__

# Eliminating Thurston obstructions and controlling dynamics on curves

Mario Bonk  Department of Mathematics, University of California, Los Angeles,
CA 90095, USA<EMAIL_ADDRESS>,  Mikhail Hlushchanka  Aix–Marseille
Université, Institut de Mathéatiques de Marseille, 13009 Marseille, France
Mathematisch Instituut, Universiteit Utrecht, 3508 TA Utrecht, The Netherlands
<EMAIL_ADDRESS>and  Annina Iseli  Department of Mathematics, University
of Fribourg, 1700 Fribourg, Switzerland  Department of Mathematics, University
of California, Los Angeles, CA 90095, USA<EMAIL_ADDRESS>
(Date: May 14, 2021)

###### Abstract.

Every Thurston map $f\colon S^{2}\rightarrow S^{2}$ on a $2$-sphere $S^{2}$
induces a pull-back operation on Jordan curves $\alpha\subset S^{2}\setminus
P_{f}$, where $P_{f}$ is the postcritical set of $f$. Here the isotopy class
$[f^{-1}(\alpha)]$ (relative to $P_{f}$) only depends on the isotopy class
$[\alpha]$. We study this operation for Thurston maps with four postcritical
points. In this case a Thurston obstruction for the map $f$ can be seen as a
fixed point of the pull-back operation.

We show that if a Thurston map $f$ with a hyperbolic orbifold and four
postcritical points has a Thurston obstruction, then one can “blow up”
suitable arcs in the underlying $2$-sphere and construct a new Thurston map
$\widehat{f}$ for which this obstruction is eliminated. We prove that no other
obstruction arises and so $\widehat{f}$ is realized by a rational map. In
particular, this allows for the combinatorial construction of a large class of
rational Thurston maps with four postcritical points.

We also study the dynamics of the pull-back operation under iteration. We
exhibit a subclass of our rational Thurston maps with four postcritical points
for which we can give positive answer to the global curve attractor problem.

###### Key words and phrases:

Thurston maps, obstructions, Lattès maps, intersection numbers, curve
attractor.

###### 2010 Mathematics Subject Classification:

Primary 37F20, 37F10

M.B. was partially supported by NSF grant DMS-1808856. M.H. was partially
supported by the ERC advanced grant “HOLOGRAM”. A.I. was partially supported
by the Swiss National Science Foundation (project no. 181898).

###### Contents

  1. 1 Introduction
    1. 1.1 Blowing up Lattès maps
    2. 1.2 The global curve attractor problem
    3. 1.3 Organization of this paper
  2. 2 Preliminaries
    1. 2.1 Notation and basic concepts
    2. 2.2 Branched covering maps
    3. 2.3 Planar embedded graphs
    4. 2.4 The Euclidean square pillow
    5. 2.5 Isotopies and intersection numbers
    6. 2.6 Jordan curves in spheres with four marked points
  3. 3 Thurston maps
    1. 3.1 The $(n\times n)$–Lattès map
    2. 3.2 Thurston’s characterization of rational maps
  4. 4 Blowing up arcs
    1. 4.1 The general construction
    2. 4.2 Blowing up the $(n\times n)$-Lattès map
  5. 5 Realizing blown-up Lattès maps
    1. 5.1 The horizontal and vertical curves
    2. 5.2 Ruling out other obstructions
  6. 6 Essential circuit length
  7. 7 Eliminating obstructions by blowing up arcs
  8. 8 Global curve attractors
  9. 9 Further discussion
    1. 9.1 Julia sets of blown-up Lattès maps
    2. 9.2 The global curve attractor problem
    3. 9.3 Twisting problems
    4. 9.4 Thurston maps with more than four postcritical points
    5. 9.5 Other combinatorial constructions of rational maps
  10. 10 Appendix: Isotopy classes of Jordan curves in spheres with four marked points

##  1\. Introduction

In this paper, we consider Thurston maps and the dynamics of the induced pull-
back operation on Jordan curves on the underlying $2$-sphere. By definition, a
Thurston map is a branched covering map $f\colon S^{2}\rightarrow S^{2}$ on a
topological $2$-sphere $S^{2}$ such that $f$ is not a homeomorphism and every
critical point of $f$ (points where $f$ is not a local homeomorphism) has a
finite orbit under iteration of $f$. These maps are named after William
Thurston who introduced them in his quest for a better understanding of the
dynamics of postcritically-finite rational maps on the Riemann sphere. We
refer to [BM17, Chapter 2] for general background on Thurston maps and related
concepts.

For a branched covering map $f\colon S^{2}\rightarrow S^{2}$ we denote by
$C_{f}$ the set of critical points of $f$ and by $f^{n}$ the $n$-th iterate of
$f$ for $n\in\mathbb{N}$. Then the postcritical set of $f$ is defined as

$P_{f}=\bigcup_{n\in\mathbb{N}}\\{f^{n}(c):c\in C_{f}\\}.$

For a Thurston map $f$ this set has finite cardinality $2\leq\\#P_{f}<\infty$
(for the first inequality see [BM17, Corollary 2.13]).

A Thurston map $f$ often admits a description in purely combinatorial-
topological terms. In this context, it is an interesting question whether $f$
can be realized (in a suitable sense) by a rational map with the same
combinatorics. Roughly speaking, this means that $f$ is conjugate to a
rational map “up to isotopy” (see Section 3 for the precise definition).

It is not hard to see that each Thurston map with two or three postcritical
points is realized. The situation is much more complicated for Thurston maps
$f$ with $\\#P_{f}\geq 4$. William Thurston found a necessary and sufficient
condition when a Thurston map can be realized by a rational map [DH93].
Namely, if $f$ has an associated hyperbolic orbifold (this is always true
apart from some well-understood exceptional maps), then $f$ is realized if and
only if $f$ has no _(Thurston) obstruction_. Such an obstruction is given by a
finite collection of disjoint Jordan curves in $S^{2}\setminus P_{f}$ (up to
isotopy) with certain invariance properties (see Section 3.2 for more
discussion).

The “if part” of this statement gives a positive criterion for $f$ to be
realized, but it is very hard to apply in practice, because, at least in
principle, it involves the verification of infinitely many conditions for the
map $f$. For this reason, in each individual case a successful verification
for a map, or a class of maps, is difficult and usually constitutes an
interesting result in its own right.

We mention two results in this direction. The first one is the “arcs
intersecting obstructions” theorem by Kevin Pilgrim and Tan Lei [PL98, Theorem
3.2] that gives control on the position of an obstruction and has many
applications in holomorphic dynamics (see, for instance, [PL98, DMRS19]). The
other one is the “mating criterion” by Tan Lei, Mary Rees, and Mitsuhiro
Shishikura that addresses the question when two postcritically-finite
quadratic polynomials can be topologically glued together to form a rational
map (see [Lei92, Ree92, Shi00]).

The investigation of obstructions of a Thurston map $f\colon S^{2}\rightarrow
S^{2}$ is closely related to the study of the _pull-back operation_ on Jordan
curves. It is easy to show that if $\alpha\subset S^{2}\setminus P_{f}$ is a
Jordan curve, then the isotopy class $[f^{-1}(\alpha)]$ (rel. $P_{f}$) only
depends on the isotopy class $[\alpha]$ (see Lemma 3.4). Intuitively, the
number of postcritical points of a Thurston map can be seen as a measure of
its combinatorial complexity. In this paper, we focus on the simplest non-
trivial case, namely Thurston maps $f$ with $\\#P_{f}=4$. In this case the
pull-back operation gives rise to a well-defined map, the slope map, on these
isotopy classes $[\alpha]$ (we will discuss this in more detail below). The
search for obstructions of $f$ amounts to understanding the fixed points of
the slope map.

There exist various natural constructions that allow one to _combine_ or
_modify_ given (rational) Thurston maps to obtain a new dynamical system. The
most studied constructions are _mating_ (see [Lei92, SL00]), _tuning_ (see
[Ree92]), and _capture_ (see [Hea88, Lei97]). In this paper, we study the
operation of _blowing up arcs_ , originally introduced by Kevin Pilgrim and
Tan Lei in [PL98]. This operation can be applied to an arbitrary Thurston map
$f$ and results in a new Thurston map $\widehat{f}$ that is of higher degree,
but combinatorially closely related to the original map $f$. In particular,
$f$ and $\widehat{f}$ have the same set of postcritical points and the same
dynamics on them. Nevertheless, the dynamical behavior of Jordan curves under
the pull-back operation for the original map $f$ and the new map $\widehat{f}$
may differ drastically.

We show that, if a Thurston map $f\colon S^{2}\rightarrow S^{2}$ with
$\\#P_{f}=4$ has an obstruction $\alpha$, then one can naturally modify $f$ by
blowing up certain arcs to produce a new Thurston map $\widehat{f}$ for which
this obstruction $\alpha$ is eliminated. The main result of this paper is the
fact that then no new obstructions arise for $\widehat{f}$ and so it is
realized by a rational map.

######  Theorem 1.1.

Let $f\colon S^{2}\to S^{2}$ be a Thurston map with $\\#P_{f}=4$ and a
hyperbolic orbifold. Suppose that $f$ has an obstruction represented by a
Jordan curve $\alpha\subset S^{2}\setminus P_{f}$, and $E\neq\emptyset$ is a
finite set of arcs in $(S^{2},f^{-1}(P_{f}))$ that satisfy the
$\alpha$-restricted blow-up conditions.

Let $\widehat{f}$ be a Thurston map obtained from $f$ by blowing up arcs in
$E$ (with some multiplicities) so that $\lambda_{\widehat{f}}(\alpha)<1$. Then
$\widehat{f}$ is realized by a rational map.

The technical verbiage and the notation in this formulation will be explained
in subsequent sections (see in particular (3.3) for the definition of the
“eigenvalue” $\lambda_{\widehat{f}}(\alpha)$ and Definition 6.5 for
$\alpha$-restricted blow-up conditions).

Recently, Dylan Thurston provided a positive characterization when a Thurston
map is realized, at least in the case when each critical point eventually
lands in a critical cycle under iteration. He proved that such a Thurston map
$f$ is realized by a rational map if and only if there is an “elastic spine”
(that is, a planar embedded graph in $S^{2}\setminus P_{f}$ with a suitable
metric on it) that gets “looser” under backwards iteration (see [Thu16, Thu20]
for more details). In concrete cases, especially for Thurston maps that should
be realized by rational maps with Julia sets homeomorphic to the Sierpiński
carpet, the application of Dylan Thurston’s criterion is not so
straightforward. Moreover, his criterion is only valid for Thurston maps with
periodic critical points. In contrast, for some maps for which Dylan
Thurston’s criterion is not applicable or hard to apply, Theorem 1.1 can be
used to verify that the maps are realized. In particular, many maps obtained
by blowing up Lattès maps (see below) are of this type.

###  1.1. Blowing up Lattès maps

We will now discuss a special case of Theorem 1.1 in detail to give the reader
some intuition for the geometric ideas behind this statement and its proof.

Let $\mathbb{P}$ be a pillow obtained from two copies of the unit square
$[0,1]^{2}\subset\mathbb{R}^{2}\cong\mathbb{C}$ glued together along their
boundaries. We consider the two copies of $[0,1]^{2}$ in $\mathbb{P}$ as the
front and back side of $\mathbb{P}$ and call them the tiles of level $0$ or
simply $0$-tiles. We denote by $A\coloneq(0,0)\in\mathbb{P}$ the lower left
corner of $\mathbb{P}$ (see the right part of Figure 1). The pillow
$\mathbb{P}$ is a topological $2$-sphere. Actually, if we consider
$\mathbb{P}$ as an abstract polyhedral surface, then $\mathbb{P}$ carries a
conformal structure making $\mathbb{P}$ conformally equivalent to the Riemann
sphere $\widehat{\mathbb{C}}$. See Section 2.4 for more discussion.

$\mathcal{L}_{4}$
---
$A$
---
$A$
---
$\mathbb{P}$
---
$\mathbb{P}$
---
Figure 1. The $(4\times 4)$-Lattès map.

We now fix $n\in\mathbb{N}$ with $n\geq 2$. We subdivide each of the two
$0$-tiles of $\mathbb{P}$ into $n^{2}$ small squares of sidelength $1/n$,
called the $1$-tiles. We color these $1$-tiles in a checkerboard fashion black
and white so that the $1$-tile in the front $0$-tile that contains the vertex
$A$ on its boundary is colored white (see the left part of Figure 1). We map
this white $1$-tile to the front $0$-tile of the right-hand pillow by an
orientation-preserving Euclidean similarity that fixes the vertex $A$. This
similarity scales distances by the factor $n$. We can uniquely extend the
similarity by a successive Schwarz reflection process to the whole pillow
$\mathbb{P}$ to obtain a continuous map
$\mathcal{L}_{n}\colon\mathbb{P}\rightarrow\mathbb{P}$. Then on each $1$-tile
$S$ the map $\mathcal{L}_{n}$ is a Euclidean similarity that sends $S$ to the
front or back $0$-tile of $\mathbb{P}$ depending on whether $S$ is white or
black. We call $\mathcal{L}_{n}$ the $(n\times n)$-Lattès map, because under a
suitable conformal equivalence $\mathbb{P}\cong\widehat{\mathbb{C}}$, the map
$\mathcal{L}_{n}$ is conjugate to a rational map obtained from
$n$-multiplication of a Weierstrass $\wp$-function. See Figure 1 for an
illustration of the map $\mathcal{L}_{4}$. Here, the marked points on the left
pillow $\mathbb{P}$ (the domain of the map) correspond to the preimage points
$\mathcal{L}_{4}^{-1}(A)$. Note that there is exactly one preimage of $A$ in
the interior of the back side of the pillow.

It is easy to see that the $(n\times n)$-Lattès map
$\mathcal{L}_{n}\colon\mathbb{P}\rightarrow\mathbb{P}$ is a Thurston map with
four postcritical points, namely, the four corners of the pillow $\mathbb{P}$.
The map $\mathcal{L}_{n}$ is realized by a rational map, because it is even
conjugate to such a map.

We now modify the map $\mathcal{L}_{n}$ by gluing in vertical or horizontal
flaps to $\mathbb{P}$. This is a special case of the more general construction
of blowing up arcs mentioned above. We will describe this in detail in Section
4, but will illustrate the procedure in Figure 2, where we show how to glue in
one horizontal flap.

We cut the pillow $\mathbb{P}$ open along a horizontal side $e$ of one of the
$1$-tiles. Note that in this process $e$ is “doubled” into two arcs
$e^{\prime}$ and $e^{\prime\prime}$ with common endpoints. We then take two
disjoint copies of the Euclidean square $[0,1/{n}]^{2}$ and identify them
along three corresponding sides to obtain a flap $F$. It has two “free” sides
on its boundary. We glue each free side to one of the arcs $e^{\prime}$ and
$e^{\prime\prime}$ of the cut in the obvious way.

choose an edge
---
cut the pillow open
---
glue in a flap
---
$e$
---
$e^{\prime\prime}$
---
$e^{\prime}$
---
$F$
---
Figure 2. Gluing in a flap.

In general, one can repeat this construction and glue several flaps at the
location given by the arc $e$. We assume that this has been done
simultaneously for $n_{h}\geq 0$ flaps along horizontal edges and $n_{v}\geq
0$ flaps along vertical edges. By this procedure we obtain a “flapped” pillow
$\widehat{\mathbb{P}}$, which is still a topological $2$-sphere (see the left
part of Figure 3). By construction it is tiled by $2n^{2}+2(n_{h}+n_{v})$
squares of sidelength $1/n$, which we consider as the $1$-tiles of
$\widehat{\mathbb{P}}$. The checkerboard coloring of the base surface
$\mathbb{P}$ extends in a unique way to the new surface
$\widehat{\mathbb{P}}$. The original $(n\times n)$-Lattès map
$\mathcal{L}_{n}\colon\mathbb{P}\rightarrow\mathbb{P}$ can naturally be
“extended” to a continuous map
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\to\mathbb{P}$ so that each
$1$-tile $S$ of $\widehat{\mathbb{P}}$ is mapped to the front or back $0$-tile
of $\mathbb{P}$ (depending on the color of $S$) by a Euclidean similarity
scaling distances by the factor $n$. See Figure 3 for an illustration of a map
$\widehat{\mathcal{L}}$ obtained from the Lattès map $\mathcal{L}_{4}$ by
gluing in flaps at a vertical and a horizontal edge. Similarly as in Figure 1,
on the left we marked the preimages of $A$ under $\widehat{\mathcal{L}}$.

$\widehat{\mathcal{L}}$
---
$A$
---
$A$
---
$\mathbb{P}$
---
$\widehat{\mathbb{P}}$
---
Figure 3. An example of a map $\widehat{\mathcal{L}}$ obtained from
$\mathcal{L}_{4}$ by gluing in flaps.

In order to obtain a Thurston map $f\colon\mathbb{P}\rightarrow\mathbb{P}$
from this construction, we need to choose a homeomorphism
$\phi\colon\widehat{\mathbb{P}}\rightarrow\mathbb{P}$. Roughly speaking,
$\phi$ is a homeomorphism that identifies $\widehat{\mathbb{P}}$ with
$\mathbb{P}$ and fixes each corner of the pillow. The precise choice of $\phi$
is somewhat technical and so we refer to Section 4.2 for the details. Then,
one easily observes that the postcritical set $P_{f}$ consists of the four
corners of the pillow $\mathbb{P}$. The map $f$ is uniquely determined up to
Thurston equivalence (see Definition 3.2) independently of the choice of
$\phi$ under suitable restrictions. We refer to $f$ as a Thurston map obtained
from the $(n\times n)$-Lattès map by gluing $n_{h}$ horizontal and $n_{v}$
vertical flaps to $\mathbb{P}$.

Now the following statement is true. As we will explain, it can be seen as a
special case of our main result.

######  Theorem 1.2.

Let $n\in\mathbb{N}$ with $n\geq 2$ and $f\colon\mathbb{P}\to\mathbb{P}$ be a
Thurston map obtained from the $(n\times n)$-Lattès map $\mathcal{L}_{n}$ by
gluing $n_{h}\geq 0$ horizontal and $n_{v}\geq 0$ vertical flaps to
$\mathbb{P}$, where $n_{h}+n_{v}>0$. Then the map $f$ has a hyperbolic
orbifold. It has an obstruction if and only if $n_{h}=0$ or $n_{v}=0$. In
particular, if $n_{h}>0$ and $n_{v}>0$, then $f$ is realized by a rational
map.

If $n_{h}=n_{v}=0$, then no flaps were glued to $\mathbb{P}$ and the map $f$
coincides with the original $(n\times n)$-Lattès map $\mathcal{L}_{n}$
(strictly speaking, only if we choose the homeomorphism $\phi$ used in the
construction above to be the identity on $\mathbb{P}$, as we may). Then
$f=\mathcal{L}_{n}$ has a parabolic orbifold. Therefore, Thurston’s criterion
as formulated in Section 3.2 does not apply.

If $n_{h}=0$ or $n_{v}=0$, but $n_{h}+n_{v}>0$ as in Theorem 1.2, then it is
immediate to see that $f$ has an obstruction (see Section 5.1) and therefore
$f$ cannot be realized by a rational map. So the interesting part of Theorem
1.2 is the claim that if $n_{h}>0$ and $n_{v}>0$, then $f$ has no obstruction.

Even though Theorem 1.2 follows from our more general statement formulated in
Theorem 1.1, we will give a complete proof. We will argue by contradiction and
assume that a map $f$ with $n_{h}>0$ and $n_{v}>0$ has an obstruction. In
principle, there are infinitely many candidates represented by essential
isotopy classes of Jordan curves $\alpha\subset\mathbb{P}\setminus P_{f}$.
These isotopy classes in turn are distinguished by different slopes in
$\widehat{\mathbb{Q}}=\mathbb{Q}\cup\\{\infty\\}$ (as will be explained in
Section 2.5). For such an isotopy class represented by $\alpha$ to be an
obstruction, it has to be $f$-invariant in the sense that $f^{-1}(\alpha)$
should contain a component $\widetilde{\alpha}$ isotopic to $\alpha$ rel.
$P_{f}$. It seems to be a very intricate problem to find all slopes in
$\widehat{\mathbb{Q}}$ that give an invariant isotopy class for $f$. Since we
have been able to decide this question only for very simple maps $f$, we
proceed in a more indirect manner.

We assume that the Jordan curve $\alpha\subset\mathbb{P}\setminus P_{f}$ is
$f$-invariant and gives an obstruction. We then investigate the mapping
degrees of $f$ on components of $f^{-1}(\alpha)$ and consider intersection
numbers of some relevant curves together with a careful counting argument. We
heavily use the fact that the horizontal and vertical Jordan curves (see
(2.4)) are $f$-invariant. Ultimately, we arrive at a contradiction. See
Section 5 for the details of this argument.

Our idea to use intersection numbers (as in Lemma 5.5) to control possible
locations of obstructions and dynamics on curves is not new (see, for example,
[PL98, Theorem 3.2], [CPL16, Section 8], and [Par18]). However, the previously
available results do not provide sharp enough estimates applicable in our
situation.

One can think of Theorem 1.2 in the following way. Suppose that instead of
directly passing from the Lattès map $\mathcal{L}_{n}$ to a map, let us now
call it $\widehat{f}$, obtained by gluing $n_{h}>0$ horizontal and $n_{v}>0$
vertical flaps to $\mathbb{P}$, we first create an intermediate map $f$
obtained by gluing $n_{h}>0$ horizontal, but no vertical flaps. Then $f$ has a
hyperbolic orbifold and an obstruction given by a “horizontal” Jordan curve
$\alpha$. In the passage from $f$ to $\widehat{f}$ we kill this obstruction,
because we glue additional vertical flaps that serve as obstacles and increase
the mapping degree on some pullbacks of $\alpha$. Theorem 1.1 then says that
no other obstructions arise for $\widehat{f}$. Therefore, Theorem 1.1
generalizes Theorem 1.2 if we interpret it in the way just described. The
proof of Theorem 1.1 is based on a refinement and extension of the ideas that
we use to establish Theorem 1.2. The proof of Theorem 1.1 is based on the
ideas that we use to establish Theorem 1.2, but substantial refinements and
extensions are required.

###  1.2. The global curve attractor problem

The mapping properties of Jordan curves play an important role in Thurston’s
characterization of rational maps. The original proof of this statement
associates with a given Thurston map $f\colon S^{2}\to S^{2}$ with a
hyperbolic orbifold a certain Teichmüller space $\mathcal{T}_{f}$ and a real-
analytic map $\sigma_{f}\colon\mathcal{T}_{f}\to\mathcal{T}_{f}$, called
_Thurston’s pullback map_. One can show that the map $f$ is realized by a
rational map if and only if $\sigma_{f}$ has a fixed point [DH93]. This
reduction to a fixed point problem in a Teichmüller space has also been
successfully applied by Thurston in the other contexts such as uniformization
problems and the theory of $3$-manifoldsDH (there is a rich literature on the
subject; see, for example, [Thu88, Thu98, Ota01, FLP12, Hub16]).

In recent years, the pullback map $\sigma_{f}$ and its dynamical properties
have been subject to deeper investigations (see, for example, [Pil12, Sel12,
Lod13, KPS16]). In particular, Nikita Selinger showed in [Sel12] that
$\sigma_{f}$ extends to the _Weil-Petersson boundary_ of $\mathcal{T}_{f}$.
The behavior of $\sigma_{f}$ on this boundary is closely related to the
behavior of Jordan curves under pull-back by $f$. This in turn leads to the
following difficult open question in holomorphic dynamics, called the _global
curve attractor problem_ (see [Lod13, Section 9]).

######  Conjecture.

Let $f\colon S^{2}\to S^{2}$ be a Thurston map with a hyperbolic orbifold that
is realized by a rational map. Then there exists a finite set $\mathscr{A}(f)$
of Jordan curves in $S^{2}\setminus P_{f}$ such that for every Jordan curve
$\gamma\subset S^{2}\setminus P_{f}$, all pullbacks $\widetilde{\gamma}$ of
$\gamma$ under $f^{n}$ are contained in $\mathscr{A}(f)$ up to isotopy rel.
$P_{f}$ for all sufficiently large $n\in\mathbb{N}$.

A set of Jordan curves $\mathscr{A}(f)$ as in this conjecture is called a
_global curve attractor_ of $f$. We will give a solution of this problem for
maps as in Theorem 1.2 with $n=2$ and $n_{h},n_{v}\geq 1$. Unfortunately, our
methods only apply for $n=2$ and not for $n\geq 3$.

######  Theorem 1.3.

Let $f\colon\mathbb{P}\rightarrow\mathbb{P}$ be a Thurston map obtained from
the $(2\times 2)$-Lattès map by gluing $n_{h}\geq 1$ horizontal and $n_{v}\geq
1$ vertical flaps to the pillow $\mathbb{P}$. Then $f$ has a global curve
attractor $\mathscr{A}(f)$.

One can show that the Julia set of a rational map as provided by Theorem 1.2
is either a Sierpiński carpet or the whole Riemann sphere depending on whether
the map has periodic critical points or not (see Proposition 9.1).
Accordingly, Theorem 1.3 provides the first examples of maps with Sierpiński
carpet Julia set for which an answer to the global curve attractor problem is
known. In fact, we obtain such maps with arbitrary large degrees.

Recently, Belk-Lanier-Margalit-Winarski proved the existence of a finite
global curve attractor for all postcritically-finite polynomials [BLMW19]. The
conjecture is also known to be true for all _critically fixed rational maps_
(that is, rational maps for which each critical point is fixed) and some
_nearly Euclidean Thurston maps_ (that is, Thurston maps with exactly four
postcritical points and only simple critical points); see [Hlu19] and [Lod13,
FKK+17]. In [KL19] Gregory Kelsey and Russell Lodge verified the conjecture
for all quadratic non-Lattès maps with four postcritical points. However, for
general postcritcally-finite rational maps the conjecture remains wide open.

Since the maps we consider have four postcritical points, it is convenient to
reformulate the global curve attractor problem by introducing the slope map
(it is closely related to the Thurston pull-back map $\sigma_{f}$ on the Weil-
Petersson boundary of $\mathcal{T}_{f}$). To define it in the special case
relevant for us, we consider the marked pillow $(\mathbb{P},V)$, where $V$ is
the set consisting of the four corners of $\mathbb{P}$, and assume that
$f\colon\mathbb{P}\to\mathbb{P}$ is a Thurston map with $P_{f}=V$. Up to
topological conjugacy, every Thurston map with four postcritical points can be
assumed to have this form.

As we already mentioned, there is a bijective correspondence between isotopy
classes $[\alpha]$ of essential Jordan curves $\alpha$ in $(\mathbb{P},V)$ and
slopes $r/s\in\widehat{\mathbb{Q}}$ (see Lemma 2.3). We introduce the
additional symbol $\odot$ to represent peripheral Jordan curves in
$(\mathbb{P},V)$. We now define the slope map
$\mu_{f}\colon\widehat{\mathbb{Q}}\cup\\{\odot\\}\rightarrow\widehat{\mathbb{Q}}\cup\\{\odot\\}$
associated with $f$ as follows. We set $\mu_{f}(\odot)\coloneq\odot$. This
corresponds to the fact that each pullback of a peripheral Jordan curve
$\alpha$ in $(\mathbb{P},V)$ under $f$ is peripheral (see Corollary 3.5 (i)).
If $r/s\in\widehat{\mathbb{Q}}$ is an arbitrary slope, then we choose a Jordan
curve $\alpha$ in $(\mathbb{P},V)$ whose isotopy class $[\alpha]$ is
represented by $r/s$. If all pullbacks of $\alpha$ under $f$ are peripheral,
we set $\mu_{f}(r/s)\coloneq\odot$. Otherwise, there exists an essential
pullback $\widetilde{\alpha}$ of $\alpha$ under $f$. Then the isotopy class
$[\widetilde{\alpha}]$ is independent of the choice of the essential pullback
$\widetilde{\alpha}$ (see Corollary 3.5 (ii)) and so it is represented by a
unique slope $r^{\prime}/s^{\prime}\in\widehat{\mathbb{Q}}$. In this case, we
set $\mu_{f}(r/s)\coloneq r^{\prime}/s^{\prime}$. In this way,
$\mu_{f}(x)\in\widehat{\mathbb{Q}}\cup\\{\odot\\}$ is defined for all
$x\in\widehat{\mathbb{Q}}\cup\\{\odot\\}$. Since the map $\mu_{f}$ has the
same source and target, we can iterate it. If $n\in\mathbb{N}_{0}$, then we
denote by $\mu^{n}_{f}$ the $n$-th iterate of $\mu_{f}$. We will then prove
the following statement.

######  Theorem 1.4.

Let $f\colon\mathbb{P}\rightarrow\mathbb{P}$ be a Thurston map obtained from
the $(2\times 2)$-Lattès map by gluing $n_{h}\geq 1$ horizontal and $n_{v}\geq
1$ vertical flaps to the pillow $\mathbb{P}$. Then there exists a finite set
$S\subset\widehat{\mathbb{Q}}\cup\\{\odot\\}$ with the following property: for
each $x\in\widehat{\mathbb{Q}}\cup\\{\odot\\}$ there exists
$N\in\mathbb{N}_{0}$ such that $\mu^{n}_{f}(x)\in S$ for all $n\geq N$.

Note that $P_{f}=V$ in this case; so our previous considerations apply and the
map $\mu_{f}$ is defined. It is clear that the previous theorem leads to the
solution of the global curve attractor problem for the maps $f$ considered:

###### Proof of Theorem 1.3 based on Theorem 1.4.

To obtain a finite attractor $\mathscr{A}(f)$, pick a Jordan curve in each
isotopy class represented by a slope in $S$ and add five Jordan curves that
represent the isotopy classes of peripheral Jordan curves in $(\mathbb{P},V)$
(one for null-homotopic curves and one for each corner of $\mathbb{P}$). ∎

For the proof of Theorem 1.4 we will establish a certain monotonicity property
of the slope map $\mu_{f}$ for a map $f$ as in the statement (see Proposition
8.1). Roughly speaking, this monotonicity means that up to isotopy rel.
$P_{f}=V$ complicated essential Jordan curves in $(\mathbb{P},V)$ get
“simpler” and “less twisted” if we take successive preimages under $f$ and
eventually end up in the global curve attractor.

Our methods again rely on the consideration of intersection numbers. The
algebraic methods for solving the global curve attractor problem developed in
[Pil12] (specifically, [Pil12, Theorem 1.4]) do not apply in general for the
maps considered in Theorem 1.3 (see the discussion in Section 9.3).

Some of our ideas can also be used for the study of the global dynamics of the
slope map for Thurston maps that are not covered by Theorem 1.4. In
particular, we are able to describe the iterative behavior of $\mu_{f}$ for a
specific obstructed Thurston map $f$ obtained by blowing up the $(2\times
2)$-Lattès map (see Section 9.2 for the details). This provides an answer to a
question by Kevin Pilgrim (see [Pil21, Question 4.4]).

While it is straightforward to compute $\mu_{f}(x)$ for individual values
$x\in\widehat{\mathbb{Q}}\cup\\{\odot\\}$, we have been unable to give an
explicit formula for $\mu_{f}$ for the maps $f$ we consider. In general, these
slope maps show very complicated behavior. Currently, very few explicit
computations of slope maps are known in the literature. Except for some very
special situations (for example, when the slope map is constant, that is, when
$\mu_{f}(x)=\odot$ for all $x\in\widehat{\mathbb{Q}}\cup\\{\odot\\}$), we are
only aware of computations of slope maps for nearly Euclidian Thurston maps in
[CFPP12, Section 5] and [Lod13, Section 6]. See also [FPP18] for some general
properties of the slope map $\mu_{f}$.

An undergraduate student at UCLA, Darragh Glynn, performed some computer
experiments to compute $\mu_{f}$ for maps $f$ as in Theorem 1.2 for $n\geq 3$
(and $n_{h},n_{v}\geq 1$ corresponding to the rational case). His results show
that in these cases the map $\mu_{f}$ does not have the monotonicity property
as for $n=2$, but indicate that these maps $f$ still have a global curve
attractor (see Section 9.2 for more discussion).

###  1.3. Organization of this paper

Our paper is organized as follows. In the next two sections we review some
background. In Section 2, we fix notation and state some basic definitions. We
also discuss isotopy classes of Jordan curves in spheres with four marked
points, how isotopy classes of such curves correspond to slopes in
$\widehat{\mathbb{Q}}$, as well as some relevant facts about intersection
numbers. Even though all of this is fairly standard, we give complete proofs
in the appendix, because it is hard to track down this material in the
literature with a detailed exposition.

In Section 3 we recall some basics about Thurston maps and the relevant
concepts for a precise formulation of Thurston’s characterization of rational
maps for Thurston maps with four postcritical points—the only case relevant
for us (see Section 3.2).

We explain the blow-up procedure for arcs in Section 4 and relate this to the
procedure of gluing flaps to the pillow $\mathbb{P}$ (see Section 4.2). The
proof of Theorem 1.2 is then given in Section 5.

The proof of our main result, Theorem 1.1, requires more preparation. This is
the purpose of Section 6. There we introduce the concept of essential circuit
length that will allow us to formulate tight estimates for the number of
essential pullbacks of a Jordan curve under a Thurston map with four
postcritical points. This is formulated in the rather technical Lemma 6.2
which is of crucial importance though. The proof of Theorem 1.1 is then given
in Section 7.

Section 8 is devoted to the proof of Theorem 1.4. In Section 9, we discuss
some further directions related to this work. As we already mentioned, the
appendix is devoted to the discussion of isotopy classes and intersection
numbers of Jordan curves in spheres with four marked points.

Acknowledgments. The authors would like to thank Kostya Drach, Dima Dudko,
Daniel Meyer, Kevin Pilgrim, and Dylan Thurston for various useful comments
and remarks. We are grateful to Darragh Glynn for allowing us to incorporate
some of his numerical findings in this paper.

##  2\. Preliminaries

In this section, we discuss background relevant for the rest of the paper.

###  2.1. Notation and basic concepts

We denote by $\mathbb{N}=\\{1,2,\dots\\}$ the set of natural numbers and by
$\mathbb{N}_{0}=\\{0,1,2,\dots\\}$ the set of natural numbers including $0$.
The sets of integers, real numbers, and complex numbers are denoted by
$\mathbb{Z}$, $\mathbb{R}$, and $\mathbb{C}$, respectively. We write $i$ for
the imaginary unit in $\mathbb{C}$, and $\text{Im}(z)$ for the imaginary part
of a complex number $z\in\mathbb{C}$.

As usual, $\mathbb{R}^{2}\coloneq\\{(x,y):x,y\in\mathbb{R}\\}$ is the
Euclidean plane and $\widehat{\mathbb{C}}\coloneq\mathbb{C}\cup\\{\infty\\}$
is the Riemann sphere. Here and elsewhere, we write $A\coloneq B$ for emphasis
when an object $A$ is defined to be another object $B$. When we consider two
objects $A$ and $B$, and there is a natural identification between them that
is clear from the context, we write $A\cong B$. For example,
$\mathbb{R}^{2}\cong\mathbb{C}$ if we identify a point
$(x,y)\in\mathbb{R}^{2}$ with $x+iy\in\mathbb{C}$. We will freely switch back
and forth between these different viewpoints of
$\mathbb{R}^{2}\cong\mathbb{C}$.

We use the notation $\mathbb{I}\coloneq[0,1]\subset\mathbb{R}$ for the closed
unit interval, $\mathbb{D}\coloneq\\{z\in\mathbb{C}:|z|<1\\}$ for the open
unit disk in $\mathbb{C}$, and
$\mathbb{Z}^{2}\coloneq\\{x+iy:x,y\in\mathbb{Z}\\}$ for the square lattice in
$\mathbb{C}$. If $z,w\in\mathbb{C}$, then we write
$[z,w]\coloneq\\{z+t(w-z):t\in\mathbb{I}\\}$ for the line segment in
$\mathbb{C}$ joining $z$ and $w$. We also use the notation
$[z_{0},w_{0})\coloneq[z_{0},w_{0}]\setminus\\{w_{0}\\}$ and
$(z_{0},w_{0})\coloneq[z_{0},w_{0}]\setminus\\{z_{0},w_{0}\\}$.

The cardinality of a set $X$ is denoted by
$\\#X\in\mathbb{N}_{0}\cup\\{\infty\\}$ and the identity map on $X$ by
$\operatorname{id}_{X}$. If $X$ is a topological space and $M\subset X$, then
$\operatorname{cl}(M)$ denotes the closure, $\operatorname{int}(M)$ the
interior, and $\partial M$ the boundary of $M$ in $X$.

Let $f\colon X\to Y$ be a map between sets $X$ and $Y$. If $M\subset X$, then
$f|M$ stands for the restriction of $f$ to $M$. If $N\subset Y$, then
$f^{-1}(N)\coloneq\\{x\in X:f(x)\in N\\}$ is the preimage of $N$ in $X$.
Similarly, $f^{-1}(y)\coloneq\\{x\in X:f(x)=y\\}$ is the preimage of a point
$y\in Y$.

Let $f\colon X\rightarrow X$ be a map. For $n\in\mathbb{N}$ we denote by

$f^{n}\coloneq\underbrace{f\circ\dots\circ f}_{\text{$n$ factors}}$

the $n$-th iterate of $f$. It is convenient to define
$f^{0}\coloneq\operatorname{id}_{X}$. For $n\in\mathbb{N}_{0}$ we denote by
$f^{-n}(M)\coloneq\\{x\in X:f^{n}(x)\in M\\}$ and $f^{-n}(p)\coloneq\\{x\in
X:f^{n}(x)=p\\}$ the preimages of a set $M\subset X$ and a point $p\in X$
under $f^{n}$, respectively.

A surface $S$ is a connected and oriented topological $2$-manifold. We denote
its Euler characteristic by $\chi(S)$. Note that
$\chi(S)\in\\{2,1,0,-1,\dots\\}\cup\\{-\infty\\}$. Throughout this paper, we
use the notation $S^{2}$ for a (topological) $2$-sphere, that is, $S^{2}$
indicates a surface homeomorphic to the Riemann sphere $\widehat{\mathbb{C}}$.
An annulus is a surface homeomorphic to $\\{z\in\mathbb{C}:1<|z|<2\\}$.

A _Jordan curve $\alpha$_ in a surface $S$ is the image
$\alpha=\eta(\partial\mathbb{D})$ of a (topological) embedding
$\eta\colon\partial\mathbb{D}\rightarrow S$ of the unit circle
$\partial\mathbb{D}=\\{z\in\mathbb{C}:|z|=1\\}$ into $S$. An _arc $e$_ in $S$
is the image $e=\iota(\mathbb{I})$ of an embedding $\iota\colon\mathbb{I}\to
S$. Then $\iota(0)$ and $\iota(1)$ are the endpoints of $e$, and we define
$\partial e\coloneq\\{\iota(0),\iota(1)\\}$. The set
$\operatorname{int}(e)\coloneq e\setminus\partial e$ is called the _interior_
of $e$. The notions of endpoints and interior of $e$ only depend on $e$ and
not on the choice of the embedding $\iota$. Note that the notation $\partial
e$ and $\operatorname{int}(e)$ is ambiguous, because it should not be confused
with the boundary and interior of $e$ as a subset of $S$. For arcs $e$ in a
surface $S$, we will only use $\partial e$ and $\operatorname{int}(e)$ with
the meaning just defined.

A subset $U$ of a surface $S$ is called an open or closed Jordan region if
there exists a topological embedding
$\eta\colon\operatorname{cl}(\mathbb{D})=\\{z\in\mathbb{C}:|z|\leq
1\\}\rightarrow S$ such that $U=\eta(\mathbb{D})$ or
$U=\eta(\operatorname{cl}(\mathbb{D}))$, respectively. In both cases,
$\partial U=\eta(\partial\mathbb{D})$ is a Jordan curve in $S$. A crosscut $e$
in an open or closed Jordan region $U$ is an arc
$e\subset\operatorname{cl}(U)$ such that
$\operatorname{int}(e)\subset\operatorname{int}(U)$ and $\partial
e\subset\partial U$.

A path $\gamma$ in a surface $S$ is a continuous map
$\gamma\colon[a,b]\rightarrow S$, where $[a,b]\subset\mathbb{R}$ is a compact
(non-degenerate) interval. As is common, we will use the same notation
$\gamma$ for the image $\gamma([a,b])$ of the path if no confusion can arise.
The path $\gamma$ joins two sets $M,N\subset S$ if $\gamma(a)\in M$ and
$\gamma(b)\in N$, or vice versa. A loop in $S$ based at $p\in S$ is a path
$\gamma\colon[a,b]\rightarrow S$ such that $\gamma(a)=\gamma(b)=p$. The loop
$\gamma$ is called simple if $\gamma$ is injective on $[a,b)$. So essentially
a simple loop is a Jordan curve run through with some parametrization.

Let $M,N,K$ be subsets of a surface $S$. We say that $K$ separates $M$ and $N$
if every path in $S$ joining $M$ and $N$ meets $K$. Note that here $K$ is not
necessarily disjoint from $M$ or $N$. We say that $K$ separates a point $p\in
S$ from a set $M\subset S$ if $K$ separates $\\{p\\}$ and $M$.

Let $Z\subset S$ be a finite set of points in a surface $S$. Then we refer to
the pair $(S,Z)$ as a _marked surface_ , and the points in $Z$ as the _marked
points_ in $S$. The most important case for us will be when $S=S^{2}$ is a
$2$-sphere and $Z\subset S^{2}$ consists of four points.

A _Jordan curve $\alpha$ in a marked surface $(S,Z)$_ is a Jordan curve
$\alpha\subset S\setminus Z$. An _arc $e$ in $(S,Z)$_ is an arc $e\subset S$
with $\partial e\subset Z$ and $\operatorname{int}(e)\subset S\setminus Z$. We
say that a Jordan curve $\alpha$ in a marked sphere $(S^{2},Z)$ is _essential_
if each of the two connected components of $S^{2}\setminus\alpha$ contains at
least two points of $Z$; otherwise, we say that $\alpha$ is _peripheral_.

Let $(S^{2},Z)$ be a marked sphere with $\\#Z=4$. A _core arc_ of an essential
Jordan curve $\alpha$ in $(S^{2},Z)$ is an arc in $(S^{2},Z)$ that is
contained in one of the two connected components of $S^{2}\setminus\alpha$ and
joins the two points in $Z$ that lie in this component.

Let $A$ be an annulus. Then a core curve of $A$ is a Jordan curve
$\beta\subset A$ such that under some homeomorphism $\varphi\colon
A\rightarrow A^{\prime}$ the curve $\beta^{\prime}=\varphi(\beta)$ separates
the boundary components of $A^{\prime}=\\{z\in\mathbb{C}:1<|z|<2\\}$.

###  2.2. Branched covering maps

Let $X$ and $Y$ be surfaces. Then a continuous map $f\colon X\to Y$ is called
a branched covering map if for each point $q\in Y$ there exists an open set
$V\subset Y$ homeomorphic to $\mathbb{D}$ with $q\in V$ that is evenly covered
in the following sense: for some index set $J\neq\emptyset$ we can write
$f^{-1}(V)$ as a disjoint union

(2.1) $f^{-1}(V)=\bigcup_{j\in J}U_{j}$

of open sets $U_{j}\subset X$ such that $U_{j}$ contains precisely one point
$p_{j}\in f^{-1}(q)$. Moreover, we require that for each $j\in J$ there exists
$d_{j}\in\mathbb{N}$ and orientation-preserving homeomorphisms
$\varphi_{j}\colon U_{j}\rightarrow\mathbb{D}$ with $\varphi_{j}(p_{j})=0$ and
$\psi_{j}\colon V\rightarrow\mathbb{D}$ with $\psi_{j}(q)=0$ such that

$(\psi_{j}\circ f\circ\varphi_{j}^{-1})(z)=z^{d_{j}}$

for all $z\in\mathbb{D}$ (see [BM17, Section A.6] for more background on
branched covering maps). For given $f$, the number $d_{j}$ is uniquely
determined by $p=p_{j}$, and called the local degree of $f$ at $p$ and denoted
by $\deg(f,p)$. A point $p\in X$ with $\deg(f,p)\geq 2$ is called a _critical
point_ of $f$. The set of all critical points of $f$ is a discrete set in $X$
and denoted by $C_{f}$. If $f$ is a branched covering map, then it is a
covering map (in the usual sense) from $X\setminus
f^{-1}\big{(}f(C_{f})\big{)}$ onto $Y\setminus f(C_{f})$.

In the following, suppose $X$ and $Y$ are compact surfaces, and $f\colon
X\rightarrow Y$ is a branched covering map. Then $C_{f}\subset X$ is a finite
set. Moreover, if $\deg(f)\in\mathbb{N}$ denotes the topological degree of
$f$, then

$\sum_{p\in f^{-1}(q)}\deg(f,p)=\deg(f)$

for each $q\in Y$.

If $\gamma\colon[a,b]\rightarrow Y$ is a path, then we call a path
$\widetilde{\gamma}\colon[a,b]\rightarrow X$ a lift of $\gamma$ (under $f$) if
$f\circ\widetilde{\gamma}=\gamma$. Every path $\gamma$ in $Y$ has a lift
$\widetilde{\gamma}$ in $X$ (see [BM17, Lemma A.18]), but in general
$\widetilde{\gamma}$ is not unique. If $\gamma([a,b))\subset Y\setminus
f(C_{f})$ and $x_{0}\in f^{-1}(\gamma(a))$, then there exists a unique lift
$\widetilde{\gamma}\colon[a,b]\rightarrow X$ of $\gamma$ under $f$ with
$\widetilde{\gamma}(a)=x_{0}$. This easily follows from standard existence and
uniqueness theorems for lifts under covering maps (see [BM17, Lemma A.6]).

If $e\subset Y$ is an arc, then an arc $\widetilde{e}\subset X$ is called a
lift of $e$ (under $f$) if $f|\widetilde{e}$ is a homeomorphism of
$\widetilde{e}$ onto $e$. It easily follows from the existence and uniqueness
statements for lifts of paths just discussed that if $e$ is an arc in
$(Y,f(C_{f}))$, $y_{0}\in\operatorname{int}(e)$, and $x_{0}\in f^{-1}(y_{0})$,
then there exists a unique lift $\widetilde{e}\subset X$ of $e$ with
$x_{0}\in\widetilde{e}$.

Let $V\subset Y$ be an open and connected set and $U\subset f^{-1}(V)$ be a
(connected) component of $f^{-1}(V)$. Then $f|U\colon U\rightarrow V$ is also
a branched covering map. Each point $q\in V$ has the same number
$d\in\mathbb{N}$ of preimages under $f|U$ counting local degrees. We set
$\deg(f|U)\coloneq d$. If the Euler characteristic $\chi(V)$ is finite, then
$\chi(U)$ is also finite and we have the Riemann-Hurwitz formula

(2.2) $\chi(U)+\sum_{p\in U\cap C_{f}}(\deg(f,p)-1)=\deg(f|U)\cdot\chi(V).$

###  2.3. Planar embedded graphs

A _planar embedded graph_ in a sphere $S^{2}$ is a pair $G=(V,E)$, where $V$
is a finite set of points in $S^{2}$ and $E$ is a finite set of arcs in
$(S^{2},V)$ with pairwise disjoint interiors. The sets $V$ and $E$ are called
the _vertex_ and _edge sets_ of $G$, respectively. Note that our notion of a
planar embedded graph does not allow _loops_ , that is, edges that connect a
vertex to itself, but it does allow _multiple edges_ , that is, distinct edges
that join the same pair of vertices. The _degree_ of a vertex $v$ in $G$,
denoted $\deg_{G}(v)$, is the number of edges of $G$ incident to $v$. Note
that $2\cdot\\#E=\sum_{v\in V}\deg_{G}(v)$.

The _realization_ of $G$ is the subset $\mathcal{G}$ of $S^{2}$ given by

$\mathcal{G}\coloneq V\cup\bigcup_{e\in E}e.$

A _face_ of $G$ is a connected component of $S^{2}\setminus\mathcal{G}$.
Usually, we conflate a planar embedded graph $G$ with its realization
$\mathcal{G}$. Then it is understood that $\mathcal{G}$ contains a finite set
$V\subset\mathcal{G}$ of distinguished points that are the vertices of the
graph. Its edges are the closures of the components of $\mathcal{G}\setminus
V$.

A _subgraph_ of a planar embedded graph $G=(V,E)$ is a planar embedded graph
$G^{\prime}=(V^{\prime},E^{\prime})$ with $V^{\prime}\subset V$ and
$E^{\prime}\subset E$. A _path of length $n$ between vertices $v$ and
$v^{\prime}$_ in $G$ is a sequence
$v_{0},e_{0},v_{1},e_{1},\dots,e_{n-1},v_{n}$, where $v_{0}=v$,
$v_{n}=v^{\prime}$, and $e_{k}$ is an edge incident to the vertices $v_{k}$
and $v_{k+1}$ for $k=0,\dots,n-1$. A path that does not repeat vertices is
called a _simple path_.

A path $v_{0},e_{0},v_{1},e_{1},\dots,e_{n-1},v_{n}$ with $v_{0}=v_{n}$ and
$n\geq 2$ is called a _circuit of length $n$_ in $G$ and is denoted by
$(e_{0},e_{1},\dots,e_{n-1})$. Such a circuit is called a _simple cycle_ if
all vertices $v_{k}$, $k=0,\dots,n-1$, are distinct.

A planar embedded graph $G$ is called _connected_ if any two distinct vertices
of $G$ can be joined by a path in $G$. Equivalently, $G$ is connected if its
realization $\mathcal{G}$ is connected as a subset of $S^{2}$. Note that if
$G$ is connected, then each face of $G$ is simply-connected.

As follows from [Die05, Lemma 4.2.2], the topological boundary $\partial U$ of
each face $U$ of $G$ may be viewed as the realization of a subgraph of $G$.
Moreover, a walk around any connected component of the boundary $\partial U$
traces a circuit $(e_{0},e_{1},\dots,e_{n-1})$ in $G$ such that each edge of
$G$ appears zero, one, or two times in the sequence
$e_{0},e_{1},\dots,e_{n-1}$. We will say that the circuit
$(e_{0},e_{1},\dots,e_{n-1})$ _traces_ (a connected component of) the boundary
$\partial U$. If $U$ is simply-connected, then $\partial U$ is connected, and
the length of the (essentially unique) circuit that bounds $U$ is called the
_circuit length_ of $U$ in $G$.

A planar embedded graph $(V,E)$ is called bipartite if we can split $V$ into
two disjoint subsets $V_{1}$ and $V_{2}$ such that each edge $e\in E$ has one
endpoint in $V_{1}$ and one in $V_{2}$.

###  2.4. The Euclidean square pillow

As discussed in the introduction, we consider a square pillow $\mathbb{P}$
obtained from gluing two identical copies of the unit square
$\mathbb{I}^{2}\subset\mathbb{R}^{2}$ along their boundaries by the identity
map. Then $\mathbb{P}$ is a topological $2$-sphere. We equip $\mathbb{P}$ with
the induced path metric that agrees with the Euclidean metric on each of the
two copies of the unit square. We call this metric space $\mathbb{P}$ the
_Euclidean square pillow_. The vertices and edges of the unit square
$\mathbb{I}^{2}$ in $\mathbb{P}$ are called the _vertices_ and _edges_ of
$\mathbb{P}$. One copy of $\mathbb{I}^{2}$ in $\mathbb{P}$ is called the front
and the other copy the back side of $\mathbb{P}$. In a dynamical context, we
also refer to these two copies of $\mathbb{I}^{2}$ as the $0$-tiles of
$\mathbb{P}$. We color the front side of $\mathbb{P}$ white, and its back side
black. Finally, we equip $\mathbb{P}$ with the orientation that agrees with
the standard orientation on the front side $\mathbb{I}^{2}$ of $\mathbb{P}$
(represented by the positively-oriented standard flag
$((0,0),\mathbb{I}\times\\{0\\},\mathbb{I}^{2})$; see [BM17, Appendix A.4]).

We label the vertices and edges of $\mathbb{P}$ in counterclockwise order by
$A,B,C,D$ and $a,b,c,d$, respectively, so that $A\in\mathbb{P}$ corresponds to
the vertex $(0,0)\in\mathbb{I}^{2}$ and the edge $a\subset\mathbb{P}$
corresponds to $[0,1]\times\\{0\\}\subset\mathbb{I}^{2}$. Then $a$ has the
endpoints $A$ and $B$. We can view the boundary $\partial\mathbb{I}^{2}$ of
$\mathbb{I}^{2}$ as a planar embedded graph in $\mathbb{P}$ with the vertex
set $V\coloneq\\{A,B,C,D\\}$ and the edge set $E\coloneq\\{a,b,c,d\\}$. We
call $a$ and $c$ the _horizontal_ edges, and $b$ and $d$ the _vertical_ edges
of $\mathbb{P}$; see Figure 4.

$C$
---
$B$
---
$a$
---
$c$
---
$b$
---
$D$
---
$A$
---
$d$
---
Figure 4. The Euclidean square pillow $\mathbb{P}$.

The pillow $\mathbb{P}$ is an example of a _Euclidean polyhedral surface_ ,
that is, a surface obtained by gluing Euclidean polygons along boundary edges
by using isometries. Note that the metric on $\mathbb{P}$ is locally flat
except at its vertices, which are Euclidean conic singularities. So
$\mathbb{P}$ is an _orbifold_ (see, for example, [Mil06a, Appendix E] and
[BM17, Appendix A.9]).

An alternative description for the pillow $\mathbb{P}$ can be given as
follows. We consider the unit square
$\mathbb{I}^{2}\subset\mathbb{R}^{2}\cong\mathbb{C}$ and map it to the upper
half-plane in $\widehat{\mathbb{C}}$ by a conformal map, normalized so that
the vertices $0,1,1+i,i$ are mapped to $0,1,\infty,-1$, respectively. By
Schwarz reflection, this map can be extended to a meromorphic function
$\wp\colon\mathbb{C}\to\widehat{\mathbb{C}}$. Then $\wp$ is a _Weierstrass
$\wp$-function_ (up to a postcomposition with a Möbius transformation) that is
doubly periodic with respect to the lattice
$2\mathbb{Z}^{2}\coloneq\\{2k+2ni:k,n\in\mathbb{Z}\\}\subset\mathbb{C}$.
Actually, for $z,w\in\mathbb{C}$ we have

(2.3) $\wp(z)=\wp(w)\text{ if and only if }z-w\in 2\mathbb{Z}^{2}\text{ or
}z+w\in 2\mathbb{Z}^{2}.$

We can push forward the Euclidean metric on $\mathbb{C}$ to the Riemann sphere
$\widehat{\mathbb{C}}$ by $\wp$. With respect to this metric, called the
_canonical orbifold metric_ for $\wp$, the sphere $\widehat{\mathbb{C}}$ is
isometric to the Euclidean square pillow $\mathbb{P}$. In the following, we
identify the pillow $\mathbb{P}$ with $\widehat{\mathbb{C}}$ by the
orientation-preserving isometry that maps the vertices $A,B,C,D$ to
$0,1,\infty,-1$, respectively. Then we can consider
$\wp\colon\mathbb{C}\to\widehat{\mathbb{C}}\cong\mathbb{P}$ as a map onto the
pillow $\mathbb{P}$. Actually, $\wp$ is the _universal orbifold covering map_
for $\mathbb{P}$ (see [BM17, Section A.9] for more background). A very
intuitive description of this map can be given if we color the squares
$[k,k+1]\times[n,n+1]$, $k,n\in\mathbb{Z}$, in checkerboard manner black and
white so that $[0,1]\times[0,1]$ is white. Restricted to such a square $S$,
the map $\wp$ is an isometry that sends $S$ to the white $0$-tile of
$\mathbb{P}$ if $S$ is white, and to the black $0$-tile $\mathbb{P}$ if $S$ is
black; see Figure 5 for an illustration. Here, the points in the complex plane
$\mathbb{C}$ marked by a black dot (on the left) are mapped to $A$ by $\wp$
and are elements of $\wp^{-1}(A)=2\mathbb{Z}^{2}$.

$\mathbb{P}$
---
$\wp$
---
$A$
---
$\mathbb{C}$
---
Figure 5. The map $\wp\colon\mathbb{C}\to\mathbb{P}$.

###  2.5. Isotopies and intersection numbers

Let $X$ and $Y$ be topological spaces. Then a continuous map $H\colon
X\times\mathbb{I}\to Y$ is called a homotopy from $X$ to $Y$. For
$t\in\mathbb{I}$ we denote by $H_{t}\coloneq H(\cdot,t)\colon X\rightarrow Y$
the time-$t$ map of the homotopy. The homotopy $H$ is called an isotopy if
$H_{t}$ is a homeomorphism from $X$ onto $Y$ for each $t\in\mathbb{I}$. If
$Z\subset X$, then a homotopy $H\colon X\times\mathbb{I}\to Y$ is said to be a
_homotopy relative to $Z$_ (abbreviated “$H$ is a homotopy rel. $Z$”) if
$H_{t}(p)=H_{0}(p)$ for all $p\in Z$ and $t\in\mathbb{I}$. In other words, the
image of each point in $Z$ remains fixed during the homotopy $H$. Isotopies
rel. $Z$ are defined in a similar way.

Two homeomorphisms $h_{0},h_{1}\colon X\rightarrow Y$ are called _isotopic_
(_rel. $Z\subset X$_) if there exists an isotopy $H\colon X\times\mathbb{I}\to
Y$ (rel. $Z$) with $H_{0}=h_{0}$ and $H_{1}=h_{1}$. Given $M,N,Z\subset X$, we
say that _$M$ is isotopic to $N$ rel. $Z$_ (or _$M$ can be isotoped into $N$
rel. $Z$_), denoted by $M\sim N$ rel. $Z$, if there exists an isotopy $H\colon
X\times\mathbb{I}\to X$ rel. $Z$ with $H_{0}=\operatorname{id}_{X}$ and
$H_{1}(M)=N$. Recall that $\operatorname{id}_{X}$ is the identity map on $X$.

Let $(S,Z)$ be a marked surface (with a finite, possibly empty set $Z\subset
S$ of marked points). If $\alpha$ is a Jordan curve in $(S,Z)$, then its
isotopy class $[\alpha]$ (with $(S,Z)$ understood) consists of all Jordan
curves $\beta$ in $(S,Z)$ such that $\alpha\sim\beta$ rel. $Z$.

The following statement gives a sufficient condition for two Jordan curves in
$(S,Z)$ to be isotopic rel. $Z$.

######  Lemma 2.1.

Let $\alpha$ and $\beta$ be disjoint Jordan curves in a marked surface
$(S,Z)$. Suppose there is an annulus $U\subset S\setminus Z$ such that
$\partial U=\alpha\cup\beta$. Then $\alpha$ and $\beta$ are isotopic rel. $Z$.

###### Proof.

This is standard and we will only give a sketch of the proof. Since Jordan
curves in surfaces are tame, one can slightly enlarge the annulus $U$ to an
annulus $U^{\prime}\subset S^{2}\setminus Z$ that contains $\alpha$ and
$\beta$. Then $\alpha$ can be isotoped into $\beta$ by an isotopy on
$U^{\prime}$ that is the identity near $\partial U^{\prime}$. This isotopy on
$U^{\prime}$ can be extended to an isotopy on $S^{2}$ rel. $Z$ that isotopes
$\alpha$ into $\beta$. ∎

Let $(S,Z)$ be a marked surface. If $\alpha$ and $\beta$ are arcs or Jordan
curves in $(S,Z)$, we define their (unsigned) _intersection number_ as

$\mathrm{i}(\alpha,\beta)\coloneq\inf\\{\\#(\alpha^{\prime}\cap\beta^{\prime}):\text{$\alpha\sim\alpha^{\prime}$
rel.\ $Z$ and $\beta\sim\beta^{\prime}$ rel.\ $Z$}\\}.$

The relevant marked surface $(S,Z)$ here will be understood from the context,
and we suppress it from our notation for intersection numbers. If we want to
emphasize it, we will say that we consider intersection numbers in $(S,Z)$.
The intersection number is always finite, because we can always reduce to the
case when $\alpha$ and $\beta$ are piecewise geodesic with respect to some
Riemannian metric on $S$ (see [Bus10, Lemma A.8]). If $\alpha$ and $\beta$
satisfy $\mathrm{i}(\alpha\cap\beta)=\\#(\alpha\cap\beta)$, then we say that
$\alpha$ and $\beta$ are in minimal position (in their isotopy classes rel.
$Z$).

Suppose $\alpha$ and $\beta$ are arcs or Jordan curves in $(S,Z)$. Then we say
that $\alpha$ and $\beta$ meet transversely at a point
$p\in\alpha\cap\beta\cap(S\setminus Z)$ (or $\alpha$ crosses $\beta$ at $p$)
if $p$ is an isolated point in $\alpha\cap\beta$ and if the following
condition is true for a (small) arc $\sigma\subset\alpha$ containing $p$ as an
interior point such that $\sigma\cap\beta=\\{p\\}$: let $\sigma^{L}$ and
$\sigma^{R}$ be the two subarcs of $\sigma$ into which $\sigma$ is split by
$p$, then with suitable orientation of $\beta$ near $p$ the arc $\sigma^{L}$
lies to the left and $\sigma^{R}$ to the right of $\beta$. We say that
$\alpha$ and $\beta$ meet transversely or have transverse intersection if the
set $\alpha\cap\beta$ is finite and if $\alpha$ and $\beta$ meet transversely
at each point $p\in\alpha\cap\beta\cap(S\setminus Z)$.

######  Lemma 2.2.

Suppose $\alpha$ and $\beta$ are Jordan curves or arcs in a marked surface
$(S,Z)$. If $\alpha$ and $\beta$ are in minimal position, then $\alpha$ and
$\beta$ meet transversely.

###### Proof.

This is essentially a standard fact (see, for example, [Bus10, pp. 416–417]),
and we will only give an outline of the proof.

Since $\\#(\alpha\cap\beta)=\mathrm{i}(\alpha\cap\beta)$, the set
$\alpha\cap\beta$ consists of finitely many isolated points. To reach a
contradiction, suppose that $\alpha$ and $\beta$ do not meet transversely at
some point $p\in\alpha\cap\beta\cap(S\setminus Z)$. Then there exists an arc
$\sigma\subset\alpha$ containing $p$ as an interior point such that
$\sigma\cap\beta=\\{p\\}$ and with the following property: if $\sigma_{1}$ and
$\sigma_{2}$ denote the two subarcs of $\sigma$ into which $\sigma$ is split
by $p$, then $\sigma_{1}$ and $\sigma_{2}$ lie on the same side of $\beta$
(equipped with some orientation locally near $p$). In other words, $\alpha$
touches $\beta$ locally near $p$ from one side and does not cross $\beta$ at
$p$.

We can then modify the curve $\alpha$ near $p$ by an isotopy that pulls the
subarc $\sigma$ away from $\beta$ so that the new curve $\alpha$ does not have
the intersection point $p$ with $\beta$ while no new intersection points of
$\alpha$ and $\beta$ arise. This contradicts our assumption that for the
original curve $\alpha$ we have
$\\#(\alpha\cap\beta)=\mathrm{i}(\alpha\cap\beta)$. ∎

###  2.6. Jordan curves in spheres with four marked points

If $(S^{2},Z)$ is a marked sphere where $Z\subset S^{2}$ consists of exactly
four points, then, up to homeomorphism, we may assume that $S^{2}$ is equal to
the pillow $\mathbb{P}$, and $Z=V=\\{A,B,C,D\\}$ consists of the four vertices
of $\mathbb{P}$. We will freely switch back and forth between a general marked
sphere $(S^{2},Z)$ with $\\#Z=4$ and $(\mathbb{P},V)$.

We need some statements about isotopy classes of Jordan curves and arcs in
$(\mathbb{P},V)$ and their intersections numbers. They are “well known”, but
unfortunately we have been unable to track down a comprehensive account in the
literature. Accordingly, we will provide a complete treatment. This may be of
independent interest apart from the main objective of the paper. We will give
the statements in this section, but will provide the details of the proofs in
the appendix.

As we will see, there is a natural way to define a bijection between the set
of isotopy classes $[\gamma]$ of essential Jordan curves $\gamma$ in
$(\mathbb{P},V)$ and the set of _extended rational numbers_
$\widehat{\mathbb{Q}}\coloneq\mathbb{Q}\cup\\{\infty\\}$. Throughout this
paper, whenever we write $r/s\in\widehat{\mathbb{Q}}$, we assume that
$r\in\mathbb{Z}$ and $s\in\mathbb{N}_{0}$ are two relatively prime integers.
We allow $s=0$ here, in which case we assume $r=1$. Then
$r/s=1/0\coloneq\infty\in\widehat{\mathbb{Q}}$.

We say that a (straight) line $\ell\subset\mathbb{C}$ has slope
$r/s\in\widehat{\mathbb{Q}}$ if it is given as

$\ell=\\{z_{0}+(s+ir)t:t\in\mathbb{R}\\}\subset\mathbb{C}$

for some $z_{0}\in\mathbb{C}$. We use the notation $\ell_{r/s}(z_{0})$ for the
unique line in $\mathbb{C}$ with slope $r/s$ passing through
$z_{0}\in\mathbb{C}$, and the notation $\ell_{r/s}$ (when the point $z_{0}$ is
not important) for any line in $\mathbb{C}$ with slope $r/s$.

Let $\ell_{r/s}\subset\mathbb{C}$ be any line with slope
$r/s\in\widehat{\mathbb{Q}}$. If $\ell_{r/s}$ does not contain any point in
the lattice $\mathbb{Z}^{2}=\wp^{-1}(V)$ and so
$\ell_{r/s}\subset\mathbb{C}\setminus\mathbb{Z}^{2}$, then
$\tau_{r/s}\coloneq\wp(\ell_{r/s})$ is a Jordan curve in $\mathbb{P}\setminus
V$. Actually, $\tau_{r/s}$ is a simple closed geodesic in the Euclidean square
pillow $\mathbb{P}$ (see Figure 6 for an illustration). If $\ell_{r/s}$
contains a point in $\mathbb{Z}^{2}$, then $\xi_{r/s}\coloneq\wp(\ell_{r/s})$
is a geodesic arc in $(\mathbb{P},V)$.

It is easy to see that every simple closed geodesic or geodesic arc $\tau$ in
$(\mathbb{P},V)$ has the form $\tau=\wp(\ell_{r/s})$ for a line
$\ell_{r/s}\subset\mathbb{C}$ with some slope $r/s\in\widehat{\mathbb{Q}}$. In
the following, we use the notation $\tau_{r/s}$ for a simple closed geodesic
and $\xi_{r/s}$ for a geodesic arc obtained in this way.

It follows from (2.3) that for fixed $r/s\in\widehat{\mathbb{Q}}$ we obtain
precisely two distinct arcs $\xi_{r/s}$ and $\xi^{\prime}_{r/s}$ of the form
$\wp(\ell_{r/s}(z_{0}))$ depending on $z_{0}\in\mathbb{Z}^{2}$. For each
simple closed geodesic $\tau_{r/s}$ the arcs $\xi_{r/s}$ and
$\xi^{\prime}_{r/s}$ are core arcs of $\tau_{r/s}$ lying in different
components of $\mathbb{P}\setminus\tau_{r/s}$ (see the appendix for more
details). In particular, $\tau_{r/s}$ is always an essential Jordan curve in
$(\mathbb{P},V)$.

It turns out that the isotopy classes of essential Jordan curves in
$(\mathbb{P},V)$ are closely related to the simple closed geodesics
$\tau_{r/s}$.

$\mathbb{P}$
---
$\wp$
---
$A$
---
$\ell_{2}$
---
$\tau_{2}$
---
$\mathbb{C}$
---
Figure 6. A line $\ell_{2}$ and the corresponding Jordan curve
$\tau_{2}=\wp(\ell_{2})$ in $\mathbb{P}$.

######  Lemma 2.3.

Let $\gamma$ be an essential Jordan curve in $(\mathbb{P},V)$. Then there
exists a unique slope $r/s\in\widehat{\mathbb{Q}}$ with the following
property. Let $\ell_{r/s}$ be any line in $\mathbb{C}$ with slope $r/s$ and
$\ell_{r/s}\subset\mathbb{C}\setminus\mathbb{Z}^{2}$, and set
$\tau_{r/s}\coloneq\wp(\ell_{r/s})$. Then $\tau_{r/s}$ is an essential Jordan
curve in $(\mathbb{P},V)$ with $\gamma\sim\tau_{r/s}$ rel. $V$. Moreover, the
map $[\gamma]\mapsto r/s$ gives a bijection between isotopy classes $[\gamma]$
of essential Jordan curves $\gamma$ in $(\mathbb{P},V)$ and slopes
$r/s\in\widehat{\mathbb{Q}}$.

While this is well known (see, for example, [FM12, Proposition 2.6] or [KS94,
Proposition 2.1]), we find the available proofs too sketchy. This is the
reason why we provide a detailed proof in the appendix. Implicit in Lemma 2.3
is the fact that the isotopy class $[\tau_{r/s}]$ of
$\tau_{r/s}=\wp(\ell_{r/s})$ only depends on $r/s$ and not on the specific
choice of the line $\ell_{r/s}$ with
$\ell_{r/s}\subset\mathbb{C}\setminus\mathbb{Z}^{2}$ (see Lemma 10.7 for an
explicit statement).

Recall that $a$, $c$ denote the horizontal, and $b$, $d$ the vertical edges of
$\mathbb{P}$. In the following, we denote by $\alpha^{h}=\tau_{0}$ a
horizontal essential Jordan curve in $(\mathbb{P},V)$ (corresponding to slope
$0$ and separating the edges $a$ and $c$ of $\mathbb{P}$) and by
$\alpha^{v}=\tau_{\infty}$ a vertical essential Jordan curve in
$(\mathbb{P},V)$ (corresponding to slope $\infty$ and separating $b$ from
$d$). To be specific, we set

(2.4) $\alpha^{h}\coloneq\wp(\mathbb{R}\times\\{1/2\\})\text{ and
}\alpha^{v}\coloneq\wp(\\{1/2\\}\times\mathbb{R}).$

The following lemma summarizes the intersection properties of essential Jordan
curves and arcs in $(\mathbb{P},V)$.

######  Lemma 2.4.

Let $\alpha$ and $\beta$ be essential Jordan curves in $(\mathbb{P},V)$ and
$r/s,r^{\prime}/s^{\prime}\in\widehat{\mathbb{Q}}$ be the unique slopes such
that $\alpha\sim\tau_{r/s}$ and $\beta\sim\tau_{r^{\prime}/s^{\prime}}$ rel.
$V$, where $\tau_{r/s}$ and $\tau_{r^{\prime}/s^{\prime}}$ are simple closed
geodesics in $(\mathbb{P},V)$ with slopes $r/s$ and $r^{\prime}/s^{\prime}$,
respectively. Let $\xi$ be a core arc of $\beta$, and
$\xi_{r^{\prime}/s^{\prime}}$ be a geodesic arc in $(\mathbb{P},V)$ with slope
$r^{\prime}/s^{\prime}$. Then the following statements are true for
intersection numbers in $(\mathbb{P},V)$:

  1. (i)

If $r/s=r^{\prime}/s^{\prime}$, then $\mathrm{i}(\alpha,\beta)=0$, and if
$r/s\neq r^{\prime}/s^{\prime}$, then
$\mathrm{i}(\alpha,\beta)=\\#(\tau_{r/s}\cap\tau_{r^{\prime}/s^{\prime}})=2|rs^{\prime}-sr^{\prime}|>0$.

  2. (ii)

$\mathrm{i}(\alpha,\xi)=\\#(\tau_{r/s}\cap\xi_{r^{\prime}/s^{\prime}})=\frac{1}{2}\mathrm{i}(\alpha,\beta)=|rs^{\prime}-sr^{\prime}|$.

  3. (iii)

$\mathrm{i}(\alpha,a)=\\#(\tau_{r/s}\cap a)=|r|,\
\mathrm{i}(\alpha,c)=\\#(\tau_{r/s}\cap c)=|r|$.

  4. (iv)

$\mathrm{i}(\alpha,b)=\\#(\tau_{r/s}\cap b)=s,\
\mathrm{i}(\alpha,d)=\\#(\tau_{r/s}\cap d)=s$.

  5. (v)

$\mathrm{i}(\alpha,\alpha^{h})=2|r|$ and $\mathrm{i}(\alpha,\alpha^{v})=2s$.

$\tau_{2}$
---
$c$
---
$a$
---
$\alpha^{h}$
---
Figure 7. Counting intersections of $\tau_{2}$ with the horizontal curve
$\alpha^{h}$ and the horizontal edges $a$ and $c$.

We will prove this lemma in the appendix. Note that Figure 7 illustrates (iii)
and (v) when $\alpha=\tau_{2}$. It follows from the statement that
$\tau_{r/s}$ for $r/s\neq 0,\infty$ is in minimal position with each of the
curves $a,b,c,d,\alpha^{h},\alpha^{v}$.

Let $\gamma$ be a Jordan curve or an arc in a surface $S$, and
$M_{1},M_{2}\subset S$ be two disjoint sets with $0<\\#(\gamma\cap
M_{j})<\infty$ for $j=1,2$. We say that the points in $\gamma\cap
M_{1}\neq\emptyset$ and $\gamma\cap M_{2}\neq\emptyset$ alternate on $\gamma$
if any two points in one of the sets are separated by the other, that is, any
subarc $\sigma\subset\gamma$ with both endpoints in either of the sets
$\gamma\cap M_{1}$ or $\gamma\cap M_{2}$ must contain a point in the other
set.

More intuitively, this situation when the points in $\gamma\cap M_{1}$ and
$\gamma\cap M_{2}$ alternate on an arc $\gamma$ can be described as follows.
Suppose we traverse $\gamma$ in some (injective) parametrization starting from
one of its endpoints. Then we will first meet a point in either $M_{1}$ or
$M_{2}$, say in $M_{1}$. Then as we continue along $\gamma$, we will meet a
point in $M_{2}$, then a point in $M_{1}$, etc. A similar remark applies when
$\gamma$ is a Jordan curve. Note that is this case $\\#(\gamma\cap
M_{1})=\\#(\gamma\cap M_{2})$.

######  Lemma 2.5.

Let $\tau=\wp(\ell_{r/s})$ be a simple closed geodesic or a geodesic arc in
$(\mathbb{P},V)$ obtained from a line $\ell_{r/s}\subset\mathbb{C}$ with slope
$r/s\in\widehat{\mathbb{Q}}$. If $r/s\neq 0$, then the sets $a\cap\tau$ and
$c\cap\tau$ are non-empty and finite, and the points in $a\cap\tau$ and
$c\cap\tau$ alternate on $\tau$.

A similar statement is true if $r/s\neq\infty$ and we replace $a,c$ with
$b,d$, respectively. Lemma 2.5 is related to a similar statement in a more
general setting that is the key to proving Lemma 2.3 (see Lemma 10.3).

###### Proof.

Define $\omega\coloneq s+ir\in\mathbb{Z}^{2}$. Suppose first that
$\tau=\wp(\ell_{r/s})$ is a simple closed geodesic. Then
$\tau=\wp([z_{0},w_{0}])$, where
$z_{0}\in\ell_{r/s}\subset\mathbb{C}\setminus\mathbb{Z}^{2}$ and
$w_{0}=z_{0}+2\omega$, and the map $u\in[0,1]\mapsto\wp(uz_{0}+(1-u)w_{0})$
provides a parametrization of $\tau$ as a simple loop as follows from (2.3).

Now the set $\wp^{-1}(a\cup c)$ consists precisely of the lines
$\ell_{0}(ni)=\\{z\in\mathbb{C}:\text{Im}(z)=n\\}$, $n\in\mathbb{Z}$. These
lines alternate in the sense that $\wp$ maps $\ell_{0}(ni)$ onto $a$ or $c$
depending on whether $n\in\mathbb{Z}$ is even or odd, respectively. Since
$r/s\neq 0$, we have $r\in\mathbb{Z}\setminus\\{0\\}$, and so
$\text{Im}(w_{0}-z_{0})=\text{Im}(2\omega)=2r$ is a non-zero even integer.
This implies that the line segment $[z_{0},w_{0}]\subset\ell_{r/s}$ has non-
empty intersections (consisting of finitely many points) with each of the sets
$\wp^{-1}(a)$ and $\wp^{-1}(c)$. Moreover, the points in these intersections
alternate on the segment $[z_{0},w_{0}]$. From this together with the fact
that

$\\#(\wp^{-1}(a)\cap[z_{0},w_{0}))=|r|=\\#(\wp^{-1}(c)\cap[z_{0},w_{0}))$

the statement follows (the latter fact is needed to argue that the points in
$a\cap\tau$ and $c\cap\tau$ alternate on the simple closed geodesic $\tau$).

If $\tau$ is a geodesic arc, then there exists $z_{0}\in\mathbb{Z}^{2}$ such
that $\tau=\wp([z_{0},w_{0}])$, where $w_{0}=z_{0}+\omega$. The map $\wp$
sends $[z_{0},w_{0}]$ homeomorphically onto $\tau$. Again, $[z_{0},w_{0}]$ has
non-empty intersections consisting of finitely many points with each of the
sets $\wp^{-1}(a)$ and $\wp^{-1}(c)$. Moreover, the points in the sets
$\wp^{-1}(a)\cap[z_{0},w_{0}]\neq\emptyset$ and
$\wp^{-1}(c)\cap[z_{0},w_{0}]\neq\emptyset$ alternate on the segment
$[z_{0},w_{0}]$. The statement also follows in this case. ∎

We conclude this section with a statement related to the previous
considerations formulated for an arbitrary sphere with four marked points.

######  Lemma 2.6.

Let $(S^{2},Z)$ be a marked sphere with $\\#Z=4$, and $\alpha,\gamma$ be
essential Jordan curves in $(S^{2},Z)$. Suppose that $a_{\alpha}$ and
$c_{\alpha}$ are core arcs of $\alpha$ that lie in different components of
$S^{2}\setminus\alpha$. Then the following statements are true:

  1. (i)

$\mathrm{i}(\alpha,\gamma)=2\mathrm{i}(a_{\alpha},\gamma)=2\mathrm{i}(c_{\alpha},\gamma)$.

  2. (ii)

If $\mathrm{i}(\alpha,\gamma)>0$, then there exists a Jordan curve
$\gamma^{\prime}$ in $(S^{2},Z)$ with $\gamma^{\prime}\sim\gamma$ rel. $Z$
such that $\gamma^{\prime}$ is in minimal position with $\alpha$,
$a_{\alpha}$, $c_{\alpha}$ and the points in
$a_{\alpha}\cap\gamma^{\prime}\neq\emptyset$ and
$c_{\alpha}\cap\gamma^{\prime}\neq\emptyset$ alternate on $\gamma^{\prime}$.

###### Proof.

We may identify the marked sphere $(S^{2},Z)$ with the pillow $(\mathbb{P},V)$
by a homeomorphism that sends $\alpha$, $a_{\alpha}$, $c_{\alpha}$ to
$\alpha^{h}$, $a$, $c$, respectively. Then, by Lemma 2.3, the curve $\gamma$
is isotopic to a simple closed geodesic $\tau_{r/s}$ with slope
$r/s\in\widehat{\mathbb{Q}}$. Statement (i) then follows from Lemma 2.4 (iii)
and (v).

If $\mathrm{i}(\alpha,\gamma)=\mathrm{i}(\alpha^{h},\tau_{r/s})=2|r|>0$, then
we can choose $\gamma^{\prime}=\tau_{r/s}$ in (ii). Indeed, then
$\gamma^{\prime}=\tau_{r/s}\sim\gamma$ rel. $V=Z$, and
$\gamma^{\prime}=\tau_{r/s}$ is in minimal position with
$\alpha^{h}=\tau_{0}$, $a_{\alpha}=a$, $c_{\alpha}=c$ as follows from Lemma
2.4 (i) and (iii). Since $r/s\neq 0$ in this case, the statement about
alternation follows from Lemma 2.5. ∎

##  3\. Thurston maps

Here we provide a very brief summary of some relevant definitions and facts.
For more details we refer the reader to [BM17, Chapter 2].

Let $f\colon S^{2}\to S^{2}$ be a branched covering map of a topological
$2$-sphere $S^{2}$. A point $p\in S^{2}$ is called _periodic_ (for $f$) if
$f^{n}(p)=p$ for some $n\in\mathbb{N}$. Recall that $C_{f}$ denotes the set of
all critical points of $f$. The union

$P_{f}=\bigcup_{n\in\mathbb{N}}{f^{n}(C_{f})}$

of the orbits of critical points is called the _postcritical set_ of $f$. Note
that

$f(P_{f})\subset P_{f}\subset f^{-1}(P_{f}).$

The map $f$ is said to be _postcritically-finite_ if its postcritical set
$P_{f}$ is finite, in other words, if every critical point of $f$ has a finite
orbit under iteration.

######  Definition 3.1.

A _Thurston map_ is a postcritically-finite branched covering map $f\colon
S^{2}\to S^{2}$ of topological degree $\deg(f)\geq 2$.

Natural examples of Thurston maps are given by _rational Thurston maps_ , that
is, postcritically-finite rational maps on the Riemann sphere
$\widehat{\mathbb{C}}$.

The _ramification function_ of a Thurston map $f\colon S^{2}\to S^{2}$ is a
function $\alpha_{f}\colon S^{2}\to\mathbb{N}\cup\\{\infty\\}$ such that
$\alpha_{f}(p)$ for $p\in S^{2}$ is the lowest common multiple of all local
degrees $\deg(f^{n},q)$, where $q\in f^{-n}(p)$ and $n\in\mathbb{N}$ are
arbitrary. In particular, $\alpha_{f}(p)=1$ for $p\in S^{2}\setminus P_{f}$
and $\alpha_{f}(p)\geq 2$ for $p\in P_{f}$.

######  Definition 3.2.

Two Thurston maps $f\colon S^{2}\to S^{2}$ and
$g\colon\widehat{S}^{2}\to\widehat{S}^{2}$, where $\widehat{S}^{2}$ is another
topological $2$-sphere, are called _Thurston equivalent_ if there are
homeomorphisms $h_{0},h_{1}\colon S^{2}\to\widehat{S}^{2}$ that are isotopic
rel. $P_{f}$ such that $h_{0}\circ f=g\circ h_{1}$.

We say that a Thurston map is _realized_ (by a rational map) if it is Thurston
equivalent to a rational map. Otherwise, we say that it is _obstructed_.

The _orbifold_ $\mathcal{O}_{f}$ associated with a Thurston map $f$ is the
pair $(S^{2},\alpha_{f})$. The _Euler characteristic_ of $\mathcal{O}_{f}$ is

(3.1) $\chi(\mathcal{O}_{f})\coloneq 2-\sum_{p\in
P_{f}}\biggl{(}1-\frac{1}{\alpha_{f}(p)}\biggr{)}.$

Here we set $1/\infty\coloneq 0$.

The Euler characteristic of the orbifold $\mathcal{O}_{f}$ satisfies
$\chi(\mathcal{O}_{f})\leq 0$. We call $\mathcal{O}_{f}$ _hyperbolic_ if
$\chi(\mathcal{O}_{f})<0$, and _parabolic_ if $\chi(\mathcal{O}_{f})=0$.

If $f\colon S^{2}\rightarrow S^{2}$ is a Thurston map, then $f(C_{f}\cup
P_{f})\subset P_{f}$, which implies $C_{f}\cup P_{f}\subset f^{-1}(P_{f})$.
The reverse inclusion is related to the parabolicity of $\mathcal{O}_{f}$.

######  Lemma 3.3.

Let $f\colon S^{2}\rightarrow S^{2}$ be a Thurston map. If $f$ has a parabolic
orbifold, then $f^{-1}(P_{f})=C_{f}\cup P_{f}$. Moreover, conversely, if
$\\#P_{f}\geq 4$ and $f^{-1}(P_{f})\subset C_{f}\cup P_{f}$, then $f$ has a
parabolic orbifold.

The second part follows from [DH93, Lemma 2], but for the convenience of the
reader we will provide the simple proof. Here the assumption $\\#P_{f}\geq 4$
cannot be omitted as some examples with $\\#P_{f}=3$ show (such as the
Thurston map arising from the “barycentric subdivision rule”; see [BM17,
Example 12.21]).

###### Proof.

Let $\alpha_{f}$ be the ramification function of $f$. Then $p\in P_{f}$ if and
only if $\alpha_{f}(p)\geq 2$.

First suppose that $f$ has a parabolic orbifold. Then
$\alpha_{f}(q)\cdot\deg(f,q)=\alpha_{f}(f(q))$ for all $q\in S^{2}$ (see
[BM17, Proposition 2.14]). So if $q\in f^{-1}(P_{f})$, then $f(q)\in P_{f}$
which implies

$\alpha_{f}(q)\cdot\deg(f,q)=\alpha_{f}(f(q))\geq 2.$

This is only possible if $\alpha_{f}(q)\geq 2$ in which case $q\in P_{f}$, or
if $\deg(f,q)\geq 2$ in which case $q\in C_{f}$. Hence $q\in C_{f}\cup P_{f}$,
and so $f^{-1}(P_{f})\subset C_{f}\cup P_{f}$. Since the reverse inclusion is
true for all Thurston maps, we see that $f^{-1}(P_{f})=C_{f}\cup P_{f}$ if $f$
has a parabolic orbifold.

For the converse suppose that $f\colon S^{2}\rightarrow S^{2}$ is an arbitrary
Thurston map with $\\#P_{f}\geq 4$ and $f^{-1}(P_{f})\subset C_{f}\cup P_{f}$.
Let $d\coloneq\deg(f)\geq 2$. Note that $f^{-1}(P_{f})\subset(C_{f}\setminus
P_{f})\cup P_{f}$ by our hypotheses.

Each point $p\in S^{2}$ has precisely $d$ preimages counting multiplicities,
that is,

$d=\sum_{q\in f^{-1}(p)}\deg(f,q).$

Furthermore, since $C_{f}\subset f^{-1}(P_{f})$ and $\deg(f,q)\geq 2$ for
$q\in S^{2}$ if and only if $q\in C_{f}$, the Riemann-Hurwitz formula implies

$\displaystyle\\#(C_{f}\setminus P_{f})\leq\\#C_{f}$
$\displaystyle\leq\sum_{c\in C_{f}}(\deg(f,c)-1)=\sum_{q\in
f^{-1}(P_{f})}(\deg(f,q)-1)$ $\displaystyle=2d-2.$

It follows that

$\displaystyle d\cdot\\#P_{f}$ $\displaystyle=\sum_{q\in
f^{-1}(P_{f})}\deg(f,q)=\sum_{q\in
f^{-1}(P_{f})}(\deg(f,q)-1)+\\#f^{-1}(P_{f})$
$\displaystyle=(2d-2)+\\#f^{-1}(P_{f})\leq(2d-2)+\\#(C_{f}\setminus
P_{f})+\\#P_{f}$ $\displaystyle\leq 4(d-1)+\\#P_{f}.$

Hence $(d-1)\cdot\\#P_{f}\leq 4(d-1)$, and so $4\leq\\#P_{f}\leq 4$. This
implies $\\#P_{f}=4$ and that all the previous inequalities must be
equalities. In particular, $\\#(C_{f}\setminus P_{f})=2d-2$, which shows that
at all critical points the local degree of $f$ is equal to $2$ and no critical
points belong to $P_{f}$.

As a consequence, under iteration of $f$ the orbit of any point $p\in S^{2}$
passes through at most one critical point. It follows that we have
$\alpha_{f}(p)=1$ for $p\in S^{2}\setminus P_{f}$ and $\alpha_{f}(p)=2$ for
$p\in P_{f}$. This implies that the Euler characteristic (see (3.1)) of the
orbifold $\mathcal{O}_{f}$ associated with $f$ is equal to

$\chi(\mathcal{O}_{f})=2-\sum_{p\in
P_{f}}\left(1-\frac{1}{\alpha_{f}(p)}\right)=2-(1/2+1/2+1/2+1/2)=0.$

We conclude that $f$ has a parabolic orbifold. ∎

###  3.1. The $(n\times n)$–Lattès map

In general, a _Lattès map_ is a rational Thurston map with parabolic orbifold
that does not have periodic critical points. Here we provide the analytic
definition for the Lattès maps that we use in this paper and interpret this
from a more geometric perspective. See [Mil06b] and [BM17, Chapter 3] for a
general discussion of Lattès maps.

Let $\mathbb{P}\cong\widehat{\mathbb{C}}$ be the Euclidean square pillow and
$\wp\colon\mathbb{C}\to\mathbb{P}$ be its universal orbifold covering map, as
discussed in Section 2.4. Fix a natural number $n\geq 2$. It follows from
$\eqref{eq:wpeq}$ that there is a unique (and well-defined) map
$\mathcal{L}_{n}\colon\mathbb{P}\to\mathbb{P}$ such that

(3.2) $\mathcal{L}_{n}(\wp(z))=\wp(nz)\text{ for }z\in\mathbb{C}.$

We call $\mathcal{L}_{n}$ the _$(n\times n)$ -Lattès map_. In fact,
$\mathcal{L}_{n}$ is a rational map under the identification
$\mathbb{P}\cong\widehat{\mathbb{C}}$ as discussed in Section 2.4.

Alternatively, we can describe the map $\mathcal{L}_{n}$ in a combinatorial
fashion as follows. Recall that the front side of $\mathbb{P}$ is colored
white, and the back side black. These are the two $0$-tiles of $\mathbb{P}$,
and we subdivide each of them into $n^{2}$ squares of sidelength $1/n$. We
refer to these small squares as $1$-tiles (with $n$ understood), and color
them in a checkerboard fashion black and white so that the $1$-tile $S$ in the
white side of $\mathbb{P}$ with the vertex $A$ on its boundary is colored
white. We map $S$ to the white side of the pillow $\mathbb{P}$ by an
orientation-preserving Euclidean similarity (that scales by the factor $n$) so
that the vertex $A$ is fixed. If we extend this map by reflection to the whole
pillow, we get the $(n\times n)$-Lattès map $\mathcal{L}_{n}$ (see Figure 1
for $n=4$). The map $\mathcal{L}_{n}$ sends each black or white $1$-tile
homeomorphically (by a similarity) onto the $0$-tile in $\mathbb{P}$ of the
same color.

Based on this combinatorial description, it is easy to see that each critical
point of $\mathcal{L}_{n}$ has local degree $2$ and that the postcritical set
of $\mathcal{L}_{n}$ coincides with the set of vertices of $\mathbb{P}$, that
is, $P_{\mathcal{L}_{n}}=\\{A,B,C,D\\}=V$. One can also check that for the
ramification function of $\mathcal{L}_{n}$ we have
$\alpha_{\mathcal{L}_{n}}(p)=2$ for each $p\in P_{\mathcal{L}_{n}}$.
Substituting this into $\eqref{eq:orb-char}$, we see that
$\chi(\mathcal{O}_{\mathcal{L}_{n}})=0$. Thus, $\mathcal{L}_{n}$ has a
parabolic orbifold.

###  3.2. Thurston’s characterization of rational maps

Thurston maps can often be described from a combinatorial viewpoint as the
Lattès map $\mathcal{L}_{n}$ in Section 3.1 (see, for instance, [CFKP03] and
[BM17, Chapter 12]). The question whether a given Thurston map $f$ can be
realized by a rational map is usually difficult to answer except in some
special cases. William Thurston provided a sharp, purely topological criterion
that answers this question. The formulation and proof of this celebrated
result can be found in [DH93]. In this section, we introduce the necessary
concepts and formulate the result only when $\\#P_{f}=4$, which is the
relevant case for this paper.

In the following, let $f\colon S^{2}\to S^{2}$ be a Thurston map. The map $f$
defines a natural pullback operation on Jordan curves in $(S^{2},P_{f})$: a
_pullback_ of a Jordan curve $\gamma\subset S^{2}\setminus P_{f}$ under $f$ is
a connected component $\widetilde{\gamma}$ of $f^{-1}(\gamma)$. Since $f$ is a
covering map from $S^{2}\setminus f^{-1}(P_{f})$ onto $S^{2}\setminus P_{f}$,
each pullback $\widetilde{\gamma}$ of $\gamma$ is a Jordan curve in
$(S^{2},P_{f})$. Moreover,
$f|\widetilde{\gamma}\colon\widetilde{\gamma}\rightarrow\gamma$ is a covering
map. For some $k\in\mathbb{N}$ with $1\leq k\leq\deg(f)$ each point
$p\in\gamma$ has precisely $k$ distinct preimages in $\widetilde{\gamma}$.
Here $k$ is the (unsigned) mapping degree of $f|\widetilde{\gamma}$ which we
denote by $\deg(f\colon\widetilde{\gamma}\to\gamma)$.

Recall that a Jordan curve $\gamma\subset S^{2}\setminus P_{f}$ is called
_essential_ if each of the two connected components of $S^{2}\setminus\gamma$
contains at least two points from $P_{f}$, and is called _peripheral_
otherwise.

We will need the following standard facts.

######  Lemma 3.4.

Let $f\colon S^{2}\rightarrow S^{2}$ be a Thurston map and let $\gamma$ and
$\gamma^{\prime}$ be Jordan curves in $(S^{2},P_{f})$ with
$\gamma^{\prime}\sim\gamma$ rel. $P_{f}$. Then there is a bijection
$\widetilde{\gamma}\leftrightarrow\widetilde{\gamma}^{\prime}$ between the
pullbacks $\widetilde{\gamma}$ of $\gamma$ and the pullbacks
$\widetilde{\gamma}^{\prime}$ of $\gamma^{\prime}$ under $f$ such that for all
pullbacks corresponding under this bijection we have
$\widetilde{\gamma}\sim\widetilde{\gamma}^{\prime}$ rel. $P_{f}$ and
$\deg(f\colon\widetilde{\gamma}\rightarrow\gamma)=\deg(f\colon\widetilde{\gamma}^{\prime}\rightarrow\gamma^{\prime})$.

For the proof see [BM17, Lemma 6.9]. A consequence of this statement is that
the isotopy classes of curves in $f^{-1}(\gamma)$ rel. $P_{f}$ only depend on
the isotopy class $[\gamma]$ rel. $P_{f}$ and not on the specific choice of
$\gamma$.

######  Corollary 3.5.

Let $f\colon S^{2}\rightarrow S^{2}$ be a Thurston map, and $\gamma$ be Jordan
curve in $(S^{2},P_{f})$.

  1. (i)

If $\gamma$ is peripheral, then every pullback of $\gamma$ under $f$ is also
peripheral.

  2. (ii)

Suppose that $\\#P_{f}=4$ and let $\widetilde{\gamma}$ be a pullback of
$\gamma$ under $f$. If $\gamma$ and $\widetilde{\gamma}$ are essential, then
the isotopy class $[\widetilde{\gamma}]$ rel. $P_{f}$ only depends on the
isotopy class $[\gamma]$ rel. $P_{f}$ and not on the specific choice of
$\gamma$ and its essential pullback $\widetilde{\gamma}$.

###### Proof.

(i) Since $\gamma$ is peripheral, $\gamma$ can be isotoped (rel. $P_{f}$) into
a Jordan curve $\gamma^{\prime}$ inside a small open Jordan region $V\subset
S^{2}$ such that $\\#(V\cap P_{f})\leq 1$ and $V$ is evenly covered by the
branched covering map $f$ as in (2.1).

Then for each component $U_{j}$ of $f^{-1}(V)$, the map $f|U_{j}\colon
U_{j}\to V$ is given by $z\in\mathbb{D}\mapsto z^{d_{j}}\in\mathbb{D}$ for
some $d_{j}\in\mathbb{N}$ after orientation-preserving homeomorphic coordinate
changes in the source and target. This implies that $\\#(U_{j}\cap P_{f})\leq
1$ and that each pullback of $\gamma^{\prime}$ in $U_{j}$ is peripheral. Hence
all pullbacks of $\gamma^{\prime}$ under $f$ are peripheral and the same is
true for the pullbacks of $\gamma$ as follows from Lemma 3.4.

(ii) Suppose $\widetilde{\gamma}$ and $\widetilde{\gamma}^{\prime}$ are two
distinct essential pullbacks of $\gamma$ under $f$. Since these are components
of $f^{-1}(\gamma)$, the Jordan curves $\widetilde{\gamma}$ and
$\widetilde{\gamma}^{\prime}$ are disjoint. Then the set
$S^{2}\setminus(\widetilde{\gamma}\cup\widetilde{\gamma}^{\prime})$ is a
disjoint union
$S^{2}\setminus(\widetilde{\gamma}\cup\widetilde{\gamma}^{\prime})=V\cup U\cup
V^{\prime}$, where $V,V^{\prime}\subset S^{2}$ are Jordan regions and
$U\subset S^{2}$ is an annulus with $\partial
U=\widetilde{\gamma}\cup\widetilde{\gamma}^{\prime}$. Since
$\widetilde{\gamma}$ and $\widetilde{\gamma}^{\prime}$ are essential, both $V$
and $V^{\prime}$ must contain at least two postcritical points. Now
$\\#P_{f}=4$, and so $U\cap P_{f}=\emptyset$. Lemma 2.1 then implies that
$\widetilde{\gamma}$ and $\widetilde{\gamma}^{\prime}$ are isotopic rel.
$P_{f}$.

It follows that the isotopy class $[\widetilde{\gamma}]$ rel. $P_{f}$ does not
depend on the choice of the essential pullback $\widetilde{\gamma}$ of
$\gamma$. At the same time, Lemma 3.4 implies that $[\widetilde{\gamma}]$ only
depends on the isotopy class $[\gamma]$, as desired. ∎

For a general Thurston map $f$ the concept of an invariant multicurve is
important to decide whether $f$ is realized or obstructed. By definition, a
multicurve is a non-empty finite family $\Gamma$ of essential Jordan curves in
$S^{2}\setminus P_{f}$ that are pairwise disjoint and pairwise non-isotopic
rel. $P_{f}$.

Suppose now that $\\#P_{f}=4$. Then any two essential Jordan curves in
$S^{2}\setminus P_{f}$ are either isotopic rel. $P_{f}$ or have a non-empty
intersection (as follows from the argument in the proof of Corollary 3.5
(ii)). Thus, in this case each multicurve $\Gamma$ consists of a single
essential Jordan curve $\gamma$ in $S^{2}\setminus P_{f}$. We say that an
essential Jordan curve $\gamma$ in $S^{2}\setminus P_{f}$ is _$f$ -invariant_
if each essential pullback of $\gamma$ under $f$ is isotopic to $\gamma$ rel.
$P_{f}$.

######  Definition 3.6.

Let $f\colon S^{2}\to S^{2}$ be a Thurston map with $\\#P_{f}=4$ and let
$\gamma\subset S^{2}\setminus P_{f}$ be an essential $f$-invariant Jordan
curve. We denote by $\gamma_{1},\dots,\gamma_{n}$ for $n\in\mathbb{N}_{0}$ all
the essential pullbacks of $\gamma$ under $f$ and define

(3.3)
$\lambda_{f}(\gamma)\coloneq\sum_{j=1}^{n}\frac{1}{\deg(f\colon\gamma_{j}\to\gamma)}.$

Then $\gamma$ is called a _(Thurston) obstruction_ for $f$ if
$\lambda_{f}(\gamma)\geq 1$.

Note that if $n=0$, then the sum in (3.3) is the empty sum and so
$\lambda_{f}(\gamma)=0$. It immediately follows from Lemma 3.4 that whether or
not $\gamma$ is an obstruction for $f$ only depends on the isotopy class
$[\gamma]$ rel. $P_{f}$.

The following theorem gives a criterion when a Thurston map $f$ with
$\\#P_{f}=4$ is realized.

######  Theorem 3.7 (Thurston’s criterion).

Let $f\colon S^{2}\rightarrow S^{2}$ be a Thurston map with $\\#P_{f}=4$ and
suppose that $f$ has a hyperbolic orbifold. Then $f$ is realized by a rational
map if and only if $f$ has no obstruction.

With a suitable definition of an obstruction (as an invariant multicurve that
satisfies certain mapping properties) this statement is also true for general
Thurston maps with a hyperbolic orbifold; see [DH93] or [BM17, Section 2.6].

The example of the $(n\times n)$-Lattès map
$\mathcal{L}_{n}\colon\mathbb{P}\to\mathbb{P}$ with $n\geq 2$ shows that
Theorem 3.7 is false if $f$ has a parabolic orbifold. Indeed, let $\gamma$ be
any essential Jordan curve in $(\mathbb{P},P_{\mathcal{L}_{n}})$, where
$P_{\mathcal{L}_{n}}=V=\\{A,B,C,D\\}$ consists of the vertices of the pillow
$\mathbb{P}$. Since only the isotopy class $[\gamma]$ rel. $V$ matters, by
Lemma 2.3 we may assume without loss of generality that
$\gamma=\tau_{r/s}=\wp(\ell_{r/s}(z_{0}))$ with $z_{0}\in\mathbb{C}$,
$r/s\in\widehat{\mathbb{Q}}$, and
$\ell_{r/s}(z_{0})\subset\mathbb{C}\setminus\mathbb{Z}^{2}$. Here $r$ and $s$
are relatively prime integers and so there exist $p,q\in\mathbb{Z}$ such that
$pr+qs=1$. Let $\widetilde{\omega}\coloneq-p+iq$. Using (2.3) and (3.2) one
can verify that under $\mathcal{L}_{n}$ the curve $\gamma$ has exactly $n$
distinct pullbacks

(3.4) $\gamma_{j}=\wp(\ell_{r/s}((z_{0}+2j\widetilde{\omega})/n)),\quad
j=1,\dots,n.$

Moreover, each curve $\gamma_{j}$ is isotopic to $\gamma$ rel.
$P_{\mathcal{L}_{n}}=V$ and $\deg(\mathcal{L}_{n}\colon\gamma_{j}\to\gamma)=n$
for all $j=1,\dots,n$; see Figure 8 for an illustration. Thus,
$\lambda_{\mathcal{L}_{n}}(\gamma)=1$ and $\gamma$ is an obstruction.

$\mathcal{L}_{2}$
---
$\tau_{2}$
---
$\mathbb{P}$
---
$\mathbb{P}$
---
Figure 8. The two pullbacks of a curve $\gamma=\tau_{2}$ under
$\mathcal{L}_{2}$.

##  4\. Blowing up arcs

Here, we describe the operation of “blowing up arcs”, originally introduced by
Kevin Pilgrim and Tan Lei in [PL98, Section 2.5]. This operation allows us to
define and modify various Thurston maps and plays a crucial role in this
paper. We will first describe the general construction and then illustrate it
for Lattès maps. As we will explain, the procedure of “gluing a flap” that we
introduced in Section 1.1 can be viewed as a special case of blowing up arcs
for Lattès maps.

The construction of blowing up arcs will involve a finite collection $E$ of
arcs with pairwise disjoint interiors in a $2$-sphere $S^{2}$. We denote by
$V$ the set of endpoints of these arcs and consider $(V,E)$ as an embedded
graph in $S^{2}$. In the construction we will make various topological
choices. The following general statement guarantees that we do not run into
topological difficulties. In the formulation we equip $S^{2}$ with a “nice”
metric $d$ so that $(S^{2},d)$ is isometric to $\widehat{\mathbb{C}}$ carrying
the spherical metric with length element $ds=2|dz|/(1+|z|^{2})$.

######  Proposition 4.1.

Let $G=(V,E)$ be a planar embedded graph in $S^{2}$ and $\mathcal{G}\subset
S^{2}$ be its realization. Then there exists a planar embedded graph
$G^{\prime}=(V,E^{\prime})$ in $S^{2}$ with the same vertex set such that its
realization $\mathcal{G}^{\prime}$ is isotopic to $\mathcal{G}$ rel. $V$ and
such that each edge of $G^{\prime}$ is a piecewise geodesic arc in
$(S^{2},d)$.

An outline of the proof is given in [Bol79, Chapter I, §4]; the proposition
also follows from [Bus10, Lemma A.8].

###  4.1. The general construction

Before we provide a formal definition, we give some rough idea of how to “blow
up” arcs. In the following, $f\colon S^{2}\to S^{2}$ is a fixed Thurston map.
Let $e$ be an arc in $S^{2}$ such that the restriction $f|e$ is a
homeomorphism onto its image. We cut the sphere $S^{2}$ open along $e$ and
glue in a closed Jordan region $D$ along the boundary. In this way we obtain a
new $2$-sphere on which we can define a branched covering map $\widehat{f}$ as
follows: $\widehat{f}$ maps the complement of $\operatorname{int}(D)$ in the
same way as the original map $f$ and it maps $\operatorname{int}(D)$ to the
complement of $f(e)$ by a homeomorphism that matches the map $f|e$. We say
that $\widehat{f}$ is obtained from $f$ by blowing up the arc $e$ with
multiplicity $1$.

Now we proceed to give a rigorous definition of the blow-up operation in the
general case, where several arcs $e$ are blown up simultaneously with possibly
different multiplicitities $m_{e}\geq 1$ resulting in a new Thurston map
$\widehat{f}$. To this end, let $E$ be a finite set of arcs in
$(S^{2},f^{-1}(P_{f}))$ with pairwise disjoint interiors such that the
restriction $f|e\colon e\to f(e)$ is a homeomorphism for each $e\in E$. In
this case, we say that $E$ satisfies the _blow-up conditions_.

We assume that each arc $e\in E$ has an assigned multiplicity
$m_{e}\in\mathbb{N}$. Since each $e\in E$ is an arc in
$(S^{2},f^{-1}(P_{f}))$, its interior $\operatorname{int}(e)$ is disjoint from
$f^{-1}(P_{f})\supset P_{f}$ and so $\operatorname{int}(e)$ does not contain
any critical or postcritical point of $f$.

For each arc $e\in E$, we choose an open Jordan region $W_{e}\subset S^{2}$ so
that the following conditions hold:

  1. (A1)

The open Jordan regions $W_{e}$, $e\in E$, are pairwise disjoint.

  2. (A2)

For distinct arcs $e_{1},e_{2}\in E$, we have
$\operatorname{cl}{(W_{e_{1}})}\cap\operatorname{cl}({W_{e_{2}})}=\partial
e_{1}\cap\partial e_{2}$.

  3. (A3)

$\operatorname{int}(e)\subset W_{e}$ and $\partial e\subset\partial W_{e}$ for
each $e\in E$.

  4. (A4)

$\operatorname{cl}(W_{e})\cap f^{-1}(P_{f})=e\cap f^{-1}(P_{f})=\partial e$
for each $e\in E$.

  5. (A5)

$f|\operatorname{cl}(W_{e})$ is a homeomorphism onto its image for each $e\in
E$.

The existence of such a choice (and also of the choices below) can easily be
justified based on Propositon 4.1 and we will skip the details.

Let $e\in E$ and $W_{e}$ be chosen as above. Then we choose a closed Jordan
region $D_{e}$ so that $e$ is a crosscut in $D_{e}$ and
$D_{e}\setminus\partial e\subset W_{e}$. The two endpoints of $e$ lie on the
Jordan curve $\partial D_{e}$ and partition it into two arcs, which we denote
by $\partial D_{e}^{+}$ and $\partial D_{e}^{-}$. One can think of $D_{e}$ as
the resulting region if $e$ has been “opened up”. This is illustrated in the
left and middle parts of Figure 9.

$e_{2}\subset W_{e_{2}}$
---
$e_{1}\subset W_{e_{1}}$
---
$D_{e_{1}}\subset W_{e_{1}}$
---
$D_{e_{2}}\subset W_{e_{2}}$
---
$D^{2}_{e_{2}}$
---
$D^{1}_{e_{1}}$
---
Choose regions $W_{e}\supset e$
---
Open up the arcs $e$
---
to regions $D_{e}\subset W_{e}$
Subdivide each $D_{e}$ into
---
regions $D_{e}^{1},\dots,D_{e}^{m}$
$D^{1}_{e_{2}}$
---
Figure 9. Setup for blowing up the arcs $e_{1}$ and $e_{2}$ (in the sphere on
the left) with the multiplicities $m_{e_{1}}=1$ and $m_{e_{2}}=2$.

In order to define the desired Thurston map $\widehat{f}$, we want to collapse
$D_{e}$ back to $e$. For this we choose a continuous map $h\colon
S^{2}\times\mathbb{I}\to S^{2}$ with the following properties:

  1. (B1)

$h$ is a pseudo-isotopy, that is, $h_{t}\coloneq h(\cdot,t)$ is a
homeomorphism on $S^{2}$ for each $t\in[0,1)$.

  2. (B2)

$h_{0}$ is the identity map on $S^{2}$.

  3. (B3)

$h_{t}$ is the identity map on $S^{2}\setminus\bigcup_{e\in E}{W_{e}}$ for
each $t\in[0,1]$.

  4. (B4)

$h_{1}$ is a homeomorphism of $S^{2}\setminus\bigcup_{e\in E}{D_{e}}$ onto
$S^{2}\setminus\bigcup_{e\in E}{e}$, and $h_{1}$ maps $\partial D_{e}^{+}$ and
$\partial D_{e}^{-}$ homeomorphically onto $e$ for each $e\in E$.

It is easy to see that if we equip $S^{2}$ with some metric, then the set
$h_{t}(D_{e})$ Hausdorff converges to $e$ as $t\to 1^{-}$. This implies that
$h_{1}(D_{e})=e$. So intuitively, the deformation process described by $h$
collapses each closed Jordan region $D_{e}$ to $e$ at time $1$ so that the
points in $S^{2}\setminus\bigcup_{e\in E}{W_{e}}$ remain fixed.

We now make yet another choice. For a fixed arc $e\in E$, let $m=m_{e}$. We
choose $m-1$ crosscuts $e^{1},\dots,e^{m-1}$ in $D_{e}$ with the same
endpoints as $e$ such that these crosscuts have pairwise disjoint interiors.
We set $e^{0}\coloneq\partial D_{e}^{+}$ and $e^{m}\coloneq\partial
D_{e}^{-}$. The arcs $e^{0},\dots,e^{m}$ subdivide the closed Jordan region
$D_{e}$ into $m$ closed Jordan regions $D_{e}^{1},\dots,D_{e}^{m}$, called
_components_ of $D_{e}$. This is illustrated in the right-hand part of Figure
9.

We may assume that the labeling is such that $\partial D_{e}^{k}=e^{k-1}\cup
e^{k}$ for $k=1,\dots,m$. For each $k=1,\dots,m$, we now choose a continuous
map $\varphi_{k}\colon D_{e}^{k}\rightarrow S^{2}$ with the following
properties:

  1. (C1)

$\varphi_{k}$ is an orientation-preserving homeomorphism of
$\operatorname{int}(D_{e}^{k})$ onto $S^{2}\setminus f(e)$ and maps $e^{k-1}$
and $e^{k}$ homeomorphically onto $f(e)$.

  2. (C2)

$\varphi_{1}|e^{0}=f\circ h_{1}|e^{0}$, $\varphi_{m}|e^{m}=f\circ
h_{1}|e^{m}$, and $\varphi_{k}|e^{k}=\varphi_{k+1}|e^{k}$ for $k=1,\dots,m-1$.

Note that, by the earlier discussion, $h_{1}$ maps $e^{0}=\partial D^{+}_{e}$
and $e^{m}=\partial D^{-}_{e}$ homeomorphically onto $e$ and $f$ is a
homeomorphism of $e$ onto $f(e)$. These choices of the maps $\varphi_{k}$
depend on $e$, but we suppress this in our notation for simplicity.

A map $\widehat{f}\colon S^{2}\to S^{2}$ can now be defined as follows:

  1. $D^{2}_{e_{2}}$
---
$D^{1}_{e_{1}}$
---
$f(e_{1})$
---
$f(e_{2})$
---
$f$
---
$\widehat{f}$
---
$h_{1}$
---
$D^{1}_{e_{2}}$
---
$e_{2}$
---
$e_{1}$
---
Figure 10. The map $\widehat{f}$ is obtained from $f$ by blowing up the arcs
$e_{1}$ and $e_{2}$ with multiplicities $m_{e_{1}}=1$ and $m_{e_{2}}=2$.

  2. (D1)

If $p\in S^{2}\setminus\bigcup_{e\in E}\operatorname{int}(D_{e})$, we set
$\widehat{f}(p)=f(h_{1}(p))$.

  3. (D2)

If $p\in D_{e}$ for some $e\in E$, then $p$ lies in one of the components
$D_{e}^{k}$ of $D_{e}$ and we set $\widehat{f}(p)=\varphi_{k}(p)$.

The matching conditions (C2) above immediately imply that $\widehat{f}$ is
well-defined and continuous.

######  Definition 4.2.

We say that the map $\widehat{f}\colon S^{2}\rightarrow S^{2}$ as described
above is obtained from the Thurston map $f$ by _blowing up each arc $e\in E$
with multiplicity $m_{e}$_.

Figure 10 illustrates the construction of $\widehat{f}$. Here, we blow up the
arcs $e_{1}$ and $e_{2}$ from Figure 9 with multiplicities $m_{e_{1}}=1$ and
$m_{e_{2}}=2$. The arcs $f(e_{1})$ and $f(e_{2})$ share an endpoint (since
$e_{1}$ and $e_{2}$ do), but in general they could have more points in common
or even coincide. For simplicity, we chose to draw them with disjoint
interiors.

By construction, $\widehat{f}$ acts in a similar way as $f$ outside the closed
Jordan regions $D_{e_{1}}=D^{1}_{e_{1}}$ and $D_{e_{2}}=D^{1}_{e_{2}}\cup
D^{2}_{e_{2}}$. More precisely, $\widehat{f}=f\circ h_{1}$ on
$S^{2}\setminus(\operatorname{int}(D_{e_{1}})\cup\operatorname{int}(D_{e_{2}}))$,
where $h_{1}$ collapses the closed Jordan regions $D_{e_{1}}$ and $D_{e_{2}}$
onto $e_{1}$ and $e_{2}$, respectively. At the same time, $\widehat{f}$ maps
$\operatorname{int}(D_{e_{1}})$ homeomorphically onto $S^{2}\setminus
f(e_{1})$, and each of the regions $\operatorname{int}(D^{1}_{e_{2}})$ and
$\operatorname{int}(D^{2}_{e_{2}})$ homeomorphically onto $S^{2}\setminus
f(e_{2})$.

$e_{1}$
---
$e_{2}$
---
$D_{e_{2}}\subset W_{e_{2}}$
---
$D_{e_{1}}\subset W_{e_{1}}$
---
$D^{1}_{e_{2}}$
---
$D^{2}_{e_{2}}$
---
$D^{1}_{e_{1}}$
---
Figure 11. Setup for blowing up the arc set $E=\\{e_{1},e_{2}\\}$ (on the left
pillow) with $m_{e_{1}}=1$ and $m_{e_{2}}=2$. $\widehat{f}$
---
$\mathbb{P}$
---
$\mathbb{P}$
---
Figure 12. The map $\widehat{f}$ obtained from $f=\mathcal{L}_{2}$ by blowing
up the arcs in the set $E=\\{e_{1},e_{2}\\}$ illustrated in Figure 11 with
$m_{e_{1}}=1$ and $m_{e_{2}}=2$.

In the next section, we want to relate “blowing up arcs” with “gluing flaps”
as discussed in the introduction. In order to set this up, we consider the
$(2\times 2)$-Lattès map $f=\mathcal{L}_{2}$. We choose two edges $e_{1}$ and
$e_{2}$ of a $1$-tile in $\mathbb{P}$ as shown in the pillow on the left in
Figure 11. Note that $f$ sends $e_{1}$ and $e_{2}$ homeomorphically onto the
edges $c$ and $b$ of $\mathbb{P}$, respectively. Thus the set
$E=\\{e_{1},e_{2}\\}$ satisfies the blow-up conditions. Figure 11 illustrates
the setup for blowing up these arcs $e_{1}$ and $e_{2}$ with the
multiplicities $m_{e_{1}}=1$ and $m_{e_{2}}=2$. The resulting map
$\widehat{f}\colon\mathbb{P}\to\mathbb{P}$ is shown in Figure 12. The points
marked by a dot on the left pillow $\mathbb{P}$ (the domain of the map)
correspond to the preimage points $\widehat{f}^{-1}(V)$. The pillow on the
left is subdivided into closed Jordan regions alternately colored black and
white. The map $\widehat{f}$ sends each of these closed Jordan regions $U$
homeomorphically onto the back side or front side of the pillow $\mathbb{P}$
depending on whether $U$ is black or white.

The following statement summarizes the main properties of maps $\widehat{f}$
as in Definition 4.2.

######  Lemma 4.3.

Let $f\colon S^{2}\rightarrow S^{2}$ be a Thurston map and $E$ be a set of
arcs in $(S^{2},f^{-1}(P_{f}))$ satisfying the blow-up conditions. Suppose
$\widehat{f}\colon S^{2}\rightarrow S^{2}$ is the map obtained by blowing up
each arc $e\in E$ with multiplicity $m_{e}\in\mathbb{N}$.

Then $\widehat{f}$ is a Thurston map with $P_{\widehat{f}}=P_{f}$. Moreover,
the map $\widehat{f}$ is uniquely determined up to Thurston equivalence
independently of the choices in the above construction. More precisely, up to
Thurston equivalence $\widehat{f}$ depends only on the original map $f$, the
isotopy classes of the arcs in $E$ rel. $f^{-1}(P_{f})$, and the
multiplicities $m_{e}$ for $e\in E$.

###### Proof.

By construction $\widehat{f}$ is an orientation-preserving local homeomorphism
near each point $p\in S^{2}\setminus f^{-1}(P_{f})$. By considering the number
of preimages of a generic point in $S^{2}$ we see that the topological degree
of $\widehat{f}$ is equal to $\deg(f)+\sum_{e\in E}m_{e}>0.$ The fact that
$\widehat{f}\colon S^{2}\to S^{2}$ is a branched covering map can now be
deduced from [BM17, Corollary A.14].

We have $\deg(\widehat{f},p)=1$ for $p\in S^{2}\setminus f^{-1}(P_{f})$ and

$\deg(\widehat{f},p)=\deg(f,p)+\sum_{\\{e\in E:\ p\in e\\}}m_{e}$

for $p\in f^{-1}(P_{f})$. This implies $C_{f}\subset C_{\widehat{f}}\subset
f^{-1}(P_{f})$. Since on the set $f^{-1}(P_{f})$ the maps $f$ and
$\widehat{f}$ agree, it follows that $P_{\widehat{f}}=P_{f}$. So $\widehat{f}$
has a finite postcritical set, and we conclude that $\widehat{f}$ is indeed a
Thurston map.

We omit a detailed justification of the second claim that $\widehat{f}$ is
uniquely determined up to Thurston equivalence by $f$, the isotopy classes of
the arcs $E$, and their multiplicities. A proof can be given along the lines
of [PL98, Proposition 2]. ∎

######  Remark 4.4.

If in the previous statement $E\neq\emptyset$ and $P_{f}\geq 3$, then
$\widehat{f}$ has a hyperbolic orbifold. To see this, pick an arc $e\in E$.
Then $f(e)$ has at most two points in common with $P_{f}$, and so we can find
a point $p\in P_{f}\setminus f(e)\subset P_{\widehat{f}}$. Then it follows
from the construction of $\widehat{f}$ that there exists a point $q$ in the
interior of the region $D_{e}$ associated with $e$ such that
$\widehat{f}(q)=p$ and $\deg(\widehat{f},q)=1$. Then $q\not\in
C_{\widehat{f}}$, but we also have

$q\in\operatorname{int}(D_{e})\subset S^{2}\setminus f^{-1}(P_{f})\subset
S^{2}\setminus P_{f}=S^{2}\setminus P_{\widehat{f}}.$

This shows that
$q\in\widehat{f}^{-1}(p)\subset\widehat{f}^{-1}(P_{\widehat{f}})$, but
$q\not\in C_{\widehat{f}}\cup P_{\widehat{f}}$, and so $\widehat{f}$ must have
a hyperbolic orbifold by the first part of Lemma 3.3.

###  4.2. Blowing up the $(n\times n)$-Lattès map

Let $\mathbb{P}$ be the Euclidean square pillow and
$\mathcal{L}_{n}\colon\mathbb{P}\to\mathbb{P}$ be the $(n\times n)$-Lattès map
for fixed $n\geq 2$. We denote by $\mathcal{C}\subset\mathbb{P}$ the common
boundary of the two sides of $\mathbb{P}$. The set $\mathcal{C}$ may be viewed
as a planar embedded graph with the vertex set $V=\\{A,B,C,D\\}$ and the edge
set $\\{a,b,c,d\\}$ in the notation from Section 2.4. Let
$\widetilde{\mathcal{C}}\coloneq\mathcal{L}_{n}^{-1}(\mathcal{C})\subset\mathbb{P}$
be the preimage of $\mathcal{C}$ under $\mathcal{L}_{n}$, viewed as a planar
embedded graph with the vertex set $\mathcal{L}_{n}^{-1}(V)$. In the next
section, we will study the question whether a Thurston map is realized by a
rational map if it is obtained from $\mathcal{L}_{n}$ by blowing up edges of
$\widetilde{\mathcal{C}}$. In order to facilitate this discussion, we will
provide a more concrete combinatorial model for these maps.

By the definition of the map $\mathcal{L}_{n}$, the graph
$\widetilde{\mathcal{C}}$ subdivides the pillow $\mathbb{P}$ into $2n^{2}$
$1$-tiles, which are squares of sidelength $1/n$. The edges of the embedded
graph $\widetilde{\mathcal{C}}$ are precisely the sides of these squares. We
call them the $1$-edges of $\mathbb{P}$ (for given $n$). The map
$\mathcal{L}_{n}$ sends each $1$-edge $e$ of $\mathbb{P}$ homeomorphically
onto one of the edges $a$, $b$, $c$, $d$ of $\mathcal{C}$. We call $e$
horizontal if $\mathcal{L}_{n}$ maps it onto $a$ or $c$, and vertical if
$\mathcal{L}_{n}$ maps it onto $b$ or $d$.

We take two disjoint copies of the Euclidean square
$[0,1/n]^{2}\subset\mathbb{R}^{2}$ and identify the points on three of their
sides, say the sides $\\{0\\}\times[0,1/n]$, $[0,1/n]\times\\{1/n\\}$, and
$\\{1/n\\}\times[0,1/n]$. We call the object obtained a _flap_ $F$. Note that
it is homeomorphic to the closed unit disk and has two “free” sides
corresponding to the two copies of $[0,1/n]\times\\{0\\}$ in $F$.

We can cut the pillow $\mathbb{P}$ open along one of the edges of
$\widetilde{\mathcal{C}}$ and glue in a flap $F$ to the pillow by identifying
each copy of $[0,1/n]\times\\{0\\}$ in the flap with one side of the slit by
an isometry (see Figure 2). In this way, we get a new polyhedral surface
homeomorphic to $S^{2}$. One can also glue multiple copies of the flap to the
slit by an isometry and obtain a polyhedral surface $\widehat{\mathbb{P}}$
homeomorphic to $S^{2}$. This can be described more concretely as follows. Let
$e$ be an edge in $\widetilde{\mathcal{C}}$ and $F_{1},\dots,F_{m}$ be $m\geq
1$ copies of the flap. For each $k=1,\dots,m$, we denote the two copies of
$[0,1/n]\times\\{0\\}$ in the flap $F_{k}$ by $e_{k}^{\prime}$ and
$e_{k}^{\prime\prime}$. We now construct a new polyhedral surface
$\widehat{\mathbb{P}}$ in the following way:

  1. (i)

First, we cut the original pillow $\mathbb{P}$ open along the edge $e$.

  2. (ii)

Then, for each $k=1,\dots,m-1$, we identify the edge $e_{k}^{\prime\prime}$ of
$F_{k}$ with the edge $e_{k+1}^{\prime}$ of $F_{k+1}$ by an isometry. We get a
polyhedral surface $D_{e}$ homeomorphic to a closed disk, whose boundary
consists of two edges $e_{1}^{\prime}$ and $e_{m}^{\prime\prime}$.

  3. (iii)

Finally, we glue the disk $D_{e}$ to the pillow $\mathbb{P}$ cut open along
$e$ by identifying the edges $e_{1}^{\prime}$ and $e_{m}^{\prime\prime}$ in
$\partial D_{e}$ with the two sides of the slit by an isometry so that
$e_{1}^{\prime}$ and $e_{m}^{\prime\prime}$ are identified with different
sides of the slit. We obtain a polyhedral surface $\widehat{\mathbb{P}}$ that
is homeomorphic to a $2$-sphere.

More generally, we can cut open $\mathbb{P}$ simultaneously along several
edges $e$ of $\widetilde{\mathcal{C}}$ and, by the method described, glue
$m_{e}\in\mathbb{N}$ copies of the flap to the slit obtained from each edge
$e$. If these edges $e$ of $\widetilde{\mathcal{C}}$ with their multiplicities
$m_{e}$ are given, then there is essentially only one way of gluing flaps so
that the resulting object is a polyhedral surface homeomorphic to $S^{2}$.

Let $\widehat{\mathbb{P}}$ be the polyhedral surface obtained from
$\mathbb{P}$ by gluing a total number of $n_{h}\geq 0$ horizontal flaps (i.e.,
flaps glued along horizontal edges of $\widetilde{\mathcal{C}}$) and a total
number of $n_{v}\geq 0$ vertical flaps (i.e., flaps glued along vertical edges
of $\widetilde{\mathcal{C}}$). We call this surface a _flapped pillow_. We
denote by $E$ the set of all edges in $\widetilde{\mathcal{C}}$ along which
flaps were glued and by $m_{e}$, $e\in E$, the corresponding multiplicities.
See the left part of Figure 13 for an example of a flapped pillow
$\widehat{\mathbb{P}}$ obtained by gluing one horizontal and two vertical
flaps at the edges $e_{1}$ and $e_{2}$ from Figure 11.

$\widehat{\mathcal{L}}$
---
$\widehat{\mathbb{P}}$
---
$\mathbb{P}$
---
Figure 13. A branched covering map
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\rightarrow\mathbb{P}$ induced
by the flapped pillow $\widehat{\mathbb{P}}$ on the left.

The polyhedral surface $\widehat{\mathbb{P}}$ is naturally subdivided into

$2(n^{2}+n_{h}+n_{v})=2n^{2}+2\sum_{e\in E}m_{e}$

squares of sidelength $1/n$, called the _$1$ -tiles_ of the flapped pillow
$\widehat{\mathbb{P}}$. The vertices and the edges of these squares are called
the _$1$ -vertices_ and _$1$ -edges_ of $\widehat{\mathbb{P}}$. There is a
natural path metric on $\widehat{\mathbb{P}}$ that agrees with the Euclidean
metric on each $1$-tile. The surface $\widehat{\mathbb{P}}$ equipped with this
metric is locally Euclidean with conic singularities at some of the
$1$-vertices. Such a conic singularity arises at a $1$-vertex
$v\in\widehat{\mathbb{P}}$ if $v$ is contained in $k_{v}\neq 4$ distinct
$1$-tiles.

We will assume that $\widehat{\mathbb{P}}$ has at least one flap, that is,
$n_{h}+n_{v}\geq 1$. Let $F_{j}$, $j=1,\dots,n_{h}+n_{v}$, be the collection
of flaps glued to $\mathbb{P}$. Each flap $F_{j}$ consists of two $1$-tiles in
$\widehat{\mathbb{P}}$. We call the four $1$-vertices that belong to $F_{j}$
the _vertices_ of the flap $F_{j}$. The boundary $\partial F_{j}$ is a Jordan
curve in $\widehat{\mathbb{P}}$ composed of two $1$-edges $e^{\prime}_{j}$ and
$e^{\prime\prime}_{j}$, which we call the _base edges_ of $F_{j}$. The
$1$-edge in $F_{j}$ that is opposite to the base edges is called the _top
edge_ of the flap $F_{j}$. Note that $\partial e^{\prime}_{j}=\partial
e^{\prime\prime}_{j}$ consists of two vertices of $F_{j}$.

We now define the base $B(\widehat{\mathbb{P}})\subset\widehat{\mathbb{P}}$ of
the flapped pillow as

(4.1)
$B(\widehat{\mathbb{P}})\coloneq\widehat{\mathbb{P}}\setminus\biggl{(}\bigcup_{j=1}^{n_{h}+n_{v}}(F_{j}\setminus\partial
e^{\prime}_{j})\biggr{)}.$

In other words, $B(\widehat{\mathbb{P}})$ is obtained from
$\widehat{\mathbb{P}}$ by removing all flaps $F_{j}$ from
$\widehat{\mathbb{P}}$, except that we keep the two vertices in $\partial
e^{\prime}_{j}\subset F_{j}$ from each flap. There is a natural identification

(4.2) $B(\widehat{\mathbb{P}})\cong\mathbb{P}\setminus\bigcup_{e\in
E}\operatorname{int}(e)\subset\mathbb{P}.$

This means that we can consider the base $B(\widehat{\mathbb{P}})$ both as a
subset of $\widehat{\mathbb{P}}$ and of $\mathbb{P}$. Figure 14 illustrates
these two viewpoints. This is slightly imprecise, but this point of view will
be extremely convenient in the following.

$\cong$
---
Figure 14. The base $B(\widehat{\mathbb{P}})$ of the flapped pillow
$\widehat{\mathbb{P}}$ from Figure 13 depicted in two different ways: as a
subset of $\widehat{\mathbb{P}}$ and as the subset
$\mathbb{P}\setminus\bigcup_{e\in E}\operatorname{int}(e)$ of $\mathbb{P}$.

We choose the orientation on $\widehat{\mathbb{P}}$ so that the induced
orientation on $B(\widehat{\mathbb{P}})$ considered as a subset of
$\widehat{\mathbb{P}}$ coincides with the orientation on
$B(\widehat{\mathbb{P}})$ considered as a subset of the oriented sphere
$\mathbb{P}$ (if we represent orientations on surfaces by flags as described
in [BM17, Section A.4], then we simply pick a positively-oriented flag
contained in $B(\widehat{\mathbb{P}})\subset\mathbb{P}$ and declare it to be
positively-oriented in $\widehat{\mathbb{P}}\supset B(\widehat{\mathbb{P}})$).

The set $B(\widehat{\mathbb{P}})\cong\mathbb{P}\setminus\bigcup_{e\in
E}\operatorname{int}(e)$ contains the vertex set
$\mathcal{L}_{n}^{-1}(V)\subset\mathbb{P}$ of the graph
$\widetilde{\mathcal{C}}=\mathcal{L}_{n}^{-1}(\mathcal{C})\subset\mathbb{P}$.
This means that we can naturally view each vertex of $\widetilde{\mathcal{C}}$
also as a $1$-vertex in $\widehat{\mathbb{P}}$. Let
$\widehat{A},\,\widehat{B},\,\widehat{C},\,\widehat{D}$ be the $1$-vertices of
$\widehat{\mathbb{P}}$ that correspond to the vertices $A,\,B,\,C,\,D$ of the
original pillow, respectively. We set
$\widehat{V}\coloneq\\{\widehat{A},\widehat{B},\widehat{C},\widehat{D}\\}$ and
call $\widehat{A},\,\widehat{B},\,\widehat{C},\,\widehat{D}$ the _vertices_ of
$\widehat{\mathbb{P}}$.

Recall from Section 3.1 that the faces of the embedded graph
$\widetilde{\mathcal{C}}\subset\mathbb{P}$ are colored black and white in a
checkerboard manner. This coloring induces a checkerboard coloring on the
$1$-tiles of the flapped pillow $\widehat{\mathbb{P}}$. The original map
$\mathcal{L}_{n}\colon\mathbb{P}\to\mathbb{P}$ can now be naturally extended
to a continuous map
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\to\mathbb{P}$ by reflection
so that it preserves the coloring: $\widehat{\mathcal{L}}$ maps each $1$-tile
of $\widehat{\mathbb{P}}$ by a Euclidean similarity (scaling distances by the
factor $n$) onto the $0$-tile of $\mathbb{P}$ with the same color; see Figure
13 for an illustration. On the base $B(\widehat{\mathbb{P}})$ the map
$\widehat{\mathcal{L}}$ agrees with the original $(n\times n)$-Lattès map
$\mathcal{L}_{n}$ (if we consider $B(\widehat{\mathbb{P}})$ as a subset of
$\mathbb{P}$ by the identification (4.2)).

It is clear that
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\rightarrow\mathbb{P}$ is a
branched covering map. This map is essentially the Thurston map obtained from
$\mathcal{L}_{n}$ by blowing up each arc $e\in E$ with muliplicity $m_{e}$. To
make this more precise, we need a suitable identification of the source
$\widehat{\mathbb{P}}$ with the target $\mathbb{P}$ of $\widehat{\mathcal{L}}$
so that we obtain a self-map on $\mathbb{P}$. For this we choose a natural
homeomorphism $\phi\colon\widehat{\mathbb{P}}\to\mathbb{P}$, which we will now
define.

We view the set
$\widehat{\mathcal{C}}\coloneq\widehat{\mathcal{L}}^{-1}(\mathcal{C})\subset\widehat{\mathbb{P}}$
as a planar embedded graph, whose vertices and edges are precisely the
$1$-vertices and the $1$-edges of the flapped pillow $\widehat{\mathbb{P}}$.
Each $1$-edge of $\widehat{\mathbb{P}}$ is homeomorphically mapped by
$\widehat{\mathcal{L}}$ onto one of the edges of $\mathbb{P}$. Similarly as
before, the $1$-edges of $\widehat{\mathbb{P}}$ that are mapped by
$\widehat{\mathcal{L}}$ onto $a$ or $c$ are called _horizontal_ , while the
$1$-edges of $\widehat{\mathbb{P}}$ that are mapped by $\widehat{\mathcal{L}}$
onto $b$ or $d$ are called _vertical_.

There is a simple path of length $n$ in the graph $\widehat{\mathcal{C}}$ that
connects the vertices $\widehat{A}$ and $\widehat{B}$. Clearly, any such path
consists only of horizontal $1$-edges in $\widehat{\mathbb{P}}$. We denote by
$\widehat{a}$ the realization of the chosen path in the sphere
$\widehat{\mathbb{P}}$, which is an arc in
$(\widehat{\mathbb{P}},\widehat{V})$. The arc $\widehat{a}$ may not be
uniquely determined (namely, if flaps have been glued to slits obtained from
edges $e\subset a$), but any two such arcs are isotopic rel. $\widehat{V}$. We
define $\widehat{b}$, $\widehat{c}$, $\widehat{d}$ in a similar way and call
$\widehat{a},\,\widehat{c}$ the _horizontal edges_ , and
$\widehat{b},\,\widehat{d}$ the _vertical edges_ of $\widehat{\mathbb{P}}$.

We now choose an orientation-preserving homeomorphism
$\phi\colon\widehat{\mathbb{P}}\to\mathbb{P}$ that sends
$\widehat{A},\,\widehat{B},\,\widehat{C},\,\widehat{D}$ to $A,\,B,\,C,\,D$,
and $\widehat{a},\,\widehat{b},\,\widehat{c},\,\widehat{d}$ to
$a,\,b,\,c,\,d$, respectively. We define
$f\coloneq\widehat{\mathcal{L}}\circ\phi^{-1}$, which is a self-map on
$\mathbb{P}$. Clearly, $f$ is a branched covering map on $\mathbb{P}$. To
refer to this map, we say that $f\colon\mathbb{P}\to\mathbb{P}$ is obtained
from the $(n\times n)$-Lattès map $\mathcal{L}_{n}$ by gluing $n_{h}$
horizontal and $n_{v}$ vertical flaps to $\mathbb{P}$. More informally, we
call both maps $f$ and $\widehat{\mathcal{L}}$ a blown-up Lattès map.

A point in $\widehat{\mathbb{P}}$ is a critical point for
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\to\mathbb{P}$ if and only if
it is on the boundary of at least four $1$-tiles subdividing
$\widehat{\mathbb{P}}$. This implies that the set $C_{\widehat{\mathcal{L}}}$
of critical points of $\widehat{\mathcal{L}}$ consists of $1$-vertices of
$\widehat{\mathbb{P}}$ and that each critical point of $\mathcal{L}_{n}$ is
also a critical point for $\widehat{\mathcal{L}}$ (recall that we can view
each point in $\mathcal{L}_{n}^{-1}(V)\supset C_{\mathcal{L}_{n}}$ as a
$1$-vertex in $\widehat{\mathbb{P}}$). Moreover, if a $1$-vertex of
$\widehat{\mathbb{P}}$ is a critical point of $\widehat{\mathcal{L}}$, but not
of $\mathcal{L}_{n}$, then it must be one of the points in $\widehat{V}$. For
example, $\widehat{A}\in\widehat{V}$ is a critical point of
$\widehat{\mathcal{L}}$ if and only if a flap was glued to an edge of
$\widetilde{\mathcal{C}}$ incident to $A\cong\widehat{A}$. In any case, since
$\widehat{\mathcal{L}}$ sends the $1$-vertices of $\widehat{\mathbb{P}}$ to
the vertices of $\mathbb{P}$, the postcritical set of
$f=\widehat{\mathcal{L}}\circ\phi^{-1}$ coincides with the vertex set $V$.
Thus, $f$ is a Thurston map.

Since we assumed that $\widehat{\mathbb{P}}$ contains at least one flap (that
is, $n_{h}+n_{v}>0$), the orbifold of the Thurston map $f$ is hyperbolic.
Indeed, each $1$-vertex of $\widehat{\mathbb{P}}$ that is a critical point of
$\mathcal{L}_{n}$ is also a critical point of $\widehat{\mathcal{L}}$ with the
same or larger local degree. Since we glued at least one flap, there is at
least one $1$-vertex $v$ contained in six or more $1$-tiles of
$\widehat{\mathbb{P}}$. Then
$\deg(\widehat{\mathcal{L}},v)=\deg(f,v^{\prime})\geq 3$, where
$v^{\prime}=\phi(v)$. Now

$X\coloneqq f(v^{\prime})=\widehat{\mathcal{L}}(v)\in V=\\{A,B,C,D\\}=P_{f},$

and so for the ramification function $\alpha_{f}$ of $f$ we have
$\alpha_{f}(X)\geq 3$. On the other hand, for all other points $Y\in V=P_{f}$
we have $\alpha_{f}(Y)\geq\alpha_{\mathcal{L}_{n}}(Y)\geq 2$. It then follows
from (3.1) that $\chi(\mathcal{O}_{f})<0$, and so $f$ has indeed a hyperbolic
orbifold.

The homeomorphism $\phi$ chosen in the definition of $f$ is not unique, but
any two such homeomorphisms are isotopic rel. $\widehat{V}$ (this easily
follows from [Bus10, Theorem A.5]). This implies that $f$ is uniquely
determined up to Thurston equivalence. This map may be viewed (up to Thurston
equivalence) as the result of the blowing up operation introduced in Section
4.1 applied to the edges $e\in E$ with the multiplicities $m_{e}$. In
particular, if we run the procedure for the map $\widehat{\mathcal{L}}$
indicated in Figure 13, then we obtain the map $\widehat{f}$ illustrated in
Figure 12 (up to Thurston equivalence).

##  5\. Realizing blown-up Lattès maps

The goal of this section is to determine when a blown-up Lattès map is
realized by a rational map. In particular, we will apply Thurston’s criterion
to prove Theorem 1.2. The strategies and techniques used in the proof will
highlight the main ideas needed for establishing the more general Theorem 1.1.

We fix $n\geq 2$, $n_{h},n_{v}\geq 0$, and follow the notation introduced in
Section 4.2. In particular, we denote by $\widehat{\mathbb{P}}$ a flapped
pillow with $n_{h}$ horizontal and $n_{v}$ vertical flaps, by
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\to\mathbb{P}$ the respective
blown-up $(n\times n)$-Lattès map, and by
$\phi\colon\widehat{\mathbb{P}}\to\mathbb{P}$ the identifying homeomorphism.
Then $f\colon\mathbb{P}\to\mathbb{P}$ given as
$f=\widehat{\mathcal{L}}\circ\phi^{-1}$ is the Thurston map under
consideration. We will assume that $n_{h}+n_{v}>0$, and so
$\widehat{\mathbb{P}}$ has at least one flap. In this case, $f$ has a
hyperbolic orbifold as we have seen, and so we can apply Thurston’s criterion.
For this we consider essential Jordan curves $\gamma$ in
$(\mathbb{P},P_{f})=(\mathbb{P},V)$ and study their (essential) pullbacks
under $f$.

If $\gamma$ is such a curve, then the homeomorphism $\phi$ sends the pullbacks
of $\gamma$ under $\widehat{\mathcal{L}}$ to the pullbacks of $\gamma$ under
$f$. So in order to understand the isotopy types and mapping properties of the
pullbacks under $f$, we will instead look at the pullbacks of $\gamma$ under
$\widehat{\mathcal{L}}$. In particular, if $\widehat{\gamma}$ is a pullback of
$\gamma$ under $\widehat{\mathcal{L}}$, then
$\deg(\widehat{\mathcal{L}}\colon\widehat{\gamma}\to\gamma)=\deg(f\colon\phi(\widehat{\gamma})\to\gamma)$
and $\phi(\widehat{\gamma})$ is essential in
$(\mathbb{P},P_{f})=(\mathbb{P},V)$ if and only if $\widehat{\gamma}$ is
essential in $(\widehat{\mathbb{P}},\widehat{V})$, where $\widehat{V}$ denotes
the vertex set of the flapped pillow $\widehat{\mathbb{P}}$.

Since the mapping
$\widehat{\mathcal{L}}\colon\widehat{\mathbb{P}}\to\mathbb{P}$ is a similarity
map on each $1$-tile of $\widehat{\mathbb{P}}$, the preimage
$\widehat{\mathcal{L}}^{-1}(\gamma)$ of a Jordan curve $\gamma$ in
$(\mathbb{P},P_{f})$ (or of any subset $\gamma$ of $\mathbb{P}$), can be
obtained in the following intuitive way: we rescale and copy the part of
$\gamma$ that belongs to the white $0$-tile of $\mathbb{P}$ into each white
$1$-tile of $\widehat{\mathbb{P}}$ and the part of $\gamma$ that belongs to
the black $0$-tile of $\mathbb{P}$ into each black $1$-tile.

###  5.1. The horizontal and vertical curves

Recall that $\alpha^{h}$ and $\alpha^{v}$ (see (2.4)) denote the Jordan curves
in $(\mathbb{P},V)=(\mathbb{P},P_{f})$ that separate the two horizontal and
the two vertical edges of $\mathbb{P}$, respectively. These two curves are
invariant under $f$ and will play a crucial role in the considerations of this
section.

$\widehat{\mathcal{L}}$
---
$\alpha^{h}$
---
$\widehat{\mathbb{P}}$
---
$\mathbb{P}$
---
Figure 15. Pullbacks of $\alpha^{h}$ for a blown-up $(4\times 4)$-Lattès map
with $n_{h}=n_{v}=1$.

######  Lemma 5.1.

Let $f=\widehat{\mathcal{L}}\circ\phi^{-1}\colon\mathbb{P}\to\mathbb{P}$ be a
Thurston map obtained from the $(n\times n)$-Lattès map, $n\geq 2$, by gluing
$n_{h}\geq 0$ horizontal and $n_{v}\geq 0$ vertical flaps to $\mathbb{P}$.
Then the following statements are true:

  1. (i)

The Jordan curve $\alpha^{h}$ has $n+n_{h}$ pullbacks under $f$. Exactly $n$
of these pullbacks are essential. Each of these essential pullbacks is
isotopic to $\alpha^{h}$ rel. $P_{f}$.

  2. (ii)

If $\widetilde{\alpha}$ is one of the $n$ essential pullbacks of $\alpha^{h}$,
then $\deg(f\colon\widetilde{\alpha}\to\alpha^{h})=n+n_{\widetilde{\alpha}}$,
where $n_{\widetilde{\alpha}}\geq 0$ is the number of distinct vertical flaps
in $\widehat{\mathbb{P}}$ that $\phi^{-1}(\widetilde{\alpha})$ meets.

  3. (iii)

We have

$\displaystyle\lambda_{f}(\alpha^{h})=\sum_{\widetilde{\alpha}}\frac{1}{n+n_{\widetilde{\alpha}}},$

where the sum is taken over all essential pullbacks $\widetilde{\alpha}$ of
$\alpha^{h}$ under $f$.

Analogous statements are true for the curve $\alpha^{v}$.

###### Proof.

Figure 15 illustrates the proof. It is obvious that the Jordan curve
$\alpha^{h}$ has exactly $n+n_{h}$ distinct pullbacks under
$\widehat{\mathcal{L}}$. Among them, there are $n$ essential pullbacks
$\widehat{\alpha}_{1},\dots,\widehat{\alpha}_{n}$ that separate the two
horizontal edges of $\widehat{\mathbb{P}}$ and thus are isotopic to each other
relative to the vertex set $\widehat{V}$ of $\widehat{\mathbb{P}}$. For each
$j=1,\dots,n$, the image $\alpha_{j}\coloneq\phi(\widehat{\alpha}_{j})$ is
isotopic to $\alpha^{h}$. Moreover, we have
$\deg(\widehat{\mathcal{L}}\colon\widehat{\alpha}_{j}\to\alpha^{h})=n+n_{{\alpha}_{j}}$.
The other $n_{h}$ pullbacks of $\alpha^{h}$ under $\widehat{\mathcal{L}}$ are
each contained in one of the horizontal flaps and thus are peripheral in
$(\widehat{\mathbb{P}},\widehat{V})$. Consequently,

$\lambda_{f}(\alpha^{h})=\sum_{j=1}^{n}\frac{1}{\deg(f\colon{\alpha}_{j}\to\alpha^{h})}=\sum_{j=1}^{n}\frac{1}{\deg(\widehat{\mathcal{L}}\colon\widehat{\alpha}_{j}\to\alpha^{h})}=\sum_{j=1}^{n}\frac{1}{n+n_{{\alpha}_{j}}}.$

This completes the proof of the lemma for the curve $\alpha^{h}$. The proof
for the curve $\alpha^{v}$ follows from similar considerations. ∎

The following corollary is an immediate consequence of the previous lemma.

######  Corollary 5.2.

Let $f=\widehat{\mathcal{L}}\circ\phi^{-1}\colon\mathbb{P}\to\mathbb{P}$ be a
Thurston map obtained from the $(n\times n)$-Lattès map, $n\geq 2$, by gluing
$n_{h}\geq 0$ horizontal and $n_{v}\geq 0$ vertical flaps to $\mathbb{P}$.
Then $\alpha^{h}$ is an obstruction (for $f$) if and only if $n_{v}=0$, and
$\alpha^{v}$ is an obstruction if and only if $n_{h}=0$.

###### Proof.

Let us first suppose that $n_{v}=0$, that is, $\widehat{\mathbb{P}}$ does not
have any vertical flaps. Then by Lemma 5.1, $\alpha^{h}$ has $n$ essential
pullbacks under $f$, each of which is mapped onto $\alpha^{h}$ with degree $n$
(this is illustrated in Figure 16 in a special case). Consequently,
$\lambda_{f}(\alpha^{h})=n\cdot(1/n)=1$, which means $\alpha^{h}$ is an
obstruction for $f$.

If $n_{v}>0$, the flapped pillow $\widehat{\mathbb{P}}$ has at least one
vertical flap. Then $n_{\widetilde{\alpha}}>0$ for at least one essential
pullback $\widetilde{\alpha}$ of $\alpha^{h}$. Lemma 5.1 implies that
$\lambda_{f}(\alpha^{h})\leq(n-1)\frac{1}{n}+\frac{1}{n+1}<1,$ and so
$\alpha^{h}$ is not an obstruction for $f$.

The proof for the vertical curve $\alpha^{v}$ is completely analogous. ∎

The above corollary can be read as follows: the obstruction $\alpha^{h}$ for
the $(n\times n)$-Lattès map can be eliminated by gluing a vertical flap to
$\mathbb{P}$. Similarly, the obstruction $\alpha^{v}$ can be eliminated by
gluing a horizontal flap. We will show momentarily that if both of these
obstructions are eliminated (that is, if there are both horizontal and
vertical flaps) then no other obstructions are present and so the map $f$ is
realized.

$\alpha^{h}$
---
$\widehat{\mathcal{L}}$
---
$\widehat{\mathbb{P}}$
---
$\mathbb{P}$
---
Figure 16. Pullbacks of $\alpha^{h}$ for a blown up $(4\times 4)$-Lattès map
with $n_{h}=1$ and $n_{v}=0$.

###  5.2. Ruling out other obstructions

Now we discuss what happens with the essential curves in $(\mathbb{P},P_{f})$
that are not isotopic to the horizontal curve $\alpha^{h}$ or the vertical
curve $\alpha^{v}$.

######  Theorem 5.3.

Let $f=\widehat{\mathcal{L}}\circ\phi^{-1}\colon\mathbb{P}\to\mathbb{P}$ be a
Thurston map obtained from the $(n\times n)$-Lattès map, $n\geq 2$, by gluing
$n_{h}\geq 0$ horizontal and $n_{v}\geq 0$ vertical flaps to $\mathbb{P}$ and
assume that $n_{h}+n_{v}>0$. If $\gamma\subset\mathbb{P}\setminus P_{f}$ is an
essential Jordan curve that is not isotopic to either $\alpha^{h}$ or
$\alpha^{v}$, then $\gamma$ is not an obstruction for $f$.

Before we turn to the proof of this theorem, we first record how it implies
Theorem 1.2 stated in the introduction.

###### Proof of Theorem 1.2.

Let $n$, $f$, $n_{h}$, and $n_{v}$ with $n_{h}+n_{v}>0$ be as in the
statement. We have seen in Section 4.2 that then $f$ has a hyperbolic
orbifold. If $n_{h}=0$ or $n_{v}=0$, then by Corollary 5.2 the curve
$\alpha^{v}$ or the curve $\alpha^{h}$ is an obstruction, respectively.

If $n_{h}>0$ and $n_{v}>0$, then $f$ has no obstruction as follows from
Corollary 5.2 and Theorem 5.3. Since $f$ has a hyperbolic orbifold, in this
case $f$ is realized by a rational map according to Theorem 3.7. ∎

Corollary 5.2 and Theorem 5.3 also imply that if $n_{h}=0$, then $\alpha^{v}$
is the only obstruction for $f$ (up to isotopy rel. $P_{f}$). Similarly,
$\alpha^{h}$ is the only obstruction if $n_{v}=0$.

Before we go into the details, we will give an outline for the proof of
Theorem 5.3. We argue by contradiction and assume that $f$ has an obstruction
given by an essential Jordan curve $\gamma$ in $(\mathbb{P},P_{f})$ that is
isotopic to neither $\alpha^{h}$ nor $\alpha^{v}$ rel. $P_{f}$. Then
$\lambda_{f}(\gamma)\geq 1$. Let $\gamma_{1},\dots,\gamma_{k}$ for some
$k\in\mathbb{N}$ be all the essential pullbacks of $\gamma$ under $f$, which
must be isotopic to $\gamma$ rel. $P_{f}$.

Using facts about intersection numbers and the mapping properties of $f$, one
can show that for the number of essential pullbacks of $\gamma$ we have $k\leq
n$ and that the corresponding mapping degrees satisfy
$\deg(f\colon\gamma_{j}\to\gamma)\geq n$ for all $j=1,\dots,k$. Since
$\lambda_{f}(\gamma)\geq 1$, it follows that there are exactly $k=n$ essential
pullbacks and that $\deg(f\colon\gamma_{j}\to\gamma)=n$ for each
$j=1,\dots,n$.

This in turn implies that none of the essential pullbacks
$\widehat{\gamma}_{j}\coloneq\phi^{-1}(\gamma_{j})$ of $\gamma$ under
$\widehat{\mathcal{L}}$ goes over a flap in $\widehat{\mathbb{P}}$. Then all
the $n$ pullbacks $\widehat{\gamma}_{1},\dots,\widehat{\gamma}_{n}$ belong to
the base $B(\widehat{\mathbb{P}})$ of $\widehat{\mathbb{P}}$. This means that
each $\widehat{\gamma}_{j}$ can be thought of as a pullback of $\gamma$ under
the original $(n\times n)$-Lattès map $\mathcal{L}_{n}$. However, there are
only $n$ pullbacks of $\gamma$ under $\mathcal{L}_{n}$, which cross all the
edges of the graph $\mathcal{L}_{n}^{-1}(\mathcal{C})$, where $\mathcal{C}$ is
the common boundary of the $0$-tiles in $\mathbb{P}$. Consequently, the
pullbacks $\widehat{\gamma}_{1},\dots,\widehat{\gamma}_{n}$ cross all the
$1$-edges in the closure of the base $B(\widehat{\mathbb{P}})$. It follows
that one of the pullbacks $\widehat{\gamma}_{j}$ must cross one of the base
edges of a flap $F$ in $\widehat{\mathbb{P}}$, which would necessarily mean
that $\widehat{\gamma}_{j}$ goes over the flap $F$. This gives the desired
contradiction and Theorem 5.3 follows.

In the remainder of this section we will fill in the details for this outline.
First, we establish several general facts about degrees and intersection
numbers.

Let $n\in\mathbb{N}$ and $f\colon X\rightarrow Y$ be a map between two sets
$X$ and $Y$. We say that $f$ is at most $n$-to-$1$ if $\\#f^{-1}(y)\leq n$ for
each $y\in Y$. We say that $f$ is $n$-to-$1$ if $\\#f^{-1}(y)=n$ for each
$y\in Y$.

######  Lemma 5.4.

Let $f\colon X\rightarrow Y$ be a map between two sets $X$ and $Y$. Suppose
$M\subset X$, $N\subset Y$, and $f|M\colon M\to f(M)$ is at most $n$-to-$1$