since
$u=v$
. So, we henceforth assume that W is non-empty.
Let m be the maximum number of incoming edges from critical vertices at the critical values other than u. Then, it is possible to partition W into disjoint non-empty subsets
$W_1,\dotsc ,W_m$
so that if
$w,w'\in W_i$
for some i with
$w\ne w'$
, then
$\tau (w)\ne \tau (w')$
. We do this so that if
$u\ne v$
, then there exists
$w\in W_m$
such that
$\tau (w)=v$
.
Now, we begin to construct
$R_1$
. We enumerate the elements of
$W_1$
and for every
${w\in W_1}$
, we construct a closed (two-dimensional) polygon
$P_w\subseteq S_1^2$
with
$\deg _\tau (w)$
sides whose interior has label
$\tau (w)$
. These polygons are disjoint from each other, except that each has exactly one vertex in common with the next. We obtain a chain (not to be confused with the chains in §3)
$C_1$
of polygons. We also construct such chains
$C_2,\dotsc ,C_m$
for
$W_2,\dotsc ,W_m$
so that the chains
$C_1,\dotsc ,C_m$
are disjoint from each other and if
$u\ne v$
, then the polygon with label v in
$C_m$
is last.
The choices of u and m imply that U contains at least m elements. We choose
$m-1$
distinct elements
$u_1,\dotsc ,u_{m-1}\in U$
. For every
$i\in \{1,\ldots ,m-1\}$
, we construct a polygon
$P_{u_i}$
as before which is disjoint from the polygons already constructed and from the other
$P_{u_j}$
polygons, except that one vertex of
$P_{u_i}$
is a vertex of
$C_i$
that is only in the last polygon of
$C_i$
, and a different vertex of
$P_{u_i}$
is a vertex of
$C_{i+1}$
that is only in the first tile of
$C_{i+1}$
. The polygons
$P_{u_1},\dotsc ,P_{u_{m-1}}$
join the chains
$C_1,\dotsc ,C_m$
to form a single chain C.
We intend to also construct similar polygons
$P_v$
for the remaining elements v of U. We intend to construct each of them in one of two ways. One way to construct
$P_v$
is to choose a vertex of C contained in only one polygon and to construct
$P_v$
so that it meets the polygons already constructed exactly in this vertex. Here is another way to construct
$P_v$
. Choose
$i\in \{1,\ldots ,m\}$
, and two consecutive polygons P and
$P'$
in
$C_i$
. Let x be the vertex common to P and
$P'$
. We modify P and
$P'$
slightly near x, pulling them apart so that they become disjoint. We then construct
$P_v$
so that it contains both of the new vertices in P and
$P'$
while being otherwise disjoint from the polygons already constructed.
The only obstacle to performing the constructions described in the previous paragraph is that C might not contain enough vertices to accommodate all the elements of U. However, it is not difficult to see that the number of elements of U that can be accommodated, including
$u_1,\dotsc ,u_{m-1}$
, is
$$ \begin{align*} 1+\sum_{w\in W}(\deg_\tau(w)-1). \end{align*} $$
Thus, line equation (5.1) shows that C does indeed have enough vertices to accommodate all the elements of U. So, we construct a polygon
$P_v$
as described in the previous paragraph for every
$v\in U\setminus \{u_1,\dotsc ,u_{m-1}\}$
.
Because U contains at least m elements, this can be done so that
$$ \begin{align} \text{the first of these polygons meets exactly one polygon in } C, \text{the first polygon in } C_1.\quad \end{align} $$
Furthermore, if u has an incoming edge from a non-critical vertex, then the inequality in line equation (5.1) is strict, so we may also construct these polygons so that
$$ \begin{align} &\text{if } u \text{ has an incoming edge from a non-critical vertex, then the last polygon in }\nonumber\\ & C_m \text{ contains a vertex not in any of these polygons.} \end{align} $$
Every vertex in the complex constructed thus far is contained in either one or two polygons. If there are two, then their labels are different. Hence, it is possible to add monogons with labeled interiors at each of these vertices so that the cyclic order of labels about each vertex agrees with the cyclic order of the labels of
$R_2$
. We have
$R_1$
. The discussion preceding the lemma describes how to construct a rose map
$g\colon \, (S_1^2,R_1) \to (S_2^2,R_2)$
from this information. Line equation (5.2) implies statement (1) of the lemma and line equation (5.3) implies statement (2).
6 Obstructed ramification portraits
Proof of Theorem 1.9
We first prove the theorem in cases (i) and (ii). Assume that
$\Gamma $
satisfies either condition (i) or (ii). We will use the following example to illustrate various constructions in the proof. Consider the abstract polynomial ramification portrait that is shown as follows.
It satisfies the conditions of case (i).
We prepare to apply Lemma 5.1. Suppose that
$\Gamma $
has n finite postcritical vertices
$v_0,\dotsc ,v_n$
with
$v_n=v_0$
such that
$\tau (v_i)=v_{i+1}$
for every
$i\in \{0,\ldots ,n-1\}$
. Of course, if
$\Gamma $
satisfies condition (i), then
$n=p^k$
. Choose a finite critical value u of
$\Gamma $
with the maximum number of incoming edges from critical vertices. We redefine
$v_0,\dotsc ,v_n$
if necessary so that
$v_0=u$
without changing the assumptions.
In particular, if
$\Gamma $
satisfies condition (i), then there exists a critical value
${v_i\in \{v_0,\dotsc ,v_{p^k-1}\}}$
such that i is not a multiple of
$p^{k-1}$
. We set
$v=v_i$
. In our example, we take
$u=v_0$
and
$v=v_2$
.
To define v in the case of condition (ii), suppose that
$\Gamma $
satisfies condition (ii). Because u is in a non-attractor cycle,
$\tau $
maps some vertex which is not critical to u. It follows that the inequality in line equation (5.1) is strict and so
$\Gamma $
has more than one finite critical value. Let v be any finite critical value other than u.
We next define a positive integer m. If
$\Gamma $
satisfies condition (i), then we set
$m=p^{k-1}$
. Suppose that
$\Gamma $
satisfies condition (ii). Let i be the index such that
$v=v_i$
. Because n is not a prime power, it is the product of two relatively prime proper divisors. Because they are relatively prime, if both of these two proper divisors divide i, then n divides i, which is not true. Hence, some positive proper divisor of n does not divide i. Let m be such a positive proper divisor of n.
So, in either case (i) or (ii), m is a positive proper divisor of n such that
$v=v_i$
and m does not divide i.
Lemma 5.1 implies that there exists a rose map
$g\colon \, (S_1^2,R_1)\to (S_2^2,R_2)$
realizing the branch data of
$\Gamma $
such that n connected components of
$S_2^2\setminus R_2$
each bounded by a petal of
$R_2$
are labeled
$v_0,\dotsc ,v_{n-1}$
in counterclockwise order, the remaining connected component is labeled
$\infty $
, and
$R_1$
has a dual graph
$R_1^*$
for which:
-
(1) the boundary of one connected component
$C_u$
of
$S_1^2\setminus R_1^*$
contains exactly two critical points: one vertex with label u and one vertex with label
$\infty $
; -
(2) the boundary of one connected component
$C_v$
of
$S_1^2\setminus R_1^*$
contains exactly two critical points: one vertex with label v and one vertex with label
$\infty $
.
By modifying
$R_1^*$
if necessary, we may assume that the restriction of g to both
$C_u$
and
$C_v$
is injective. We identify every postcritical vertex w of
$\Gamma $
with a point in the connected component of
$S_2^2\setminus R_2$
with label w. These serve as the vertices of a graph
$R_2^*$
dual to
$R_2$
. Their g-pullbacks serve as the vertices of
$R_1^*$
. Figure 11 depicts important features of the rose map g for our example.
Figure 11 Rose map
$g:S_{1}^2\to S_{2}^2$
.
Now, we choose m disjoint closed topological disks
$D_i\subseteq S_2^2\setminus \{\infty \}$
such that: (i)
$v_j$
is in the interior of
$D_i$
if
$j\equiv i\text { (mod }m)$
for
$i\in \{0,\dotsc ,m-1\}$
and
$j\in \{0,\dotsc ,n-1\}$
; (ii)
$D_0$
is in the open disk
$g(C_v)$
; and (iii) for
$i\in \{1,\dotsc ,m-1\}$
,
$D_i$
is in the open disk
$g(C_u)$
. The restriction of g to
$C_v$
is a homeomorphism, so there exists a unique lift
$\widetilde {D}_0$
of
$D_0$
to
$C_v$
. We denote the lift of
$v_j$
to
$C_v$
by
$\widetilde {v}_j$
for every index
$j\equiv 0\text { (mod } m)$
. There likewise exist unique lifts
$\widetilde {D}_i$
of
$D_i$
to
$C_u$
for every
$i\in \{1,\dotsc ,m-1\}$
. We denote the lift of
$v_j$
to
$C_u$
by
$\widetilde {v}_j$
for every index
$j\not \equiv 0\text { (mod }m)$
. Figure 12 shows all of these points and disks for our example.
Figure 12 Rose map
$g:S_{1}^2\to S_{2}^2$
.
We next construct an orientation-preserving homeomorphism
$h\colon \, S_2^2\to S_1^2$
such that: (i)
$h(v_j)=\widetilde {v}_{j+1}$
for
$j\in \{0,\dotsc ,n-1\}$
; (ii)
$h(\infty )=g^{-1}(\infty )$
; and (iii)
$h(D_i)=\widetilde {D}_{i+1}$
for
$i\in \{0,\dotsc ,m-1\}$
, where
$D_m=D_0$
. See Figure 13.
Finally, define
$f:=g\circ h$
. The map f is a Thurston map whose ramification portrait is isomorphic to
$\Gamma $
and the boundaries of the disks
$D_i$
form a degenerate Levy cycle. This proves Theorem 1.9 in cases (i) and (ii).
Now, suppose that
$\Gamma $
satisfies condition (iii). Let C be a non-attractor cycle of length
$\ell \ge 2$
that does not contain all of the finite critical values of
$\Gamma $
. Let
$v_1,\dotsc ,v_n$
be the finite postcritical vertices of
$\Gamma $
indexed so that
$v_1,\dotsc ,v_\ell $
are the vertices of C,
$\tau (v_\ell )=v_1$
, and
$\tau (v_i)=v_{i+1}$
for
$i\in \{1,\ldots ,\ell -1\}$
. By Lemma 5.1, there exists a critical value
$v\notin C$
and a rose map
$g\colon \, (S_1^2, R_1)\to (S_2^2,R_2)$
realizing the branch data of
$\Gamma $
such that n connected components of
$S_2^2\setminus R_2$
each bounded by a petal of
$R_2$
are labeled
$v_1,\dotsc ,v_n$
in counterclockwise order, the remaining connected component is labeled
$\infty $
, and the boundary of one connected component
$C_v$
of
$S_1\setminus R_1^*$
contains exactly two critical points: one vertex with label v and one vertex with label
$\infty $
. We may assume that the restriction of g to
$C_v$
is injective. As in cases (i) and (ii), we identify every postcritical vertex w of
$\Gamma $
with a point in the connected component of
$S_2^2\setminus R_2$
with label w.
Let D be a closed topological disk in the open disk
$g(C_v)$
whose interior contains
$v_1,\dotsc ,v_\ell $
, but such that D contains no other postcritical vertex of
$\Gamma $
. Let
$\widetilde {D}$
be the lift of D to
$C_v$
. Let
$\widetilde {v}_i$
be the lift of
$v_i$
to
$C_v$
for
$i\in \{1,\ldots ,\ell \}$
. Now, construct an orientation-preserving homeomorphism
$h\colon \, S_2^2\to S_1^2$
such that: (i)
$h(v_i)=\widetilde {v}_{i+1}$
for
$i\in \{1,\ldots ,\ell -1\}$
; (ii)
$h(v_\ell )=v_1$
; (iii)
$h(\infty )=g^{-1}(\infty )$
; (iv)
$h(D)=\widetilde {D}$
; and (v) the ramification portrait of
$f:=g\circ h$
is isomorphic to
$\Gamma $
. Since
$\partial D$
is by itself a degenerate Levy cycle for f, this proves Theorem 1.9 in case (iii).
Finally, we consider case (iv), where
$\Gamma $
has at least two non-attractor cycles of length one. Let a and b denote the vertices of two non-attractor cycles of
$\Gamma $
of length
$1$
. Define the abstract ramification portrait
$\Gamma '$
as follows:
$V(\Gamma ')=V(\Gamma )$
,
$\tau '(a)=b$
,
$\tau '(b)=a$
, and
$\tau '(x)=\tau (x)$
for all
$x\in V(\Gamma ')\setminus \{a,b\}$
. So,
$\Gamma '$
satisfies condition (iii). The proof for case (iii) shows that there is a Thurston map f realizing
$\Gamma '$
that has a degenerate Levy cycle consisting of a single curve which bounds a disk D such that
$D\cap P_f = \{a,b\}$
. Postcompose f with an orientation-preserving homeomorphism
$h:S^{2}\to S^{2}$
such that h is the identity in the complement of D,
$h(b)=a$
, and
$h(a)=b$
. Then,
$h\circ f$
is an obstructed Thurston map whose ramification portrait is isomorphic to
$\Gamma $
.
Figure 13 The obstructed Thurston map
$f:S^{2}_{2}\to S^{2}_{2}$
.
7 Classification of completely unobstructed ramification portraits
Proof of Theorem 1.10
For the forward direction, we will prove the contrapositive. Suppose that
$\Gamma $
is an abstract polynomial ramification portrait that does not satisfy any of the conditions (i)–(iv). Since
$\Gamma $
does not satisfy conditions (i) and (ii),
$\Gamma $
has at least four postcritical vertices and there is a non-attractor cycle. Since
$\Gamma $
does not satisfy condition (iii), either
$\Gamma $
has at least two non-attractor cycles or there is a non-attractor cycle of length at least two. If
$\Gamma $
has at least two non-attractor cycles, then it satisfies conditions (iii) or (iv) of Theorem 1.9 and so
$\Gamma $
can be realized by an obstructed Thurston map. Now, suppose that
$\Gamma $
has a single non-attractor cycle and it has length at least two. If this cycle does not contain all of the finite postcritical vertices, then
$\Gamma $
can be realized by an obstructed map by condition (iii) of Theorem 1.9. So, we may suppose that all of the finite postcritical vertices are in a single non-attractor cycle. If its length is not a prime power, then
$\Gamma $
can be realized by an obstructed map by condition (ii) of Theorem 1.9. If its length is a prime power, then since
$\Gamma $
does not satisfy condition (iv), it satisfies condition (i) of Theorem 1.9 and
$\Gamma $
can be realized by an obstructed map.
For the reverse direction, suppose
$\Gamma $
is an abstract polynomial ramification portrait and f is a Thurston map whose ramification portrait is isomorphic to
$\Gamma $
. If
$\Gamma $
satisfies condition (i), then f is unobstructed by Thurston’s characterization theorem since there are not enough postcritical points to have an obstruction. If
$\Gamma $
satisfies condition (ii), then f is unobstructed by the Levy–Berstein theorem. If
$\Gamma $
satisfies condition (iii) or (iv), then f is unobstructed by Theorem 1.8.
Acknowledgements
The research of S. Koch was supported in part by the NSF. We would like to thank the anonymous referee for their comments and suggestions.
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