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# Dynamical approximations of postsingularly finite entire maps

Malavika Mukundan, Nikolai Prochorov, Bernhard Reinke

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Malavika Mukundan, Nikolai Prochorov, Bernhard Reinke. Dynamical approximations of postsingularly finite entire maps. 2024. hal-04526931

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# Dynamical approximations of postsingularly finite entire maps

Malavika Mukundan, Nikolai Prochorov, Bernhard Reinke

<figure><img src="image_2.png" /></figure>

## To cite this version:

Malavika Mukundan, Nikolai Prochorov, Bernhard Reinke. Dynamical approximations of postsingularly finite entire maps. 2024. hal-04526931

HAL Id: hal-04526931

https://hal.science/hal-04526931v1

Preprint submitted on 29 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
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# DYNAMICAL APPROXIMATIONS OF POSTSINGULARLY FINITE ENTIRE MAPS

MALAVIKA MUKUNDAN, NIKOLAI PROCHOROV, AND BERNHARD REINKE

ABSTRACT. We prove that every postsingularly finite entire map $g$ can be approximated by a sequence of postcritically finite complex polynomials ($g_n$) such that their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every $n \in \mathbb{N}$. To establish this result, we introduce the notion of combinatorial convergence for sequences of entire Thurston maps defined on the topological plane $\mathbb{R}^2$ and having the same marked set $A$. We prove that if such a sequence ($f_n$) converges combinatorially to a Thurston map $f$, then the sequence of Thurston pullback maps ($\sigma_{f_n}$) converges to $\sigma_f$ locally uniformly on the Teichmüller space $\text{Teich}(\mathbb{R}^2, A)$.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Main results.</td><td>3</td></tr><tr><td>1.2. Organization of the paper.</td><td>5</td></tr><tr><td>2. Background</td><td>6</td></tr><tr><td>2.1. Notation and basic concepts</td><td>6</td></tr><tr><td>2.2. Planar embedded graphs</td><td>7</td></tr><tr><td>2.3. Topologically holomorphic maps</td><td>9</td></tr><tr><td>2.4. Entire Thurston maps</td><td>12</td></tr><tr><td>2.5. Quasicoformal maps and Teichmüller spaces</td><td>14</td></tr><tr><td>2.6. Thurston pullback map</td><td>15</td></tr><tr><td>2.7. Admissible quadruples</td><td>17</td></tr><tr><td>3. Approximations of entire Thurston maps</td><td>21</td></tr><tr><td>3.1. Combinatorial convergence</td><td>22</td></tr><tr><td>3.2. Topological convergence</td><td>24</td></tr><tr><td>3.3. Polynomial approximations</td><td>26</td></tr><tr><td>3.4. Convergence of $\sigma$-maps</td><td>31</td></tr><tr><td>4. Approximations of postsingularly finite entire maps</td><td>34</td></tr><tr><td>4.1. Topological properties of locally uniform convergence</td><td>35</td></tr><tr><td>4.2. Dynamical approximations</td><td>40</td></tr><tr><td>References</td><td>42</td></tr></table>

Date: January 17, 2024.

2020 Mathematics Subject Classification. Primary 37F20; Secondary 37F10, 37F34.

Key words and phrases. Thurston maps, postsingularly finite entire maps, Teichmüller spaces, Thurston pullback maps, planar embedded graphs, locally uniform convergence.
# DYNAMICAL APPROXIMATIONS OF POSTSINGULARLY FINITE ENTIRE MAPS

MALAVIKA MUKUNDAN, NIKOLAI PROCHOROV, AND BERNHARD REINKE

ABSTRACT. We prove that every postsingularly finite entire map $g$ can be approximated by a sequence of postcritically finite complex polynomials $(g_n)$ such that their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every $n \in \mathbb{N}$. To establish this result, we introduce the notion of combinatorial convergence for sequences of entire Thurston maps defined on the topological plane $\mathbb{R}^2$ and having the same marked set $A$. We prove that if such a sequence $(f_n)$ converges combinatorially to a Thurston map $f$, then the sequence of Thurston pullback maps $(\sigma_{f_n})$ converges to $\sigma_f$ locally uniformly on the Teichmüller space $\text{Teich}(\mathbb{R}^2, A)$.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Main results.</td><td>3</td></tr><tr><td>1.2. Organization of the paper.</td><td>5</td></tr><tr><td>2. Background</td><td>6</td></tr><tr><td>2.1. Notation and basic concepts</td><td>6</td></tr><tr><td>2.2. Planar embedded graphs</td><td>7</td></tr><tr><td>2.3. Topologically holomorphic maps</td><td>9</td></tr><tr><td>2.4. Entire Thurston maps</td><td>12</td></tr><tr><td>2.5. Quasicoformal maps and Teichmüller spaces</td><td>14</td></tr><tr><td>2.6. Thurston pullback map</td><td>15</td></tr><tr><td>2.7. Admissible quadruples</td><td>17</td></tr><tr><td>3. Approximations of entire Thurston maps</td><td>21</td></tr><tr><td>3.1. Combinatorial convergence</td><td>22</td></tr><tr><td>3.2. Topological convergence</td><td>24</td></tr><tr><td>3.3. Polynomial approximations</td><td>26</td></tr><tr><td>3.4. Convergence of $\sigma$-maps</td><td>31</td></tr><tr><td>4. Approximations of postsingularly finite entire maps</td><td>34</td></tr><tr><td>4.1. Topological properties of locally uniform convergence</td><td>35</td></tr><tr><td>4.2. Dynamical approximations</td><td>40</td></tr><tr><td>References</td><td>42</td></tr></table>

Date: January 17, 2024.

2020 Mathematics Subject Classification. Primary 37F20; Secondary 37F10, 37F34.

Key words and phrases. Thurston maps, postsingularly finite entire maps, Teichmüller spaces, Thurston pullback maps, planar embedded graphs, locally uniform convergence.
# 1. INTRODUCTION

The family of complex polynomials has some of the most well-understood dynamics among all entire maps. At the same time, every entire map $g$ can be approximated by a sequence of polynomials $(g_n)$ that are easier to handle compared with $g$. A priori, we cannot expect such approximations to be helpful in understanding the dynamics of $g$; even if $g$ and $g_n$ are close from the numerical point of view, their dynamical properties can still be drastically different. For instance, it is unclear to what extent Taylor approximations preserve dynamics. This raises the question: can we produce "dynamically meaningful" approximations of entire maps by sequences of complex polynomials? This problem is especially natural for families of maps that share several crucial dynamical properties with polynomials, such as entire maps of *finite type* (i.e., maps with finitely many *singular values*).

The problem of dynamical approximation was first studied for the exponential family {$z \mapsto \lambda \exp(z), \lambda \in \mathbb{C}^*$} in $[\mathrm{BDH}^+00]$. The authors considered approximations provided by unicritical polynomials of the form $\lambda(1+z/n)^n$, and showed that hyperbolic components in the parameter spaces of these polynomials of degree $n$ converge to those in the exponential family as the degree $n$ tends to infinity. They also established that certain *hairs* in the parameter plane for the exponential family are limits of corresponding *external rays* for the polynomial ones. In other words, these results show that the classical approximation of $\lambda \exp(z)$ by the sequence $(\lambda(1+z/n)^n)$ is meaningful from the dynamical point of view. Further developments on this topic include [MB12], where the author introduced a "dynamical" metric on the space of all non-constant non-linear entire maps and showed that convergence in this metric implies kernel convergence of hyperbolic components in the corresponding parameter spaces.

In this paper, we consider *postsingularly finite* (or, shortly, *psf*) entire maps, i.e., maps of finite type for which all singular values are (pre-)periodic. Psf polynomials are also called *postcritically finite* (pscf in short) since all their singular values are critical values. While postsingularly finite entire maps might seem a rather special subset of the space of all entire maps, they play an important role in modern holomorphic dynamics owing to their utility in studying the structure of the class of finite type functions. For instance, the complicated structure of the Mandelbrot set can be described in terms of pcf polynomials [DH84] (see also [Sch04]). Moreover, in the rational setting, McMullen conjectured the existence of a strong dynamical similarity between a dense subset of the space of all rational maps with connected *Julia sets* and postcritically finite rational maps (see [McM94, Conjecture 1.1, Section 4]).

One of the crucial properties of postcritically finite polynomials is that they admit a complete description in purely combinatorial terms, for instance, by *Hubbard trees* or *external angles* (see [Poi93], [Poi10]). Extending such combinatorial descriptions to the family of psf entire maps is an active area of research (see [LSV08], [PRS21] for the exponential family, and [Pfr19], [PPS21] for the general case). However, many questions still remain open. Dynamically meaningful polynomial approximations for postsingularly finite entire maps would provide intuition and potentially chalk out a pathway for studying the combinatorics of such maps.

One of the reasons for the successful development of the theory of postcritically finite polynomials is Thurston theory. This theory is based on the idea that holomorphic maps are too rigid, and one should abstract from the underlying complex structure and consider the more general setup of *branched self-coverings* of the 2-sphere $\mathbb{S}^2$ or *topologically holomorphic maps* defined on the plane $\mathbb{R}^2$ (see Definition 2.2 for further details). When such a map is
# 1. INTRODUCTION

The family of complex polynomials has some of the most well-understood dynamics among all entire maps. At the same time, every entire map $g$ can be approximated by a sequence of polynomials $(g_n)$ that are easier to handle compared with $g$. A priori, we cannot expect such approximations to be helpful in understanding the dynamics of $g$; even if $g$ and $g_n$ are close from the numerical point of view, their dynamical properties can still be drastically different. For instance, it is unclear to what extent Taylor approximations preserve dynamics. This raises the question: can we produce "dynamically meaningful" approximations of entire maps by sequences of complex polynomials? This problem is especially natural for families of maps that share several crucial dynamical properties with polynomials, such as entire maps of *finite type* (i.e., maps with finitely many *singular values*).

The problem of dynamical approximation was first studied for the exponential family {$z \mapsto \lambda \exp(z), \lambda \in \mathbb{C}^*$} in $[\mathrm{BDH}^+00]$. The authors considered approximations provided by unicritical polynomials of the form $\lambda(1+z/n)^n$, and showed that hyperbolic components in the parameter spaces of these polynomials of degree $n$ converge to those in the exponential family as the degree $n$ tends to infinity. They also established that certain *hairs* in the parameter plane for the exponential family are limits of corresponding *external rays* for the polynomial ones. In other words, these results show that the classical approximation of $\lambda \exp(z)$ by the sequence $(\lambda(1+z/n)^n)$ is meaningful from the dynamical point of view. Further developments on this topic include [MB12], where the author introduced a "dynamical" metric on the space of all non-constant non-linear entire maps and showed that convergence in this metric implies kernel convergence of hyperbolic components in the corresponding parameter spaces.

In this paper, we consider *postsingularly finite* (or, shortly, *psf*) entire maps, i.e., maps of finite type for which all singular values are (pre-)periodic. Psf polynomials are also called *postcritically finite* (pscf in short) since all their singular values are critical values. While postsingularly finite entire maps might seem a rather special subset of the space of all entire maps, they play an important role in modern holomorphic dynamics owing to their utility in studying the structure of the class of finite type functions. For instance, the complicated structure of the Mandelbrot set can be described in terms of pcf polynomials [DH84] (see also [Sch04]). Moreover, in the rational setting, McMullen conjectured the existence of a strong dynamical similarity between a dense subset of the space of all rational maps with connected *Julia sets* and postcritically finite rational maps (see [McM94, Conjecture 1.1, Section 4]).

One of the crucial properties of postcritically finite polynomials is that they admit a complete description in purely combinatorial terms, for instance, by *Hubbard trees* or *external angles* (see [Poi93], [Poi10]). Extending such combinatorial descriptions to the family of psf entire maps is an active area of research (see [LSV08], [PRS21] for the exponential family, and [Pfr19], [PPS21] for the general case). However, many questions still remain open. Dynamically meaningful polynomial approximations for postsingularly finite entire maps would provide intuition and potentially chalk out a pathway for studying the combinatorics of such maps.

One of the reasons for the successful development of the theory of postcritically finite polynomials is Thurston theory. This theory is based on the idea that holomorphic maps are too rigid, and one should abstract from the underlying complex structure and consider the more general setup of *branched self-coverings* of the 2-sphere $\mathbb{S}^2$ or *topologically holomorphic maps* defined on the plane $\mathbb{R}^2$ (see Definition 2.2 for further details). When such a map is
postsingularly finite, it is called a *Thurston map* (for a precise version see Definition 2.14). An important application of Thurston theory is to generate holomorphic psf (or pcf) maps on $\mathbb{C}$ defined, for instance, by combinatorial objects such as *spiders*; see [Hub16, Chapter 10].

The core of this theory is a method of determining whether a given Thurston map is *realized* by a postsingularly finite holomorphic map (see Definition 2.15). This involves the dynamics of an operator called the *Thurston pullback map*, defined on a suitably chosen Teichmüller space (see Sections 2.5 and 2.6). When a Thurston map is not realised holomorphically, it is said to be *obstructed*. Crucially, it is known when a *polynomial* Thurston map is realized by a postcritically finite complex polynomial. The Levy-Berstein theorem [Hub16, Theorem 10.3.8], and Thurston’s characterization theorem [DH93] (which applies in the more general setting of pcf branched self-covers of $\mathbb{S}^2$) provide a complete answer to this question.

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. The last few decades have seen the development of this theory in the transcendental setting, but several unsolved questions remain. The biggest of these is the nature of obstructions for a *transcendental Thurston map* (in other words, necessary and sufficient conditions for such a map to be obstructed). Obstructions are understood in special classes such as *exponential Thurston maps* (see [HSS09]), and have recently been studied for *structurally finite Thurston maps* in [She22], but the general case remains wide open.

This discussion raises, in this topological setting, the same approximation question mentioned earlier for entire maps: can we view entire Thurston maps as dynamically meaningful limits of polynomial Thurston maps? As a natural extension, we can ask approximation-type questions for other constructions in Thurston theory such as Thurston pullback maps. This article contributes to transcendental Thurston theory by providing a positive answer to the above questions. It broadens our understanding of the relationships between transcendental and polynomial Thurston maps. Furthermore, it translates these results in the Thurston theory regime back to the holomorphic one, using them to construct meaningful approximations of postsingularly finite entire maps on the dynamical plane. Concurrent work by the first author [Muk23] uses spiders to prove similar results for the special case of postsingularly finite exponential maps $z \mapsto \lambda \exp(z)$ and gives further information on the approximating polynomials in terms of their location in the parameter space of unicritical polynomials of a given degree.

**1.1. Main results.** One of the main theorems of this paper shows that all postsingularly finite entire maps can be viewed as dynamically meaningful limits of postcritical finite complex polynomials.

**Main Theorem A.** Let $g$ be an arbitrary postsingularly finite entire map. Then there exists a sequence of postcritically finite polynomials $(g_n)$ converging locally uniformly to $g$ such that $g$ and $g_n$ have the same postsingular portrait for all $n \in \mathbb{N}$.

See Definition 2.18 for the precise notion of a *postsingular portrait*. In particular, Main Theorem A implies that for each $n \in \mathbb{N}$, there is a map $\varphi_n: P_g \to P_{g_n}$ (where $P_g$ and $P_{g_n}$ are the corresponding *postsingular sets*), that conjugates $g|P_g$ and $g_n|P_{g_n}$ and converges to $\text{id}_{P_g}$ as $n$ tends to infinity. Note that the behavior of singular values under iteration is one of the most important dynamical features of an entire map. This behavior, in fact, controls the global dynamics of an entire map of finite type [Sch10]. Roughly speaking, in the setting of Main Theorem A, the “dynamical cores” of the polynomials $g_n$ look more and more like the
postsingularly finite, it is called a *Thurston map* (for a precise version see Definition 2.14). An important application of Thurston theory is to generate holomorphic psf (or pcf) maps on $\mathbb{C}$ defined, for instance, by combinatorial objects such as *spiders*; see [Hub16, Chapter 10].

The core of this theory is a method of determining whether a given Thurston map is *realized* by a postsingularly finite holomorphic map (see Definition 2.15). This involves the dynamics of an operator called the *Thurston pullback map*, defined on a suitably chosen Teichmüller space (see Sections 2.5 and 2.6). When a Thurston map is not realised holomorphically, it is said to be *obstructed*. Crucially, it is known when a *polynomial* Thurston map is realized by a postcritically finite complex polynomial. The Levy-Berstein theorem [Hub16, Theorem 10.3.8], and Thurston’s characterization theorem [DH93] (which applies in the more general setting of pcf branched self-covers of $\mathbb{S}^2$) provide a complete answer to this question.

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. The last few decades have seen the development of this theory in the transcendental setting, but several unsolved questions remain. The biggest of these is the nature of obstructions for a *transcendental Thurston map* (in other words, necessary and sufficient conditions for such a map to be obstructed). Obstructions are understood in special classes such as *exponential Thurston maps* (see [HSS09]), and have recently been studied for *structurally finite Thurston maps* in [She22], but the general case remains wide open.

This discussion raises, in this topological setting, the same approximation question mentioned earlier for entire maps: can we view entire Thurston maps as dynamically meaningful limits of polynomial Thurston maps? As a natural extension, we can ask approximation-type questions for other constructions in Thurston theory such as Thurston pullback maps. This article contributes to transcendental Thurston theory by providing a positive answer to the above questions. It broadens our understanding of the relationships between transcendental and polynomial Thurston maps. Furthermore, it translates these results in the Thurston theory regime back to the holomorphic one, using them to construct meaningful approximations of postsingularly finite entire maps on the dynamical plane. Concurrent work by the first author [Muk23] uses spiders to prove similar results for the special case of postsingularly finite exponential maps $z \mapsto \lambda \exp(z)$ and gives further information on the approximating polynomials in terms of their location in the parameter space of unicritical polynomials of a given degree.

**1.1. Main results.** One of the main theorems of this paper shows that all postsingularly finite entire maps can be viewed as dynamically meaningful limits of postcritical finite complex polynomials.

**Main Theorem A.** Let $g$ be an arbitrary postsingularly finite entire map. Then there exists a sequence of postcritically finite polynomials $(g_n)$ converging locally uniformly to $g$ such that $g$ and $g_n$ have the same postsingular portrait for all $n \in \mathbb{N}$.

See Definition 2.18 for the precise notion of a *postsingular portrait*. In particular, Main Theorem A implies that for each $n \in \mathbb{N}$, there is a map $\varphi_n: P_g \to P_{g_n}$ (where $P_g$ and $P_{g_n}$ are the corresponding *postsingular sets*), that conjugates $g|P_g$ and $g_n|P_{g_n}$ and converges to $\text{id}_{P_g}$ as $n$ tends to infinity. Note that the behavior of singular values under iteration is one of the most important dynamical features of an entire map. This behavior, in fact, controls the global dynamics of an entire map of finite type [Sch10]. Roughly speaking, in the setting of Main Theorem A, the “dynamical cores” of the polynomials $g_n$ look more and more like the
“dynamical core” of the limiting map $g$ as $n$ tends to infinity. This supplies a contrast with Taylor approximations ($h_n$) for $g$, where it is possible that the number of critical values of $h_n$ tends to infinity as $n \to \infty$. This results in the sets $P_{h_n}$ having a structure very different from $P_g$.

The combinatorics of the pcf polynomials $g_n$ from Main Theorem A also approach the combinatorics of $g$ in the following sense: let $\gamma \subset \mathbb{C}\backslash P_g$ be a simple closed curve and $\tilde{\gamma}$ be a connected component of $g^{-1}(\gamma)$. If $\tilde{\gamma}$ is also a simple closed curve, then, as it is shown in Corollary 4.7, for any sufficiently small $\varepsilon > 0$ and sufficiently large $n$, there exists a unique simple closed curve $\tilde{\gamma}_n \subset g_n^{-1}(\gamma)$ that lies in the $\varepsilon$-neighbourhood of $\tilde{\gamma}$. In particular, for sufficiently large $n$, the curves $\tilde{\gamma}_n$ and $\tilde{\gamma}$ are free-homotopic relative $P_{g_n} \cup P_g$ and, moreover, $\deg(g|\tilde{\gamma}) = \deg(g_n|\tilde{\gamma}_n)$. If $\tilde{\gamma}$ is not a simple closed curve (i.e., $\deg(g|\tilde{\gamma})$ is infinite), there exists a unique (at least for $n$ large enough) simple closed curve $\tilde{\gamma}_n \subset g_n^{-1}(\gamma)$ so that $d(z, \tilde{\gamma}_n) \to 0$ for every $z \in \tilde{\gamma}$ and, moreover, $\deg(g_n|\tilde{\gamma}_n) \to \infty$ as $n$ tends to $\infty$. For a more detailed version see Corollary 4.7. The above remarks provide a summary of the various points of view on the approximating maps we construct, along with the implications for their dynamics.

The main engine of the proof of Main Theorem A is Thurston theory. As it is common in the study of postsingularly finite maps, we start by establishing a desired result in the topological context, i.e., for Thurston maps. But talking about pointwise or locally uniform convergence for Thurston maps is not meaningful since they are usually considered up to isotopy (see Definition 2.16) relative to their postsingular sets. To sidestep this issue, we introduce the notion of *combinatorial convergence*.

Let $f_n: \mathbb{R}^2 \to \mathbb{R}^2$, $n \in \mathbb{N}$ and $f: \mathbb{R}^2 \to \mathbb{R}^2$ be Thurston maps with the same postsingular set $A$ for all $n \in \mathbb{N}$. Further assume that there exist points $b, t \in \mathbb{R}^2 \setminus A$ such that $f(b) = f_n(b) = t$ for all $n \in \mathbb{N}$. Let $\gamma \subset \mathbb{R}^2 \setminus A$ be a loop based at $t$, and let $\gamma\uparrow(f, b)$ and $\gamma\downarrow(f_n, b)$ respectively denote the $f$- and $f_n$-lifts of $\gamma$ beginning at $b$. We say that the sequence $(f_n)$ *converges combinatorially* to $f$ if for every such loop $\gamma$, the following conditions are satisfied for all $n$ sufficiently large:

* the closing behavior of $\gamma\uparrow(f, b)$ (i.e., whether it is a loop or a non-closed path) is the same as that of $\gamma\downarrow(f_n, b)$, and

* if $\gamma\uparrow(f, b)$ is a loop, then it is path-homotopic in $\mathbb{R}^2 \setminus A$ to $\gamma\downarrow(f_n, b)$.

See Definition 3.3 for a more general and precise formulation. In particular, if $(f_n)$ converges combinatorially to $f$, then the functions $f|A$ and $f_n|A$ coincide for all $n$ large enough.

We investigate more profound properties of this type of convergence and establish a topological version of Main Theorem A.

**Proposition 1.1.** Suppose that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is an entire Thurston map, then there exists a sequence $f_n: \mathbb{R}^2 \to \mathbb{R}^2$, $n \in \mathbb{N}$ of polynomial Thurston maps converging combinatorially to $f$.

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a Thurston map. Associated with $f$ is a holomorphic function $\sigma_f: \text{Teich}(\mathbb{R}^2, P_f) \to \text{Teich}(\mathbb{R}^2, P_f)$ called the Thurston pullback map (or the $\sigma$-map), where $\text{Teich}(\mathbb{R}^2, P_f)$ is a complex manifold called the *Teichmüller space of $\mathbb{R}^2$ with the marked set* $P_f$ or the Teichmüller space *modelled on* $\mathbb{R}^2\backslash P_f$ (see Sections 2.5, 2.6 for further details). Pullback maps provide a powerful tool to study the associated Thurston maps. In particular, $\sigma_f$ has a (unique) fixed point $\tau \in \text{Teich}(\mathbb{R}^2, P_f)$ if and only if $f$ is realized by a postsingularly finite entire map (see Proposition 2.24). Our strategy in proving Main Theorem A is to first solve the approximation problem for Thurston pullback maps. In a sense, we translate the original question to the Teichmüller space setting. The following theorem forms the
“dynamical core” of the limiting map $g$ as $n$ tends to infinity. This supplies a contrast with Taylor approximations ($h_n$) for $g$, where it is possible that the number of critical values of $h_n$ tends to infinity as $n \to \infty$. This results in the sets $P_{h_n}$ having a structure very different from $P_g$.

The combinatorics of the pcf polynomials $g_n$ from Main Theorem A also approach the combinatorics of $g$ in the following sense: let $\gamma \subset \mathbb{C}\backslash P_g$ be a simple closed curve and $\tilde{\gamma}$ be a connected component of $g^{-1}(\gamma)$. If $\tilde{\gamma}$ is also a simple closed curve, then, as it is shown in Corollary 4.7, for any sufficiently small $\varepsilon > 0$ and sufficiently large $n$, there exists a unique simple closed curve $\tilde{\gamma}_n \subset g_n^{-1}(\gamma)$ that lies in the $\varepsilon$-neighbourhood of $\tilde{\gamma}$. In particular, for sufficiently large $n$, the curves $\tilde{\gamma}_n$ and $\tilde{\gamma}$ are free-homotopic relative $P_{g_n} \cup P_g$ and, moreover, $\deg(g|\tilde{\gamma}) = \deg(g_n|\tilde{\gamma}_n)$. If $\tilde{\gamma}$ is not a simple closed curve (i.e., $\deg(g|\tilde{\gamma})$ is infinite), there exists a unique (at least for $n$ large enough) simple closed curve $\tilde{\gamma}_n \subset g_n^{-1}(\gamma)$ so that $d(z, \tilde{\gamma}_n) \to 0$ for every $z \in \tilde{\gamma}$ and, moreover, $\deg(g_n|\tilde{\gamma}_n) \to \infty$ as $n$ tends to $\infty$. For a more detailed version see Corollary 4.7. The above remarks provide a summary of the various points of view on the approximating maps we construct, along with the implications for their dynamics.

The main engine of the proof of Main Theorem A is Thurston theory. As it is common in the study of postsingularly finite maps, we start by establishing a desired result in the topological context, i.e., for Thurston maps. But talking about pointwise or locally uniform convergence for Thurston maps is not meaningful since they are usually considered up to isotopy (see Definition 2.16) relative to their postsingular sets. To sidestep this issue, we introduce the notion of combinatorial convergence.

Let $f_n: \mathbb{R}^2 \to \mathbb{R}^2$, $n \in \mathbb{N}$ and $f: \mathbb{R}^2 \to \mathbb{R}^2$ be Thurston maps with the same postsingular set $A$ for all $n \in \mathbb{N}$. Further assume that there exist points $b, t \in \mathbb{R}^2 \setminus A$ such that $f(b) = f_n(b) = t$ for all $n \in \mathbb{N}$. Let $\gamma \subset \mathbb{R}^2 \setminus A$ be a loop based at $t$, and let $\gamma\uparrow(f, b)$ and $\gamma\downarrow(f_n, b)$ respectively denote the $f$- and $f_n$-lifts of $\gamma$ beginning at $b$. We say that the sequence $(f_n)$ converges combinatorially to $f$ if for every such loop $\gamma$, the following conditions are satisfied for all $n$ sufficiently large:

* the closing behavior of $\gamma\uparrow(f, b)$ (i.e., whether it is a loop or a non-closed path) is the same as that of $\gamma\downarrow(f_n, b)$, and

* if $\gamma\uparrow(f, b)$ is a loop, then it is path-homotopic in $\mathbb{R}^2 \setminus A$ to $\gamma\downarrow(f_n, b)$.

See Definition 3.3 for a more general and precise formulation. In particular, if $(f_n)$ converges combinatorially to $f$, then the functions $f|A$ and $f_n|A$ coincide for all $n$ large enough.

We investigate more profound properties of this type of convergence and establish a topological version of Main Theorem A.

**Proposition 1.1.** Suppose that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is an entire Thurston map, then there exists a sequence $f_n: \mathbb{R}^2 \to \mathbb{R}^2$, $n \in \mathbb{N}$ of polynomial Thurston maps converging combinatorially to $f$.

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a Thurston map. Associated with $f$ is a holomorphic function $\sigma_f: \text{Teich}(\mathbb{R}^2, P_f) \to \text{Teich}(\mathbb{R}^2, P_f)$ called the Thurston pullback map (or the $\sigma$-map), where $\text{Teich}(\mathbb{R}^2, P_f)$ is a complex manifold called the *Teichmüller space of $\mathbb{R}^2$ with the marked set* $P_f$ or the Teichmüller space *modelled on* $\mathbb{R}^2\backslash P_f$ (see Sections 2.5, 2.6 for further details). Pullback maps provide a powerful tool to study the associated Thurston maps. In particular, $\sigma_f$ has a (unique) fixed point $\tau \in \text{Teich}(\mathbb{R}^2, P_f)$ if and only if $f$ is realized by a postsingularly finite entire map (see Proposition 2.24). Our strategy in proving Main Theorem A is to first solve the approximation problem for Thurston pullback maps. In a sense, we translate the original question to the Teichmüller space setting. The following theorem forms the
second main result of this paper and illustrates the effect of combinatorial convergence on
the behavior of the corresponding $\sigma$-maps.

**Main Theorem B.** Let $f_n: \mathbb{R}^2 \to \mathbb{R}^2, n \in \mathbb{N}$ and $f: \mathbb{R}^2 \to \mathbb{R}^2$ be Thurston maps with the same postsingular set $A$ for all $n \in \mathbb{N}$. If the sequence $(f_n)$ converges combinatorially to $f$, then the sequence $(\sigma_{f_n})$ converges locally uniformly on $\text{Teich}(\mathbb{R}^2, A)$ to $\sigma_f$.

It is easy to see that Main Theorem A follows from the points below:

(1) Main Theorem B, taken together with Proposition 1.1, tells us that for any Thurston map $f$ with $P_f = A$, there exists a sequence of polynomial Thurston maps $(f_n)$ with $P_{f_n} = A$ that converges combinatorially to $f$ so that the sequence $(\sigma_{f_n})$ converges locally uniformly on $\text{Teich}(\mathbb{R}^2, A)$ to $\sigma_f$.

(2) If $f$ is realised by a postsingularly finite entire map $g$, we use the weak contraction properties of $\sigma$-maps (see Proposition 2.22) and further analysis to show that for sufficiently large $n$, the map $f_n$ is realised by a postcritically finite complex polynomial $g_n$ such that

• $(g_n)$ converges to $g$ locally uniformly on $\mathbb{C}$, and

• $g_n$ has the same dynamical portrait as $g$, in particular, $g_n|P_{g_n}$ is conjugate to $g|P_g$. This statement is proved in Theorem 4.12.

Proposition 1.1 and Main Theorem B allow us to think of entire Thurston maps and their $\sigma$-maps as limits of polynomial Thurston maps and the corresponding $\sigma$-maps. These results form the first direct link between polynomial and entire Thurston theory, and present a new viewpoint for studying other problems in the latter setting. In particular, Main Theorem B as a standalone result relates to the realizability problem for entire Thurston maps. For $f$ and $f_n$'s as in point (1) above, we have shown that if $f$ is realised, then so are the $f_n$'s for $n$ large enough (see Corollary 3.21). Conversely, if the $f_n$'s are realised for sufficiently large $n$, then $\sigma_{f_n}$ has a unique fixed point $\tau_n \in \text{Teich}(\mathbb{R}^2, A)$ for all $n$ large enough; in this case $f$ is obstructed if and only if the sequence $(\tau_n)$ leaves every compact set of $\text{Teich}(\mathbb{R}^2, A)$ as $n \to \infty$. By understanding the ways this divergence might occur, we may hope to unravel information about the obstructions for $f$. We may also attempt to see if the results in this paper hold for transcendental meromorphic Thurston maps, if we wish to approximate them by rational ones.

**1.2. Organization of the paper.** Our paper is organized as follows. Section 2.1 is an introduction to basic definitions and our notation. In Section 2.2, we discuss planar embedded graphs, and in Section 2.3, we introduce topologically holomorphic maps and discuss their basic properties. Section 2.4 provides the necessary background in Thurston theory, and Section 2.5 discusses quasiconformal maps and Teichmüller spaces. We define Thurston pullback maps in Section 2.6, and provide a framework for defining topologically holomorphic maps using planar embedded graphs in Section 2.7.

Section 3 is an investigation of different notions of convergence for sequences of entire Thurston maps. In Section 3.1, we introduce the idea of combinatorial convergence and study its properties. We define *topological convergence* in Section 3.2 and show that the above two notions of convergence are equivalent. In Section 3.3, we prove that every entire Thurston map can be approximated by a sequence of polynomial Thurston maps and provide tools for constructing such approximations. Finally, in Section 3.4, we establish Main Theorem B, i.e., show that combinatorial convergence of entire Thurston maps implies convergence of the corresponding $\sigma$-maps in Teichmüller space.
second main result of this paper and illustrates the effect of combinatorial convergence on
the behavior of the corresponding $\sigma$-maps.

**Main Theorem B.** Let $f_n: \mathbb{R}^2 \to \mathbb{R}^2, n \in \mathbb{N}$ and $f: \mathbb{R}^2 \to \mathbb{R}^2$ be Thurston maps with the same postsingular set $A$ for all $n \in \mathbb{N}$. If the sequence $(f_n)$ converges combinatorially to $f$, then the sequence $(\sigma_{f_n})$ converges locally uniformly on $\text{Teich}(\mathbb{R}^2, A)$ to $\sigma_f$.

It is easy to see that Main Theorem A follows from the points below:

(1) Main Theorem B, taken together with Proposition 1.1, tells us that for any Thurston map $f$ with $P_f = A$, there exists a sequence of polynomial Thurston maps $(f_n)$ with $P_{f_n} = A$ that converges combinatorially to $f$ so that the sequence $(\sigma_{f_n})$ converges locally uniformly on $\text{Teich}(\mathbb{R}^2, A)$ to $\sigma_f$.

(2) If $f$ is realised by a postsingularly finite entire map $g$, we use the weak contraction properties of $\sigma$-maps (see Proposition 2.22) and further analysis to show that for sufficiently large $n$, the map $f_n$ is realised by a postcritically finite complex polynomial $g_n$ such that

* $(g_n)$ converges to $g$ locally uniformly on $\mathbb{C}$, and

* $g_n$ has the same dynamical portrait as $g$, in particular, $g_n|P_{g_n}$ is conjugate to $g|P_g$.

This statement is proved in Theorem 4.12.

Proposition 1.1 and Main Theorem B allow us to think of entire Thurston maps and their $\sigma$-maps as limits of polynomial Thurston maps and the corresponding $\sigma$-maps. These results form the first direct link between polynomial and entire Thurston theory, and present a new viewpoint for studying other problems in the latter setting. In particular, Main Theorem B as a standalone result relates to the realizability problem for entire Thurston maps. For $f$ and $f_n$'s as in point (1) above, we have shown that if $f$ is realised, then so are the $f_n$'s for $n$ large enough (see Corollary 3.21). Conversely, if the $f_n$'s are realised for sufficiently large $n$, then $\sigma_{f_n}$ has a unique fixed point $\tau_n \in \text{Teich}(\mathbb{R}^2, A)$ for all $n$ large enough; in this case $f$ is obstructed if and only if the sequence $(\tau_n)$ leaves every compact set of $\text{Teich}(\mathbb{R}^2, A)$ as $n \to \infty$. By understanding the ways this divergence might occur, we may hope to unravel information about the obstructions for $f$. We may also attempt to see if the results in this paper hold for transcendental meromorphic Thurston maps, if we wish to approximate them by rational ones.

**1.2. Organization of the paper.** Our paper is organized as follows. Section 2.1 is an introduction to basic definitions and our notation. In Section 2.2, we discuss planar embedded graphs, and in Section 2.3, we introduce topologically holomorphic maps and discuss their basic properties. Section 2.4 provides the necessary background in Thurston theory, and Section 2.5 discusses quasiconformal maps and Teichmüller spaces. We define Thurston pullback maps in Section 2.6, and provide a framework for defining topologically holomorphic maps using planar embedded graphs in Section 2.7.

Section 3 is an investigation of different notions of convergence for sequences of entire Thurston maps. In Section 3.1, we introduce the idea of combinatorial convergence and study its properties. We define topological convergence in Section 3.2 and show that the above two notions of convergence are equivalent. In Section 3.3, we prove that every entire Thurston map can be approximated by a sequence of polynomial Thurston maps and provide tools for constructing such approximations. Finally, in Section 3.4, we establish Main Theorem B, i.e., show that combinatorial convergence of entire Thurston maps implies convergence of the corresponding $\sigma$-maps in Teichmüller space.
In Section 4, we upgrade the topological results of Section 3 to the holomorphic setting. In Section 4.1, we provide a way of analyzing the convergence of a sequence of entire maps of finite type based on their combinatorial and topological properties. Then we prove Main Theorem A and discuss its implications in Section 4.2.

**Acknowledgments.** This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930, while the first and the second authors were in residence at the Mathematical Sciences Research Institute (now Simons Laufer Mathematical Sciences Institute) in Berkeley, California, during the Spring 2022 semester. The second and the third authors were partially supported by the ERC advanced grant “HOLOGRAM”. The third author was also partially supported by the ERC advanced grant “EMERGENCE”.

We would like to thank Dierk Schleicher for proposing this topic and his continued guidance, John Hubbard for introducing the problem of dynamical approximations, and Sarah Koch for providing insightful comments and supporting us through this process. We are also grateful to Walter Bergweiler, Kostiantyn Drach, Alexandre Eremenko, Núria Fagella, Lukas Geyer, Kevin Pilgrim, Lasse Rempe, and Dylan Thurston for inspiring discussions.

## 2. BACKGROUND

**2.1. Notation and basic concepts.** The sets of positive integers, integers, real and complex numbers are denoted by $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, and $\mathbb{C}$, respectively. We use the notation $\mathbb{I} := [0, 1]$ for the closed unit interval on the real line, $\mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}$ for the open unit disk in $\mathbb{C}$, $\mathbb{D}^* := \mathbb{D}\setminus\{0\}$ for the punctured unit disk, $\mathbb{H} := \{z \in \mathbb{C} : \mathrm{Re}(z) < 0\}$ for the left half-plane, and $\hat{\mathbb{C}} := \mathbb{C} \cup \{\infty\}$ for the Riemann sphere. The open disk of radius $r > 0$ centered at $z_0 \in \mathbb{C}$ is denoted by $\mathbb{D}(z_0, r) := \{z \in \mathbb{C} : |z - z_0| < r\}$, and, for simplicity, $\mathbb{D}_r$ is the disk of radius $r$ centered at 0. We denote the 2-dimensional plane and the 2-dimensional sphere by $\mathbb{R}^2$ and $\mathbb{S}^2$, respectively. In this article, we treat them as purely topological objects. In particular, our convention is to write $f : \mathbb{C} \to \mathbb{C}$ or $f : \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ to indicate that $f$ is holomorphic, and $f : \mathbb{R}^2 \to \mathbb{R}^2$ or $f : \mathbb{S}^2 \to \mathbb{S}^2$ if $f$ is continuous, but not necessarily holomorphic or smooth (cf. Definition 2.2).

The cardinality of a set $X$ is denoted by $|X|$ and the identity map on $X$ by $\mathrm{id}_X$. If $f: X \to Y$ is a map and $U \subset X$, then $f|_U$ stands for the restriction of $f$ to $U$. If $X$ is a topological space and $U \subset X$, then $\bar{U}$ denotes the closure, $\mathrm{int}(U)$ the interior, and $\partial U$ the boundary of $U$ in $X$. By $(f_n)$, we denote a sequence of maps $f_n$ (similarly, numbers, points, sets, groups, etc.) indexed by the set $\mathbb{N}$.

A subset $U \subset \mathbb{R}^2$ is called an *open* or *closed Jordan region* (in $\mathbb{R}^2$) if there exists an injective continuous map $\eta: \bar{\mathbb{D}} \to \mathbb{R}^2$ such that $U = \eta(\bar{\mathbb{D}})$ or $U = \eta(\bar{\mathbb{D}})$, respectively.

We assume that every topological surface $X$ is oriented. We say that $\mathcal{A}$ is a *complex structure* on $X$ if $\mathcal{A}$ is a *maximal* (under inclusion) *complex atlas* on $X$, i.e., a collection of complex charts compatible with the orientation on $X$ and covering $X$, where any two of them are holomorphically compatible with each other. We write $(X, \mathcal{A})$ to emphasize that $\mathcal{A}$ is a complex structure on $X$, and we say that $f: (X, \mathcal{A}) \to (Y, \mathcal{B})$ is holomorphic to emphasize that $f$ is holomorphic with respect to the complex structures $\mathcal{A}$ and $\mathcal{B}$ on the topological surfaces $X$ and $Y$, respectively.

If $X$ is a topological space, then a *path* $\alpha$ in $X$ is a continuous map $\alpha: \mathbb{I} \to X$. Points $x = \alpha(0)$ and $y = \alpha(1)$ are called *endpoints* of the path $\alpha$ and we say that $\alpha$ *joins* $x$ with $y$. The interior of the path $\alpha$ is the set $\alpha((0,1)) := \mathrm{int}(\alpha)$. The path $\alpha$ is called a *loop* if $\alpha(0) = \alpha(1)$, otherwise we say that $\alpha$ is a *non-closed path*. We say that the path $\alpha$ *starts* at
In Section 4, we upgrade the topological results of Section 3 to the holomorphic setting. In Section 4.1, we provide a way of analyzing the convergence of a sequence of entire maps of finite type based on their combinatorial and topological properties. Then we prove Main Theorem A and discuss its implications in Section 4.2.

**Acknowledgments.** This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930, while the first and the second authors were in residence at the Mathematical Sciences Research Institute (now Simons Laufer Mathematical Sciences Institute) in Berkeley, California, during the Spring 2022 semester. The second and the third authors were partially supported by the ERC advanced grant “HOLOGRAM”. The third author was also partially supported by the ERC advanced grant “EMERGENCE”.

We would like to thank Dierk Schleicher for proposing this topic and his continued guidance, John Hubbard for introducing the problem of dynamical approximations, and Sarah Koch for providing insightful comments and supporting us through this process. We are also grateful to Walter Bergweiler, Kostiantyn Drach, Alexandre Eremenko, Núria Fagella, Lukas Geyer, Kevin Pilgrim, Lasse Rempe, and Dylan Thurston for inspiring discussions.

## 2. BACKGROUND

**2.1. Notation and basic concepts.** The sets of positive integers, integers, real and complex numbers are denoted by $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, and $\mathbb{C}$, respectively. We use the notation $\mathbb{I} := [0, 1]$ for the closed unit interval on the real line, $\mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}$ for the open unit disk in $\mathbb{C}$, $\mathbb{D}^* := \mathbb{D}\setminus\{0\}$ for the punctured unit disk, $\mathbb{H} := \{z \in \mathbb{C} : \mathrm{Re}(z) < 0\}$ for the left half-plane, and $\hat{\mathbb{C}} := \mathbb{C} \cup \{\infty\}$ for the Riemann sphere. The open disk of radius $r > 0$ centered at $z_0 \in \mathbb{C}$ is denoted by $\mathbb{D}(z_0, r) := \{z \in \mathbb{C} : |z - z_0| < r\}$, and, for simplicity, $\mathbb{D}_r$ is the disk of radius $r$ centered at 0. We denote the 2-dimensional plane and the 2-dimensional sphere by $\mathbb{R}^2$ and $\mathbb{S}^2$, respectively. In this article, we treat them as purely topological objects. In particular, our convention is to write $f : \mathbb{C} \to \mathbb{C}$ or $f : \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ to indicate that $f$ is holomorphic, and $f : \mathbb{R}^2 \to \mathbb{R}^2$ or $f : \mathbb{S}^2 \to \mathbb{S}^2$ if $f$ is continuous, but not necessarily holomorphic or smooth (cf. Definition 2.2).

The cardinality of a set $X$ is denoted by $|X|$ and the identity map on $X$ by $\mathrm{id}_X$. If $f: X \to Y$ is a map and $U \subset X$, then $f|_U$ stands for the restriction of $f$ to $U$. If $X$ is a topological space and $U \subset X$, then $\bar{U}$ denotes the closure, $\mathrm{int}(U)$ the interior, and $\partial U$ the boundary of $U$ in $X$. By $(f_n)$, we denote a sequence of maps $f_n$ (similarly, numbers, points, sets, groups, etc.) indexed by the set $\mathbb{N}$.

A subset $U \subset \mathbb{R}^2$ is called an *open* or *closed Jordan region* (in $\mathbb{R}^2$) if there exists an injective continuous map $\eta: \bar{\mathbb{D}} \to \mathbb{R}^2$ such that $U = \eta(\bar{\mathbb{D}})$ or $U = \eta(\bar{\mathbb{D}})$, respectively.

We assume that every topological surface $X$ is oriented. We say that $\mathcal{A}$ is a *complex structure* on $X$ if $\mathcal{A}$ is a *maximal* (under inclusion) *complex atlas* on $X$, i.e., a collection of complex charts compatible with the orientation on $X$ and covering $X$, where any two of them are holomorphically compatible with each other. We write $(X, \mathcal{A})$ to emphasize that $\mathcal{A}$ is a complex structure on $X$, and we say that $f: (X, \mathcal{A}) \to (Y, \mathcal{B})$ is holomorphic to emphasize that $f$ is holomorphic with respect to the complex structures $\mathcal{A}$ and $\mathcal{B}$ on the topological surfaces $X$ and $Y$, respectively.

If $X$ is a topological space, then a *path* $\alpha$ in $X$ is a continuous map $\alpha: \mathbb{I} \to X$. Points $x = \alpha(0)$ and $y = \alpha(1)$ are called *endpoints* of the path $\alpha$ and we say that $\alpha$ *joins* $x$ with $y$. The interior of the path $\alpha$ is the set $\alpha((0, 1)) := \mathrm{int}(\alpha)$. The path $\alpha$ is called a *loop* if $\alpha(0) = \alpha(1)$, otherwise we say that $\alpha$ is a *non-closed path*. We say that the path $\alpha$ *starts* at
$x \in X$ if $\alpha(0) = x$. When $\alpha$ is a loop, we also say that $\alpha$ is based at $x$. A non-closed path $\alpha$ is called *simple* if it has no self-intersections (i.e., $\alpha: I \to X$ is injective). A loop $\alpha$ is called a *simple* if it has no self-intersections except at endpoints (i.e., $\alpha|_{(0,1)}$ is injective). Loop $\alpha$ is called *constant* if the map $\alpha: I \to X$ is constant. Sometimes we are going to conflate paths and their images. For instance, we write $\alpha \subset Y$, where $\alpha: I \to X$ is a path and $Y \subset X$, to indicate that $\alpha(\mathbb{I}) \subset Y$. We say that $\gamma \subset X$ is a *simple closed curve* if $\gamma = \alpha(\mathbb{I})$ for some simple loop $\alpha$ in $X$.

If $\alpha$ and $\beta$ are two paths in $X$, then we denote by $\alpha \cdot \beta$ their concatenation, i.e., the path that first traverses $\alpha$ and then $\beta$. By $\bar{\alpha}$ we denote the path in $X$ such that $\bar{\alpha}(t) = \alpha(1-t)$ for all $t \in \mathbb{I}$. For $n \in \mathbb{Z}$, we define $\alpha^n$ to be a constant loop based at $\alpha(0)$ if $n = 0$, a concatenation of $\alpha$ with itself $n$ times, if $n > 0$, and concatenation of $\bar{\alpha}$ with itself $-n$ times, if $n < 0$.

Two paths $\alpha: \mathbb{I} \to X$ and $\beta: \mathbb{I} \to X$ are called *path-homotopic* (or simply *homotopic*) if there exists a continuous map $H: \mathbb{I} \times \mathbb{I} \to X$ called a *homotopy* so that $H(t, 0) = \alpha(t)$, $H(t, 1) = \beta(t)$ for all $t \in \mathbb{I}$, and $H(0, s) = \alpha(0) = \beta(0)$, $H(1, s) = \alpha(1) = \beta(1)$ for all $s \in \mathbb{I}$. Let $A$ be a subset of $X$. We say that two paths $\alpha$ and $\beta$ are *path-homotopic relative A* (abbreviated as “$\alpha$ and $\beta$ are homotopic rel. $A$” and denoted $\alpha \sim \beta$ rel. $A$) if $\alpha \subset X \setminus A$ and $\beta \subset X \setminus A$ are homotopic in $X \setminus A$, i.e., $H(s, t) \in X \setminus A$ for all $(s, t) \in \mathbb{I} \times \mathbb{I}$, where $H$ is the corresponding homotopy.

If $X$ is a topological space and $x \in X$, then by $\pi_1(X, x)$ we denote the fundamental group of $X$ at $x$, i.e., the set of all homotopy equivalence classes of loops in $X$ based at $x$ endowed with the operation of path concatenation. If $f: X \to Y$ is a continuous map between topological spaces $X$ and $Y$ such that $f(x) = y$, then the group homomorphism $f_*: \pi_1(X, x) \to \pi_1(Y, y)$ is defined as $f_*([\alpha]) = [f \circ \alpha]$ for any loop $\alpha \subset X$ based at $x$, where $[\cdot]$ denotes a homotopy equivalence class. Finally, we say that path $\alpha \subset Y$ lifts under $f$ to a path $\beta \subset X$, if $\alpha = f \circ \beta$, and $\beta$ is called the *f-lift* (or *lift under f*) of $\alpha$. If $f: X \to Y$ is a covering map and $f(x) = y$, then every path $\alpha$ starting at $y$ has a unique $f$-lift starting at $x$, which is denoted by $\alpha\uparrow(f, x)$. By definition, the path $\alpha\uparrow(f, x)$ is a loop if and only if $[\alpha] \in f_*\pi_1(X, x)$.

Suppose that $X$ and $Y$ are topological spaces. We say that homeomorphisms $\varphi: X \to Y$ and $\psi: X \to Y$ are *isotopic* if there exists a continuous map $H: X \times \mathbb{I} \to Y$ called an *isotopy* such that $H(x, 0) = \varphi(x)$ and $H(x, 1) = \psi(x)$ for all $x \in X$, and $H(\cdot, t): X \to Y$ is a homeomorphism for every $t \in \mathbb{I}$. We say that $\varphi$ and $\psi$ are *isotopic rel. A*, where $A \subset X$, if additionally $H(x, t) = x$ for all $(x, t) \in A \times \mathbb{I}$.

Let $X$ be a topological surface. We denote by $\text{Homeo}^+(X)$ and $\text{Homeo}^+(X, A)$ 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 $\text{Homeo}_0^+(X, A)$ for the subgroup of $\text{Homeo}^+(X, A)$ consisting of all homeomorphisms isotopic rel. $A$ to $\text{id}_X$.

We use the following notion of convergence, in the sense of [Cha50], of sequences of subgroups of a given group endowed with the discrete topology.

**Definition 2.1.** Let $G$ be a group and $(H_n)$ be a sequence of its subgroups. We say that the sequence $(H_n)$ converges to a subgroup $H$ of $G$ and write $\lim_{n\to\infty} H_n = H$ if for every $g \in G$, there exists $N = N(g)$ so that for each $n \ge N$, $g \in H_n$ if and only if $g \in H$.

## 2.2. Planar embedded graphs.
We refer the reader to [Die05] for classical background in graph theory. A *planar embedded graph* is a pair $G = (V, E)$, where
$x \in X$ if $\alpha(0) = x$. When $\alpha$ is a loop, we also say that $\alpha$ is based at $x$. A non-closed path $\alpha$ is called *simple* if it has no self-intersections (i.e., $\alpha: I \to X$ is injective). A loop $\alpha$ is called a *simple* if it has no self-intersections except at endpoints (i.e., $\alpha|_{(0,1)}$ is injective). Loop $\alpha$ is called *constant* if the map $\alpha: I \to X$ is constant. Sometimes we are going to conflate paths and their images. For instance, we write $\alpha \subset Y$, where $\alpha: I \to X$ is a path and $Y \subset X$, to indicate that $\alpha(\mathbb{I}) \subset Y$. We say that $\gamma \subset X$ is a *simple closed curve* if $\gamma = \alpha(\mathbb{I})$ for some simple loop $\alpha$ in $X$.

If $\alpha$ and $\beta$ are two paths in $X$, then we denote by $\alpha \cdot \beta$ their concatenation, i.e., the path that first traverses $\alpha$ and then $\beta$. By $\bar{\alpha}$ we denote the path in $X$ such that $\bar{\alpha}(t) = \alpha(1-t)$ for all $t \in \mathbb{I}$. For $n \in \mathbb{Z}$, we define $\alpha^n$ to be a constant loop based at $\alpha(0)$ if $n = 0$, a concatenation of $\alpha$ with itself $n$ times, if $n > 0$, and concatenation of $\bar{\alpha}$ with itself $-n$ times, if $n < 0$.

Two paths $\alpha: \mathbb{I} \to X$ and $\beta: \mathbb{I} \to X$ are called *path-homotopic* (or simply *homotopic*) if there exists a continuous map $H: \mathbb{I} \times \mathbb{I} \to X$ called a *homotopy* so that $H(t, 0) = \alpha(t)$, $H(t, 1) = \beta(t)$ for all $t \in \mathbb{I}$, and $H(0, s) = \alpha(0) = \beta(0)$, $H(1, s) = \alpha(1) = \beta(1)$ for all $s \in \mathbb{I}$. Let $A$ be a subset of $X$. We say that two paths $\alpha$ and $\beta$ are *path-homotopic relative A* (abbreviated as “$\alpha$ and $\beta$ are homotopic rel. $A$” and denoted $\alpha \sim \beta$ rel. $A$) if $\alpha \subset X \setminus A$ and $\beta \subset X \setminus A$ are homotopic in $X \setminus A$, i.e., $H(s, t) \in X \setminus A$ for all $(s, t) \in \mathbb{I} \times \mathbb{I}$, where $H$ is the corresponding homotopy.

If $X$ is a topological space and $x \in X$, then by $\pi_1(X, x)$ we denote the *fundamental group* of $X$ at $x$, i.e., the set of all homotopy equivalence classes of loops in $X$ based at $x$ endowed with the operation of path concatenation. If $f: X \to Y$ is a continuous map between topological spaces $X$ and $Y$ such that $f(x) = y$, then the group homomorphism $f_*: \pi_1(X, x) \to \pi_1(Y, y)$ is defined as $f_*([\alpha]) = [f \circ \alpha]$ for any loop $\alpha \subset X$ based at $x$, where $[\cdot]$ denotes a homotopy equivalence class. Finally, we say that path $\alpha \subset Y$ *lifts under f* to a path $\beta \subset X$, if $\alpha = f \circ \beta$, and $\beta$ is called the *f-lift* (or *lift under f*) of $\alpha$. If $f: X \to Y$ is a covering map and $f(x) = y$, then every path $\alpha$ starting at $y$ has a unique $f$-lift starting at $x$, which is denoted by $\alpha\uparrow(f, x)$. By definition, the path $\alpha\uparrow(f, x)$ is a loop if and only if $[\alpha] \in f_*\pi_1(X, x)$.

Suppose that $X$ and $Y$ are topological spaces. We say that homeomorphisms $\varphi: X \to Y$ and $\psi: X \to Y$ are *isotopic* if there exists a continuous map $H: X \times \mathbb{I} \to Y$ called an *isotopy* such that $H(x, 0) = \varphi(x)$ and $H(x, 1) = \psi(x)$ for all $x \in X$, and $H(\cdot, t): X \to Y$ is a homeomorphism for every $t \in \mathbb{I}$. We say that $\varphi$ and $\psi$ are *isotopic rel. A*, where $A \subset X$, if additionally $H(x, t) = x$ for all $(x, t) \in A \times \mathbb{I}$.

Let $X$ be a topological surface. We denote by $\text{Homeo}^+(X)$ and $\text{Homeo}^+(X, A)$ 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 $\text{Homeo}_0^+(X, A)$ for the subgroup of $\text{Homeo}^+(X, A)$ consisting of all homeomorphisms isotopic rel. $A$ to $\text{id}_X$.

We use the following notion of convergence, in the sense of [Cha50], of sequences of subgroups of a given group endowed with the discrete topology.

**Definition 2.1.** Let $G$ be a group and $(H_n)$ be a sequence of its subgroups. We say that the sequence $(H_n)$ *converges* to a subgroup $H$ of $G$ and write $\lim_{n \to \infty} H_n = H$ if for every $g \in G$, there exists $N = N(g)$ so that for each $n \ge N$, $g \in H_n$ if and only if $g \in H$.

## 2.2. Planar embedded graphs.
We refer the reader to [Die05] for classical background in graph theory. A *planar embedded graph* is a pair $G = (V, E)$, where
(1) $V$ is a discrete (in particular, countable) set of points in $\mathbb{R}^2$, and

(2) $E$ is a set of simple paths and simple loops (viewed as subsets of $\mathbb{R}^2$, i.e., as images of the corresponding maps) such that their endpoints belong to $V$, their interiors are pairwise disjoint and lie in $\mathbb{R}^2 \setminus V$, and every compact set $K \subset \mathbb{R}^2$ intersects finitely many elements of $E$.

The sets $V$ and $E$ are called the *vertex set* and *edge set* of $G$, respectively. Note that our notion of a planar embedded graph allows *multi-edges*, i.e., distinct edges that connect the same pair of vertices, and *loop-edges*, i.e., edges that connect a vertex to itself.

In the following, suppose $G = (V, E)$ is a planar embedded graph. We say that $G$ is *finite* if $V$ and $E$ are finite sets. The *degree* of a vertex $v$ in $G$, denoted by $\deg_G(v)$, is the number $n_1 + 2n_2$, where $n_1$ and $n_2$ are the numbers of simple paths and simple loops in $E$ incident to $v$, respectively. We say that $G$ is *k-regular* if $\deg_G(v) = k$ for every $v \in V$. A *subgraph* of $G$ is a planar embedded graph $G' = (V', E')$ such that $V' \subset V$ and $E' \subset E$.

The subset $\mathcal{G} := V \cup \bigcup_{e \in E} e$ of $\mathbb{R}^2$ is called the *realization* of $G$. A *face* of the graph $G$ is a connected component of $\mathbb{R}^2 \setminus \mathcal{G}$. The set of all faces of $G$ is denoted by $F(G)$. Note that due to the definition of a planar embedded graph, the set $\mathcal{G}$ is closed in $\mathbb{R}^2$, and thus, every face $F$ of $G$ is open. If $F$ is a face of $G$, then we denote by $\partial F$ the subgraph of $G$ forming the (topological) boundary of $F$ in $\mathbb{R}^2$.

The graph $G$ is called connected if its realization $\mathcal{G}$ is connected (or, equivalently, path-connected). It follows that $G$ is connected if and only if each face of $G$ is simply connected. We will often conflate a planar embedded graph $G$ with its realization $\mathcal{G}$.

Let $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ be two planar embedded graphs. We say that $G_1$ is *isomorphic* to $G_2$ if there exists a homeomorphism $\varphi \in \text{Homeo}^+(\mathbb{R}^2)$ that maps vertices and edges of $G_1$ into vertices and edges of $G_2$, that is, $\varphi(V_1) = V_2$ and $\varphi(E_1) = E_2$. In this case, we call $\varphi$ an *isomorphism* between $G_1$ and $G_2$. If $\varphi$ is isotopic rel. $A$ to $\text{id}_{\mathbb{R}^2}$ for some set $A \subset \mathbb{R}^2$, we say that $G_1$ is *isotopic* rel. $A$ to $G_2$.

Suppose that $f: U \to W$ is a covering map, where $U$ and $W$ are open subsets of $\mathbb{R}^2$. Additionally, assume that the realization $\mathcal{G}$ of a graph $G = (V, E)$ lies in $U$. Then we can define a new graph $f^{-1}(G)$, called the *preimage* of $G$ under $f$, where $V(f^{-1}(G)) := f^{-1}(V(G))$ and the set $E(f^{-1}(G))$ consists of all $f$-lifts of the edges of $G$.

Examples of planar embedded graphs are *cycles*, *chains* and *rose graphs*. We say that $G = (V, E)$ is a cycle if $G$ is finite, connected, and 2-regular. The graph $G$ is a finite chain if it consists of a single vertex with no edges, or it is connected and has exactly two vertices of degree one and all other vertices have degree two. In the case when $G$ is a finite chain, all vertices of $G$ of degree less than 2 are called *endpoints* of $G$. If $G$ is infinite, connected, and 2-regular, then it is called an infinite chain. Finally, if $|V| = 1$ and for every edge $e$ of $G$, the bounded connected component of $\mathbb{R}^2 \setminus e$ does not intersect the realization $\mathcal{G}$ of $G$, then we say that $G$ is a *rose graph*. If $\mathcal{R}$ is a rose graph, then we call its unique vertex the *center* of $\mathcal{R}$ and its edges the *petals* of the rose graph $\mathcal{R}$. Given a finite set $A \subset \mathbb{R}^2$ and a rose graph $\mathcal{R}$, we say that $\mathcal{R}$ *surrounds* $A$ if every bounded face of $\mathcal{R}$ contains a unique point of $A$, and every point of $A$ is contained in some bounded face of $\mathcal{R}$.

We say that graph $G' = (V', E')$ is the result of *subdivision* of an edge $e \in E(G)$ of the planar embedded graph $G = (V, E)$ if $G'$ is obtained from $G$ by adding a new vertex in the interior of $e$. More precisely, $V' = V \cup \{v\}$ and $E = (E \setminus \{e\}) \cup \{e_1, e_2\}$, where $v \in \text{int}(e)$, and $e_1, e_2$ are the closures of the connected components of $e \setminus \{v\}$. In particular, subdivision of an
(1) $V$ is a discrete (in particular, countable) set of points in $\mathbb{R}^2$, and

(2) $E$ is a set of simple paths and simple loops (viewed as subsets of $\mathbb{R}^2$, i.e., as images of the corresponding maps) such that their endpoints belong to $V$, their interiors are pairwise disjoint and lie in $\mathbb{R}^2 \setminus V$, and every compact set $K \subset \mathbb{R}^2$ intersects finitely many elements of $E$.

The sets $V$ and $E$ are called the *vertex set* and *edge set* of $G$, respectively. Note that our notion of a planar embedded graph allows *multi-edges*, i.e., distinct edges that connect the same pair of vertices, and *loop-edges*, i.e., edges that connect a vertex to itself.

In the following, suppose $G = (V, E)$ is a planar embedded graph. We say that $G$ is *finite* if $V$ and $E$ are finite sets. The *degree* of a vertex $v$ in $G$, denoted by $\deg_G(v)$, is the number $n_1 + 2n_2$, where $n_1$ and $n_2$ are the numbers of simple paths and simple loops in $E$ incident to $v$, respectively. We say that $G$ is *k-regular* if $\deg_G(v) = k$ for every $v \in V$. A *subgraph* of $G$ is a planar embedded graph $G' = (V', E')$ such that $V' \subset V$ and $E' \subset E$.

The subset $\mathcal{G} := V \cup \bigcup_{e \in E} e$ of $\mathbb{R}^2$ is called the *realization* of $G$. A *face* of the graph $G$ is a connected component of $\mathbb{R}^2 \setminus \mathcal{G}$. The set of all faces of $G$ is denoted by $F(G)$. Note that due to the definition of a planar embedded graph, the set $\mathcal{G}$ is closed in $\mathbb{R}^2$, and thus, every face $F$ of $G$ is open. If $F$ is a face of $G$, then we denote by $\partial F$ the subgraph of $G$ forming the (topological) boundary of $F$ in $\mathbb{R}^2$.

The graph $G$ is called connected if its realization $\mathcal{G}$ is connected (or, equivalently, path-connected). It follows that $G$ is connected if and only if each face of $G$ is simply connected. We will often conflate a planar embedded graph $G$ with its realization $\mathcal{G}$.

Let $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ be two planar embedded graphs. We say that $G_1$ is *isomorphic* to $G_2$ if there exists a homeomorphism $\varphi \in \text{Homeo}^+(\mathbb{R}^2)$ that maps vertices and edges of $G_1$ into vertices and edges of $G_2$, that is, $\varphi(V_1) = V_2$ and $\varphi(E_1) = E_2$. In this case, we call $\varphi$ an *isomorphism* between $G_1$ and $G_2$. If $\varphi$ is isotopic rel. $A$ to $\text{id}_{\mathbb{R}^2}$ for some set $A \subset \mathbb{R}^2$, we say that $G_1$ is *isotopic* rel. $A$ to $G_2$.

Suppose that $f: U \to W$ is a covering map, where $U$ and $W$ are open subsets of $\mathbb{R}^2$. Additionally, assume that the realization $\mathcal{G}$ of a graph $G = (V, E)$ lies in $U$. Then we can define a new graph $f^{-1}(G)$, called the *preimage* of $G$ under $f$, where $V(f^{-1}(G)) := f^{-1}(V(G))$ and the set $E(f^{-1}(G))$ consists of all $f$-lifts of the edges of $G$.

Examples of planar embedded graphs are *cycles*, *chains* and *rose graphs*. We say that $G = (V, E)$ is a cycle if $G$ is finite, connected, and 2-regular. The graph $G$ is a finite chain if it consists of a single vertex with no edges, or it is connected and has exactly two vertices of degree one and all other vertices have degree two. In the case when $G$ is a finite chain, all vertices of $G$ of degree less than 2 are called *endpoints* of $G$. If $G$ is infinite, connected, and 2-regular, then it is called an infinite chain. Finally, if $|V| = 1$ and for every edge $e$ of $G$, the bounded connected component of $\mathbb{R}^2 \setminus e$ does not intersect the realization $\mathcal{G}$ of $G$, then we say that $G$ is a *rose graph*. If $\mathcal{R}$ is a rose graph, then we call its unique vertex the *center* of $\mathcal{R}$ and its edges the *petals* of the rose graph $\mathcal{R}$. Given a finite set $A \subset \mathbb{R}^2$ and a rose graph $\mathcal{R}$, we say that $\mathcal{R}$ *surrounds* $A$ if every bounded face of $\mathcal{R}$ contains a unique point of $A$, and every point of $A$ is contained in some bounded face of $\mathcal{R}$.

We say that graph $G' = (V', E')$ is the result of *subdivision* of an edge $e \in E(G)$ of the planar embedded graph $G = (V, E)$ if $G'$ is obtained from $G$ by adding a new vertex in the interior of $e$. More precisely, $V' = V \cup \{v\}$ and $E = (E \setminus \{e\}) \cup \{e_1, e_2\}$, where $v \in \text{int}(e)$, and $e_1, e_2$ are the closures of the connected components of $e \setminus \{v\}$. In particular, subdivision of an
edge does not change the realization of the graph, and the result is uniquely defined up to an isotopy rel. V.

Let $G = (V, E)$ be a planar embedded graph and $e \in E$ be one of its edges. We say that a continuous map $\alpha: \mathbb{I} \to e$ is a *parametrization* of $e$ if $\alpha|_{(0,1)}$ is bijective onto $\text{int}(e)$. We say that two parametrizations $\alpha_1$ and $\alpha_2$ of the same edge $e$ are equivalent if $\alpha_1^{-1} \circ \alpha_2|_{(0,1)}: (0,1) \to (0,1)$ is an increasing function. One can easily see that for every edge $e \in E(G)$ there are two distinct equivalence classes of parametrizations. We call each of these equivalence classes a *direction* of the edge $e$. We say that the graph $G$ is *directed* if each of its edges is endowed with a unique direction (called the *forward direction*). A choice of forward directions for all edges of a graph $G$ is also called an *orientation* of $G$. Directions that are omitted from the orientation of $G$ are called *backward*. In a similar way, we introduce notions of forward and backward parametrizations of the edges of the directed graph $G$. If $v$ is a vertex of $G$ and $e$ is an edge incident to $v$, then there is a natural way to call $e$ *incoming* or *outgoing* at $v$ unless $e$ is a loop-edge.

Suppose that $\Phi: G_1 \to G_2$ is a continuous map, where $G_1$ is a planar embedded graph and $G_2$ is a directed planar embedded graph, such that $\Phi^{-1}(V(G_2)) = V(G_1)$ and $\Phi|_{\text{int}(e)}$ is injective for each edge $e \in E(G_1)$. The map $\Phi$ and the orientation of the graph $G_2$ naturally induce an orientation of the graph $G_1$. Indeed, we choose a forward direction for $e \in E(G_1)$ so that if $\alpha$ is a parametrization of $e$, then $\alpha$ is forward if and only if $\Phi \circ \alpha$ is a forward parametrization of the edge $\Phi(e) \in E(G_2)$. Similarly, there is a natural way to define forward directions for a subgraph and graph obtained by a subdivision of an edge of a directed planar embedded graph.

We say that a directed planar embedded graph $G = (V, E)$ is *unilaterally connected* if for every pair of vertices $u, v \in V$, there exists a path $\alpha$ with endpoints at $u$ and $v$ that is obtained by concatenation of forward parametrizations of edges of $G$.

We say that a rose graph $\mathcal{R}$ is *counterclockwise directed* if for each edge $e \in E(\mathcal{R})$ any forward parametrization $\alpha$ orients $e$ counterclockwise if viewed from the interior of the bounded connected component of $\mathbb{R}^2 \setminus e$. Graphs at the top in Figures 2, 3, 4, and 5 provide examples of counterclockwise directed rose graphs with two or three petals (in particular, arrows on their petals indicate forward directions of the corresponding edges). In a similar way we can define a *counterclockwise directed cycle*.

## 2.3. Topologically holomorphic maps.

**Definition 2.2.** Let $X$ and $Y$ be topological surfaces. Then a map $f: X \to Y$ is called *topologically holomorphic* if for every $x \in X$ there exists $d \in \mathbb{N}$, an open neighbourhood $U$ of $x$, and two orientation-preserving homeomorphisms $\psi: U \to \mathbb{D}$ and $\varphi: f(U) \to \mathbb{D}$ such that $\psi(x) = \varphi(f(x)) = 0$ and $(\varphi \circ f \circ \psi^{-1})(z) = z^d$ for all $z \in \mathbb{D}$.

In particular, the integer $\text{deg}(f, x) := d$ is independent of $U, \psi$ and $\varphi$ and is called the *local degree* of $f$ at $x$.

*Remark 2.3.* The notion of a topologically holomorphic map is close in spirit to the notion of *branched covering map*. However, these two notions are generally not the same (cf. [BM17, Definition A.7]).

**Definition 2.4.** Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is called *discrete* if $f^{-1}(y)$ is a discrete subset of $X$ for every $y \in Y$.

Theorem [LP20, Theorem 1.3, Remark 1.4] (see also [Sto28], [Sto56]) imply that we can give an alternative definition of a topologically holomorphic map.
edge does not change the realization of the graph, and the result is uniquely defined up to an isotopy rel. V.

Let $G = (V, E)$ be a planar embedded graph and $e \in E$ be one of its edges. We say that a continuous map $\alpha: \mathbb{I} \to e$ is a *parametrization* of $e$ if $\alpha|_{(0,1)}$ is bijective onto $\text{int}(e)$. We say that two parametrizations $\alpha_1$ and $\alpha_2$ of the same edge $e$ are equivalent if $\alpha_1^{-1} \circ \alpha_2|_{(0,1)}: (0,1) \to (0,1)$ is an increasing function. One can easily see that for every edge $e \in E(G)$ there are two distinct equivalence classes of parametrizations. We call each of these equivalence classes a *direction* of the edge $e$. We say that the graph $G$ is *directed* if each of its edges is endowed with a unique direction (called the *forward direction*). A choice of forward directions for all edges of a graph $G$ is also called an *orientation* of $G$. Directions that are omitted from the orientation of $G$ are called *backward*. In a similar way, we introduce notions of forward and backward parametrizations of the edges of the directed graph $G$. If $v$ is a vertex of $G$ and $e$ is an edge incident to $v$, then there is a natural way to call $e$ *incoming* or *outgoing* at $v$ unless $e$ is a loop-edge.

Suppose that $\Phi: G_1 \to G_2$ is a continuous map, where $G_1$ is a planar embedded graph and $G_2$ is a directed planar embedded graph, such that $\Phi^{-1}(V(G_2)) = V(G_1)$ and $\Phi|_{\text{int}(e)}$ is injective for each edge $e \in E(G_1)$. The map $\Phi$ and the orientation of the graph $G_2$ naturally induce an orientation of the graph $G_1$. Indeed, we choose a forward direction for $e \in E(G_1)$ so that if $\alpha$ is a parametrization of $e$, then $\alpha$ is forward if and only if $\Phi \circ \alpha$ is a forward parametrization of the edge $\Phi(e) \in E(G_2)$. Similarly, there is a natural way to define forward directions for a subgraph and graph obtained by a subdivision of an edge of a directed planar embedded graph.

We say that a directed planar embedded graph $G = (V, E)$ is *unilaterally connected* if for every pair of vertices $u, v \in V$, there exists a path $\alpha$ with endpoints at $u$ and $v$ that is obtained by concatenation of forward parametrizations of edges of $G$.

We say that a rose graph $\mathcal{R}$ is *counterclockwise directed* if for each edge $e \in E(\mathcal{R})$ any forward parametrization $\alpha$ orients $e$ counterclockwise if viewed from the interior of the bounded connected component of $\mathbb{R}^2 \setminus e$. Graphs at the top in Figures 2, 3, 4, and 5 provide examples of counterclockwise directed rose graphs with two or three petals (in particular, arrows on their petals indicate forward directions of the corresponding edges). In a similar way we can define a *counterclockwise directed cycle*.

## 2.3. Topologically holomorphic maps.

**Definition 2.2.** Let $X$ and $Y$ be topological surfaces. Then a map $f: X \to Y$ is called *topologically holomorphic* if for every $x \in X$ there exists $d \in \mathbb{N}$, an open neighbourhood $U$ of $x$, and two orientation-preserving homeomorphisms $\psi: U \to \mathbb{D}$ and $\varphi: f(U) \to \mathbb{D}$ such that $\psi(x) = \varphi(f(x)) = 0$ and $(\varphi \circ f \circ \psi^{-1})(z) = z^d$ for all $z \in \mathbb{D}$.

In particular, the integer $\text{deg}(f, x) := d$ is independent of $U, \psi$ and $\varphi$ and is called the *local degree* of $f$ at $x$.

*Remark 2.3.* The notion of a topologically holomorphic map is close in spirit to the notion of *branched covering map*. However, these two notions are generally not the same (cf. [BM17, Definition A.7]).

**Definition 2.4.** Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is called *discrete* if $f^{-1}(y)$ is a discrete subset of $X$ for every $y \in Y$.

Theorem [LP20, Theorem 1.3, Remark 1.4] (see also [Sto28], [Sto56]) imply that we can give an alternative definition of a topologically holomorphic map.
**Proposition 2.5.** Let $X$ and $Y$ be topological surfaces. Then a map $f: X \to Y$ is topologically holomorphic if and only if it is continuous, open, discrete, and for every $x \in X$ such that $f$ is locally injective at $x$, there exists an open neighbourhood $U$ of $x$ for which $f|_U: U \to f(U)$ is an orientation-preserving homeomorphism.

It is known that we can pull back complex structures under topologically holomorphic maps (see [LP20, Stoïlow's factorisation theorem] and [BM17, Lemma A.12]) as the following proposition states.

**Proposition 2.6.** Let $f: X \to Y$ be topologically holomorphic. Then for every complex structure $\mathcal{A}$ on $Y$ there exists a unique complex structure $f^*\mathcal{A}$ on $X$ such that the map $f: (X, f^*\mathcal{A}) \to (Y, \mathcal{A})$ is holomorphic.

In particular, for any orientation-preserving homeomorphism $\varphi: Y \to S_Y$, where $S_Y$ is a Riemann surface, there exists a Riemann surface $S_X$ and an orientation-preserving homeomorphism $\psi: X \to S_X$ such that $\varphi \circ f \circ \psi^{-1}: S_X \to S_Y$ is holomorphic.

In particular, the second part of Proposition 2.6 provides the third equivalent definition of a topologically holomorphic map.

Several classical notions of one-dimensional complex analysis can be extended to the setting of topologically holomorphic maps.

**Definition 2.7.** Let $X$ and $Y$ be topological surfaces and $f: X \to Y$ be a topologically holomorphic map.

(1) A point $y \in Y$ is a regular value of $f$ if there exists a neighbourhood $U$ of $y$ such that $f^{-1}(U) = \bigsqcup_{i \in I} U_i$ and $f|_{U_i}: U_i \to U$ is a homeomorphism for each $i \in I$. If such a neighborhood $U$ does not exist, we call $y$ a singular value of $f$. The set of all singular values of $f$ is denoted by $S_f$. If $S_f$ is finite, then we say that $f$ is a map of finite type or $f$ belongs to the *Speiser class* $S$.

(2) We say that $x \in X$ is a critical point of $f$ if $\deg(f, x) > 1$. A point $y \in Y$ is called a critical value of $f$ if there exists a critical point $x \in X$ such that $f(x) = y$.

(3) We say that $y \in Y$ is an asymptotic value of $f$ if there exists a continuous curve $\gamma: [0, \infty) \to X$ such that $\gamma$ leaves every compact set of $X$ and $f(\gamma(t)) \to y$ as $t \to \infty$.

**Remark 2.8.** (1) Critical and asymptotic values are always singular.

(2) If $f: X \to Y$ is topologically holomorphic, then the restriction

$$f|_{X \setminus f^{-1}(S_f)}: X \setminus f^{-1}(S_f) \to Y \setminus S_f$$

is a covering map.

Every locally non-constant holomorphic function between two Riemann surfaces is an obvious example of a topologically holomorphic map. Suppose that $f: X \to Y$ is topologically holomorphic, we say that $f$ is *entire* if $X = Y = \mathbb{R}^2$, *meromorphic* if $X = \mathbb{R}^2, Y = \mathbb{S}^2$ and if $X = Y = \mathbb{S}^2$, then $f$ is a branched covering map. Thus, topologically holomorphic maps serve as topological models for one-dimensional holomorphic functions, and the types of maps listed above are the underlying models for the corresponding families of holomorphic maps.

Our focus in this article is on entire topologically holomorphic maps. Interesting phenomena can occur when we pull back complex structures in this setting. Suppose that $\mathcal{A}$ is a complex structure on $\mathbb{R}^2$ such that $(\mathbb{R}^2, \mathcal{A})$ is biholomorphic to $\mathbb{D}$. It is evident that $(\mathbb{R}^2, f^*\mathcal{A})$ is biholomorphic to $\mathbb{D}$, since there is no non-constant map from $\mathbb{C}$ to $\mathbb{D}$. However, if $(\mathbb{R}^2, \mathcal{A})$
**Proposition 2.5.** Let $X$ and $Y$ be topological surfaces. Then a map $f: X \to Y$ is topologically holomorphic if and only if it is continuous, open, discrete, and for every $x \in X$ such that $f$ is locally injective at $x$, there exists an open neighbourhood $U$ of $x$ for which $f|_U: U \to f(U)$ is an orientation-preserving homeomorphism.

It is known that we can pull back complex structures under topologically holomorphic maps (see [LP20, Stoïlow's factorisation theorem] and [BM17, Lemma A.12]) as the following proposition states.

**Proposition 2.6.** Let $f: X \to Y$ be topologically holomorphic. Then for every complex structure $\mathcal{A}$ on $Y$ there exists a unique complex structure $f^*\mathcal{A}$ on $X$ such that the map $f: (X, f^*\mathcal{A}) \to (Y, \mathcal{A})$ is holomorphic.

In particular, for any orientation-preserving homeomorphism $\varphi: Y \to S_Y$, where $S_Y$ is a Riemann surface, there exists a Riemann surface $S_X$ and an orientation-preserving homeomorphism $\psi: X \to S_X$ such that $\varphi \circ f \circ \psi^{-1}: S_X \to S_Y$ is holomorphic.

In particular, the second part of Proposition 2.6 provides the third equivalent definition of a topologically holomorphic map.

Several classical notions of one-dimensional complex analysis can be extended to the setting of topologically holomorphic maps.

**Definition 2.7.** Let $X$ and $Y$ be topological surfaces and $f: X \to Y$ be a topologically holomorphic map.

(1) A point $y \in Y$ is a regular value of $f$ if there exists a neighbourhood $U$ of $y$ such that $f^{-1}(U) = \bigsqcup_{i \in I} U_i$ and $f|_{U_i}: U_i \to U$ is a homeomorphism for each $i \in I$. If such a neighborhood $U$ does not exist, we call $y$ a singular value of $f$. The set of all singular values of $f$ is denoted by $S_f$. If $S_f$ is finite, then we say that $f$ is a map of finite type or $f$ belongs to the Speiser class $\mathcal{S}$.

(2) We say that $x \in X$ is a critical point of $f$ if $\deg(f, x) > 1$. A point $y \in Y$ is called a critical value of $f$ if there exists a critical point $x \in X$ such that $f(x) = y$.

(3) We say that $y \in Y$ is an asymptotic value of $f$ if there exists a continuous curve $\gamma: [0, \infty) \to X$ such that $\gamma$ leaves every compact set of $X$ and $f(\gamma(t)) \to y$ as $t \to \infty$.

**Remark 2.8.** (1) Critical and asymptotic values are always singular.

(2) If $f: X \to Y$ is topologically holomorphic, then the restriction

$$f|_{X \setminus f^{-1}(S_f)}: X \setminus f^{-1}(S_f) \to Y \setminus S_f$$

is a covering map.

Every locally non-constant holomorphic function between two Riemann surfaces is an obvious example of a topologically holomorphic map. Suppose that $f: X \to Y$ is topologically holomorphic, we say that $f$ is entire if $X = Y = \mathbb{R}^2$, meromorphic if $X = \mathbb{R}^2, Y = \mathbb{S}^2$ and if $X = Y = \mathbb{S}^2$, then $f$ is a branched covering map. Thus, topologically holomorphic maps serve as topological models for one-dimensional holomorphic functions, and the types of maps listed above are the underlying models for the corresponding families of holomorphic maps.

Our focus in this article is on entire topologically holomorphic maps. Interesting phenomena can occur when we pull back complex structures in this setting. Suppose that $\mathcal{A}$ is a complex structure on $\mathbb{R}^2$ such that $(\mathbb{R}^2, \mathcal{A})$ is biholomorphic to $\mathbb{D}$. It is evident that $(\mathbb{R}^2, f^*\mathcal{A})$ is biholomorphic to $\mathbb{D}$, since there is no non-constant map from $\mathbb{C}$ to $\mathbb{D}$. However, if $(\mathbb{R}^2, \mathcal{A})$
is biholomorphic to $\mathbb{C}$, then by the Uniformization Theorem, $(\mathbb{R}^2, f^*A)$ is biholomorphic to either $\mathbb{C}$ or $\mathbb{D}$. This ambiguity in the biholomorphism class of $(\mathbb{R}^2, f^*A)$ is known as the *type problem* [Ere04].

**Definition 2.9.** We say that a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ has *stable parabolic type*, if for every complex structure $A$ on $\mathbb{R}^2$ such that $(\mathbb{R}^2, A)$ is biholomorphic to $\mathbb{C}$, $(\mathbb{R}^2, f^*A)$ is biholomorphic to $\mathbb{C}$, as well.

If for every such complex structure $A$, $(\mathbb{R}^2, f^*A)$ is biholomorphic to $\mathbb{D}$, we say that $f$ has *stable hyperbolic type*.

*Remark 2.10.* It is shown in [Ere04, pp. 3-4] that every topologically holomorphic map of finite type has either stable parabolic or stable hyperbolic type (see also [Tei37]).

If $f: \mathbb{R}^2 \to \mathbb{R}^2$ is a topologically holomorphic map having stable parabolic type, then by Proposition 2.6, for any orientation-preserving homeomorphism $\varphi: \mathbb{R}^2 \to \mathbb{C}$, there exist another orientation-preserving homeomorphism $\psi: \mathbb{R}^2 \to \mathbb{C}$ and entire map $g$ such that $f = \varphi^{-1} \circ g \circ \psi$. Note that $f \in S$ if and only if $g \in S$.

In the rest of the paper, we consider only entire topologically holomorphic maps of finite type. By Remark 2.10, the type problem in this class of maps has only two possible answers, and we simply call such a map parabolic or hyperbolic.

We say that a topologically holomorphic map $f$ of finite type is a *topological polynomial* of degree $d$ if the covering map $f|\mathbb{R}^2\setminus f^{-1}(S_f)$ has a finite degree $d =: \deg(f)$. Otherwise, we say that $f$ is *transcendental*. It is easy to see that any topological polynomial is parabolic.

The following propositions lay out the structure of the preimage of a neighborhood of an isolated singular value (or infinity) under an entire topologically holomorphic map.

**Proposition 2.11.** Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a topologically holomorphic map, $U \subset \mathbb{R}^2$ be a bounded simply connected domain, and $V$ be a connected component of $f^{-1}(U)$. If $U \cap S_f = \{y\}$, then $V$ is simply connected and exactly one of the following is true:

(1) there exist orientation-preserving homeomorphisms $\varphi: U \to \mathbb{D}$ and $\psi: V \to \mathbb{D}$ and an integer $d \in \mathbb{N}$ such that $\psi(y) = 0$ and $(\varphi \circ (f|V) \circ \psi^{-1})(z) = z^d$ for all $z \in \mathbb{D}$. In particular, $V \setminus f^{-1}(y)$ is an annulus and $f|V \setminus f^{-1}(y): V \setminus f^{-1}(y) \to U \setminus \{y\}$ is a covering map of degree $d$. Additionally, if $\partial U \cap S_f = \emptyset$, then $V$ is bounded;

(2) there exist orientation-preserving homeomorphisms $\varphi: U \to \mathbb{D}$ and $\psi: V \to \mathbb{H}$ such that $\varphi(y) = 0$ and $(\varphi \circ (f|V) \circ \psi^{-1})(z) = \exp(z)$ for all $z \in \mathbb{H}$. In particular, $V$ is unbounded, and $f|V: V \to U \setminus \{y\}$ is a universal covering map.

If $U$ is a bounded simply connected domain such that $U \cap S_f = \emptyset$, then each connected component $V$ of $f^{-1}(U)$ is simply connected and $f|V: V \to U$ is an orientation-preserving homeomorphism. If additionally $\partial U \cap S_f = \emptyset$, then $V$ is bounded.

*Proof.* This proposition follows from [For91, Theorem 5.10, 5.11] in the case when $f$ is an entire holomorphic map. The general case follows easily by applying Proposition 2.6. $\square$

**Proposition 2.12.** Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a topologically holomorphic map, $U$ be an unbounded simply connected domain, and $V$ be a connected component of $f^{-1}(U)$. If $U \cap S_f = \emptyset$, then $V$ is an unbounded simply connected domain, and $f|V: V \to U$ is an orientation-preserving homeomorphism. If $U = \mathbb{R}^2\setminus W$, where $W$ is a compact simply connected set containing $S_f$, then $V$ is an unbounded domain and one of the following is true:
is biholomorphic to $\mathbb{C}$, then by the Uniformization Theorem, $(\mathbb{R}^2, f^*A)$ is biholomorphic to either $\mathbb{C}$ or $\mathbb{D}$. This ambiguity in the biholomorphism class of $(\mathbb{R}^2, f^*A)$ is known as the *type problem* [Ere04].

**Definition 2.9.** We say that a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ has *stable parabolic type*, if for every complex structure $A$ on $\mathbb{R}^2$ such that $(\mathbb{R}^2, A)$ is biholomorphic to $\mathbb{C}$, $(\mathbb{R}^2, f^*A)$ is biholomorphic to $\mathbb{C}$, as well.

If for every such complex structure $A$, $(\mathbb{R}^2, f^*A)$ is biholomorphic to $\mathbb{D}$, we say that $f$ has *stable hyperbolic type*.

*Remark 2.10.* It is shown in [Ere04, pp. 3-4] that every topologically holomorphic map of finite type has either stable parabolic or stable hyperbolic type (see also [Tei37]).

If $f: \mathbb{R}^2 \to \mathbb{R}^2$ is a topologically holomorphic map having stable parabolic type, then by Proposition 2.6, for any orientation-preserving homeomorphism $\varphi: \mathbb{R}^2 \to \mathbb{C}$, there exist another orientation-preserving homeomorphism $\psi: \mathbb{R}^2 \to \mathbb{C}$ and entire map $g$ such that $f = \varphi^{-1} \circ g \circ \psi$. Note that $f \in S$ if and only if $g \in S$.

In the rest of the paper, we consider only entire topologically holomorphic maps of finite type. By Remark 2.10, the type problem in this class of maps has only two possible answers, and we simply call such a map parabolic or hyperbolic.

We say that a topologically holomorphic map $f$ of finite type is a *topological polynomial* of degree $d$ if the covering map $f|\mathbb{R}^2\setminus f^{-1}(S_f)$ has a finite degree $d =: \deg(f)$. Otherwise, we say that $f$ is *transcendental*. It is easy to see that any topological polynomial is parabolic.

The following propositions lay out the structure of the preimage of a neighborhood of an isolated singular value (or infinity) under an entire topologically holomorphic map.

**Proposition 2.11.** Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a topologically holomorphic map, $U \subset \mathbb{R}^2$ be a bounded simply connected domain, and $V$ be a connected component of $f^{-1}(U)$. If $U \cap S_f = \{y\}$, then $V$ is simply connected and exactly one of the following is true:

(1) there exist orientation-preserving homeomorphisms $\varphi: U \to \mathbb{D}$ and $\psi: V \to \mathbb{D}$ and an integer $d \in \mathbb{N}$ such that $\psi(y) = 0$ and $(\varphi \circ (f|V) \circ \psi^{-1})(z) = z^d$ for all $z \in \mathbb{D}$. In particular, $V \setminus f^{-1}(y)$ is an annulus and $f|V \setminus f^{-1}(y): V \setminus f^{-1}(y) \to U \setminus \{y\}$ is a covering map of degree $d$. Additionally, if $\partial U \cap S_f = \emptyset$, then $V$ is bounded;

(2) there exist orientation-preserving homeomorphisms $\varphi: U \to \mathbb{D}$ and $\psi: V \to \mathbb{H}$ such that $\varphi(y) = 0$ and $(\varphi \circ (f|V) \circ \psi^{-1})(z) = \exp(z)$ for all $z \in \mathbb{H}$. In particular, $V$ is unbounded, and $f|V: V \to U \setminus \{y\}$ is a universal covering map.

If $U$ is a bounded simply connected domain such that $U \cap S_f = \emptyset$, then each connected component $V$ of $f^{-1}(U)$ is simply connected and $f|V: V \to U$ is an orientation-preserving homeomorphism. If additionally $\partial U \cap S_f = \emptyset$, then $V$ is bounded.

*Proof.* This proposition follows from [For91, Theorem 5.10, 5.11] in the case when $f$ is an entire holomorphic map. The general case follows easily by applying Proposition 2.6. $\square$

**Proposition 2.12.** Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a topologically holomorphic map, $U$ be an unbounded simply connected domain, and $V$ be a connected component of $f^{-1}(U)$. If $U \cap S_f = \emptyset$, then $V$ is an unbounded simply connected domain, and $f|V: V \to U$ is an orientation-preserving homeomorphism. If $U = \mathbb{R}^2\setminus W$, where $W$ is a compact simply connected set containing $S_f$, then $V$ is an unbounded domain and one of the following is true:
(1) if $f$ is a topological polynomial of degree $d$, then there exist orientation-preserving homeomorphisms $\varphi: U \to D^*$ and $\psi: V \to D^*$ such that $(\varphi \circ (f|_V) \circ \psi^{-1})(z) = z^d$ for all $z \in D^*$. In particular, $V$ is an annulus, $\mathbb{R}^2 \setminus V$ is compact, and $f|_V: V \to U$ is a covering map of degree $d$;

(2) if $f$ is transcendental, then there exist orientation-preserving homeomorphisms $\varphi: U \to D^*$ and $\psi: V \to \mathbb{H}$ such that $(\varphi \circ (f|_V) \circ \psi^{-1})(z) = \exp(z)$ for all $z \in \mathbb{H}$. In particular, $f|_V: V \to U$ is a universal covering map.

*Proof.* The proof is similar to that of Proposition 2.11. □

<figure><img src="image_2.png" /></figure>

Note that Proposition 2.11 implies that every singular value of a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ of finite type is either critical or asymptotic.

We say that an unbounded simply connected domain $V \subset \mathbb{R}^2$ is an *asymptotic tract* of $f$ over $y \in S_f$ if there exists a bounded simply connected domain $U \subset \mathbb{R}^2$ such that $U \cap S_f = \bar{U} \cap S_f = \{y\}$ and $V$ is a connected component of $f^{-1}(U)$. Similarly, $V$ is an asymptotic tract of $f$ over $\infty$ if there exists a simply connected compact set $W$ such that $S_f \subset \text{int}(W)$ and $V$ is a connected component of $f^{-1}(\mathbb{R}^2 \setminus W)$. Note that if $V$ is a tract of $f$, then by Propositions 2.11 and 2.12, $f|_V: V \to f(V)$ is a universal covering map.

Analogous to [ERG15, Proposition 2.3] and proven in a similar way, we have the following isotopy lifting property for entire topologically holomorphic maps of finite type.

**Proposition 2.13.** Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ and $\hat{f}: \mathbb{R}^2 \to \mathbb{R}^2$ be topologically holomorphic maps of finite type with $\varphi_0 \circ f = \hat{f} \circ \psi_0$ for some $\varphi_0, \psi_0 \in \text{Homeo}^+(\mathbb{R}^2)$.

Let $A \subset \mathbb{R}^2$ be a finite set containing $S_f$, and $\varphi_1 \in \text{Homeo}^+(\mathbb{R}^2)$ be isotopic rel. $A$ to $\varphi_0$. Then $\varphi_1 \circ f = \hat{f} \circ \psi_1$ for some $\psi_1 \in \text{Homeo}^+(\mathbb{R}^2)$ isotopic rel. $f^{-1}(A)$ to $\psi_0$.

**2.4. Entire Thurston maps.** Let $f: X \to X$ be a topologically holomorphic map, where $X$ is a topological surface. The *postsingular set* of $f$ is defined as

$$P_f := \bigcup_{n \ge 0} f^{\circ n}(S_f),$$

and its elements are called *postsingular values*. We say that $f$ is *postsingularly finite* (or, shortly, *psf*) if $P_f$ is finite, i.e., $f \in S$ and every singular value is (pre-)periodic under $f$. Postsingularly finite topological polynomials are also called *postcritically finite* (pcf in short), and their postsingular values are called *postcritical* since all their singular values are critical values.

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

**Definition 2.14.** A non-injective topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ is called an *entire Thurston map* if it is parabolic and postsingularly finite.

Given a finite set $A \subset \mathbb{R}^2$ with $P_f \subset A$ and $f(A) \subset A$, we call the pair $(f, A)$ a *marked entire Thurston map* and $A$ its *marked set*.

We often treat marked Thurston maps the same as usual Thurston maps and use the notation $f: (\mathbb{R}^2, A) \hookrightarrow \mathbb{R}^2$. If no marked set is mentioned, we assume it to be $P_f$.

Similarly to Definition 2.14, we can introduce the notion of a *rational Thurston map* $f: \mathbb{S}^2 \to \mathbb{S}^2$ and a *meromorphic Thurston map* $f: \mathbb{R}^2 \to \mathbb{S}^2$. In the literature, the phrase “Thurston map” usually refers to a rational Thurston map, but we use it as shorthand for an entire Thurston map instead. Natural examples of entire Thurston maps are provided by
(1) if $f$ is a topological polynomial of degree $d$, then there exist orientation-preserving homeomorphisms $\varphi: U \to D^*$ and $\psi: V \to D^*$ such that $(\varphi \circ (f|V) \circ \psi^{-1})(z) = z^d$ for all $z \in D^*$. In particular, $V$ is an annulus, $\mathbb{R}^2 \setminus V$ is compact, and $f|V: V \to U$ is a covering map of degree $d$;

(2) if $f$ is transcendental, then there exist orientation-preserving homeomorphisms $\varphi: U \to D^*$ and $\psi: V \to \mathbb{H}$ such that $(\varphi \circ (f|V) \circ \psi^{-1})(z) = \exp(z)$ for all $z \in \mathbb{H}$. In particular, $f|V: V \to U$ is a universal covering map.

*Proof.* The proof is similar to that of Proposition 2.11. $\square$

<figure><img src="image_3.png" /></figure>

Note that Proposition 2.11 implies that every singular value of a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ of finite type is either critical or asymptotic.

We say that an unbounded simply connected domain $V \subset \mathbb{R}^2$ is an *asymptotic tract* of $f$ over $y \in S_f$ if there exists a bounded simply connected domain $U \subset \mathbb{R}^2$ such that $U \cap S_f = \bar{U} \cap S_f = \{y\}$ and $V$ is a connected component of $f^{-1}(U)$. Similarly, $V$ is an asymptotic tract of $f$ over $\infty$ if there exists a simply connected compact set $W$ such that $S_f \subset \text{int}(W)$ and $V$ is a connected component of $f^{-1}(\mathbb{R}^2 \setminus W)$. Note that if $V$ is a tract of $f$, then by Propositions 2.11 and 2.12, $f|V: V \to f(V)$ is a universal covering map.

Analogous to [ERG15, Proposition 2.3] and proven in a similar way, we have the following isotopy lifting property for entire topologically holomorphic maps of finite type.

**Proposition 2.13.** Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ and $\hat{f}: \mathbb{R}^2 \to \mathbb{R}^2$ be topologically holomorphic maps of finite type with $\varphi_0 \circ f = \hat{f} \circ \psi_0$ for some $\varphi_0, \psi_0 \in \text{Homeo}^+(\mathbb{R}^2)$.

Let $A \subset \mathbb{R}^2$ be a finite set containing $S_f$, and $\varphi_1 \in \text{Homeo}^+(\mathbb{R}^2)$ be isotopic rel. $A$ to $\varphi_0$. Then $\varphi_1 \circ f = \hat{f} \circ \psi_1$ for some $\psi_1 \in \text{Homeo}^+(\mathbb{R}^2)$ isotopic rel. $f^{-1}(A)$ to $\psi_0$.

**2.4. Entire Thurston maps.** Let $f: X \to X$ be a topologically holomorphic map, where $X$ is a topological surface. The *postsingular set* of $f$ is defined as

$$P_f := \bigcup_{n \ge 0} f^{\circ n}(S_f),$$

and its elements are called *postsingular values*. We say that $f$ is *postsingularly finite* (or, shortly, *psf*) if $P_f$ is finite, i.e., $f \in S$ and every singular value is (pre-)periodic under $f$. Postsingularly finite topological polynomials are also called *postcritically finite* (pcf in short), and their postsingular values are called *postcritical* since all their singular values are critical values.

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

**Definition 2.14.** A non-injective topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ is called an *entire Thurston map* if it is parabolic and postsingularly finite.

Given a finite set $A \subset \mathbb{R}^2$ with $P_f \subset A$ and $f(A) \subset A$, we call the pair $(f, A)$ a *marked entire Thurston map* and $A$ its *marked set*.

We often treat marked Thurston maps the same as usual Thurston maps and use the notation $f: (\mathbb{R}^2, A) \hookrightarrow \mathbb{R}^2$. If no marked set is mentioned, we assume it to be $P_f$.

Similarly to Definition 2.14, we can introduce the notion of a *rational Thurston map* $f: \mathbb{S}^2 \to \mathbb{S}^2$ and a *meromorphic Thurston map* $f: \mathbb{R}^2 \to \mathbb{S}^2$. In the literature, the phrase “Thurston map” usually refers to a rational Thurston map, but we use it as shorthand for an entire Thurston map instead. Natural examples of entire Thurston maps are provided by
(holomorphic) postsingularly finite entire maps, particularly by postcritically finite complex
polynomials.

**Definition 2.15.** We say that two Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ are Thurston (or combinatorially) equivalent, if there exist homeomorphisms $\varphi, \psi \in \mathrm{Homeo}^+(\mathbb{R}^2)$ such that $\varphi(A) = \psi(A) = B$, $\varphi$ and $\psi$ are isotopic rel. A, and $\varphi \circ f = \hat{f} \circ \psi$.

A Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ is said to be *realized* if it is combinatorially equivalent to a postsingularly finite entire map $g$. If $f$ is not realized, we say that it is *obstructed*.

Frequently, Thurston maps we work with are defined uniquely only up to the following
notions of equivalence (for instance, when we define Thurston maps using combinatorial
constructions).

**Definition 2.16.** Two Thurston maps $f_1: (\mathbb{R}^2, A) \hookrightarrow$ and $f_2: (\mathbb{R}^2, A) \hookrightarrow$ are called *isotopic* (rel. $A$) if there exists $\varphi \in \mathrm{Homeo}_0^+(\mathbb{R}^2, A)$ such that $f_1 = f_2 \circ \varphi$.

The following proposition provides a simple classification of topologically holomorphic maps with only one singular value, as well as Thurston maps with a single postsingular value.

**Proposition 2.17.** *Suppose that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is a topologically holomorphic map such that $|S_f| = 1$. Then $f = \varphi^{-1} \circ g \circ \psi$, for some orientation-preserving homeomorphisms $\varphi, \psi: \mathbb{R}^2 \to \mathbb{C}$ and a unique $g \in \{z \mapsto z^d | d \ge 2\} \cup \{z \mapsto \exp(z)\}$.*

If $f: (\mathbb{R}^2, A) \hookrightarrow$ is a Thurston map with $|A| = 1$, then $f$ is combinatorially equivalent to $z \mapsto z^d$ for some $d \ge 2$. In particular, $A$ consists of a unique fixed critical point of $f$.

*Proof*. The first part essentially follows from Proposition 2.12.

Suppose now that $f: (\mathbb{R}^2, A) \hookrightarrow$ is a Thurston map with $|A| = 1$. Thus, $f$ has a fixed
singular value, so it cannot be of the form $\varphi^{-1} \circ \exp \circ \psi$ for some orientation-preserving
homeomorphisms $\varphi: \mathbb{R}^2 \to \mathbb{C}$ and $\psi: \mathbb{R}^2 \to \mathbb{C}$. Therefore, $(\varphi \circ f \circ \psi^{-1})(z) = z^d$ for all $z \in \mathbb{C}$
and $d = \deg(f)$, where $A = \{\varphi(0)\} = \{\psi(0)\}$. In particular, $\varphi$ and $\psi$ are isotopic rel. $A$ and
$f$ is combinatorially equivalent to $z \mapsto z^d$. $\square$

The dynamics of a Thurston map on its marked set can also be represented visually, in a
way that turns out to be useful in study. The following definition makes this precise.

**Definition 2.18.** Let $f: (\mathbb{R}^2, A) \hookrightarrow$ be a Thurston map. The *marked portrait* (rel. $A$) of $f$ is a weighted directed abstract graph $\mathcal{P}_{f,A}$ such that the vertex set of $\mathcal{P}_{f,A}$ equals $A$, and for each vertex $v \in A$ there exists a unique directed edge from $v$ to $f(v)$ with weight $\deg(f, v)$. Additionally, among all vertices of $\mathcal{P}_{f,A}$, we label the ones that are singular values of $f$.

We say that two Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ have *the same marked portrait* if there exists a bijective map $\varphi: A \to B$ such that

• $\varphi(S_f) = S_{\hat{f}}$, and

• $\varphi$ is an *isomorphism* between the weighted directed abstract graphs $\mathcal{P}_{f,A}$ and $\mathcal{P}_{\hat{f},B}$, i.e., there is an edge $e_{u,v}$ of $\mathcal{P}_{f,A}$ joining $u$ with $v$ if and only if there is an edge $\hat{e}_{\varphi(u),\varphi(v)}$ of $\mathcal{P}_{\hat{f},B}$ joining $\varphi(u)$ with $\varphi(v)$. The weights of $e_{u,v}$ and $\hat{e}_{\varphi(u),\varphi(v)}$ coincide.

In the setting of Definition 2.18, if $A = P_f$, then, for simplicity, we denote by $\mathcal{P}_f$ the marked portrait rel. $P_f$ of $f$ and call it the *postsingular portrait* of $f$.
(holomorphic) postsingularly finite entire maps, particularly by postcritically finite complex
polynomials.

**Definition 2.15.** We say that two Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ are Thurston (or combinatorially) equivalent, if there exist homeomorphisms $\varphi, \psi \in \mathrm{Homeo}^+(\mathbb{R}^2)$ such that $\varphi(A) = \psi(A) = B$, $\varphi$ and $\psi$ are isotopic rel. A, and $\varphi \circ f = \hat{f} \circ \psi$.

A Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ is said to be *realized* if it is combinatorially equivalent to a postsingularly finite entire map $g$. If $f$ is not realized, we say that it is *obstructed*.

Frequently, Thurston maps we work with are defined uniquely only up to the following
notions of equivalence (for instance, when we define Thurston maps using combinatorial
constructions).

**Definition 2.16.** Two Thurston maps $f_1: (\mathbb{R}^2, A) \hookrightarrow$ and $f_2: (\mathbb{R}^2, A) \hookrightarrow$ are called *isotopic* (rel. $A$) if there exists $\varphi \in \mathrm{Homeo}_0^+(\mathbb{R}^2, A)$ such that $f_1 = f_2 \circ \varphi$.

The following proposition provides a simple classification of topologically holomorphic
maps with only one singular value, as well as Thurston maps with a single postsingular
value.

**Proposition 2.17.** *Suppose that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is a topologically holomorphic map such that $|S_f| = 1$. Then $f = \varphi^{-1} \circ g \circ \psi$, for some orientation-preserving homeomorphisms $\varphi, \psi: \mathbb{R}^2 \to \mathbb{C}$ and a unique $g \in \{z \mapsto z^d | d \ge 2\} \cup \{z \mapsto \exp(z)\}$.*

If $f: (\mathbb{R}^2, A) \hookrightarrow$ is a Thurston map with $|A| = 1$, then $f$ is combinatorially equivalent to $z \mapsto z^d$ for some $d \ge 2$. In particular, $A$ consists of a unique fixed critical point of $f$.

*Proof*. The first part essentially follows from Proposition 2.12.

Suppose now that $f: (\mathbb{R}^2, A) \hookrightarrow$ is a Thurston map with $|A| = 1$. Thus, $f$ has a fixed
singular value, so it cannot be of the form $\varphi^{-1} \circ \exp \circ \psi$ for some orientation-preserving
homeomorphisms $\varphi: \mathbb{R}^2 \to \mathbb{C}$ and $\psi: \mathbb{R}^2 \to \mathbb{C}$. Therefore, $(\varphi \circ f \circ \psi^{-1})(z) = z^d$ for all $z \in \mathbb{C}$
and $d = \deg(f)$, where $A = \{\varphi(0)\} = \{\psi(0)\}$. In particular, $\varphi$ and $\psi$ are isotopic rel. $A$ and
$f$ is combinatorially equivalent to $z \mapsto z^d$. $\square$

The dynamics of a Thurston map on its marked set can also be represented visually, in a
way that turns out to be useful in study. The following definition makes this precise.

**Definition 2.18.** Let $f: (\mathbb{R}^2, A) \hookrightarrow$ be a Thurston map. The *marked portrait* (rel. $A$) of $f$ is a weighted directed abstract graph $\mathcal{P}_{f,A}$ such that the vertex set of $\mathcal{P}_{f,A}$ equals $A$, and for each vertex $v \in A$ there exists a unique directed edge from $v$ to $f(v)$ with weight $\deg(f, v)$. Additionally, among all vertices of $\mathcal{P}_{f,A}$, we label the ones that are singular values of $f$.

We say that two Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ have *the same marked portrait* if there exists a bijective map $\varphi: A \to B$ such that

$\varphi(S_f) = S_{\hat{f}}$, and

* $\varphi$ is an *isomorphism* between the weighted directed abstract graphs $\mathcal{P}_{f,A}$ and $\mathcal{P}_{\hat{f},B}$, i.e., there is an edge $e_{u,v}$ of $\mathcal{P}_{f,A}$ joining $u$ with $v$ if and only if there is an edge $\hat{e}_{\varphi(u),\varphi(v)}$ of $\mathcal{P}_{\hat{f},B}$ joining $\varphi(u)$ with $\varphi(v)$. The weights of $e_{u,v}$ and $\hat{e}_{\varphi(u),\varphi(v)}$ coincide.

In the setting of Definition 2.18, if $A = P_f$, then, for simplicity, we denote by $\mathcal{P}_f$ the
marked portrait rel. $P_f$ of $f$ and call it the *postsingular portrait* of $f$.
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Examples of postsingular portraits</figcaption></figure>

If Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ have the same marked portrait, then it is clear that $f|A$ and $\hat{f}|B$ are conjugate dynamical systems. Combinatorial equivalence of $f$ and $\hat{f}$ is a sufficient condition for them to have the same marked portrait.

*Example 2.19.* Given below are two postsingularly finite entire maps that will be used as prototypical examples throughout this paper.

(1) The map $G_1(z) = \pi \cos(z)/2$ has no asymptotic values and two critical values $\pm \pi/2$, with $P_{G_1} = \{0, -\pi/2, \pi/2\}$;

(2) The map $G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$ has a unique critical value 0 and a unique asymptotic value $\sqrt{\ln 2}$, with $P_{G_2} = \{0, -\sqrt{\ln 2}, \sqrt{\ln 2}\}$.

Figure 1 illustrates $P_{G_1}$ and $P_{G_2}$. Singular and non-singular vertices of the corresponding graphs are labeled by solid and hollow squares, respectively.

**2.5. Quasincoformal maps and Teichmüller spaces.** We say that an orientation-preserving homeomorphism $\varphi: \mathbb{C} \to \mathbb{C}$ is $K$-quasiconformal for $K \ge 1$ if for every annulus $V \subset \mathbb{C}$ of finite modulus, we have

$$ \mathrm{mod}(V)/K \le \mathrm{mod}(\varphi(V)) \le K\mathrm{mod}(V), $$

where $\mathrm{mod}(V)$ and $\mathrm{mod}(\varphi(V))$ are the moduli of the annuli $V$ and $\varphi(V)$, respectively (see [Hub06, Proposition 3.2.1] for the definition of modulus).

The infimum of all values $K$ such that $\varphi$ is $K$-quasiconformal is called the *dilatation* of $\varphi$, and it is denoted by $K(\varphi)$ (in particular, $\varphi$ is $K(\varphi)$-quasiconformal). For further properties and equivalent definitions of quasiconformal maps, see [Ahl06], [Hub06], or [BF14].

**Proposition 2.20.** Let $\varphi_n: \mathbb{C} \to \mathbb{C}, n \in \mathbb{N}$ be a sequence of $K_n$-quasiconformal maps fixing two distinct points $a, b \in \mathbb{C}$. Suppose that $(K_n)$ converges to 1 as $n \to \infty$, then the sequence $(\varphi_n)$ converges locally uniformly to $\mathrm{id}_\mathbb{C}$.

*Proof.* Let us consider an arbitrary subsequence $(\varphi_{n_k})$ of the sequence $(\varphi_n)$. Theorem [BF14, Theorem 1.26] implies that from $(\varphi_{n_k})$, we can extract a further subsequence converging locally uniformly to the quasiconformal limit $\varphi: \mathbb{C} \to \mathbb{C}$. Moreover, from the same theorem, one can easily derive that the dilatation of $\varphi$ equals 1. Thus, by Weyl's Lemma [BF14, Theorem 1.14], we see that $\varphi$ is holomorphic; since it fixes two distinct points of $\mathbb{C}$, it coincides with $\mathrm{id}_\mathbb{C}$. This argument applies to an arbitrary subsequence of $(\varphi_n)$. Therefore, we can conclude that the full sequence $(\varphi_n)$ converges locally uniformly (on $\mathbb{C}$) to $\mathrm{id}_\mathbb{C}$. $\square$
<figure><img src="image_4.png" /><figcaption>FIGURE 1. Examples of postsingular portraits</figcaption></figure>

If Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ have the same marked portrait, then it is clear that $f|A$ and $\hat{f}|B$ are conjugate dynamical systems. Combinatorial equivalence of $f$ and $\hat{f}$ is a sufficient condition for them to have the same marked portrait.

*Example 2.19.* Given below are two postsingularly finite entire maps that will be used as prototypical examples throughout this paper.

(1) The map $G_1(z) = \pi \cos(z)/2$ has no asymptotic values and two critical values $\pm \pi/2$, with $P_{G_1} = \{0, -\pi/2, \pi/2\}$;

(2) The map $G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$ has a unique critical value 0 and a unique asymptotic value $\sqrt{\ln 2}$, with $P_{G_2} = \{0, -\sqrt{\ln 2}, \sqrt{\ln 2}\}$.

Figure 1 illustrates $P_{G_1}$ and $P_{G_2}$. Singular and non-singular vertices of the corresponding graphs are labeled by solid and hollow squares, respectively.

**2.5. Quasincoformal maps and Teichmüller spaces.** We say that an orientation-preserving homeomorphism $\varphi: \mathbb{C} \to \mathbb{C}$ is $K$-quasiconformal for $K \ge 1$ if for every annulus $V \subset \mathbb{C}$ of finite modulus, we have

$$ \mathrm{mod}(V)/K \le \mathrm{mod}(\varphi(V)) \le K\mathrm{mod}(V), $$

where $\mathrm{mod}(V)$ and $\mathrm{mod}(\varphi(V))$ are the moduli of the annuli $V$ and $\varphi(V)$, respectively (see [Hub06, Proposition 3.2.1] for the definition of modulus).

The infimum of all values $K$ such that $\varphi$ is $K$-quasiconformal is called the *dilatation* of $\varphi$, and it is denoted by $K(\varphi)$ (in particular, $\varphi$ is $K(\varphi)$-quasiconformal). For further properties and equivalent definitions of quasiconformal maps, see [Ahl06], [Hub06], or [BF14].

**Proposition 2.20.** Let $\varphi_n: \mathbb{C} \to \mathbb{C}, n \in \mathbb{N}$ be a sequence of $K_n$-quasiconformal maps fixing two distinct points $a, b \in \mathbb{C}$. Suppose that $(K_n)$ converges to 1 as $n \to \infty$, then the sequence $(\varphi_n)$ converges locally uniformly to $\mathrm{id}_\mathbb{C}$.

*Proof.* Let us consider an arbitrary subsequence $(\varphi_{n_k})$ of the sequence $(\varphi_n)$. Theorem [BF14, Theorem 1.26] implies that from $(\varphi_{n_k})$, we can extract a further subsequence converging locally uniformly to the quasiconformal limit $\varphi: \mathbb{C} \to \mathbb{C}$. Moreover, from the same theorem, one can easily derive that the dilatation of $\varphi$ equals 1. Thus, by Weyl's Lemma [BF14, Theorem 1.14], we see that $\varphi$ is holomorphic; since it fixes two distinct points of $\mathbb{C}$, it coincides with $\mathrm{id}_\mathbb{C}$. This argument applies to an arbitrary subsequence of $(\varphi_n)$. Therefore, we can conclude that the full sequence $(\varphi_n)$ converges locally uniformly (on $\mathbb{C}$) to $\mathrm{id}_\mathbb{C}$. $\square$
Let $A \subset \mathbb{R}^2$ be a finite set. Then Teichmüller space of $\mathbb{R}^2$ with the marked set $A$ (or Teichmüller space modelled on $\mathbb{R}^2\setminus A$) is defined as

$$ \text{Teich}(\mathbb{R}^2, A) := \{\varphi|\varphi: \mathbb{R}^2 \to \mathbb{C} \text{ is an orientation-preserving homeomorphism}\} / \sim $$
where $\varphi \sim \psi$ if there exists an affine map $M$ such that $\varphi$ is isotopic rel. $A$ to $M \circ \psi$.

Note that Teich($\mathbb{R}^2$, $A$) is essentially the same space as more commonly known Teichmüller space of the topological sphere $S^2 = \mathbb{R}^2 \cup \{\infty\}$ with the marked set $A \cup \{\infty\}$ (see, for instance, [BCT14, Section 1.1]). However, since transcendental Thurston maps cannot be extended to $S^2$, the point $\infty$ plays a special role. Hence, for every $\tau = [\varphi] \in \text{Teich}(S^2, A \cup \{\infty\})$, we make a choice of representative $\varphi$ so that $\varphi(\infty) = \infty$ leading to the definition of Teich($\mathbb{R}^2$, $A$) above.

The space Teich($\mathbb{R}^2$, $A$) has a natural structure of a complex ($|A|-2$)-dimensional manifold [Hub06, Theorem 6.5.1]. It is also possible to define a complete metric, called the Teichmüller metric, on Teich($\mathbb{R}^2$, $A$) by setting

$$ d([\varphi_1], [\varphi_2]) := \inf_{\psi} \log K(\psi), $$

where the infimum is taken over all quasiconformal homeomorphisms $\psi: \mathbb{C} \to \mathbb{C}$ such that $\varphi_1$ and $\psi \circ \varphi_2$ are isotopic rel. $A$ (see [Hub06, Section 6.4] for details).

## 2.6. Thurston pullback map.
In this section, we introduce the notion of Thurston pullback map for an (entire) Thurston map and provide its essential properties. This notion is classical for rational Thurston maps (e.g., [Hub06, Definition 10.6.1]), but it is less well-known in the transcendental setting (see [HSS09] for the case of exponential Thurston maps and [Pfr19, Section 5.1], [She22] for the general case).

**Proposition 2.21.** Let $f: (\mathbb{R}^2, A) \subsetneq \mathcal{B}$ be a Thurston map and $\varphi: \mathbb{R}^2 \to \mathbb{C}$ be an orientation-preserving homeomorphism. Then there exists an orientation-preserving homeomorphism $\psi: \mathbb{R}^2 \to \mathbb{C}$ such that $g_\varphi := \varphi \circ f \circ \psi^{-1}: \mathbb{C} \to \mathbb{C}$ is an entire holomorphic map. In other words, the following diagram commutes

$$ \begin{tikzcd} (\mathbb{R}^2, A) \arrow[r, "\psi"] & (\mathbb{C}, \psi(A)) \arrow[d, "f"] \\ (\mathbb{R}^2, A) \arrow[r, "\varphi"] & (\mathbb{C}, \varphi(A)) \arrow[d, "g_\varphi := \varphi \circ f \circ \psi^{-1}"] \end{tikzcd} $$

The homeomorphism $\psi$ is unique up to post-composition with an affine map. Different choices of $\varphi$ that represent the same point in Teich($\mathbb{R}^2$, $A$) yield maps $\psi$ that represent the same point in Teich($\mathbb{R}^2$, $A$).

In other words, we have a well-defined map $\sigma_f: \text{Teich}(\mathbb{R}^2, A) \to \text{Teich}(\mathbb{R}^2, A)$ such that $\sigma_f([\varphi]) = [\psi]$, called the Thurston pullback map (or simply the $\sigma$-map) associated with $f$. As $\varphi$ ranges across all maps representing a single point in Teich($\mathbb{R}^2$, $A$), the function $g_\varphi$ is uniquely defined up to pre- and post-composition with affine maps.

**Proof.** The existence of a homeomorphism $\psi$ follows from Proposition 2.6 and the Unification theorem. It is also clear that $\psi$ is unique up to post-composition with an affine map and that post-composing $\varphi$ with an affine map does not affect $\psi$. Due to Proposition 2.13, changing $\varphi$ by isotopy rel. $A$ does not change $[\psi]$. Thus, changing $\varphi$ within its equivalence class in Teich($\mathbb{R}^2$, $A$) does not affect $[\psi]$, showing that $\sigma_f$ introduced as above
Let $A \subset \mathbb{R}^2$ be a finite set. Then Teichmüller space of $\mathbb{R}^2$ with the marked set $A$ (or Teichmüller space modelled on $\mathbb{R}^2\setminus A$) is defined as

$$ \text{Teich}(\mathbb{R}^2, A) := \{\varphi|\varphi : \mathbb{R}^2 \to \mathbb{C} \text{ is an orientation-preserving homeomorphism}\} / \sim $$
where $\varphi \sim \psi$ if there exists an affine map $M$ such that $\varphi$ is isotopic rel. $A$ to $M \circ \psi$.

Note that Teich($\mathbb{R}^2$, $A$) is essentially the same space as more commonly known Teichmüller space of the topological sphere $S^2 = \mathbb{R}^2 \cup \{\infty\}$ with the marked set $A \cup \{\infty\}$ (see, for instance, [BCT14, Section 1.1]). However, since transcendental Thurston maps cannot be extended to $S^2$, the point $\infty$ plays a special role. Hence, for every $\tau = [\varphi] \in \text{Teich}(S^2, A \cup \{\infty\})$, we make a choice of representative $\varphi$ so that $\varphi(\infty) = \infty$ leading to the definition of Teich($\mathbb{R}^2$, $A$) above.

The space Teich($\mathbb{R}^2$, $A$) has a natural structure of a complex ($|A|-2$)-dimensional manifold [Hub06, Theorem 6.5.1]. It is also possible to define a complete metric, called the Teichmüller metric, on Teich($\mathbb{R}^2$, $A$) by setting

$$ d([\varphi_1], [\varphi_2]) := \inf_{\psi} \log K(\psi), $$

where the infimum is taken over all quasiconformal homeomorphisms $\psi: \mathbb{C} \to \mathbb{C}$ such that $\varphi_1$ and $\psi \circ \varphi_2$ are isotopic rel. $A$ (see [Hub06, Section 6.4] for details).

## 2.6. Thurston pullback map.
In this section, we introduce the notion of Thurston pullback map for an (entire) Thurston map and provide its essential properties. This notion is classical for rational Thurston maps (e.g., [Hub06, Definition 10.6.1]), but it is less well-known in the transcendental setting (see [HSS09] for the case of exponential Thurston maps and [Pfr19, Section 5.1], [She22] for the general case).

**Proposition 2.21.** Let $f: (\mathbb{R}^2, A) \subsetneq \mathcal{B}$ be a Thurston map and $\varphi: \mathbb{R}^2 \to \mathbb{C}$ be an orientation-preserving homeomorphism. Then there exists an orientation-preserving homeomorphism $\psi: \mathbb{R}^2 \to \mathbb{C}$ such that $g_\varphi := \varphi \circ f \circ \psi^{-1}: \mathbb{C} \to \mathbb{C}$ is an entire holomorphic map. In other words, the following diagram commutes

$$ \begin{tikzcd} (\mathbb{R}^2, A) \arrow[r, "\psi"] & (\mathbb{C}, \psi(A)) \arrow[d, "f"] \\ (\mathbb{R}^2, A) \arrow[r, "\varphi"] & (\mathbb{C}, \varphi(A)) \arrow[d, "g_\varphi := \varphi \circ f \circ \psi^{-1}"] \end{tikzcd} $$

The homeomorphism $\psi$ is unique up to post-composition with an affine map. Different choices of $\varphi$ that represent the same point in Teich($\mathbb{R}^2$, $A$) yield maps $\psi$ that represent the same point in Teich($\mathbb{R}^2$, $A$).

In other words, we have a well-defined map $\sigma_f: \text{Teich}(\mathbb{R}^2, A) \to \text{Teich}(\mathbb{R}^2, A)$ such that $\sigma_f([\varphi]) = [\psi]$, called the Thurston pullback map (or simply the $\sigma$-map) associated with $f$. As $\varphi$ ranges across all maps representing a single point in Teich($\mathbb{R}^2$, $A$), the function $g_\varphi$ is uniquely defined up to pre- and post-composition with affine maps.

**Proof.** The existence of a homeomorphism $\psi$ follows from Proposition 2.6 and the Unification theorem. It is also clear that $\psi$ is unique up to post-composition with an affine map and that post-composing $\varphi$ with an affine map does not affect $\psi$. Due to Proposition 2.13, changing $\varphi$ by isotopy rel. $A$ does not change $[\psi]$. Thus, changing $\varphi$ within its equivalence class in Teich($\mathbb{R}^2$, $A$) does not affect $[\psi]$, showing that $\sigma_f$ introduced as above
is well-defined. These arguments also show that $g_\varphi$ is uniquely defined up to pre- and post-
composition with affine maps.
$\square$

It is important to note that $\sigma_f$ is a holomorphic map with respect to the natural complex structure on Teich($\mathbb{R}^2$, $A$) (for a proof, see, for instance, [BCT14, Section 1.3]). Moreover, it is well-behaved with respect to the Teichmüller metric, as the following proposition suggests.

**Proposition 2.22.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{be a Thurston map, then}$

(1) $\sigma_f$ is 1-Lipschitz, i.e., $d(\sigma_f(\tau_1), \sigma_f(\tau_2)) \le d(\tau_1, \tau_2)$ for every $\tau_1, \tau_2 \in \text{Teich}(\mathbb{R}^2, A)$;

(2) if $f$ is transcendental, then $\sigma_f$ is locally uniformly contracting, i.e., for any compact $K \subset \text{Teich}(\mathbb{R}^2, A)$ there exists $\varepsilon_K > 0$ such that $d(\sigma_f(\tau_1), \sigma_f(\tau_2)) \le (1 - \varepsilon_K) d(\tau_1, \tau_2)$ for every $\tau_1, \tau_2 \in K$;

(3) if $f$ is polynomial, then $\sigma_f^{\circ 2}$ is locally uniformly contracting.

*Proof.* The first item follows from the fact that every holomorphic map on Teich($\mathbb{R}^2$, $A$) is 1-Lipschitz [Hub06, Corollary 6.10.7]. The second item follows from [HSS09, Section 3.2] (see also [Pfr19, Chapter 5.1] and [She22]). The last item follows from [Hub16, Corollary 10.7.8].
$\square$

<figure><img src="image_4.png" /></figure>

*Remark 2.23.* If Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow \hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ are Thurston equivalent, then $\sigma_f: \text{Teich}(\mathbb{R}^2, A) \to \text{Teich}(\mathbb{R}^2, A)$ and $\sigma_{\hat{f}}: \text{Teich}(\mathbb{R}^2, B) \to \text{Teich}(\mathbb{R}^2, B)$ are conjugate by a biholomorphism. In the special case where $A=B$ and $f$ is isotopic rel. $A$ to $\hat{f}$, we have $\sigma_f = \sigma_{\hat{f}}$.

One of the most crucial properties of Thurston pullback maps is summarized by the
following proposition, and follows from Definition 2.15 and Proposition 2.21 (also cf. [Hub16,
Theorem 10.6.4] and [HSS09, Theorem 3.1]).

**Proposition 2.24.** A Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \sigma_f$ is realized if and only if the Thurston pullback map $\sigma_f: \text{Teich}(\mathbb{R}^2, A) \to \text{Teich}(\mathbb{R}^2, A)$ has a fixed point $\tau \in \text{Teich}(\mathbb{R}^2, A)$.

*Proof.* Suppose that $f$ is realized by a postsingularly finite entire map $g: (\mathbb{C}, B) \hookrightarrow$. Then by Definition 2.15, there exist orientation-preserving homeomorphisms $\varphi, \psi: \mathbb{R}^2 \to \mathbb{C}$ such that $\varphi(A) = \psi(A) = B$, $\varphi$ and $\psi$ are isotopic rel. $A$, and $\varphi \circ f = g \circ \psi$. Clearly, $\tau = [\varphi] = [\psi] \in \text{Teich}(\mathbb{R}^2, A)$ is a fixed point of $\sigma_f$.

Now suppose that $\tau = [\varphi] \in \text{Teich}(\mathbb{R}^2, A)$ is a fixed point of $\sigma_f$. Let $\psi: \mathbb{R}^2 \to \mathbb{C}$ be an orientation-preserving homeomorphism so that $g_\varphi = \varphi \circ f \circ \psi^{-1}$ is entire. Since $[\varphi] = [\psi]$, by post-composing $\psi$ with an affine map we can assume that $\varphi|_A = \psi|_A$, which in turn implies that $\varphi$ and $\psi$ are isotopic rel. $A$. Thus, $g_\varphi: (\mathbb{C}, \varphi(A)) \hookrightarrow$ is a postsingularly finite entire map Thurston equivalent to $f$. $\square$

<figure><img src="image_5.png" /></figure>

*Remark 2.25.* Since Teich($\mathbb{R}^2$, $A$) consists of one point for $|A| = 2$, Proposition 2.24 immediately implies that every Thurston map with two postsingular values is realized.

Finally, Propositions 2.22 and 2.24 lead to the following result called *Thurston's rigidity* (cf. [Hub06, Corollary 10.7.8]).

**Proposition 2.26.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{be a Thurston map, then}$

(1) $\sigma_f$ can have at most one fixed point;

(2) if $g_1: (\mathbb{C}, A_1) \hookrightarrow$ and $g_2: (\mathbb{C}, A_2) \hookrightarrow$ are postsingularly finite entire maps Thurston equivalent to $f$, then $g_1$ and $g_2$ are conjugate by an affine map.
is well-defined. These arguments also show that $g_\varphi$ is uniquely defined up to pre- and post-
composition with affine maps.
$\square$

It is important to note that $\sigma_f$ is a holomorphic map with respect to the natural complex structure on Teich($\mathbb{R}^2$, $A$) (for a proof, see, for instance, [BCT14, Section 1.3]). Moreover, it is well-behaved with respect to the Teichmüller metric, as the following proposition suggests.

**Proposition 2.22.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{be a Thurston map, then}$

(1) $\sigma_f$ is 1-Lipschitz, i.e., $d(\sigma_f(\tau_1), \sigma_f(\tau_2)) \le d(\tau_1, \tau_2)$ for every $\tau_1, \tau_2 \in \text{Teich}(\mathbb{R}^2, A)$;

(2) if $f$ is transcendental, then $\sigma_f$ is locally uniformly contracting, i.e., for any compact $K \subset \text{Teich}(\mathbb{R}^2, A)$ there exists $\varepsilon_K > 0$ such that $d(\sigma_f(\tau_1), \sigma_f(\tau_2)) \le (1 - \varepsilon_K) d(\tau_1, \tau_2)$ for every $\tau_1, \tau_2 \in K$;

(3) if $f$ is polynomial, then $\sigma_f^{\circ 2}$ is locally uniformly contracting.

*Proof*. The first item follows from the fact that every holomorphic map on Teich($\mathbb{R}^2$, $A$) is 1-Lipschitz [Hub06, Corollary 6.10.7]. The second item follows from [HSS09, Section 3.2] (see also [Pfr19, Chapter 5.1] and [She22]). The last item follows from [Hub16, Corollary 10.7.8].
$\square$

<figure><img src="image_5.png" /></figure>

*Remark 2.23.* If Thurston maps $f: (\mathbb{R}^2, A) \hookrightarrow \hat{f}: (\mathbb{R}^2, B) \hookrightarrow$ are Thurston equivalent, then $\sigma_f$: Teich($\mathbb{R}^2$, $A) \to \text{Teich}(\mathbb{R}^2$, $A)$ and $\sigma_{\hat{f}}$: Teich($\mathbb{R}^2$, $B) \to \text{Teich}(\mathbb{R}^2$, $B)$ are conjugate by a biholomorphism. In the special case where $A = B$ and $f$ is isotopic rel. $A$ to $\hat{f}$, we have $\sigma_f = \sigma_{\hat{f}}$.

One of the most crucial properties of Thurston pullback maps is summarized by the
following proposition, and follows from Definition 2.15 and Proposition 2.21 (also cf. [Hub16,
Theorem 10.6.4] and [HSS09, Theorem 3.1]).

**Proposition 2.24.** A Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \sigma_f$ is realized if and only if the Thurston pullback map $\sigma_f: \text{Teich}(\mathbb{R}^2, A) \to \text{Teich}(\mathbb{R}^2, A)$ has a fixed point $\tau \in \text{Teich}(\mathbb{R}^2, A)$.

*Proof.* Suppose that $f$ is realized by a postsingularly finite entire map $g: (\mathbb{C}, B) \hookrightarrow$. Then by Definition 2.15, there exist orientation-preserving homeomorphisms $\varphi, \psi: \mathbb{R}^2 \to \mathbb{C}$ such that $\varphi(A) = \psi(A) = B$, $\varphi$ and $\psi$ are isotopic rel. $A$, and $\varphi \circ f = g \circ \psi$. Clearly, $\tau = [\varphi] = [\psi] \in \text{Teich}(\mathbb{R}^2, A)$ is a fixed point of $\sigma_f$.

Now suppose that $\tau = [\varphi] \in \text{Teich}(\mathbb{R}^2, A)$ is a fixed point of $\sigma_f$. Let $\psi: \mathbb{R}^2 \to \mathbb{C}$ be an orientation-preserving homeomorphism so that $g_\varphi = \varphi \circ f \circ \psi^{-1}$ is entire. Since $[\varphi] = [\psi]$, by post-composing $\psi$ with an affine map we can assume that $\varphi|_A = \psi|_A$, which in turn implies that $\varphi$ and $\psi$ are isotopic rel. $A$. Thus, $g_\varphi: (\mathbb{C}, \varphi(A)) \hookrightarrow$ is a postsingularly finite entire map Thurston equivalent to $f$. $\square$

<figure><img src="image_6.png" /></figure>

*Remark 2.25.* Since Teich($\mathbb{R}^2$, $A$) consists of one point for $|A| = 2$, Proposition 2.24 immediately implies that every Thurston map with two postsingular values is realized.

Finally, Propositions 2.22 and 2.24 lead to the following result called *Thurston's rigidity* (cf. [Hub06, Corollary 10.7.8]).

**Proposition 2.26.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{be a Thurston map, then}$

(1) $\sigma_f$ can have at most one fixed point;

(2) if $g_1: (\mathbb{C}, A_1) \hookrightarrow$ and $g_2: (\mathbb{C}, A_2) \hookrightarrow$ are postsingularly finite entire maps Thurston equivalent to $f$, then $g_1$ and $g_2$ are conjugate by an affine map.
**2.7. Admissible quadruples.** This section discusses a combinatorial construction of entire topologically holomorphic maps of finite type. More precisely, we introduce combinatorial models based on planar embedded graphs called *admissible quadruples* (Definition 2.27) and show that every such quadruple defines an entire topologically holomorphic map of finite type and vice versa (Propositions 2.29 and 2.31). We also introduce a notion of equivalence for admissible quadruples and show that equivalent quadruples correspond to equivalent maps (see Definition 2.32 and Proposition 2.33 for details). Although the framework introduced in this section is non-dynamical, in Section 3.3 we will showcase its utility in exploring dynamical properties of Thurston maps.

The constructions in this section are very similar to *line complexes* (see [Nev70, Section 2] and [GO08, Section XI]) and *rose maps* used for defining topological polynomials in $\mathrm{FKK}^+22$. We refrain from borrowing the above objects directly for the sake of continuity. Additionally, we provide short proofs for all the necessary statements, even though they are mostly folklore in the context of the objects mentioned above.

Let $\mathcal{R}$ be a counterclockwise directed rose graph that surrounds a finite set $A \subset \mathbb{R}^2$ (for a definition, see Section 2.2) with $m$ petals $p_1, p_2, \dots, p_m$ written in the counterclockwise order around the center $t$ of $\mathcal{R}$. Denote by $P_j$ the face of $\mathcal{R}$ surrounded by $p_j$ for each $j = 1, 2, \dots, m$, and by $P_\infty$ the unique unbounded face of $\mathcal{R}$. Label the points $A$ as $a_1, a_2, \dots, a_m$, such that $a_j \in P_j$ for each $j = 1, 2, \dots, m$.

Let $\Gamma$ be a $2m$-regular connected planar embedded graph, and $\Phi: \Gamma \to \mathcal{R}$ be a covering map such that $\Phi(v) = t$ for all $v \in V(\Gamma)$ and $\Phi(e) \in E(\mathcal{R})$ for all $e \in E(\Gamma)$. We view the graph $\Gamma$ as a directed graph since the map $\Phi$ naturally induces an orientation on $\Gamma$ (see Section 2.2).

Suppose that $m \ge 2$. Consider an arbitrary face $F \in F(\Gamma)$. We label $F$ by $P_j$ if $\Phi(E(\partial F)) = \{p_j\}$, and if no such $j$ exists (i.e., $|\Phi(E(\partial F))| > 1$) we label $F$ by $P_\infty$. We denote by $\Gamma^*$ the directed planar embedded graph obtained by subdividing each edge of $\Gamma$.

**Definition 2.27.** Consider a quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$, where $A, \mathcal{R}, \Gamma$, and $\Phi$ are as above. We say that $\Delta$ is an *admissible quadruple* if $m = 1$ or if the following conditions are satisfied:

(1) every face $F \in F(\Gamma)$ labeled by $P_\infty$ is unbounded, and

(2) for each $v \in V(\Gamma)$, the set of edges of $\Gamma^*$ incident to $v$ can be written in the counterclockwise order as $e_1 = e_{2m+1}, e_2, e_3, \dots, e_{2m}$ such that the following conditions are satisfied for each $j = 1, 2, \dots, m$:

* $e_{2j-1}$ is outgoing at $v$ and $e_{2j}$ is incoming at $v$;

* the edges $e_{2j-1}$ and $e_{2j}$ belong to the boundary of the same face $F \in F(\Gamma)$ labelled by $P_j$;

* the edges $e_{2j}$ and $e_{2j+1}$ belong to the boundary of the same face $F \in F(\Gamma)$ labelled by $P_\infty$.

The set $A$ is called the *marked set* of $\Delta$.

*Remark 2.28.* Suppose that $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ is an admissible quadruple and $F$ is a face of $\Gamma$. If $F$ is not labelled by $P_\infty$, then clearly, $\partial F$ is a counterclockwise directed cycle if $F$ is bounded, and an infinite unilaterally connected chain, otherwise. If $F$ is labelled by $P_\infty$, then $\partial F$ is a unilaterally connected graph.

Natural examples of admissible quadruples are provided by preimages of rose graphs under entire topologically holomorphic maps of finite type. More precisely, suppose that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is topologically holomorphic and rose graph $\mathcal{R}$ surrounds the set $A$, where
**2.7. Admissible quadruples.** This section discusses a combinatorial construction of entire topologically holomorphic maps of finite type. More precisely, we introduce combinatorial models based on planar embedded graphs called *admissible quadruples* (Definition 2.27) and show that every such quadruple defines an entire topologically holomorphic map of finite type and vice versa (Propositions 2.29 and 2.31). We also introduce a notion of equivalence for admissible quadruples and show that equivalent quadruples correspond to equivalent maps (see Definition 2.32 and Proposition 2.33 for details). Although the framework introduced in this section is non-dynamical, in Section 3.3 we will showcase its utility in exploring dynamical properties of Thurston maps.

The constructions in this section are very similar to *line complexes* (see [Nev70, Section 2] and [GO08, Section XI]) and *rose maps* used for defining topological polynomials in $\mathrm{FKK}^+22$. We refrain from borrowing the above objects directly for the sake of continuity. Additionally, we provide short proofs for all the necessary statements, even though they are mostly folklore in the context of the objects mentioned above.

Let $\mathcal{R}$ be a counterclockwise directed rose graph that surrounds a finite set $A \subset \mathbb{R}^2$ (for a definition, see Section 2.2) with $m$ petals $p_1, p_2, \dots, p_m$ written in the counterclockwise order around the center $t$ of $\mathcal{R}$. Denote by $P_j$ the face of $\mathcal{R}$ surrounded by $p_j$ for each $j = 1, 2, \dots, m$, and by $P_\infty$ the unique unbounded face of $\mathcal{R}$. Label the points $A$ as $a_1, a_2, \dots, a_m$, such that $a_j \in P_j$ for each $j = 1, 2, \dots, m$.

Let $\Gamma$ be a $2m$-regular connected planar embedded graph, and $\Phi: \Gamma \to \mathcal{R}$ be a covering map such that $\Phi(v) = t$ for all $v \in V(\Gamma)$ and $\Phi(e) \in E(\mathcal{R})$ for all $e \in E(\Gamma)$. We view the graph $\Gamma$ as a directed graph since the map $\Phi$ naturally induces an orientation on $\Gamma$ (see Section 2.2).

Suppose that $m \ge 2$. Consider an arbitrary face $F \in F(\Gamma)$. We label $F$ by $P_j$ if $\Phi(E(\partial F)) = \{p_j\}$, and if no such $j$ exists (i.e., $|\Phi(E(\partial F))| > 1$) we label $F$ by $P_\infty$. We denote by $\Gamma^*$ the directed planar embedded graph obtained by subdividing each edge of $\Gamma$.

**Definition 2.27.** Consider a quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$, where $A, \mathcal{R}, \Gamma$, and $\Phi$ are as above. We say that $\Delta$ is an *admissible quadruple* if $m = 1$ or if the following conditions are satisfied:

(1) every face $F \in F(\Gamma)$ labeled by $P_\infty$ is unbounded, and

(2) for each $v \in V(\Gamma)$, the set of edges of $\Gamma^*$ incident to $v$ can be written in the counterclockwise order as $e_1 = e_{2m+1}, e_2, e_3, \dots, e_{2m}$ such that the following conditions are satisfied for each $j = 1, 2, \dots, m$:

* $e_{2j-1}$ is outgoing at $v$ and $e_{2j}$ is incoming at $v$;

* the edges $e_{2j-1}$ and $e_{2j}$ belong to the boundary of the same face $F \in F(\Gamma)$ labelled by $P_j$;

* the edges $e_{2j}$ and $e_{2j+1}$ belong to the boundary of the same face $F \in F(\Gamma)$ labelled by $P_\infty$.

The set $A$ is called the *marked set* of $\Delta$.

*Remark 2.28.* Suppose that $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ is an admissible quadruple and $F$ is a face of $\Gamma$. If $F$ is not labelled by $P_\infty$, then clearly, $\partial F$ is a counterclockwise directed cycle if $F$ is bounded, and an infinite unilaterally connected chain, otherwise. If $F$ is labelled by $P_\infty$, then $\partial F$ is a unilaterally connected graph.

Natural examples of admissible quadruples are provided by preimages of rose graphs under entire topologically holomorphic maps of finite type. More precisely, suppose that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is topologically holomorphic and rose graph $\mathcal{R}$ surrounds the set $A$, where
$S_f \subset A$. Denote by $\Delta(A, \mathcal{R}, f)$ the quadruple $(A, \mathcal{R}, f^{-1}(\mathcal{R}), \Phi_{\mathcal{R},f})$, where $\Phi_{\mathcal{R},f}(x) = f(x)$ for each $x \in f^{-1}(\mathcal{R})$.

**Proposition 2.29.** Let $f, A$, and $\mathcal{R}$ be as above. Then $\Delta(A, \mathcal{R}, f)$ is an admissible quadruple. Moreover, if $m \ge 2$, then for each face $F$ of $f^{-1}(\mathcal{R})$ the following properties hold:

(1) if $F$ is bounded and labeled by $P_j$ for some $j \in \{1, 2, ..., m\}$, then $f(F) = P_j$ and $|F \cap f^{-1}(A)| = 1$. If $|V(\partial F)| = 1$, then $F$ does not contain any critical points of $f$ and $f|_F$ is injective; otherwise, $F$ contains a unique critical point $x_F$ with $\deg(f, x_F) = |V(\partial F)|$;

(2) if $F$ is unbounded and labeled by $P_j$ for some $j \in \{1, 2, ..., m\}$, then $f(F) = P_j \setminus \{a_j\}$ and $F \cap f^{-1}(A) = \emptyset$. In particular, $a_j \in S_f$ and $F$ is an asymptotic tract of $f$ over $a_j$;

(3) if $F$ is labeled by $P_\infty$, then $f(F) = P_\infty$ and $F \cap f^{-1}(A) = \emptyset$. In particular, $f$ restricts to a universal covering map from $F$ to $P_\infty$.

*Proof.* First, we show that $f^{-1}(\mathcal{R})$ is connected. Consider any two distinct vertices $u$ and $v$ of $f^{-1}(\mathcal{R})$. There exists a path $\alpha: \mathbb{I} \to \mathbb{R}^2 \setminus f^{-1}(A)$ joining $u$ and $v$. Note that $f \circ \alpha$ is a loop in $\mathbb{R}^2 \setminus A$ based at $t$. Assuming that each petal $p_j$ is parameterized by a loop $\alpha_j: \mathbb{I} \to p_j$ based at $t$, it follows that $f \circ \alpha$ is homotopic rel. $A$ to a loop $\gamma = \gamma_1 \cdot \gamma_2 \cdots \gamma_k$, where $\gamma_l \in \{\alpha_1, \alpha_2, \dots, \alpha_m, \overline{\alpha_1}, \overline{\alpha_2}, \dots, \overline{\alpha_m}\}$ for each $l \in \{1, 2, \dots, k\}$. By the homotopy lifting property, the path $\alpha$ is homotopic rel. $f^{-1}(A)$ to the $f$-lift $\tilde{\gamma}$ of $\gamma$ starting at $u$. In particular, $\tilde{\gamma}$ joins $u$ with $v$. Since $\tilde{\gamma} \subset f^{-1}(\mathcal{R})$, the vertices $u$ and $v$ belong to the same connected component of $f^{-1}(\mathcal{R})$.

Proposition 2.12 implies that every face of $f^{-1}(\mathcal{R})$ labeled by $P_\infty$ is unbounded. Finally, the regularity of $f^{-1}(\mathcal{R})$ and condition (2) follow from the fact that $f$ is locally injective and orientation-preserving at every $v \in V(f^{-1}(\mathcal{R}))$. Thus, the quadruple $\Delta(A, \mathcal{R}, f)$ is admissible.

The rest of the statement easily follows from Propositions 2.11 and 2.12. $\square$

<figure><img src="image_6.png" /></figure>

*Remark 2.30.* Let $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ be an admissible quadruple with $m = |E(\mathcal{R})| = 1$. In this case, $\Gamma$ is a counterclockwise directed cycle or an infinite unilaterally connected chain. It is easy to see that there exists a map $f$ such that $\Delta(A, \mathcal{R}, f) = \Delta$ satisfying $f = \varphi^{-1} \circ g \circ \psi$ for some orientation-preserving homeomorphisms $\varphi, \psi: \mathbb{R}^2 \to \mathbb{C}$, where $g(z) = z^d$ if $d = |V(\Gamma)| < \infty$, and $g(z) = \exp(z)$ if $\Gamma$ is infinite. In particular, using Proposition 2.17, we can formulate a statement close in spirit to Proposition 2.29 for the case $m = 1$.

Now assume that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is an arbitrary topologically holomorphic map such that $S_f \subset A$ and $\Delta(A, \mathcal{R}, f) = \Delta$. Since $\mathcal{R}$ is a deformation retract of $\mathbb{R}^2 \setminus A$ and $\Gamma$ is a deformation retract of $\mathbb{R}^2 \setminus f^{-1}(A)$ as Proposition 2.29 and Remark 2.30 suggest, for any $v \in V(\Gamma)$, we see that $f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), v) = \Phi_*\pi_1(\Gamma, v)$, and

$$ \Phi_*\pi_1(\Gamma, v) = \{[\Phi(\delta_1) \cdot \Phi(\delta_2) \cdots \Phi(\delta_l)] : \text{ each } \delta_i \text{ is a path parameterizing an edge of } \Gamma, \\ \text{ and } \delta_1 \cdot \delta_2 \cdots \delta_l \text{ is a loop in } \Gamma \text{ based at } v\}. $$

Now suppose that $\gamma$ is a loop based at the center of the rose graph $\mathcal{R}$ such that $\gamma$ is homotopic rel. $A$ to $\gamma_1 \cdot \gamma_2 \cdots \gamma_k$, where $\gamma_j \in \{\alpha_1, \alpha_2, \dots, \alpha_m, \overline{\alpha_1}, \overline{\alpha_2}, \dots, \overline{\alpha_m}\}$ and $\alpha_j$ is a parametrization of $p_j$ with $\alpha_j(0) = t$ for each $j = 1, 2, \dots, m$. If we know $\Phi$, we can reconstruct $f$-lift of $\gamma$ starting at any vertex $v \in V(\Gamma)$ up to homotopy rel. $f^{-1}(A)$.

The following result is the converse to Proposition 2.29:
$S_f \subset A$. Denote by $\Delta(A, \mathcal{R}, f)$ the quadruple $(A, \mathcal{R}, f^{-1}(\mathcal{R}), \Phi_{\mathcal{R},f})$, where $\Phi_{\mathcal{R},f}(x) = f(x)$ for each $x \in f^{-1}(\mathcal{R})$.

**Proposition 2.29.** Let $f, A$, and $\mathcal{R}$ be as above. Then $\Delta(A, \mathcal{R}, f)$ is an admissible quadruple. Moreover, if $m \ge 2$, then for each face $F$ of $f^{-1}(\mathcal{R})$ the following properties hold:

(1) if $F$ is bounded and labeled by $P_j$ for some $j \in \{1, 2, ..., m\}$, then $f(F) = P_j$ and $|F \cap f^{-1}(A)| = 1$. If $|V(\partial F)| = 1$, then $F$ does not contain any critical points of $f$ and $f|_F$ is injective; otherwise, $F$ contains a unique critical point $x_F$ with $\deg(f, x_F) = |V(\partial F)|$;

(2) if $F$ is unbounded and labeled by $P_j$ for some $j \in \{1, 2, ..., m\}$, then $f(F) = P_j \setminus \{a_j\}$ and $F \cap f^{-1}(A) = \emptyset$. In particular, $a_j \in S_f$ and $F$ is an asymptotic tract of $f$ over $a_j$;

(3) if $F$ is labeled by $P_\infty$, then $f(F) = P_\infty$ and $F \cap f^{-1}(A) = \emptyset$. In particular, $f$ restricts to a universal covering map from $F$ to $P_\infty$.

*Proof.* First, we show that $f^{-1}(\mathcal{R})$ is connected. Consider any two distinct vertices $u$ and $v$ of $f^{-1}(\mathcal{R})$. There exists a path $\alpha: \mathbb{I} \to \mathbb{R}^2 \setminus f^{-1}(A)$ joining $u$ and $v$. Note that $f \circ \alpha$ is a loop in $\mathbb{R}^2 \setminus A$ based at $t$. Assuming that each petal $p_j$ is parameterized by a loop $\alpha_j: \mathbb{I} \to p_j$ based at $t$, it follows that $f \circ \alpha$ is homotopic rel. $A$ to a loop $\gamma = \gamma_1 \cdot \gamma_2 \cdots \gamma_k$, where $\gamma_l \in \{\alpha_1, \alpha_2, \dots, \alpha_m, \overline{\alpha_1}, \overline{\alpha_2}, \dots, \overline{\alpha_m}\}$ for each $l \in \{1, 2, \dots, k\}$. By the homotopy lifting property, the path $\alpha$ is homotopic rel. $f^{-1}(A)$ to the $f$-lift $\tilde{\gamma}$ of $\gamma$ starting at $u$. In particular, $\tilde{\gamma}$ joins $u$ with $v$. Since $\tilde{\gamma} \subset f^{-1}(\mathcal{R})$, the vertices $u$ and $v$ belong to the same connected component of $f^{-1}(\mathcal{R})$.

Proposition 2.12 implies that every face of $f^{-1}(\mathcal{R})$ labeled by $P_\infty$ is unbounded. Finally, the regularity of $f^{-1}(\mathcal{R})$ and condition (2) follow from the fact that $f$ is locally injective and orientation-preserving at every $v \in V(f^{-1}(\mathcal{R}))$. Thus, the quadruple $\Delta(A, \mathcal{R}, f)$ is admissible.

The rest of the statement easily follows from Propositions 2.11 and 2.12. $\square$

<figure><img src="image_7.png" /></figure>

*Remark 2.30.* Let $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ be an admissible quadruple with $m = |E(\mathcal{R})| = 1$. In this case, $\Gamma$ is a counterclockwise directed cycle or an infinite unilaterally connected chain. It is easy to see that there exists a map $f$ such that $\Delta(A, \mathcal{R}, f) = \Delta$ satisfying $f = \varphi^{-1} \circ g \circ \psi$ for some orientation-preserving homeomorphisms $\varphi, \psi: \mathbb{R}^2 \to \mathbb{C}$, where $g(z) = z^d$ if $d = |V(\Gamma)| < \infty$, and $g(z) = \exp(z)$ if $\Gamma$ is infinite. In particular, using Proposition 2.17, we can formulate a statement close in spirit to Proposition 2.29 for the case $m = 1$.

Now assume that $f: \mathbb{R}^2 \to \mathbb{R}^2$ is an arbitrary topologically holomorphic map such that $S_f \subset A$ and $\Delta(A, \mathcal{R}, f) = \Delta$. Since $\mathcal{R}$ is a deformation retract of $\mathbb{R}^2 \setminus A$ and $\Gamma$ is a deformation retract of $\mathbb{R}^2 \setminus f^{-1}(A)$ as Proposition 2.29 and Remark 2.30 suggest, for any $v \in V(\Gamma)$, we see that $f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), v) = \Phi_*\pi_1(\Gamma, v)$, and

$$ \Phi_*\pi_1(\Gamma, v) = \{[\Phi(\delta_1) \cdot \Phi(\delta_2) \cdots \Phi(\delta_l)] : \text{ each } \delta_i \text{ is a path parameterizing an edge of } \Gamma, \\ \text{ and } \delta_1 \cdot \delta_2 \cdots \delta_l \text{ is a loop in } \Gamma \text{ based at } v\}. $$

Now suppose that $\gamma$ is a loop based at the center of the rose graph $\mathcal{R}$ such that $\gamma$ is homotopic rel. $A$ to $\gamma_1 \cdot \gamma_2 \cdots \gamma_k$, where $\gamma_j \in \{\alpha_1, \alpha_2, \dots, \alpha_m, \overline{\alpha_1}, \overline{\alpha_2}, \dots, \overline{\alpha_m}\}$ and $\alpha_j$ is a parametrization of $p_j$ with $\alpha_j(0) = t$ for each $j = 1, 2, \dots, m$. If we know $\Phi$, we can reconstruct $f$-lift of $\gamma$ starting at any vertex $v \in V(\Gamma)$ up to homotopy rel. $f^{-1}(A)$.

The following result is the converse to Proposition 2.29:
**Proposition 2.31.** Let $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ be an admissible quadruple. Then there exists a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ of finite type such that $S_f \subset A$ and $\Delta(A, \mathcal{R}, f) = \Delta$.

*Proof.* When $|A| = 1$, the desired result follows from Remark 2.30. Thus, we assume $|A| \ge 2$ and give an outline of the construction.

First, we define $f$ on $\Gamma$ simply by setting $f|\Gamma := \Phi$. Choose orientation-preserving homeomorphisms $\varphi_{P_j}: \mathbb{D} \to P_j$ and $\varphi_{P_\infty}: \mathbb{H} \to P_\infty$. Similarly, since each face $F$ of $\Gamma$ is simply connected, choose an orientation-preserving homeomorphism $\psi_F: X \to F$ where $X = \mathbb{D}$ if $F$ is bounded, and $X = \mathbb{H}$ if $F$ is unbounded.

Given a face $F$ with label $P \in F(\mathcal{R})$, we define $f|F$ so that

(1) if $F$ is bounded, we let $f|F := \varphi_P \circ g_d \circ \psi_F^{-1}$, where $d = |V(\partial F)|$ and $g_d(z) = z^d$,

(2) if $F$ is unbounded, let $f|F := \varphi_P \circ \exp \circ \psi_F^{-1}$.

Due to admissibility conditions and Remark 2.28, the sets of homeomorphisms $\{\varphi_P\}_{P \in F(\mathcal{R})}$ and $\{\psi_F\}_{F \in F(\Gamma)}$ can be chosen so that the map $f$ we construct above is continuous. Finally, one can show that $f$ acts locally as a power map (in the sense of Definition 2.2) and $S_f \subset A$. Thus, $f$ is topologically holomorphic, has finite type, and satisfies $\Delta(A, \mathcal{R}, f) = \Delta$. $\square$

**Definition 2.32.** Two admissible quadruples $\Delta_1 = (A, \mathcal{R}_1, \Gamma_1, \Phi_1)$ and $\Delta_2 = (A, \mathcal{R}_2, \Gamma_2, \Phi_2)$ are said to be equivalent if there exist $\theta \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ and $\varphi \in \text{Homeo}^+(\mathbb{R}^2, A)$ such that

(1) $\theta$ is an isomorphism between $\mathcal{R}_1$ and $\mathcal{R}_2$;

(2) $\varphi$ is an isomorphism between $\Gamma_1$ and $\Gamma_2$;

(3) $\theta \circ \Phi_1 = \Phi_2 \circ \varphi$.

It is easy to see that Definition 2.32 provides an equivalence relation on the set of all admissible quadruples with a fixed marked set.

**Proposition 2.33.** Let $A \subset \mathbb{R}^2$ be finite, and $\mathcal{R}_1, \mathcal{R}_2$ be rose graphs surrounding $A$ such that $\mathcal{R}_1$ is isotopic rel. $A$ to $\mathcal{R}_2$. Let $f_1: \mathbb{R}^2 \to \mathbb{R}^2$ and $f_2: \mathbb{R}^2 \to \mathbb{R}^2$ be topologically holomorphic maps such that $S_{f_1} \subset A$ and $S_{f_2} \subset A$. Then $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are equivalent if and only if there exists a homeomorphism $\psi \in \text{Homeo}^+(\mathbb{R}^2)$ such that $f_1 = f_2 \circ \psi$.

Moreover, if $f_1$ and $f_2$ are holomorphic, and $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are equivalent, then the map $\psi$ is an affine transformation.

*Proof.* Suppose that the admissible quadruples $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are equivalent. Due to Proposition 2.13 we can assume that $\mathcal{R}_1 = \mathcal{R}_2 = \mathcal{R}$ and that the equivalence between $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ is provided by $\theta = \text{id}_{\mathbb{R}^2}$ and an orientation-preserving homeomorphism $\varphi$. By pre-composing $f_2$ with $\varphi$ we can further assume $\theta = \varphi = \text{id}_{\mathbb{R}^2}$, i.e., $\Delta(A, f_1, \mathcal{R}_1) = \Delta(A, f_2, \mathcal{R}_2) = (A, \mathcal{R}, \Gamma, \Phi)$. Then by the previous discussions, we have that

$$ (f_1)_*\pi_1(\mathbb{R}^2\setminus f_1^{-1}(A), v) = \Phi_*\pi_1(\Gamma, v) = (f_2)_*\pi_1(\mathbb{R}^2\setminus f_2^{-1}(A), v) $$

for every $v \in V(\Gamma)$. By the classical theory of covering maps, there exists an orientation-preserving homeomorphism $\psi: \mathbb{R}^2\setminus f_1^{-1}(A) \to \mathbb{R}^2\setminus f_2^{-1}(A)$ such that $f_1 = f_2 \circ \psi$ on $\mathbb{R}^2\setminus f_1^{-1}(A)$. Since $f_1^{-1}(A)$ is a discrete subset of $\mathbb{R}^2$, we can extend $\psi$ to $\mathbb{R}^2$, still satisfying $f_1 = f_2 \circ \psi$.

Conversely, let us suppose there exists $\psi \in \text{Homeo}^+(\mathbb{R}^2)$ such that $f_1 = f_2 \circ \psi$. By our assumptions on $\mathcal{R}_1$ and $\mathcal{R}_2$, we can find $\theta \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ such that $\theta(\mathcal{R}_1) = \mathcal{R}_2$. Then by Proposition 2.13, there exists $\varphi \in \text{Homeo}^+(\mathbb{R}^2)$ such that $\theta \circ f_1 = f_2 \circ \varphi$, and the rest easily follows.
**Proposition 2.31.** Let $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ be an admissible quadruple. Then there exists a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ of finite type such that $S_f \subset A$ and $\Delta(A, \mathcal{R}, f) = \Delta$.

*Proof.* When $|A| = 1$, the desired result follows from Remark 2.30. Thus, we assume $|A| \ge 2$ and give an outline of the construction.

First, we define $f$ on $\Gamma$ simply by setting $f|\Gamma := \Phi$. Choose orientation-preserving homeomorphisms $\varphi_{P_j}: \mathbb{D} \to P_j$ and $\varphi_{P_\infty}: \mathbb{H} \to P_\infty$. Similarly, since each face $F$ of $\Gamma$ is simply connected, choose an orientation-preserving homeomorphism $\psi_F: X \to F$ where $X = \mathbb{D}$ if $F$ is bounded, and $X = \mathbb{H}$ if $F$ is unbounded.

Given a face $F$ with label $P \in F(\mathcal{R})$, we define $f|F$ so that

(1) if $F$ is bounded, we let $f|F := \varphi_P \circ g_d \circ \psi_F^{-1}$, where $d = |V(\partial F)|$ and $g_d(z) = z^d$,

(2) if $F$ is unbounded, let $f|F := \varphi_P \circ \exp \circ \psi_F^{-1}$.

Due to admissibility conditions and Remark 2.28, the sets of homeomorphisms $\{\varphi_P\}_{P \in F(\mathcal{R})}$ and $\{\psi_F\}_{F \in F(\Gamma)}$ can be chosen so that the map $f$ we construct above is continuous. Finally, one can show that $f$ acts locally as a power map (in the sense of Definition 2.2) and $S_f \subset A$. Thus, $f$ is topologically holomorphic, has finite type, and satisfies $\Delta(A, \mathcal{R}, f) = \Delta$. $\square$

**Definition 2.32.** Two admissible quadruples $\Delta_1 = (A, \mathcal{R}_1, \Gamma_1, \Phi_1)$ and $\Delta_2 = (A, \mathcal{R}_2, \Gamma_2, \Phi_2)$ are said to be equivalent if there exist $\theta \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ and $\varphi \in \text{Homeo}^+(\mathbb{R}^2, A)$ such that

(1) $\theta$ is an isomorphism between $\mathcal{R}_1$ and $\mathcal{R}_2$;

(2) $\varphi$ is an isomorphism between $\Gamma_1$ and $\Gamma_2$;

(3) $\theta \circ \Phi_1 = \Phi_2 \circ \varphi$.

It is easy to see that Definition 2.32 provides an equivalence relation on the set of all admissible quadruples with a fixed marked set.

**Proposition 2.33.** Let $A \subset \mathbb{R}^2$ be finite, and $\mathcal{R}_1, \mathcal{R}_2$ be rose graphs surrounding $A$ such that $\mathcal{R}_1$ is isotopic rel. $A$ to $\mathcal{R}_2$. Let $f_1: \mathbb{R}^2 \to \mathbb{R}^2$ and $f_2: \mathbb{R}^2 \to \mathbb{R}^2$ be topologically holomorphic maps such that $S_{f_1} \subset A$ and $S_{f_2} \subset A$. Then $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are equivalent if and only if there exists a homeomorphism $\psi \in \text{Homeo}^+(\mathbb{R}^2)$ such that $f_1 = f_2 \circ \psi$.

Moreover, if $f_1$ and $f_2$ are holomorphic, and $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are equivalent, then the map $\psi$ is an affine transformation.

*Proof.* Suppose that the admissible quadruples $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are equivalent. Due to Proposition 2.13 we can assume that $\mathcal{R}_1 = \mathcal{R}_2 = \mathcal{R}$ and that the equivalence between $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ is provided by $\theta = \text{id}_{\mathbb{R}^2}$ and an orientation-preserving homeomorphism $\varphi$. By pre-composing $f_2$ with $\varphi$ we can further assume $\theta = \varphi = \text{id}_{\mathbb{R}^2}$, i.e., $\Delta(A, f_1, \mathcal{R}_1) = \Delta(A, f_2, \mathcal{R}_2) = (A, \mathcal{R}, \Gamma, \Phi)$. Then by the previous discussions, we have that

$$ (f_1)_*\pi_1(\mathbb{R}^2\setminus f_1^{-1}(A), v) = \Phi_*\pi_1(\Gamma, v) = (f_2)_*\pi_1(\mathbb{R}^2\setminus f_2^{-1}(A), v) $$

for every $v \in V(\Gamma)$. By the classical theory of covering maps, there exists an orientation-preserving homeomorphism $\psi: \mathbb{R}^2\setminus f_1^{-1}(A) \to \mathbb{R}^2\setminus f_2^{-1}(A)$ such that $f_1 = f_2 \circ \psi$ on $\mathbb{R}^2\setminus f_1^{-1}(A)$. Since $f_1^{-1}(A)$ is a discrete subset of $\mathbb{R}^2$, we can extend $\psi$ to $\mathbb{R}^2$, still satisfying $f_1 = f_2 \circ \psi$.

Conversely, let us suppose there exists $\psi \in \text{Homeo}^+(\mathbb{R}^2)$ such that $f_1 = f_2 \circ \psi$. By our assumptions on $\mathcal{R}_1$ and $\mathcal{R}_2$, we can find $\theta \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ such that $\theta(\mathcal{R}_1) = \mathcal{R}_2$. Then by Proposition 2.13, there exists $\varphi \in \text{Homeo}^+(\mathbb{R}^2)$ such that $\theta \circ f_1 = f_2 \circ \varphi$, and the rest easily follows.
If $f_1$ and $f_2$ are holomorphic, any homeomorphism $\psi$ satisfying $f_1 = f_2 \circ \psi$ is holomorphic, and therefore affine. $\square$

We say that a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ realizes an admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ or, equivalently, $\Delta$ defines $f$ if $S_f \subset A$ and $\Delta(A, \mathcal{R}, f)$ is equivalent to $\Delta$. In particular, Propositions 2.31 and 2.33 imply that every admissible quadruple $\Delta$ defines an entire topologically holomorphic map $f$ of finite type, which is unique up to pre-composition by an orientation-preserving homeomorphism.

Note that an admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ is a combinatorial object even though $\Phi$ is a continuous map. In fact, to define the map $\Phi$ uniquely (up to a certain notion of equivalence introduced below), it is sufficient to know the images $\Phi(e), e \in E(\Gamma)$ and the orientation of the graph $\Gamma$ induced by $\Phi$. Indeed, suppose that $\Psi: \Gamma \to \mathcal{R}$ is a covering map such that $\Psi(v) = t$ for each vertex $v \in V(\Gamma)$, $\Phi(e) = \Psi(e)$ for each edge $e \in E(\Gamma)$, and the orientations of $\Gamma$ induced by the maps $\Phi$ and $\Psi$ coincide. In this case, it is clear that there exists a homeomorphism $\varphi: \Gamma \to \Gamma$ isotopic rel. $V(\Gamma)$ to $\text{id}_\Gamma$ such that $\Psi = \Phi \circ \varphi$. In particular, the orientation of the graph $\Gamma$ and the images of its edges under the map $\Phi$ uniquely define the equivalence class of the admissible quadruple $\Delta$.

The language of admissible quadruples provides a convenient way of thinking about entire topologically holomorphic maps of finite type, which we demonstrate in the following example.

*Example 2.34.* Let $A = \{-1, 1\}$ be the set represented by solid black squares at the top of Figure 2. Denote by $\mathcal{R}_1$ and $\Gamma_1$ the planar embedded graphs shown at the top and bottom of Figure 2, respectively. The map $\Phi_1: \Gamma_1 \to \mathcal{R}_1$ is a covering map such that maps each edge of $\Gamma_1$ to the unique edge of $\mathcal{R}_1$ of the same color. Arrows on the edges of the graphs $\mathcal{R}_1$ and $\Gamma_1$ indicate the orientations of the corresponding graphs. It is straightforward to check that $\Delta_1 = (A, \mathcal{R}_1, \Gamma_1, \Phi_1)$ is an admissible quadruple, which is realized by the map $g_1(z) = \cos(z)$. Thus, any entire (holomorphic) map realizing $\Delta_1$ has the form $\cos(az + b)$ for some constants $a, b \in \mathbb{C}$ with $a \neq 0$.

Figure 3 is analogous to Figure 2, and provides another example $\Delta_2 = (A, \mathcal{R}_2, \Gamma_2, \Phi_2)$ of an admissible quadruple. This quadruple can be shown to be realized by the map $g_2(z) = 2\exp(z^2) - 1$. In particular, the planar embedded graph $\Gamma_2$ has two unbounded faces that correspond to the asymptotic tracts of $g_2$ over $w_0 = 1$, two unbounded faces that correspond to the asymptotic tracts of $g_2$ over $\infty$, and the only face with the boundary consisting of two edges, corresponding to the unique critical point of $g_2$ (see Proposition 2.29).

<figure><img src="image_7.png" /><figcaption>FIGURE 2. Admissible quadruple realized by $g_1(z) = \cos(z)$.</figcaption></figure>
If $f_1$ and $f_2$ are holomorphic, any homeomorphism $\psi$ satisfying $f_1 = f_2 \circ \psi$ is holomorphic, and therefore affine. $\square$

We say that a topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ realizes an admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ or, equivalently, $\Delta$ defines $f$ if $S_f \subset A$ and $\Delta(A, \mathcal{R}, f)$ is equivalent to $\Delta$. In particular, Propositions 2.31 and 2.33 imply that every admissible quadruple $\Delta$ defines an entire topologically holomorphic map $f$ of finite type, which is unique up to pre-composition by an orientation-preserving homeomorphism.

Note that an admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ is a combinatorial object even though $\Phi$ is a continuous map. In fact, to define the map $\Phi$ uniquely (up to a certain notion of equivalence introduced below), it is sufficient to know the images $\Phi(e), e \in E(\Gamma)$ and the orientation of the graph $\Gamma$ induced by $\Phi$. Indeed, suppose that $\Psi: \Gamma \to \mathcal{R}$ is a covering map such that $\Psi(v) = t$ for each vertex $v \in V(\Gamma)$, $\Phi(e) = \Psi(e)$ for each edge $e \in E(\Gamma)$, and the orientations of $\Gamma$ induced by the maps $\Phi$ and $\Psi$ coincide. In this case, it is clear that there exists a homeomorphism $\varphi: \Gamma \to \Gamma$ isotopic rel. $V(\Gamma)$ to $\text{id}_\Gamma$ such that $\Psi = \Phi \circ \varphi$. In particular, the orientation of the graph $\Gamma$ and the images of its edges under the map $\Phi$ uniquely define the equivalence class of the admissible quadruple $\Delta$.

The language of admissible quadruples provides a convenient way of thinking about entire topologically holomorphic maps of finite type, which we demonstrate in the following example.

*Example 2.34.* Let $A = \{-1, 1\}$ be the set represented by solid black squares at the top of Figure 2. Denote by $\mathcal{R}_1$ and $\Gamma_1$ the planar embedded graphs shown at the top and bottom of Figure 2, respectively. The map $\Phi_1: \Gamma_1 \to \mathcal{R}_1$ is a covering that maps each edge of $\Gamma_1$ to the unique edge of $\mathcal{R}_1$ of the same color. Arrows on the edges of the graphs $\mathcal{R}_1$ and $\Gamma_1$ indicate the orientations of the corresponding graphs. It is straightforward to check that $\Delta_1 = (A, \mathcal{R}_1, \Gamma_1, \Phi_1)$ is an admissible quadruple, which is realized by the map $g_1(z) = \cos(z)$. Thus, any entire (holomorphic) map realizing $\Delta_1$ has the form $\cos(az + b)$ for some constants $a, b \in \mathbb{C}$ with $a \neq 0$.

Figure 3 is analogous to Figure 2, and provides another example $\Delta_2 = (A, \mathcal{R}_2, \Gamma_2, \Phi_2)$ of an admissible quadruple. This quadruple can be shown to be realized by the map $g_2(z) = 2\exp(z^2) - 1$. In particular, the planar embedded graph $\Gamma_2$ has two unbounded faces that correspond to the asymptotic tracts of $g_2$ over $w_0 = 1$, two unbounded faces that correspond to the asymptotic tracts of $g_2$ over $\infty$, and the only face with the boundary consisting of two edges, corresponding to the unique critical point of $g_2$ (see Proposition 2.29).

<figure><img src="image_8.png" /><figcaption>FIGURE 2. Admissible quadruple realized by $g_1(z) = \cos(z)$.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>FIGURE 3. Admissible quadruple realized by $g_2(z) = 2 \exp(z^2) - 1$.</figcaption></figure>

**Definition 2.35.** Let $\Delta$ be an admissible quadruple. If every topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ realizing $\Delta$ is parabolic, then we say that $\Delta$ is *parabolic*. If every such map $f$ is hyperbolic, then $\Delta$ is called *hyperbolic*.

Proposition 2.33 implies that every admissible quadruple is either parabolic or hyperbolic. Moreover, from Propositions 2.6 and 2.33, it follows that every parabolic admissible quadruple $\Delta$ with a marked set $A$ defines an entire holomorphic map of finite type with $S_f \subset A$, which is unique up to pre-composition with an affine transformation.

### 3. APPROXIMATIONS OF ENTIRE THURSTON MAPS

In this section, we introduce and study different notions of convergence for sequences of entire Thurston maps having the same marked set $A$. In fact, we provide two different points of view (one combinatorial, and one topological, see Definitions 3.3 and 3.6), and then show that these notions are equivalent (see Proposition 3.7). Afterward, we provide a construction that allows us to approximate, in an appropriate sense, an arbitrary transcendental Thurston map by a sequence of polynomial Thurston maps (see Proposition 3.14). Finally, we study
<figure><img src="image_9.png" /><figcaption>FIGURE 3. Admissible quadruple realized by $g_2(z) = 2 \exp(z^2) - 1$.</figcaption></figure>

**Definition 2.35.** Let $\Delta$ be an admissible quadruple. If every topologically holomorphic map $f: \mathbb{R}^2 \to \mathbb{R}^2$ realizing $\Delta$ is parabolic, then we say that $\Delta$ is *parabolic*. If every such map $f$ is hyperbolic, then $\Delta$ is called *hyperbolic*.

Proposition 2.33 implies that every admissible quadruple is either parabolic or hyperbolic. Moreover, from Propositions 2.6 and 2.33, it follows that every parabolic admissible quadruple $\Delta$ with a marked set $A$ defines an entire holomorphic map of finite type with $S_f \subset A$, which is unique up to pre-composition with an affine transformation.

### 3. APPROXIMATIONS OF ENTIRE THURSTON MAPS

In this section, we introduce and study different notions of convergence for sequences of entire Thurston maps having the same marked set $A$. In fact, we provide two different points of view (one combinatorial, and one topological, see Definitions 3.3 and 3.6), and then show that these notions are equivalent (see Proposition 3.7). Afterward, we provide a construction that allows us to approximate, in an appropriate sense, an arbitrary transcendental Thurston map by a sequence of polynomial Thurston maps (see Proposition 3.14). Finally, we study
how this impacts the convergence of the corresponding Thurston pullback maps and establish Main Theorem B.

**3.1. Combinatorial convergence.** Suppose that $f: (\mathbb{R}^2, A) \hookrightarrow$ is a Thurston map with $f(b) = t$ for some $b, t \in \mathbb{R}^2 \setminus A$. If $\gamma$ is a loop in $\mathbb{R}^2 \setminus A$ based at $t$, then we use the notation $\gamma \uparrow(f, b)$ as a shorthand for the $f$-lift of $\gamma$ starting at $b$.

Then for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$ we can record

(1) whether $\gamma$ lifts under $f$ to a loop based at $b$, i.e., whether $[\gamma] \in f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$;

(2) if $[\gamma] \in f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$, homotopy class rel. $A$ of $\gamma \uparrow(f, b)$, i.e., $[\gamma \uparrow(f, b)] \in \pi_1(\mathbb{R}^2 \setminus A, b)$.

We now establish that this information is enough to define the Thurston map $f$ uniquely up to isotopy rel. $A$ (cf. [Kam01] and [BD17, Theorem II.3] for rational Thurston maps).

**Proposition 3.1.** Let $f_1: (\mathbb{R}^2, A) \hookrightarrow$ and $f_2: (\mathbb{R}^2, A) \hookrightarrow$ be Thurston maps such that $f_1(b_1) = t$ for some $b_1 \in \mathbb{R}^2$ and $t \in \mathbb{R}^2 \setminus A$. Then $f_1$ and $f_2$ are isotopic rel. $A$ if and only if there exists $b_2 \in \mathbb{R}^2$ and a path $p \subset \mathbb{R}^2 \setminus A$ joining $b_1$ with $b_2$ such that $f_2(b_2) = t$ and for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$ the following holds:

(1) $\gamma$ lifts under $f_1$ to a loop based at $b_1$ if and only if it lifts under $f_2$ to a loop based at $b_2$, in other words, $(f_1)_*\pi_1(\mathbb{R}^2 \setminus f_1^{-1}(A), b_1) = (f_2)_*\pi_1(\mathbb{R}^2 \setminus f_2^{-1}(A), b_2)$;

(2) if $[\gamma] \in (f_1)_*\pi_1(\mathbb{R}^2 \setminus f_1^{-1}(A), b_1)$, then loops $\gamma \uparrow(f_1, b_1)$ and $p \cdot \gamma \uparrow(f_2, b_2) \cdot \bar{p}$ are homotopic rel. $A$.

Before proving the above, we formulate a simple observation.

**Lemma 3.2.** Suppose $A \subset \mathbb{R}^2$ is finite, and $\varphi$ is an orientation-preserving homeomorphism such that $\varphi(b) = t$ for some $b, t \in \mathbb{R}^2 \setminus A$. Then $\varphi$ is isotopic rel. $A$ to $\text{id}_{\mathbb{R}^2}$ if and only if there exists a path $p \subset \mathbb{R}^2 \setminus A$ joining $b$ with $t$ such that every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b$ is homotopic rel. $A$ to $p \cdot (\varphi \circ \gamma) \cdot \bar{p}$.

*Proof.* Suppose that $\varphi$ is isotopic rel. $A$ to $\text{id}_{\mathbb{R}^2}$ via an isotopy $(\varphi_s)_{s \in I}$, where $\varphi_0 = \text{id}_{\mathbb{R}^2}$ and $\varphi_1 = \varphi$. Define $p: I \to \mathbb{R}^2 \setminus A$ as $p(s) = \varphi_s(b)$ for each $s \in I$. Then it is evident that any loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b$ and $p \cdot (\varphi \circ \gamma) \cdot \bar{p}$ are homotopic rel. $A$ via a homotopy $H_s := p_s \cdot (\varphi_s \circ \gamma) \cdot \bar{p}_s$, where $p_s$ is a subpath of $p$ joining $b$ with $\varphi_s(b)$ for every $s \in I$.

Now suppose that there exists a path $p \subset \mathbb{R}^2 \setminus A$ joining $b$ with $t$ such that every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b$ is homotopic rel. $A$ to $p \cdot (\varphi \circ \gamma) \cdot \bar{p} \subset \mathbb{R}^2 \setminus A$. Taking $\gamma$ to be a loop separating a unique point $a \in A$ from the other points of $A$ and using continuity of $\varphi$, we can show that $\varphi(a) = a$ for each $a \in A$. The rest easily follows: for instance, it is a particular case of the much stronger Dehn-Neilsen-Baer theorem [FM12, Theorem 8.1, 8.8]. $\square$

*Proof of Proposition 3.1.* Suppose that conditions (1) and (2) are satisfied under some choice of $b_2$ and $p$. Since the restrictions

$$f_1|\mathbb{R}^2\setminus f_1^{-1}(A): \mathbb{R}^2\setminus f_1^{-1}(A) \to \mathbb{R}^2\setminus A \text{ and } f_2|\mathbb{R}^2\setminus f_2^{-1}(A): \mathbb{R}^2\setminus f_2^{-1}(A) \to \mathbb{R}^2\setminus A$$

are both covering maps, condition (1) implies that there exists an orientation-preserving homeomorphism $\varphi: \mathbb{R}^2\setminus f_1^{-1}(A) \to \mathbb{R}^2\setminus f_2^{-1}(A)$ such that $f_1 = f_2 \circ \varphi$ on $\mathbb{R}^2\setminus f_1^{-1}(A)$ and $\varphi(b_1) = b_2$. Since $f_1^{-1}(A)$ is discrete, we can extend $\varphi$ uniquely to a homeomorphism of $\mathbb{R}^2$ such that $f_1 = f_2 \circ \varphi$.

Let $\gamma$ be an arbitrary loop in $\mathbb{R}^2\setminus f_1^{-1}(A)$ based at $b_1$. Then $\varphi \circ \gamma$ is a $f_2$-lift (based at $b_2$) of the loop $f_1 \circ \gamma$. In particular, due to condition (2), the loops $\gamma$ and $p \cdot (\varphi \circ \gamma) \cdot \bar{p}$ are homotopic rel. $A$. Now Lemma 3.2 implies that $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$.
how this impacts the convergence of the corresponding Thurston pullback maps and establish Main Theorem B.

**3.1. Combinatorial convergence.** Suppose that $f: (\mathbb{R}^2, A) \hookrightarrow$ is a Thurston map with $f(b) = t$ for some $b, t \in \mathbb{R}^2 \setminus A$. If $\gamma$ is a loop in $\mathbb{R}^2 \setminus A$ based at $t$, then we use the notation $\gamma \uparrow(f, b)$ as a shorthand for the $f$-lift of $\gamma$ starting at $b$.

Then for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$ we can record

(1) whether $\gamma$ lifts under $f$ to a loop based at $b$, i.e., whether $[\gamma] \in f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$;

(2) if $[\gamma] \in f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$, homotopy class rel. $A$ of $\gamma \uparrow(f, b)$, i.e., $[\gamma \uparrow(f, b)] \in \pi_1(\mathbb{R}^2 \setminus A, b)$.

We now establish that this information is enough to define the Thurston map $f$ uniquely up to isotopy rel. $A$ (cf. [Kam01] and [BD17, Theorem II.3] for rational Thurston maps).

**Proposition 3.1.** Let $f_1: (\mathbb{R}^2, A) \hookrightarrow$ and $f_2: (\mathbb{R}^2, A) \hookrightarrow$ be Thurston maps such that $f_1(b_1) = t$ for some $b_1 \in \mathbb{R}^2$ and $t \in \mathbb{R}^2 \setminus A$. Then $f_1$ and $f_2$ are isotopic rel. $A$ if and only if there exists $b_2 \in \mathbb{R}^2$ and a path $p \subset \mathbb{R}^2 \setminus A$ joining $b_1$ with $b_2$ such that $f_2(b_2) = t$ and for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$ the following holds:

(1) $\gamma$ lifts under $f_1$ to a loop based at $b_1$ if and only if it lifts under $f_2$ to a loop based at $b_2$, in other words, $(f_1)_*\pi_1(\mathbb{R}^2 \setminus f_1^{-1}(A), b_1) = (f_2)_*\pi_1(\mathbb{R}^2 \setminus f_2^{-1}(A), b_2)$;

(2) if $[\gamma] \in (f_1)_*\pi_1(\mathbb{R}^2 \setminus f_1^{-1}(A), b_1)$, then loops $\gamma \uparrow(f_1, b_1)$ and $p \cdot \gamma \uparrow(f_2, b_2) \cdot \bar{p}$ are homotopic rel. $A$.

Before proving the above, we formulate a simple observation.

**Lemma 3.2.** Suppose $A \subset \mathbb{R}^2$ is finite, and $\varphi$ is an orientation-preserving homeomorphism such that $\varphi(b) = t$ for some $b, t \in \mathbb{R}^2 \setminus A$. Then $\varphi$ is isotopic rel. $A$ to $\mathrm{id}_{\mathbb{R}^2}$ if and only if there exists a path $p \subset \mathbb{R}^2 \setminus A$ joining $b$ with $t$ such that every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b$ is homotopic rel. $A$ to $p \cdot (\varphi \circ \gamma) \cdot \bar{p}$.

*Proof.* Suppose that $\varphi$ is isotopic rel. $A$ to $\mathrm{id}_{\mathbb{R}^2}$ via an isotopy $(\varphi_s)_{s \in I}$, where $\varphi_0 = \mathrm{id}_{\mathbb{R}^2}$ and $\varphi_1 = \varphi$. Define $p: I \to \mathbb{R}^2 \setminus A$ as $p(s) = \varphi_s(b)$ for each $s \in I$. Then it is evident that any loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b$ and $p \cdot (\varphi \circ \gamma) \cdot \bar{p}$ are homotopic rel. $A$ via a homotopy $H_s := p_s \cdot (\varphi_s \circ \gamma) \cdot \bar{p}_s$, where $p_s$ is a subpath of $p$ joining $b$ with $\varphi_s(b)$ for every $s \in I$.

Now suppose that there exists a path $p \subset \mathbb{R}^2 \setminus A$ joining $b$ with $t$ such that every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b$ is homotopic rel. $A$ to $p \cdot (\varphi \circ \gamma) \cdot \bar{p} \subset \mathbb{R}^2 \setminus A$. Taking $\gamma$ to be a loop separating a unique point $a \in A$ from the other points of $A$ and using continuity of $\varphi$, we can show that $\varphi(a) = a$ for each $a \in A$. The rest easily follows: for instance, it is a particular case of the much stronger Dehn-Neilsen-Baer theorem [FM12, Theorem 8.1, 8.8]. $\square$

*Proof of Proposition 3.1.* Suppose that conditions (1) and (2) are satisfied under some choice of $b_2$ and $p$. Since the restrictions

$$f_1|\mathbb{R}^2\setminus f_1^{-1}(A): \mathbb{R}^2\setminus f_1^{-1}(A) \to \mathbb{R}^2\setminus A \text{ and } f_2|\mathbb{R}^2\setminus f_2^{-1}(A): \mathbb{R}^2\setminus f_2^{-1}(A) \to \mathbb{R}^2\setminus A$$

are both covering maps, condition (1) implies that there exists an orientation-preserving homeomorphism $\varphi: \mathbb{R}^2\setminus f_1^{-1}(A) \to \mathbb{R}^2\setminus f_2^{-1}(A)$ such that $f_1 = f_2 \circ \varphi$ on $\mathbb{R}^2\setminus f_1^{-1}(A)$ and $\varphi(b_1) = b_2$. Since $f_1^{-1}(A)$ is discrete, we can extend $\varphi$ uniquely to a homeomorphism of $\mathbb{R}^2$ such that $f_1 = f_2 \circ \varphi$.

Let $\gamma$ be an arbitrary loop in $\mathbb{R}^2\setminus f_1^{-1}(A)$ based at $b_1$. Then $\varphi \circ \gamma$ is a $f_2$-lift (based at $b_2$) of the loop $f_1 \circ \gamma$. In particular, due to condition (2), the loops $\gamma$ and $p \cdot (\varphi \circ \gamma) \cdot \bar{p}$ are homotopic rel. $A$. Now Lemma 3.2 implies that $\varphi \in \mathrm{Homeo}_0^+(\mathbb{R}^2, A)$.
Now suppose that $f_1$ and $f_2$ are isotopic rel. A, i.e., there exists $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ such that $f_1 = f_2 \circ \varphi$. Let us choose $b_2$ to be equal $\varphi(b_1)$. Condition (1) is then obviously satisfied, and condition (2) follows from Lemma 3.2. □

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

**Definition 3.3.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps. We say that $(f_n)$ converges combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ if there exist $b_n, b, t \in \mathbb{R}^2 \setminus A$, where $f_n(b_n) = f(b) = t$ for every $n \in \mathbb{N}$, and a sequence $(p_n)$ of paths in $\mathbb{R}^2 \setminus A$ joining $b$ with $b_n$ such that for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$, there exists $N = N(\gamma)$ so that for all $n \ge N$, the following conditions hold:

(1) $\gamma$ lifts under $f_n$ to a loop starting at $b_n$ if and only if $\gamma$ lifts under $f$ to a loop starting at $b$. In other words,

$$ \lim_{n \to \infty} (f_n)_* \pi_1(\mathbb{R}^2 \setminus f_n^{-1}(A), b_n) = f_* \pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b); $$

(2) if $[\gamma] \in f_* \pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$, then the loops $\gamma\uparrow(f, b)$ and $p_n \cdot \gamma\uparrow(f_n, b_n) \cdot \overline{p_n}$ are homotopic rel. A.

**Proposition 3.4.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps converging combinatorially to a polynomial Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$. Then $f_n$ is isotopic rel. A to $f$ for all sufficiently large $n$.

*Proof.* Suppose that $(f_n)$ converges combinatorially to $f$ with respect to some choice of $(b_n), b, t$ and $(p_n)$. Since $f$ is a topological polynomial, then $f^{-1}(A)$ is finite and, in particular, the group $f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$ is finitely generated. Thus, if $n$ is large enough, $(f_n)_*\pi_1(\mathbb{R}^2 \setminus f_n^{-1}(A), b_n)$ contains $f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$ as a subgroup. From the classical theory of covering maps, it follows that $f = f_n \circ \varphi_n$ on $\mathbb{R}^2 \setminus f^{-1}(A)$, where $\varphi_n: \mathbb{R}^2 \setminus f^{-1}(A) \to \mathbb{R}^2 \setminus f_n^{-1}(A)$ is a covering map sending $b$ to $b_n$. In particular, the topological degrees of the maps $f_n$ and $\varphi_n$ are finite and do not exceed $\text{deg}(f)$ for all sufficiently large $n$.

Let $\alpha$ be a simple loop in $\mathbb{R}^2$ based at $t$ such that the unique bounded component of $\mathbb{R}^2 \setminus \alpha$ contains the set A. Proposition 2.12 implies that $\alpha^k, k \in \mathbb{Z}$ lifts to a loop based at $b$ under the topological polynomial $f$ if and only if $k$ is divisible by $\text{deg}(f)$; a similar statement holds for the topological polynomials $f_n$, as well. Thus, condition (1) implies that $\text{deg}(f) = \text{deg}(f_n)$ for all sufficiently large $n$. Hence, the covering map $\varphi_n: \mathbb{R}^2 \setminus f^{-1}(A) \to \mathbb{R}^2 \setminus f_n^{-1}(A)$ is an orientation-preserving homeomorphism and, therefore, can be extended to $\mathbb{R}^2$ so that $f = f_n \circ \varphi_n$. Finally, similarly to the proof of Proposition 3.1, one can show that that $\varphi_n$ is isotopic rel. A to $\text{id}_{\mathbb{R}^2}$ using condition(2) and Lemma 3.2. □

The next proposition shows that the notion of combinatorial convergence is independent of choices of representatives within the isotopy classes of considered Thurston maps.

**Proposition 3.5.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps converging combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$. Suppose that Thurston maps $\hat{f}_n: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, A) \hookrightarrow$ are isotopic rel. A to $f_n$ and $f$, respectively, for every $n \in \mathbb{N}$. Then the sequence $(\hat{f}_n)$ converges combinatorially to $\hat{f}$.

*Proof.* First of all, we note that if $\hat{f}_n = f_n \circ \varphi^{-1}$ and $\hat{f} = f \circ \varphi^{-1}$, where $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$, then the statement is obviously true with respect to $\hat{t} := t, \hat{b} := \varphi(b), \hat{b}_n := \varphi(b_n)$, and
Now suppose that $f_1$ and $f_2$ are isotopic rel. A, i.e., there exists $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ such that $f_1 = f_2 \circ \varphi$. Let us choose $b_2$ to be equal $\varphi(b_1)$. Condition (1) is then obviously satisfied, and condition (2) follows from Lemma 3.2. $\square$

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

**Definition 3.3.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps. We say that $(f_n)$ converges combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ if there exist $b_n, b, t \in \mathbb{R}^2 \setminus A$, where $f_n(b_n) = f(b) = t$ for every $n \in \mathbb{N}$, and a sequence $(p_n)$ of paths in $\mathbb{R}^2 \setminus A$ joining $b$ with $b_n$ such that for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$, there exists $N = N(\gamma)$ so that for all $n \ge N$, the following conditions hold:

(1) $\gamma$ lifts under $f_n$ to a loop starting at $b_n$ if and only if $\gamma$ lifts under $f$ to a loop starting at $b$. In other words,

$$ \lim_{n \to \infty} (f_n)_* \pi_1(\mathbb{R}^2 \setminus f_n^{-1}(A), b_n) = f_* \pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b); $$

(2) if $[\gamma] \in f_* \pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$, then the loops $\gamma\uparrow(f, b)$ and $p_n \cdot \gamma\uparrow(f_n, b_n) \cdot \bar{p}_n$ are homotopic rel. A.

**Proposition 3.4.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps converging combinatorially to a polynomial Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$. Then $f_n$ is isotopic rel. A to $f$ for all sufficiently large $n$.

*Proof.* Suppose that $(f_n)$ converges combinatorially to $f$ with respect to some choice of $(b_n), b, t$ and $(p_n)$. Since $f$ is a topological polynomial, then $f^{-1}(A)$ is finite and, in particular, the group $f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$ is finitely generated. Thus, if $n$ is large enough, $(f_n)_*\pi_1(\mathbb{R}^2 \setminus f_n^{-1}(A), b_n)$ contains $f_*\pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b)$ as a subgroup. From the classical theory of covering maps, it follows that $f = f_n \circ \varphi_n$ on $\mathbb{R}^2 \setminus f^{-1}(A)$, where $\varphi_n: \mathbb{R}^2 \setminus f^{-1}(A) \to \mathbb{R}^2 \setminus f_n^{-1}(A)$ is a covering map sending $b$ to $b_n$. In particular, the topological degrees of the maps $f_n$ and $\varphi_n$ are finite and do not exceed $\text{deg}(f)$ for all sufficiently large $n$.

Let $\alpha$ be a simple loop in $\mathbb{R}^2$ based at $t$ such that the unique bounded component of $\mathbb{R}^2 \setminus \alpha$ contains the set A. Proposition 2.12 implies that $\alpha^k, k \in \mathbb{Z}$ lifts to a loop based at $b$ under the topological polynomial $f$ if and only if $k$ is divisible by $\text{deg}(f)$; a similar statement holds for the topological polynomials $f_n$, as well. Thus, condition (1) implies that $\text{deg}(f) = \text{deg}(f_n)$ for all sufficiently large $n$. Hence, the covering map $\varphi_n: \mathbb{R}^2 \setminus f^{-1}(A) \to \mathbb{R}^2 \setminus f_n^{-1}(A)$ is an orientation-preserving homeomorphism and, therefore, can be extended to $\mathbb{R}^2$ so that $f = f_n \circ \varphi_n$. Finally, similarly to the proof of Proposition 3.1, one can show that that $\varphi_n$ is isotopic rel. A to $\text{id}_{\mathbb{R}^2}$ using condition(2) and Lemma 3.2. $\square$

The next proposition shows that the notion of combinatorial convergence is independent of choices of representatives within the isotopy classes of considered Thurston maps.

**Proposition 3.5.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps converging combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$. Suppose that Thurston maps $\hat{f}_n: (\mathbb{R}^2, A) \hookrightarrow$ and $\hat{f}: (\mathbb{R}^2, A) \hookrightarrow$ are isotopic rel. A to $f_n$ and $f$, respectively, for every $n \in \mathbb{N}$. Then the sequence $(\hat{f}_n)$ converges combinatorially to $\hat{f}$.

*Proof.* First of all, we note that if $\hat{f}_n = f_n \circ \varphi^{-1}$ and $\hat{f} = f \circ \varphi^{-1}$, where $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$, then the statement is obviously true with respect to $\hat{t} := t, \hat{b} := \varphi(b), \hat{b}_n := \varphi(b_n)$, and
$\hat{p}_n := \varphi \circ p_n$. This observation allows us to reduce the verification to the case when $\hat{f}_n = f_n \circ \varphi_n^{-1}$, where $\varphi_n \in \text{Homeo}_0^+(\mathbb{R}^2, A)$, and $\hat{f} = f$.

One can easily see that condition (1) is satisfied for the sequence $(\hat{f}_n)$ and the map $\hat{f}$ if we take $\hat{b} := b$, $\hat{t} = t$, and $\hat{b}_n := \varphi_n(b_n)$ for every $n \in \mathbb{N}$.

Lemma 3.2 implies that for every $n \in \mathbb{N}$ there exists a path $\ell_n \subset \mathbb{R}^2 \setminus A$ joining $b_n$ with $\hat{b}_n = \varphi_n(b_n)$ such that any loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b_n$ is homotopic rel. $A$ to $\ell_n \cdot (\varphi_n \circ \gamma) \cdot \overline{\ell_n}$. Now let us take $\hat{p}_n := p_n \cdot \ell_n$ and show that $(\hat{f}_n)$ converges combinatorially to $\hat{f}$. Indeed, let $\delta \subset \mathbb{R}^2 \setminus A$ be a loop based at $\hat{t}$ such that $[\delta] \in \hat{f}_*\pi_1(\mathbb{R}^2 \setminus \hat{f}^{-1}(A), b)$, then for sufficiently large $n$ we have

$$ \delta\uparrow(\hat{f}, \hat{b}) \sim p_n \cdot \delta\uparrow(f_n, b_n) \cdot \overline{p_n} \text{ rel. } A, $$

$$ \delta\uparrow(f_n, b_n) \sim \ell_n \cdot \delta\uparrow(\hat{f}_n, \hat{b}_n) \cdot \overline{\ell_n} \text{ rel. } A. $$

In other words, $\delta\uparrow(\hat{f}, \hat{b})$ is homotopic rel. $A$ to $\hat{p}_n \cdot \delta\uparrow(f_n, b_n) \cdot \overline{\hat{p}_n}$ and this completes the proof. □

**3.2. Topological convergence.** The following definition provides another way to think of convergence of Thurston maps.

**Definition 3.6.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps. We say that $(f_n)$ converges topologically to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ if for every bounded set $D \subset \mathbb{R}^2$ and for all sufficiently large $n$, the inequality $f_n|_D = f|_D$ holds.

Unlike for combinatorial convergence, note that the limit in Definition 3.6 is uniquely defined. The following proposition shows the relationship between Definitions 3.3 and 3.6.

**Proposition 3.7.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps. Then $(f_n)$ converges combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ if and only if there exists a sequence of Thurston maps $\hat{f}_n: (\mathbb{R}^2, A) \hookrightarrow$ converging topologically to $f$, where $\hat{f}_n$ is isotopic rel. $A$ to $f_n$.

In other words, the two notions of convergence - combinatorial and topological, are, in a certain sense, equivalent. Before we prove Proposition 3.7, we obtain the following result of a non-dynamical nature.

**Proposition 3.8.** Suppose that $f_n: \mathbb{R}^2 \to \mathbb{R}^2$, $n \in \mathbb{N}$ and $f: \mathbb{R}^2 \to \mathbb{R}^2$ are topologically holomorphic maps of finite type such that $S_{f_n} \subset A$ and $S_f \subset A$ for every $n \in \mathbb{N}$ and some finite set $A$. Let $b, b_n \in \mathbb{R}^2$ and $t \in \mathbb{R}^2 \setminus A$ be points such that $f_n(b_n) = f(b) = t$ for every $n \in \mathbb{N}$.

Further assume that for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$, for all sufficiently large $n$, $\gamma$ lifts under $f_n$ to a loop based at $b_n$ if and only if it lifts under $f$ to a loop based at $b$. In other words,

$$ \lim_{n \to \infty} (f_n)_* \pi_1(\mathbb{R}^2 \setminus f_n^{-1}(A), b_n) = f_* \pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b). $$

Then, given any bounded set $D \subset \mathbb{R}^2$ containing $b$, for all sufficiently large $n$, there exists a continuous injective map $\varphi_n: D \to \mathbb{R}^2$ such that $f|_D = f_n \circ \varphi_n$ and $\varphi_n(b) = b_n$.

*Proof.* Without loss of generality we can assume that $D$ is an open Jordan region such that $b \in D$ and $\partial D \cap f^{-1}(A) = \emptyset$. Choose a spider $S$ that consists of $|A|$ disjoint continuous simple curves $L_a: [0, +\infty) \to \mathbb{R}^2$, $a \in A$ such that $L_a$ joins $a$ with $\infty$, i.e., $L_a(0) = a$ and $\lim_{t \to +\infty} L_a(t) = \infty$ for every $a \in A$.
$\hat{p}_n := \varphi \circ p_n$. This observation allows us to reduce the verification to the case when $\hat{f}_n = f_n \circ \varphi_n^{-1}$, where $\varphi_n \in \text{Homeo}_0^+(\mathbb{R}^2, A)$, and $\hat{f} = f$.

One can easily see that condition (1) is satisfied for the sequence $(\hat{f}_n)$ and the map $\hat{f}$ if we take $\hat{b} := b$, $\hat{t} = t$, and $\hat{b}_n := \varphi_n(b_n)$ for every $n \in \mathbb{N}$.

Lemma 3.2 implies that for every $n \in \mathbb{N}$ there exists a path $\ell_n \subset \mathbb{R}^2 \setminus A$ joining $b_n$ with $\hat{b}_n = \varphi_n(b_n)$ such that any loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $b_n$ is homotopic rel. $A$ to $\ell_n \cdot (\varphi_n \circ \gamma) \cdot \overline{\ell_n}$. Now let us take $\hat{p}_n := p_n \cdot \ell_n$ and show that $(\hat{f}_n)$ converges combinatorially to $\hat{f}$. Indeed, let $\delta \subset \mathbb{R}^2 \setminus A$ be a loop based at $\hat{t}$ such that $[\delta] \in \hat{f}_*\pi_1(\mathbb{R}^2 \setminus \hat{f}^{-1}(A), b)$, then for sufficiently large $n$ we have

$$ \delta\uparrow(\hat{f}, \hat{b}) \sim p_n \cdot \delta\uparrow(f_n, b_n) \cdot \overline{p_n} \text{ rel. } A, $$

$$ \delta\uparrow(f_n, b_n) \sim \ell_n \cdot \delta\uparrow(\hat{f}_n, \hat{b}_n) \cdot \overline{\ell_n} \text{ rel. } A. $$

In other words, $\delta\uparrow(\hat{f}, \hat{b})$ is homotopic rel. $A$ to $\hat{p}_n \cdot \delta\uparrow(f_n, b_n) \cdot \overline{\hat{p}_n}$ and this completes the proof. $\square$

**3.2. Topological convergence.** The following definition provides another way to think of convergence of Thurston maps.

**Definition 3.6.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps. We say that $(f_n)$ converges topologically to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ if for every bounded set $D \subset \mathbb{R}^2$ and for all sufficiently large $n$, the inequality $f_n|_D = f|_D$ holds.

Unlike for combinatorial convergence, note that the limit in Definition 3.6 is uniquely defined. The following proposition shows the relationship between Definitions 3.3 and 3.6.

**Proposition 3.7.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps. Then $(f_n)$ converges combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow$ if and only if there exists a sequence of Thurston maps $\hat{f}_n: (\mathbb{R}^2, A) \hookrightarrow$ converging topologically to $f$, where $\hat{f}_n$ is isotopic rel. $A$ to $f_n$.

In other words, the two notions of convergence - combinatorial and topological, are, in a certain sense, equivalent. Before we prove Proposition 3.7, we obtain the following result of a non-dynamical nature.

**Proposition 3.8.** Suppose that $f_n: \mathbb{R}^2 \to \mathbb{R}^2$, $n \in \mathbb{N}$ and $f: \mathbb{R}^2 \to \mathbb{R}^2$ are topologically holomorphic maps of finite type such that $S_{f_n} \subset A$ and $S_f \subset A$ for every $n \in \mathbb{N}$ and some finite set $A$. Let $b, b_n \in \mathbb{R}^2$ and $t \in \mathbb{R}^2 \setminus A$ be points such that $f_n(b_n) = f(b) = t$ for every $n \in \mathbb{N}$.

Further assume that for every loop $\gamma \subset \mathbb{R}^2 \setminus A$ based at $t$, for all sufficiently large $n$, $\gamma$ lifts under $f_n$ to a loop based at $b_n$ if and only if it lifts under $f$ to a loop based at $b$. In other words,

$$ \lim_{n \to \infty} (f_n)_* \pi_1(\mathbb{R}^2 \setminus f_n^{-1}(A), b_n) = f_* \pi_1(\mathbb{R}^2 \setminus f^{-1}(A), b). $$

Then, given any bounded set $D \subset \mathbb{R}^2$ containing $b$, for all sufficiently large $n$, there exists a continuous injective map $\varphi_n: D \to \mathbb{R}^2$ such that $f|_D = f_n \circ \varphi_n$ and $\varphi_n(b) = b_n$.

*Proof.* Without loss of generality we can assume that $D$ is an open Jordan region such that $b \in D$ and $\partial D \cap f^{-1}(A) = \emptyset$. Choose a spider $S$ that consists of $|A|$ disjoint continuous simple curves $L_a: [0, +\infty) \to \mathbb{R}^2$, $a \in A$ such that $L_a$ joins $a$ with $\infty$, i.e., $L_a(0) = a$ and $\lim_{t \to +\infty} L_a(t) = \infty$ for every $a \in A$.
*Claim 1.* Let $K \subset \mathbb{R}^2$ be a bounded set, then there are only finitely many connected components of $\mathbb{R}^2\setminus f^{-1}(S)$ intersecting $K$.

*Proof of Claim 1.* Suppose that there exist distinct connected components $E_1, E_2, \dots, E_n, \dots$ of $\mathbb{R}^2\setminus f^{-1}(S)$ intersecting $K$. Pick an arbitrary point $x_n \in E_n \cap K$ for each $n \in \mathbb{N}$. We may assume without loss of generality that $(x_n)$ converges to $x \in \bar{K}$. Thus, any neighbourhood of $x$ intersects infinitely many connected components of $\mathbb{R}^2\setminus f^{-1}(S)$, which leads to a contradiction since $f$ is a topologically holomorphic map. ■

*Claim 2.* There exists a bounded domain $D' \subset \mathbb{R}^2$ such that $D \subset D'$ and for every connected component $F$ of $\mathbb{R}^2\setminus f^{-1}(S)$, the set $D' \cap F$ is connected (possibly empty).

*Proof of Claim 2.* We will only give an outline of the proof, leaving some straightforward details to the reader. For each $a \in A^* := A \cup \{\infty\}$ we choose a set $U_a$ such that

(1) $U_a$ is a closed Jordan region containing $a$ for each $a \in A$ and $U_\infty = \mathbb{R}^2\setminus V_\infty$, where $V_\infty$ is an open Jordan region containing $A$;

(2) $U_{a_1} \cap U_{a_2} = \emptyset$ for distinct $a_1, a_2 \in A^*$;

(3) $U_a \cap S$ is connected for each $a \in A$ and $S \cup \bigcup_{a \in A^*} U_a$ is simply connected;

(4) for each $a \in A^*$, every connected component of $f^{-1}(U_a)$ is either contained in $D$ or disjoint from it.

The last condition can be satisfied if we take $U_a, a \in A^*$ disjoint from $f(\partial D) \subset \mathbb{R}^2\setminus A$.

Let $U := \bigcup_{a \in A^*} U_a$ and $W_F := F\setminus f^{-1}(U)$ for each connected component $F$ of $\mathbb{R}^2\setminus f^{-1}(S)$. It is easy to see that $F$ can be written as the following disjoint union

$$F = F\setminus f^{-1}(U) \sqcup (F \cap f^{-1}(U)) = W_F \sqcup \bigsqcup_{a \in A^*} (F \cap f^{-1}(U_a)).$$

By Proposition 2.12, the map $f|F: F \to \mathbb{R}^2\setminus S$ is a homeomorphism. Therefore, conditions (1) and (3) imply that the set $F \cap f^{-1}(U_a) = (f|F)^{-1}(U_a\setminus S)$ is connected for each $a \in A$, so it is either contained in $D$ or disjoint from it by condition (4). Thus,

$$(D \cup W_F) \cap F = W_F \sqcup \bigsqcup_{a \in A_F} (F \cap f^{-1}(U_a)) = (f|F)^{-1} \left( \mathbb{R}^2 \setminus \left( S \cup \bigcup_{a \in A^* \setminus A_F} U_a \right) \right),$$

where $A_F := \{a \in A^*: F \cap f^{-1}(U_a) \subset D\}$. Due to condition (3), the set $\mathbb{R}^2\setminus(S \cup \bigcup_{a \in A^* \setminus A_F} U_a)$ is a domain. Therefore, from Proposition 2.12, it follows that the set $(D \cup W_F) \cap F$ is connected.

Thus, $D' := D \cup \bigcup_{F: F \cap D \neq \emptyset} W_F$ is an open set such that $F \cap D'$ is connected for each connected component $F$ of $\mathbb{R}^2\setminus f^{-1}(S)$. The set $D'$ is connected since $W_F \cap D \neq \emptyset$ if and only if $F \cap D \neq \emptyset$. At the same time, $D'$ is bounded since each $W_F$ is bounded and there are finitely many connected components of $\mathbb{R}^2\setminus f^{-1}(S)$ intersecting $D$ due to Claim 1. ■

Due to the claims above, there exists a positive integer $m$ such that any $x \in D'\setminus f^{-1}(A)$ can be joined with the point $b$ by a simple path contained in $D'\setminus f^{-1}(A)$ and intersecting $f^{-1}(S)$ at most $m$ times. Denote by $P_{S,k}$ the set of elements of $\pi_1(\mathbb{R}^2\setminus A, t)$ that can be represented by a loop $\alpha: I \to \mathbb{R}^2\setminus A$ based at $t$ such that $|\alpha^{-1}(S)| \le k$. It is clear that $P_{S,k}$ is a finite subset of $\pi_1(\mathbb{R}^2\setminus A, t)$.

Now we construct a continuous map $\varphi_n: D' \to \mathbb{R}^2$ and prove its injectivity as required. Let $D''$ be an open Jordan region containing $D'$. Since the group $f_*\pi_1(D''\setminus f^{-1}(A), b)$ is