2.1 Notation and basic concepts

We denote by $\mathbb {N}=\{1,2,\ldots \}$ the set of natural numbers and by $\mathbb {N}_0=\{0,1, 2, \ldots \}$ 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, is the Euclidean plane and is the Riemann sphere. Here and elsewhere, we write 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 for the closed unit interval, for the open unit disk in $\mathbb {C}$ , and for the square lattice in $\mathbb {C}$ . If $z,w\in ~\mathbb {C}$ , then we write for the line segment in $\mathbb {C}$ joining z and w. We also use the notation and .

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 is the preimage of N in X. Similarly, 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

the nth iterate of f. It is convenient to define . For $n\in \mathbb {N}_0$ , we denote by and 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 $\chi (S)\in \{2, 1,0, -1, \ldots \} \cup \{-\infty \}$ . 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 . The set 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 $\eta \colon \operatorname {cl}(\mathbb {D})=\{z\in \mathbb {C}: |z|\le 1\}\rightarrow S$ 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 parameterization.

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 $\alpha \subset S\smallsetminus Z$ . An arc e in $(S,Z)$ is an arc $e\subset S$ with $\partial e\subset Z$ and $\operatorname {int}(e)\subset S\smallsetminus Z$ . 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}}\smallsetminus \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}}\smallsetminus \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 $\varphi \colon A\rightarrow A'$ , the curve $\beta '=\varphi (\beta )$ separates the boundary components of $A'=\{z\in \mathbb {C}: 1<|z|<2\}$ .

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\ne \emptyset $ , we can write $f^{-1}(V)$ as a disjoint union

(2.1) $$ \begin{align} f^{-1}(V)= \bigcup_{j\in J}U_j \end{align} $$

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

$$ \begin{align*} (\psi_j \circ f \circ \varphi_j^{-1})(z)=z^{d_j}\end{align*} $$

for all $z\in \mathbb {D}$ (see [Reference Bonk and MeyerBM17, §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 $X\smallsetminus f^{-1}(f({C_f}))$ onto $Y\smallsetminus f({C_f})$ .

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

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

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 [Reference Bonk and MeyerBM17, Lemma A.18]), but in general, $\widetilde \gamma $ is not unique. If $\gamma ([a,b))\subset Y\smallsetminus f({C_f})$ 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 [Reference Bonk and MeyerBM17, 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 . If the Euler characteristic $\chi (V)$ is finite, then $\chi (U)$ is also finite and we have the Riemann–Hurwitz formula

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

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

A face of G is a connected component of ${S^{2}}\smallsetminus \mathcal {G}$ . 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 $\mathcal {G}\smallsetminus V$ .

A subgraph of a planar embedded graph $G=(V,E)$ is a planar embedded graph ${G'=(V',E')}$ with $V'\subset V$ and $E'\subset E$ . A path of length n between vertices v and $v'$ in G is a sequence $v_0,e_0,v_1,e_1,\ldots ,e_{n-1},v_n$ , where $v_0=v$ , $v_n=v'$ , and $e_k$ is an edge incident to the vertices $v_{k}$ and $v_{k+1}$ for $k=0,\ldots ,n-1$ . A path that does not repeat vertices is called a simple path.

A path $v_0,e_0, v_1,e_1,\ldots ,e_{n-1},v_n$ with $v_0=v_n$ and $n\geq 2$ is called a circuit of length n in G and is denoted by $(e_0,e_1,\ldots ,e_{n-1})$ . Such a circuit is called a simple cycle if all vertices $v_k$ , $k = 0,\ldots ,n-1$ , 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 [Reference DiestelDie05, 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 $(e_0,e_1,\ldots ,e_{n-1})$ in G such that each edge of G appears zero, one, or two times in the sequence $e_0,e_1,\ldots ,e_{n-1}$ . We will say that the circuit $(e_0,e_1,\ldots ,e_{n-1})$ 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 [Reference Bonk and MeyerBM17, 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 and the edge set . We call a and c the horizontal edges, and b and d the vertical edges of $\mathbb {P}$ ; see Figure 4.

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, [Reference MilnorMil06a, Appendix E] and [Reference Bonk and MeyerBM17, 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 . Actually, for $z, w\in \mathbb {C}$ , we have

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

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 [Reference Bonk and MeyerBM17, §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 $\wp ^{-1}(A)= 2\mathbb {Z}^2$ .

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 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 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 relative to Z are defined in a similar way.

Two homeomorphisms $h_0, h_1 \colon X \rightarrow Y $ are called isotopic (relative to $Z\subset X$ ) if there exists an isotopy $H\colon X\times \mathbb {I}\to Y$ (relative to 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 relative to Z (or M can be isotoped into N relative to Z), denoted by $M\sim N$ relative to Z, if there exists an isotopy $H\colon X\times \mathbb {I} \to X$ relative to 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 $ relative to Z.

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

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

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 [Reference BuserBus10, 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 relative to 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 $p\in \alpha \cap \beta \cap (S\smallsetminus Z)$ (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 $p\in \alpha \cap \beta \cap (S\smallsetminus Z)$ .

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 . 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 .

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

$$ \begin{align*}\ell =\{ z_0+(s+ ir)t: t\in \mathbb{R}\}\subset \mathbb{C}\end{align*} $$

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 $\ell _{r/s}\subset \mathbb {C} \smallsetminus \mathbb {Z}^2$ , then is a Jordan curve in $\mathbb {P}\smallsetminus V$ . 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 is a geodesic arc in $(\mathbb {P}, V)$ .

Figure 6

A line $\ell _{2}$ and the corresponding Jordan curve $\tau _2=\wp (\ell _2)$ in $\mathbb {P}$ .

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 $\mathbb {P}\smallsetminus \tau _{r/s}$ (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 $\tau _{r/s}$ .

While this is well known (see, for example, [Reference Farb and MargalitFM12, Proposition 2.6] or [Reference Keen and SeriesKS94, 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 $\ell _{r/s} \subset \mathbb {C}\smallsetminus \mathbb {Z}^2$ (see Lemma A.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)

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

We will prove this lemma in the appendix. Note that Figure 7 illustrates statements (iii) and (v) when $\alpha =\tau _2$ . It follows from the lemma that $\tau _{r/s}$ for $r/s\ne 0, \infty $ is in minimal position with each of the curves $a,b,c,d, \alpha ^h, \alpha ^v$ .

Figure 7

Counting intersections of $\tau _2$ with the horizontal curve $\alpha ^h$ and the horizontal edges a and c.

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\ne \emptyset $ and $\gamma \cap M_2\ne \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) parameterization 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 $\#(\gamma \cap M_1)= \#(\gamma \cap M_2)$ .

A similar statement is true if $r/s\ne \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 A.3).

Proof. Define . Suppose first that $\tau =\wp (\ell _{r/s})$ is a simple closed geodesic. Then $ \tau =\wp ([z_0, w_0])$ , where $z_0\in \ell _{r/s} \subset \mathbb {C}\smallsetminus \mathbb {Z}^2$ and $w_0=z_0+2\omega $ , and the map $u\in [0,1]\mapsto \wp (uz_0+ (1-u)w_0)$ provides a parameterization 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 $r/s\ne 0$ , we have $r\in \mathbb {Z}\smallsetminus \{0\}$ , 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

$$ \begin{align*}\#(\wp^{-1}(a)\cap [z_0, w_0))=|r|=\#(\wp^{-1}(c)\cap [z_0, w_0)), \end{align*} $$

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]\ne \emptyset $ and ${\wp ^{-1}(c)\cap [z_0, w_0]\ne \emptyset }$ 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.

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 [Reference Milnor, Hjorth and PetersenMil06b] and [Reference Bonk and MeyerBM17, Ch. 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 §2.4. Fix a natural number $n\geq 2$ . It follows from (2.3) that there is a unique (and well-defined) map $\mathcal {L}_n\colon \mathbb {P}\to \mathbb {P}$ such that

(3.2) $$ \begin{align} \mathcal{L}_n(\wp(z))=\wp(nz) \text{ for } z\in \mathbb{C}. \end{align} $$

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 §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 equation (3.1), 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 §3.1 (see, for instance, [Reference Cannon, Floyd, Kenyon and ParryCFKP03] and [Reference Bonk and MeyerBM17, Ch. 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 [Reference Douady and HubbardDH93]. 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 ${\gamma \subset {S^{2}}\smallsetminus {P_f}}$ under f is a connected component $\widetilde {\gamma }$ of $f^{-1}(\gamma )$ . Since f is a covering map from $S^2\smallsetminus f^{-1}({P_f})$ onto $S^2\smallsetminus {P_f}$ , 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 $1\le k\le \deg (f)$ , 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 $\gamma \subset {S^{2}}\smallsetminus {P_f}$ is called essential if each of the two connected components of ${S^{2}}\smallsetminus \gamma $ contains at least two points from ${P_f}$ , and is called peripheral otherwise.

We will need the following standard facts.

For the proof, see [Reference Bonk and MeyerBM17, Lemma 6.9]. A consequence of this statement is that the isotopy classes of curves in $f^{-1}(\gamma )$ relative to ${P_f}$ only depend on the isotopy class $[\gamma ]$ relative to $ {P_f}$ and not on the specific choice of $\gamma $ .

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 ${S^{2}}\smallsetminus {P_f}$ that are pairwise disjoint and pairwise non-isotopic relative to ${P_f}$ .

Suppose now that $\#{P_f}=4$ . Then any two essential Jordan curves in ${S^{2}}\smallsetminus {P_f}$ are either isotopic relative to ${P_f}$ 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 ${S^{2}}\smallsetminus {P_f}$ . We say that an essential Jordan curve $\gamma $ in ${S^{2}}\smallsetminus {P_f}$ is f-invariant if each essential pullback of $\gamma $ under f is isotopic to $\gamma $ relative to ${P_f}$ .

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}}\smallsetminus {P_f} $ be an essential f-invariant Jordan curve. We denote by $\gamma _1,\ldots ,\gamma _n$ for $n\in \mathbb {N}_0$ all the essential pullbacks of $\gamma $ under f and define

(3.3)

Then $\gamma $ is called a (Thurston) obstruction for f if $\unicode{x3bb} _f(\gamma ) \geq 1$ .

Note that if $n=0$ , then the sum in equation (3.3) is the empty sum and so $\unicode{x3bb} _f(\gamma )=0$ . It immediately follows from Lemma 3.4 that whether or not $\gamma $ is an obstruction for f only depends on the isotopy class $[\gamma ]$ relative to ${P_f}$ .

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.

Figure 8

The two pullbacks of a curve $\gamma =\tau _2$ under $\mathcal {L}_2$ .

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 [Reference Douady and HubbardDH93] or [Reference Bonk and MeyerBM17, §2.6].

The example of the $(n\times n)$ -Lattès map $\mathcal {L}_n\colon \mathbb {P}\to \mathbb {P}$ with $n\ge 2$ 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 $[\gamma ]$ relative to 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 $\ell _{r/s}(z_0)\subset \mathbb {C} \smallsetminus \mathbb {Z}^2$ . Here r and s are relatively prime integers and so there exist $p,q\in \mathbb {Z}$ such that $pr+qs=1$ . Let . Using (2.3) and (3.2), one can verify that under $\mathcal {L}_n$ , the curve $\gamma $ has exactly n distinct pullbacks

(3.4) $$ \begin{align} \gamma_j=\wp( \ell_{r/s}((z_0+2j\widetilde \omega)/n)), \quad j=1,\ldots,n. \end{align} $$

Moreover, each curve $\gamma _j$ is isotopic to $\gamma $ relative to $P_{\mathcal {L}_n}=V$ and $\deg (\mathcal {L}_n\colon \gamma _j\to \gamma )=n$ for all $j=1,\ldots , n$ ; see Figure 8 for an illustration. Thus, $\unicode{x3bb} _{\mathcal {L}_n}(\gamma )=1$ and $\gamma $ is an obstruction.

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 multiplicities $m_e\ge 1$ 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 Proposition 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 $D_e\smallsetminus \partial e \subset W_e$ . 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.

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 $m_{e_2}=2$ .

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, 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 ${S^{2}}\smallsetminus \bigcup _{e\in E}{W_e}$ for each $t\in [0,1]$ ;

4. (B4) $h_1$ is a homeomorphism of $S^2\smallsetminus \bigcup _{e\in E}{D_e}$ onto $S^2\smallsetminus \bigcup _{e\in E}{e}$ , 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 ${S^{2}}\smallsetminus \bigcup _{e\in E}{W_e}$ remain fixed.

We now make yet another choice. For a fixed arc $e\in E$ , let $m=m_e$ . We choose ${m-1}$ crosscuts $e^1,\ldots ,e^{m-1}$ in $D_e$ with the same endpoints as e such that these crosscuts have pairwise disjoint interiors. We set and . The arcs $e^0,\ldots ,e^{m}$ subdivide the closed Jordan region $D_e$ into m closed Jordan regions $D_e^1, \ldots , D_e^m$ , 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 $k=1, \ldots , m$ . For each $k=1, \ldots , m$ , we now choose a continuous map $\varphi _k\colon D_e^k \rightarrow S^2$ with the following properties: