Finite and infinite degree Thurston maps with a small postsingular set

In the one-dimensional rational dynamics, the crucial role is played by the family of postcritically finite (or pcf in short) rational maps, i.e., maps with all critical points being periodic or pre-periodic. In this context, one of the most influential ideas has been to abstract from the rigid underlying complex structure and consider the more general setup of postcritically finite branched self-coverings of the topological $2$-sphere $S^{2}$. Nowadays, orientation-preserving pcf branched covering maps $f\colon S^{2}\to S^{2}$ of topological degree $\deg(f)\geq 2$ are called Thurston maps (of finite degree), in honor of William Thurston, who introduced them to deepen the understanding of the dynamics of postcritically finite rational maps on $\widehat{{\mathbb{C}}}$.

These ideas can be extended to the transcendental setting to explore the dynamics of postsingularly finite (psf in short) meromorphic maps. A meromorphic map is called postsingularly finite if it has finitely many singular values, and each of them eventually becomes periodic or lands on the essential singularity under the iteration. We can generalize the notion of a Thurston map to include postsingularly finite topologically holomorphic non-injective maps $f\colon X\to S^{2}$, where $X$ is a punctured topological sphere, $f$ does not extend continuously to the entire $S^{2}$ and meets a technical condition of being a parabolic type map; see Sections 2.2 and 2.3. Note that in this case the map $f$ must be transcendental, meaning that it has infinite topological degree. For simplicity, we will use the notation $f\colon S^{2}\dashrightarrow S^{2}$ to indicate that the Thurston map $f$, whether finite or infinite degree, might not be defined at a single point of $S^{2}$.

For a Thurston map $f\colon S^{2}\dashrightarrow S^{2}$, the postsingular set $P_{f}$ is defined as the union of all orbits of the singular values of the map $f$. It is important to note that some of these orbits might terminate after several iterations, if a singular value reaches the point where the map $f$ is not defined. The elements of the postsingular set $P_{f}$ are called the postsingular values of the Thurston map $f$. If the map $f$ is defined on the entire sphere $S^{2}$, we simply refer to its postcritical set and postcritical values, as the set of singular values of $f$ coincides with the set of its critical values. Two Thurston maps are called combinatorially (or Thurston) equivalent if they are conjugate up to isotopy relative to their postsingular sets; see Definition 2.7.

A fundamental question in this context is whether a given Thurston map $f$ can be realized by a psf meromorphic map with the same combinatorics, that is, if $f$ is combinatorially equivalent to a psf meromorphic map. If the Thurston map $f$ is not realized, then we say that $f$ is obstructed. William Thurston answered this question for Thurston maps of finite degree in his celebrated characterization of rational maps: if a finite degree Thurston map $f\colon S^{2}\to S^{2}$ has a hyperbolic orbifold (this is always true, except for some well-understood special maps), then $f$ is realized by a pcf rational map if and only if $f$ has no Thurston obstruction [DH93]. Such an obstruction is given by a finite collection of disjoint simple closed curves in $S^{2}-P_{f}$ with certain invariance properties under the map $f$. In many instances, it suffices to restrict to simpler types of Thurston obstructions provided by Levy cycles or even Levy fixed curves; see Definition 2.10, and [Hub16, Theorem 10.3.8], [HP22, Corollary 1.5], or [Par23, Theorems 7.6 and 8.6] for examples of such cases.

The same characterization question can be also asked in the transcendental setting. The first breakthrough in this area was obtained in [HSS09], where it was shown that an exponential Thurston map is realized if and only if it has no Levy cycle. In this context, the exponential Thurston map is defined as a Thurston map with two singular values, both of which are omitted, and one of which is the only essential singularity of the considered Thurston map. Furthermore, the results of [She22] and [MPR24] suggest that a Thurston-like criterion for realizability may hold in a greater generality. However, the characterization question in the transcendental setting remains largely open, as many of the techniques used in Thurston theory for finite degree maps do not extend to this context.

Thurston theory lays out the relationship between the topological properties of a map, its dynamics, and its geometry in terms of the existence of a holomorphic realization. Further- more, it is strongly connected with the combinatorial and algebraic aspects of the dynamics of pcf rational and psf meromorphic maps. The results mentioned above have substantial applications for both rational and transcendental dynamics. For instance, Thurston’s characterization result has allowed to classify various families of postcritically finite rational maps or finite degree Thurston maps in terms of combinatorial models [Poi93, Poi10, BLMW22], [DMRS19, LMS22], [Hlu19, HP22]. Building on the result of [HSS09], similar classifications were obtained in [LSV08] in terms of kneading sequences and in [PRS21] in terms of homotopic Hubbard treed for the family of postsingularly finite exponential maps. Moreover, the concept of a homotopic Hubbard tree was extended to general postsingularly finite entire maps in [Pfr19] (see also [PPS21]). It is plausible that a Thurston-like criterion is the final missing ingredient for the complete classification of the family of all psf entire maps.

The key method in determining whether a given finite or infinite degree Thurston map $f\colon S^{2}\dashrightarrow S^{2}$ with the postsingular set $P_{f}=A$ is realized by a psf meromorphic map is the analysis of the dynamics of a holomorphic operator $\sigma_{f}$, known as the pullback map, defined on a complex manifold called the Teichmüller space $\mathcal{T}_{A}$; see Sections 2.4 and 2.5. Crucially, the Thurston map $f$ is realized if and only if the pullback map $\sigma_{f}$ has a fixed point in $\mathcal{T}_{A}$; see 2.17. Moreover, the dynamics of the pullback map encodes many other properties of the corresponding Thurston map; see, for instance, [BEKP09], [KPS16], and [BDP24].

Instead of working directly in the Teichmüller space $\mathcal{T}_{A}$, it is often more convenient to work in a simpler complex manifold $\mathcal{M}_{A}$, known as a moduli space. This space, roughly speaking, encodes all possible complex structures biholomorhic to a punctured Riemann sphere that can be put on the punctured topological sphere $S^{2}-A$ (see Section 2.4 for the precise definition). There is a natural projection map $\pi\colon\mathcal{T}_{A}\to\mathcal{M}_{A}$ that is a holomorphic universal covering. However, the map $\sigma_{f}$ rarely descends to a map on the moduli space $\mathcal{M}_{A}$. Nevertheless, for a finite degree Thurston map $f\colon S^{2}\to S^{2}$, there exists a complex manifold $\mathcal{W}_{f}$, known as the Hurwitz space of the Thurston map $f$, along with the holomorphic $X$-map $X_{f}\colon\mathcal{W}_{f}\to\mathcal{M}_{A}$, a holomorphic covering map $Y_{f}\colon\mathcal{W}_{f}\to\mathcal{M}_{A}$, called the $Y$-map, and a holomorphic covering $\omega_{f}\colon\mathcal{T}_{A}\to\mathcal{W}_{f}$, such that the following diagram commutes [Koc13, Section 2]:

In other words, the pullback map $\sigma_{f}$ is semi-conjugate to a self-correspondence $X_{f}\circ Y_{f}^{-1}$ of the moduli space $\mathcal{M}_{A}$. If $f\colon S^{2}\to S^{2}$ with $P_{f}=A$ is a finite degree Thurston map, then the $Y$-map $Y_{f}\colon\mathcal{W}_{f}\to\mathcal{M}_{A}$ has a finite topological degree; see [Koc13, Theorem 2.6]. This observation plays a crucial role in the proof of Thurston’s characterization of rational maps. In fact, it allows to conclude that for a Thurston map $f$ with a hyperbolic orbifold, the $\sigma_{f}$-orbit $(\sigma_{f}^{\circ n}(\tau))$ of $\tau\in\mathcal{T}_{A}$ converges (indicating that $f$ is realized), if the projection $(\pi(\sigma_{f}^{\circ n}(\tau)))$ of this orbit visits some compact set of the moduli space $\mathcal{M}_{A}$ infinitely many times; see [Hub16, Section 10.9 and Lemma 10.11.9] and [Sel12, Proof of Theorem 2.3, p. 20].

The objects introduced above, along with commutative diagram (1) and the fact that $Y_{f}\colon\mathcal{W}_{f}\to\mathcal{M}_{A}$ has a finite degree, have broad applications beyond the proof of Thurston’s characterization of rational maps; see, for example, [BN06, BEKP09, Sel12, Sel13, Koc13, Lod13, KPS16, FKK⁺17, Smi24a, Smi24b, BDP24]. These tools, for instance, allow to simultaneously study the entire Hurwitz (equivalence) class ${\mathcal{H}}_{f}$ of the finite degree Thurston map $f$. Here, two Thurston maps $f_{1}\colon S^{2}\dashrightarrow S^{2}$ and $f_{2}\colon S^{2}\dashrightarrow S^{2}$ with $P_{f_{1}}=P_{f_{2}}$ are said to be Hurwitz equivalent if there exist orientation-preserving homeomorphisms $\phi_{1},\phi_{2}\colon S^{2}\to S^{2}$ such that $\phi_{1}|P_{f_{1}}=\phi_{2}|P_{f_{1}}$ and $\phi_{1}\circ f_{1}=f_{2}\circ\phi_{2}$; see [BN06, Lod13, KPS16] for examples of results on Hurwitz classes. For instance, one can pose a question whether a given Thurston map is totally unobstructed, i.e., ${\mathcal{H}}_{f}$ consists of only realized Thurston maps.

Commutative diagram (1) is particularly powerful in two specific cases. The first is when the Thurston map $f\colon S^{2}\to S^{2}$ has the postcritical set $P_{f}=A$ consisting of exactly four points. In this situation, the spaces $\mathcal{T}_{A}$, $\mathcal{M}_{A}$, and $\mathcal{W}_{f}$ are simply Riemann surfaces. In fact, the Teichmüller space is biholomorphic to the unit disk ${\mathbb{D}}$ and the moduli space $\mathcal{M}_{A}$ is biholomorphic to the three punctured Riemann sphere $\Sigma=\widehat{{\mathbb{C}}}-\{0,1,\infty\}$. This allows the use of powerful machinery of one-dimensional holomorphic dynamics to study pullback maps. For example, this approach was utilized in [Smi24a] to derive an alternative proof of Thurston’s characterization of rational maps in the case of four postcritical values, as well as in [Smi24a, Smi24b] to investigate the global curve attractor conjecture, which was ultimately resolved in [BDP24] for all pcf rational maps with four postcritical values. Secondly, when the $X$-map $X_{f}$ is injective, the “inverse” of $\sigma_{f}$ descends to the so-called $g$-map $g_{f}:=Y_{f}\circ X_{f}^{-1}$, which is defined on the subset $X_{f}(\mathcal{W}_{f})$ of the moduli space $\mathcal{M}_{A}$; see [Koc13, Section 5] for examples of finite degree Thurston maps that satisfy this condition.

The theory of moduli maps, as outlined above, is developed for finite degree Thurston maps but has not yet been established in the context of transcendental Thurston maps. In this paper, we consider a certain family of Thurston maps, that includes maps of both finite and infinite degree, with four postsingular values and show that the corresponding pullback maps admit an analogue of commutative diagram (1), where the $X$-map is always injective, but the $Y$-map has infinite degree if the initial Thurston map $f\colon S^{2}\dashrightarrow S^{2}$ is transcendental. Using tools of one-dimensional holomorphic dynamics and hyperbolic geometry, we establish a Thurston-like realizability criterion for this family of maps. In particular, we demonstrate that the obstacle of the $Y$-map having infinite degree can be finessed. Afterward, we illustrate how the developed machinery can be used to investigate the properties of the corresponding Hurwitz classes.

In this paper, we study the family of Thurston maps $f\colon S^{2}\dashrightarrow S^{2}$ satisfying the following conditions:

1. (A)

   the map $f$ has at most three singular values;

2. (B)

   the postsingular set $P_{f}$ consists of exactly four points;

3. (C)

   there exists a set $B\subset P_{f}$ such that $|B|=3$, $S_{f}\subset B$, and $|\overline{f^{-1}(B)}\cap P_{f}|=3$.

Here, $\overline{f^{-1}(B)}$ is the closure of the set $f^{-1}(B)$ in the topology of $S^{2}$. In particular, it coincides with $f^{-1}(B)$ if the Thurston map $f$ has a finite degree; otherwise, it also includes the point where $f$ is not defined.

Clearly, conditions (B) and (C) are independent from the function-theoretical properties of the map $f$. More specifically, if the map $f$ has at most three singular values, then these conditions can be verified by analyzing the dynamics of the map $f$ on the finite set $P_{f}$. For instance, in the case of entire Thurston maps (those that can be restricted to self-maps of ${\mathbb{R}}^{2}$; see Section 2.2) with three singular and four postsingular values, three out of seven possible postsingular portraits satisfy condition (C); see Example 4.13. More examples of families of Thurston maps that meet these conditions can be found in Section 4.3.

Although conditions (A)–(C) are quite restrictive, there are still uncountably many pairwise combinatorially inequivalent both realized and obstructed Thurston maps that meet them; see Remark 4.11. Notably, these conditions are preserved under Hurwitz equivalence of Thurston maps. Finally, it is worth mentioning that all of our further results work in a slightly more general setting of marked Thurston maps satisfying analogous properties to (A)–(C); see Sections 2.3 and 4.

Our paper is organized as follows. In Section 2, we review some general background. In Section 2.1, we fix the notation and state some basic definitions. We discuss topologically holomorphic maps in Section 2.2. The necessary background on Thurston maps is covered in Section 2.3. Section 2.4 introduces the Teichmüller and moduli spaces of a marked topological sphere. Finally, in Section 2.5, we define pullback maps, discuss their basic properties and their relations with the associated Thurston maps.

In Section 3, we present several results concerning the hyperbolic geometry and dynamics of holomorphic self-maps of the unit disk. Section 3.1 provides tools for identifying obstructions for Thurston maps with four postsingular values. We establish some estimates for hyperbolic contraction of inclusion maps between two hyperbolic Riemann surfaces in Section 3.2. In Section 3.3, we investigate dynamics of holomorphic self-maps of the unit disk satisfying certain additional assumptions.

Further, in Section 4, we develop the Thurston theory for a family of Thurston maps satisfying condition (A)–(C). In particular, in Section 4.1, we address the characterization problem for this class of Thurston maps and prove Main Theorem A. We study properties of Hurwitz classes and prove Main Theorem B and Corollary 1.1 in Section 4.2. Finally, in Section 4.3, we provide and analyze various examples.

Acknowledgments. I would like to express my deep gratitude to my thesis advisor, Dierk Schleicher, for introducing me to the fascinating world of Transcendental Thurston Theory. I am also profoundly thankful to Kevin Pilgrim and Lasse Rempe for the their valuable suggestions and for many helpful and inspiring discussions. I would like to thank Centre National de la Recherche Scientifique (CNRS) for supporting my visits to the University of Saarland and University of Liverpool, where these conversations took place. Special thanks go to Anna Jové and Zachary Smith for the discussions on the dynamics of holomorphic self-maps of the unit disk and their diverse applications.

The sets of positive integers, non-zero integers, integers, real and complex numbers are denoted by ${\mathbb{N}}$, ${\mathbb{Z}}^{*}$, ${\mathbb{Z}}$, ${\mathbb{R}}$, and ${\mathbb{C}}$, respectively. We use the notation ${\mathbb{I}}:=[0,1]$ for the closed unit interval on the real line, ${\mathbb{D}}:=\{z\in{\mathbb{C}}\colon|z|<1\}$ for the open unit disk in the complex plane, ${\mathbb{D}}^{*}:={\mathbb{D}}-\{0\}$ for the punctured unit disk, ${\mathbb{C}}^{*}:={\mathbb{C}}-\{0\}$ for the punctured complex plane, ${\mathbb{H}}:=\{z\in{\mathbb{C}}:\operatorname{Im}(z)>0\}$ for the upper half-plane, $\widehat{{\mathbb{C}}}:={\mathbb{C}}\cup\{\infty\}$ for the Riemann sphere, and $\Sigma$ for the three-punctured Riemann sphere $\widehat{{\mathbb{C}}}-\{0,1,\infty\}$. The open and closed disks of radius $r>0$ centered at $0$ are denoted by ${\mathbb{D}}_{r}$ and $\overline{{\mathbb{D}}}_{r}$, respectively. Finally, $\arg(z)\in[0,2\pi)$ and $|z|$ denote the argument and the absolute value, respectively, of the complex number $z$.

We denote the oriented 2-dimensional sphere by $S^{2}$. In this paper, we treat it as a purely topological object. In particular, our convention is to write $g\colon{\mathbb{C}}\to\widehat{{\mathbb{C}}}$ or $g\colon\widehat{{\mathbb{C}}}\to\widehat{{\mathbb{C}}}$ to indicate that $g$ is holomorphic, and $f\colon S^{2}\to S^{2}$ if $f$ is only continuous. The same rule applies to the notation $g\colon\widehat{{\mathbb{C}}}\dashrightarrow\widehat{{\mathbb{C}}}$ and $f\colon S^{2}\dashrightarrow S^{2}$ (see Section 2.2 for the details).

The cardinality of a set $A$ is denoted by $|A|$ and the identity map on $A$ by $\operatorname{id}_{A}$. If $f\colon U\to V$ is a map and $W\subset U$, then $f|W$ stands for the restriction of $f$ to $W$. If $U$ is a topological space and $W\subset U$, then $\overline{W}$ denotes the closure and $\partial W$ the boundary of $W$ in $U$.

A subset $D$ of $\widehat{{\mathbb{C}}}$ is called an open Jordan region if there exists an injective continuous map $\varphi\colon\overline{{\mathbb{D}}}\to\widehat{{\mathbb{C}}}$ such that $D=\varphi({\mathbb{D}})$. In this case, $\partial D=\varphi(\partial{\mathbb{D}})$ is a simple closed curve in $\widehat{{\mathbb{C}}}$. A domain $U\subset\widehat{{\mathbb{C}}}$ is called an annulus if $\widehat{{\mathbb{C}}}-U$ has two connected components. The modulus of an annulus $U$ is denoted by $\mathrm{mod}(U)$ (see [Hub06, Proposition 3.2.1] for the definition).

Let $U$ and $V$ be topological spaces. A continuous map $H\colon U\times{\mathbb{I}}\to V$ is called a homotopy from $U$ to $V$. When $U=V$, we simply say that $H$ is a homotopy in $U$. Given a homotopy $H\colon U\times{\mathbb{I}}\to V$, for each $t\in{\mathbb{I}}$, we associate the time-$t$ map $H_{t}:=H(t,\cdot)\colon U\to V$. Sometimes it is convenient to think of the homotopy $H$ as a continuous family of its time maps $(H_{t})_{t\in{\mathbb{I}}}$. The homotopy $H$ is called an (ambient) isotopy if the map $H_{t}\colon U\to V$ is a homeomorphism for each $t\in{\mathbb{I}}$. Suppose $A$ is a subset of $U$. An isotopy $H\colon U\times{\mathbb{I}}\to V$ is said to be an isotopy relative to $A$ (abbreviated “$H$ is an isotopy rel. $A$”) if $H_{t}(p)=H_{0}(p)$ for all $p\in A$ and $t\in{\mathbb{I}}$.

Given $M,N\subset U$, we say that $M$ is homotopic (in $U$) to $N$ if there exists a homotopy $H\colon U\times{\mathbb{I}}\to U$ with $H_{0}=\operatorname{id}_{U}$ and $H_{1}(M)=N$. Two homeomorphisms $\varphi_{0},\varphi_{1}\colon U\to V$ are called isotopic (rel. $A\subset U$) if there exists an isotopy $H\colon U\times{\mathbb{I}}\to V$ (rel. $A$) with $H_{0}=\varphi_{0}$ and $H_{1}=\varphi_{1}$.

We assume that every topological surface $X$ is oriented. We denote by $\operatorname{Homeo}^{+}(X)$ and $\operatorname{Homeo}^{+}(X,A)$ the group of all orientation-preserving self-homeomorphisms of $X$ and the group of all orientation-preserving self-homeomorphisms of $X$ fixing $A$ pointwise, respectively. We use the notation $\operatorname{Homeo}^{+}_{0}(X,A)$ for the subgroup of $\operatorname{Homeo}^{+}(X,A)$ consisting of all homeomorphisms isotopic rel. $A$ to $\operatorname{id}_{X}$.

In this section, we briefly recall the definition of a topologically holomorphic map and some of its basic properties (for more detailed discussion see [MPR24, Section 2.3]; see also [LP20]).

Note that in condition (4) of Definition 2.1, the homeomorphism $\psi$ is defined uniquely up to post-composition with a conformal automorphism of $S_{X}$ for fixed $\varphi$ and $S_{X}$.

It is straightforward to define the concepts of regular, singular, critical, and asymptotic values, as well as regular and critical points and their local degrees (denoted by $\deg(f,\cdot)$) for the topologically holomorphic map $f$ (see [MPR24, Definition 2.7]). We denote by $S_{f}\subset Y$ the singular set of $f$, i.e., the union of all singular values of the topologically holomorphic map $f\colon X\to Y$. We say that the map $f$ is of finite type or belongs to the Speiser class $\mathcal{S}$ if the set $S_{f}$ is finite.

In this paper, we study topologically holomorphic maps $f\colon X\to S^{2}$, where $X$ is either the sphere $S^{2}$ or the punctured sphere $S^{2}-\{e\}$. In the latter case, we assume that $f$ cannot be extended as a topologically holomorphic map to the entire sphere $S^{2}$. For the sake of simplicity, we are going to use the notation $f\colon S^{2}\dashrightarrow S^{2}$ in order to indicate that $f$ might not be defined at a single point $e\in S^{2}$. Similar to the holomorphic case, the point $e$ is referred as the essential singularity of the map $f$. Likewise, for a holomorphic map $g$ defined everywhere on $\widehat{{\mathbb{C}}}$ with the possible exception of a single essential singularity, we use the notation $g\colon\widehat{{\mathbb{C}}}\dashrightarrow\widehat{{\mathbb{C}}}$, and we say that $g\colon\widehat{{\mathbb{C}}}\dashrightarrow\widehat{{\mathbb{C}}}$ is holomorphic.

It is possible to derive the following isotopy lifting property for topologically holomorphic maps as above in the case when they are of finite type (cf. [ERG15, Propostion 2.3]).

Due to the Uniformization Theorem and item (4) of Definition 2.1, in the case when $X=S^{2}$, a topologically holomorphic map $f\colon X\to S^{2}$ can be written as $f=\varphi\circ g\circ\psi^{-1}$, where $g\colon\widehat{{\mathbb{C}}}\to\widehat{{\mathbb{C}}}$ is a non-constant rational map and $\varphi,\psi\colon S^{2}\to\widehat{{\mathbb{C}}}$ are orientation-preserving homeomorphisms. In fact, in this case $f\colon S^{2}\to S^{2}$ is simply a branched self-covering of $S^{2}$, which is always of finite type and has finite topological degree.

Similarly, in the case when $X=S^{2}-\{e\}$, we can write $f$ as $\varphi\circ g\circ\psi^{-1}$ such that $g\colon R\to\widehat{{\mathbb{C}}}$ is a non-constant meromorphic map, $\varphi\colon S^{2}\to\widehat{{\mathbb{C}}}$ and $\psi\colon S^{2}-\{e\}\to R$ are orientation-preserving homeomorphisms, where $R={\mathbb{C}}$ or $R={\mathbb{D}}$. Suppose that the map $f$ is of finite type. Then the image of $\psi$ above does not depend on the choice of the homeomorphism $\varphi$ (see [Ere04, pp. 3-4]; it essentially follows from Proposition 2.2 and some well-known facts from the theory of quasiconformal mappings). Thus, finite-type topologically holomorphic maps for which the image of $\psi$ is ${\mathbb{C}}$ are referred to as parabolic type maps, while those for which the image of $\psi$ is ${\mathbb{D}}$ are called hyperbolic type maps.

Further, we assume that every topologically holomorphic map $f\colon S^{2}\dashrightarrow S^{2}$ we consider either has no essential singularities or is a finite-type topologically holomorphic map of parabolic type. Definition 2.1 and Great Picard’s Theorem imply that in any neighborhood of an essential singularity $e$ of such a map $f$, every value is attained infinitely often with at most two exceptions. In particular, $f$ can have at most two omitted values, i.e., points $p$ in $S^{2}$ such that the preimage $f^{-1}(p)$ is empty. Furthermore, each omitted value is an asymptotic value of $f$. Additionally, observe that if $A\subset S^{2}$ is a finite set with $|A|\geq 3$ and $S_{f}\subset A$, then the restriction

is a covering map. Note that the closure $\overline{f^{-1}(A)}$ equals $f^{-1}(A)$ if $f\colon S^{2}\to S^{2}$ has no essential singularity. Otherwise, $\overline{f^{-1}(A)}$ consists of $f^{-1}(A)$ and the essential singularity $e\in S^{2}$ of $f$ due to Great Picard’s Theorem and the assumption $|A|\geq 3$.

We say that a topologically holomorphic map $f\colon S^{2}\dashrightarrow S^{2}$ is transcendental if it has an essential singularity. Given our previous assumptions on the map $f$, this is equivalent to saying that $f$ has infinite topological degree. The map $f\colon S^{2}\dashrightarrow S^{2}$ is called entire if either $f$ has finite topological degree and there exists a point $p\in S^{2}$ such that $f^{-1}(p)=\{p\}$ (in which case $f$ is called a topological polynomial), or $f$ has infinite topological degree and $f^{-1}(e)=\emptyset$, where $e$ is the essential singularity of $f$. We can view entire topologically holomorphic maps as topologically holomorphic self-maps of ${\mathbb{R}}^{2}$.

Let $f\colon S^{2}\dashrightarrow S^{2}$ be a topologically holomorphic map. Then the postsingular set $P_{f}$ of the map $f$ is defined as

In other words, the postsingular set $P_{f}$ is the union of all forward orbits of the singular values of $f$. It is worth noting that some of these orbits might terminate after several iterations if a singular value reaches the essential singularity of the map $f$.

We say that $f\colon S^{2}\dashrightarrow S^{2}$ is postsingularly finite (psf in short) if the set $P_{f}$ is finite, i.e., $f$ has finitely many singular values and each of them eventually becomes periodic or lands on the essential singularity of $f$ under the iteration. Postsingularly finite topologically holomorphic maps of finite degree are also called postcritically finite (pcf in short), and their postsingular values are called postcritical, as their singular values are always critical.

Now we are ready to state one of the key definitions of this section.

We often consider marked Thurston maps in the same way as usual Thurston maps and use the notation $f\colon(S^{2},A)\righttoleftarrow$ while still assuming that $f$ might not be defined on the entire sphere $S^{2}$. If no specific marked set is mentioned, we assume it to be $P_{f}$. Note that when the marked set $A$ contains the essential singularity of the map $f$, the set $A$ is not forward invariant with respect to $f$ in the usual sense. However, if $|A|\geq 3$, then $A\subset\overline{f^{-1}(A)}$.

The dynamics of a Thurston map on its marked set or some other finite subsets of $S^{2}$ can also be represented graphically, in a way that turns out to be useful in study. Suppose that $f\colon S^{2}\dashrightarrow S^{2}$ is a Thurston map and $A\subset S^{2}$ is a finite set such that every $a\in A$ is either the essential singularity of $f$ or $f(a)\in A$. Then the dynamical portrait of the map $f$ on the set $A$ is a directed abstract graph with the vertex set $A$, where for each vertex $v\in A$ that is not the essential singularity of $f$, there is a unique directed edge from $v$ to $f(v)$, and if $v\in A$ is the essential singularity of $f$, there are no outgoing edges from $v$. If the set $A$ coincides with the postsingular set $P_{f}$, the dynamical portrait of $f$ on the set $A$ is called the postsingular portrait of the Thurston map $f$.

The notion of isotopy for Thurston maps depends on their common marked set. Consequently, we sometimes refer to isotopy relative $A$ (or rel. $A$ for short) to specify which marked set is being considered. This applies to other notions introduced below that also depend on the choice of the marked set.

We say that two (marked) Thurston maps are combinatorially equivalent if they are “topologically conjugate up to isotopy”:

A Thurston map $f\colon(S^{2},A)\righttoleftarrow$ is said to be realized if it is combinatorially equivalent to a postsingularly finite holomorphic map $g\colon(\widehat{{\mathbb{C}}},P)\righttoleftarrow$. If $f\colon(S^{2},A)\righttoleftarrow$ is not realized, we say that it is obstructed.

Let $A\subset S^{2}$ be a finite set. We say that a simple closed curve $\gamma\subset S^{2}-A$ is essential in $S^{2}-A$ if each connected component of $S^{2}-\gamma$ contains at least two points of the set $A$. In other words, $\gamma$ is essential in $S^{2}-A$ if it cannot be shrinked to a point via a homotopy in $S^{2}-A$.

If $n=1$ in Definition 2.10, then $\gamma$ is called a Levy fixed (or Levy invariant) curve. Levy fixed curve $\gamma$ is called weakly degenerate if $f$ is injective on one of the connected components of $S^{2}-\widetilde{\gamma}$. If additionally the image of this connected component $U$ under $f$ contains the same points of the set $A$ as $U$, i.e., $U\cap A=f(U)\cap A$, we say that $\gamma$ is a degenerate Levy fixed curve for the Thurston map $f\colon(S^{2},A)\righttoleftarrow$.

The following observation is widely known in the context of finite degree Thurston maps [Hub16, Exercise 10.3.6], and its proof extends to the case of transcendental Thurston maps as well (see Section 3.1 for the proof).

We require one more notion of equivalence between Thurston maps.

If $f\colon(S^{2},A)\righttoleftarrow$ is a Thurston map, then the Hurwitz class ${\mathcal{H}}_{f,A}$ of $f$ is the union of all Thurston maps with the marked set $A$ that are Hurwitz equivalent to $f$. If $A$ coincides with the postsingular set $P_{f}$ of $f$, we simply use the notation ${\mathcal{H}}_{f}$. We say that a Thurston map $f\colon(S^{2},A)\righttoleftarrow$ is totally unobstructed if every Thurston map in ${\mathcal{H}}_{f,A}$ is unobstructed.

Let $A\subset S^{2}$ be a finite set containing at least three points. Then the Teichmüller space of the sphere $S^{2}$ with the marked set $A$ is defined as

where $\varphi_{1}\sim\varphi_{2}$ if there exists a Möbius transformation $M$ such that $\varphi_{1}$ is isotopic rel. $A$ to $M\circ\varphi_{2}$.

Similarly, we define the moduli space of the sphere $S^{2}$ with the marked set $A$:

where $\eta_{1}\sim\eta_{2}$ if there exists a Möbius transformation $M$ such that $\eta_{1}=M\circ\eta_{2}$.

Further, $[\cdot]$ denotes an equivalence class corresponding to a point of either the Teichmüller space $\mathcal{T}_{A}$ or the moduli space $\mathcal{M}_{A}$. Note that there is an obvious map $\pi\colon\mathcal{T}_{A}\to\mathcal{M}_{A}$ defined as $\pi([\varphi])=[\varphi|A]$. According to [FM12, Proposition 2.3], when $|A|=3$, both the Teichmüller space $\mathcal{T}_{A}$ and the moduli space $\mathcal{M}_{A}$ are just single points. Therefore, for the rest of this section, we assume that $|A|\geq 4$.

It is known that the Teichmüller space $\mathcal{T}_{A}$ admits a complete metric $d_{T}$, known as the Teichmüller metric [Hub06, Proposition 6.4.4]. Moreover, with respect to the topology induced by this metric, $\mathcal{T}_{A}$ is a contractible space [Hub06, Corollary 6.7.2]. At the same time, both $\mathcal{T}_{A}$ and $\mathcal{M}_{A}$ admit structures of $(|A|-3)$-complex manifolds (see [Hub06, Theorem 6.5.1]) so that the map $\pi\colon\mathcal{T}_{A}\to\mathcal{M}_{A}$ becomes a holomorphic universal covering map [Hub16, Section 10.9].

Moreover, the complex structure of $\mathcal{M}_{A}$ is quite explicit in the general case. Let $A=\{a_{1},a_{2},\dots,a_{k},a_{k+1},a_{k+2},a_{k+3}\}$, $k\geq 1$, where the indexing of the points of $A$ is chosen arbitrarily. Define the map $h\colon\mathcal{M}_{A}\to{\mathbb{C}}^{k}-\mathcal{L}_{k}$ by

where the representative $\varphi\colon S^{2}\to\widehat{{\mathbb{C}}}$ is chosen so that $\varphi(a_{k+1})=0,\varphi(a_{k+2})=1$, and $\varphi(a_{k+3})=\infty$, and where $\mathcal{L}_{k}$ is the subset of ${\mathbb{C}}^{k}$ defined by

It is known that the map $h$ provides a biholomorphism between $\mathcal{M}_{A}$ and ${\mathbb{C}}^{k}-\mathcal{L}_{k}$ (see [Hub16, Section 10.9]).

Our focus in this paper is on the case when $|A|=4$. In this situation, the Teichmüller space $\mathcal{T}_{A}$ is biholomorphic to ${\mathbb{D}}$, with the metric $d_{T}$ coinciding with the usual hyperbolic metric on ${\mathbb{D}}$; see [Hub06, Corollary 6.10.3 and Theorem 6.10.6]. Furthermore, the moduli space $\mathcal{M}_{A}$ is biholomorphic to the three-punctured Riemann sphere $\Sigma$.