*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
finitely generated, by our initial assumptions, it is a subgroup of $(f_n)_*\pi_1(\mathbb{R}^2\backslash f_n^{-1}(A), b_n)$ for all $n$ sufficiently large. Therefore, there exists a continuous map $\varphi_n: D''\backslash f^{-1}(A) \to \mathbb{R}^2\backslash f_n^{-1}(A)$ such that $f = f_n \circ \varphi_n$ on $D''\backslash f^{-1}(A)$ and $\varphi_n(b) = b_n$.

We prove that the map $\varphi_n$ is injective on $D'\backslash f^{-1}(A)$. Suppose that it is not true for some $n$, i.e., there exist two distinct points $x_n, y_n \in D'\backslash f^{-1}(A)$ such that $\varphi_n(x_n) = \varphi_n(y_n)$. We join $b$ with $x_n$ and $y_n$ by two simple paths $\alpha_n$ and $\beta_n$ in $D'\backslash f^{-1}(A)$, respectively. Additionally, assume that both paths intersect $f^{-1}(S)$ at most $m$ times. It is clear that the loop $\gamma_n := f \circ \alpha_n \cdot \overline{f \circ \beta_n}$ based at $t$ does not lift under $f$ to a loop based at $b$ but it lifts under $f_n$ to the loop $\varphi_n \circ \alpha_n \cdot \overline{\varphi_n \circ \beta_n}$ based at $b_n$. Thus,

$$[\gamma_n] \in G_n := P_{S,2m} \cap \left( (f_n)_*\pi_1(\mathbb{R}^2\backslash f_n^{-1}(A), b_n) \setminus f_*\pi(\mathbb{R}^2\backslash f^{-1}(A), b) \right).$$

Since $P_{S,2m}$ is a finite set, if $\varphi_n$ fails to be injective for infinitely many $n$, then there exists $[\gamma] \in \pi_1(\mathbb{R}^2\backslash A, t)$ such that $[\gamma] \in G_n$ for infinitely many $n$. This, however, is not possible: since $[\gamma] \notin f_*\pi_1(\mathbb{R}^2\backslash f^{-1}(A), b)$, then for all $n$ sufficiently large, $[\gamma] \notin (f_n)_*\pi_1(\mathbb{R}^2\backslash f_n^{-1}(A), t)$.

Finally, since $f^{-1}(A)$ and $f_n^{-1}(A)$ are discrete subsets of $\mathbb{R}^2$, we can extend $\varphi_n$ to a continuous injective map defined on $D'$ satisfying $f|_{D'} = f_n \circ \varphi_n$ and $\varphi_n(b) = b_n$ for all $n$ large enough. $\square$

**Proof of Proposition 3.7.** If the sequence $(\hat{f}_n)$ converges topologically to $f$, then it does so combinatorially with respect to $t, b, (b_n), p, (p_n)$, where $t \in \mathbb{R}^2\backslash A$ and $b \in f^{-1}(t)$ are arbitrary, $b_n := b$, and $p, p_n$ are the constant loops based at $b$ for every $n \in \mathbb{N}$. Hence, Proposition 3.5 implies that the sequence $(f_n)$ converges combinatorially to $f$.

Now suppose that the sequence $(f_n)$ converges combinatorially to $f$ with respect to some choice of $t, b, (b_n), p$ and $(p_n)$. Let $(D_m)$ be an exhaustion of $\mathbb{R}^2$ by closed Jordan regions containing $A$ and $b$ in their interiors for each $m \in \mathbb{N}$. Then for $n$ sufficiently large, there exists a maximal index $m = m(n)$ such that

(1) $f|_{D_m} = f_n \circ \varphi_n$ for some continuous injective map $\varphi_n: D_m \to \mathbb{R}^2$ that satisfies $\varphi_n(b) = b_n$;

(2) for every loop $\gamma \subset \mathbb{R}^2\backslash A$ based at $t$ such that $[\gamma] \in f_*\pi_1(D_m\backslash f^{-1}(A), b)$, loops $\gamma\uparrow(f, b)$ and $p_n \cdot \gamma\uparrow(f_n, b_n) \cdot \overline{p_n}$ are homotopic rel. $A$.

These conditions hold for sufficiently large $n$ because of Definition 3.3, Proposition 3.8 and the fact that $f_*\pi_1(D_m\backslash f^{-1}(A), b)$ is finitely generated. Moreover, it is easy to show that $m(n)$ converges to infinity as $n$ tends to infinity.

Now we can extend $\varphi_n$ to an orientation-preserving homeomorphism $\hat{\varphi}_n$ of $\mathbb{R}^2$. In fact, by Alexander's trick, all such extensions of $\varphi_n$ are isotopic rel. $D_m$ to each other. Clearly, if $\gamma$ is a loop in $D_m\backslash f^{-1}(A)$ based at $b$, then $\varphi_n \circ \gamma$ is a $f_n$-lift (based at $b_n$) of $f \circ \gamma$. In particular, $\gamma$ is homotopic rel. $A$ to $p_n \cdot (\hat{\varphi}_n \circ \gamma) \cdot \overline{p_n}$ for every loop $\gamma \subset \mathbb{R}^2\backslash A$ based at $b$. Thus, the homeomorphism $\hat{\varphi}_n$ is isotopic rel. $A$ to $\mathrm{id}_{\mathbb{R}^2}$ by Lemma 3.2.

Now consider the sequence $(\hat{f}_n)$ of Thurston maps, where $\hat{f}_n := f_n \circ \hat{\varphi}_n$ for all $n \in \mathbb{N}$. Clearly, $(\hat{f}_n)$ converges topologically to $f$. $\square$

### 3.3. Polynomial approximations.
In Section 2.7, we show how to define topologically holomorphic maps of finite type using the language of admissible quadruples. Now we apply this framework to study Thurston maps, as this point of view is helpful while working with the notion of combinatorial convergence. In particular, we establish criteria for combinatorial convergence of a given sequence of Thurston maps (Proposition 3.13) and show that every
finitely generated, by our initial assumptions, it is a subgroup of $(f_n)_*\pi_1(\mathbb{R}^2\backslash f_n^{-1}(A), b_n)$ for all $n$ sufficiently large. Therefore, there exists a continuous map $\varphi_n: D''\backslash f^{-1}(A) \to \mathbb{R}^2\backslash f_n^{-1}(A)$ such that $f = f_n \circ \varphi_n$ on $D''\backslash f^{-1}(A)$ and $\varphi_n(b) = b_n$.

We prove that the map $\varphi_n$ is injective on $D'\backslash f^{-1}(A)$. Suppose that it is not true for some $n$, i.e., there exist two distinct points $x_n, y_n \in D'\backslash f^{-1}(A)$ such that $\varphi_n(x_n) = \varphi_n(y_n)$. We join $b$ with $x_n$ and $y_n$ by two simple paths $\alpha_n$ and $\beta_n$ in $D'\backslash f^{-1}(A)$, respectively. Additionally, assume that both paths intersect $f^{-1}(S)$ at most $m$ times. It is clear that the loop $\gamma_n := f \circ \alpha_n \cdot \overline{f \circ \beta_n}$ based at $t$ does not lift under $f$ to a loop based at $b$ but it lifts under $f_n$ to the loop $\varphi_n \circ \alpha_n \cdot \overline{\varphi_n \circ \beta_n}$ based at $b_n$. Thus,

$$[\gamma_n] \in G_n := P_{S,2m} \cap \left( (f_n)_*\pi_1(\mathbb{R}^2\backslash f_n^{-1}(A), b_n) \setminus f_*\pi(\mathbb{R}^2\backslash f^{-1}(A), b) \right).$$

Since $P_{S,2m}$ is a finite set, if $\varphi_n$ fails to be injective for infinitely many $n$, then there exists $[\gamma] \in \pi_1(\mathbb{R}^2\backslash A, t)$ such that $[\gamma] \in G_n$ for infinitely many $n$. This, however, is not possible: since $[\gamma] \notin f_*\pi_1(\mathbb{R}^2\backslash f^{-1}(A), b)$, then for all $n$ sufficiently large, $[\gamma] \notin (f_n)_*\pi_1(\mathbb{R}^2\backslash f_n^{-1}(A), t)$.

Finally, since $f^{-1}(A)$ and $f_n^{-1}(A)$ are discrete subsets of $\mathbb{R}^2$, we can extend $\varphi_n$ to a continuous injective map defined on $D'$ satisfying $f|_{D'} = f_n \circ \varphi_n$ and $\varphi_n(b) = b_n$ for all $n$ large enough. $\square$

**Proof of Proposition 3.7.** If the sequence $(\hat{f}_n)$ converges topologically to $f$, then it does so combinatorially with respect to $t, b, (b_n), p, (p_n)$, where $t \in \mathbb{R}^2\backslash A$ and $b \in f^{-1}(t)$ are arbitrary, $b_n := b$, and $p, p_n$ are the constant loops based at $b$ for every $n \in \mathbb{N}$. Hence, Proposition 3.5 implies that the sequence $(f_n)$ converges combinatorially to $f$.

Now suppose that the sequence $(f_n)$ converges combinatorially to $f$ with respect to some choice of $t, b, (b_n), p$ and $(p_n)$. Let $(D_m)$ be an exhaustion of $\mathbb{R}^2$ by closed Jordan regions containing $A$ and $b$ in their interiors for each $m \in \mathbb{N}$. Then for $n$ sufficiently large, there exists a maximal index $m = m(n)$ such that

(1) $f|_{D_m} = f_n \circ \varphi_n$ for some continuous injective map $\varphi_n: D_m \to \mathbb{R}^2$ that satisfies $\varphi_n(b) = b_n$;

(2) for every loop $\gamma \subset \mathbb{R}^2\backslash A$ based at $t$ such that $[\gamma] \in f_*\pi_1(D_m\backslash f^{-1}(A), b)$, loops $\gamma\uparrow(f, b)$ and $p_n \cdot \gamma\uparrow(f_n, b_n) \cdot \overline{p_n}$ are homotopic rel. $A$.

These conditions hold for sufficiently large $n$ because of Definition 3.3, Proposition 3.8 and the fact that $f_*\pi_1(D_m\backslash f^{-1}(A), b)$ is finitely generated. Moreover, it is easy to show that $m(n)$ converges to infinity as $n$ tends to infinity.

Now we can extend $\varphi_n$ to an orientation-preserving homeomorphism $\hat{\varphi}_n$ of $\mathbb{R}^2$. In fact, by Alexander's trick, all such extensions of $\varphi_n$ are isotopic rel. $D_m$ to each other. Clearly, if $\gamma$ is a loop in $D_m\backslash f^{-1}(A)$ based at $b$, then $\varphi_n \circ \gamma$ is a $f_n$-lift (based at $b_n$) of $f \circ \gamma$. In particular, $\gamma$ is homotopic rel. $A$ to $p_n \cdot (\hat{\varphi}_n \circ \gamma) \cdot \overline{p_n}$ for every loop $\gamma \subset \mathbb{R}^2\backslash A$ based at $b$. Thus, the homeomorphism $\hat{\varphi}_n$ is isotopic rel. $A$ to $\mathrm{id}_{\mathbb{R}^2}$ by Lemma 3.2.

Now consider the sequence $(\hat{f}_n)$ of Thurston maps, where $\hat{f}_n := f_n \circ \hat{\varphi}_n$ for all $n \in \mathbb{N}$. Clearly, $(\hat{f}_n)$ converges topologically to $f$. $\square$

### 3.3. Polynomial approximations.
In Section 2.7, we show how to define topologically holomorphic maps of finite type using the language of admissible quadruples. Now we apply this framework to study Thurston maps, as this point of view is helpful while working with the notion of combinatorial convergence. In particular, we establish criteria for combinatorial convergence of a given sequence of Thurston maps (Proposition 3.13) and show that every
<figure><img src="image_9.png" /><figcaption>FIGURE 4. Dynamically admissible quadruple realized by the psf entire map
$G_1(z) = \pi \cos(z)/2$, where $P_{G_1} = \{a_1, a_2, a_3\} = \{-\pi/2, 0, \pi/2\}$.</figcaption></figure>

Thurston map can be approximated combinatorially by a sequence of polynomial Thurston maps (Proposition 3.14).

**Definition 3.9.** An admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ is called *dynamically admissible* if $\Delta$ is parabolic, $\Gamma \cap A = \emptyset$, and every face $F$ of the graph $\Gamma$ contains at most one point of $A$ if $F$ is bounded and no points of $A$, otherwise.

Proposition 2.29 implies that $\Delta(A, \mathcal{R}, f)$ is dynamically admissible if $f: (\mathbb{R}^2, A) \subsetneq$ is a Thurston map. Also, analogous to Proposition 2.31, we can establish the following statement.

**Proposition 3.10.** Let $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ be a dynamically admissible quadruple. Then there exists a Thurston map $f: (\mathbb{R}^2, A) \subsetneq$ such that $\Delta(A, \mathcal{R}, f) = \Delta$.

In a similar way we define a dynamical equivalence relation on the set of all dynamically admissible quadruples with a fixed marked set. We say that two dynamically admissible quadruples $\Delta_1 = (A, \mathcal{R}_1, \Gamma_1, \Phi_1)$ and $\Delta_2 = (A, \mathcal{R}_2, \Gamma_2, \Phi_2)$ are dynamically equivalent if there exist $\theta, \varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ 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) $\theta \circ \Phi_1 = \Phi_2 \circ \varphi$.

Finally, the following observation provides a dynamical analog of Proposition 2.33.

**Proposition 3.11.** Let $f_1: (\mathbb{R}^2, A) \subsetneq$ and $f_2: (\mathbb{R}^2, A) \subsetneq$ be Thurston maps, and $\mathcal{R}_1, \mathcal{R}_2$ be rose graphs surrounding $A$ such that $\mathcal{R}_1$ is isotopic rel. $A$ to $\mathcal{R}_2$. Then $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are dynamically equivalent if and only if $f_1$ and $f_2$ are isotopic rel. $A$.

*Proof*. First suppose that $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are dynamically equivalent via $\theta, \varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$. As in the proof of Proposition 2.33, we can assume that $\mathcal{R}_1 = \mathcal{R}_2 = \mathcal{R}$ and $\theta = \varphi = \text{id}_{\mathbb{R}^2}$. The rest easily follows from Proposition 3.1 and the discussion of Section 2.7.

Conversely, suppose that there exists $\psi \in \text{Homeo}_0^+(\mathbb{R}^2)$ such that $f_1 = f_2 \circ \psi$. Choose $\theta \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ with $\theta(\mathcal{R}_1) = \mathcal{R}_2$. By Proposition 2.13, there exists $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$
<figure><img src="image_10.png" /><figcaption>FIGURE 4. Dynamically admissible quadruple realized by the psf entire map
$G_1(z) = \pi \cos(z)/2$, where $P_{G_1} = \{a_1, a_2, a_3\} = \{-\pi/2, 0, \pi/2\}$.</figcaption></figure>

Thurston map can be approximated combinatorially by a sequence of polynomial Thurston maps (Proposition 3.14).

**Definition 3.9.** An admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ is called *dynamically admissible* if $\Delta$ is parabolic, $\Gamma \cap A = \emptyset$, and every face $F$ of the graph $\Gamma$ contains at most one point of $A$ if $F$ is bounded and no points of $A$, otherwise.

Proposition 2.29 implies that $\Delta(A, \mathcal{R}, f)$ is dynamically admissible if $f: (\mathbb{R}^2, A) \subsetneq$ is a Thurston map. Also, analogous to Proposition 2.31, we can establish the following statement.

**Proposition 3.10.** Let $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$ be a dynamically admissible quadruple. Then there exists a Thurston map $f: (\mathbb{R}^2, A) \subsetneq$ such that $\Delta(A, \mathcal{R}, f) = \Delta$.

In a similar way we define a dynamical equivalence relation on the set of all dynamically admissible quadruples with a fixed marked set. We say that two dynamically admissible quadruples $\Delta_1 = (A, \mathcal{R}_1, \Gamma_1, \Phi_1)$ and $\Delta_2 = (A, \mathcal{R}_2, \Gamma_2, \Phi_2)$ are dynamically equivalent if there exist $\theta, \varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ 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) $\theta \circ \Phi_1 = \Phi_2 \circ \varphi$.

Finally, the following observation provides a dynamical analog of Proposition 2.33.

**Proposition 3.11.** Let $f_1: (\mathbb{R}^2, A) \subsetneq$ and $f_2: (\mathbb{R}^2, A) \subsetneq$ be Thurston maps, and $\mathcal{R}_1, \mathcal{R}_2$ be rose graphs surrounding $A$ such that $\mathcal{R}_1$ is isotopic rel. $A$ to $\mathcal{R}_2$. Then $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are dynamically equivalent if and only if $f_1$ and $f_2$ are isotopic rel. $A$.

*Proof*. First suppose that $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are dynamically equivalent via $\theta, \varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$. As in the proof of Proposition 2.33, we can assume that $\mathcal{R}_1 = \mathcal{R}_2 = \mathcal{R}$ and $\theta = \varphi = \text{id}_{\mathbb{R}^2}$. The rest easily follows from Proposition 3.1 and the discussion of Section 2.7.

Conversely, suppose that there exists $\psi \in \text{Homeo}_0^+(\mathbb{R}^2)$ such that $f_1 = f_2 \circ \psi$. Choose $\theta \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ with $\theta(\mathcal{R}_1) = \mathcal{R}_2$. By Proposition 2.13, there exists $\varphi \in \text{Homeo}_0^+(\mathbb{R}^2, A)$
<figure><img src="image_10.png" /><figcaption>FIGURE 5. Dynamically admissible quadruple realized by the psf entire map
$G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$, where $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.</figcaption></figure>

such that $\theta \circ f_1 = f_2 \circ \varphi$. It directly implies that $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are dynamically equivalent. □

Given dynamically admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$, we say that a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \Delta$ (or equivalently, $\Delta$ *defines* $f$), if $\Delta(A, \mathcal{R}, f)$ is dynamically equivalent to $\Delta$. Propositions 3.10 and 3.11 imply that every dynamically admissible $\Delta$ with a marked set $A$ defines a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \Delta$, unique up to isotopy rel. $A$.

*Example 3.12.* Let $G_1: (\mathbb{C}, P_{G_1}) \hookrightarrow \Delta$ and $G_2: (\mathbb{C}, P_{G_2}) \hookrightarrow \Delta$ be the postsingularly finite entire maps defined in Example 2.19. The graphs in Figures 4 and 5 describe dynamically admissible quadruples realized by $G_1$ and $G_2$ respectively (compare with Figures 2 and 3 referenced in Example 2.34). The solid black squares in these figures represent the postsingular values of $G_1$ and $G_2$, respectively.

We shall use the framework of dynamically admissible quadruples to construct Thurston maps with required combinatorial convergence properties.

**Proposition 3.13.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow \Delta, n \in \mathbb{N}$ and $f: (\mathbb{R}^2, A) \hookrightarrow \mathcal{R}$ be Thurston maps, and $\mathcal{R}$ be a rose graph that surrounds $A$. Then the sequence $(f_n)$ converges combinatorially to $f$ if
<figure><img src="image_11.png" /><figcaption>FIGURE 5. Dynamically admissible quadruple realized by the psf entire map
$G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$, where $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.</figcaption></figure>

such that $\theta \circ f_1 = f_2 \circ \varphi$. It directly implies that $\Delta(A, \mathcal{R}_1, f_1)$ and $\Delta(A, \mathcal{R}_2, f_2)$ are dynamically equivalent. □

Given dynamically admissible quadruple $\Delta = (A, \mathcal{R}, \Gamma, \Phi)$, we say that a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \Delta$ (or equivalently, $\Delta$ *defines* $f$), if $\Delta(A, \mathcal{R}, f)$ is dynamically equivalent to $\Delta$. Propositions 3.10 and 3.11 imply that every dynamically admissible $\Delta$ with a marked set $A$ defines a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \Delta$, unique up to isotopy rel. $A$.

*Example 3.12.* Let $G_1: (\mathbb{C}, P_{G_1}) \hookrightarrow \Delta$ and $G_2: (\mathbb{C}, P_{G_2}) \hookrightarrow \Delta$ be the postsingularly finite entire maps defined in Example 2.19. The graphs in Figures 4 and 5 describe dynamically admissible quadruples realized by $G_1$ and $G_2$ respectively (compare with Figures 2 and 3 referenced in Example 2.34). The solid black squares in these figures represent the postsingular values of $G_1$ and $G_2$, respectively.

We shall use the framework of dynamically admissible quadruples to construct Thurston maps with required combinatorial convergence properties.

**Proposition 3.13.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow \Delta, n \in \mathbb{N}$ and $f: (\mathbb{R}^2, A) \hookrightarrow \Delta$ be Thurston maps, and $\mathcal{R}$ be a rose graph that surrounds $A$. Then the sequence $(f_n)$ converges combinatorially to $f$ if
and only if for every finite subgraph $K$ of $f^{-1}(\mathcal{R})$ and all sufficiently large $n$, there exists a homeomorphism $\varphi_{K,n} \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ such that $\varphi_{K,n}(K)$ is a subgraph of $f_n^{-1}(\mathcal{R})$ and $\Phi_{f,\mathcal{R}}|_K = \Phi_{f_n,\mathcal{R}} \circ \varphi_{K,n}|_K$.

*Proof.* Sufficiency easily follows from Definition 3.3, Proposition 3.5, and the discussion of Section 2.7. Necessity can be obtained by applying Proposition 3.7. □

**Proposition 3.14.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \mathcal{R}$ be an arbitrary Thurston map. Then there exists a sequence $f_n: (\mathbb{R}^2, A) \hookrightarrow \mathcal{R}, n \in \mathbb{N}$ of polynomial Thurston maps converging combinatorially to $f$.

*Proof.* Due to Proposition 2.17 and Remark 2.30, the case when $|A| = 1$ is trivial and, therefore, we can assume that $|A| \ge 2$. Let us choose a rose graph $\mathcal{R}$ surrounding the set $A$ and consider the dynamically admissible quadruple $\Delta(A, \mathcal{R}, f) = (A, \mathcal{R}, \Gamma, \Phi)$. Next, we choose an arbitrary exhaustion of $\Gamma$ by finite connected subgraphs $K_n = (V_n, E_n)$. We shall construct a sequence of eventually dynamically admissible quadruples $\Delta_n = (A, \mathcal{R}, \Gamma_n, \Phi_n)$, where $\Gamma_n$ is obtained from $K_n$ by adding several new edges, and the maps $\Phi_n$ and $\Phi$ coincide on $K_n$. We describe this more precisely by defining $\Gamma_n$ and $\Phi_n$ algorithmically. For each $n \in \mathbb{N}$, initialize $\Gamma_n$ as $K_n$, and $\Phi_n$ as $\Phi|_{K_n}$. Let $F$ be an arbitrary face of $\Gamma$ labelled by a bounded face $P$ of $\mathcal{R}$ with the property that $\partial F$ intersects $K_n$ but it is not a proper subset of $K_n$.

<figure><img src="image_11.png" /><figcaption>FIGURE 6. Constructing the directed edge $e_F$. The diagram on the left represents the face $F$ and the graph $K_n$, and the diagram on the right demonstrates the newly added edge $e_F$. Black dots and colored (solid and dashed) arcs represent the vertices and directed edges of $\Gamma$, respectively. The solid blue arcs are edges of $K_n$, while the dashed ones are in $\Gamma \setminus K_n$.</figcaption></figure>

*Claim.* $C := \partial F \cap K_n$ is a directed finite chain.

*Proof of Claim.* First we consider the case when $F$ is bounded: here, $\partial F$ is a counterclockwise directed cycle by Remark 2.28. It suffices to show that $C$ (or equivalently $\partial F \setminus C$) is connected. Supposing the contrary, let $e_1$ and $e_2$ be two edges in disjoint components of $\partial F \setminus C$. It is easy to see that each edge $e \in E(\Gamma)$ is a boundary of exactly two faces and one of them is always unbounded. Therefore, for $i = 1, 2$, there exists a continuous curve $L_i: [0, +\infty) \to \mathbb{R}^2$ that joins an interior point $x_i := L_i(0)$ of $e_i$ with $\infty$ (i.e., $\lim_{t \to +\infty} L_i(t) = \infty$) and intersects $\Gamma$ only at the point $x_i$. Let $y_1$ and $y_2$ be vertices of $C$ lying in disjoint components of $\partial F \setminus \{\text{int}(e_1), \text{int}(e_2)\}$. Any path in $\mathbb{R}^2$ joining $y_1$ to $y_2$ has to intersect $F \cup \{\text{int}(e_1), \text{int}(e_2)\}$ or $L_1 \cup L_2$. In particular, there is no path in $K_n$ joining $y_1$ to $y_2$. This forms a contradiction since $K_n$ is connected. The case when $F$ is unbounded is analogous. ■

Let $u$ and $v$ be the endpoints of $C$ (it is possible that $u = v$). Then there exist unique edges $e_u, e_v \in E(\partial F \setminus K_n)$ such that $e_u$ is incident to $u$, and $e_v$ is incident to $v$ (again, $e_u$ and
and only if for every finite subgraph $K$ of $f^{-1}(\mathcal{R})$ and all sufficiently large $n$, there exists a homeomorphism $\varphi_{K,n} \in \text{Homeo}_0^+(\mathbb{R}^2, A)$ such that $\varphi_{K,n}(K)$ is a subgraph of $f_n^{-1}(\mathcal{R})$ and $\Phi_{f,\mathcal{R}}|_K = \Phi_{f_n,\mathcal{R}} \circ \varphi_{K,n}|_K$.

*Proof.* Sufficiency easily follows from Definition 3.3, Proposition 3.5, and the discussion of Section 2.7. Necessity can be obtained by applying Proposition 3.7. □

**Proposition 3.14.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \mathcal{R}$ be an arbitrary Thurston map. Then there exists a sequence $f_n: (\mathbb{R}^2, A) \hookrightarrow \mathcal{R}, n \in \mathbb{N}$ of polynomial Thurston maps converging combinatorially to $f$.

*Proof.* Due to Proposition 2.17 and Remark 2.30, the case when $|A| = 1$ is trivial and, therefore, we can assume that $|A| \ge 2$. Let us choose a rose graph $\mathcal{R}$ surrounding the set $A$ and consider the dynamically admissible quadruple $\Delta(A, \mathcal{R}, f) = (A, \mathcal{R}, \Gamma, \Phi)$. Next, we choose an arbitrary exhaustion of $\Gamma$ by finite connected subgraphs $K_n = (V_n, E_n)$. We shall construct a sequence of eventually dynamically admissible quadruples $\Delta_n = (A, \mathcal{R}, \Gamma_n, \Phi_n)$, where $\Gamma_n$ is obtained from $K_n$ by adding several new edges, and the maps $\Phi_n$ and $\Phi$ coincide on $K_n$. We describe this more precisely by defining $\Gamma_n$ and $\Phi_n$ algorithmically. For each $n \in \mathbb{N}$, initialize $\Gamma_n$ as $K_n$, and $\Phi_n$ as $\Phi|_{K_n}$. Let $F$ be an arbitrary face of $\Gamma$ labelled by a bounded face $P$ of $\mathcal{R}$ with the property that $\partial F$ intersects $K_n$ but it is not a proper subset of $K_n$.

<figure><img src="image_12.png" /><figcaption>FIGURE 6. Constructing the directed edge $e_F$. The diagram on the left represents the face $F$ and the graph $K_n$, and the diagram on the right demonstrates the newly added edge $e_F$. Black dots and colored (solid and dashed) arcs represent the vertices and directed edges of $\Gamma$, respectively. The solid blue arcs are edges of $K_n$, while the dashed ones are in $\Gamma\setminus K_n$.</figcaption></figure>

*Claim.* $C := \partial F \cap K_n$ is a directed finite chain.

*Proof of Claim.* First we consider the case when $F$ is bounded: here, $\partial F$ is a counterclockwise directed cycle by Remark 2.28. It suffices to show that $C$ (or equivalently $\partial F\setminus C$) is connected. Supposing the contrary, let $e_1$ and $e_2$ be two edges in disjoint components of $\partial F\setminus C$. It is easy to see that each edge $e \in E(\Gamma)$ is a boundary of exactly two faces and one of them is always unbounded. Therefore, for $i = 1, 2$, there exists a continuous curve $L_i: [0, +\infty) \to \mathbb{R}^2$ that joins an interior point $x_i := L_i(0)$ of $e_i$ with $\infty$ (i.e., $\lim_{t \to +\infty} L_i(t) = \infty$) and intersects $\Gamma$ only at the point $x_i$. Let $y_1$ and $y_2$ be vertices of $C$ lying in disjoint components of $\partial F\setminus\{\text{int}(e_1), \text{int}(e_2)\}$. Any path in $\mathbb{R}^2$ joining $y_1$ to $y_2$ has to intersect $F \cup \{\text{int}(e_1), \text{int}(e_2)\}$ or $L_1 \cup L_2$. In particular, there is no path in $K_n$ joining $y_1$ to $y_2$. This forms a contradiction since $K_n$ is connected. The case when $F$ is unbounded is analogous. ■

Let $u$ and $v$ be the endpoints of $C$ (it is possible that $u = v$). Then there exist unique edges $e_u, e_v \in E(\partial F \setminus K_n)$ such that $e_u$ is incident to $u$, and $e_v$ is incident to $v$ (again, $e_u$ and
$e_v$ might coincide). We shall construct an edge $e_F$ with endpoints at $v$ and $u$ (see Figure 6) with $e_F \subset e_u \cup e_v \cup F$, so that $e_F$ coincides with the edges $e_u$ and $e_v$ in small neighborhoods of $u$ and $v$, respectively. In particular, $\text{int}(e_F)$ does not intersect $K_n$. We add $e_F$ to $\Gamma_n$ and define $\Phi_n|_{e_F}$ so that $\Phi_n$ and $\Phi$ coincide on $e_F \cap e_u$ and $e_F \cap e_v$. Then we repeat the above procedure for all faces $F \in F(\Gamma)$ satisfying previously listed properties.

Now consider the sequence of constructed quadruples $\Delta_n = (A, \mathcal{R}, \Gamma_n, \Phi_n)$. Since $(K_n)$ is an exhaustion of $\Gamma$, there exists $N \in \mathbb{N}$ such that for each $n \ge N$, if $F \in F(\Gamma)$ is bounded and contains a point of $A$, then $\partial F \subset K_n$. For each $n \ge N$, it is straightforward to check that the conditions of Definitions 2.27 and 3.9 are satisfied. In particular, $\Delta_n$ is parabolic since $\Gamma_n$ is a finite graph. By Proposition 3.10, we can construct a polynomial Thurston map $f_n: (\mathbb{R}^2, A) \hookrightarrow A$ such that $\Delta(A, \mathcal{R}, f_n) = \Delta_n$ for each $n \ge N$. Thus, the sequence $(f_n)$ converges combinatorially to $f$ due to Proposition 3.13. $\square$

*Remark 3.15.* Suppose that we are in the setting of the proof of Proposition 3.14. Let $F$ be a face of $\Gamma_n$ labelled by a bounded face $P$ of $\mathcal{R}$ (with respect to the quadruple $\Delta_n$) for some $n \ge N$. Then there exists a unique face $F' \in F(\Gamma)$ having the same label $P$ (with respect to the quadruple $\Delta$) such that $F \subsetneq F'$. Moreover, exactly one of the following is true:

(1) $\deg(f_n|F) = \deg(f_n|F')$. Then Proposition 2.29 implies that $F$ contains a (unique) critical point $z_n$ of $f_n$ if and only if $F'$ contains a (unique) critical point $z$ of $f$ and, moreover, $\deg(f, z) = \deg(f_n, z_n)$.

(2) $\deg(f_n|F) < \deg(f_n|F')$. If $F$ contains a critical point of $f_n$, then it is unique in $F$. In this case $F'$ is either an asymptotic tract of $f$ or it contains a unique critical point $z$ of $f$ and, moreover, $\deg(f, z) > \deg(f_n, z_n)$.

With this discussion and the following example, we show that dynamically admissible quadruples provide a convenient way for constructing “combinatorial” approximations and thinking about combinatorial convergence.

*Example 3.16.* Consider the postsingularly finite entire map $G_2$ from Example 2.19, realizing the dynamically admissible quadruple $\Delta_{G_2} = \Delta(P_{G_2}, \mathcal{R}, G_2) = (P_{G_2}, \mathcal{R}, \Gamma, \Phi)$ depicted in Figure 5 along with a chosen point $b \in V(\Gamma)$. Define the graph $K_n \subset \Gamma$ to be a collection of all vertices and edges of $\Gamma$ accessible from $b$ via a path in $\Gamma$ intersecting interiors of at most $n$ edges. It is clear that the sequence $(K_n)$ is an exhaustion of $\Gamma$ by finite connected graphs. Starting with $\Delta_{G_2}$ and $(K_n)$ and applying the construction from the proof of Proposition 3.14, we obtain a sequence of polynomial Thurston maps $f_n: (\mathbb{R}^2, P_{G_2}) \hookrightarrow A$, $n \in \mathbb{N}$ converging combinatorially to $G_2: (\mathbb{C}, P_{G_2}) \hookrightarrow A$, with each $f_n$ defined by a dynamically admissible quadruple $\Delta_n = (P_{G_2}, \mathcal{R}, \Gamma_n, \Phi_n)$. Figure 7 illustrates the graphs $\Gamma_n$ and the maps $\Phi_n$ for $n=1, 2, 3$ (from left to right). As usual, $\Phi_n: \Gamma_n \to \mathcal{R}$ maps each edge of $\Gamma_n$ to the unique edge of $\mathcal{R}$ of the same color, and the set $P_{G_2}$ is represented by solid black squares.

However, combinatorial approximations need not to be polynomial. Consider the sequence of infinite graphs $(\hat{\Gamma}_n)$ depicted in Figure 8. For each $n$, we can similarly define a map $\hat{\Phi}_n: \hat{\Gamma}_n \to \mathcal{R}$ such that $\hat{\Delta}_n = (P_{G_2}, \mathcal{R}, \hat{\Gamma}_n, \hat{\Phi}_n)$ is a dynamically admissible quadruple (parabolicity of $\hat{\Delta}_n$ follows from [Cui21, Theorem 4.1]). Then $\hat{\Delta}_n$ determines a transcendental Thurston map $\hat{f}_n$ such that the sequence $(\hat{f}_n)$ converges combinatorially to $G_2: (\mathbb{C}, P_{G_2}) \hookrightarrow A$ as $n$ tends to $\infty$.

There is also no canonical choice for a sequence of polynomial Thurston maps converging combinatorially to a given transcendental Thurston map. For instance, the sequence of polynomial Thurston maps illustrated in Figure 9 (with respect to the rose graph showed at
$e_v$ might coincide). We shall construct an edge $e_F$ with endpoints at $v$ and $u$ (see Figure 6) with $e_F \subset e_u \cup e_v \cup F$, so that $e_F$ coincides with the edges $e_u$ and $e_v$ in small neighborhoods of $u$ and $v$, respectively. In particular, $\text{int}(e_F)$ does not intersect $K_n$. We add $e_F$ to $\Gamma_n$ and define $\Phi_n|_{e_F}$ so that $\Phi_n$ and $\Phi$ coincide on $e_F \cap e_u$ and $e_F \cap e_v$. Then we repeat the above procedure for all faces $F \in F(\Gamma)$ satisfying previously listed properties.

Now consider the sequence of constructed quadruples $\Delta_n = (A, \mathcal{R}, \Gamma_n, \Phi_n)$. Since $(K_n)$ is an exhaustion of $\Gamma$, there exists $N \in \mathbb{N}$ such that for each $n \ge N$, if $F \in F(\Gamma)$ is bounded and contains a point of $A$, then $\partial F \subset K_n$. For each $n \ge N$, it is straightforward to check that the conditions of Definitions 2.27 and 3.9 are satisfied. In particular, $\Delta_n$ is parabolic since $\Gamma_n$ is a finite graph. By Proposition 3.10, we can construct a polynomial Thurston map $f_n: (\mathbb{R}^2, A) \subsetneq F$ such that $\Delta(A, \mathcal{R}, f_n) = \Delta_n$ for each $n \ge N$. Thus, the sequence $(f_n)$ converges combinatorially to $f$ due to Proposition 3.13. $\square$

*Remark 3.15.* Suppose that we are in the setting of the proof of Proposition 3.14. Let $F$ be a face of $\Gamma_n$ labelled by a bounded face $P$ of $\mathcal{R}$ (with respect to the quadruple $\Delta_n$) for some $n \ge N$. Then there exists a unique face $F' \in F(\Gamma)$ having the same label $P$ (with respect to the quadruple $\Delta$) such that $F \subsetneq F'$. Moreover, exactly one of the following is true:

(1) $\deg(f_n|F) = \deg(f_n|F')$. Then Proposition 2.29 implies that $F$ contains a (unique) critical point $z_n$ of $f_n$ if and only if $F'$ contains a (unique) critical point $z$ of $f$ and, moreover, $\deg(f, z) = \deg(f_n, z_n)$.

(2) $\deg(f_n|F) < \deg(f_n|F')$. If $F$ contains a critical point of $f_n$, then it is unique in $F$. In this case $F'$ is either an asymptotic tract of $f$ or it contains a unique critical point $z$ of $f$ and, moreover, $\deg(f, z) > \deg(f_n, z_n)$.

With this discussion and the following example, we show that dynamically admissible quadruples provide a convenient way for constructing “combinatorial” approximations and thinking about combinatorial convergence.

*Example 3.16.* Consider the postsingularly finite entire map $G_2$ from Example 2.19, realizing the dynamically admissible quadruple $\Delta_{G_2} = \Delta(P_{G_2}, \mathcal{R}, G_2) = (P_{G_2}, \mathcal{R}, \Gamma, \Phi)$ depicted in Figure 5 along with a chosen point $b \in V(\Gamma)$. Define the graph $K_n \subset \Gamma$ to be a collection of all vertices and edges of $\Gamma$ accessible from $b$ via a path in $\Gamma$ intersecting interiors of at most $n$ edges. It is clear that the sequence $(K_n)$ is an exhaustion of $\Gamma$ by finite connected graphs. Starting with $\Delta_{G_2}$ and $(K_n)$ and applying the construction from the proof of Proposition 3.14, we obtain a sequence of polynomial Thurston maps $f_n: (\mathbb{R}^2, P_{G_2}) \subsetneq F, n \in \mathbb{N}$ converging combinatorially to $G_2: (\mathbb{C}, P_{G_2}) \subsetneq F$, with each $f_n$ defined by a dynamically admissible quadruple $\Delta_n = (P_{G_2}, \mathcal{R}, \Gamma_n, \Phi_n)$. Figure 7 illustrates the graphs $\Gamma_n$ and the maps $\Phi_n$ for $n=1, 2, 3$ (from left to right). As usual, $\Phi_n: \Gamma_n \to \mathcal{R}$ maps each edge of $\Gamma_n$ to the unique edge of $\mathcal{R}$ of the same color, and the set $P_{G_2}$ is represented by solid black squares.

However, combinatorial approximations need not to be polynomial. Consider the sequence of infinite graphs $(\hat{\Gamma}_n)$ depicted in Figure 8. For each $n$, we can similarly define a map $\hat{\Phi}_n: \hat{\Gamma}_n \to \mathcal{R}$ such that $\hat{\Delta}_n = (P_{G_2}, \mathcal{R}, \hat{\Gamma}_n, \hat{\Phi}_n)$ is a dynamically admissible quadruple (parabolicity of $\hat{\Delta}_n$ follows from [Cui21, Theorem 4.1]). Then $\hat{\Delta}_n$ determines a transcendental Thurston map $\hat{f}_n$ such that the sequence $(\hat{f}_n)$ converges combinatorially to $G_2: (\mathbb{C}, P_{G_2}) \subsetneq F$ as $n$ tends to $\infty$.

There is also no canonical choice for a sequence of polynomial Thurston maps converging combinatorially to a given transcendental Thurston map. For instance, the sequence of polynomial Thurston maps illustrated in Figure 9 (with respect to the rose graph showed at
<figure><img src="image_12.png" /><figcaption>FIGURE 7. Sequence of graphs that define polynomial Thurston maps converging combinatorially to $G_2$: $(\mathbb{C}, P_{G_2}) \subsetneq$, where $G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$ and $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.</figcaption></figure>

the top of Figure 4) converges combinatorially the map $G_1$: $(\mathbb{C}, P_{G_1}) \subsetneq$ from Example 2.19, which realizes dynamically admissible quadruple $\Delta_{G_1}$ as in Figure 4. However, the property of Remark 3.15 cannot be satisfied for these combinatorial approximations.

**3.4. Convergence of $\sigma$-maps.** We start this section by introducing the following result on quasiconformal maps [Kru87, Theorem 1].

**Theorem 3.17.** Let $\varphi: \mathbb{D} \to \mathbb{C}$ be an injective holomorphic map. Then for every $t \in \mathbb{D}^*$ the map $\varphi_t: \mathbb{D} \to \mathbb{C}$, where $\varphi_t(z) := \varphi(tz)/t$ for all $z \in \mathbb{D}$, admits a $K_t$-quasiconformal extension $\hat{\varphi}_t: \mathbb{C} \to \mathbb{C}$, where $K_t = (1 + |t|)/(1 - |t|)$.

The following corollary provides a slight reformulation of Theorem 3.17.

**Corollary 3.18.** Let $\varphi: \mathbb{D}_R \to \mathbb{C}$ be an injective holomorphic map. Then for every $r, 0 < r < R$, the map $\psi_r := \varphi|_{\mathbb{D}_r}: \mathbb{D}_r \to \mathbb{C}$ admits a $K_{r/R}$-quasiconformal extension $\hat{\psi}_r: \mathbb{C} \to \mathbb{C}$, where $K_{r/R} = (1 + r/R)/(1 - r/R)$.

*Proof.* Applying Theorem 3.17 with $t = r/R$ to the map $z \mapsto \varphi(Rz)$, where $z \in \mathbb{D}$, we see that the map $z \mapsto \frac{R}{r}\varphi(rz)$ defined on $\mathbb{D}$ admits a $K_{r/R}$-quasiconformal extension $\hat{\varphi}_{r/R}: \mathbb{C} \to \mathbb{C}$. Define $\hat{\psi}_r: \mathbb{C} \to \mathbb{C}$ as follows:

$$ \hat{\psi}_r(z) := \frac{r}{R} \hat{\varphi}_{r/R} \left( \frac{z}{r} \right) $$

Then one can see that $\hat{\psi}_r|_{\mathbb{D}_r} = \varphi|_{\mathbb{D}_r} = \psi_r$, and $\hat{\psi}_r$ is $K_{r/R}$-quasiconformal. $\square$

<figure><img src="image_13.png" /></figure>

Theorem 3.17 and Corollary 3.18 allow us to establish the following upper bound on the distance between two points of Teich$(\mathbb{R}^2, A)$ that satisfy certain properties.
<figure><img src="image_13.png" /><figcaption>FIGURE 7. Sequence of graphs that define polynomial Thurston maps converging combinatorially to $G_2$: $(\mathbb{C}, P_{G_2}) \subsetneq$, where $G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$ and $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.</figcaption></figure>

the top of Figure 4) converges combinatorially the map $G_1$: $(\mathbb{C}, P_{G_1}) \subsetneq$ from Example 2.19, which realizes dynamically admissible quadruple $\Delta_{G_1}$ as in Figure 4. However, the property of Remark 3.15 cannot be satisfied for these combinatorial approximations.

**3.4. Convergence of $\sigma$-maps.** We start this section by introducing the following result on quasiconformal maps [Kru87, Theorem 1].

**Theorem 3.17.** Let $\varphi: \mathbb{D} \to \mathbb{C}$ be an injective holomorphic map. Then for every $t \in \mathbb{D}^*$ the map $\varphi_t: \mathbb{D} \to \mathbb{C}$, where $\varphi_t(z) := \varphi(tz)/t$ for all $z \in \mathbb{D}$, admits a $K_t$-quasiconformal extension $\hat{\varphi}_t: \mathbb{C} \to \mathbb{C}$, where $K_t = (1 + |t|)/(1 - |t|)$.

The following corollary provides a slight reformulation of Theorem 3.17.

**Corollary 3.18.** Let $\varphi: \mathbb{D}_R \to \mathbb{C}$ be an injective holomorphic map. Then for every $r, 0 < r < R$, the map $\psi_r := \varphi|_{\mathbb{D}_r}: \mathbb{D}_r \to \mathbb{C}$ admits a $K_{r/R}$-quasiconformal extension $\hat{\psi}_r: \mathbb{C} \to \mathbb{C}$, where $K_{r/R} = (1 + r/R)/(1 - r/R)$.

*Proof.* Applying Theorem 3.17 with $t = r/R$ to the map $z \mapsto \varphi(Rz)$, where $z \in \mathbb{D}$, we see that the map $z \mapsto \frac{R}{r}\varphi(rz)$ defined on $\mathbb{D}$ admits a $K_{r/R}$-quasiconformal extension $\hat{\varphi}_{r/R}: \mathbb{C} \to \mathbb{C}$. Define $\hat{\psi}_r: \mathbb{C} \to \mathbb{C}$ as follows:

$$ \hat{\psi}_r(z) := \frac{r}{R} \hat{\varphi}_{r/R} \left( \frac{z}{r} \right) $$

Then one can see that $\hat{\psi}_r|_{\mathbb{D}_r} = \varphi|_{\mathbb{D}_r} = \psi_r$, and $\hat{\psi}_r$ is $K_{r/R}$-quasiconformal. $\square$

<figure><img src="image_14.png" /></figure>

Theorem 3.17 and Corollary 3.18 allow us to establish the following upper bound on the distance between two points of Teich$(\mathbb{R}^2, A)$ that satisfy certain properties.
<figure><img src="image_14.png" /><figcaption>FIGURE 8. Sequence of graphs that define transcendental Thurston maps converging combinatorially to $G_2: (\mathbb{C}, P_{G_2}) \subsetneq$, where
$G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$ and $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>FIGURE 8. Sequence of graphs that define transcendental Thurston maps converging combinatorially to $G_2: (\mathbb{C}, P_{G_2}) \subsetneq$, where
$G_2(z) = \sqrt{\ln 2}(1 - \exp(z^2))$ and $P_{G_2} = \{a_1, a_2, a_3\} = \{-\sqrt{\ln 2}, 0, \sqrt{\ln 2}\}$.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>FIGURE 9. Sequence of graphs that define polynomial Thurston maps converging combinatorially to $G_1$: $(\mathbb{C}, P_{G_1}) \subsetneq$, where $G_1(z) = \pi \cos(z)/2$ and $P_{G_1} = \{a_1, a_2, a_3\} = \{0, -\pi/2, \pi/2\}$.</figcaption></figure>

**Proposition 3.19.** Let $\tau_1 = [\varphi_1], \tau_2 = [\varphi_2]$ be points in Teich($\mathbb{R}^2, A$). Suppose that there exist positive real numbers $r$ and $R$ such that $0 < r < R$, $\varphi_1(A) \subset \mathbb{D}_r$, and $\varphi_2 \circ \varphi_1^{-1}$ is holomorphic on $\mathbb{D}_R$. Then

$$d(\tau_1, \tau_2) \le \log \left( \frac{1+r/R}{1-r/R} \right).$$

*Proof.* Let $\varphi := (\varphi_2 \circ \varphi_1^{-1})|_{\mathbb{D}_R}: \mathbb{D}_R \to \mathbb{C}$ be a holomorphic injective map. By Corollary 3.18, $\psi_r := \varphi|_{\mathbb{D}_r}$ extends $K_{r/R}$-quasiconformally to a map $\hat{\psi}_r$ on $\mathbb{C}$ (with $K_{r/R} = \frac{1+r/R}{1-r/R}$). One can see that $\hat{\psi}_r \circ \varphi_1$ and $\varphi_2$ coincide on $\varphi_1^{-1}(\mathbb{D}_r)$. In particular, these homeomorphisms are isotopic rel. A by Alexander's trick. Therefore, the inequality $d(\tau_1, \tau_2) \le \log K(\hat{\psi}_r)$ is satisfied. Since $\hat{\psi}_r$ is $K_{r/R}$-quasiconformal, we are done. $\square$

Now we are ready to establish Main Theorem B.

*Proof of Main Theorem B.* Since $(f_n)$ converges combinatorially to $f$, by Proposition 3.7 and Remark 2.23, we can assume without loss of generality that $(f_n)$ converges topologically to $f$.

Given $\tau \in \text{Teich}(\mathbb{R}^2, A)$, we first show that $\sigma_{f_n}(\tau) \to \sigma_f(\tau)$ as $n \to \infty$. Let $\tau = [\varphi], \sigma_f(\tau) = [\psi]$, and $\sigma_{f_n}(\tau) = [\psi_n]$, where $\varphi, \psi, \psi_n: \mathbb{R}^2 \to \mathbb{C}$ are orientation-preserving homeomorphisms for every $n \in \mathbb{N}$. By Definition 3.6, given any bounded open set $D \subset \mathbb{R}^2$, the equality $f_n|_D = f|_D$ holds for sufficiently large $n$. In particular, $\psi_n \circ \psi^{-1}$ is holomorphic on $\psi(D)$. Then Proposition 3.19 implies that $d(\sigma_f(\tau), \sigma_{f_n}(\tau)) \to 0$ as $n \to \infty$.
<figure><img src="image_16.png" /><figcaption>FIGURE 9. Sequence of graphs that define polynomial Thurston maps converging combinatorially to $G_1$: $(\mathbb{C}, P_{G_1}) \subsetneq$, where $G_1(z) = \pi \cos(z)/2$ and $P_{G_1} = \{a_1, a_2, a_3\} = \{0, -\pi/2, \pi/2\}$.</figcaption></figure>

**Proposition 3.19.** Let $\tau_1 = [\varphi_1], \tau_2 = [\varphi_2]$ be points in Teich($\mathbb{R}^2, A$). Suppose that there exist positive real numbers $r$ and $R$ such that $0 < r < R$, $\varphi_1(A) \subset \mathbb{D}_r$, and $\varphi_2 \circ \varphi_1^{-1}$ is holomorphic on $\mathbb{D}_R$. Then

$$d(\tau_1, \tau_2) \le \log \left( \frac{1 + r/R}{1 - r/R} \right).$$

*Proof.* Let $\varphi := (\varphi_2 \circ \varphi_1^{-1})|_{\mathbb{D}_R}: \mathbb{D}_R \to \mathbb{C}$ be a holomorphic injective map. By Corollary 3.18, $\psi_r := \varphi|_{\mathbb{D}_r}$ extends $K_{r/R}$-quasiconformally to a map $\hat{\psi}_r$ on $\mathbb{C}$ (with $K_{r/R} = \frac{1+r/R}{1-r/R}$). One can see that $\hat{\psi}_r \circ \varphi_1$ and $\varphi_2$ coincide on $\varphi_1^{-1}(\mathbb{D}_r)$. In particular, these homeomorphisms are isotopic rel. A by Alexander's trick. Therefore, the inequality $d(\tau_1, \tau_2) \le \log K(\hat{\psi}_r)$ is satisfied. Since $\hat{\psi}_r$ is $K_{r/R}$-quasiconformal, we are done. $\square$

Now we are ready to establish Main Theorem B.

*Proof of Main Theorem B.* Since $(f_n)$ converges combinatorially to $f$, by Proposition 3.7 and Remark 2.23, we can assume without loss of generality that $(f_n)$ converges topologically to $f$.

Given $\tau \in \text{Teich}(\mathbb{R}^2, A)$, we first show that $\sigma_{f_n}(\tau) \to \sigma_f(\tau)$ as $n \to \infty$. Let $\tau = [\varphi], \sigma_f(\tau) = [\psi]$, and $\sigma_{f_n}(\tau) = [\psi_n]$, where $\varphi, \psi, \psi_n: \mathbb{R}^2 \to \mathbb{C}$ are orientation-preserving homeomorphisms for every $n \in \mathbb{N}$. By Definition 3.6, given any bounded open set $D \subset \mathbb{R}^2$, the equality $f_n|_D = f|_D$ holds for sufficiently large $n$. In particular, $\psi_n \circ \psi^{-1}$ is holomorphic on $\psi(D)$. Then Proposition 3.19 implies that $d(\sigma_f(\tau), \sigma_{f_n}(\tau)) \to 0$ as $n \to \infty$.
Now we show that the convergence $\sigma_{f_n} \to \sigma_f$ is in fact locally uniform on $\text{Teich}(\mathbb{R}^2, A)$. Let $K \subset \text{Teich}(\mathbb{R}^2, A)$ be a compact set. Choose arbitrary $\varepsilon > 0$ and cover $K$ by open balls $B_1, B_2, \dots, B_k$ each of radius $\varepsilon/3$, with $B_j$ centered at $\mu_j \in \text{Teich}(\mathbb{R}^2, A)$ for each $j = 1, 2, \dots, k$. As we proved earlier, for sufficiently large $n$, $d(\sigma_{f_n}(\mu_j), \sigma_f(\mu_j)) \le \varepsilon/3$ for each $j = 1, 2, \dots, k$. Hence, for all $n$ large enough and any $\mu \in K$, we have

$$d(\sigma_{f_n}(\mu), \sigma_f(\mu)) \le d(\sigma_{f_n}(\mu), \sigma_{f_n}(\mu_j)) + d(\sigma_{f_n}(\mu_j), \sigma_f(\mu_j)) + d(\sigma_f(\mu_j), \sigma_f(\mu)) \le \varepsilon,$$

where $j$ is chosen so that $d(\mu, \mu_j) \le \varepsilon/3$. We also use the fact that all $\sigma$-maps above are 1-Lipschitz (see Proposition 2.22). $\square$

**Corollary 3.20.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{Thurston map}$. Then there exists a sequence of polynomial Thurston maps $f_n: (\mathbb{R}^2, A) \hookrightarrow \mathbb{N}$ such that the sequence $(\sigma_{f_n})$ converges locally uniformly on $\text{Teich}(\mathbb{R}^2, A)$ to $\sigma_f$.

*Proof.* This follows directly from Proposition 3.14 and Main Theorem B. $\square$

**Corollary 3.21.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{Thurston map}$ and $f_n: (\mathbb{R}^2, A) \hookrightarrow \mathbb{N}$ be a sequence of Thurston maps converging combinatorially to $f$. Then $f_n$ is realized for sufficiently large $n$. Moreover, letting $\tau_n \in \text{Teich}(\mathbb{R}^2, A)$ be a unique fixed point of $\sigma_{f_n}$, we have $\tau_n \to \tau$, where $\tau$ is the fixed point of $\sigma_f$.

*Proof.* If $f$ is a polynomial Thurston map, by Remark 2.23 and Proposition 3.4, the equality $\sigma_{f_n} = \sigma_f$ holds for sufficiently large $n$.

Now suppose that $f$ is transcendental. Let $B \subset \text{Teich}(\mathbb{R}^2, A)$ be a closed ball of radius $r > 0$ centered at $\tau$, the fixed point of $\sigma_f$. Since $\text{Teich}(\mathbb{R}^2, A)$ is a manifold, $B$ is compact for sufficiently small $r$. In fact, it is compact for any $r > 0$ since $\text{Teich}(\mathbb{R}^2, A)$ is a path metric space in the sense of [Gro07, Definition 1.7] (for the proof see [FM12, Section 11.8] and [Gro07, Sections 1.8bis and 1.8bis+]). By the metric version of Hopf-Rinow theorem [Gro07, page 8], every bounded closed set of $\text{Teich}(\mathbb{R}^2, A)$ is compact.

Proposition 2.22 implies existence of $\varepsilon_B > 0$ such that the inequality $d(\sigma_f(\mu_1), \sigma_f(\mu_2)) \le (1 - \varepsilon_B)d(\mu_1, \mu_2)$ is satisfied for all $\mu_1, \mu_2 \in B$. Thus, for any $\mu \in B$ we have the following:

$$(1) \qquad d(\sigma_{f_n}(\mu), \tau) \le d(\sigma_{f_n}(\mu), \sigma_f(\mu)) + d(\sigma_f(\mu), \tau) \le \varepsilon_n + (1 - \varepsilon_B)r,$$

where we know by Main Theorem B that $\varepsilon_n \to 0$ as $n \to \infty$. In other words, $\sigma_{f_n}(B) \subset B$ for all sufficiently large $n$. By Proposition 2.22, $\sigma_{f_n}^{\circ 2}$ is uniformly contracting on $B$ and, therefore, by the Banach fixed point theorem, when $n$ is large enough, the map $\sigma_{f_n}$ has a fixed point $\tau_n \in B$.

Lastly, similar to inequality (1), we have

$$d(\tau_n, \tau) \le d(\sigma_{f_n}(\tau_n), \sigma_f(\tau_n)) + d(\sigma_f(\tau_n), \sigma_f(\tau)) \le \varepsilon_n + (1 - \varepsilon_B)d(\tau_n, \tau).$$

This shows that the sequence $(\tau_n)$ converges to $\tau$. $\square$

## 4. APPROXIMATIONS OF POSTSINGULARLY FINITE ENTIRE MAPS

The main goal of this section is to prove Theorem 4.12, which is a stronger version of Main Theorem A. To do this, we first investigate combinatorial and topological properties of locally uniform convergence of sequences of finite type maps. In particular, we develop techniques of combinatorial nature for finding the limit of such a sequence, where all maps have the same number of singular values (see Theorem 4.9). This result and the methods developed in Section 3 are the key ingredients for proving the desired theorem.
Now we show that the convergence $\sigma_{f_n} \to \sigma_f$ is in fact locally uniform on $\text{Teich}(\mathbb{R}^2, A)$. Let $K \subset \text{Teich}(\mathbb{R}^2, A)$ be a compact set. Choose arbitrary $\varepsilon > 0$ and cover $K$ by open balls $B_1, B_2, \dots, B_k$ each of radius $\varepsilon/3$, with $B_j$ centered at $\mu_j \in \text{Teich}(\mathbb{R}^2, A)$ for each $j = 1, 2, \dots, k$. As we proved earlier, for sufficiently large $n$, $d(\sigma_{f_n}(\mu_j), \sigma_f(\mu_j)) \le \varepsilon/3$ for each $j = 1, 2, \dots, k$. Hence, for all $n$ large enough and any $\mu \in K$, we have

$$d(\sigma_{f_n}(\mu), \sigma_f(\mu)) \le d(\sigma_{f_n}(\mu), \sigma_{f_n}(\mu_j)) + d(\sigma_{f_n}(\mu_j), \sigma_f(\mu_j)) + d(\sigma_f(\mu_j), \sigma_f(\mu)) \le \varepsilon,$$

where $j$ is chosen so that $d(\mu, \mu_j) \le \varepsilon/3$. We also use the fact that all $\sigma$-maps above are 1-Lipschitz (see Proposition 2.22). $\square$

**Corollary 3.20.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{Thurston map}$. Then there exists a sequence of polynomial Thurston maps $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ such that the sequence $(\sigma_{f_n})$ converges locally uniformly on $\text{Teich}(\mathbb{R}^2, A)$ to $\sigma_f$.

*Proof.* This follows directly from Proposition 3.14 and Main Theorem B. $\square$

**Corollary 3.21.** Let $f: (\mathbb{R}^2, A) \hookrightarrow \text{Thurston map}$ and $f_n: (\mathbb{R}^2, A) \hookrightarrow, n \in \mathbb{N}$ be a sequence of Thurston maps converging combinatorially to $f$. Then $f_n$ is realized for sufficiently large $n$. Moreover, letting $\tau_n \in \text{Teich}(\mathbb{R}^2, A)$ be a unique fixed point of $\sigma_{f_n}$, we have $\tau_n \to \tau$, where $\tau$ is the fixed point of $\sigma_f$.

*Proof.* If $f$ is a polynomial Thurston map, by Remark 2.23 and Proposition 3.4, the equality $\sigma_{f_n} = \sigma_f$ holds for sufficiently large $n$.

Now suppose that $f$ is transcendental. Let $B \subset \text{Teich}(\mathbb{R}^2, A)$ be a closed ball of radius $r > 0$ centered at $\tau$, the fixed point of $\sigma_f$. Since $\text{Teich}(\mathbb{R}^2, A)$ is a manifold, $B$ is compact for sufficiently small $r$. In fact, it is compact for any $r > 0$ since $\text{Teich}(\mathbb{R}^2, A)$ is a path metric space in the sense of [Gro07, Definition 1.7] (for the proof see [FM12, Section 11.8] and [Gro07, Sections 1.8bis and 1.8bis+]). By the metric version of Hopf-Rinow theorem [Gro07, page 8], every bounded closed set of $\text{Teich}(\mathbb{R}^2, A)$ is compact.

Proposition 2.22 implies existence of $\varepsilon_B > 0$ such that the inequality $d(\sigma_f(\mu_1), \sigma_f(\mu_2)) \le (1 - \varepsilon_B)d(\mu_1, \mu_2)$ is satisfied for all $\mu_1, \mu_2 \in B$. Thus, for any $\mu \in B$ we have the following:

$$(1) \qquad d(\sigma_{f_n}(\mu), \tau) \le d(\sigma_{f_n}(\mu), \sigma_f(\mu)) + d(\sigma_f(\mu), \tau) \le \varepsilon_n + (1 - \varepsilon_B)r,$$

where we know by Main Theorem B that $\varepsilon_n \to 0$ as $n \to \infty$. In other words, $\sigma_{f_n}(B) \subset B$ for all sufficiently large $n$. By Proposition 2.22, $\sigma_{f_n}^{\circ 2}$ is uniformly contracting on $B$ and, therefore, by the Banach fixed point theorem, when $n$ is large enough, the map $\sigma_{f_n}$ has a fixed point $\tau_n \in B$.

Lastly, similar to inequality (1), we have

$$d(\tau_n, \tau) \le d(\sigma_{f_n}(\tau_n), \sigma_f(\tau_n)) + d(\sigma_f(\tau_n), \sigma_f(\tau)) \le \varepsilon_n + (1 - \varepsilon_B)d(\tau_n, \tau).$$

This shows that the sequence $(\tau_n)$ converges to $\tau$. $\square$

## 4. APPROXIMATIONS OF POSTSINGULARLY FINITE ENTIRE MAPS

The main goal of this section is to prove Theorem 4.12, which is a stronger version of Main Theorem A. To do this, we first investigate combinatorial and topological properties of locally uniform convergence of sequences of finite type maps. In particular, we develop techniques of combinatorial nature for finding the limit of such a sequence, where all maps have the same number of singular values (see Theorem 4.9). This result and the methods developed in Section 3 are the key ingredients for proving the desired theorem.
**4.1. Topological properties of locally uniform convergence.** Let $f: X \to Y$ be a map, then we denote its domain of definition $X$ by $\text{Dom}(f)$ and its image $f(X)$ by $\text{Rg}(f)$. Also, we define the *kernel* of a family $\{U_j\}_{j \in J}$ of subsets of $\mathbb{C}$ as the set of all points $z \in \mathbb{C}$ having an open neighborhood $V$ such that $V \subset U_j$ for all but finitely many $j \in J$.

Suppose that $X \subset \mathbb{C}$ and $(g_n)$ is a sequence of maps, where $\text{Dom}(g_n)$ and $\text{Rg}(g_n)$ are subsets of $\mathbb{C}$ for all $n \in \mathbb{N}$. The sequence $(g_n)$ is said to converge locally uniformly on $X$ if for every $z \in X$, there exists a neighbourhood $U$ of $z$ such that $U \subset \text{Dom}(g_n)$ for all $n \ge N = \text{N}(U)$ and $(g_n)_{n \ge N}$ converges uniformly on $U$. In particular, we require that $X$ is a subset of the kernel of $(\text{Dom}(g_n))$.

For the rest of the paper, when we refer to a holomorphic covering map $g$, we always assume that $\text{Dom}(g)$ and $\text{Rg}(g)$ are open subsets of $\mathbb{C}$. The following proposition is based on [Bar01, Theorem 1] and provides a convenient way of analyzing the convergence of holomorphic coverings.

**Proposition 4.1.** Let $(g_n)$ be a sequence of holomorphic covering maps. Suppose that $(g_n)$ is uniformly convergent on a neighbourhood of a point $z \in \mathbb{C}$ such that the limit function is not locally constant at $z$. Denote the connected component of $z$ in the kernel of $(\text{Dom}(g_n))$ by $X$ and the connected component of $w := \lim_{n \to \infty} g_n(z)$ in the kernel of $(\text{Rg}(g_n))$ by $Y$. Then $(g_n)$ converges locally uniformly on $X$ to a holomorphic covering map $g: X \to Y$.

The following result due to [Kis95, Theorem 1] and [KK97, Lemma 1] explains the behaviour of singular values of finite type maps that converge locally uniformly.

**Proposition 4.2.** Let $g_n, n \in \mathbb{N}$ and $g$ be entire maps of finite type such that the sequence $(g_n)$ converges locally uniformly to $g$. Suppose that $w$ is a singular value of $g$. Then for all $n$ sufficiently large, there exists a singular value $w_n$ of $g_n$ such that the sequence $(w_n)$ converges to $w$ as $n \to \infty$.

Let $g: X \to Y$ be a holomorphic map, where $X, Y \subset \mathbb{C}$ are open subsets. Suppose that $U \subset Y$ is an open set and $v \in X$ satisfies $g(v) \in U$. Then define $V$ as the connected component of $v$ in $g^{-1}(U)$. In the case when $g|V: V \to U$ is bijective, we denote its inverse by $\tilde{g}_{U,v}: U \to V$, i.e., $\tilde{g}_{U,v} = (g|V)^{-1}$.

Further assume that $g: X \to Y$ is a covering map and $U$ is an open Jordan region such that $\bar{U} \subset Y$. In particular, $g|V: V \to U$ is bijective and the map $\tilde{g}_{U,v}: U \to V$ is correctly defined, where $V$ is an open Jordan region satisfying $\bar{V} \subset X$.

Suppose that $(g_n)$ is a sequence of holomorphic covering maps converging locally uniformly on $X$ to $g$. Similarly to the previous case, we can define maps $\tilde{g}_{n,U,v}: V_n \to U$ that are inverses of $g|V_n: U \to V_n$, where $V_n$ is the connected component of $v$ in $g_n^{-1}(U)$. Now we are going to investigate relations between the maps $\tilde{g}_{n,U,v}$ and $\tilde{g}_{U,v}$.

**Lemma 4.3.** The sequence $(\tilde{g}_{n,U,v})$ converges uniformly on $U$ to $\tilde{g}_{U,v}$.

*Proof.* Choose an open Jordan region $U'$ such that $\bar{U} \subset U'$ and $\bar{U}' \subset Y$. Let $V'$ be a connected component of $v$ in $g^{-1}(U')$. Once again, $V'$ is an open Jordan region such that $\bar{V} \subset V'$ and $\bar{V}' \subset X$. Denote by $\gamma$ the simple closed curve that forms the boundary of $V'$. Since $g|\bar{V}'$ is injective, then $g(\gamma) \cap \bar{U} = \emptyset$. Therefore, for sufficiently large $n$, $g_n(\gamma) \cap U = \emptyset$. It follows that $\gamma \cap V_n = \emptyset$. Since $v \in V_n$ and the set $V_n$ is connected, we must have $V_n \subset V'$.

Now suppose that $(\tilde{g}_{n,U,v})$ does not converge uniformly on $U$ to $\tilde{g}_{U,v}$. Then we can find a real number $\varepsilon > 0$ such that for infinitely many $n$, there exists $u_n \in U$ with
**4.1. Topological properties of locally uniform convergence.** Let $f: X \to Y$ be a map, then we denote its domain of definition $X$ by $\text{Dom}(f)$ and its image $f(X)$ by $\text{Rg}(f)$. Also, we define the *kernel* of a family $\{U_j\}_{j \in J}$ of subsets of $\mathbb{C}$ as the set of all points $z \in \mathbb{C}$ having an open neighborhood $V$ such that $V \subset U_j$ for all but finitely many $j \in J$.

Suppose that $X \subset \mathbb{C}$ and $(g_n)$ is a sequence of maps, where $\text{Dom}(g_n)$ and $\text{Rg}(g_n)$ are subsets of $\mathbb{C}$ for all $n \in \mathbb{N}$. The sequence $(g_n)$ is said to converge locally uniformly on $X$ if for every $z \in X$, there exists a neighbourhood $U$ of $z$ such that $U \subset \text{Dom}(g_n)$ for all $n \ge N = \text{N}(U)$ and $(g_n)_{n \ge N}$ converges uniformly on $U$. In particular, we require that $X$ is a subset of the kernel of $(\text{Dom}(g_n))$.

For the rest of the paper, when we refer to a holomorphic covering map $g$, we always assume that $\text{Dom}(g)$ and $\text{Rg}(g)$ are open subsets of $\mathbb{C}$. The following proposition is based on [Bar01, Theorem 1] and provides a convenient way of analyzing the convergence of holomorphic coverings.

**Proposition 4.1.** Let $(g_n)$ be a sequence of holomorphic covering maps. Suppose that $(g_n)$ is uniformly convergent on a neighbourhood of a point $z \in \mathbb{C}$ such that the limit function is not locally constant at $z$. Denote the connected component of $z$ in the kernel of $(\text{Dom}(g_n))$ by $X$ and the connected component of $w := \lim_{n \to \infty} g_n(z)$ in the kernel of $(\text{Rg}(g_n))$ by $Y$. Then $(g_n)$ converges locally uniformly on $X$ to a holomorphic covering map $g: X \to Y$.

The following result due to [Kis95, Theorem 1] and [KK97, Lemma 1] explains the behaviour of singular values of finite type maps that converge locally uniformly.

**Proposition 4.2.** Let $g_n, n \in \mathbb{N}$ and $g$ be entire maps of finite type such that the sequence $(g_n)$ converges locally uniformly to $g$. Suppose that $w$ is a singular value of $g$. Then for all $n$ sufficiently large, there exists a singular value $w_n$ of $g_n$ such that the sequence $(w_n)$ converges to $w$ as $n \to \infty$.

Let $g: X \to Y$ be a holomorphic map, where $X, Y \subset \mathbb{C}$ are open subsets. Suppose that $U \subset Y$ is an open set and $v \in X$ satisfies $g(v) \in U$. Then define $V$ as the connected component of $v$ in $g^{-1}(U)$. In the case when $g|V: V \to U$ is bijective, we denote its inverse by $\tilde{g}_{U,v}: U \to V$, i.e., $\tilde{g}_{U,v} = (g|V)^{-1}$.

Further assume that $g: X \to Y$ is a covering map and $U$ is an open Jordan region such that $\bar{U} \subset Y$. In particular, $g|V: V \to U$ is bijective and the map $\tilde{g}_{U,v}: U \to V$ is correctly defined, where $V$ is an open Jordan region satisfying $\bar{V} \subset X$.

Suppose that $(g_n)$ is a sequence of holomorphic covering maps converging locally uniformly on $X$ to $g$. Similarly to the previous case, we can define maps $\tilde{g}_{n,U,v}: V_n \to U$ that are inverses of $g|V_n: U \to V_n$, where $V_n$ is the connected component of $v$ in $g_n^{-1}(U)$. Now we are going to investigate relations between the maps $\tilde{g}_{n,U,v}$ and $\tilde{g}_{U,v}$.

**Lemma 4.3.** The sequence $(\tilde{g}_{n,U,v})$ converges uniformly on $U$ to $\tilde{g}_{U,v}$.

*Proof.* Choose an open Jordan region $U'$ such that $\bar{U} \subset U'$ and $\bar{U}' \subset Y$. Let $V'$ be a connected component of $v$ in $g^{-1}(U')$. Once again, $V'$ is an open Jordan region such that $\bar{V} \subset V'$ and $\bar{V}' \subset X$. Denote by $\gamma$ the simple closed curve that forms the boundary of $V'$. Since $g|\bar{V}'$ is injective, then $g(\gamma) \cap \bar{U} = \emptyset$. Therefore, for sufficiently large $n$, $g_n(\gamma) \cap U = \emptyset$. It follows that $\gamma \cap V_n = \emptyset$. Since $v \in V_n$ and the set $V_n$ is connected, we must have $V_n \subset V'$.

Now suppose that $(\tilde{g}_{n,U,v})$ does not converge uniformly on $U$ to $\tilde{g}_{U,v}$. Then we can find a real number $\varepsilon > 0$ such that for infinitely many $n$, there exists $u_n \in U$ with
$|\tilde{g}_{n,U,v}(u_n) - \tilde{g}_{U,v}(u_n)| > \varepsilon$. Without loss of generality, we can assume that $(u_n)$ converges to a point $u \in \bar{U}$, and due to the previous observation, that $(\tilde{g}_{n,U,v}(u_n))$ converges to some point $z \in \bar{V}'$. Then the inequality $|z - \tilde{g}_{U',v}(u)| \ge \varepsilon$ holds. Since $g$ is injective on $\bar{V}'$, we have $g(z) \ne g(\tilde{g}_{U',v}(u))$, but at the same time,

$$g(z) = \lim_{n \to \infty} g_n(\tilde{g}_{n,U,v}(u_n)) = \lim_{n \to \infty} u_n = u = g(\tilde{g}_{U',v}(u)),$$

resulting in a contradiction. □

**Lemma 4.4.** The map $g_n$ is injective on $V$ for sufficiently large $n$. Moreover, there exists an open set $W$ containing $g(v)$ such that $W \subset g_n(V)$ for $n$ large enough.

*Proof.* As in the proof of Lemma 4.3, we choose an open Jordan region $U'$ such that $\bar{U} \subset U'$ and $\bar{U}' \subset Y$. Lemma 4.3 states that the sequence of maps $\tilde{g}_{n,U',v}: U' \to V'_n, n \in \mathbb{N}$ converges uniformly on $U'$ to the map $\tilde{g}_{U',v}: U' \to V'$, where $V'_n$ and $V'$ are the connected components of $v$ in $g^{-1}(U')$ and $g_n^{-1}(U')$, respectively. Therefore, since $\bar{V} \subset V'$, then $V$ is a subset of $V'_n$ for sufficiently large $n$, i.e., $g_n|_V$ is injective. Further take $W$ to be an arbitrary open set containing $g(v)$ such that $\bar{W} \subset U$. Then the desired statement follows from the locally uniform (on $V$) convergence of $(g_n)$ to $g$. □

Lemmas 4.3 and 4.4 allow us to obtain the following proposition that states convergence of lifts of the same loop under converging holomorphic covering maps.

**Proposition 4.5.** Let $(g_n)$ be a sequence of holomorphic covering maps converging locally uniformly on $X \subset \mathbb{C}$ to a holomorphic covering map $g: X \to Y$. Suppose that $g_n(z_0) = g(z_0) = w_0$ for some $z_0 \in X$ and $w_0 \in Y$. Let $\alpha$ be a loop in $Y$ based at $w_0$ and $\beta$ be its g-lift starting at $z_0$. Then $\alpha \subset \text{Rg}(g_n)$ for all sufficiently large $n$. In particular, the $g_n$-lift $\beta_n$ of $\alpha$ starting at $z_0$ is well-defined and, moreover, the following conditions hold:

(1) the sequence $(\beta_n)$ converges uniformly on $\mathbb{I}$ to $\beta$, i.e., for every $\varepsilon > 0$ we have $|\beta(t) - \beta_n(t)| < \varepsilon$ for all $t \in \mathbb{I}$ and all sufficiently large $n$;

(2) there exists $N = N(\alpha)$ such that for each $n \ge N$, $\beta_n$ is a loop if and only if $\beta$ is a loop.

*Proof.* Lemma 4.4 implies that for any $w \in Y$ there exists a neighbourhood of $w$ belonging to $\text{Rg}(g_n)$ for all but finitely many $n \in \mathbb{N}$. In other words, $Y$ is a subset of the kernel of $(\text{Rg}(g_n))$. In particular, there exists a neighbourhood of $\alpha$ belonging to $\text{Rg}(g_n) \cap Y$ for all sufficiently large $n$.

We choose open Jordan regions $V_1, V_2, \dots, V_k$ covering $\beta$ and a strictly increasing finite sequence $t_0 := 0, t_1, t_2, \dots, t_{k-1}, t_k := 1$ of points of $\mathbb{I}$ satisfying the following properties for each $j = 1, 2, \dots, k$:

* $\bar{V}_j \subset X$ and $g|_{\bar{V}_j}$ is injective;

* $\beta([t_{j-1}, t_j]) \subset V_j$.

Let $U_j := g(V_j)$ for each $j = 1, 2, \dots, k$. Assume that $n$ is large enough so that $\bar{V}_j \subset \text{Dom}(g_n)$ and $\bar{U}_j \subset \text{Rg}(g_n)$ for each $j = 1, 2, \dots, k$.

Next, we prove by induction on $l$ that $(\beta_n)$ converges to $\beta$ uniformly on $[0, t_l]$. The base case $l = 0$ is obvious since $\beta_n(0) = \beta(0) = z_0$. Now we prove that if $(\beta_n)$ converges to $\beta$ uniformly on $[0, t_l]$, then it does so uniformly on $[0, t_{l+1}]$. Indeed, since $\beta_n(t_l) \to \beta(t_l)$, due
$|\tilde{g}_{n,U,v}(u_n) - \tilde{g}_{U,v}(u_n)| > \varepsilon$. Without loss of generality, we can assume that $(u_n)$ converges to a point $u \in \bar{U}$, and due to the previous observation, that $(\tilde{g}_{n,U,v}(u_n))$ converges to some point $z \in \bar{V}'$. Then the inequality $|z - \tilde{g}_{U',v}(u)| \ge \varepsilon$ holds. Since $g$ is injective on $\bar{V}'$, we have $g(z) \ne g(\tilde{g}_{U',v}(u))$, but at the same time,

$$g(z) = \lim_{n \to \infty} g_n(\tilde{g}_{n,U,v}(u_n)) = \lim_{n \to \infty} u_n = u = g(\tilde{g}_{U',v}(u)),$$

resulting in a contradiction. □

**Lemma 4.4.** The map $g_n$ is injective on $V$ for sufficiently large $n$. Moreover, there exists an open set $W$ containing $g(v)$ such that $W \subset g_n(V)$ for $n$ large enough.

*Proof.* As in the proof of Lemma 4.3, we choose an open Jordan region $U'$ such that $\bar{U} \subset U'$ and $\bar{U}' \subset Y$. Lemma 4.3 states that the sequence of maps $\tilde{g}_{n,U',v}: U' \to V'_n, n \in \mathbb{N}$ converges uniformly on $U'$ to the map $\tilde{g}_{U',v}: U' \to V'$, where $V'_n$ and $V'$ are the connected components of $v$ in $g^{-1}(U')$ and $g_n^{-1}(U')$, respectively. Therefore, since $\bar{V} \subset V'$, then $V$ is a subset of $V'_n$ for sufficiently large $n$, i.e., $g_n|_V$ is injective. Further take $W$ to be an arbitrary open set containing $g(v)$ such that $\bar{W} \subset U$. Then the desired statement follows from the locally uniform (on $V$) convergence of $(g_n)$ to $g$. □

Lemmas 4.3 and 4.4 allow us to obtain the following proposition that states convergence of lifts of the same loop under converging holomorphic covering maps.

**Proposition 4.5.** Let $(g_n)$ be a sequence of holomorphic covering maps converging locally uniformly on $X \subset \mathbb{C}$ to a holomorphic covering map $g: X \to Y$. Suppose that $g_n(z_0) = g(z_0) = w_0$ for some $z_0 \in X$ and $w_0 \in Y$. Let $\alpha$ be a loop in $Y$ based at $w_0$ and $\beta$ be its g-lift starting at $z_0$. Then $\alpha \subset \text{Rg}(g_n)$ for all sufficiently large $n$. In particular, the $g_n$-lift $\beta_n$ of $\alpha$ starting at $z_0$ is well-defined and, moreover, the following conditions hold:

(1) the sequence $(\beta_n)$ converges uniformly on $\mathbb{I}$ to $\beta$, i.e., for every $\varepsilon > 0$ we have $|\beta(t) - \beta_n(t)| < \varepsilon$ for all $t \in \mathbb{I}$ and all sufficiently large $n$;

(2) there exists $N = N(\alpha)$ such that for each $n \ge N$, $\beta_n$ is a loop if and only if $\beta$ is a loop.

*Proof.* Lemma 4.4 implies that for any $w \in Y$ there exists a neighbourhood of $w$ belonging to $\text{Rg}(g_n)$ for all but finitely many $n \in \mathbb{N}$. In other words, $Y$ is a subset of the kernel of $(\text{Rg}(g_n))$. In particular, there exists a neighbourhood of $\alpha$ belonging to $\text{Rg}(g_n) \cap Y$ for all sufficiently large $n$.

We choose open Jordan regions $V_1, V_2, \dots, V_k$ covering $\beta$ and a strictly increasing finite sequence $t_0 := 0, t_1, t_2, \dots, t_{k-1}, t_k := 1$ of points of $\mathbb{I}$ satisfying the following properties for each $j = 1, 2, \dots, k$:

* $\bar{V}_j \subset X$ and $g|_{\bar{V}_j}$ is injective;

* $\beta([t_{j-1}, t_j]) \subset V_j$.

Let $U_j := g(V_j)$ for each $j = 1, 2, \dots, k$. Assume that $n$ is large enough so that $\bar{V}_j \subset \text{Dom}(g_n)$ and $\bar{U}_j \subset \text{Rg}(g_n)$ for each $j = 1, 2, \dots, k$.

Next, we prove by induction on $l$ that $(\beta_n)$ converges to $\beta$ uniformly on $[0, t_l]$. The base case $l = 0$ is obvious since $\beta_n(0) = \beta(0) = z_0$. Now we prove that if $(\beta_n)$ converges to $\beta$ uniformly on $[0, t_l]$, then it does so uniformly on $[0, t_{l+1}]$. Indeed, since $\beta_n(t_l) \to \beta(t_l)$, due
to Lemma 4.4, $\varphi_{n,U_l,\beta_n(t_l)} = \varphi_{n,U_l,\beta(t_l)}$ for sufficiently large $n$. Thus,

$$ \begin{aligned} \beta|[t_l, t_{l+1}] &= \varphi_{U_l, \beta(t_l)}(\alpha|[t_l, t_{l+1}]), \\ \beta_n|[t_l, t_{l+1}] &= \varphi_{n, U_l, \beta_n(t_l)}(\alpha|[t_l, t_{l+1}]) = \varphi_{n, U_l, \beta(t_l)}(\alpha|[t_l, t_{l+1}]). \end{aligned} $$

Finally, it follows from Lemma 4.3 that $\beta_n$ converges to $\beta$ uniformly on $[0, t_{l+1}]$, and this completes the proof of item (1).

Now assume that $\beta$ is a loop. By item (1), the point $\beta_n(1)$ converge to $z_0$ as $n \to \infty$. But by Lemma 4.4, there exists a neighbourhood $V$ of $z_0$ such that $g_n|V$ is injective for all sufficiently large $n$. Thus, it is not possible that $\beta_n(0)$ and $\beta_n(1)$ are different for any sufficiently large $n$, i.e., $\beta_n$ is a loop. If $\beta$ is a non-closed path, then similarly to the previous case, we have $\beta_n(1) \to \beta(1) \neq z_0$. Thus, we have $z_0 = \beta_n(0) \neq \beta_n(1)$ for $n$ large enough, and the proof of item (2) follows. $\square$

**Remark 4.6.** Note that in Lemmas 4.3, 4.4, and Proposition 4.5 we never used the fact that the maps $g_n$ and $g$ are holomorphic. This is why the corresponding results still hold when $g_n$ and $g$ are covering maps, where $\text{Dom}(g)$, $\text{Rg}(g)$, $\text{Dom}(g_n)$, and $\text{Rg}(g_n)$ are open subsets of $\mathbb{R}^2$.

**Corollary 4.7.** Let $g_n, n \in \mathbb{N}$ and $g$ be entire maps of finite type having the same number of singular values. Let $\gamma \subset \mathbb{C}\setminus S_g$ be a simple closed curve and $\tilde{\gamma}$ be a connected component of $g^{-1}(\gamma)$.

Suppose that $(g_n)$ converges locally uniformly to $g$. Then for any $z \in \tilde{\gamma}$ there exists $\varepsilon_{z,\gamma} > 0$ such that for all $n$ sufficiently large, there is a unique connected component $\tilde{\gamma}_n$ of $g_n^{-1}(\gamma)$ satisfying $d(z, \tilde{\gamma}_n) < \varepsilon_{z,\gamma}$. Moreover, the following is true for all $n$ large enough:

(1) if the degree of the covering $g|\tilde{\gamma}$ is finite, then

* $\tilde{\gamma}_n$ is a simple closed curve,

* $\text{deg}(g_n|\tilde{\gamma}_n) = \text{deg}(g|\tilde{\gamma})$, and

* there exists $N = N(\varepsilon)$ such that for $n \ge N$, we have $\tilde{\gamma}_n \subset U_\varepsilon(\tilde{\gamma})$, where $U_\varepsilon(X) := \{z \in \mathbb{C} : d(z, X) < \varepsilon\}$ is the $\varepsilon$-neighbourhood of the set $X \subset \mathbb{C}$.

(2) if the degree of the covering $g|\tilde{\gamma}$ is infinite (i.e., $\tilde{\gamma}$ is an unbounded curve), then $\text{deg}(g_n|\tilde{\gamma}_n) \to \infty$ as $n \to \infty$.

**Proof.** Since $g$ and $g_n$ have the same number of singular values, Proposition 4.2 implies that $S_{g_n}$ converges to $S_g$ in the sense of Hausdorff distance. We define holomorphic covering maps $h, h_n, n \in \mathbb{N}$, as $h := g|\mathbb{C}\setminus g^{-1}(S_g)$ and $h_n := g_n|\mathbb{C}\setminus g_n^{-1}(S_{g_n})$. Then it is clear that $(h_n)$ converges locally uniformly on $\mathbb{C}\setminus g^{-1}(S_g)$ to $h$.

Let $w := g(z)$ and $\alpha: \mathbb{I} \to \mathbb{R}^2$ be a parametrization of $\gamma$, i.e., a simple loop based at $w$ such that $\alpha(\mathbb{I}) = \gamma$. By Lemma 4.4 applied to the maps $h, h_n, n \in \mathbb{N}$, there exists an open bounded neighbourhood $V$ of $z$ such that for sufficiently large $n$, both $h$ and $h_n$ are injective on $V$. In particular, there exists a unique point $z_n \in V \cap g_n^{-1}(w)$. For all $n$ large enough, we take $\tilde{\gamma}_n$ to be the connected component of $g_n^{-1}(\gamma)$ intersecting $V$, which is unique due to Lemma 4.4. In particular, it implies the existence of $\varepsilon_{z,\gamma}$ with the desired properties. By Lemma 4.3, the sequence $(z_n)$ converges to $z$ as $n \to \infty$, and, for simplicity, after pre-composing $h_n$ with an appropriate translation, we can assume that $z = z_n$.

Let us consider the case when the degree $d := \text{deg}(g|\tilde{\gamma})$ is finite. It follows that in this case the loops $\alpha^j, 1 \le j < d$ do not lift under $g$ to a loop based at $z$, but $\alpha^d$ does so. Note that by Proposition 4.5, $\alpha \subset \text{Rg}(h_n)$ for all sufficiently large $n$. Therefore, we can define $\tilde{\alpha}$ and $\tilde{\alpha}_n$ as $g$- and $g_n$-lifts of $\alpha^d$ based at $z$, respectively. Due to Proposition 4.5, $\tilde{\alpha}_n$ is a simple loop based
to Lemma 4.4, $\varphi_{n,U_l,\beta_n(t_l)} = \varphi_{n,U_l,\beta(t_l)}$ for sufficiently large $n$. Thus,

$$ \beta|[t_l, t_{l+1}] = \varphi_{U_l, \beta(t_l)}(\alpha|[t_l, t_{l+1}]), \\ \beta_n|[t_l, t_{l+1}] = \varphi_{n, U_l, \beta_n(t_l)}(\alpha|[t_l, t_{l+1}]) = \varphi_{n, U_l, \beta(t_l)}(\alpha|[t_l, t_{l+1}]). $$

Finally, it follows from Lemma 4.3 that $\beta_n$ converges to $\beta$ uniformly on $[0, t_{l+1}]$, and this completes the proof of item (1).

Now assume that $\beta$ is a loop. By item (1), the point $\beta_n(1)$ converge to $z_0$ as $n \to \infty$. But by Lemma 4.4, there exists a neighbourhood $V$ of $z_0$ such that $g_n|V$ is injective for all sufficiently large $n$. Thus, it is not possible that $\beta_n(0)$ and $\beta_n(1)$ are different for any sufficiently large $n$, i.e., $\beta_n$ is a loop. If $\beta$ is a non-closed path, then similarly to the previous case, we have $\beta_n(1) \to \beta(1) \neq z_0$. Thus, we have $z_0 = \beta_n(0) \neq \beta_n(1)$ for $n$ large enough, and the proof of item (2) follows. $\square$

**Remark 4.6.** Note that in Lemmas 4.3, 4.4, and Proposition 4.5 we never used the fact that the maps $g_n$ and $g$ are holomorphic. This is why the corresponding results still hold when $g_n$ and $g$ are covering maps, where $\text{Dom}(g)$, $\text{Rg}(g)$, $\text{Dom}(g_n)$, and $\text{Rg}(g_n)$ are open subsets of $\mathbb{R}^2$.

**Corollary 4.7.** Let $g_n, n \in \mathbb{N}$ and $g$ be entire maps of finite type having the same number of singular values. Let $\gamma \subset \mathbb{C}\setminus S_g$ be a simple closed curve and $\tilde{\gamma}$ be a connected component of $g^{-1}(\gamma)$.

Suppose that $(g_n)$ converges locally uniformly to $g$. Then for any $z \in \tilde{\gamma}$ there exists $\varepsilon_{z,\gamma} > 0$ such that for all $n$ sufficiently large, there is a unique connected component $\tilde{\gamma}_n$ of $g_n^{-1}(\gamma)$ satisfying $d(z, \tilde{\gamma}_n) < \varepsilon_{z,\gamma}$. Moreover, the following is true for all $n$ large enough:

(1) if the degree of the covering $g|\tilde{\gamma}$ is finite, then

* $\tilde{\gamma}_n$ is a simple closed curve,

* $\deg(g_n|\tilde{\gamma}_n) = \deg(g|\tilde{\gamma})$, and

* there exists $N = N(\varepsilon)$ such that for $n \ge N$, we have $\tilde{\gamma}_n \subset U_\varepsilon(\tilde{\gamma})$, where $U_\varepsilon(X) := \{z \in \mathbb{C} : d(z, X) < \varepsilon\}$ is the $\varepsilon$-neighbourhood of the set $X \subset \mathbb{C}$.

(2) if the degree of the covering $g|\tilde{\gamma}$ is infinite (i.e., $\tilde{\gamma}$ is an unbounded curve), then $\deg(g_n|\tilde{\gamma}_n) \to \infty$ as $n \to \infty$.

**Proof.** Since $g$ and $g_n$ have the same number of singular values, Proposition 4.2 implies that $S_{g_n}$ converges to $S_g$ in the sense of Hausdorff distance. We define holomorphic covering maps $h, h_n, n \in \mathbb{N}$, as $h := g|\mathbb{C}\setminus g^{-1}(S_g)$ and $h_n := g_n|\mathbb{C}\setminus g_n^{-1}(S_{g_n})$. Then it is clear that $(h_n)$ converges locally uniformly on $\mathbb{C}\setminus g^{-1}(S_g)$ to $h$.

Let $w := g(z)$ and $\alpha: \mathbb{I} \to \mathbb{R}^2$ be a parametrization of $\gamma$, i.e., a simple loop based at $w$ such that $\alpha(\mathbb{I}) = \gamma$. By Lemma 4.4 applied to the maps $h, h_n, n \in \mathbb{N}$, there exists an open bounded neighbourhood $V$ of $z$ such that for sufficiently large $n$, both $h$ and $h_n$ are injective on $V$. In particular, there exists a unique point $z_n \in V \cap g_n^{-1}(w)$. For all $n$ large enough, we take $\tilde{\gamma}_n$ to be the connected component of $g_n^{-1}(\gamma)$ intersecting $V$, which is unique due to Lemma 4.4. In particular, it implies the existence of $\varepsilon_{z,\gamma}$ with the desired properties. By Lemma 4.3, the sequence $(z_n)$ converges to $z$ as $n \to \infty$, and, for simplicity, after pre-composing $h_n$ with an appropriate translation, we can assume that $z = z_n$.

Let us consider the case when the degree $d := \deg(g|\tilde{\gamma})$ is finite. It follows that in this case the loops $\alpha^j, 1 \le j < d$ do not lift under $g$ to a loop based at $z$, but $\alpha^d$ does so. Note that by Proposition 4.5, $\alpha \subset \text{Rg}(h_n)$ for all sufficiently large $n$. Therefore, we can define $\tilde{\alpha}$ and $\tilde{\alpha}_n$ as $g$- and $g_n$-lifts of $\alpha^d$ based at $z$, respectively. Due to Proposition 4.5, $\tilde{\alpha}_n$ is a simple loop based
at $z$ such that $(\tilde{\alpha}_n)$ converges uniformly on $\mathbb{I}$ to $\tilde{\alpha}$. In particular, $\tilde{\gamma}_n = \tilde{\alpha}_n(\mathbb{I})$ and, therefore, $\tilde{\gamma}_n \subset U_\varepsilon(\gamma)$ for any $\varepsilon > 0$ and sufficiently large $n$. Finally, item (2) of Proposition 4.5 implies that for $n$ large enough, the loops $\alpha^j, 1 \le j < d$ do not lift under $g_n$ to a closed loop based at $z$. Therefore, $\deg(g_n|\tilde{\gamma}_n) = \deg(g|\tilde{\gamma})$.

Now consider the case when the degree of the covering $g|\tilde{\gamma}$ is infinite. In this case, the loops $\alpha^j, j \in \mathbb{N}$ do not lift under $g$ to a loop based at $z$. Therefore, by item (2) of Proposition 4.5, the loop $\alpha^j$ does not lift under $g_n$ to a loop based at $z$ for sufficiently large $n$. In particular, $\deg(g_n|\tilde{\gamma}_n) \to \infty$ as $n \to \infty$. $\square$

**Remark 4.8.** Similarly to Remark 4.6, the result of Corollary 4.7 still holds if we assume that $g_n$ and $g$ are topologically holomorphic (and not necessarily holomorphic) maps of finite type.

Propositions 4.1 and 4.5 allow us to obtain the following topological criteria for the locally uniform convergence of finite type entire maps, all with the same number of singular values (the authors of [BPM10, Theorem 1] prove a result of the same nature in the setting of log-Riemann surfaces).

**Theorem 4.9.** Let $g_n, n \in \mathbb{N}$ and $g$ be entire maps of finite type and $B_n \supset S_{g_n}, B \supset S_g$ be finite subsets all of the same cardinality such that $(B_n)$ converges to $B$ in the sense of Hausdorff distance. Let $z_0, w_0 \in \mathbb{C}$ be points such that $w_0 \notin B \cup \bigcup_{n \in \mathbb{N}} B_n$ and $g(z_0) = g_n(z_0) = w_0$ for all $n \in \mathbb{N}$. Then $(g_n)$ converges locally uniformly to $g$ if and only if the following conditions hold:

(1) $\lim_{n \to \infty} g'_n(z_0) = g'(z_0)$;

(2) for every loop $\alpha \subset \mathbb{C}\setminus B$ based at $w_0$, for all sufficiently large $n$, $\alpha$ lifts under $g_n$ to a loop based at $z_0$ if and only if it lifts under $g$ to a loop based at $z_0$.

*Proof.* If the sequence $(g_n)$ converges locally uniformly to $g$, then condition (1) is obvious and condition (2) easily follows from Proposition 4.5 applied to the sequence of holomorphic covering maps $h_n := g_n|\mathbb{C}\setminus g_n^{-1}(B_n)$ converging locally uniformly on $\mathbb{C}\setminus g^{-1}(B)$ to the holomorphic covering map $g|\mathbb{C}\setminus g^{-1}(B)$.

Now suppose that conditions (1) and (2) are satisfied. It suffices to show that an arbitrary subsequence of $(g_n)$ contains a further subsequence converging locally uniformly to $g$. For simplicity's sake, we will call the initial subsequence $(g_n)$.

**Claim 1.** There exist an open neighborhood $W$ of $z_0$ and a subsequence $(g_{n_k})$ of $(g_n)$ converging uniformly on $W$ to a limiting function that is not locally constant at $z_0$.

*Proof of Claim 1.* Let $\mathbb{D}(w_0, r)$ be a disk contained in $\mathbb{C}\setminus B_n$ for all sufficiently large $n$, and $V_n$ be the connected component of $z_0$ in $g_n^{-1}(\mathbb{D}(w_0, r))$. By Proposition 2.11, $g_n|_{V_n}: V_n \to \mathbb{D}(w_0, r)$ is a biholomorphism and has an inverse $\varphi_n: \mathbb{D}(w_0, r) \to V_n$ satisfying $\varphi_n(w_0) = z_0$. Note that $\varphi'_n(w_0) = 1/g'_n(z_0)$ and, since $g'_n(z_0)$ converges to $g'(z_0)$, we can assume that there exists $\lambda > 0$ such that $|\varphi'_n(w_0)| > \lambda$ for all sufficiently large $n$. Now the Koebe 1/4-theorem implies that the disk $W := \mathbb{D}(z_0, r\lambda/4)$ is contained in $V_n$ for all sufficiently large $n$. In particular, the map $g_n$ is injective on $W$ for all $n$ large enough.

Since $g_n(W) \subset \mathbb{D}(w_0, r)$, thus, the sequence $(g_n|W)$ forms a normal family by Montel's theorem. Hence, we can extract a converging subsequence $(g_{n_k}|W)$ from $(g_n|W)$. Clearly, the limiting function cannot be locally constant at $z_0$ since $\lim_{n \to \infty} g'_n(z_0) = g'(z_0) \neq 0$. $\blacksquare$

Once again, we relabel $(g_{n_k})$ as $(g_n)$ to assume that $(g_n)$ converges uniformly in a neighborhood of $z_0$ to a function that is non-constant at $z_0$. Now we apply Proposition 4.1 to the
at $z$ such that $(\tilde{\alpha}_n)$ converges uniformly on $\mathbb{I}$ to $\tilde{\alpha}$. In particular, $\tilde{\gamma}_n = \tilde{\alpha}_n(\mathbb{I})$ and, therefore, $\tilde{\gamma}_n \subset U_\epsilon(\gamma)$ for any $\epsilon > 0$ and sufficiently large $n$. Finally, item (2) of Proposition 4.5 implies that for $n$ large enough, the loops $\alpha^j, 1 \le j < d$ do not lift under $g_n$ to a closed loop based at $z$. Therefore, $\deg(g_n|\tilde{\gamma}_n) = \deg(g|\tilde{\gamma})$.

Now consider the case when the degree of the covering $g|\tilde{\gamma}$ is infinite. In this case, the loops $\alpha^j, j \in \mathbb{N}$ do not lift under $g$ to a loop based at $z$. Therefore, by item (2) of Proposition 4.5, the loop $\alpha^j$ does not lift under $g_n$ to a loop based at $z$ for sufficiently large $n$. In particular, $\deg(g_n|\tilde{\gamma}_n) \to \infty$ as $n \to \infty$. $\square$

**Remark 4.8.** Similarly to Remark 4.6, the result of Corollary 4.7 still holds if we assume that $g_n$ and $g$ are topologically holomorphic (and not necessarily holomorphic) maps of finite type.

Propositions 4.1 and 4.5 allow us to obtain the following topological criteria for the locally uniform convergence of finite type entire maps, all with the same number of singular values (the authors of [BPM10, Theorem 1] prove a result of the same nature in the setting of log-Riemann surfaces).

**Theorem 4.9.** Let $g_n, n \in \mathbb{N}$ and $g$ be entire maps of finite type and $B_n \supset S_{g_n}, B \supset S_g$ be finite subsets all of the same cardinality such that $(B_n)$ converges to $B$ in the sense of Hausdorff distance. Let $z_0, w_0 \in \mathbb{C}$ be points such that $w_0 \notin B \cup \bigcup_{n \in \mathbb{N}} B_n$ and $g(z_0) = g_n(z_0) = w_0$ for all $n \in \mathbb{N}$. Then $(g_n)$ converges locally uniformly to $g$ if and only if the following conditions hold:

(1) $\lim_{n \to \infty} g'_n(z_0) = g'(z_0)$;

(2) for every loop $\alpha \subset \mathbb{C}\setminus B$ based at $w_0$, for all sufficiently large $n$, $\alpha$ lifts under $g_n$ to a loop based at $z_0$ if and only if it lifts under $g$ to a loop based at $z_0$.

*Proof.* If the sequence $(g_n)$ converges locally uniformly to $g$, then condition (1) is obvious and condition (2) easily follows from Proposition 4.5 applied to the sequence of holomorphic covering maps $h_n := g_n|\mathbb{C}\setminus g_n^{-1}(B_n)$ converging locally uniformly on $\mathbb{C}\setminus g^{-1}(B)$ to the holomorphic covering map $g|\mathbb{C}\setminus g^{-1}(B)$.

Now suppose that conditions (1) and (2) are satisfied. It suffices to show that an arbitrary subsequence of $(g_n)$ contains a further subsequence converging locally uniformly to $g$. For simplicity's sake, we will call the initial subsequence $(g_n)$.

**Claim 1.** There exist an open neighborhood $W$ of $z_0$ and a subsequence $(g_{n_k})$ of $(g_n)$ converging uniformly on $W$ to a limiting function that is not locally constant at $z_0$.

*Proof of Claim 1.* Let $\mathbb{D}(w_0, r)$ be a disk contained in $\mathbb{C}\setminus B_n$ for all sufficiently large $n$, and $V_n$ be the connected component of $z_0$ in $g_n^{-1}(\mathbb{D}(w_0, r))$. By Proposition 2.11, $g_n|_{V_n}: V_n \to \mathbb{D}(w_0, r)$ is a biholomorphism and has an inverse $\varphi_n: \mathbb{D}(w_0, r) \to V_n$ satisfying $\varphi_n(w_0) = z_0$. Note that $\varphi'_n(w_0) = 1/g'_n(z_0)$ and, since $g'_n(z_0)$ converges to $g'(z_0)$, we can assume that there exists $\lambda > 0$ such that $|\varphi'_n(w_0)| > \lambda$ for all sufficiently large $n$. Now the Koebe 1/4-theorem implies that the disk $W := \mathbb{D}(z_0, r\lambda/4)$ is contained in $V_n$ for all sufficiently large $n$. In particular, the map $g_n$ is injective on $W$ for all $n$ large enough.

Since $g_n(W) \subset \mathbb{D}(w_0, r)$, thus, the sequence $(g_n|W)$ forms a normal family by Montel's theorem. Hence, we can extract a converging subsequence $(g_{n_k}|W)$ from $(g_n|W)$. Clearly, the limiting function cannot be locally constant at $z_0$ since $\lim_{n \to \infty} g'_n(z_0) = g'(z_0) \neq 0$. $\blacksquare$

Once again, we relabel $(g_{n_k})$ as $(g_n)$ to assume that $(g_n)$ converges uniformly in a neighbourhood of $z_0$ to a function that is non-constant at $z_0$. Now we apply Proposition 4.1 to the
sequence of holomorphic coverings $h_n = g_n|\mathbb{C}\g_n^{-1}(B_n)$: letting $X \subset \mathbb{C}$ be the connected component of $z_0$ in the kernel of $(\mathbb{C}\g_n^{-1}(B_n))$, we see that $(h_n)$ converges locally uniformly on $X$ to a holomorphic covering map $h: X \to \mathbb{C}\setminus B$. We will now show that $g$ and $h$ coincide on $X$.

To begin with, we observe that $h_*\pi_1(X, z_0) = g_*\pi_1(\mathbb{C}\g_n^{-1}(B), z_0) \subset \pi_1(\mathbb{C}\setminus B, w_0)$, which follows from condition (2) and Proposition 4.5. By the classical theory of covering maps and condition (1), there exists a biholomorphism $\varphi: \mathbb{C}\g_n^{-1}(B) \to X$ such that $g = h \circ \varphi$ on $\mathbb{C}\g_n^{-1}(B)$, $\varphi(z_0) = z_0$, and $\varphi'(z_0) = 1$.

**Claim 2.** The map $\varphi: \mathbb{C}\g_n^{-1}(B) \to X$ extends to a Möbius transformation.

*Proof of Claim 2.* As $g_n^{-1}(B)$ is discrete and $\varphi$ is injective, every point in $g_n^{-1}(B)$ is a removable singularity or pole of $\varphi$. Hence, we can extend $\varphi$ to a meromorphic map from $\mathbb{C}$ to $\hat{\mathbb{C}}$. It is easy to see that $\varphi$ still has to be injective. But the only domains in $\hat{\mathbb{C}}$ that are biholomorphic to $\mathbb{C}$ are obtained from $\hat{\mathbb{C}}$ by removing one point. Thus, by the classification of biholomorphisms of $\mathbb{C}$, we see that $\varphi$ has to be a Möbius transformation. ■

Note that Claim 2 implies that $X$ is obtained from $\hat{\mathbb{C}}$ by removing countably many points which have at most one accumulation point.

**Claim 3.** The map $\varphi$ equals $\text{id}_{\hat{\mathbb{C}}}$. ■

*Proof of Claim 3.* Let us first prove that $\varphi(\infty) = \infty$. Suppose that $\varphi(\infty) = z \neq \infty$. We choose a compact set $K \subset \mathbb{C}$ such that $z \in \text{int}(K)$ and $\partial K \subset X$. Note that $(g_n)$ converges to $h$ uniformly on $\partial K$. Let $m \in (0, +\infty)$ be the maximum of $|h|$ on $\partial K$. Then by the maximum modulus principle, for any $\varepsilon > 0$, sufficiently large $n$, and any $z \in K$, we have $|g_n(z)| \le m + \varepsilon$. Thus, $|h|$ is bounded by $m$ on $K \cap X$.

On the other hand, one can find a sequence $(z_n) \subset \varphi^{-1}(K)$ such that $z_n \to \infty$ and $g(z_n) \to \infty$ as $n \to \infty$. At the same time, $(\varphi(z_n)) \subset X \cap K$ is a sequence converging to $z$. This means that $|g(z_n)| = |h(\varphi(z_n))|$ is bounded by $m$ for all $n \in \mathbb{N}$, leading to a contradiction. Hence, we have shown that $\varphi(\infty) = \infty$.

Thus, $\varphi$ is an affine map satisfying $\varphi(z_0) = z_0$ and $\varphi'(z_0) = 1$, and hence, $\varphi = \text{id}_{\hat{\mathbb{C}}}$. ■

Claim 3 implies that $g$ coincides with $h$ on $X = \mathbb{C}\g_n^{-1}(B)$. In particular, $(g_n)$ converges locally uniformly on $\mathbb{C}\g_n^{-1}(B)$ to $g$. This is enough to conclude that $(g_n)$ converges locally uniformly (on $\mathbb{C}$) to $g$. □

*Remark 4.10.* Note that existence of the map $g$ in Theorem 4.9 is essential. For instance, consider an arbitrary sequence of polynomials $(g_n)$ satisfying the following properties:

(1) $g_n(0) = 0$ and $g'_n(0) = 1$ for every $n \in \mathbb{N}$;

(2) $S_{g_n} = \{-1, 1\}$ for every $n \in \mathbb{N}$;

(3) $\lim_{n \to \infty} (g_n)_* \pi_1(\mathbb{C}\g_n^{-1}(S_{g_n}), 0)$ is trivial.

Suppose that the sequence $(g_n)$ converges locally uniformly (at least up to a subsequence) on $\mathbb{C}$ to an entire map $g$. Proposition 4.2 implies that $S_g \subset B := \{-1, 1\}$. Then one can easily observe that the sequence of holomorphic covering maps $(g_n|\mathbb{C}\g_n^{-1}(B))$ converges to the holomorphic covering map $g|\mathbb{C}\g_n^{-1}(B)$. Therefore, it follows from Proposition 4.5 that $g_*\pi_1(\mathbb{C}\g_n^{-1}(B), 0)$ is trivial and $g|\mathbb{C}\g_n^{-1}(B)$ is a universal covering of $\mathbb{C}\setminus B$. It leads to a contradiction since $\mathbb{C}\g_n^{-1}(B)$ cannot be biholomorphic to $\mathbb{D}$.

Now we outline the construction of such a sequence $(g_n)$ as above but leave technical details to the reader. We can start from a holomorphic universal covering map $h: \mathbb{D} \to \mathbb{C}\setminus\{-1, 1\}$.
sequence of holomorphic coverings $h_n = g_n|\mathbb{C}\g_n^{-1}(B_n)$: letting $X \subset \mathbb{C}$ be the connected component of $z_0$ in the kernel of $(\mathbb{C}\g_n^{-1}(B_n))$, we see that $(h_n)$ converges locally uniformly on $X$ to a holomorphic covering map $h: X \to \mathbb{C}\setminus B$. We will now show that $g$ and $h$ coincide on $X$.

To begin with, we observe that $h_*\pi_1(X, z_0) = g_*\pi_1(\mathbb{C}\g_n^{-1}(B), z_0) \subset \pi_1(\mathbb{C}\setminus B, w_0)$, which follows from condition (2) and Proposition 4.5. By the classical theory of covering maps and condition (1), there exists a biholomorphism $\varphi: \mathbb{C}\g_n^{-1}(B) \to X$ such that $g = h \circ \varphi$ on $\mathbb{C}\g_n^{-1}(B)$, $\varphi(z_0) = z_0$, and $\varphi'(z_0) = 1$.

**Claim 2.** The map $\varphi: \mathbb{C}\g_n^{-1}(B) \to X$ extends to a Möbius transformation.

*Proof of Claim 2.* As $\g_n^{-1}(B)$ is discrete and $\varphi$ is injective, every point in $\g_n^{-1}(B)$ is a removable singularity or pole of $\varphi$. Hence, we can extend $\varphi$ to a meromorphic map from $\mathbb{C}$ to $\hat{\mathbb{C}}$. It is easy to see that $\varphi$ still has to be injective. But the only domains in $\hat{\mathbb{C}}$ that are biholomorphic to $\mathbb{C}$ are obtained from $\hat{\mathbb{C}}$ by removing one point. Thus, by the classification of biholomorphisms of $\mathbb{C}$, we see that $\varphi$ has to be a Möbius transformation. ■

Note that Claim 2 implies that $X$ is obtained from $\hat{\mathbb{C}}$ by removing countably many points which have at most one accumulation point.

**Claim 3.** The map $\varphi$ equals $\text{id}_{\hat{\mathbb{C}}}$. ■

*Proof of Claim 3.* Let us first prove that $\varphi(\infty) = \infty$. Suppose that $\varphi(\infty) = z \neq \infty$. We choose a compact set $K \subset \mathbb{C}$ such that $z \in \text{int}(K)$ and $\partial K \subset X$. Note that $(g_n)$ converges to $h$ uniformly on $\partial K$. Let $m \in (0, +\infty)$ be the maximum of $|h|$ on $\partial K$. Then by the maximum modulus principle, for any $\varepsilon > 0$, sufficiently large $n$, and any $z \in K$, we have $|g_n(z)| \le m + \varepsilon$. Thus, $|h|$ is bounded by $m$ on $K \cap X$.

On the other hand, one can find a sequence $(z_n) \subset \varphi^{-1}(K)$ such that $z_n \to \infty$ and $g(z_n) \to \infty$ as $n \to \infty$. At the same time, $(\varphi(z_n)) \subset X \cap K$ is a sequence converging to $z$. This means that $|g(z_n)| = |h(\varphi(z_n))|$ is bounded by $m$ for all $n \in \mathbb{N}$, leading to a contradiction. Hence, we have shown that $\varphi(\infty) = \infty$.

Thus, $\varphi$ is an affine map satisfying $\varphi(z_0) = z_0$ and $\varphi'(z_0) = 1$, and hence, $\varphi = \text{id}_{\hat{\mathbb{C}}}$. ■

Claim 3 implies that $g$ coincides with $h$ on $X = \mathbb{C}\g_n^{-1}(B)$. In particular, $(g_n)$ converges locally uniformly on $\mathbb{C}\g_n^{-1}(B)$ to $g$. This is enough to conclude that $(g_n)$ converges locally uniformly (on $\mathbb{C}$) to $g$. $\square$

*Remark 4.10.* Note that existence of the map $g$ in Theorem 4.9 is essential. For instance, consider an arbitrary sequence of polynomials $(g_n)$ satisfying the following properties:

(1) $g_n(0) = 0$ and $g'_n(0) = 1$ for every $n \in \mathbb{N}$;

(2) $S_{g_n} = \{-1, 1\}$ for every $n \in \mathbb{N}$;

(3) $\lim_{n \to \infty} (g_n)_* \pi_1(\mathbb{C}\g_n^{-1}(S_{g_n}), 0)$ is trivial.

Suppose that the sequence $(g_n)$ converges locally uniformly (at least up to a subsequence) on $\mathbb{C}$ to an entire map $g$. Proposition 4.2 implies that $S_g \subset B := \{-1, 1\}$. Then one can easily observe that the sequence of holomorphic covering maps $(g_n|\mathbb{C}\g_n^{-1}(B))$ converges to the holomorphic covering map $g|\mathbb{C}\g_n^{-1}(B)$. Therefore, it follows from Proposition 4.5 that $g_*\pi_1(\mathbb{C}\g_n^{-1}(B), 0)$ is trivial and $g|\mathbb{C}\g_n^{-1}(B)$ is a universal covering of $\mathbb{C}\setminus B$. It leads to a contradiction since $\mathbb{C}\g_n^{-1}(B)$ cannot be biholomorphic to $\mathbb{D}$.

Now we outline the construction of such a sequence $(g_n)$ as above but leave technical details to the reader. We can start from a holomorphic universal covering map $h: \mathbb{D} \to \mathbb{C}\setminus\{-1, 1\}$.
Using the framework of admissible quadruples (Sections 2.7 and 3.3) and following the proof of Proposition 3.14 applied to $h$ (viewed as a topologically holomorphic map from $\mathbb{R}^2$ to $\mathbb{C}$), one can construct a sequence of topological polynomials $f_n : \mathbb{R}^2 \to \mathbb{C}$ such that $S_{f_n} = \{-1, 1\}$ and $\lim_{n \to \infty} (f_n)_* \pi_1(\mathbb{R}^2 \setminus f_n^{-1}(S_{f_n})), 0)$ is trivial. Finally, due to Proposition 2.6 we can find a sequence of orientation-preserving homeomorphisms $\varphi_n : \mathbb{R}^2 \to \mathbb{C}$ so that $g_n := f_n \circ \varphi_n^{-1}$ is a complex polynomial satisfying the properties (1), (2), and (3).

**Example 4.11.** In this example, we apply Theorem 4.9 to a particular sequence of entire maps of finite type. Let $g_n(z) = (1+z/n)^n, n \in \mathbb{N}$ and $g(z) = \exp(z)$. Note that $S_{g_n} = S_g = \{0\}$, $g_n(0) = g(0) = 1$, and $g'_n(0) = g'(0) = 1$ for all $n \in \mathbb{N}$. Let $\alpha$ be a simple loop based at $w_0 = 1$ such that 0 is contained in the bounded connected component of $\mathbb{C}\setminus\alpha$. Then $(g_n)_* \pi_1(\mathbb{C}\setminus\{-n\}, 0)$ is an infinite cyclic subgroup of $\pi_1(\mathbb{C}\setminus\{0\}, w_0)$ generated by $[\alpha^n]$ and $g_* \pi_1(\mathbb{C}, 0)$ is trivial. In other words, we can apply Theorem 4.9 to obtain the classical locally uniform limit $\exp(z) = \lim_{n \to \infty} (1+z/n)^n$.

## 4.2. Dynamical approximations.
Now we are ready to formulate and prove the main result of this section, which we will then use to prove Main Theorem A.

**Theorem 4.12.** Let $f_n: (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$, $n \in \mathbb{N}$ be a sequence of Thurston map converging combinatorially to a Thurston map $f: (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$. If $g: (\mathbb{C}, B) \hookrightarrow \mathbb{C}$ is a postsingularly finite entire map Thurston equivalent to $f: (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$, then there exists a sequence of postsingularly finite entire maps $g_n: (\mathbb{R}^2, B_n) \hookrightarrow \mathbb{C}$ such that

(1) $g_n$ converges locally uniformly to $g$;

(2) $B_n$ converges to $B$ in the sense of Hausdorff distance;

(3) $g_n: (\mathbb{C}, B_n) \hookrightarrow \mathbb{C}$ is Thurston equivalent to $f_n: (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$ for sufficiently large $n$.

*Proof.* We can assume that $(f_n)$ converges topologically to $f$ due to Remark 2.23 and Proposition 3.7. In particular, without loss of generality we suppose that there are two points $b \in \mathbb{R}^2$ and $t \in \mathbb{R}^2 \setminus A$ such that $f(b) = f_n(b) = t$ for all $n \in \mathbb{N}$.

Since $f$ is Thurston equivalent to $g$, it follows from Definition 2.15 and Proposition 2.26 that there exists a unique point $\tau \in \text{Teich}(\mathbb{R}^2, A)$ such that $\tau = [\varphi] = [\psi]$, $g = \varphi \circ f \circ \psi^{-1}$, and $B = \varphi(A) = \psi(A)$, where $\varphi: \mathbb{R}^2 \to \mathbb{C}$ and $\psi: \mathbb{R}^2 \to \mathbb{C}$ are orientation-preserving homeomorphisms isotopic rel. $A$ to each other.

By Corollary 3.21, there exists $N \in \mathbb{N}$ such that for all $n \ge N$ the map $\sigma_{f_n}$ has a unique fixed point $\tau_n \in \text{Teich}(\mathbb{R}^2, A)$ and, moreover, the sequence $(\tau_n)$ converges to $\tau$. Without loss of generality, we assume $N=1$. Let $\varphi_n: \mathbb{R}^2 \to \mathbb{C}$ and $\psi_n: \mathbb{R}^2 \to \mathbb{C}$ be orientation-preserving homeomorphisms such that $\tau_n = [\varphi_n] = [\psi_n]$. Then $h_n := \varphi_n \circ f_n \circ \psi_n^{-1}$ is an entire map and the set $B_n := \varphi_n(A)$ contains its singular set $S_{h_n}$. Due to the definition of Teichmüller distance (see Section 2.5) and Proposition 2.21, we can choose $\varphi_n$ and $\psi_n$ so that they satisfy the following conditions for all $n \in \mathbb{N}$:

* $\varphi_n \circ \varphi^{-1}: \mathbb{C} \to \mathbb{C}$ is quasiconformal, and the dilations $K(\varphi_n \circ \varphi^{-1})$ converge to 1 as $n \to \infty$,

* $\varphi_n(t) = \varphi(t)$ and $\varphi_n(y) = \varphi(y)$ for some point $y \ne t$,

* $\psi_n(b) = \psi(b)$ and $h'_n(\psi_n(b)) = g'(\psi(b))$.

We note that $h_n$ is not necessarily postsingularly finite under this choice of $\varphi_n$ and $\psi_n$.

It is evident that for each $n$, the homeomorphism $\varphi_n \circ \varphi^{-1}$ fixes two distinct points $\varphi(t)$ and $\varphi(y)$. Therefore, by Proposition 2.20 the sequence $(\varphi_n)$ converges locally uniformly to $\varphi$ as $n$ tends to infinity.
Using the framework of admissible quadruples (Sections 2.7 and 3.3) and following the proof of Proposition 3.14 applied to $h$ (viewed as a topologically holomorphic map from $\mathbb{R}^2$ to $\mathbb{C}$), one can construct a sequence of topological polynomials $f_n : \mathbb{R}^2 \to \mathbb{C}$ such that $S_{f_n} = \{-1, 1\}$ and $\lim_{n \to \infty} (f_n)_* \pi_1(\mathbb{R}^2 \setminus f_n^{-1}(S_{f_n})), 0)$ is trivial. Finally, due to Proposition 2.6 we can find a sequence of orientation-preserving homeomorphisms $\varphi_n : \mathbb{R}^2 \to \mathbb{C}$ so that $g_n := f_n \circ \varphi_n^{-1}$ is a complex polynomial satisfying the properties (1), (2), and (3).

**Example 4.11.** In this example, we apply Theorem 4.9 to a particular sequence of entire maps of finite type. Let $g_n(z) = (1+z/n)^n, n \in \mathbb{N}$ and $g(z) = \exp(z)$. Note that $S_{g_n} = S_g = \{0\}$, $g_n(0) = g(0) = 1$, and $g'_n(0) = g'(0) = 1$ for all $n \in \mathbb{N}$. Let $\alpha$ be a simple loop based at $w_0 = 1$ such that 0 is contained in the bounded connected component of $\mathbb{C}\setminus\alpha$. Then $(g_n)_* \pi_1(\mathbb{C}\setminus\{-n\}, 0)$ is an infinite cyclic subgroup of $\pi_1(\mathbb{C}\setminus\{0\}, w_0)$ generated by $[\alpha^n]$ and $g_* \pi_1(\mathbb{C}, 0)$ is trivial. In other words, we can apply Theorem 4.9 to obtain the classical locally uniform limit $\exp(z) = \lim_{n \to \infty} (1+z/n)^n$.

**4.2. Dynamical approximations.** Now we are ready to formulate and prove the main result of this section, which we will then use to prove Main Theorem A.

**Theorem 4.12.** Let $f_n : (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$, $n \in \mathbb{N}$ be a sequence of Thurston map converging combinatorially to a Thurston map $f : (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$. If $g : (\mathbb{C}, B) \hookrightarrow \mathbb{C}$ is a postsingularly finite entire map Thurston equivalent to $f : (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$, then there exists a sequence of postsingularly finite entire maps $g_n : (\mathbb{R}^2, B_n) \hookrightarrow \mathbb{C}$ such that

(1) $g_n$ converges locally uniformly to $g$;

(2) $B_n$ converges to $B$ in the sense of Hausdorff distance;

(3) $g_n : (\mathbb{C}, B_n) \hookrightarrow \mathbb{C}$ is Thurston equivalent to $f_n : (\mathbb{R}^2, A) \hookrightarrow \mathbb{C}$ for sufficiently large $n$.

*Proof.* We can assume that $(f_n)$ converges topologically to $f$ due to Remark 2.23 and Proposition 3.7. In particular, without loss of generality we suppose that there are two points $b \in \mathbb{R}^2$ and $t \in \mathbb{R}^2 \setminus A$ such that $f(b) = f_n(b) = t$ for all $n \in \mathbb{N}$.

Since $f$ is Thurston equivalent to $g$, it follows from Definition 2.15 and Proposition 2.26 that there exists a unique point $\tau \in \text{Teich}(\mathbb{R}^2, A)$ such that $\tau = [\varphi] = [\psi]$, $g = \varphi \circ f \circ \psi^{-1}$, and $B = \varphi(A) = \psi(A)$, where $\varphi : \mathbb{R}^2 \to \mathbb{C}$ and $\psi : \mathbb{R}^2 \to \mathbb{C}$ are orientation-preserving homeomorphisms isotopic rel. $A$ to each other.

By Corollary 3.21, there exists $N \in \mathbb{N}$ such that for all $n \ge N$ the map $\sigma_{f_n}$ has a unique fixed point $\tau_n \in \text{Teich}(\mathbb{R}^2, A)$ and, moreover, the sequence $(\tau_n)$ converges to $\tau$. Without loss of generality, we assume $N=1$. Let $\varphi_n : \mathbb{R}^2 \to \mathbb{C}$ and $\psi_n : \mathbb{R}^2 \to \mathbb{C}$ be orientation-preserving homeomorphisms such that $\tau_n = [\varphi_n] = [\psi_n]$. Then $h_n := \varphi_n \circ f_n \circ \psi_n^{-1}$ is an entire map and the set $B_n := \varphi_n(A)$ contains its singular set $S_{h_n}$. Due to the definition of Teichmüller distance (see Section 2.5) and Proposition 2.21, we can choose $\varphi_n$ and $\psi_n$ so that they satisfy the following conditions for all $n \in \mathbb{N}$:

* $\varphi_n \circ \varphi^{-1} : \mathbb{C} \to \mathbb{C}$ is quasiconformal, and the dilations $K(\varphi_n \circ \varphi^{-1})$ converge to 1 as $n \to \infty$,

* $\varphi_n(t) = \varphi(t)$ and $\varphi_n(y) = \varphi(y)$ for some point $y \ne t$,

* $\psi_n(b) = \psi(b)$ and $h'_n(\psi_n(b)) = g'(\psi(b))$.

We note that $h_n$ is not necessarily postsingularly finite under this choice of $\varphi_n$ and $\psi_n$.

It is evident that for each $n$, the homeomorphism $\varphi_n \circ \varphi^{-1}$ fixes two distinct points $\varphi(t)$ and $\varphi(y)$. Therefore, by Proposition 2.20 the sequence $(\varphi_n)$ converges locally uniformly to $\varphi$ as $n$ tends to infinity.
*Claim 1.* The sequence $(h_n)$ converges locally uniformly to $g$.

*Proof of Claim 1.* We prove that the sequence $(h_n)$ and the map $g$ satisfy all conditions of Theorem 4.9 with respect to the points $z_0 := \psi(b) = \psi_n(b)$, $w_0 := \varphi(t) = \varphi_n(t)$, and the sets $B_n$ and $B$.

Indeed, $B_n \supset S_{h_n}$ converges to $B \supset S_g$ in the Hausdorff topology of $\mathbb{C}$ since $\varphi_n \to \varphi$ as $n \to \infty$. Due to our choices of $\varphi_n$ and $\psi_n$, the equalities $h'_n(z_0) = g'(z_0)$ and $h_n(z_0) = g(z_0) = w_0$ are satisfied, and we have $w_0 \notin B \cup \bigcup_{n \in \mathbb{N}} B_n$. Hence, condition (1) of Theorem 4.9 is satisfied for the maps $h_n$. The condition (2) required in Theorem 4.9 easily follows from the topological convergence of $(f_n)$ to $f$ and the locally uniform convergence of $(\varphi_n)$ to $\varphi$. ■

*Claim 2.* The sequence $(\psi_n)$ converges locally uniformly to $\psi$.

*Proof of Claim 2.* We prove that the sequence $(\psi_n^{-1})$ converges locally uniformly to $\psi^{-1}$. Let $z \in \mathbb{C}g^{-1}(B)$ be a point such that $\psi_n^{-1}(z) \to \psi^{-1}(z)$ as $n \to \infty$ (for instance, the point $z_0 := \psi_n(b) = \psi(b)$ satisfies this property).

Take an arbitrary bounded open neighbourhood $V \subset \mathbb{C}g^{-1}(B)$ of $z$ such that $g$ is injective on $V$. Due to the previous claim and Lemma 4.4 applied to the sequence $(h_n|\mathbb{C}h_n^{-1}(B_n))$ and the map $g|\mathbb{C}g^{-1}(B)$, we have that $V \subset \mathbb{C}h_n^{-1}(B_n)$ and $h_n|V$ is injective for $n$ large enough. Therefore, the maps $f|\psi^{-1}(V): \psi^{-1}(V) \to \varphi^{-1}(g(V))$ and $f_n|\psi_n^{-1}(V): \psi_n^{-1}(V) \to \varphi_n^{-1}(h_n(V))$ are homeomorphisms. Note that for sufficiently large $n$, sets $\psi^{-1}(V)$ and $\psi_n^{-1}(V)$ intersect, since $\psi^{-1}(V)$ contain $\psi_n^{-1}(z)$ for $n$ large enough.

Thus,

$$ \begin{aligned} \psi^{-1}|V &= (f|\psi^{-1}(V))^{-1} \circ \varphi^{-1} \circ (g|V), \\ \psi_n^{-1}|V &= (f|\psi_n^{-1}(V))^{-1} \circ \varphi_n^{-1} \circ (h_n|V). \end{aligned} $$

Since $h_n \to g$ and $\varphi_n \to \varphi$ locally uniformly (on $\mathbb{C}$ and $\mathbb{R}^2$, respectively) and $f_n \to f$ topologically as $n \to \infty$, we have that $(\psi_n^{-1})$ converges locally uniformly on $V$ to $\psi^{-1}$. Applying the same argument repeatedly (starting from $z = z_0$), we show that $(\psi_n^{-1})$ converges to $\psi^{-1}$ locally uniformly on $\mathbb{C}g^{-1}(B)$. It is enough to conclude that $\psi_n \to \psi$ locally uniformly on $\mathbb{R}^2$ as $n \to \infty$. ■

Since $[\varphi_n] = [\psi_n]$, there exists an affine map $M_n$ such that $\varphi_n|A$ and $M_n \circ \psi_n|A$ coincide. Then $g_n := h_n \circ M_n^{-1}: (\mathbb{C}, B_n) \subsetneq \mathbb{C}$ is a postsingularly finite entire map. It suffices to show that $(M_n)$ converges locally uniformly to $\mathrm{id}_{\mathbb{C}}$ to prove that the sequence $(g_n)$ converges locally uniformly to $g$.

Further we can assume that $|A| \ge 2$; otherwise, by Proposition 2.17 the maps $f_n$ and $f$ are Thurston equivalent to $z \mapsto z^d$ with the same $d \ge 2$ for all sufficiently large $n$. Note that $M_n$ is an affine map satisfying $M_n(\psi_n(a)) = \varphi_n(a)$ for every $n \in \mathbb{N}$ and $a \in A$. Now the desired statement follows from the fact that for every $a \in A$, $\psi_n(a) \to \psi(a) = \varphi(a)$ and $\varphi_n(a) \to \varphi(a) = \psi(a)$ as $n \to \infty$. □

The next corollary is an immediate consequence of the proof of Theorem 4.12.

**Corollary 4.13.** Suppose that we are in the setting of Theorem 4.12. Assume that the sequence $(f_n)$ converges topologically to the map $f$ and $g = \varphi \circ f \circ \psi^{-1}$, where $\varphi: \mathbb{R}^2 \to \mathbb{C}$ and $\psi: \mathbb{R}^2 \to \mathbb{C}$ are orientation-preserving homeomorphisms isotopic rel. A to each other. Then for sufficiently large $n$ there exist orientation-preserving homeomorphisms $\varphi_n: \mathbb{R}^2 \to \mathbb{C}$ and $\psi_n: \mathbb{R}^2 \to \mathbb{C}$ such that $g_n = \varphi_n \circ f_n \circ \psi_n^{-1}$, $\varphi_n$ is isotopic rel. A to $\psi_n$, $\varphi_n \to \varphi$ and $\psi_n \to \psi$ locally uniformly as $n \to \infty$.
*Claim 1.* The sequence $(h_n)$ converges locally uniformly to $g$.

*Proof of Claim 1.* We prove that the sequence $(h_n)$ and the map $g$ satisfy all conditions of Theorem 4.9 with respect to the points $z_0 := \psi(b) = \psi_n(b)$, $w_0 := \varphi(t) = \varphi_n(t)$, and the sets $B_n$ and $B$.

Indeed, $B_n \supset S_{h_n}$ converges to $B \supset S_g$ in the Hausdorff topology of $\mathbb{C}$ since $\varphi_n \to \varphi$ as $n \to \infty$. Due to our choices of $\varphi_n$ and $\psi_n$, the equalities $h'_n(z_0) = g'(z_0)$ and $h_n(z_0) = g(z_0) = w_0$ are satisfied, and we have $w_0 \notin B \cup \bigcup_{n \in \mathbb{N}} B_n$. Hence, condition (1) of Theorem 4.9 is satisfied for the maps $h_n$. The condition (2) required in Theorem 4.9 easily follows from the topological convergence of $(f_n)$ to $f$ and the locally uniform convergence of $(\varphi_n)$ to $\varphi$. ■

*Claim 2.* The sequence $(\psi_n)$ converges locally uniformly to $\psi$.

*Proof of Claim 2.* We prove that the sequence $(\psi_n^{-1})$ converges locally uniformly to $\psi^{-1}$. Let $z \in \mathbb{C}\g^{-1}(B)$ be a point such that $\psi_n^{-1}(z) \to \psi^{-1}(z)$ as $n \to \infty$ (for instance, the point $z_0 := \psi_n(b) = \psi(b)$ satisfies this property).

Take an arbitrary bounded open neighbourhood $V \subset \mathbb{C}\g^{-1}(B)$ of $z$ such that $g$ is injective on $V$. Due to the previous claim and Lemma 4.4 applied to the sequence $(h_n|\mathbb{C}\hbar_n^{-1}(B_n))$ and the map $g|\mathbb{C}\g^{-1}(B)$, we have that $V \subset \mathbb{C}\hbar_n^{-1}(B_n)$ and $h_n|V$ is injective for $n$ large enough. Therefore, the maps $f|\psi^{-1}(V): \psi^{-1}(V) \to \varphi^{-1}(g(V))$ and $f_n|\psi_n^{-1}(V): \psi_n^{-1}(V) \to \varphi_n^{-1}(h_n(V))$ are homeomorphisms. Note that for sufficiently large $n$, sets $\psi^{-1}(V)$ and $\psi_n^{-1}(V)$ intersect, since $\psi^{-1}(V)$ contain $\psi_n^{-1}(z)$ for $n$ large enough.

Thus,

$$ \begin{aligned} \psi^{-1}|V &= (f|\psi^{-1}(V))^{-1} \circ \varphi^{-1} \circ (g|V), \\ \psi_n^{-1}|V &= (f|\psi_n^{-1}(V))^{-1} \circ \varphi_n^{-1} \circ (h_n|V). \end{aligned} $$

Since $h_n \to g$ and $\varphi_n \to \varphi$ locally uniformly (on $\mathbb{C}$ and $\mathbb{R}^2$, respectively) and $f_n \to f$ topologically as $n \to \infty$, we have that $(\psi_n^{-1})$ converges locally uniformly on $V$ to $\psi^{-1}$. Applying the same argument repeatedly (starting from $z = z_0$), we show that $(\psi_n^{-1})$ converges to $\psi^{-1}$ locally uniformly on $\mathbb{C}\g^{-1}(B)$. It is enough to conclude that $\psi_n \to \psi$ locally uniformly on $\mathbb{R}^2$ as $n \to \infty$. ■

Since $[\varphi_n] = [\psi_n]$, there exists an affine map $M_n$ such that $\varphi_n|A$ and $M_n \circ \psi_n|A$ coincide. Then $g_n := h_n \circ M_n^{-1}: (\mathbb{C}, B_n) \subsetneqno$ is a postsingularly finite entire map. It suffices to show that $(M_n)$ converges locally uniformly to $\mathrm{id}_{\mathbb{C}}$ to prove that the sequence $(g_n)$ converges locally uniformly to $g$.

Further we can assume that $|A| \ge 2$; otherwise, by Proposition 2.17 the maps $f_n$ and $f$ are Thurston equivalent to $z \mapsto z^d$ with the same $d \ge 2$ for all sufficiently large $n$. Note that $M_n$ is an affine map satisfying $M_n(\psi_n(a)) = \varphi_n(a)$ for every $n \in \mathbb{N}$ and $a \in A$. Now the desired statement follows from the fact that for every $a \in A$, $\psi_n(a) \to \psi(a) = \varphi(a)$ and $\varphi_n(a) \to \varphi(a) = \psi(a)$ as $n \to \infty$. □

The next corollary is an immediate consequence of the proof of Theorem 4.12.

**Corollary 4.13.** Suppose that we are in the setting of Theorem 4.12. Assume that the sequence $(f_n)$ converges topologically to the map $f$ and $g = \varphi \circ f \circ \psi^{-1}$, where $\varphi: \mathbb{R}^2 \to \mathbb{C}$ and $\psi: \mathbb{R}^2 \to \mathbb{C}$ are orientation-preserving homeomorphisms isotopic rel. A to each other. Then for sufficiently large $n$ there exist orientation-preserving homeomorphisms $\varphi_n: \mathbb{R}^2 \to \mathbb{C}$ and $\psi_n: \mathbb{R}^2 \to \mathbb{C}$ such that $g_n = \varphi_n \circ f_n \circ \psi_n^{-1}$, $\varphi_n$ is isotopic rel. A to $\psi_n$, $\varphi_n \to \varphi$ and $\psi_n \to \psi$ locally uniformly as $n \to \infty$.
The following result establishes Main Theorem A from the introduction and easily follows
from Proposition 3.14 and Theorem 4.12.

**Corollary 4.14.** Let $g$ be a postsingularly finite entire map. Then there exists a sequence of postcritically finite polynomials $(g_n)$ converging locally uniformly to $g$ such that $g_n$ has the same singular portrait as $g$ for every $n \in \mathbb{N}$.

*Remark 4.15.* Note that usually there is no canonical choice for a sequence of polynomials $(g_n)$ in Corollary 4.14. Constructing different “combinatorial approximations” by Proposition 3.14 and then applying Theorem 4.12 lead to different sequences of polynomials.

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[BF14] B. Branner and N. Fagella. *Quasiconformal surgery in holomorphic dynamics*, volume 141 of *Cambridge Studies in Advanced Mathematics*. Cambridge University Press, Cambridge, 2014. With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Haïssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky.

[BM17] Mario Bonk and Daniel Meyer. *Expanding Thurston maps*, volume 225 of *Mathematical Surveys and Monographs*. American Mathematical Society, Providence, RI, 2017.

[BPM10] Kingshook Biswas and Ricardo Perez-Marco. Caratheodory convergence of log-Riemann surfaces and Euler's formula. *Preprint arXiv:1011.0535*, 2010.

[Cha50] Claude Chabauty. Limite d'ensembles et géométrie des nombres. *Bull. Soc. Math. France*, 78:143–151, 1950.

[Cui21] Weiwei Cui. Entire functions arising from trees. *Sci. China Math.*, 64(10):2231–2248, 2021.

[DH84] A. Douady and J.H. Hubbard. *Étude dynamique des polynômes complexes. Partie I*, volume 84 of *Publications Mathématiques d'Orsay [Mathematical Publications of Orsay]*. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.

[DH93] A. Douady and J.H. Hubbard. A proof of Thurston's topological characterization of rational functions. *Acta Mathematica*, 171(2):263–297, 1993.

[Die05] R. Diestel. *Graph theory*. Graduate Texts in Math. 173. Springer, Berlin, 3rd edition, 2005.

[Ere04] A. Eremenko. Geometric theory of meromorphic functions. In *In the tradition of Ahlfors and Bers, III*, volume 355 of *Contemp. Math.*, pages 221–230. Amer. Math. Soc., Providence, RI, 2004.

[ERG15] Adam Epstein and Lasse Rempe-Gillen. On invariance of order and the area property for finite-type entire functions. *Ann. Acad. Sci. Fenn. Math.*, 40(2):573–599, 2015.

[FKK$^{+22}$] W. Floyd, D. Kim, S. Koch, W. Parry, and E. Saenz. Realizing polynomial portraits. *Preprint arXiv:2105.10055*, 2022.

[FM12] B. Farb and D. Margalit. *A primer on mapping class groups*. Princeton Math. Ser. 49. Princeton Univ. Press, Princeton, NJ, 2012.

[For91] Otto Forster. *Lectures on Riemann surfaces*, volume 81 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 1991. Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation.
The following result establishes Main Theorem A from the introduction and easily follows
from Proposition 3.14 and Theorem 4.12.

**Corollary 4.14.** Let $g$ be a postsingularly finite entire map. Then there exists a sequence of postcritically finite polynomials $(g_n)$ converging locally uniformly to $g$ such that $g_n$ has the same singular portrait as $g$ for every $n \in \mathbb{N}$.

*Remark 4.15.* Note that usually there is no canonical choice for a sequence of polynomials $(g_n)$ in Corollary 4.14. Constructing different “combinatorial approximations” by Proposition 3.14 and then applying Theorem 4.12 lead to different sequences of polynomials.

REFERENCES

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[BM17] Mario Bonk and Daniel Meyer. *Expanding Thurston maps*, volume 225 of *Mathematical Surveys and Monographs*. American Mathematical Society, Providence, RI, 2017.

[BPM10] Kingshook Biswas and Ricardo Perez-Marco. Caratheodory convergence of log-Riemann surfaces and Euler's formula. *Preprint arXiv:1011.0535*, 2010.

[Cha50] Claude Chabauty. Limite d'ensembles et géométrie des nombres. *Bull. Soc. Math. France*, 78:143–151, 1950.

[Cui21] Weiwei Cui. Entire functions arising from trees. *Sci. China Math.*, 64(10):2231–2248, 2021.

[DH84] A. Douady and J.H. Hubbard. *Étude dynamique des polynômes complexes. Partie I*, volume 84 of *Publications Mathématiques d'Orsay [Mathematical Publications of Orsay]*. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.

[DH93] A. Douady and J.H. Hubbard. A proof of Thurston's topological characterization of rational functions. *Acta Mathematica*, 171(2):263–297, 1993.

[Die05] R. Diestel. *Graph theory*. Graduate Texts in Math. 173. Springer, Berlin, 3rd edition, 2005.

[Ere04] A. Eremenko. Geometric theory of meromorphic functions. In *In the tradition of Ahlfors and Bers, III*, volume 355 of *Contemp. Math.*, pages 221–230. Amer. Math. Soc., Providence, RI, 2004.

[ERG15] Adam Epstein and Lasse Rempe-Gillen. On invariance of order and the area property for finite-type entire functions. *Ann. Acad. Sci. Fenn. Math.*, 40(2):573–599, 2015.

[FKK$^{+22}$] W. Floyd, D. Kim, S. Koch, W. Parry, and E. Saenz. Realizing polynomial portraits. *Preprint arXiv:2105.10055*, 2022.

[FM12] B. Farb and D. Margalit. *A primer on mapping class groups*. Princeton Math. Ser. 49. Princeton Univ. Press, Princeton, NJ, 2012.

[For91] Otto Forster. *Lectures on Riemann surfaces*, volume 81 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 1991. Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation.
[GO08] Anatoly A. Goldberg and Iossif V. Ostrovskii. *Value distribution of meromorphic functions*, volume 236 of *Translations of Mathematical Monographs*. American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii, With an appendix by Alexandre Eremenko and James K. Langley.

[Gro07] Misha Gromov. *Metric structures for Riemannian and non-Riemannian spaces*. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, english edition, 2007. Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.

[HSS09] John Hubbard, Dierk Schleicher, and Mitsuhiro Shishikura. Exponential Thurston maps and limits of quadratic differentials. *J. Amer. Math. Soc.*, 22(1):77–117, 2009.

[Hub06] John Hamal Hubbard. *Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1*. Matrix Editions, Ithaca, NY, 2006. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle.

[Hub16] J.H. Hubbard. *Teichmüller theory and applications to geometry, topology, and dynamics, Volume 2: Surface homeomorphisms and rational functions*. Matrix Editions, Ithaca, NY, 2016.

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[KK97] Bernd Krauskopf and Hartje Kriete. Kernel convergence of hyperbolic components. *Ergodic Theory Dynam. Systems*, 17(5):1137–1146, 1997.

[Kru87] S. L. Krushkal'. An interpolating family of a univalent analytic function. *Siberian Mathematical Journal*, 28(5):762–767, 1987.

[LP20] Rami Luisto and Pekka Pankka. Stoïlow's theorem revisited. *Expo. Math.*, 38(3):303–318, 2020.

[LSV08] Bastian Laubner, Dierk Schleicher, and Vlad Vicol. A combinatorial classification of postsingularly finite complex exponential maps. *Discrete Contin. Dyn. Syst.*, 22(3):663–682, 2008.

[MB12] Helena Mihaljević-Brandt. Dynamical approximation and kernels of non-escaping hyperbolic components. *Ergodic Theory Dynam. Systems*, 32(4):1418–1434, 2012.

[McM94] C.T. McMullen. *Complex dynamics and renormalization*. Annals Math. Studies 135. Princeton Univ. Press, Princeton, NJ, 1994.

[Muk23] Malavika Mukundan. Dynamical approximation of postsingularly finite exponentials, 2023.

[Nev70] Rolf Nevanlinna. *Analytic functions*. Die Grundlehren der mathematischen Wissenschaften, Band 162. Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig.

[Pfr19] D. Pfrang. *Homotopy Hubbard Trees for Post-Singularly Finite Transcendental Entire Maps*. PhD thesis, Jacobs University, Bremen, 2019.

[Poi93] A. Poirier. *On post-critically finite polynomials*. PhD thesis, State University at New York at Stony Brook, 1993.

[Poi10] A. Poirier. Hubbard trees. *Fundamenta Mathematicae*, 208(3):193–248, 2010.

[PPS21] David Pfrang, Sören Petrat, and Dierk Schleicher. Dreadlock pairs and dynamic partitions for post-singularly finite entire functions, 2021.

[PRS21] D. Pfrang, M. Rothgang, and D. Schleicher. Homotopy Hubbard trees for post-singularly finite exponential maps. *Preprint arXiv:1812.11831*, 2021.

[Sch04] Dierk Schleicher. On fibers and local connectivity of Mandelbrot and Multibrot sets. In *Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1*, volume 72 of *Proc. Sympos. Pure Math.*, pages 477–517. Amer. Math. Soc., Providence, RI, 2004.

[Sch10] Dierk Schleicher. Dynamics of entire functions. In *Holomorphic dynamical systems*, volume 1998 of *Lecture Notes in Math.*, pages 295–339. Springer, Berlin, 2010.

[She22] S. Shemyakov. *A topological characterization of certain postsingularly finite entire functions: transcendental dynamics and Thurston theory*. PhD thesis, Université d'Aix-Marseille, 2022.

[Sto28] S. Stoïlow. Sur les transformations continues et la topologie des fonctions analytiques. *Ann. Sci. École Norm. Sup. (3)*, 45:347–382, 1928.
[GO08] Anatoly A. Goldberg and Iossif V. Ostrovskii. *Value distribution of meromorphic functions*, volume 236 of *Translations of Mathematical Monographs*. American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii, With an appendix by Alexandre Eremenko and James K. Langley.

[Gro07] Misha Gromov. *Metric structures for Riemannian and non-Riemannian spaces*. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, english edition, 2007. Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.

[HSS09] John Hubbard, Dierk Schleicher, and Mitsuhiro Shishikura. Exponential Thurston maps and limits of quadratic differentials. *J. Amer. Math. Soc.*, 22(1):77–117, 2009.

[Hub06] John Hamal Hubbard. *Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1*. Matrix Editions, Ithaca, NY, 2006. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle.

[Hub16] J.H. Hubbard. *Teichmüller theory and applications to geometry, topology, and dynamics, Volume 2: Surface homeomorphisms and rational functions*. Matrix Editions, Ithaca, NY, 2016.

[Kam01] Atsushi Kameyama. The Thurston equivalence for postcritically finite branched coverings. *Osaka J. Math.*, 38(3):565–610, 2001.

[Kis95] Masashi Kisaka. Local uniform convergence and convergence of Julia sets. *Nonlinearity*, 8(2):273–281, 1995.

[KK97] Bernd Krauskopf and Hartje Kriete. Kernel convergence of hyperbolic components. *Ergodic Theory Dynam. Systems*, 17(5):1137–1146, 1997.

[Kru87] S. L. Krushkal'. An interpolating family of a univalent analytic function. *Siberian Mathematical Journal*, 28(5):762–767, 1987.

[LP20] Rami Luisto and Pekka Pankka. Stoïlow's theorem revisited. *Expo. Math.*, 38(3):303–318, 2020.

[LSV08] Bastian Laubner, Dierk Schleicher, and Vlad Vicol. A combinatorial classification of postsingularly finite complex exponential maps. *Discrete Contin. Dyn. Syst.*, 22(3):663–682, 2008.

[MB12] Helena Mihaljević-Brandt. Dynamical approximation and kernels of non-escaping hyperbolic components. *Ergodic Theory Dynam. Systems*, 32(4):1418–1434, 2012.

[McM94] C.T. McMullen. *Complex dynamics and renormalization*. Annals Math. Studies 135. Princeton Univ. Press, Princeton, NJ, 1994.

[Muk23] Malavika Mukundan. Dynamical approximation of postsingularly finite exponentials, 2023.

[Nev70] Rolf Nevanlinna. *Analytic functions*. Die Grundlehren der mathematischen Wissenschaften, Band 162. Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig.

[Pfr19] D. Pfrang. *Homotopy Hubbard Trees for Post-Singularly Finite Transcendental Entire Maps*. PhD thesis, Jacobs University, Bremen, 2019.

[Poi93] A. Poirier. *On post-critically finite polynomials*. PhD thesis, State University at New York at Stony Brook, 1993.

[Poi10] A. Poirier. Hubbard trees. *Fundamenta Mathematicae*, 208(3):193–248, 2010.

[PPS21] David Pfrang, Sören Petrat, and Dierk Schleicher. Dreadlock pairs and dynamic partitions for post-singularly finite entire functions, 2021.

[PRS21] D. Pfrang, M. Rothgang, and D. Schleicher. Homotopy Hubbard trees for post-singularly finite exponential maps. *Preprint arXiv:1812.11831*, 2021.

[Sch04] Dierk Schleicher. On fibers and local connectivity of Mandelbrot and Multibrot sets. In *Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1*, volume 72 of *Proc. Sympos. Pure Math.*, pages 477–517. Amer. Math. Soc., Providence, RI, 2004.

[Sch10] Dierk Schleicher. Dynamics of entire functions. In *Holomorphic dynamical systems*, volume 1998 of *Lecture Notes in Math.*, pages 295–339. Springer, Berlin, 2010.

[She22] S. Shemyakov. *A topological characterization of certain postsingularly finite entire functions: transcendental dynamics and Thurston theory*. PhD thesis, Université d'Aix-Marseille, 2022.

[Sto28] S. Stoïlow. Sur les transformations continues et la topologie des fonctions analytiques. *Ann. Sci. École Norm. Sup. (3)*, 45:347–382, 1928.
[Sto56] S. Stoïlow. *Leçons sur les principes topologiques de la théorie des fonctions analytiques. Deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann.* Gauthier-Villars, Paris, 1956.

[Tei37] O. Teichmüller. Eine Anwendung quasikonformer Abbildungen auf das Typenproblem. *Deutsche Math.*, 2:321–327, 1937. Gesammelte Abhandlungen, Springer, Berlin, 1982, 171–177.

DEPT. OF MATHEMATICS, 530 CHURCH STREET, UNIVERSITY OF MICHIGAN ANN ARBOR, MI, 48109
*Email address: malavim@umich.edu*

AIX-MARSEILLE UNIV., INSTITUT DE MATHÉMATIQUES DE MARSEILLE, 13003 MARSEILLE, FRANCE
*Email address: nikolai.prochorov@etu.univ-amu.fr*

MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES, INSELSTRASSE 22, LEIPZIG, 04107
*Email address: bernhard.reinke@mis.mpg.de*
[Sto56] S. Stoïlow. *Leçons sur les principes topologiques de la théorie des fonctions analytiques. Deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann.* Gauthier-Villars, Paris, 1956.

[Tei37] O. Teichmüller. Eine Anwendung quasikonformer Abbildungen auf das Typenproblem. *Deutsche Math.*, 2:321–327, 1937. Gesammelte Abhandlungen, Springer, Berlin, 1982, 171–177.

DEPT. OF MATHEMATICS, 530 CHURCH STREET, UNIVERSITY OF MICHIGAN ANN ARBOR, MI, 48109
*Email address: malavim@umich.edu*

AIX-MARSEILLE UNIV., INSTITUT DE MATHÉMATIQUES DE MARSEILLE, 13003 MARSEILLE, FRANCE
*Email address: nikolai.prochorov@etu.univ-amu.fr*

MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES, INSELSTRASSE 22, LEIPZIG, 04107
*Email address: bernhard.reinke@mis.mpg.de*