2.1 Degenerate rational maps

As mentioned in §1, the space $\mathrm {Rat}_d$ is naturally identified as an open and dense subset of $\mathbb {P}^{2d+1}$ . We say each element $f\in \mathbb {P}^{2d+1}\setminus \mathrm {Rat}_d$ is a degenerate rational map of degree d. For such f, there exist two homogeneous polynomials $F(X,Y)$ and $G(X,Y)$ of degree d in $\mathbb {C}[X,Y]$ such that $f=[F:G]$ is in homogeneous coordinates and $H_f:=\gcd [F,G]$ is a polynomial in $\mathbb {C}[X,Y]$ of degree at least $1$ . We can rewrite

$$ \begin{align*} f=H_f\hat f, \end{align*} $$

where $\hat f$ is a rational map of degree less than d. We say each zero of $H_f$ is a hole of f and denote $\mathrm {Hole}(f)$ the set of holes of f. Moreover, we call $\hat f$ the reduction of f. For convenience, if f is a rational map of degree d, we define $H_f=1$ and then $\hat f=f$ .

Let $\{f_n\}_{n\ge 1}$ be a sequence of rational maps of degree $d\ge 1$ . We say $f_n$ converges semi-algebraically to a (degenerate) rational map f if the coefficients of $f_n$ converge to the coefficients of f in $\mathbb {P}^{2d+1}$ . Compare to the algebraically convergence in [Reference Buff and Tan1]. Without special emphasis, we mean semi-algebraically convergence when we consider convergence for a sequence in $\mathbb {P}^{2d+1}$ . The semi-algebraical convergence implies locally uniform convergence away from holes.

Suppose that each $f_n$ possesses a cycle of fixed period. If the limit of these cycles is away from the holes of f, Lemma 2.1 immediately implies that this limit is also a cycle for $\hat f$ . We state as follows and omit the proof.

If the limit intersects the holes of f, we have the following basins shrinking result.

Now we state a straightforward result about the perturbations of periodic points.

Proof. By pre and post composition of Möbius transformations, we can assume $z_0,\ldots ,z_j\in {\mathbb C}$ . For $g=H_g\hat g\in \mathbb {P}^{2d+1}$ close to f, we have $\deg \hat g\ge 1$ . Then for $0\le i\le j-1$ , the iteration $g^{j-i}$ is well defined, see [Reference DeMarco2, Lemma 2.2]. Consider the holomorphic function

$$ \begin{align*} F_i(g,z):=g^{j-i}(z)-z_j(g) \end{align*} $$

on $\Lambda _f\times D(z_j)$ , where $\Lambda _f\subseteq {\mathbb P}^{2d+1}$ is a neighborhood of f and $D(z_j)\subseteq {\mathbb C}$ is a neighborhood of $z_j$ . By the assumptions, we have that $F_i(f,z_i)=0$ and

$$ \begin{align*} \frac{\partial F_i}{\partial z}|_{(f, z_i)}=(\hat f^{j-i})'(z_i)\not=0. \end{align*} $$

Then the implicit function theorem implies there exists a holomorphic function $z_i(g)$ near f such that $\hat g^{j-i}(z_i(g))=z_j(g)$ . If $\{z_0,\ldots ,z_{j-1}\}\cap \mathrm {Hole}(f)=\emptyset $ , the function $\hat g^{j-i}(z)$ is holomorphic in z in a fixed neighborhood of $z_i$ for each g close to f. It follows from Hurwitz’s theorem (see [Reference Gamelin7]) that $g^{j-i}(z)-z_j(g)$ has a unique root near $z_i$ for g close to f. Thus statement (1) follows.

For statement (2), note that the cycle $\mathcal {O}\cap \mathrm {Hole}(f)=\emptyset $ . Applying the implicit function theorem on $G(g,z):= g^k(z)-z$ , we obtain the expected cycle $\mathcal {O}(g)$ of $ g$ for g close to f.

For $f=H_f\hat f\in \mathbb {P}^{2d+1}$ , assume $\hat f$ has an attracting cycle $\mathcal {O}$ and denote $\Omega $ the immediate basin of $\mathcal {O}$ . If $\Omega \cap \mathrm {Hole}(f)=\emptyset $ , Lemma 2.4 implies that for g close to f, the map $\hat g$ has an attracting cycle $\mathcal {O}(g)$ . Denote by $\Omega (g)$ the immediate basin of $\mathcal {O}(g)$ . Then we have the following lemma.

This above result is well known in the case where f is a rational map of degree d, see [Reference Douady3, Lemma 6.3]. Our assumption $\Omega \cap \mathrm {Hole}(f)=\emptyset $ guarantees that the argument in the non-degenerate case also works in our case. Here we omit the proof.

2.2 Newton maps

For a degree $d\ge 2$ complex polynomial $P(z)$ with simple roots, its Newton map

$$ \begin{align*} f_P(z):=z-\frac{P(z)}{P'(z)} \end{align*} $$

is a degree d rational map having d superattracting fixed points at the roots of P. The only other fixed point is at $\infty $ . The holomorphic index formula (see [Reference Milnor18, Theorem 12.4]) asserts that the point $\infty $ is the unique repelling fixed point of $f_P$ . The critical points of $f_P$ are the roots of P and the zeros of $P"$ . Moreover, the poles of $f_p$ are the zeros of $P'$ .

Recall that $\mathrm {NM}_d$ is the space of degree d Newton maps and $\overline {\mathrm {NM}}_d$ is the closure of $\mathrm {NM}_d$ in $\mathbb {P}^{2d+1}$ . Then for each $f=H_f\hat f\in \overline {\mathrm {NM}}_d$ , there exists a polynomial Q degree at most d with possible multiple roots such that $\hat f$ is the Newton map of Q. Each root r of Q is a (super)attracting fixed point of $\hat f$ with multiplier $1-1/n_r$ , where $n_r$ is the multiplicity of r as a zero of Q. Moreover, again $\hat f$ has only one more fixed point at $\infty $ , which is repelling. It follows that each hole of f is either a multiple root of Q or $\infty $ . Furthermore, $\infty \in \mathrm {Hole}(f)$ if and only if $\deg Q<d$ . For more details about degenerate Newton maps, we refer the reader to [Reference Nie21].

For $f=H_f\hat f\in \overline {\mathrm {NM}}_d$ with $\deg \hat f\ge 2$ , the Fatou components of $\hat f$ have well-studied topological structure. By a result of Shishikura [Reference Shishikura27], all Fatou components of $\hat f$ are simply connected, and hence the Julia set of $\hat f$ is connected. Moreover, the boundary of each component of the basins of roots is locally connected, see [Reference Drach and Schleicher5, Reference Wang, Yin and Zeng29].

3.1 Perturbation of Böttcher coordinates

Let $f=H_f\hat f\in \overline {\mathrm {NM}}_d$ with $\deg \hat f\ge 2$ and denote by $\Omega _{\hat f}$ the union of basins of the roots of $\hat f$ . Let $\mathcal {U}$ be a finite subset of components of $\Omega _{\hat f}$ such that if $U\in \mathcal {U}$ , then $\hat f(U)\in \mathcal {U}$ . Recall that $\hat f$ is post-critically finite on $\bigcup _{U\in \mathcal {U}}U$ if the critical points in any $U\in \mathcal {U}$ have finite orbits. For such $\hat f$ and $\mathcal {U}$ , one can choose a system of Böttcher coordinates $\{\phi _U:U\to \mathbb D\}_{U\in \mathcal {U}}$ satisfying

$$ \begin{align*} \phi_{\hat{f}(U)}\circ \hat{f}\circ \phi_U^{-1}(z)=z^{d_U}\!,\quad z\in{\mathbb D}, \text{ where }d_U:=\mathrm{ deg}(\hat{f}|_U), \end{align*} $$

and $u:= \phi _U^{-1}(0)$ is called the center of U. Moreover, we have the following proposition.

To abuse notation, we denote the set of pointed sets $(U,u)$ with $U\in \mathcal {U}$ also by $\mathcal {U}$ . Let $\{f_n\}_{n\ge 1}$ be a sequence in $\mathrm {NM}_d$ such that $f_n$ converges to f. By Proposition 3.1 and Lemma 2.5, for $(U,u)\in \mathcal {U}$ , the point u belongs to the unique component $U_n$ of $\Omega _{f_n}$ for all large n; and combining Proposition 2.4, the component $U_n$ contains a preperiodic point $u_n$ with the same preperiod and period as that of u such that $u_n\to u$ as $n\to \infty $ . We call such $(U_n,u_n)$ a deformation of $(U,u)$ at $f_n$ . Note that such $u_n$ is not necessarily unique. For example, if u is a preperiodic critical point, it possibly splits into two preperiodic points of $f_n$ contained in $U_n$ .

To the end of this subsection, under natural assumptions, we define a Böttcher coordinate $\phi _{U_n}$ on the deformations $U_n$ of U and show a convergence result of $\phi _{U_n}$ . First, we consider the convergence of deformations.

Recall the definition of Carathéodory topology on a set of pointed sets and a set of holomorphic functions respectively, following [Reference McMullen16, §5.1]. Let $\mathcal {V}$ be a set of open simply connected pointed sets $(V,v)$ in $\mathbb {C}$ . The Carathéodory topology on $\mathcal {V}$ is defined by the following convergence: $(V_n, v_n)$ converges to $(V,v)$ if and only if:

1. (i) $v_n$ converges to v;

2. (ii) for any compact $K\subset V$ , we have $K\subset V_n$ for all large n; and

3. (iii) for any domain W containing v, if $W\subset V_n$ for infinitely many n, then $W\subset V$ .

Denote $\mathcal {G}$ the set of holomorphic functions defined on $(V, v)\in \mathcal {V}$ . Then the Carathéodory topology on $\mathcal {G}$ is defined as follows. Let $g:(V, v)\to \mathbb {C}$ and $g_n:(V_n, v_n)\to \mathbb {C}$ be functions in $\mathcal {G}$ . We say $g_n$ converges to g if: (1) $(V_n, v_n)$ converges to $(V, v)$ in $\mathcal {V}$ ; and (2) $g_n$ converges to g uniformly on any compact subset of V as $n\to \infty $ .

For our propose, we use the following definition.

If $f_n\xrightarrow {\deg }f$ on $\mathcal {U}$ , it follows immediately that any $(U,u)\in {\mathcal U}$ has a unique deformation $(U_n,u_n)$ . We call such $u_n$ a center of $U_n$ . In this case, set

$$ \begin{align*} \mathcal{U}_n:=\{(U_n,u_n):(U_n,u_n)\text{ as the deformation of }(U,u)\in\mathcal{U}\}. \end{align*} $$

We mention here that the set $U_n$ may contain several distinct centers.

Remark 3.4. If a critical point c of $\hat {f}$ is contained in the boundaries of distinct $(U,u)$ and $(U',u')$ in $\mathcal {U}$ , it is possible that $U_n$ coincides with $U_n'$ and it contains the critical point of $f_n$ perturbed from c (see Figure 2). In this case, both $u_n$ and $u_n'$ are centers of $U_n=U_n'$ , and hence $f_n$ is not post-critically finite on the union of $U_n$ with $(U_n,u_n)\in \mathcal {U}_n$ .

Figure 2

(a) The dynamical plane of the Newton map f for the polynomial $z^3-1$ . The letters indicate Fatou components $U, U', U"$ , and $U"'$ with centers $u,u',u"$ , and $u"'$ respectively. The arcs indicate internal rays. The critical point $c=0$ is contained in $\partial U\cap \partial U'$ . (b) The dynamical plane of the Newton map $f_n$ for the polynomial $z^3+z/n-1$ with indicated Fatou component $U_n$ . The critical point $c_n\in U_n$ . The points $u_n, u_n', u_n"$ , and $u_n"'$ are all in $U_n$ and centers of $U_n$ . The set $(U_n,u_n)$ is the deformation of $(U,u)$ ; the set $(U_n,u^{\prime }_n)$ is the deformation of $(U',u')$ ; the set $(U_n,u^{\prime \prime }_n)$ is the deformation of $(U",u")$ ; and the set $(U_n,u^{\prime \prime \prime }_n)$ is the deformation of $(U"',u"')$ . The corresponding rays in $U_n$ either land on $\partial U_n$ or terminate at the iterated preimages of $c_n$ .

The following result states a natural sufficient condition for the convergence $f_n\xrightarrow {\deg }f$ , which we will use repeatedly in §5. The proof is straightforward, so we omit it.

From now on, we assume that $f_n\xrightarrow {\deg }f$ . Since $\hat f$ is post-critically finite on $\bigcup _{U\in \mathcal {U}}U$ , by Proposition 3.1 and Lemma 2.1, we have the following straightforward result and again omit the proof.

Lemma 3.6 suggests that for each $(U_n,u_n)\in \mathcal {U}_n$ , we have a Böttcher coordinate $\phi _{(U_n,u_n)}$ near $u_n$ such that

(3.1) $$ \begin{align} \phi_{(U_n,u_n)}(z)^{d_U}=\phi_{(f_n(U_n),f_n(u_n))}\circ f_n(z) \end{align} $$

for z near $u_n$ , and that

(3.2) $$ \begin{align} \phi_{(U_n,u_n)}'(u_n)\to \phi_{(U,u)}'(u)\text{ as }n\to\infty. \end{align} $$

The map $\phi _{(U_n,u_n)}$ extends conformally until meeting an iterated preimage of critical points of $f_n$ . Then there exists a maximum $r_n\le 1$ such that $\psi _{(U_n,u_n)}:=\phi _{(U_n,u_n)}^{-1}: \mathbb {D}_{r_n}\to U_n$ is well defined.

Denote by $\psi _{(U,u)}$ the inverse of $\phi _{(U,u)}$ . The following result asserts that $\psi _{(U_n,u_n)}$ converges to $\psi _{(U,u)}$ locally uniformly on $\mathbb {D}$ , which is well known in Carathéodory topology, see e.g. [Reference McMullen16, Theorem 5.1].

3.2 Perturbation of internal rays

In §3.1, we perturb a Böttcher coordinate in $(U,u)\in \mathcal {U}$ to obtain a Böttcher coordinate $\phi _{(U_n,u_n)}$ in $(U_n,u_n)\in \mathcal {U}_n$ . In this subsection, we use the inverse map $\psi _{(U_n,u_n)}$ to define the internal rays in $(U_n,u_n)$ and prove a convergence result on internal rays.

Now we define internal rays of $f_n$ in $(U_n,u_n)$ as follows. For each $\theta \in {\mathbb R}/{\mathbb Z}$ , let $r_{\theta }$ be the maximal radius such that $\psi _{(U_n,u_n)}$ extends along $(0,r_{\theta })e^{2\pi i\theta }$ . If $r_{\theta }<1$ , then arc $\psi _{(U_n,u_n)}((0,r_{\theta })e^{2\pi i\theta })$ terminates at an iterated preimage of critical points of $f_n$ , and if $r_{\theta }=1$ , the arc $\psi _{(U_n,u_n)}((0,1)e^{2\pi i\theta })$ accumulates and factually lands on $\partial U_n$ . In the latter case, we call

$$ \begin{align*} I_{(U_n,u_n)}(\theta):=\psi_{(U_n,u_n)}([0,1]e^{2\pi i\theta}) \end{align*} $$

the landed internal ray in $(U_n,u_n)$ of angle $\theta $ . Note that $f_n$ sends a landed internal ray of $(U_n,u_n)$ to a landed internal ray of $(f(U_n),f(u_n))$ . Also, since $U_n$ may contain more than one center, it may possess several groups of landed interval rays. In this case, each such ray starts from a center of $U_n$ and rays from distinct groups are disjoint (see Figure 2).

The following result asserts that the internal rays of eventually periodic angles converge.

Proof. To ease notation, we write $I(\theta ),I_n(\theta ),\psi $ , and $\psi _n$ for $I_{(U,u)}(\theta ), I_{(U_n,u_n)}(\theta ),\psi _{(U,u)}$ , and $\psi _{(U_n,u_n)}$ , respectively. Set $\delta :=\mathrm {deg}(\hat {f}|_U)$ and let $z_0$ be the landing point of $I(\theta )$ . It is sufficient to show that, given any $\eta>0$ , for all large n, we have $d_H(I(\theta ),I_n(\theta ))<\eta $ , where $d_H$ is the Hausdorff metric.

First assume that $I(\theta )$ is periodic of period $p\ge 1$ . Then u is a super-attracting fixed point of $\hat {f}$ . Define

$$ \begin{align*} D_{\epsilon}:=\{z\in\widehat{\mathbb{C}}:\rho(z,z_0)<\epsilon\}, \end{align*} $$

where $\rho $ is the spherical metric. Shrinking $\epsilon $ if necessary, we may assume $\hat {f}|_{D_{\epsilon }}$ is injective and $\overline {D}_{\epsilon }\subseteq \hat f^p(D_{\epsilon })$ . We claim that for any sufficiently large n and any component $D^{\prime }_n$ of $f^{-p}_n(D_{\epsilon })$ , either $\overline {D_n'}\subseteq D_{\epsilon }$ or $\overline {D_n'}\subseteq \widehat {{\mathbb C}}\setminus \overline {D}_{\epsilon }$ . Indeed, if $p>1$ , the landing point $z_0$ of $I(\theta )$ is not a hole of $f^p$ , see §2.1 and [Reference DeMarco2, Lemma 2.2]. It follows from Lemma 2.1 that $f_n^p$ converges uniformly to $\hat {f}^p$ near $z_0$ , and hence $f_n^p|_{D_{\epsilon }}$ is injective and $\overline {D}_{\epsilon }\subseteq f_n^p(D_{\epsilon })$ for all large n. Then in this case, the claim follows. Now we consider the case where $p=1$ . Then $z_0=\infty $ . If $z_0=\infty $ is not a hole of f, the claim follows by previous argument. If $z_0=\infty $ is a hole of f, then $f_n$ fails to converge uniformly to $\hat {f}$ near $\infty $ . In this case, we prove the claim by contradiction. Suppose that the claim fails. Then there exists a subsequence, denoted also by $\{f_n\}$ , such that for each $f_n$ , there exists a component $D_n'$ of $f^{-1}_n(D_{\epsilon })$ with $\overline {D}^{\prime }_n\cap \partial D_{\epsilon }\not =\emptyset $ . Choose a point $w_n\in \overline {D}_n'\cap \partial D_{\epsilon }$ . Passing to subsequence if necessary, we may assume $w_n\to w$ . Then $w\in \partial D_{\epsilon }$ . By Lemma 2.1, the sequence $f_n$ converges uniformly to $\hat f$ on $\partial D_{\epsilon }$ . It follows that as $n\to \infty $ ,

$$ \begin{align*} f_n(w_n)\to \hat f(w). \end{align*} $$

Note that $\overline {D}_{\epsilon }\subseteq f(D_{\epsilon })$ . Then

$$ \begin{align*} \hat f(\partial D_{\epsilon})\cap\overline{D}_{\epsilon}=\emptyset. \end{align*} $$

We have that $f(w)\not \in \overline {D}_{\epsilon }$ . However,

$$ \begin{align*} f_n(w_n)\in f_n(\overline{D}_n')=\overline{D}_{\epsilon}, \end{align*} $$

which implies $f(w)\in \overline {D}_{\epsilon }$ . It is a contradiction. Therefore, the claim holds.

Since $I(\theta )$ lands at $z_0$ , there exists $0<r<1$ such that

$$ \begin{align*}\psi((r,1)e^{2\pi i\theta})\subseteq U\cap D_{\epsilon}.\end{align*} $$

Pick $0<s<1$ such that $s^{\delta ^p}>r$ . Then the segment $\psi ([s^{\delta ^p},s]e^{2\pi i \theta })\subseteq I(\theta )$ belongs to $U\cap D_{\epsilon }$ . It follows from Proposition 3.7 that for all large n,

(3.3) $$ \begin{align} d_H(\psi_n\large([0,s]e^{2\pi i\theta}\large),\psi\large([0,s]e^{2\pi i\theta}\large))<\epsilon. \end{align} $$

Define

$$ \begin{align*} \gamma_{n,0}:[0,1]\to\psi_n([s^{\delta^p},s]e^{2\pi i\theta}) \end{align*} $$

to be an arc such that $\gamma _{n,0}(0)=\psi _n(s^{\delta ^p}e^{2\pi i\theta })$ and $\gamma _{n,0}(1)=\psi _n(s e^{2\pi i\theta })$ . Then,

$$ \begin{align*} \gamma_{n,0}([0,1])\subseteq D_{\epsilon}\cap U_n. \end{align*} $$

Note that $f_n^p(\gamma _{n,0}(1))=\gamma _{n,0}(0)$ . Lift $\gamma _{n,0}$ to an arc $\gamma _{n,1}$ based at $\gamma _{n,0}(1)$ . Since $I_n(\theta )$ is landed, inductively we obtain a sequence of arcs $\gamma _{n,k}$ such that $\gamma _{n,k+1}$ is a lift by $f_n$ of $\gamma _{n,k}$ based at the endpoint of $\gamma _{n,k}$ which is not in $\gamma _{n,k-1}$ .

Now we claim that for sufficiently large n, the arc $\gamma _{n,k}\subset D_{\epsilon }$ . We prove the claim by induction on k. The claim holds for $k=0$ by the definition of $\gamma _{n,0}$ . Suppose that for $k\geq 0$ , the arc $\gamma _{n,k}\subseteq D$ . Since $\gamma _{n,k+1}$ is a preimage of $\gamma _{n,k}$ under $f_n$ , there exists a component $D'$ of $f_n^{-1}(D_{\epsilon })$ containing $\gamma _{n,k+1}$ . Since the intersection point of $\gamma _{n,k+1}\subseteq D'$ and $\gamma _{n,k}\subseteq D_{\epsilon }$ belongs to $D_{\epsilon }$ , it follows that $D'\cap D_{\epsilon }\not =\emptyset .$ By the previous claim, we have $D'\subseteq D_{\epsilon }$ , and hence $\gamma _{n,k+1}\subseteq D_{\epsilon }$ , which completes the induction.

Note that for all large n,

$$ \begin{align*} I_n(\theta)=\psi_n([0,s]e^{2\pi i\theta})\bigcup(\cup_{k\geq0}\gamma_{n,k})\cup\{z_n\}, \end{align*} $$

where $z_n$ is the landing point of $I_n(\theta )$ . According to the estimate in equation (3.3) and the fact that $\gamma _{n,k}\subseteq D_{\epsilon }$ , we have

$$ \begin{align*} d_H(I(\theta),I_n(\theta))<\epsilon. \end{align*} $$

By choosing $\epsilon <\eta $ , we prove the proposition under the periodicity assumption.

In the strictly preperiodic case, we set $(V,v):=\hat {f}(U,u)$ and $I_V(\theta ')=\hat f(I(\theta ))$ . Let $(V_n,v_n)$ be the deformation of $(V,v)$ with $f_n(U_n,u_n)=(V_n,v_n)$ . Inductively, it is sufficient to prove $d_H(I(\theta ),I_n(\theta ))<\epsilon $ under the assumption that $\lim \nolimits _{n\to \infty }d_H(I_V(\theta '), I_{V_n}(\theta '))=0$ .

Define $D_{\epsilon }$ as above. By Proposition 3.7, there exists $0<s<1$ such that for all large n,

$$ \begin{align*} d_H(\psi_n([0,s]e^{2\pi it}),\psi([0,s]e^{2\pi it}))<\epsilon \text{ and } \psi_n(se^{2\pi it})\in D_{\epsilon}. \end{align*} $$

Set $L^{\prime }_n:=\psi _{(V_n,v_n)}([s^{\delta },1]e^{2\pi \theta '})$ and $L':=\psi _{(V,v)}([s^{\delta },1]e^{2\pi \theta '})$ . Since $I_{V_n}(\theta ')\to I_V(\theta ')$ , $L^{\prime }_n$ and $L'$ are contained in $\hat f(D_{\epsilon })$ for large n. Since $I_n(\theta )$ is a landed internal ray for all large n, there is a lift $L_n$ of $L^{\prime }_n$ based at the point $\psi _n(se^{2\pi i\theta })$ . Denote by L the lift of $L'$ based at the point $\psi (se^{2\pi i\theta })$ . Note that in this case, we have $z_0\not \in \mathrm {Hole}(f)$ . Then $f_n$ converges uniformly to $\hat {f}$ on $D_{\epsilon }$ . Thus for sufficiently large n,

$$ \begin{align*} f(D_{\epsilon})\subset f_n(D_{2\epsilon}). \end{align*} $$

Hence we have $L_n\subset D_{2\epsilon }$ and $L\subseteq D_{2\epsilon }$ . Note $I(\theta )=\psi ([0,s]e^{2\pi it})\cup L$ and $I_n(\theta )=\psi _n([0,s]e^{2\pi it})\cup L_n$ . It follows that

$$ \begin{align*} d_H(I(\theta),I_n(\theta))<2\epsilon. \end{align*} $$

Choose $\epsilon <\eta /2$ . This completes the proof.

3.3 Proof of Theorem 1.1

Now we begin to prove Theorem 1.1 and state a sufficient condition for the assumption (ii) in Theorem 1.1.

To end this section, we state a sufficient condition to guarantee that assumption (ii) in Theorem 1.1 holds. Note that for any internal ray $I\subset \Gamma $ , there exists an smallest integer $k_I\ge 0$ such that $\hat f^{k_I}(I)$ is contained in the immediate basin of a root of $\hat f$ .

Proof. Consider an internal ray $I\subset \Gamma $ and it’s perturbation $I_n\subset \Gamma _n$ . Let $U\in {\mathcal U}$ be the component such that $I\subset U$ , and let $U_n$ be the deformation of U at $f_n$ . We show $I_n$ lands on $\partial U_n$ . If U is the immediate basin of a root of $\hat {f}$ , the conclusion follows immediately since in this case, $k_I=0$ .

Now we consider the case where U is not the immediate basin of a root of $\hat {f}$ . We set $V=\hat f(U)$ and $I'=\hat f(I)$ . By an induction argument, it suffices to show $I_n$ lands on $\partial U_n$ under the assumption that $I^{\prime }_n$ lands on $\partial V_n$ , where $I^{\prime }_n$ is the perturbation of $I'$ and $V_n$ is the deformation of V at $f_n$ . Fix notation as in the proof of Proposition 3.8. Since the orbit of the landing point of the internal ray I avoids the critical points of $\hat f$ , we can apply a similar argument in Proposition 3.8 and obtain that

$$ \begin{align*} I_n=I_{(U_n,u_n)}(t)=\psi_{(U_n,u_n)}([0,s]e^{2\pi it})\cup L_n, \end{align*} $$

where $L_n$ is a lift of $\psi _{(V_n,v_n)}([s^{\delta },1]e^{2\pi it'})$ based at $\psi _{(U_n,u_n)}(se^{2\pi it})$ . Note that

$$ \begin{align*} \psi_{(V_n,v_n)}(e^{2\pi it'})\in\partial V_n. \end{align*} $$

It follows that $I_n$ land on $\partial U_n$ .

If $\deg f_n|_{U^{\prime }_n}=\deg \hat f|_{U'}$ for all immediate basins $U'\in {\mathcal U}$ , since $f_n\xrightarrow {\deg }f$ and $\hat f$ is post-critically finite on $U'$ , it follows that for the center $u^{\prime }_n$ of $U^{\prime }_n$ , the degrees $\deg _{u^{\prime }_n}f_n=\deg f_n|_{U^{\prime }_n}$ . Hence $u^{\prime }_n$ is the unique critical point of $f_n$ . It follows that all internal rays in $U^{\prime }_n$ are landed. Thus the conclusion follows.

4.1 Newton graphs

Let $f\in \mathrm {NM}_d$ with $d\ge 2$ . Recall that $\Omega _{ f}$ is the union of basins of its roots. Assume that f is post-critically finite on $\Omega _{f}$ . The dynamics of $ f$ can be characterized by an invariant graph that is the so-called Newton graph. Such a graph was first constructed in [Reference Drach, Mikukich, Rückert and Schleicher4] and then applied to study the dynamics of corresponding maps, see [Reference Drach and Schleicher5, Reference Gao8, Reference Gao9, Reference Lodge, Mikulich and Schleicher14, Reference Lodge, Mikulich and Schleicher13, Reference Wang, Yin and Zeng29]. In this subsection, we state briefly the construction of Newton graphs and list some properties.

Let r be a root of $ f$ and denote by $\Omega _{ f}(r)$ its immediate attracting basin. The fixed internal rays in $\Omega _{ f}(r)$ land at fixed points in $\partial \Omega _{ f}(r)$ . Since the only Julia fixed point of $ f$ is at $\infty $ , all fixed internal rays in $\Omega _{ f}$ have a common landing point at $\infty $ . We denote by $\Delta _0$ the union of all fixed internal rays in $\Omega _{ f}$ together with $\infty $ . Then $ f(\Delta _0)=\Delta _0$ . For any $m\geq 0$ , denote by $\Delta _m$ the connected component of $ f^{-m}(\Delta _0)$ that contains $\infty $ . Following [Reference Drach, Mikukich, Rückert and Schleicher4], we call $\Delta _m$ the Newton graph of f at level m. The vertex set of $\Delta _n$ consists of iterated preimages of fixed points of $ f$ contained in $\Delta _n$ .

A crucial property for Newton graphs is the following.

The Newton graphs induce naturally a puzzle structure for $ f$ on $\widehat {{\mathbb C}}$ . Let $\Delta _{ f}$ denote the Newton graph of $ f$ with the least level such that $\Delta _{ f}$ contains all poles and all critical points that map to fixed points under iteration. Set $X_0$ the complement of the union of the disks $\{z\in U:\phi _U(z)<1/2\}$ for all connected components U of $\Omega _{ f}$ with $U\cap \Delta _f\not =\emptyset $ , where $\phi _U$ is the Böttcher coordinate on U. Define $G_0:=(\Delta _f\cap X_0)\cup \partial X_0$ . Then $G_0$ is a finite graph consisting of segments of internal rays and equipotential lines in $\Omega _{ f}$ . For each $m\geq 0$ , we define $X_m:={f}^{-m}(X_0)$ and $G_m:={f}^{-m}(G_0)$ . Then each $X_m$ is connected and the interior $\mathrm {int}(X_m)$ contains the Julia set $J_{ f}$ of $ f$ . For each $m\geq 0$ , the closures of the components of $X_m\setminus G_m$ are called puzzle pieces of level m. It follows that the puzzle pieces of different levels have a nested structure. For each $z\in J_{ f}$ , denote $E_m(z)$ the union of puzzle pieces of level m which contains z. Then $z\in \mathrm {int}(E_m(z))$ . Moreover, $E_m(z)$ are puzzle pieces for all m if and only if z is not an iterated preimage of $\infty $ .

Proposition 4.2. [Reference Drach and Schleicher5, Corollary 1.2] and [Reference Wang, Yin and Zeng29, Theorem 1.1]

If z is on the boundary of a component of $\Omega _{{f}}$ , then

$$ \begin{align*} \bigcap_{m\geq0}E_m(z)=\{z\}. \end{align*} $$

In particular, the boundary of any component of basins of the roots is locally connected.

4.2 An alternative graph for cubic Newton maps

In this subsection, we focus on the case where $f\in \mathrm {NM}_3$ . Except for some special cases, we construct an invariant graph away from the unique non-fixed critical point. Our graph is based on Roesch’s work in [Reference Roesch25, §3] and differs from the Newton graphs introduced above.

Let $r_1,r_2$ , and $r_3$ be the roots of f and let $\Omega _1$ , $\Omega _2$ , and $\Omega _3$ be the corresponding immediate basins respectively. Note that f has another critical point denoted by c. In this subsection, we always assume $c\not \in \Omega _1\cup \Omega _2\cup \Omega _3$ and c is not a pole, that is, $f(c)\not =\infty $ .

Under the assumptions, we have that f has two distinct poles, denoted by $\xi _1$ and $\xi _2$ . An orientation argument implies that $\partial \Omega _1$ , $\partial \Omega _2$ , and $\partial \Omega _3$ cannot intersect at a common pole. By counting the preimages of $\Omega _i$ basins, we have that there is a unique pole at which exactly two $\partial \Omega _i$ basins intersect. Up to reindexing, we can assume $\xi _1\in \partial \Omega _1\cap \partial \Omega _2$ . It follows that $\xi _2\in \partial \Omega _3$ and $\xi _2\not \in \partial \Omega _1\cup \partial \Omega _2$ .

For $i=1,2$ , and $3$ , denote by $I_i(\theta )$ the internal ray in $\Omega _i$ of angle $\theta \in \mathbb {R}/\mathbb {Z}$ . Following Roesch [Reference Roesch25], we say an angle $\theta $ is a cut angle in $\Omega _1$ if there exists $\theta '\in \mathbb {R}/\mathbb {Z}$ such that $I_1(\theta )$ and $I_2(\theta ')$ land at a common point. It turns out that $\theta $ is a cut angle in $\Omega _1$ if and only if $1-\theta $ is a cut angle in $\Omega _2$ . For the basin $\Omega _3$ , the only cut angle is $0$ . Let $\Theta $ be the set of cut angles in $\Omega _1$ . It follows immediately that $0,1/2\in \Theta $ . Label $\Omega _1$ such that $\Omega _3$ and $I_{\Omega _1}(\theta )$ are in the same complementary component of the curve

(4.1) $$ \begin{align} \gamma(0,1/2):=I_1(0)\cup I_1(1/2)\cup I_2(0)\cup I_2(1/2) \end{align} $$

for any $\theta \in (0,1/2)$ , and define

$$ \begin{align*}\alpha:=\inf\{\theta:\theta\in\Theta\},\end{align*} $$

where $\inf $ is obtained under the order by identifying $\mathbb {R}/\mathbb {Z}$ with $(0,1]$ . In fact, the local connectivity of $\partial \Omega _1$ and $\partial \Omega _2$ implies that $\alpha \in \Theta $ .

Now we summarize the properties of the cut angles for later use. We use the following notation. Let $\Omega _i^{(1)}$ be the preimage of $\Omega _i$ disjoint from $\Omega _i$ . Then $c\not \in \Omega _i^{(1)}$ . For $j\ge 1$ , if $\Omega _i^{(j)}$ is a domain such that $f^j:\Omega _i^{(j)}\to \Omega _i$ is a homeomorphism, then an internal ray $I_i(\theta )$ in $\Omega _i$ deduces an internal ray $I_i^{(j)}(\theta )$ in $\Omega _i^{(j)}$ satisfying $I_i^{(j)}(\theta )=f^{-j}(I_i(\theta ))$ .

Figure 3 provides an example to illuminate the curves in the above lemma.

Figure 3

The dynamical plane of the Newton map for the polynomial $z^3/3-z^2/2+1$ . The curve $\gamma (0,1/2)$ consists of the internal rays $I_1(0),I_1(\tfrac 12), I_2(0)$ , and $I_2(\tfrac 12)$ . The angle $\theta \not \in \Theta $ but $2\theta \in \Theta $ . A curve in Lemma 4.3(5) consists of indicated internal rays except those in $\gamma (0,1/2)$ . In this section, we continue to use this example in the subsequent figures.

Let $\gamma (0,1/2)$ be as in equation (4.1). Then the complement of $\gamma (0,1/2)$ in $\widehat {\mathbb {C}}$ contains two components. Denote by D the one that is disjoint with $\Omega _3$ . It follows from Lemma 4.3(4) that $\Omega _3^{(1)}\subset D$ .

By Lemma 4.3(2), we can choose a rational angle $\theta \in (0,1/2)$ satisfying:

1. (i) $\theta \not \in \Theta ,$ but $2\theta \in \Theta $ ;

2. (ii) there exists $k\geq 1$ such that $\eta :=2^k\theta \in (1/2,1)$ ; and

3. (iii) the orbit of the landing point of $I_1(\theta )$ avoids c and $\infty $ .

Define

$$ \begin{align*}\mathcal{L} &:= I_3(0)\cup I_3(1/2)\cup I_1(0)\cup I_1(\theta)\cup I_2(0)\cup I_2(1-\theta)\\&\quad\cup I_2^{(1)}(1-2\theta)\cup I_2^{(1)}(0)\cup I_1^{(1)}(0)\cup I_1^{(1)}(2\theta).\end{align*} $$

Then Lemma 4.3(3) implies that $\mathcal {L}$ is a connected graph. Moreover, $\widehat {{\mathbb C}}\setminus \mathcal {\mathcal {L}}$ has three components. We label W the one disjoint with $\Omega _3$ . In the remaining two components, we label $W_-$ the one intersecting with $\Omega _1$ and label $W_+$ the one intersecting with $\Omega _2$ (see Figure 4). By Lemma 4.3(5), it immediately follows that $D\cup \overline {\Omega }_3^{(1)}\subseteq W$ and $c\in W\setminus \overline {D}$ . In particular, $\xi _1\in W$ . Moreover, we have $I_3(3/4)\subseteq W_-$ and $I_3(1/4)\subseteq W_+$ .

Figure 4

The curve $\mathcal {L}$ consists of the indicated internal rays except $I_1(1/2)$ and $I_2(1/2)$ . The boundary of D consists of $I_1(0), I_1(1/2), I_2(1/2)$ , and $I_2(0)$ .

Now consider the components of $f^{-1}(\Omega _1^{(1)})$ and $f^{-1}(\Omega _2^{(1)})$ . Note that $f^{-1}(\Omega _2^{(1)})$ has a component whose boundary contains the landing point of $I_1((1+\theta )/2)$ . Since $I_1((1+\theta )/2)\subset D$ , this component is also contained in D. Hence it does not contain c since $c\in W\setminus \overline {D}$ . Note that the landing points of $I_3(1/4)$ and $I_3(3/4)$ are contained in the boundaries of the two remaining components of $f^{-1}(\Omega _2^{(1)})$ respectively. We denote by $\Omega _2^{(2)}$ the component whose boundary contains the landing point of $I_3(3/4)$ . Then $I_2^{(2)}(0)$ and $I_3(3/4)$ land at a common point. Moreover, $\Omega _2^{(2)}\subset W_-$ since $I_3(3/4)\subseteq W_-$ . It follows that $c\not \in \Omega _2^{(2)}$ . By Lemma 4.3(3), we have $I_1(\theta )$ and $I_2^{(1)}(1-2\theta )$ land at a common point. It follows that $I_1(\theta /2)$ and $I_{2}^{(2)}(1-2\theta )$ land at a common point since $I_1(\theta /2)\subseteq W_-$ . Similarly, denote by $\Omega _1^{(2)}$ the component of $f^{-1}(\Omega _1^{(1)})$ contained in $W_+$ . Then $c\not \in \Omega _1^{(2)}$ . Moreover, $I_1^{(2)}(0)$ and $I_3(1/4)$ land at a common point, as well as $I_1^{(2)}(2\theta )$ and $I_2(1-\theta /2)$ . Define the Jordan curve

(4.2) $$ \begin{align} \mathcal{C}&:= I_3(1/4)\cup I_3(3/4)\cup I_2^{(2)}(0)\cup I_2^{(2)}(1-2\theta)\cup I_1(\theta/2)\cup I_1(\eta) \\ \nonumber &\cup I_2(1-\eta)\cup I_2(1-\theta/2)\cup I_1^{(2)}(2\theta)\cup I_1^{(2)}(0). \end{align} $$

See Figure 5 for an illustration for the curve $\mathcal {C}$ .

Figure 5

The curve $\mathcal {C}$ consists of the indicated internal rays. For this $\theta $ , we have $\eta =2\theta $ .

We show that the critical point c is not in the iterations of $\mathcal {C}$ and separated by $\mathcal {C}$ from $\infty $ . More precisely, we have the following lemma.

Proof. The Julia points in $\mathcal {C}$ are the landing points of $I_3(1/4),I_3(3/4), I_1(\theta /2),I_1(\eta )$ , and $I_2(1-\theta /2)$ . By the choice of $\theta $ , the orbits of the landing points of $I_1(\theta /2),I_1(\eta )$ , and $I_2(1-\theta /2)$ are away from c. Since

$$ \begin{align*} c\in W\setminus\{\infty,\xi_2\}\subseteq \widehat{{\mathbb C}}\setminus \overline{\Omega}_3, \end{align*} $$

it follows that $c\not \in \partial {\Omega }_3 $ , and hence the orbits of the landing points of $I_3(1/4)$ and $I_3(3/4)$ are disjoint with c. Then statement (1) holds.

Statement (2) follows immediately from the construction of $\mathcal {C}$ and Lemma 4.3 (4), (5).

Since $\theta $ is rational, there is a positive integer $k>1$ such that the graph

$$ \begin{align*} G:=\bigcup_{j=0}^kf^j(\mathcal{C}) \end{align*} $$

is invariant. Lemma 4.4 immediately implies that $c\not \in G$ . Moreover, obviously our graph G is distinct from the Newton graphs of f.

4.3 Cut angles for quartic Newton maps

In this subsection, we generalize part of results in [Reference Roesch25, §3] from the cubic case to a quartic case. Throughout this subsection, we assume that $f\in \mathrm {NM}_4$ has degree $2$ in the immediate basin of each root.

Let $r_1,r_2,r_3$ , and $r_4$ be the roots of f and denote by $\Omega _1,\Omega _2,\Omega _3$ , and $\Omega _4$ the corresponding immediate basins. Then there exist $1\le i<j \le 4$ such that $\partial \Omega _i\cap \partial \Omega _j$ contains a pole. Hence the internal rays $I_i(1/2)$ and $I_j(1/2)$ land at a common point. We say that f is of separable type if there exist $1\le i<j\le 4$ such that $I_i(1/2)$ and $I_j(1/2)$ land at a common pole and each component of $\widehat {{\mathbb C}}\setminus \gamma (0,1/2)$ contains a pole of f, where

$$ \begin{align*} \gamma(0,1/2):=I_i(0)\cup I_i(1/2)\cup I_j(0)\cup I_j(1/2). \end{align*} $$

If f is not of separable type, we can choose $1\le i<j\le 4$ such that $I_i(1/2)$ and $I_j(1/2)$ land at a common pole, but a component D of $\widehat {{\mathbb C}}\setminus \gamma (0,1/2)$ does not contain a pole of f. Relabeling the roots of f, we set $i=1$ and $j=2$ . Furthermore, we can set $I_1(\theta )\in D$ if and only if $\theta \in (1/2,1)$ . Hence $I_2(\theta ')\in D$ if and only if $\theta '\in (0,1/2)$ . We now consider the cut angles in $\Omega _1$ . An angle $\theta \in \mathbb {R}/\mathbb {Z}$ is a cut angle in $\Omega _1$ if there exists $\theta '\in \mathbb {R}/\mathbb {Z}$ such that $I_1(\theta )$ and $I_2(\theta ')$ land at a common point. If $\theta $ is a cut angle in $\Omega _1$ , then the corresponding $\theta '=1-\theta $ . Denote $\Theta $ the set of all cut angles in $\Omega _1$ and set

$$ \begin{align*} \alpha:=\inf\{\theta:\theta\in\Theta\}, \end{align*} $$

where $\inf $ is obtained under the order by identifying $\mathbb {R}/\mathbb {Z}$ with $(0,1]$ . Since $\widehat {{\mathbb C}}\setminus \overline {D}$ contains $\Omega _3\cup \Omega _4$ , it follows that $\alpha>0$ . By the local connectivity of $\partial \Omega _1$ and $\partial \Omega _2$ , we have $\alpha \in \Theta $ and $\Theta $ is a closed set in ${\mathbb R}/{\mathbb Z}$ .

Now we state some properties of the cut angles. Since we are interested in hyperbolic maps, see §5, we further assume that f is hyperbolic in the following result.

Proof. For statement (1), since $(\theta +1)/2>1/2$ , the internal rays $I_1((\theta +1)/2)\subseteq D$ . Suppose $(\theta +1)/2\not \in \Theta $ . Since $f(I_1((\theta +1)/2))=I_1(\theta )$ and $\theta \in \Theta $ , there exists a component $\Omega _2^{(1)}$ of $f^{-1}(\Omega _2)$ disjoint with $\Omega _2$ such that $\Omega _2^{(1)}$ contains the landing point of $I_1((\theta +1)/2)$ . Note that f is hyperbolic and hence the landing point of $I_2(1/2)$ is not a critical point. It follows that $\overline {\Omega }_2^{(1)}\subseteq D$ . Hence D contains a pole of f. It contradicts the choice of D.

To prove statement (2), let p be the period of the angle $\theta $ . Under the assumptions of f, the unique fixed angle is $0$ . It follows that $p>1$ . Define

$$ \begin{align*} \gamma(0,\alpha):=I_1(0)\cup I_1(\alpha)\cup I_2(0)\cup I_2(1-\alpha). \end{align*} $$

Since $\alpha \leq 1/2$ , there exists a component of $\widehat {{\mathbb C}}\setminus \gamma (0,\alpha )$ containing D. Denote this component by W. It follows that the only possible pole of f contained in W is the common landing point of $I_1(1/2)$ and $I_2(1/2)$ . Hence the only component of $f^{-1}(\Omega _1)$ (respectively $f^{-1}(\Omega _2)$ ) intersecting with W is $\Omega _1$ (respectively $\Omega _2$ ) itself.

For each $0\leq i\leq p$ , denote $z_i$ the landing point of $I_1(2^i\theta )$ and by $w_i$ the landing point of $I_2(2^i(1-\theta ))=I_2(1-2^i\theta )$ . Since $\theta $ is p-periodic, the points $z_0,\ldots ,z_{p-1}$ (respectively $w_0,\ldots ,w_{p-1}$ ) are pairwise disjoint and $z_0=z_p$ (respectively $w_0=w_p$ ). Moreover, the assumption of $\theta $ implies that $z_0,\ldots ,z_{p-1}, w_0,\ldots ,w_{p-1}\in W$ . Suppose $\theta \not \in \Theta $ . Then $z_0\not =w_0$ . As $\Theta $ is closed, we can choose an arc $\ell _0$ in $W\setminus \{I_1(t)\cup I_2(1-t):t\in \Theta \}$ joining the points $z_0=z_p$ and $w_0=w_p$ such that $\ell _0$ is disjoint with $\Omega _1\cup \Omega _2$ . Let $\ell _1$ be the lift of $\ell _0$ based at $z_{p-1}$ . By the choice of $\ell _0$ , we have

$$ \begin{align*} \ell_1\subset W\setminus \{I_1(t)\cup I_2(1-t):t\in\Theta\} \end{align*} $$

and

$$ \begin{align*} \ell_1\cap(\Omega_1\cup\Omega_2)=\emptyset. \end{align*} $$

Note that the endpoint of $\ell _1$ is on the boundary of a preimage of $\Omega _2$ . By the previous paragraph, this preimage is $\Omega _2$ itself. Note also that $w_{p-1}$ is the unique preimage of $w_p$ on $\partial \Omega _2$ such that $w_{p-1}$ and $z_{p-1}$ are in the same component of $W\setminus (I_1(1/2)\cup I_2(1/2))$ . Hence the endpoint of $\ell _1$ is $w_{p-1}$ .

Inductively, for each $m\geq 1$ , we get an arc $\ell _{mp}\subseteq W$ joining $z_0$ and $w_0$ which is a lift of $\ell _0$ by $f^{pm}$ . Choose $\ell _0$ such that it does not intersect the closure of the forward orbits of the critical points of f. Since f is hyperbolic, it is uniformly expanding near its Julia set. It follows that the length of $\ell _{mp}$ converges to $0$ as $m\to \infty $ . Then $z_0=w_0$ , a contradiction. Hence $\theta \in \Theta $ and statement (2) follows.

Now we prove statement $(3)$ . Note that $\alpha \in (0,1/2]$ . Suppose, on the contrary, that $\alpha =1/2$ . According to statement (1), the angles $1-1/2^n\in \Theta $ for all $n\geq 1$ . Choose an angle $\eta \in \Theta $ close to $1$ and define

$$ \begin{align*} \gamma(0,\eta):=I_1(0)\cup I_1(\eta)\cup I_2(1-\eta)\cup I_2(0). \end{align*} $$

Let $D_{\eta }$ be a component of $\widehat {{\mathbb C}}\setminus \gamma (0,\eta )$ contained in D. We can choose $\eta $ sufficiently close to $1$ such that $D_{\eta }$ contains no critical values of f. Since $\alpha =1/2$ , then $I_1(\eta /2)$ and $I_2(1-\eta /2)$ land at distinct points. Denote by $\Omega _1^{(1)}$ the component of $f^{-1}(\Omega _1)$ such that $I_2(1-\eta /2)$ and $I_1^{(1)}(\eta )$ land at a common point and denote $\Omega _2^{(1)}$ the component of $f^{-1}(\Omega _2)$ such that $I_1(\eta /2)$ and $I_2^{(1)}(1-\eta )$ land at a common point. Since f is hyperbolic, its Julia set contains no critical points. It follows that there exists a component $D_{\eta }'$ of $f^{-1}(D_{\eta })$ whose boundary contains the arc

$$ \begin{align*} I_1(1/2)\cup I_2(1/2)\cup I_1(\eta/2)\cup I_2(1-\eta/2)\cup I_2^{(1)}(1-\eta)\cup I_1^{(1)}(\eta). \end{align*} $$

Note that the two arcs $I_1(\eta /2)\cup I_2^{(1)}(1-\eta )$ and $I_2(1-\eta /2)\cup I_1^{(1)}(\eta )$ are disjoint and mapped to the same arc $I_1(\eta )\cup I_2(1-\eta )$ under f. Then the proper map $f:D_{\eta }'\to D_{\eta }$ has degree at least $2$ . It implies that $D_{\eta }'$ contains at least one critical point. Hence $D_{\eta }$ contains a critical value. It contradicts the choice of $D_{\eta }$ .

For the second part of statement (3), let $\theta _n:=1-1/(2^n-1)$ . Then $\theta _n$ is periodic with period n. If $0\le i<n-1$ , we have

$$ \begin{align*} 2^i\theta_n=1-2^i/(2^n-1)\in (1/2,1). \end{align*} $$

For $i=n-1$ , we have

$$ \begin{align*} 2^{n-1}\theta_n=\frac{1}{2}\bigg(1-\frac{1}{2^n-1}\bigg)\in(0,1/2). \end{align*} $$

Since $\alpha <1/2$ , it follows that $2^{n-1}\theta _n\in (\alpha ,1)$ for sufficiently large n. Then $\theta _n\in \Theta $ by statement (2), and hence $2^{n-1}\theta _n$ is as required.

5.1 Classification of hyperbolic components and known results

Let $f\in \mathrm {NM}_4$ be the Newton map of the quartic polynomial P. Then the finite fixed points of f are the zeros of P, and the critical points of f are the zeros of P and zeros of $P"$ . Hence zeros of P are the superattracting fixed points of f. We call any other (super)attracting cycles of f a free (super)attracting cycle. Then any free (super)attracting cycle has period at least $2$ . Moreover, we say a critical point c of f is additional if $P"(c)=0$ . Hence f has two additional critical points, counted with multiplicity. According to the orbits of the additional critical points, the hyperbolic components in the moduli space $\mathrm {nm}_4:=\mathrm {NM}_4/\mathrm {Aut}(\mathbb {C})$ belong to the following seven types, see [Reference Nie and Pilgrim22]. The same classification is also for hyperbolic components in $\mathrm {NM}_4$ .

Type A. Adjacent critical points. The two additional critical points belong to the same component of the immediate basin of a free (super)attracting cycle.

Type B. Bitransitive. Each of the two additional critical points belongs to the immediate basin of a free (super)attracting period cycle, with two distinct components.

Type C. Capture. Only one additional critical point belongs to the immediate basin of a free (super)attracting cycle, but the orbit of the other additional critical point eventually lies in this immediate basin.

Type D. Disjoint (super)attracting orbits. The two additional critical points belong to the immediate basins of two distinct free (super)attracting cycles.

Type IE. Immediate escape. Some additional critical point in the immediate basin of a root.

Type FE1. One future escape. Only one additional critical point in the basin (but not immediate basin) of a root, while the other additional critical point is in the immediate basin of a free (super)attracting cycle.

Type FE2. Two future escapes. The two additional critical points belong to the basins (but not immediate basins) of one or two roots.

The above classification is an analogy of that for quadratic rational maps [Reference Milnor17] and for cubic polynomials [Reference Milnor19].

Recall that a hyperbolic component in $\mathrm {nm}_4$ is bounded if it has a compact closure in $\mathrm {nm}_4$ . Since the type-D hyperbolic components have semi-algebraic boundaries, an arithmetic argument shows that such components are bounded.

In contrast, all hyperbolic components of type IE are unbounded.

In the remainder of this section, we give more bounded hyperbolic components in $\mathrm {nm}_4$ . In fact, we show the condition in Proposition 5.2 is also necessary.

5.2 Key lemmas

To prove Theorem 1.3, we need two key lemmas.

Let $\{f_n\}\subset \mathrm {NM}_4$ be a sequence converging to $f=H_f\hat f\in \overline {\mathrm {NM}}_4$ such that $\mathrm {Hole}(f)=\{\infty \}$ and $\deg \hat f=3$ . Then $f_n$ has a unique additional critical point $c_n$ converging to $\infty $ as $n\to \infty $ . We suppose that all maps $f_n$ are in a same hyperbolic component in $\mathrm {NM}_4$ and assume that $f_n$ has an attracting cycle $\mathcal {O}_n=\{w_n^{(0)},\ldots , w_n^{(m-1)}\}$ of period $m\ge 2$ . Our first lemma states the orbit of $c_n$ and the limit of $\mathcal {O}_n$ .

Proof. Denote by $r_{1,n},r_{2,n},r_{3,n}$ , and $r_{4,n}$ the roots of $f_n$ . Since $\mathrm {Hole}(f)=\{\infty \}$ and $\deg \hat f=3$ , we may assume $r_{4,n}\to \infty $ , as $n\to \infty $ , and for $1\le i\le 3$ , the point $r_{i,n}$ is outside the $\epsilon $ -neighborhood of $\infty $ for all large n. Define $M_n(z):=r_{4,n}z$ and let $g_n:=M_n^{-1}\circ f_n\circ M_n$ . Then $g_n\in \mathrm {NM}_4$ with roots at $r_{1,n}/r_{4,n},r_{2,n}/r_{4,n},r_{3,n}/r_{4,n}$ , and $1$ . Let $g=H_g\hat g$ be the degenerate Newton map of the polynomial $z^3(z-1)$ . Then $g_n$ converges locally uniformly to $\hat g$ away from $\mathrm {Hole}(g)=\{0\}$ . Note that $\hat g$ has a critical point at $\tilde c=1/2$ and $\tilde {c}$ is attracted to the attracting fixed point $0$ . Given any $k\geq 0$ , the point $\tilde c$ is not in $\mathrm { Hole}(g^k)=\bigcup _{i=0}^{k-1}\hat g^{-i}(0)$ . It follows that there exists $\epsilon _0=\epsilon _0(k)>0$ such that $|\hat g^j(\tilde c)|>\epsilon _0$ for all $0\le j\le k$ . By Lemma 2.1, we have $|g_n^j(\tilde c_n)|>\epsilon _0$ for all large n. Note that for the maps $f_n$ , we have $f^j_n(c_n)=M_n(g^j_n(\tilde {c}_n))$ . It follows that $|f^j_n(c_n)|>r_{4,n}\epsilon _0$ for all $0\le j\le k$ . Thus, statement (1) follows.

For statement (2), suppose in contrast that $\mathcal {O}=\{\infty \}$ . Then all $w_i^{(0)}$ s converge to $\infty $ . In the following argument, we may pass to subsequences if necessary to obtain limits. Relabeling the indices, we may assume $w_n^{(i)}/w_n^{(0)}$ does not converge to $0$ for all $0\le i\le m-1$ . Write $L_n(z)=w_n^{(0)}z$ . Then

$$ \begin{align*} \mathcal{O}^{\prime}_n:=\{1,w_n^{(1)}/w_n^{(0)},\ldots, w_n^{(m-1)}/w_n^{(0)}\} \end{align*} $$

is an attracting cycle of $h_n:=L_n^{-1}\circ f_n\circ L_n\in \mathrm {NM}_4$ . Denote by $\mathcal {O}'$ the limit of $\mathcal {O}_n'$ . Then $0\not \in \mathcal {O}'$ . Assume that $h_n\to h=H_h\hat h\in \overline {\mathrm {NM}}_4$ . Note that $\mathrm {Hole}(h)\subset \{0,\infty \}$ and $1\le \deg \hat h\le 2$ .

If $\deg \hat h=1$ , then at least three roots of $h_n$ collide to $0$ as $n\to \infty $ and the remaining root either collides to $0$ or diverges to $\infty $ . For otherwise, $\hat h$ would have degree $2$ . It follows that $\hat h(z)=3z/4$ or $h(z)=2z/3$ . Thus, $\hat h$ has an attracting fixed point at $0$ and a repelling fixed point at $\infty $ . Moreover, $\mathrm {Hole}(h^j)=\mathrm {Hole}(h)\subset \{0,\infty \}$ for all $j\ge 1$ . It follows that $\mathcal {O}'\cap \mathrm {Hole}(h)=\emptyset $ . Then by Lemma 2.2, the set $\mathcal {O}'$ is a non-repelling cycle of $\hat h$ . Note $1\in \mathcal {O}'$ is not a fixed point of $\hat h$ . It is a contradiction since all the periodic points of $\hat h$ are fixed points.

If $\deg \hat h=2$ , then $\mathrm {Hole}(h)=\{0\}$ . Moreover, $\hat h$ has an attracting fixed point at $0$ , a superattracting fixed point at the limit r of $r_4^{(n)}/w_n^{(0)}$ , and a repelling fixed point at $\infty $ . Since $0\not \in \mathcal {O}'$ , then $\mathcal {O}'\cap \mathrm {Hole}(h)=\emptyset $ . By Lemma 2.2, the set $\mathcal {O}'$ is a non-repelling cycle of $\hat h$ of period at least $2$ . It follows that $\hat h$ has at least three non-repelling cycles: two (super)attracting fixed points $0$ and r, and one non-repelling cycle $\mathcal {O}'$ . It contradicts the Fatou–Shishikura inequality (see [Reference Shishikura26]) which asserts that $\hat f$ has at most two non-repelling cycles. Therefore, we have $\mathcal {O}\not =\{\infty \}$ and the conclusion follows.

Now we prove statement (3). For $0\le j\le m-1$ , denote by $U(w_n^{(j)})$ the Fatou component of $f_n$ containing $w_n^{(j)}$ . Suppose in contrast that $c_n\in \bigcup _{j=0}^{m-1}U(w_n^{(j)})$ . Then $c_n\in U(w_n^{(j_0)})$ for some $0\le j_0\le m-1$ . By relabeling the index, we can assume that $j_0=0$ . If $w_n^{(0)}\to \infty $ , by statement (2), there exists $1\le j\le m-1$ such that $w_n^{(j)}\not \to \infty $ . It follows from Lemma 2.3 that the basin $U(w_n^{(j)})$ stays outside a neighborhood of $\infty $ for all large n. Since $f_n^{j}(c_n)\in U(w_n^{(j)})$ , statement (1) implies that $c_n\not \to \infty $ . If $w_n^{(0)}\not \to \infty $ , some $w_n^{(\ell )}$ with $1\le \ell \le m-1$ must converge to $\infty $ since $\infty \in \mathcal {O}$ . Again by Lemma 2.3, the basin $U(w_n^{(0)})$ stays outside a neighborhood of $\infty $ for all large n. Hence $c_n\not \to \infty $ . It contradicts the assumption that $c_n\to \infty $ . Hence $c_n\not \in \bigcup _{j=0}^{m-1}U(w_n^{(j)})$ .