#  From Hyperbolic to Parabolic Parameters
along Internal Rays

Yi-Chiuan Chen and Tomoki Kawahira  2010 Mathematics Subject Classification.
Primary 37F45; Secondary 37F99.

###### Abstract

For the quadratic family $f_{c}(z)=z^{2}+c$ with $c$ in a hyperbolic component
of the Mandelbrot set, it is known that every point in the Julia set moves
holomorphically. In this paper we give a uniform derivative estimate of such a
motion when the parameter $c$ converges to a parabolic parameter ${\hat{c}}$
radially; in other words, it stays within a bounded Poincaré distance from the
internal ray that lands on ${\hat{c}}$. We also show that the motion of each
point in the Julia set is uniformly one-sided Hölder continuous at ${\hat{c}}$
with exponent depending only on the petal number.

This paper is a parabolic counterpart of the authors’ paper “From Cantor to
semi-hyperbolic parameters along external rays” (Trans. Amer. Math. Soc. 372
(2019) pp. 7959–7992).

###### Contents

  1. 1 Introduction and main results
  2. 2 Radial access condition and S-cycles
  3. 3 Proof of the main theorem assuming three lemmas
  4. 4 Postcritical set and hyperbolic metric
  5. 5 Proof of Lemma A assuming Lemmas G and H
  6. 6 Proof of Lemma B
  7. 7 Proofs of Propositions 2.2 and 2.3
  8. 8 Proof of Lemma G
  9. 9 Proof of Proposition 2.4
  10. 10 Proof of Lemma H
  11. 11 Proof of Lemma D
  12. 12 Proof of Lemma C
  13. 13 Proofs of Theorems 1.2 and 1.3

##  1 Introduction and main results

### Hyperbolic components.

Let $\mathbb{M}$ be the Mandelbrot set, the connectedness locus of the
quadratic family

${\left\\{f_{c}:z\mapsto z^{2}+c\right\\}}_{c\,\in\,\mathbb{C}}.$

That is, the Julia set $J(f_{c})$ is connected if and only if
$c\in\mathbb{M}$. A parameter $c\in\mathbb{M}$ is called hyperbolic if $f_{c}$
has a (super-)attracting periodic point. Equivalently, there exist positive
numbers $\gamma_{c}$ and $\varepsilon_{c}$ such that
$|Df_{c}^{n}(z)|\geq\gamma_{c}(1+\varepsilon_{c})^{n}$ for any $n\geq 0$ and
$z\in J(f_{c})$. The set of hyperbolic parameters in $\mathbb{M}$ is an open
subset and its connected components are called hyperbolic components of the
Mandelbrot set. (The complement of $\mathbb{M}$ is also called a hyperbolic
component, but in this paper we only consider those contained in the
Mandelbrot set.)

Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ and $\mathbb{X}$ a
hyperbolic component of $\mathbb{M}$. Sullivan and Douady-Hubbard (see [DH,
Exposés XIV & XIX] and [Mi, Thm.6.5]) gave a uniformization of $\mathbb{X}$,
which is a canonical homeomorphism
$\Phi=\Phi_{\mathbb{X}}:\overline{\mathbb{D}}\to\overline{\mathbb{X}}$ such
that $\Phi|_{\mathbb{D}}:\mathbb{D}\to\mathbb{X}$ is an conformal isomorphism;
and for any $c=\Phi(\mu)$ with $\mu\in\overline{\mathbb{D}}-\\{1\\}$, the map
$f_{c}$ has a periodic point of multiplier $\mu$ with a common period. The
parameters $\Phi(0)$ and $\Phi(1)$ in $\overline{\mathbb{X}}$ are called the
center and the root of $\mathbb{X}$ respectively. Note that the Poincaré
distance in $\mathbb{X}$ is defined by pulling-back the Poincaré (hyperbolic)
metric $|dz|/(1-|z|^{2})$ on $\mathbb{D}$ by the isomorphism $\Phi$.

### Internal rays and thick internal rays.

For a given hyperbolic component $\mathbb{X}$ and a given real number
$\theta$, we define the internal ray $I(\theta)$ of angle $\theta$ by

$I(\theta)=I_{\mathbb{X}}(\theta):={\left\\{\Phi(re^{2\pi
i\theta})\in\mathbb{X}\,:\,0\leq r<1\right\\}}.$

The point

${\hat{c}}:=\Phi(e^{2\pi i\theta})\in\partial\mathbb{X}$

is called the landing point of $I(\theta)$. For a given $\delta\geq 0$, we
define the $\delta$-thick internal ray ${\mathcal{I}}(\theta,\delta)$ of angle
$\theta$ by the closed $\delta$-neighborhood of $I(\theta)$ in $\mathbb{X}$
with respect to the Poincaré distance. We say the parameter $c$ tends to
${\hat{c}}$ along a thick internal ray if there exists a $\delta\geq 0$ such
that $c$ stays in the $\delta$-thick internal ray
${\mathcal{I}}(\theta,\delta)$ while it tends to ${\hat{c}}$. It is rather
common to say that such a $c$ converges to ${\hat{c}}$ radially (after
McMullen [Mc2]) or non-tangentially. Indeed, for any angle
$A_{0}\in[0,\pi/2)$, if $c$ stays in the $\delta(A_{0})$-thick internal ray
with

$\delta(A_{0})=\frac{1}{2}\log\frac{1+\tan(A_{0}/2)}{1-\tan(A_{0}/2)}\in[0,\infty),$
(1.1)

then by letting $\mu_{c}:=\Phi^{-1}(c)$ and $\hat{\mu}:=\Phi^{-1}({\hat{c}})$
we have

${\left|\arg{\left(1-\frac{\mu_{c}}{\hat{\mu}}\right)}\right|}\leq A_{0}$

for $c$ sufficiently close to ${\hat{c}}$. In other words, $\mu_{c}$ stays in
the Stolz angle at $\hat{\mu}\in\partial\mathbb{D}$ with opening angle
$2A_{0}$. (See [P, p.7] and Figure 4 in the next section.)

### Holomorphic motion of the hyperbolic Julia sets.

It is well-known that there exists a holomorphic motion ([BR, L, Mc1, MSS]) of
the Julia sets over any hyperbolic component $\mathbb{X}$ of $\mathbb{M}$.
Indeed, we have an equivariant holomorphic motion as follows. For any base
point $\sigma\in\mathbb{X}$, there exists a unique map $H:\mathbb{X}\times
J(f_{\sigma})\to\mathbb{C}$ such that

  1. (1)

$H(\sigma,z)=z$ for any $z\in J(f_{\sigma})$.

  2. (2)

For any $c\in\mathbb{X}$, the map $z\mapsto{H}(c,z)$ is injective on
$J(f_{\sigma})$.

  3. (3)

For any $z\in J(f_{\sigma})$, the map $c\mapsto H(c,{z})$ is holomorphic on
$\mathbb{X}$.

  4. (4)

For any $c\in\mathbb{X}$, the map $h_{c}(z):=H(c,z)$ satisfies
$h_{c}(J(f_{\sigma}))=J(f_{c})$ and $f_{c}\circ h_{c}=h_{c}\circ f_{\sigma}$
on $J(f_{\sigma})$.

See [Mc1, §4] for more details. In this paper we choose the center
$\sigma:=\Phi(0)$ of $\mathbb{X}$ as the base point of the motion. We are
concerned with boundary behavior of such an equivariant holomorphic motion of
the Julia set $J(f_{\sigma})$ when $c\in\mathbb{X}$ tends to some
${\hat{c}}\in\partial\mathbb{X}$ along a thick internal ray.

### Parabolic parameters.

Now suppose that the angle $\theta$ of the internal ray $I(\theta)$ is a
rational number. Then for the landing point ${\hat{c}}$ of $I(\theta)$,
$f_{\hat{c}}$ has a parabolic periodic point, that is, a periodic point whose
multiplier is a root of unity. We say such a parameter ${\hat{c}}$ is
parabolic, and a parabolic periodic point ${\hat{b}}$ of $f_{\hat{c}}$ has $q$
petals if the local dynamics of $f_{\hat{c}}^{k}$ near ${\hat{b}}$ is of the
form $\zeta\mapsto\zeta+\zeta^{q+1}+O(\zeta^{q+2})$ for some $k$ in an
appropriate local coordinate. In our setting, it is known that ${\hat{b}}$ has
$q$ petals if and only if the multiplier of ${\hat{b}}$ is a primitive $q$-th
root of unity (since $f_{\hat{c}}$ is quadratic and has only one critical
point in $\mathbb{C}$).

Figure 1: Internal rays (a) – (f) of the Mandelbrot set.

######  Example 1 (Period one, the main cardioid).

For the hyperbolic component $\mathbb{X}_{1}$ containing $0$ (the main
cardioid), the map $\Phi=\Phi_{\mathbb{X}_{1}}$ is explicitly given by
$\Phi(\mu):=-\mu^{2}/4+\mu/2~{}(\mu\in\overline{\mathbb{D}})$ and
$f_{\Phi(\mu)}$ has a fixed point with multiplier $\mu$. The internal ray of
angle $\theta$ is given by
$I_{\mathbb{X}_{1}}(\theta)={\left\\{-r^{2}e^{4\pi\theta i}/4+re^{2\pi\theta
i}/2\,:\,0\leq r<1\right\\}}.$ In Figure 1, $I_{\mathbb{X}_{1}}(\theta)$ for
$\theta=0,1/2$ and $1/3$ are depicted as paths (a), (b), and (c) respectively.
The corresponding holomorphic motions along $I_{\mathbb{X}_{1}}(0)$ and
$I_{\mathbb{X}_{1}}(1/2)$ are illustrated in Figure 2 and in Figures 3(a) and
3(b). The motion along $I_{\mathbb{X}_{1}}(1/3)$ is depicted in Figure 3(c).

######  Example 2 (Period two).

Similarly the hyperbolic component $\mathbb{X}_{2}$ containing $-1$ consists
of hyperbolic parameters $c$ such that $f_{c}$ has an attracting cycle of
period two. The map $\Phi=\Phi_{\mathbb{X}_{2}}$ is explicitly given by
$\Phi(\mu):=\mu/4-1$ for $\mu\in\overline{\mathbb{D}}$, and $f_{\Phi(\mu)}$
with $\mu\in\overline{\mathbb{D}}-\\{1\\}$ has a periodic point of period two
with multiplier $\mu$. The internal ray of angle $\theta$ is given by
$I_{\mathbb{X}_{2}}(\theta)={\left\\{re^{2\pi\theta i}/4-1\,:\,0\leq
r<1\right\\}}.$ In Figure 1, $I_{\mathbb{X}_{2}}(\theta)$ for $\theta=0$ and
$1/3$ are depicted as paths (d) and (e) respectively. The internal ray
$I_{\mathbb{X}_{2}}(0)$ lands at the root ${\hat{c}}=\Phi(1)=-3/4$ of
$\mathbb{X}_{2}$, where the map $f_{\hat{c}}$ has a parabolic fixed point of
multiplier $-1$ that has two petals. Note that ${\hat{c}}$ is the landing
point of another internal ray $I_{\mathbb{X}_{1}}(1/2)$ of $\mathbb{X}_{1}$.
See Figures 3(d) and 3(e) for the motions along $I_{\mathbb{X}_{2}}(0)$ and
$I_{\mathbb{X}_{2}}(1/3)$.

######  Example 3 (Period three).

There is a hyperbolic component $\mathbb{X}_{3}$ attached to the main cardioid
$\mathbb{X}_{1}$ that consists of parameters $c$ with $\mathrm{Im}\,c>0$ for
which $f_{c}$ has an attracting cycle of period three. The internal ray
$I_{\mathbb{X}_{3}}(0)$ (depicted as path (f) in Figure 1) joins the center
(so-called “rabbit”) and the root ${\hat{c}}=\Phi(1)$ (“the fat rabbit”),
where the map $f_{\hat{c}}$ has a parabolic fixed point of multiplier $e^{2\pi
i/3}$ that has three petals. (See Figure 3(f) for the holomorphic motion along
$I_{\mathbb{X}_{3}}(0)$.) Again ${\hat{c}}$ is the landing point of another
internal ray $I_{\mathbb{X}_{1}}(1/3)$ of $\mathbb{X}_{1}$.

Figure 2: Real analytic motion of the preimages of the repelling fixed point
along the internal rays $I(1/2)$ (left) and $I(0)$ (right) in
$\mathbb{X}_{1}$.

### Main results.

Let $\mathbb{X}$ be a hyperbolic component in the Mandelbrot set and $\sigma$
be its center. For any $z_{\ast}$ in $J(f_{\sigma})$, the map $c\mapsto
z(c):=H(c,z_{\ast})$ is holomorphic over $\mathbb{X}$. Let
${\hat{c}}\in\partial\mathbb{X}$ be the landing point of the internal ray
$I_{\mathbb{X}}(\theta)$ of rational angle $\theta$. Our main theorem states
that the speed of $z(c)=H(c,z_{\ast})$ is uniformly bounded by a function of
$|c-{\hat{c}}|$ as $c$ tends to ${\hat{c}}$ along a thick internal ray:

######  Theorem 1.1 (Main Theorem).

Suppose that $f_{\hat{c}}$ has a parabolic periodic point with $q$ petals and
$c$ tends to ${\hat{c}}$ along a thick internal ray
${\mathcal{I}}(\theta,\delta)$ in $\mathbb{X}$. Then there exists a constant
$K>0$ depending only on ${\hat{c}}$ and $\delta$ such that for any $z=z(c)\in
J(f_{c})$, the point $z(c)$ moves holomorphically with

${\left|\frac{d}{dc}z(c)\right|}\leq\frac{K}{{\left|\,c-{\hat{c}}\,\right|}^{1-1/Q}},$

where $Q=\max\\{2,q\\}$.

By this theorem we obtain one-sided Hölder continuity of the holomorphic
motion along thick internal rays landing on parabolic parameters:

######  Theorem 1.2 (One-sided Hölder Continuity).

Under the same assumption as Theorem 1.1 above, the point $z=z(c)$ in
$J(f_{c})$ tends to a limit $z({\hat{c}})$ in $J(f_{\hat{c}})$ as $c$ tends to
${\hat{c}}$ along the thick internal ray ${\mathcal{I}}(\theta,\delta)$.
Moreover, there exists a constant $K^{\prime}$ depending only on ${\hat{c}}$
and $\delta$ such that

$|z(c)-z({\hat{c}})|\leq K^{\prime}|\,c-{\hat{c}}\,|^{1/Q}$ (1.2)

for any $c$ in ${\mathcal{I}}(\theta,\delta)$.

As an immediate consequence the holomorphic motion of each point $z(c)\in
J(f_{c})$ lands when $c$ moves along the internal ray
$I(\theta)={\mathcal{I}}(\theta,0)$. This theorem yields a precise description
of the degeneration of the dynamics on the Julia sets along the internal rays
of rational angles:

######  Theorem 1.3 (Pinching Semiconjugacy).

Under the same assumption as the theorems above, the conjugacy
$H(c,\cdot)=h_{c}:J(f_{\sigma})\to J(f_{c})$ converges uniformly to a
semiconjugacy $h_{{\hat{c}}}:J(f_{\sigma})\to J(f_{\hat{c}})$ from
$f_{\sigma}$ to $f_{{\hat{c}}}$ as $c$ tends to ${\hat{c}}$ along the thick
internal ray $I(\theta,\delta)$. Moreover, $h_{\hat{c}}$ satisfies the
following:

  1. (1)

If ${\hat{c}}$ is the root of $\mathbb{X}$, then $h_{{\hat{c}}}$ is injective
and thus a conjugacy.

  2. (2)

If ${\hat{c}}$ is not the root of $\mathbb{X}$ (hence $q\geq 2)$, then the
preimage $h_{\hat{c}}^{-1}(\\{w\\})$ of any $w\in J(f_{\hat{c}})$ consists of
one or $q$ distinct points, and the latter holds if and only if $w$ eventually
lands on a parabolic periodic point of $f_{\hat{c}}$.

  3. (3)

The semiconjugacy $\eta_{c}:=h_{\hat{c}}\circ h_{c}^{-1}:J(f_{c})\to
J(f_{\hat{c}})$ satisfies

$|\eta_{c}(z)-z|\leq K^{\prime}|c-{\hat{c}}|^{1/Q}$ (1.3)

for any $c$ in the thick internal ray ${\mathcal{I}}(\theta,\delta)$.

By (3) of this theorem we obtain:

######  Corollary 1.4 (Hausdorff Convergence).

The Hausdorff distance between $J(f_{c})$ and $J(f_{\hat{c}})$ is
$O(|c-{\hat{c}}|^{1/Q})$ as $c$ tends to ${\hat{c}}$ along a thick internal
ray.

######  Remark 1.5.

  * •

These results are parabolic counterparts of the authors’ results in [CK1]
about parameter rays (external rays) landing on semi-hyperbolic parameters of
the Mandelbrot set.

  * •

For any $c\in\mathbb{X}$ and $z=z(c)\in J(f_{c})$, we have

${\left|\frac{d}{dc}{z}(c)\right|}\leq\frac{\,1+\sqrt{1+6\,|c|\,}\,}{\mathrm{dist}\,(c,\partial\mathbb{X})}$
(1.4)

by Proposition 3.1 in [CK1]. However, this will only give

${\left|\frac{d}{dc}{z}(c)\right|}=O{\left(\frac{1}{|c-{\hat{c}}|}\right)}$

as $c$ tends to ${\hat{c}}$.

  * •

In [CK2], the authors showed that for any $c\in[0,1/4)=I_{\mathbb{X}_{1}}(0)$
and $z=z(c)\in J(f_{c})$, we have an optimal estimate

${\left|\frac{d}{dc}{z}(c)\right|}\leq\frac{1}{2\sqrt{1/4-c}}.$

In particular, the Hausdorff distance between $J(f_{c})$ and $J(f_{1/4})$ is
exactly $\sqrt{1/4-c}$.

  * •

The existence of the semiconjugacy in Theorem 1.3 and the Hausdorff
convergence of the Julia sets in Corollary 1.4 are previously shown in a more
general context by the second author [K1] and McMullen [Mc2] respectively. The
novel part of our results is the quantitative estimate
$O(|c-{\hat{c}}|^{1/Q})$.

### Structure of the paper.

In Section 2, we define a parametrization of $c$ with a complex parameter
$t\in\mathbb{C}$ such that $c=c_{t}$ converges to ${\hat{c}}$ along a thick
internal ray as $t\to 0$. Also in Section 2, we state three propositions,
Propositions 2.2, 2.3 and 2.4, which concern the local dynamics of $f_{c}$ in
a neighborhood $U_{0}$ of a parabolic point of $f_{\hat{c}}$ when $c$ is near
${\hat{c}}$, and will be employed to prove Theorems 1.2 and 1.3 as well as
some lemmas in the paper. Then, we introduce the notion of “S-cycle” to
describe how an orbit of $f_{c}$ repeatedly (infinite or finite times or
never) enters and leaves a fixed subset of $U_{0}$. In Section 3, by assuming
Lemmas A, B and C, we prove our main theorem, Theorem 1.1. It is well-known
(for example [Mc1, §3.2]) that the Julia set is expanding with respect to the
hyperbolic metric on $\mathbb{C}-P(f_{c})$, where $P(f_{c})$ denotes the
postcritical set of $f_{c}$. In order to estimate the expansion of the Julia
set with respect to the Euclidean metric, we give an estimate of the distance
between $z\in J(f_{c})$ and $P(f_{\hat{c}})$ in Lemma D in Section 4 for $z$
not too close to the parabolic cycle of $f_{\hat{c}}$. Lemma A is proved in
Section 5 by assuming another two lemmas, Lemmas G and H. Both lemmas rely on
local dynamics of perturbed parabolic cycle. We prove Lemma B in Section 6,
and Lemma G in Section 8. Section 7 is devoted to the proofs of Propositions
2.2 and 2.3. We use a branched coordinate to prove Proposition 2.4 in Section
9. We also employ the branched coordinate to prove Lemma H in Section 10.
Then, using some results presented in Section 10, we are able to prove Lemma D
in Section 11. Some arguments in the proofs of Lemmas H and D are used to
prove Lemma C in Section 12. Finally, in Section 13 we prove Theorems 1.2 and
1.3 simultaneously.

(a): $I(0)$ in $\mathbb{X}_{1}$

(b): $I(1/2)$ in $\mathbb{X}_{1}$

(c): $I(1/3)$ in $\mathbb{X}_{1}$

(d): $I(0)$ in $\mathbb{X}_{2}$

(e): $I(1/3)$ in $\mathbb{X}_{2}$

(f): $I(0)$ in $\mathbb{X}_{3}$

Figure 3:  Holomorphic motions along the internal rays depicted in Figure 1.

##  2 Radial access condition and S-cycles

In this section we introduce the notion of S-cycles for a given orbit in the
Julia set. The idea of S-cycles was introduced in [CK1] to describe orbits
that repeatedly come close to the postcritical set. Here we present a modified
version where the postcritical set is replaced by the parabolic cycle.

### Notation.

We start with some notation and the terminology that will be used in what
follows.

  * •

Let $\mathbb{N}$ denote the set of positive integers. We denote the set of
non-negative integers by $\mathbb{N}_{0}:=\\{0\\}\cup\mathbb{N}$.

  * •

Let $\mathbb{D}(a,r)$ denote the disk in $\mathbb{C}$ centered at $a$ and of
radius $r>0$. When $a=0$ we denote it by $\mathbb{D}(r)$.

  * •

For non-negative variables $X$ and $Y$, by $X\asymp Y$ we mean there exists an
implicit constant $C>1$ independent of $X$ and $Y$ such that $X/C\leq Y\leq
CX$.

  * •

When we say “for any $X\ll 1$” it means that “for any sufficiently small
$X>0$”. More precisely, we mean there exists an implicit constant $C>0$ such
that $0<X<C$.

### Hyperbolic components and internal rays.

Let ${\hat{c}}\in\partial\mathbb{M}$ be a parabolic parameter having a
parabolic periodic point ${\hat{b}}$ of period exactly $p$. Let
$\hat{\lambda}:=Df_{{\hat{c}}}^{p}({\hat{b}})$ be the multiplier of this
cycle, and assume that it is a primitive $q$-th root of unity. We specify an
internal ray $I(\theta)=I_{\mathbb{X}}(\theta)$ of the hyperbolic component
$\mathbb{X}$ that lands at ${\hat{c}}$ as follows. (See [DH] or [Mi] for
details on the hyperbolic components of $\mathbb{M}$.)

### Case 1.

If $q=1$, then there is only one hyperbolic component $\mathbb{X}$ such that
${\hat{c}}=\Phi_{\mathbb{X}}(1)\in\partial\mathbb{X}$, where
$\Phi_{\mathbb{X}}$ is the uniformizing map of $\mathbb{X}$. Hence by letting
$\theta=0$ the internal ray $I(\theta)$ of $\mathbb{X}$ lands at ${\hat{c}}$.

### Case 2.

If $q\geq 2$, then there are exactly two hyperbolic components
$\mathbb{X}^{-}$ and $\mathbb{X}^{+}$ such that

  * •

$\partial\mathbb{X}^{-}\cap\partial\mathbb{X}^{+}=\\{{\hat{c}}\\}$.

  * •

${\hat{c}}=\Phi_{\mathbb{X}^{-}}(\hat{\lambda})$ and
${\hat{c}}=\Phi_{\mathbb{X}^{+}}(1)$, where $\Phi_{\mathbb{X}^{\pm}}$ is the
uniformizing map of $\mathbb{X}^{\pm}$.

Hence $\mathbb{X}$ can be either $\mathbb{X}^{-}$ or $\mathbb{X}^{+}$, and the
case of $q\geq 2$ is divided into two sub-cases.

  * •

Case 2-: If $\mathbb{X}=\mathbb{X}^{-}$, then we let
$\theta=(\arg\hat{\lambda})/(2\pi)$; and

  * •

Case 2+: If $\mathbb{X}=\mathbb{X}^{+}$, then we let $\theta=0$

in such a way that the internal ray $I(\theta)=I_{\mathbb{X}}(\theta)$ of
$\mathbb{X}$ lands at ${\hat{c}}$.

Note that ${\hat{c}}$ is the root of $\mathbb{X}$ if and only if it is as Case
1 or Case 2+. Hence,

$\theta=\left\\{\begin{array}[]{ll}0&\text{for {Case 1} or {Case
2${}^{+}$},}\\\\[3.01385pt] (\arg\hat{\lambda})/(2\pi)&\text{for {Case
2${}^{-}$}.}\end{array}\right.$

######  Example 4.

Case 1 holds when ${\hat{c}}=1/4$ with $\mathbb{X}=\mathbb{X}_{1}$ (Example
1), or ${\hat{c}}=-7/4$ with $\mathbb{X}$ whose center is a unique real
parameter $\sigma<0$ with $f_{\sigma}^{3}(0)=0$ (“the airplane”). If
$\hat{c}=-3/4$, Case 2- holds when $\mathbb{X}^{-}=\mathbb{X}_{1}$, and Case
2+ holds when $\mathbb{X}^{+}=\mathbb{X}_{2}$ (Example 2).

Figure 4: The Stolz angle $S(\Delta)$ with opening angle $2A_{0}$ equal to
$\pi/3$ at $\hat{\lambda}\in\partial\mathbb{D}$. The dark gray region is the
image of $\delta$-thick internal ray ${\mathcal{I}}(\theta,\delta)$ under
$\Phi^{-1}$ with $\delta=\delta(A_{0})$.

### Radial convergence and thick internal rays.

Let $A_{0}$ and $T_{0}$ be constants with $0\leq A_{0}<\pi/2$ and
$0<T_{0}<2\cos A_{0}$, and let

$\Delta=\Delta(A_{0},T_{0}):=\\{t\in\mathbb{C}\,:\,0<|t|\leq T_{0},~{}|\arg
t\,|\leq A_{0}\\}.$

The Stolz angle at $\hat{\lambda}$ with opening angle $2A_{0}$ is given by

$S(\Delta):=\\{\mu=(1-t)\hat{\lambda}\in\mathbb{D}\,:\,t\in\Delta\\}\subset\mathbb{D}.$
(2.1)

Let $\Phi:=\Phi_{\mathbb{X}}$. If the parameter $c\in\mathbb{X}$ tends to
${\hat{c}}$ satisfying $\Phi(c)\in S(\Delta)$, we say $c\to{\hat{c}}$ radially
after McMullen [Mc2].

For a given $\delta$-thick internal ray
${\mathcal{I}}(\theta,\delta)\subset\mathbb{X}$ of angle $\theta$, one can
easily check that

$\Phi^{-1}({\mathcal{I}}(\theta,\delta))\cap\mathbb{D}(\hat{\lambda},T_{0})\subset
S(\Delta)$

if $\delta\leq\delta(A_{0})$, which is given in (1.1). See Figure 4. Hence in
what follows it is enough to consider the parameters of the form $c=\Phi(\mu)$
with $\mu\in S(\Delta)\cup\\{\hat{\lambda}\\}$.

######  Remark 2.1.

Conversely, the Stolz angle $S(\Delta)$ is contained in
$\Phi^{-1}({\mathcal{I}}(\theta,\delta))$ with $\delta=\delta(A_{0})+1$ by
taking a sufficiently small $T_{0}$. This implies that the convergence in a
thick internal ray is equivalent to the radial convergence for some angle.

### Parametrization and notation.

For a technical reason, instead of (2.1), it is convenient to re-parametrize
$\mu\in S(\Delta)$ by

$\mu=\mu_{t}:=\left\\{\begin{array}[]{ll}1-q\,t&\text{in {Case 1} or {Case
2${}^{+}$}}\\\\[3.01385pt] (1-t/q)\hat{\lambda}&\text{in {Case
2${}^{-}$}}\end{array}\right.$

for $t\in\Delta=\Delta(A_{0},T_{0})$ with sufficiently small $T_{0}$.

### The radial access condition.

In what follows, by

$c\approx{\hat{c}}\quad\text{or}\quad c=c_{t}\approx{\hat{c}}$

we mean the parameter $c$ is of the form

$c=c_{t}=\Phi(\mu_{t})$

for some $t\in\Delta=\Delta(A_{0},T_{0})$, where we take a smaller $T_{0}$ in
the definition of $\Delta$ if necessary. We say such a parameter $c$ satisfies
the radial access condition or $c$ is in a thick internal ray.

### Perturbation of parabolic points.

Let $c_{0}:={\hat{c}}$ and $\lambda_{0}:=\hat{\lambda}$. In Section 7, we will
show the following two propositions under the radial access condition:

######  Proposition 2.2.

The function $t\mapsto c=c_{t}$ is holomorphic on $\Delta$, and there exists a
constant $B_{0}\neq 0$ such that

  * •

Case 1 ($q=1$): $c_{t}={\hat{c}}+B_{0}t^{2}+O(t^{3})$

  * •

Case 2± ($q\geq 2$): $c_{t}={\hat{c}}+B_{0}t+O(t^{2})$

as $t\in\Delta$ tends to $0$. In particular, we have
$\sqrt{|c_{t}-{\hat{c}}|}\asymp|t|$ or $|c_{t}-{\hat{c}}|\asymp|t|$ for
$t\in\Delta$ according to Case 1 or Case 2±.

######  Proposition 2.3.

There exists a continuous map $t\mapsto b_{t}$ defined for
$t\in\Delta\cup\\{0\\}$ such that

  1. (1)

$b_{0}=\hat{b}$ and $b_{t}$ is a periodic point of $f_{c_{t}}$ with the same
period $p$ as ${\hat{b}}$.

  2. (2)

Let $\lambda_{t}:=Df_{c_{t}}^{p}(b_{t})$. Then there exist two families of
holomorphic local coordinates
${\left\\{\zeta=\varphi_{t}(z)\right\\}}_{t\,\in\,\Delta\cup\\{0\\}}$ and
${\left\\{w=\psi_{t}(z)\right\\}}_{t\,\in\,\Delta\cup\\{0\\}}$ defined on a
disk $\hat{U}:=\mathbb{D}({\hat{b}},\hat{R})$ such that: For each
$z\in\hat{U}$, $\varphi_{t}(z)$ and $\psi_{t}(z)$ are holomorphic in
$t\in\Delta$ and continuous at $t=0$; $\varphi_{t}(b_{t})=\psi_{t}(b_{t})=0$;
and

$\displaystyle\varphi_{t}\circ f_{c_{t}}^{p}\circ\varphi_{t}^{-1}(\zeta)$
$\displaystyle=\lambda_{t}\zeta+\zeta^{q+1}+O(\zeta^{2q+1}),~{}\text{and}$
(2.2) $\displaystyle\psi_{t}\circ f_{c_{t}}^{pq}\circ\psi_{t}^{-1}(w)$
$\displaystyle=\lambda_{t}^{q}w\,(1+w^{q}+O(w^{2q})).$ (2.3)

In particular, both $D\varphi_{t}^{-1}(0)$ and $D\psi_{t}^{-1}(0)$ are
uniformly bounded away from zero.

  3. (3)

In Case 1 or Case 2+, $b_{t}$ is repelling for $t\in\Delta$, and the
multiplier satisfies

$\lambda_{t}=(1+t/q+o(t))\,\hat{\lambda}.$

Moreover, there are $q$ distinct attracting fixed points
$\alpha_{t}^{1},\,\cdots,\,\alpha_{t}^{q}$ of $f_{c_{t}}^{pq}$ satisfying
$Df_{c_{t}}^{pq}(\alpha_{t}^{j})=\mu_{t}=1-qt$ and
$\left(\psi_{t}(\alpha_{t}^{j})\right)^{q}=-t+o(t)$ for $j=1,\cdots,q$.

  4. (4)

In Case 2-, $b_{t}$ is attracting for $t\in\Delta$ and the multiplier
satisfies

$\lambda_{t}=\mu_{t}=(1-t/q)\,\hat{\lambda}.$

Moreover, there are $q$ distinct repelling fixed points
$\beta_{t}^{1},\,\cdots,\,\beta_{t}^{q}$ of $f_{c_{t}}^{pq}$ satisfying
$Df_{c_{t}}^{pq}(\beta_{t}^{j})=1+qt+o(t)$ and
$\left(\psi_{t}(\beta_{t}^{j})\right)^{q}=t+o(t)$ for $j=1,\cdots,q$.

The local dynamics of $f_{c_{t}}^{pq}$ observed as (2.3) behaves quite similar
to that of $f_{{\hat{c}}}^{pq}$. See Figure 5. In particular, in the domain
$\hat{U}=\mathbb{D}({\hat{b}},\hat{R})$ of $\psi_{t}$, the map
$f_{c_{t}}^{pq}$ has exactly

  * •

one repelling fixed point $b_{t}$ in Case 1 or Case 2+; and

  * •

$q$ repelling fixed points in Case 2- that are symmetrically arrayed near
$b_{t}$.

Figure 5:  The middle frame depicts the local dynamics of $f_{{\hat{c}}}^{pq}$
near $\hat{U}$ (indicated by the big circle) that can be mildly perturbed in
two ways: Case 2- on the left, and Case 2+ on the right. The white, black, and
gray dots indicate attracting, repelling, and parabolic fixed points
respectively. The red spiky curves indicate the Julia sets.

### Definition of $U_{0}$.

We fix a small $R_{0}\in(0,\hat{R})$ such that the disk

$U_{0}:=\mathbb{D}({\hat{b}},R_{0})\quad(\Subset\hat{U})$

possesses the property that $f_{\hat{c}}^{p}:U^{\prime}\to U_{0}$ is univalent
for each connected component $U^{\prime}$ of $f_{\hat{c}}^{-p}(U_{0})$. Such
an $R_{0}$ exists because the orbit $f_{\hat{c}}^{n}(0)$ of $0$ (the critical
point) keeps a definite distance from the parabolic cycle for $0\leq n\leq p$.
The next proposition will be proved in Section 9:

######  Proposition 2.4.

For each parameter $c_{t}$ with $t\in\Delta$ if $z_{0}\in J(f_{c_{t}})\cap
U_{0}$ is none of the repelling fixed points of $f_{c_{t}}^{pq}$ described as
above, then the orbit $z_{kpq}:=f_{c_{t}}^{kpq}(z_{0})~{}(k\in\mathbb{N}_{0})$
leaves $U_{0}$ for some $k>0$.

We choose some $\xi\in(0,1]$ such that

$\mathrm{dist}\,(0,f_{\hat{c}}^{k}(U_{0}))\geq\frac{3\xi}{4}$

for any integer $k$ with $-p\leq k\leq p$. Note that by continuity, we have

$\mathrm{dist}\,(0,f_{c}^{k}(U_{0}))\geq\frac{\xi}{2}$

for any $c\approx{\hat{c}}$ (taking a smaller $T_{0}$ in the definition of
$\Delta$ if necessary) and any $k$ with $-p\leq k\leq p$.

######  Remark 2.5.

We will frequently use the following property: If $z\in f_{c}^{k}(U_{0})$ for
some $c\approx{\hat{c}}$ and $k$ with $-p\leq k\leq p$, then
$|Df_{c}(z)|=2|z|\geq\xi$.

### Definition of $V_{0}$ and ${\mathcal{V}}(c)$.

Now we take a small $\nu\in(0,R_{0})$ and let

$V_{0}:=\mathbb{D}({\hat{b}},\nu)\quad(\Subset U_{0}).$

We also define

${\mathcal{V}}(c):=\bigcup_{j=0}^{p-1}f_{c}^{-j}(V_{0})$

for each $c\approx{\hat{c}}$. By Remark 2.5 above,
$f_{c}(z)\in{\mathcal{V}}(c)$ implies $|Df_{c}(z)|\geq\xi$ for
$c\approx{\hat{c}}$.

### S-cycles.

For $c\approx{\hat{c}}$, let $z_{0}$ be any point in the Julia set $J(f_{c})$.
The orbit $z_{n}:=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$ may land on $V_{0}$
($\Subset U_{0}$), and leave $U_{0}$ by Proposition 2.4 (unless it lands
exactly on the repelling cycle), then it may come back to $V_{0}$ again. To
describe the behavior of such an orbit, we introduce the notion of “S-cycle”
for the orbit of $z_{0}$, where “S” indicates that orbit stays near the
“singularity” of the hyperbolic metric $\rho(z)|dz|$ on the complement of the
postcritical set of $f_{\hat{c}}$ to be defined in Section 4.

### Definition (S-cycle).

A finite S-cycle of the orbit $z_{n}=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$
is a finite subset of $\mathbb{N}_{0}$ of the form

${\sf S}={\left\\{n\in\mathbb{N}_{0}\,:\,M\leq
n<M^{\prime}\right\\}}=[M,M^{\prime})\cap\mathbb{N}_{0}$

with the following properties:

  * (S1)

$z_{M}\in V_{0}$, and if $M>0$ then $z_{M-1}\notin V_{0}$.

  * (S2)

There exists a minimal $m\geq 1$ such that for $n=M+mpq$, $z_{n-pq}\in U_{0}$
but $z_{n}\notin U_{0}$.

  * (S3)

$M^{\prime}=M+mpq+L$ for some $L\in[1,\infty)$ such that $z_{n}\notin V_{0}$
for $n=M+mpq+i~{}(0\leq i<L)$ and $z_{M^{\prime}}\in V_{0}$.

An infinite S-cycle ${\sf S}$ of the orbit
$z_{n}=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$ is an infinite subset of
$\mathbb{N}_{0}$ of the form

${\sf S}={\left\\{n\in\mathbb{N}_{0}\,:\,M\leq
n<\infty\right\\}}=[M,\infty)\cap\mathbb{N}_{0},$

satisfying either

  * •

Type (I): (S1), (S2), and

    * (S3)’

$z_{n}\notin V_{0}$ for all $n\geq M+mpq$;

or

  * •

Type (II): (S1) and

    * (S2)’

$z_{M+kpq}\in V_{0}$ for any $k\in\mathbb{N}$. Equivalently, $z_{M}$ is a
repelling periodic point of period $p$ or period $pq$ in $V_{0}$ (by
Proposition 2.4).

By an S-cycle we mean a finite or infinite S-cycle. In both cases, we denote
them by ${\sf S}=[M,M^{\prime})$ or ${\sf S}=[M,\infty)$ for brevity.

######  Remark 2.6.

We may assume without loss of generality that $L$ of the finite S-cycle is at
least $p$ by shrinking the radius of the disk $U_{0}$. Indeed, after the orbit
$z_{n}~{}(n\in\mathbb{N}_{0})$ leaves $U_{0}$ when $n=M+mpq$, the orbit
follows the parabolic cycle for a while and it cannot land immediately on
$V_{0}$ by the local dynamics near the perturbed cycle. See Figure 6.

Figure 6: A finite S-cycle $[M,M^{\prime})$ with $M^{\prime}=M+mpq+L$, $p=3$.
The dotted arrow indicates there is another connected component of
$f_{c}^{-p}(U_{0})$ intersecting with $U_{0}$ which is not drawn.

### Decomposition of the orbit by S-cycles.

For a given orbit $z_{n}=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$ of $z_{0}\in
J(f_{c})$, the set $\mathbb{N}_{0}$ of indices is uniquely decomposed by using
finite or infinite S-cycles in one of the following three types:

  * •

The first type is of the form

$\mathbb{N}_{0}=[0,M_{1})\sqcup{\sf S}_{1}\sqcup{\sf S}_{2}\sqcup\cdots,$
(2.4)

where $z_{n}\notin V_{0}$ for $n\in[0,M_{1})$ and ${\sf
S}_{k}:=[M_{k},M_{k+1})$ is a finite S-cycle for each $k\geq 1$.

  * •

The second type is of the form

$\mathbb{N}_{0}=[0,M_{1})\sqcup{\sf S}_{1}\sqcup{\sf
S}_{2}\sqcup\cdots\sqcup{\sf
S}_{k_{0}}\sqcup\emptyset\sqcup\emptyset\sqcup\cdots$ (2.5)

with $k_{0}\geq 1$, where $z_{n}\notin V_{0}$ for $n\in[0,M_{1})$; ${\sf
S}_{k}:=[M_{k},M_{k+1})$ is a finite S-cycle for each $1\leq k\leq k_{0}-1$;
${\sf S}_{k_{0}}=[M_{k_{0}},\infty)$ is an infinite S-cycle; and ${\sf
S}_{k}=\emptyset$ for $k\geq k_{0}+1$.

  * •

The third type is

$\mathbb{N}_{0}=[0,M_{1})\sqcup\emptyset\sqcup\emptyset\sqcup\cdots$ (2.6)

with $M_{1}=\infty$ and ${\sf S}_{k}=\emptyset$ for $k\geq 1$, where
$z_{n}\notin V_{0}$ for all $n\in\mathbb{N}$.

In the first and second types it is possible that $M_{1}=0$ and $[0,M_{1})$ is
empty.

##  3 Proof of the main theorem assuming three lemmas

### The derivative formula.

Let $z_{\ast}$ be any point in $J(f_{\sigma})$ (where $\sigma$ is the center
of $\mathbb{X}$), and consider its motion $z=z(c)=H(c,z_{\ast})\in J(f_{c})$
for $c\in\mathbb{X}$. The estimate of the main theorem is based on the
following formula:

######  Proposition 3.1 (The Derivative Formula).

For any $c\in\mathbb{X}$ and $z=z(c)\in J(f_{c})$, we have

$\frac{d}{dc}{z}(c)=-\sum_{n=1}^{\infty}\frac{1}{Df_{c}^{{n}}({z}(c))}.$

See [CK1, Proposition 3.2] for the proof. Since $f_{c}$ is hyperbolic, the
convergence of the series above is absolute and it is enough to show

$\left|\sum_{n=1}^{\infty}\frac{1}{Df_{c}^{{n}}({z}(c))}\right|\leq\frac{K}{|c-{\hat{c}}|^{1-1/Q}}$

for some constant $K>0$ independent of $c$ in a thick internal ray, where
$Q:=\max\\{2,\,q\\}$ and $q$ is the petal number of the parabolic fixed point
${\hat{b}}$.

Now we present three principal lemmas about S-cycles that are valid for
sufficiently small $\nu$ (the radius of $V_{0}$) under the radial access
condition $c=c_{t}\approx{\hat{c}}$. That is, we only consider $c=c_{t}$ with
$t\in\Delta=\Delta(A_{0},T_{0})$ as in the previous section.

### Lemma A.

There exists a constant $K_{\mathrm{A}}>0$ such that for any
$c\approx{\hat{c}}$, any $z_{0}\in J(f_{c})$, and for any S-cycle ${\sf
S}=[M,M^{\prime})$ of the orbit
$z_{n}=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$, we have

$\left|\sum_{i=1}^{M^{\prime}-M}\frac{1}{Df_{c}^{i}(z_{M})}\right|\leq\frac{K_{\mathrm{A}}}{|c-{\hat{c}}|^{1-1/Q}},$
(3.1)

where we set $M^{\prime}-M:=\infty$ if $M^{\prime}=\infty$.

### Lemma B.

There exists a constant $K_{\mathrm{B}}>0$ such that for any
$c\approx{\hat{c}}$ and any $M\leq\infty$, if $z_{0}\in J(f_{c})$ satisfies
$z_{n}\notin V_{0}$ for any $n\in[0,M)$, then we have

$\sum_{i=1}^{M}\frac{1}{|Df_{c}^{i}(z_{0})|}\leq K_{\mathrm{B}}.$ (3.2)

An immediate consequence of Lemma B is:

######  Corollary 3.2.

For $c\approx{\hat{c}}$, if the orbit of $z=z(c)\in J(f_{c})$ by $f_{c}$ never
lands on $V_{0}$, then the derivative satisfies

${\left|\frac{d}{dc}z(c)\right|}\leq\sum_{n=1}^{\infty}\frac{1}{|Df_{c}^{n}(z(c))|}\leq
K_{\mathrm{B}}.$ (3.3)

### Lemma C (S-cycles Expand Uniformly).

There exists a constant $\Lambda>1$ such that for any $c\approx{\hat{c}}$, any
$z_{0}\in J(f_{c})$, and for any finite S-cycle ${\sf S}=[M,M^{\prime})$ of
the orbit $z_{n}=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$, we have

$|Df_{c}^{M^{\prime}-M}(z_{M})|\geq\Lambda.$ (3.4)

The constants $K_{\mathrm{A}}$, $K_{\mathrm{B}}$, and $\Lambda$ above depends
only on the choice of ${\hat{c}}$, $\nu$, and the thickness $\delta$ of
${\mathcal{I}}(\theta,\delta)$ (equivalently, the angle $A_{0}$ of
$\Delta=\Delta(A_{0},T_{0})$). The proofs of these lemmas will be given later.

By assuming these three lemmas, we can give a proof of the main theorem:

### Proof of the main theorem assuming Lemmas A, B, and C.

It is enough to show the theorem for $c\approx{\hat{c}}$. (Indeed, if $c$
stays a uniform distance away from $\partial\mathbb{X}$, the derivative is
bounded above by the inequality (1.4).)

For a given $c\approx{\hat{c}}$ and $z_{\ast}\in J(f_{\sigma})$, let
$z_{0}=z(c)=H(c,z_{\ast})\in J(f_{c})$. We consider the decomposition
$\mathbb{N}_{0}=[0,M_{1})\sqcup{\sf S}_{1}\sqcup{\sf S}_{2}\sqcup\cdots$ as in
(2.4), (2.5) or (2.6). Then we have

$\displaystyle{\left|\frac{d}{dc}z(c)\right|}$
$\displaystyle=\left|\sum_{n=1}^{\infty}\frac{1}{Df_{c}^{n}(z_{0})}\right|\leq\sum_{n=1}^{M_{1}}\frac{1}{|Df_{c}^{n}(z_{0})|}+\left|\sum_{k\geq
1}\sum_{n\in{\sf S}_{k}}\frac{1}{Df_{c}^{n+1}(z_{0})}\right|$
$\displaystyle\leq\sum_{n=1}^{M_{1}}\frac{1}{|Df_{c}^{n}(z_{0})|}+\sum_{k\geq
1,{\sf
S}_{k}\neq\emptyset}\frac{1}{|Df_{c}^{M_{k}}(z_{0})|}\left|\sum_{i=1}^{M_{k+1}-M_{k}}\frac{1}{Df_{c}^{i}(z_{M_{k}})}\right|.$

By Lemma B, we obviously have $1/|Df_{c}^{M_{1}}(z_{0})|\leq K_{\mathrm{B}}$.
By Lemma C, we have

$|Df_{c}^{M_{k}}(z_{0})|=|Df_{c}^{M_{k}-M_{k-1}}(z_{M_{k-1}})|\cdots|Df_{c}^{M_{2}-M_{1}}(z_{M_{1}})|~{}|Df_{c}^{M_{1}}(z_{0})|\geq\Lambda^{k-1}/K_{\mathrm{B}}$

as long as ${\sf S}_{k}\neq\emptyset$. Hence by Lemma A, we have

$\left|\sum_{n=1}^{\infty}\frac{1}{Df_{c}^{n}(z_{0})}\right|\leq
K_{\mathrm{B}}+\sum_{k\geq
1}\frac{K_{\mathrm{B}}}{\Lambda^{k-1}}\cdot\frac{K_{\mathrm{A}}}{|c-{\hat{c}}|^{1-1/Q}}=K_{\mathrm{B}}+\frac{K_{\mathrm{B}}\Lambda}{\Lambda-1}\cdot\frac{K_{\mathrm{A}}}{|c-{\hat{c}}|^{1-1/Q}}.$

We may assume that ${|c-{\hat{c}}|}\leq 1$ such that $K_{\mathrm{B}}\leq
K_{\mathrm{B}}/|c-{\hat{c}}|^{1-1/Q}$. Hence by setting
$K:=K_{\mathrm{B}}+\dfrac{K_{\mathrm{B}}K_{\mathrm{A}}\Lambda}{\Lambda-1}$, we
have ${\left|\dfrac{dz}{dc}\right|}\leq\dfrac{K}{|c-{\hat{c}}|^{1-1/Q}}$ for
any $c\approx{\hat{c}}$. $\blacksquare$

##  4 Postcritical set and hyperbolic metric

In this section we show some properties of the hyperbolic metric on the
complement of the postcritical set of $f_{\hat{c}}$.

### Postcritical sets.

The postcritical set $P(f_{c})\subset\mathbb{C}$ of the polynomial
$f_{c}(z)=z^{2}+c$ is defined by

$P(f_{c}):=\overline{{\left\\{f_{c}(0),\,f_{c}^{2}(0),\,f_{c}^{3}(0),\,\cdots\right\\}}}.$

In this paper we only consider $P(f_{\hat{c}})$, which is a countable set and
accumulates only on the parabolic cycle of $f_{\hat{c}}$. In particular, the
universal covering of $\mathbb{C}-P(f_{\hat{c}})$ is the unit disk
$\mathbb{D}$.

Let $\mathrm{dist}\,(z,P(f_{\hat{c}}))$ be the Euclidean distance between $z$
and $P(f_{\hat{c}})$. The following lemma provides an estimate of
$\mathrm{dist}\,(z,P(f_{\hat{c}}))$ for $z\in J(f_{c})-{\mathcal{V}}(c)$ for
$c\approx{\hat{c}}$, and plays a very important role.

### Lemma D.

There exists a constant $K_{\mathrm{D}}\in(0,1]$ such that for any
sufficiently small $\nu\in(0,R_{0})$, we have
$\mathrm{dist}\,(z,P(f_{\hat{c}}))\geq K_{\mathrm{D}}\nu$ for any $z\in
J(f_{c})-{\mathcal{V}}(c)$ with $c=c_{t}\approx{\hat{c}}~{}(t\in\Delta)$ by
taking a sufficiently small $T_{0}$ (which depends on $\nu$) in the definition
of $\Delta=\Delta(A_{0},T_{0})$.

The proof will be given later in Section 11.

### Hyperbolic metric.

For $c={\hat{c}}$, let $\rho(z)|dz|$ denote the hyperbolic metric of
$\mathbb{C}-P(f_{\hat{c}})$, which is induced by the metric $|dz|/(1-|z|^{2})$
of constant curvature $-4$ on the unit disk. The metric $\rho(z)|dz|$ has the
following properties:

  1. (i)

$\rho:\mathbb{C}-P(f_{\hat{c}})\to\mathbb{R}_{+}$ is real analytic and
diverges on $P(f_{\hat{c}})\cup\\{\infty\\}$.

  2. (ii)

if both $z$ and $f_{\hat{c}}(z)$ are in $\mathbb{C}-P(f_{\hat{c}})$, we have

$\frac{\rho(f_{\hat{c}}(z))}{\rho(z)}|Df_{\hat{c}}(z)|>1.$

See [Mc1, §3.2] for example.

There is an important observation: Fix any compact subset $\Gamma$ of
$\mathbb{C}-P(f_{\hat{c}})$, and suppose that both $z$ and $f_{c}(z)$ are
contained in $\Gamma$ for some $c\approx{\hat{c}}$. Then by continuity of
$\rho(f_{c}(z))|Df_{c}(z)|$ with respect to $c$, we have

$\frac{\rho(f_{c}(z))}{\rho(z)}|Df_{c}(z)|\geq A,\quad\mbox{or
equvalently~{}}|Df_{c}(z)|\geq\frac{\rho(z)}{\rho(f_{c}(z))}\cdot A,$

for some uniform constant $A=A_{\Gamma}>1$ independent of $c\approx{\hat{c}}$.

To give some estimates of the function $\rho(z)$, we need the following:

######  Proposition 4.1.

The hyperbolic metric $\rho(z)|dz|$ of $\mathbb{C}-P(f_{\hat{c}})$ satisfies

$\rho(z)\leq\frac{1}{\mathrm{dist}\,(z,P(f_{\hat{c}}))}.$

This is just an application of a standard fact on hyperbolic metrics. See [A,
Theorems 1.10 & 1.11] for example.

###

Now we are ready to show:

### Lemma E.

If the constant $\nu$ is sufficiently small, there exists a constant
$K_{\mathrm{E}}\asymp\nu$ with the following property: For any
$c\approx{\hat{c}}$, we have

$\frac{\rho(z)}{\rho(\zeta)}\geq K_{\mathrm{E}}$

if $z,\zeta\in J(f_{c})-{\mathcal{V}}(c)$.

### Proof.

Note that $J(f_{c})\subset\overline{\mathbb{D}(2)}$ for $c\in\mathbb{M}$.
Since $\rho$ diverges only at the postcritical set $P(f_{\hat{c}})$ in
$\overline{\mathbb{D}(3)}$, there exists a constant $C_{1}>0$ such that
$\rho(w)\geq C_{1}$ for any $w\in\overline{\mathbb{D}(3)}-P(f_{{\hat{c}}})$.
In particular, we have $\rho(z)\geq C_{1}$. By Lemma D, for $\nu\ll 1$ and
$c\approx{\hat{c}}$ we have

$\mathrm{dist}\,(\zeta,P(f_{\hat{c}}))\geq K_{\mathrm{D}}\nu$

and thus Propositon 4.1 implies $\rho(\zeta)\leq 1/(K_{\mathrm{D}}\nu)$ for
sufficiently small $\nu$. Now we have $\rho(z)/\rho(\zeta)\geq
C_{1}K_{\mathrm{D}}\nu=:K_{\mathrm{E}}$. $\blacksquare$

By Lemma E we obtain a kind of uniform expansion of $f_{c}$ with respect to
$\rho$:

### Lemma F.

There exists a constant $A>1$ such that for $c\approx{\hat{c}}$, if
$z,f_{c}(z),\ldots,f_{c}^{n}(z)$ are all contained in
$J(f_{c})-{\mathcal{V}}(c)$, we have

$|Df_{c}^{n}(z)|\geq K_{\mathrm{E}}A^{n}.$

### Proof.

By Lemma D, $J(f_{c})-{\mathcal{V}}(c)$ is contained in a compact set $\Gamma$
in $\mathbb{C}-P(f_{\hat{c}})$ independent of $c\approx{\hat{c}}$. As
mentioned in the observation above, there exists a constant $A=A_{\Gamma}>1$
such that for any $c\approx{\hat{c}}$,

$\frac{\rho(f_{c}(w))}{\rho(w)}|Df_{c}(w)|\geq A$

if both $w,\,f_{c}(w)\in J(f_{c})-{\mathcal{V}}(c)$.

By the chain rule, we have

$|Df_{c}^{n}(z)|=\prod_{i=0}^{n-1}|Df_{c}(f_{c}^{i}(z))|\geq\prod_{i=0}^{n-1}\frac{\rho(f_{c}^{i}(z))}{\rho(f_{c}^{i+1}(z))}A\geq\frac{\rho(z)}{\rho(f_{c}^{n}(z))}A^{n}.$
(4.1)

By applying Lemma E with $\zeta=f_{c}^{n}(z)$, we obtain the desired
inequality. $\blacksquare$

##  5 Proof of Lemma A assuming Lemmas G and H

For any $c=c_{t}\approx{\hat{c}}$, we choose an arbitrary $z_{0}\in J(f_{c})$
and let $z_{n}:=f_{c}^{n}(z_{0})~{}(n\in\mathbb{N}_{0})$. For a given S-cycle
${\sf S}=[M,M^{\prime})$ of this orbit, we may assume that $M=0$ without loss
of generality. We divide the proof into two cases.

First we suppose that ${\sf S}$ is either a finite S-cycle or an infinite
S-cycle of type (I). Then there exist $m\in\mathbb{N}$ and
$L\in\mathbb{N}\cup{\left\\{\infty\right\\}}$ such that

  * •

$z=z_{0}\in V_{0}$;

  * •

$z_{(m-1)pq}\in U_{0}$ but $z_{mpq}\notin U_{0}$;

  * •

$z_{mpq+i}\notin V_{0}$ if $0\leq i<L$; and

  * •

$M^{\prime}<\infty$ iff $L<\infty$ and $M^{\prime}=mpq+L$.

As in Remark 2.6, by shrinking $U_{0}$ (that is, by taking a smaller $R_{0}$)
if necessary, each $z_{mpq+i}$ with $0\leq i\leq p$ remains near
$f^{i}(U_{0})$ and we may assume that $L>p$ for any S-cycle.

Recall that under the radial access condition $c=c_{t}\approx{\hat{c}}$, we
have a periodic point $b_{t}$ of period $p$ in $U_{0}$ such that
$b_{t}\to{\hat{b}}$ as $t\in\Delta$ tends to $0$ (Proposition 2.3). For
$c=c_{t}\approx{\hat{c}}$, let $b_{0}(c),b_{1}(c),\ldots,b_{p-1}(c)$ denote
the periodic points $b_{t},f_{c_{t}}(b_{t}),\cdots,f_{c_{t}}^{p-1}(b_{t})$
respectively.

The next two lemmas rely on local dynamics of perturbed parabolic cycle.

### Lemma G.

Suppose that $q\geq 2$. There exists a constant $K_{\mathrm{G}}>0$ with the
following property: For any $c\approx{\hat{c}}$ and $z_{0}\in J(f_{c})\cap
U_{0}$ such that $z_{p}$, $z_{2p},\ldots,z_{qp}\in U_{0}$, we have

$\left|\sum_{l=0}^{q-1}\frac{1}{Df_{c}^{lp}(z_{j})}\right|\leq
K_{\mathrm{G}}\big{(}|c-{\hat{c}}|+|z_{j}-b_{j}(c)|\big{)}.$

for each $0\leq j\leq p-1$.

### Lemma H.

There exists a constant $K_{\mathrm{H}}>0$ with the following property: For
any $c\approx{\hat{c}}$, $z_{0}\in J(f_{c})\cap V_{0}$, and $m\in\mathbb{N}$
such that $z_{kpq}\in U_{0}$ for $0\leq k\leq m-1$, we have for each $0\leq
j\leq p-1$ that

$\sum_{k=0}^{m-1}\frac{1}{|Df_{c}^{kpq}(z_{j})|}\leq\frac{K_{\mathrm{H}}}{|\,c-{\hat{c}}\,|^{1/2}}\qquad\mbox{if
}q=1,$

and

$\sum_{k=0}^{m-1}\frac{|c-{\hat{c}}|+|f_{c}^{kpq}(z_{j})-b_{j}(c)|}{|Df_{c}^{kpq}(z_{j})|}\leq\frac{K_{\mathrm{H}}}{|\,c-{\hat{c}}\,|^{1-1/q}}\qquad\mbox{if
}q\geq 2.$

The proofs of Lemmas G and H will be given later.

### Finite S-cycles.

Set $f=f_{c}$. For the finite S-cycle ${\sf S}=[0,M^{\prime})$ with
$M^{\prime}=mpq+L$, we have

$\displaystyle\sum_{i=1}^{M^{\prime}}\frac{1}{Df^{i}(z_{0})}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{p}\frac{1}{Df^{j}(z_{0})}\sum_{k=0}^{m-1}\frac{1}{Df^{kpq}(z_{j})}\sum_{l=0}^{q-1}\frac{1}{Df^{lp}(z_{kpq+j})}$
(5.1)
$\displaystyle~{}~{}~{}~{}+\sum_{i=1}^{L-p}\frac{1}{Df^{mpq}(z_{0})\,Df^{i}(z_{mpq})}+\sum_{i=1}^{p}\frac{1}{Df^{mpq+(L-p)}(z_{0})\,Df^{i}(z_{mpq+(L-p)})}.$

When $1\leq j\leq p$, we have $z_{j}\in f^{j}(U_{0})$ and thus

$|Df^{j}(z_{0})|\geq\xi^{j}$

in (5.1) by Remark 2.5. Note that by the Koebe distortion theorem,

$|Df^{mpq}(z_{j})|\geq
C_{2}\cdot\frac{\mathrm{diam}\,U_{0}}{\mathrm{diam}\,V_{0}}\geq\frac{C_{3}}{\nu}$

for some constants $C_{2}$ and $C_{3}$ independent of $c$ and
$j\in\\{0,1,\ldots,p-1\\}$. Consequently,

$\displaystyle\sum_{j=1}^{p}\frac{1}{|Df^{j}(z_{0})|}\sum_{k=0}^{m-1}\frac{1}{|Df^{kpq}(z_{j})|}\left|\sum_{l=0}^{q-1}\frac{1}{Df^{lp}(z_{kpq+j})}\right|$
$\displaystyle\leq$
$\displaystyle\begin{cases}\displaystyle\sum_{j=1}^{p}\frac{1}{\xi^{j}}\sum_{k=0}^{m-1}\frac{1}{|Df^{kpq}(z_{j})|}&\mbox{if
}q=1\\\
\displaystyle\sum_{j=1}^{p}\frac{1}{\xi^{j}}\sum_{k=0}^{m-1}\frac{K_{\mathrm{G}}\,{\left(|c-{\hat{c}}|+\left|z_{kpq+j}-b_{j}(c)\right|\right)}}{|Df^{kpq}(z_{j})|}&\mbox{if
$q\geq 2$ (by using Lemma G)}\end{cases}$ $\displaystyle\leq$
$\displaystyle\frac{p}{\xi^{p}}\cdot\max\\{1,K_{\mathrm{G}}\\}\cdot\frac{K_{\mathrm{H}}}{|c-{\hat{c}}|^{1-1/Q}}\qquad\mbox{(by
Lemma H)}$

where $Q=\max\\{2,\,q\\}$.

When $n=mpq+i$ with $1\leq i\leq L-p$, we have
$z_{mpq},\,z_{mpq+i}\notin{\mathcal{V}}(c)$ and thus

$\displaystyle|Df^{n}(z_{0})|$
$\displaystyle=|Df^{mpq}(z_{0})|\,|Df^{i}(z_{mpq})|$
$\displaystyle\geq|Df^{mpq}(z_{0})|\cdot K_{\mathrm{E}}\cdot A^{i}.$

Here the constant $A$ above is the same as that of Lemma F. Consequently,

$\displaystyle\sum_{i=1}^{L-p}\frac{1}{|Df^{mpq}(z_{0})|\,|Df^{i}(z_{mpq})|}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{L-p}\frac{1}{|Df^{mpq}(z_{0})|~{}K_{\mathrm{E}}~{}A^{i}}$
$\displaystyle\leq$
$\displaystyle\frac{\nu}{C_{3}~{}K_{\mathrm{E}}}\frac{1}{A-1}.$

When $n=mpq+(L-p)+i$ with $1\leq i\leq p$, $f(z_{n})\in{\mathcal{V}}(c)$ and
thus

$\displaystyle|Df^{n}(z)|$
$\displaystyle=|Df^{mpq+(L-p)}(z_{0})|\,|Df^{i}(z_{mpq+(L-p)})|$
$\displaystyle\geq|Df^{mpq}(z_{0})|\cdot K_{\mathrm{E}}\cdot
A^{L-p}\cdot\xi^{i}$ $\displaystyle\geq|Df^{mpq}(z_{0})|\cdot
K_{\mathrm{E}}\cdot 1\cdot\xi^{p}.$

Consequently,

$\displaystyle\sum_{i=1}^{p}\frac{1}{|Df^{mpq+(L-p)}(z_{0})|\,|Df^{i}(z_{mpq+(L-p)})|}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{p}\frac{1}{|Df^{mpq}(z_{0})|~{}K_{\mathrm{E}}~{}\xi^{p}}$
$\displaystyle\leq$
$\displaystyle\frac{p\nu}{C_{3}~{}K_{\mathrm{E}}~{}\xi^{p}}.$

By these estimates, when $M^{\prime}<\infty$, we have:

$\displaystyle\left|\sum_{i=1}^{M^{\prime}}\frac{1}{Df^{i}(z_{0})}\right|$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{p}\frac{1}{|Df^{j}(z_{0})|}\sum_{k=0}^{m-1}\frac{1}{|Df^{kpq}(z_{j})|}\left|\sum_{l=0}^{q-1}\frac{1}{Df^{lp}(z_{kpq+j})}\right|$
$\displaystyle\quad+\sum_{i=1}^{L-p}\frac{1}{|Df^{mpq}(z_{0})|\,|Df^{i}(z_{mpq})|}$
$\displaystyle\quad+\sum_{i=1}^{p}\frac{1}{|Df^{mpq+(L-p)}(z_{0})|\,|Df^{i}(z_{mpq+(L-p)})|}$
$\displaystyle\leq$
$\displaystyle\frac{p}{\xi^{p}}\cdot\max\\{1,K_{\mathrm{G}}\\}\cdot\frac{K_{\mathrm{H}}}{|c-{\hat{c}}|^{1-1/Q}}+\frac{\nu}{C_{3}~{}K_{\mathrm{E}}}{\left(\frac{1}{A-1}+\frac{p}{\xi^{p}}\right)}.$

Hence by setting

$K_{\mathrm{A}}:=\max{\left\\{1,K_{\mathrm{G}}\right\\}}\cdot\frac{2p}{\xi^{p}}\cdot
K_{\mathrm{H}},$

we obtain our desired estimate

$\left|\sum_{i=1}^{M^{\prime}}\frac{1}{Df^{i}(z_{0})}\right|\leq\frac{K_{\mathrm{A}}}{|\,c-{\hat{c}}\,|^{1-1/Q}}$

when $|c-{\hat{c}}|$ is sufficiently small. Note that $K_{\mathrm{A}}$ does
not depend on $m$ and $L$.

### Infinite S-cycles, Type (I).

If $M^{\prime}=\infty$, then $L=\infty$ and one can easily check

$\displaystyle\left|\sum_{i=1}^{\infty}\frac{1}{Df^{i}(z_{0})}\right|\leq$
$\displaystyle\frac{p}{\xi^{p}}\cdot\max\\{1,K_{\mathrm{G}}\\}\cdot\frac{K_{\mathrm{H}}}{|c-{\hat{c}}|^{1-1/Q}}+\frac{\nu}{C_{3}~{}K_{\mathrm{E}}}{\left(\frac{1}{A-1}\right)}$
$\displaystyle<$
$\displaystyle\frac{K_{\mathrm{A}}}{|\,c-{\hat{c}}\,|^{1-1/Q}}$

by the same argument as the case of finite S-cycles.

### Infinite S-cycles, Type (II).

Next we suppose that ${\sf S}=[0,\infty)$ is an infinite S-cycle of Type (II).
As we have seen in Section 2 (see in particular Propositions 2.3 and 2.4),
$z_{0}$ must be a repelling periodic point of period $p$ or $pq$ of $f_{c}$
contained in $V_{0}$ for $c=c_{t}\approx{\hat{c}}~{}(t\in\Delta)$ by taking a
smaller $T_{0}>0$ in the definition of $\Delta$ if necessary. Note that Lemmas
G and H are valid even in such a case.

### Case 1 ($q=1$).

Since the assumption of Lemma H is valid for any large $m$, we have

$\sum_{k=0}^{\infty}\frac{1}{|Df_{c}^{kp}(z_{j})|}\leq\frac{K_{\mathrm{H}}}{|\,c-{\hat{c}}\,|^{1/2}}$

for $q=1$ and $0\leq j\leq p-1$. Hence for $c\approx{\hat{c}}$, we have

$\displaystyle{\left|\sum_{i=1}^{\infty}\frac{1}{Df^{i}(z_{0})}\right|}$
$\displaystyle\leq\sum_{k=0}^{\infty}\sum_{j=1}^{p}\frac{1}{|Df^{kp}(z_{j})||Df^{j}(z_{0})|}$
$\displaystyle\leq\sum_{k=0}^{\infty}\sum_{j=1}^{p}\frac{1}{|Df^{kp}(z_{j})|\cdot\xi^{j}}\leq\frac{pK_{\mathrm{H}}}{\xi^{p}|\,c-{\hat{c}}\,|^{1/2}}.$

### Case 2±.

As in Case 1, Lemma H implies

$\sum_{k=0}^{\infty}\frac{|c-{\hat{c}}|+|f_{c}^{kpq}(z_{j})-b_{j}(c)|}{|Df_{c}^{kpq}(z_{j})|}\leq\frac{K_{\mathrm{H}}}{|\,c-{\hat{c}}\,|^{1-1/q}}$

for $q\geq 2$ and $0\leq j\leq p-1$. (In Case 2+, $b_{j}(c)$ is a repelling
periodic point and we have $f_{c}^{kpq}(z_{j})=b_{j}(c)$ for any $k\geq 0$.)
Hence for $c\approx{\hat{c}}$, we have

$\displaystyle{\left|\sum_{i=1}^{\infty}\frac{1}{Df^{i}(z_{0})}\right|}$
$\displaystyle\leq\sum_{j=1}^{p}\frac{1}{|Df^{j}(z_{0})|}\sum_{k=0}^{\infty}\frac{1}{|Df^{kpq}(z_{j})|}\left|\sum_{l=0}^{q-1}\frac{1}{Df^{lp}(z_{kpq+j})}\right|$
$\displaystyle\leq\sum_{j=1}^{p}\frac{1}{\xi^{j}}\sum_{k=0}^{\infty}\frac{1}{|Df^{kpq}(z_{j})|}\cdot
K_{\mathrm{G}}\big{(}|c-{\hat{c}}|+\left|z_{kpq+j}-b_{j}(c)\right|\big{)}$
$\displaystyle\leq\frac{p}{\xi^{p}}\cdot
K_{\mathrm{G}}\cdot\frac{K_{\mathrm{H}}}{|c-{\hat{c}}|^{1-1/q}}$

by Lemma G. In both Case 1 and Case 2±, we conclude that Lemma A is valid with
the same constant $K_{\rm A}>0$ defined as above. $\blacksquare$

##  6 Proof of Lemma B

Without loss of generality, we may assume that either

  * •

$M<\infty$ and $z_{M}\in V_{0}$; or

  * •

$M=\infty$.

For the first case, if $M\leq p$, then
$z_{0}\in\bigcup_{k=0}^{p}f_{c}^{-k}(V_{0})$ since $z_{M}\in V_{0}$. Hence

$|Df_{c}^{i}(z_{0})|\geq\xi^{i}\geq\xi^{p}$

for $i\in[1,M]$. This implies

$\sum_{i=0}^{M}\frac{1}{|Df_{c}^{i}(z_{0})|}\leq\frac{M}{\xi^{p}}\leq\frac{p}{\xi^{p}}.$

If $M>p$, then $z_{M-p}\in f_{c}^{-p}(V_{0})$ and
$z_{i}\notin\bigcup_{k=0}^{p}f_{c}^{-k}(V_{0})$ for $i\in[0,M-p)$. Then

  * •

$|Df_{c}^{i}(z_{0})|\geq K_{\mathrm{E}}A^{i}$ for $i\in[1,M-p]$ (by Lemma F)

  * •

$|Df_{c}^{M-p+j}(z_{0})|\geq K_{\mathrm{E}}A^{M-p}\cdot\xi^{j}\geq
K_{\mathrm{E}}\xi^{p}$ for $j\in[1,p]$.

This implies:

$\sum_{i=1}^{M}\dfrac{1}{|Df_{c}^{i}(z_{0})|}\leq\sum_{i=1}^{M-p}\dfrac{1}{K_{\mathrm{E}}A^{i}}+\sum_{j=1}^{p}\dfrac{1}{K_{\mathrm{E}}\xi^{p}}\leq\frac{1}{K_{\mathrm{E}}}{\left(\frac{1}{A-1}+\frac{p}{\xi^{p}}\right)}.$

By defining

$K_{\mathrm{B}}:=\frac{1}{K_{\mathrm{E}}}{\left(\frac{1}{A-1}+\frac{p}{\xi^{p}}\right)}$

we have the desired estimate.

For the second case ($M=\infty$), the orbit never land on ${\mathcal{V}}(c)$
and by Lemma F, we have

$|Df_{c}^{i}(z_{0})|\geq K_{\mathrm{E}}A^{i}$

for any $i\in\mathbb{N}$. Hence

$\sum_{i=1}^{\infty}\dfrac{1}{|Df_{c}^{i}(z_{0})|}\leq\sum_{i=1}^{\infty}\dfrac{1}{K_{\mathrm{E}}A^{i}}\leq\frac{1}{K_{\mathrm{E}}}\cdot\frac{1}{A-1}<K_{\mathrm{B}}.$

$\blacksquare$

##  7 Proofs of Propositions 2.2 and 2.3

This section is devoted to the proofs of Propositions 2.2 and 2.3, which
describe the local properties of parabolic periodic points and its
perturbations along thick internal rays. The argument here relies on some
well-known facts originally due to Douady and Hubbard [DH, Exposé XIV] on
perturbation of parabolic cycles. Here we will adopt Milnor’s formulation in
[Mi, §4]. (See also [T, Thm. 1.1 and Thm. A.1].)

###

Let ${\hat{c}}$ be a parabolic parameter and ${\hat{b}}$ be a parabolic
periodic point of period $p$ of $f_{\hat{c}}$. Let $\hat{\lambda}$ be the
multiplier $Df_{\hat{c}}^{p}({\hat{b}})$ of ${\hat{b}}$ that is a primitive
$q$-th root of unity.

######  Proposition 7.1 (Lemma 4.5 of [Mi]).

There exist unique single valued functions $c(\lambda)$ and $z(\lambda)$
defined on a neighborhood of $\hat{\lambda}$ so that $z(\lambda)$ is a
periodic point of period $p$ and multiplier $\lambda$ for the map
$f_{c(\lambda)}$, with ${\hat{c}}=c(\hat{\lambda})$ and
${\hat{b}}=z(\hat{\lambda})$. This function $c(\lambda)$ has a single critical
point at $\hat{\lambda}$ when $q=1$ (Case 1) but is univalent when $q\geq 2$
(Case 2±).

By applying the argument of [K2, Appendix A.4] one can find a holomorphic
family of local coordinates $\zeta=\varphi_{\lambda}(z)$ and
$w=\psi_{\lambda}(z)$ defined near $z(\lambda)$ such that
$\varphi_{\lambda}(z(\lambda))=\psi_{\lambda}(z(\lambda))=0$,

$\displaystyle\varphi_{\lambda}\circ
f_{c(\lambda)}^{p}\circ\varphi_{\lambda}^{-1}(\zeta)$
$\displaystyle=\lambda\zeta+\zeta^{q+1}+O(\zeta^{2q+1}),~{}\text{and}$ (7.1)
$\displaystyle\psi_{\lambda}\circ
f_{c(\lambda)}^{pq}\circ\psi_{\lambda}^{-1}(w)$
$\displaystyle=\lambda^{q}w\,(1+w^{q}+O(w^{2q}))$ (7.2)

where

$w={\left(\frac{1+\lambda^{q}+\lambda^{2q}+\cdots+\lambda^{(q-1)q}}{\lambda}\right)}^{1/q}\zeta.$

(Note that the error terms in (7.1) and (7.2) are slightly refined compared
with the similar coordinates given in [DH, Prop. 11.1] and [Mi, Lem.4.2].)
Then we take a sufficiently small $\hat{R}>0$ such that the domains of these
local coordinates contain $\hat{U}=\mathbb{D}({\hat{b}},\hat{R})$ for
$\lambda$ sufficiently close to $\hat{\lambda}$. In particular, both
$D\varphi^{-1}_{\lambda}(0)$ and $D\psi^{-1}_{\lambda}(0)$ are uniformly
bounded away from zero.

######  Proposition 7.2.

For $\lambda\neq\hat{\lambda}$, one can find exactly $q$ non-zero fixed points
of $\psi_{\lambda}\circ
f_{c(\lambda)}^{pq}\circ\psi_{\lambda}^{-1}(w)=\lambda^{q}w\,(1+w^{q}+O(w^{2q}))$
of the form $w=w_{\lambda}(1+o(1))$ with multiplier
$1+q(1-\lambda^{q})+o(|\lambda-\hat{\lambda}|)$, where $w_{\lambda}$ is a
$q$-th root of $(1-\lambda^{q})/\lambda^{q}$.

### Proof.

Since the equation of the form $\lambda^{q}w\,(1+w^{q}+O(w^{2q}))=w$ is
regarded as a perturbation of the $\lambda=\hat{\lambda}$ case, it has exactly
$q$ non-zero roots by Hurwitz’s theorem. To obtain the estimate of the
solution, we apply Rouché’s theorem000 Let
$F(w)=w^{q}-w_{\lambda}^{q}=w^{q}-(1-\lambda^{q})/\lambda^{q}$ and
$G(w)=O(w^{2q})$. Let $\ell:=|\lambda-\hat{\lambda}|$. Then consider the
circle $|w-w_{\lambda}|=\ell^{s}$ with $1/q<s<1+1/q$. For example, one can
take $s:=(2q+1)/(2q)$. Now one can check $|w_{\lambda}|\asymp\ell^{1/q}$ and
thus $|F(w)|\asymp\ell^{1-1/q+s}$ and $|G(w)|=O(\ell^{2})$ on this circle.
Since $|F(w)|>|G(w)|$ for $\ell\ll 1$, $F(w)+G(w)=0$ and $F(w)=0$ has the same
number of zeroes, which is one. Hence the solution is of the form
$w=w_{\lambda}+O(\ell^{s})=w_{\lambda}(1+o(1))$. . The multiplier comes from
$D(\psi_{\lambda}\circ
f_{c(\lambda)}^{pq}\circ\psi_{\lambda}^{-1})(w)=\lambda^{q}(1+(q+1)w^{q}+O(w^{2q}))$
and $|w_{\lambda}|\asymp|\lambda-\hat{\lambda}|^{1/q}$. $\blacksquare$

###

Now we fix a hyperbolic component $\mathbb{X}$ attached to ${\hat{c}}$ with
uniformization
$\Phi=\Phi_{\mathbb{X}}:\overline{\mathbb{D}}\to\overline{\mathbb{X}}$, and a
Stolz angle $S(\Delta)\subset\mathbb{D}$ at $\hat{\lambda}$ with
$\Delta=\Delta(A_{0},T_{0})$. In what follows we will take a sufficiently
small $T_{0}$ if necessary. Let us start with Case 2-.

### Proofs of Propositions 2.2 and 2.3 for Case 2-.

We obtain Case 2- by letting $\lambda=\lambda_{t}:=(1-t/q)\hat{\lambda}$ with
$t\in\Delta$ in Proposition 7.1. Indeed, the hyperbolic component $\mathbb{X}$
is specified by the family of attracting fixed points $z(\lambda)$ of
$f_{c(\lambda)}^{p}$ with $\lambda=\lambda_{t}$. Proposition 7.1 implies that
$|c_{t}-{\hat{c}}|\asymp|\lambda_{t}-\hat{\lambda}|\asymp|t|$ by setting
$c_{t}=c(\lambda_{t})$. This proves Proposition 2.2 for this case.

Items (1), (2) and (4) of Proposition 2.3 are straightforward by applying
Proposition 7.2 with $b_{t}:=z(\lambda_{t})$,
$\varphi_{t}:=\varphi_{\lambda_{t}}$, and $\psi_{t}:=\psi_{\lambda_{t}}$ for
$t\in\Delta$.

### Proofs of Propositions 2.2 and 2.3 for Case 1 and Case 2+.

In these cases, $q$ non-zero fixed points of $f_{\lambda}^{pq}$ given in
Proposition 7.2 should be attracting with multiplier $1-qt$ for $t\in\Delta$,
and this specifies the hyperbolic component $\mathbb{X}$ in Case 1 or Case 2+.
Hence by the equality $1-qt=1+q(1-\lambda^{q})+o(|\lambda-\hat{\lambda}|)$ we
obtain $\lambda=\lambda_{t}:=(1+t/q+o(t))\hat{\lambda}$ for $t\in\Delta$. By
setting $c_{t}:=c(\lambda_{t})$, Proposition 7.1 implies that
$|c_{t}-{\hat{c}}|\asymp|\lambda_{t}-\hat{\lambda}|^{2}\asymp|t|^{2}$ in Case
1, and $|c_{t}-{\hat{c}}|\asymp|\lambda_{t}-\hat{\lambda}|\asymp|t|$ in Case
2+. This proves Proposition 2.2 for these cases.

Items (1), (2) and (3) of Proposition 2.3 are straightforward by applying
Proposition 7.2 with $b_{t}:=z(\lambda_{t})$,
$\varphi_{t}:=\varphi_{\lambda_{t}}$, and $\psi_{t}:=\psi_{\lambda_{t}}$ for
$t\in\Delta$. $\blacksquare$

##  8 Proof of Lemma G

### Preliminary.

Lemma G comes from symmetric behavior of the local dynamics of $f_{c}^{p}$
near the periodic point $b(c)$. Indeed, it holds without the radial access
condition and the assumption that $z_{0}\in J(f_{c})$. To describe its
principle in a generalized form, we adopt Milnor’s formulation as above.

Let $\lambda$ range over a neighborhood $\hat{S}$ of $\hat{\lambda}$ such that
the holomorphic maps $c(\lambda)$ and $z(\lambda)$ in Proposition 7.1 are
defined. Note that the map $\lambda\mapsto c(\lambda)$ gives an isomorphism
between $\hat{S}$ and a neighborhood of ${\hat{c}}=c(\hat{\lambda})$ since we
are in the case of $q\geq 2$ (Case 2±). We claim:

######  Proposition 8.1.

Suppose that $q\geq 2$. For any $\lambda\in\hat{S}$ and
$z_{0}\in\hat{U}=\mathbb{D}(\hat{b},\hat{R})$, suppose that
$z_{lp}:=f_{c(\lambda)}^{lp}(z_{0})\in\hat{U}$ for $0\leq l\leq q$. Then we
have

$\left|\sum_{l=0}^{q-1}\frac{1}{Df_{c(\lambda)}^{lp}(z_{0})}\right|=O(|c(\lambda)-{\hat{c}}|)+O(|z_{0}-z(\lambda)|)$

by taking smaller $\hat{S}$ and $\hat{U}$ if necessary.

### Proof of Proposition 8.1.

Let $\epsilon:=\lambda/\hat{\lambda}-1~{}(\lambda\in\hat{S})$ be an
alternative parameter such that

$\lambda=(1+\epsilon)\hat{\lambda}\in\hat{S}.$

Since we are in the case of $q\geq 2$ (Case 2±), we have $\hat{\lambda}\neq 1$
and $\lambda\mapsto c(\lambda)$ is univalent (Proposition 7.1). Hence we
obtain $|c(\lambda)-{\hat{c}}|\asymp|\lambda-{\hat{\lambda}}|\asymp|\epsilon|$
for $\lambda\in\hat{S}$.

Recall that we have a local coordinate $\zeta=\varphi_{\lambda}(z)$ defined on
$\hat{U}$ such that $\varphi_{\lambda}(z(\lambda))=0$ and

$F_{\lambda}(\zeta):=\varphi_{\lambda}\circ
f_{c(\lambda)}^{p}\circ\varphi_{\lambda}^{-1}(\zeta)=\lambda\zeta+\zeta^{q+1}+O(\zeta^{2q+1})$

as in (7.1). For a given $z_{0}\in\hat{U}$, let
$\varphi_{\lambda}^{-1}(\zeta_{0})=z_{0}$ and
$\zeta_{lp}:=\varphi_{\lambda}(z_{lp})$.

Then it is easy to check that

$\zeta_{lp}=F_{\lambda}^{l}(\zeta_{0})=\lambda^{l}\zeta_{0}+\lambda^{l-1}\left(1+\lambda^{q}+\cdots+\lambda^{(l-1)q}\right)\zeta_{0}^{q+1}+O(\zeta_{0}^{2q+1})$
(8.1)

for any $1\leq l\leq q$. Since the local coordinate $\varphi_{\lambda}(z)$
depends holomorphically on $\lambda$, we have

$\varphi_{\lambda}^{-1}(\zeta)=z(\lambda)+C_{4}\zeta+C_{5}\zeta^{2}+O(\zeta^{3})$
(8.2)

for some $C_{4}=C_{4}(\lambda)$ and $C_{5}=C_{5}(\lambda)$ with $C_{4}$
bounded away from zero for $\lambda\in\hat{S}$.

Let

$\mathcal{A}(z_{0}):=\sum_{l=0}^{q-1}\frac{1}{Df_{c(\lambda)}^{lp}(z_{0})}.$

Now,

$Df_{c(\lambda)}^{lp}(z_{0})=D\varphi_{\lambda}^{-1}(\zeta_{lp})~{}DF^{l}_{\lambda}(\zeta_{0})~{}D\varphi_{\lambda}(z_{0}),$

and

$\mathcal{A}(z_{0})=1+\frac{D\varphi^{-1}_{\lambda}(\zeta_{0})}{D\varphi_{\lambda}^{-1}(\zeta_{p})}\frac{1}{DF_{\lambda}(\zeta_{0})}+\frac{D\varphi_{\lambda}^{-1}(\zeta_{0})}{D\varphi_{\lambda}^{-1}(\zeta_{2p})}\frac{1}{DF_{\lambda}^{2}(\zeta_{0})}+\cdots+\frac{D\varphi^{-1}_{\lambda}(\zeta_{0})}{D\varphi_{\lambda}^{-1}(\zeta_{(q-1)p})}\frac{1}{DF_{\lambda}^{q-1}(\zeta_{0})}.$

By (8.1) and (8.2),

$\displaystyle D\varphi_{\lambda}^{-1}(\zeta_{lp})$ $\displaystyle=$
$\displaystyle C_{4}+2C_{5}\zeta_{lp}+O(\zeta^{2}_{lp})$ $\displaystyle=$
$\displaystyle C_{4}+2C_{5}\lambda^{l}\zeta_{0}+O(\zeta_{0}^{2}),$

thus

$\displaystyle\frac{D\varphi_{\lambda}^{-1}(\zeta_{0})}{D\varphi_{\lambda}^{-1}(\zeta_{lp})}$
$\displaystyle=$
$\displaystyle\frac{C_{4}+2C_{5}\zeta_{0}+O(\zeta_{0}^{2})}{C_{4}+2C_{5}\zeta_{lp}+O(\zeta_{lp}^{2})}$
$\displaystyle=$ $\displaystyle
1+\frac{2C_{5}}{C_{4}}(1-\lambda^{l})~{}\zeta_{0}+O(\zeta_{0}^{2}).$

Formula (8.1) also gives

$\frac{1}{DF_{\lambda}^{l}(\zeta_{0})}=\frac{1}{\lambda^{l}}\left(1+O(\zeta_{0}^{2})\right).$

Therefore,

$\displaystyle\mathcal{A}(z_{0})$ $\displaystyle=$ $\displaystyle
1+\sum_{l=1}^{q-1}\bigg{\\{}\left(1+\frac{2C_{5}}{C_{4}}(1-\lambda^{l})\zeta_{0}+O(\zeta_{0}^{2})\right)\cdot\frac{1}{\lambda^{l}}\left(1+O(\zeta_{0}^{2})\right)\bigg{\\}}$
$\displaystyle=$ $\displaystyle
1+\sum_{l=1}^{q-1}\frac{1}{\lambda^{l}}\left(1+\frac{2C_{5}}{C_{4}}(1-\lambda^{l})\zeta_{0}+O(\zeta_{0}^{2})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{q}}\left(\lambda+\lambda^{2}+\cdots+\lambda^{q}+\frac{2C_{5}}{C_{4}}(\lambda+\lambda^{2}+\cdots+\lambda^{q-1}+\lambda^{q}-q\lambda^{q})\zeta_{0}\right)+O(\zeta_{0}^{2})$
$\displaystyle=$
$\displaystyle\frac{\lambda+\lambda^{2}+\cdots+\lambda^{q}}{\lambda^{q}}\left(1+\frac{2C_{5}}{C_{4}}\zeta_{0}\right)-\frac{2qC_{5}}{C_{4}}\zeta_{0}+O(\zeta_{0}^{2}).$

Clearly,

$\hat{\lambda}+\hat{\lambda}^{2}+\cdots+\hat{\lambda}^{q}=0.$

Thus, by using the alternative parameter $\epsilon$, we have

$\displaystyle\lambda+\lambda^{2}+\cdots+\lambda^{q}$
$\displaystyle~{}=~{}\hbox
to0.0pt{$\displaystyle(1+\epsilon)\hat{\lambda}+(1+\epsilon)^{2}\hat{\lambda}^{2}+\cdots+(1+\epsilon)^{q}\hat{\lambda}^{q}$\hss}$
$\displaystyle~{}=~{}(\hat{\lambda}+2\hat{\lambda}^{2}+\cdots+q\hat{\lambda}^{q})~{}\epsilon+O(\epsilon^{2})$
$\displaystyle~{}=~{}\frac{q\hat{\lambda}}{\hat{\lambda}-1}\epsilon+O(\epsilon^{2}).$

Hence,

$\displaystyle\mathcal{A}(z_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{(1+\epsilon)^{q}}{\left\\{\frac{q\hat{\lambda}}{\hat{\lambda}-1}\epsilon+O(\epsilon^{2})\right\\}}\left(1+\frac{2C_{5}}{C_{4}}\zeta_{0}\right)-\frac{2q\,C_{5}}{C_{4}}\zeta_{0}{\left(1+O(\zeta_{0})\right)}$
$\displaystyle=$ $\displaystyle O(\epsilon)+O(\zeta_{0})$

where the implicit constants are independent of
$\lambda=(1+\epsilon)\hat{\lambda}\in\hat{S}$ and
$\zeta_{0}\in\varphi_{\lambda}(\hat{U})$.

Since $C_{4}$ is uniformly bounded away from zero for $\lambda\in\hat{S}$,
(8.2) implies $|\zeta_{0}|\asymp|z_{0}-z(\lambda)|.$ Since
$|\epsilon|\asymp|c(\lambda)-{\hat{c}}|$, we conclude that

$|\mathcal{A}(z_{0})|=O(|c(\lambda)-{\hat{c}}|)+O(|z_{0}-z(\lambda)|).$

$\blacksquare$

### Proof of Lemma G.

When $j=0$, we apply Proposition 8.1 by letting $\lambda:=\lambda_{t}$,
$c=c_{t}:=c(\lambda_{t})$, $b_{0}(c)=b_{t}:=z(\lambda_{t})$ as in the proof of
statements (3) and (4) of Proposition 2.3 given in the previous section. (Note
that we have $U_{0}\subset\hat{U}$ and $S(\Delta)\subset\hat{S}$ by taking a
smaller $T_{0}$.) Then we immediately obtain

$\left|\sum_{l=0}^{q-1}\frac{1}{Df_{c}^{lp}(z_{0})}\right|=O(|c-{\hat{c}}|)+O(|z_{0}-b_{0}(c)|).$

When $1\leq j\leq p-1$, we can apply the same argument as Proposition 8.1 by
replacing $\hat{U}$ with $\hat{U}_{j}:=f_{c}^{j}(\hat{U})$ and $b_{0}(c)$ with
$b_{j}(c)$. More precisely, having the local coordinate
$\varphi_{0}=\varphi_{\lambda_{t}}$ on $\hat{U}$ for $c=c_{t}=c(\lambda_{t})$,
we can define a local coordinate $\varphi_{j}:=\varphi_{c}\circ
f_{c}^{-j}|_{\hat{U}_{j}}$ on $\hat{U}_{j}$ for each $1\leq j\leq p-1$ by a
branch such that $\varphi_{j}(z_{lp+j})=\zeta_{lp}$ for $0\leq l\leq q$. Since
$\varphi_{j}^{-1}$ is of the form

$\varphi^{-1}_{j}(\zeta)=b_{j}(c)+C_{4,j}\zeta+C_{5,j}\zeta^{2}+O(\zeta^{3})$

where $C_{4,j}$ is a constant bounded away from zero, we have
$|z_{j}-b_{j}(c)|\asymp|\zeta_{0}|$ and the same argument as Proposition 8.1
yields

$\left|\sum_{l=0}^{q-1}\frac{1}{Df_{c}^{lp}(z_{j})}\right|=O(|c-{\hat{c}}|)+O(|z_{j}-b_{j}(c)|).$

Hence there exists a constant $K_{\mathrm{G}}>0$ independent of
$c=c_{t}\approx{\hat{c}}$ and $z_{0}\in U_{0}$ such that

$\left|\sum_{l=0}^{q-1}\frac{1}{Df_{c}^{lp}(z_{j})}\right|\leq
K_{\mathrm{G}}\big{(}|c-{\hat{c}}|+|z_{j}-b_{j}(c)|\big{)}$

for $0\leq j\leq p-1$. $\blacksquare$

##  9 Proof of Proposition 2.4

This section is devoted to the proof of Proposition 2.4.

### Branched coordinates near infinity.

Let $c=c_{t}=c(\lambda_{t})\in\mathbb{X}\cup\\{{\hat{c}}\\}$ with
$\lambda_{t}\in S(\Delta)\cup\\{\hat{\lambda}\\}$ and $t\in\Delta\cup\\{0\\}$.
It is convenient to use a branched coordinate
$\Psi_{t}:\psi_{t}(\hat{U})\to\overline{\mathbb{C}}_{W}$ given by
$W=\Psi_{t}(w):=-\lambda_{t}^{q^{2}}/(q\,w^{q})$ where
$\overline{\mathbb{C}}_{W}$ denotes the Riemann sphere in $W$-coordinate. Let
$W=\Phi_{t}(z):=\Psi_{t}\circ\psi_{t}(z)$. We may assume that
$\Phi_{t}(\hat{U})$ is always contained in
${\left\\{W\in\overline{\mathbb{C}}_{W}\,:\,|W|\geq\hat{r}\right\\}}$ for some
constant $\hat{r}>0$ independent of $t\in\Delta$, with
$\hat{r}\hat{R}^{q}\asymp 1$ by taking a smaller $\hat{R}$ if necessary.

In this coordinate we observe the map $f_{c_{t}}^{pq}$ as $G:=\Phi_{t}\circ
f_{c_{t}}^{pq}\circ\Phi_{t}^{-1}$ (taking an appropriate branch of
$\Phi_{t}^{-1}$) of the form

$G(W)=G_{t}(W)=\tau\,W+1+O(1/W),$

where

$\tau:=\lambda_{t}^{-q^{2}}=\left\\{\begin{array}[]{ll}1-qt+O(t^{2})&\text{for
Case 1 and Case 2${}^{+}$},\\\\[5.0pt] 1+qt+O(t^{2})&\text{for Case
2${}^{-}$}\end{array}\right.$

by Proposition 2.3. For $t\in\Delta$, let

$B:=\frac{1}{1-\tau}=\pm\frac{1}{qt}+O(1).$

Note that $B$ is a fixed point of $W\mapsto\tau W+1$ and
$G(W)-B=\tau(W-B)+O(1/W)$. By Rouché’s theorem, there is a fixed point
$B^{\ast}$ of $G$ of the form $B^{\ast}=B+O(1)=O(1/t)$ with multiplier
$\tau^{\ast}:=DG(B^{\ast})=\tau+O(t^{2})$.

Figure 7 illustrates the Julia set in $\overline{\mathbb{C}}_{W}$.

Figure 7: The image of the Julia set (red) by $\Phi_{t}$ in $W$-coordinate for
Cases 1 and 2+ (left) and Case 2- (right). The origin of the $W$-plane is the
center of the dotted (or black) circle. The radius of the black circle equals
$\hat{r}$. The annulus between the dotted circle and the black one encloses
the region where the dynamics by $G$ is relatively close to the translation
$W\mapsto W+1$ (see Proposition 10.1).

### Dynamics of $G$ near infinity.

Let

$d_{0}:=\frac{\cos A_{0}}{4}>0,$

where $A_{0}\in[0,\pi/2)$ is given in the definition of $\Delta$ (hence it is
determined by the given thickness of the thick internal ray).

######  Proposition 9.1.

When $|W|>1/(qd_{0}|t|)$, we have

$\displaystyle|G(W)|$ $\displaystyle<|W|\quad\text{in Cases 1 and 2${}^{+}$,
and}$ $\displaystyle|G(W)|$ $\displaystyle>|W|\quad\text{in Case 2${}^{-}$}$

by taking sufficiently small $\hat{R}$ and $T_{0}$ in the definitions of
$\hat{U}$ and $\Delta$.

### Proof.

Since $G(W)/W=\tau+W^{-1}+O(W^{-2})=1\mp qt+O(t^{2})+W^{-1}+O(W^{-2})$, we
have

$\log{\left|\frac{G(W)}{W}\right|}={\mathrm{Re}\,}\log\frac{G(W)}{W}={\mathrm{Re}\,}{\left(\mp
qt+O(t^{2})+W^{-1}+O(W^{-2})\right)}.$

Now suppose that $|W|>1/(qd_{0}|t|)$. By taking a smaller $T_{0}$ if
necessary, for any $t\in\Delta=\Delta(A_{0},T_{0})$ we have

$\displaystyle{\mathrm{Re}\,}(qt)\geq q|t|\cos A_{0}=4qd_{0}|t|,$
$\displaystyle{\mathrm{Re}\,}(W^{-1})\leq|W^{-1}|\leq
qd_{0}|t|,\quad\text{and}\quad{\mathrm{Re}\,}\left(O(t^{2})+O(W^{-2})\right)\leq
qd_{0}|t|.$

Hence

$\log{\left|\frac{G(W)}{W}\right|}\leq(-4+1+1)qd_{0}|t|<0$

in Cases 1 and 2+, and

$\log{\left|\frac{G(W)}{W}\right|}\geq(4-1-1)qd_{0}|t|>0$

in Case 2-. $\blacksquare$

###

We first prove Proposition 2.4 for Case 2-, then prove it for Case 1 and Case
2+.

### Proof of Proposition 2.4: Case 2-.

In this case $\tau=1+qt+O(t^{2})$ and thus $B=-1/(qt)+O(1)$. Note that we may
assume $|B|>7\hat{r}$ for any $t\in\Delta$ by taking a smaller $T_{0}$ if
necessary. We first claim:

######  Proposition 9.2.

In Case 2-, for any $W$ with $|W|\geq\hat{r}$ and $|W-B|\geq|B|/2$, we have

$|G(W)-B|\geq|W-B|+d_{0}$

by taking sufficiently small $\hat{R}$ and $T_{0}$ in the definitions of
$\hat{U}$ and $\Delta$.

### Proof.

Since $G(W)-B=\tau(W-B)+O(W^{-1})$ and $|W-B|\geq|B|/2$,

$|G(W)-B|=|\tau|\,|W-B|+|O(W^{-1})|\geq|W-B|+\frac{|\tau|-1}{2|\tau-1|}+|O(W^{-1})|.$

Taking smaller $\hat{R}$ (hence a larger $\hat{r}$) and $T_{0}$ if necessary,
we obtain

$\frac{|\tau|-1}{2|\tau-1|}=\frac{{\mathrm{Re}\,}(qt)+O(|t^{2}|)}{2|qt+O(|t^{2}|)|}\geq\frac{q|t|\cos
A_{0}\cdot(1+O(|t|))}{2q|t|\cdot(1+O(|t|))}\geq\frac{\cos
A_{0}}{2}\cdot\frac{3}{4}$

and $|O(W^{-1})|\leq(\cos A_{0})/8$. Hence

$|G(W)-B|\geq|W-B|+{\left(\frac{3}{8}-\frac{1}{8}\right)}\cos
A_{0}=|W-B|+d_{0}.$

$\blacksquare$

######  Proposition 9.3.

In Case 2-, there exists a unique linearizing coordinate
$L:\mathbb{D}(B,3|B|/4)\to\mathbb{C}$ of the repelling fixed point $B^{\ast}$
of $G$ such that $L(B^{\ast})=0$, $DL(B^{\ast})=1$, and if
$G(W)\in\mathbb{D}(B,3|B|/4)$,

$L\circ G(W)=\tau^{\ast}L(W)$

where $\tau^{\ast}=DG(B^{\ast})$.

### Proof.

By the previous proposition, the disk $\mathbb{D}(B,3|B|/4)$ is compactly
contained in $G(\mathbb{D}(B,3|B|/4))$. Hence the univalent branch of $G^{-1}$
on $G(\mathbb{D}(B,3|B|/4))$ is strictly contracting. By the Riemann mapping
theorem and the Schwarz lemma, there exists a unique attracting fixed point of
$G^{-1}$ in $\mathbb{D}(B,3|B|/4)$, which must be $B^{\ast}$. Hence there is a
unique linearizing coordinate $L$ near $B^{\ast}$ that satisfies
$L(B^{\ast})=0$, $DL(B^{\ast})=1$, and the relation $L\circ
G^{-1}(W)=(\tau^{\ast})^{-1}L(W)$. We obtain the desired linearizing
coordinate by extending $L$ to $\mathbb{D}(B,3|B|/4)$ by this relation.
$\blacksquare$

###

Now we are ready to finish the proof of Proposition 2.4 in Case 2-. Take any
$z_{0}\in J(f_{c_{t}})\cap U_{0}$ for a given $t\in\Delta$. Let
$W_{0}:=\Phi_{t}(z_{0})$ and $W_{k}:=G^{k}(W)$ for $k\geq 0$.

If $|W_{0}-B|\leq|B|/2$, Proposition 9.3 implies that $|W_{k}-B|>|B|/2$ for
some $k\geq 1$ unless $W_{0}=B^{\ast}$. If $W_{0}=B^{\ast}$, then $z_{0}$ is a
repelling fixed point of $f_{c}^{pq}$ in $U_{0}$. If $W_{0}\neq B^{\ast}$, by
Proposition 9.2, either $|W_{k+j}-B|\geq|W_{k}-B|+jd_{0}\to\infty$ as
$j\to\infty$ or $|W_{k+j}|\leq\hat{r}$ for some $j\geq 0$. The former implies
that the orbit of $z_{0}$ is attracted to an attracting fixed point of
$f_{c}^{pq}$ (see Proposition 9.1) and thus a contradiction. The latter
implies that $z_{pq(k+j)}$ is not contained in $U_{0}$ any more.
$\blacksquare$

### Proof of Proposition 2.4: Cases 1 and 2+.

In this case $\tau=1-qt+O(t^{2})$ and thus $B=1/(qt)+O(1)$. The proof is
analogous to Case 2- and we only give an outline.

Since $G^{-1}(W)-B=\tau^{-1}(W-B)+O(W^{-1})$, we have

$|G^{-1}(W)-B|\geq|W-B|+d_{0}$ (9.1)

when $|W|\geq\hat{r}$ and $|W-B|\geq|B|/2$. The fixed point $B^{\ast}$ of
$G^{-1}$ is repelling and we can find a linearizing coordinate $L$ of $G^{-1}$
in $\mathbb{D}(B,3|B|/4)$.

Let $z_{0}\in J(f_{c_{t}})\cap U_{0}$ for a given $t\in\Delta$. Let
$W_{0}:=\Phi_{t}(z_{0})$ and $W_{k}:=G^{k}(W_{0})$ for $k\geq 0$. If
$W_{0}=\infty$, then $z_{0}$ is the repelling fixed point of $f_{c_{t}}^{pq}$
(see Proposition 9.1). Suppose that $W_{0}\neq\infty$. Then $W_{0}$ is not
contained in $\mathbb{D}(B,|B|/2)$, otherwise $W_{k}$ tends to $B^{\ast}$ as
$k\to\infty$. By (9.1), we have $|W_{0}-B|\geq|W_{k}-B|+kd_{0}$ and thus
either $|W_{k}-B|<3|B|/4$ or $|W_{k}|\leq\hat{r}$ for some $k\geq 0$. The
former implies that $z_{pqk}$ is contained in the attracting basin of an
attracting fixed point of $f_{c}^{pq}$, and thus a contradiction. The latter
implies that $z_{pqk}$ is not contained in $U_{0}$ any more. $\blacksquare$

##  10 Proof of Lemma H

In this section we give a proof of Lemma H. The proof employs the branched
coordinates and the dynamics of the form $G(W)=\tau W+1+O(1/W)$ as in the
previous section.

### Dynamics of $G$ near the origin.

Since $\tau\to 1$ as $t\in\Delta$ tends to $0$, the dynamics of $G$
(relatively) near the origin becomes closer to the translation $W\mapsto W+1$:

######  Proposition 10.1.

For any $t\in\Delta$ and any $W$ with $\hat{r}\leq|W|\leq 1/(5q|t|)$, we have
$|G(W)-(W+1)|\leq 1/2$ by taking sufficiently small $\hat{R}$ and $T_{0}$ in
the definitions of $\hat{U}$ and $\Delta$. In particular, we have

$1/2\leq{\mathrm{Re}\,}G(W)-{\mathrm{Re}\,}W\leq
3/2\quad\text{and}\quad|\arg(W-G(W))|\leq\pi/6.$ (10.1)

### Proof.

We have $|(\tau-1)W|=|(\mp qt+O(t^{2}))W|\leq 1/4$ when $|W|\leq(5q|t|)^{-1}$
by taking a smaller $T_{0}$. We also have $|G(W)-(\tau W+1)|=|O(1/W)|\leq 1/4$
when $|W|\geq\hat{r}$ by taking a smaller $\hat{R}$ (hence a larger
$\hat{r}$). It follows that $|G(W)-(W+1)|=|(\tau-1)W+O(1/W)|\leq 1/2$ and
(10.1) is an immediate consequence. $\blacksquare$

### Some additional conditions on $U_{0}$ in $W$-coordinate.

In what follows we suppose that $z_{0}\in J(f_{c_{t}})\cap U_{0}$, and there
exists an $m\in\mathbb{N}$ such that $z_{(m-1)pq}\in U_{0}$ but $z_{mpq}\notin
U_{0}$. Let $W_{k}:=\Phi_{t}(z_{kpq})$. Then we may assume that
$W_{m-1}\in\Phi_{t}(U_{0})$ but $W_{m}\notin\Phi_{t}(U_{0})$.

By taking smaller and appropriate $\hat{R}$ and $R_{0}$ in the definitions of
$\hat{U}$ and $U_{0}$ (and taking a smaller $T_{0}$ if necessary), we may
assume that the set $\Phi_{t}(\partial U_{0})$ is contained in the annulus

$\hat{A}:={\left\\{W\in\overline{\mathbb{C}}_{W}\,:\,5\hat{r}<|W|<6\hat{r}\right\\}}$

(where $\hat{r}\asymp\hat{R}^{-q}$ is introduced in the previous section) for
any $t\in\Delta$. Indeed, we can apply the Koebe distortion theorem to the
family of local coordinates $\\{\psi_{t}\\}$ to control the shape
(eccentricity) of the image $\Phi_{t}(\partial U_{0})$. Because
${\mathrm{Re}\,}W_{m}-{\mathrm{Re}\,}W_{m-1}\leq 3/2$ by Proposition 10.1, we
may assume in addition that $W_{m}$ is contained in this $\hat{A}$.

Now we claim that $3\pi/4\leq\arg W_{m}\leq 5\pi/4$, equivalently,
$|\arg(-W_{m})|\leq\pi/4$. Indeed, since $z_{0}$ is contained in the Julia
set, the same argument as Proposition 2.4 yields that the orbit of $W_{0}$ by
$G$ must leave the domain of $G$ (by taking a larger $\hat{r}$ if necessary).
In other words, there exists an integer $k>m$ such that $|W_{k}|>\hat{r}$ and
$|W_{k+1}|\leq\hat{r}$. Since $|\arg(W_{k}-W_{m})|\leq\pi/6$ by the
proposition above, the condition $|W_{m}|>5\hat{r}$ implies
$|\arg(-W_{m})|\leq\pi/4$. (Here we have used $\cos(5\pi/12)>1/5$. See Figure
8.)

Figure 8: The orbit leaving the domain $\Phi_{t}(U_{0})$ of $G$.

###

The lemma below is a very important conclusion from the radial access
condition:

### Lemma I.

There exists a constant $K_{\mathrm{I}}\in(0,1/(5q)]$ such that for any
$t\in\Delta$, $z_{0}\in J(f_{c_{t}})\cap U_{0}$ and $k\geq 0$, the condition
$|W_{k}|\leq K_{\mathrm{I}}/|t|$ implies $|\arg(-W_{k})|\leq\pi/4$. Moreover,
for such a $W_{k}$, we have $|W_{k+1}|<|W_{k}|$.

###

Hence we can only find $W_{k}$ near the negative real axis in the disk
$\mathbb{D}(0,K_{\mathrm{I}}/|t|)$.

### Proof.

By the assumption on $|W_{m}|$ as above, we may assume that there exists
$j\in\mathbb{N}$ such that

$W_{m-j},\,W_{m-j+1},\cdots,W_{m}\in{\left\\{W\in\overline{\mathbb{C}}_{W}\,:\,|W|\leq(5q|t|)^{-1},\,|\arg(-W)|\leq\pi/4\right\\}}$

and that $|W_{m-j}|\geq(6q|t|)^{-1}$. Note that the condition
$|\arg(-W)|\leq\pi/4$ implies $|W|/\sqrt{2}\leq-{\mathrm{Re}\,}W\leq|W|.$

By Proposition 10.1, we have ${\mathrm{Re}\,}W_{m}-{\mathrm{Re}\,}W_{m-j}\leq
3j/2$ and thus

$3j/2\geq-|W_{m}|+|W_{m-j}|/\sqrt{2}\geq-|W_{m}|+(6\sqrt{2}q|t|)^{-1}.$

Since the assumption $5\hat{r}<|W_{m}|<6\hat{r}$ implies $|W_{m}|\asymp 1$, we
conclude that $j\geq C_{6}/|t|$ for some constant $C_{6}\asymp 1$ independent
of $t\in\Delta$ (taking a smaller $T_{0}$ if necessary).

In Case 2-, as in the proof of Proposition 2.4, there is $k^{\prime}>j\geq 1$
for which $|W_{m-k^{\prime}}-B|>|B|/2$ (otherwise $W_{0}=B^{*}$). We have

$|W_{m}-B|\geq|W_{m-k^{\prime}}-B|+k^{\prime}d_{0}>|W_{m-k^{\prime}}-B|+C_{6}d_{0}/|t|$

by Proposition 9.2. Hence

$|W_{m-k^{\prime}}|\geq|B|-|W_{m-k^{\prime}}-B|>|B|-|W_{m}-B|+C_{6}d_{0}/|t|\geq-|W_{m}|+C_{6}d_{0}/|t|.$

Since $|W_{m}|\asymp 1$, there exists a constant $K_{\mathrm{I}}\asymp 1$ such
that $|W_{m-k^{\prime}}|>K_{\mathrm{I}}/|t|$ for any such $k^{\prime}$ and
$t\in\Delta$ (again taking a smaller $T_{0}$ if necessary). Hence if we have
$|W_{k}|\leq K_{\mathrm{I}}/|t|$ for some $k$, then $m-j\leq k\leq m$ and thus
$|\arg(-W_{k})|\leq\pi/4$. Since $|W_{k}|\leq K_{\mathrm{I}}/|t|$ also implies
$|W_{k}|\leq 1/(5q|t|)$, we conclude $K_{\mathrm{I}}\leq 1/(5q)$.

The proof for Cases 1 and 2+ is analogous: By (9.1), we have

$|W_{m-k^{\prime}}-B|\geq|W_{m}-B|+k^{\prime}d_{0}>|W_{m}-B|+C_{6}d_{0}/|t|$

instead, and this implies

$|W_{m-k^{\prime}}|\geq|W_{m-k^{\prime}}-B|-|B|>|W_{m}-B|-|B|+C_{6}d_{0}/|t|\geq-|W_{m}|+C_{6}d_{0}/|t|.$

Then we repeat the same argument as above.

Now, we show the condition $|W_{k}|\leq K_{\mathrm{I}}/|t|$ implies
$|W_{k+1}|<|W_{k}|$. By Proposition 10.1, we get $|W_{k+1}-(W_{k}+1)|\leq 1/2$
and thus

$\left|\frac{W_{k+1}}{W_{k}}\right|\leq\left|1+\frac{1}{W_{k}}\right|+\frac{1}{2|W_{k}|}.$

Since $|\arg(-W_{k})|\leq\pi/4$ implies $3\pi/4\leq\arg(1/W_{k})\leq 5\pi/4$,
we obtain

$\left|1+\frac{1}{W_{k}}\right|\leq\sqrt{{\left(1-\frac{1}{\sqrt{2}~{}|W_{k}|}\right)}^{2}+{\left(\frac{1}{\sqrt{2}~{}|W_{k}|}\right)}^{2}}$

and thus

$\displaystyle\left|\frac{W_{k+1}}{W_{k}}\right|$ $\displaystyle\leq$
$\displaystyle
1-\frac{1}{\sqrt{2}~{}W_{k}}+\frac{1}{2~{}|W_{k}|}+O\left(\frac{1}{|W_{k}|^{2}}\right)$
$\displaystyle=$ $\displaystyle
1-\frac{\sqrt{2}-1}{2~{}|W_{k}|}+O\left(\frac{1}{|W_{k}|^{2}}\right)$
$\displaystyle<$ $\displaystyle 1$

by taking a sufficiently large $\hat{r}$ if necessary. $\blacksquare$

######  Remark 10.2.

Once we have $|W_{k}|\leq K_{\mathrm{I}}/|t|$, then $|W_{j}|\leq
K_{\mathrm{I}}/|t|$ for $k\leq j\leq m$.

######  Proposition 10.3.

For any $t\in\Delta$, any $z_{0}\in J(f_{c_{t}})\cap U_{0}$ that is not a
fixed point of $f_{c_{t}}^{pq}$, and any $k$ with $0\leq k\leq m$, we have

$|DG^{k}(W_{0})|\asymp|\tau|^{k}$

in Case 1 or Case 2+, and

$|DG^{k}(W_{0})|\asymp|\tau^{\ast}|^{k}$

in Case 2-, where $W_{0}=\Phi_{t}(z_{0})$. The implicit constants are
independent of $t,z_{0}$, and $k$.

### Proof for Case 1 and Case 2+.

Since $DG(W)=\tau+O(W^{-2})$,

$DG^{k}(W_{0})=\tau^{k}\,\prod_{j=0}^{k-1}{\left(1+O({W_{j}}^{-2})\right)}.$
(10.2)

Hence it is enough to show that $\sum_{j=0}^{k-1}|W_{j}|^{-2}$ is uniformly
bounded in $t$, $W_{0}$, and $k$.

As in the proof of Proposition 2.4, $\mathbb{D}(B,3|B|/4)$ is contained in the
attracting basin of $B^{*}$. Therefore, we may always assume
$|W_{j}|\geq\hat{r}$ and $|W_{j}-B|>|B|/2$ for $j=0,1,\ldots,k$. By (9.1),
$|W_{j}-B|$ is strictly decreasing in $j$. This implies that if
$|W_{k_{1}}-B|<4|B|$ for some $0\leq k_{1}\leq k$, then $|W_{j}-B|<4|B|$ for
all $k_{1}\leq j\leq k$. On the other hand, by Lemma I, if $|W_{k_{2}}|\leq
K_{\mathrm{I}}/|t|$ for some $0\leq k_{2}\leq k$, then $|W_{j}|\leq
K_{\mathrm{I}}/|t|$ and $|\arg(-W_{j})|\leq\pi/4$ for all $k_{2}\leq j\leq k$
$(\leq m)$. Consequently, we can assume the location of $W_{0},\ldots,W_{k}$
to be

  1. (i+)

$|W_{j}-B|\geq 4|B|$ for $j=0,\ldots,k_{1}-1$,

  2. (ii+)

$|W_{j}-B|<4|B|$ and $|W_{j}|>K_{\mathrm{I}}/|t|$ for
$j=k_{1},\ldots,k_{2}-1$,

  3. (iii+)

$|W_{j}-B|<4|B|$ and $|W_{j}|\leq K_{\mathrm{I}}/|t|$ for $j=k_{2},\ldots,k$.

For the case (i+), by (9.1) again we have

$|W_{j}|\geq|W_{j}-B|-|B|\geq|W_{k_{1}-1}-B|-|B|+(k_{1}-1-j)d_{0}\geq|B|+(k_{1}-1-j)d_{0}.$

Hence

$\displaystyle\sum_{j=0}^{k_{1}-1}|W_{j}|^{-2}=O{\left(\sum_{j=0}^{k_{1}-2}\frac{1}{(k_{1}-1-j)^{2}}\right)}+\frac{1}{|B|^{2}}=O{\left(\sum_{j=1}^{\infty}\frac{1}{j^{2}}\right)}+O{\left(|t|^{2}\right)}=O(1).$

The condition $|W_{j}-B|<4|B|$ implies $|W_{j}|<5|B|=5/(q|t|)+O(1)$. Hence the
case (ii+) implies $|W_{j}|\asymp|t|^{-1}$ and thus
$|W_{j}|^{-2}\asymp|t|^{2}$. From (9.1), we obtain

$4|B|>|W_{k_{1}}-B|\geq|W_{k_{2}-1}-B|+(k_{2}-1-k_{1})d_{0}\geq(k_{2}-1-k_{1})d_{0}$

and thus $k_{2}-1-k_{1}=O(|t|^{-1})$. This gives

$\sum_{j=k_{1}}^{k_{2}-1}|W_{j}|^{-2}=O(|t|).$

In the (iii+) case, we have $|\arg(-W_{j})|\leq\pi/4$ and thus
$|W_{j}|/\sqrt{2}\leq-{\mathrm{Re}\,}W_{j}\leq|W_{j}|$. By Proposition 10.1,
we have $-{\mathrm{Re}\,}W_{j}\geq-{\mathrm{Re}\,}W_{k}+(k-j)/2.$ Hence, we
get $|W_{j}|\geq|W_{k}|/\sqrt{2}+(k-j)/2\geq(k-j)/2$, and

$\displaystyle\sum_{j=k_{2}}^{k-1}|W_{j}|^{-2}=O{\left(\sum_{j=k_{2}}^{k-1}\frac{1}{(k-j)^{2}}\right)}=O{\left(\sum_{j=1}^{\infty}\frac{1}{j^{2}}\right)}=O(1).$

### Proof for Case 2-.

First note by Proposition 9.1 that $|W_{j}|\leq 1/(qd_{0}|t|)$ for $0\leq
j\leq k$ (otherwise $W_{j}$ must be attracted by $\infty$). Second, by
Proposition 9.2, if $|W_{k_{1}}-B|\geq|B|/2$ for some least $k_{1}\leq k$,
then $|W_{j}-B|$ is strictly increasing for $j$ with $k_{1}\leq j\leq k$.
Hence, $|W_{j}-B|\geq|B|/2$ for such a $j$. Furthermore, if $|W_{k_{2}}|\leq
K_{\mathrm{I}}/|t|$ for some $0\leq k_{2}\leq k$, then as in the Case 1 and
Case 2+, the location of $W_{0},\ldots,W_{k}$ must be

  1. (i-)

$|W_{j}-B|<|B|/2$ for $j=0,\ldots,k_{1}-1$,

  2. (ii-)

$|W_{j}-B|\geq|B|/2$ and $|W_{j}|>K_{\mathrm{I}}/|t|$ for
$j=k_{1},\ldots,k_{2}-1$,

  3. (iii-)

$|W_{j}-B|\geq|B|/2$ and $|W_{j}|\leq K_{\mathrm{I}}/|t|$ for
$j=k_{2},\ldots,k$.

Now,

$\displaystyle\frac{DG^{k}(W_{0})}{(\tau^{\ast})^{k}}$ $\displaystyle=$
$\displaystyle\frac{DG^{k_{1}-1}(W_{0})}{(\tau^{\ast})^{k_{1}-1}}\cdot\frac{DG^{k-k_{1}+1}(W_{k_{1}-1})}{(\tau^{\ast})^{k-k_{1}+1}}$
$\displaystyle\asymp$ $\displaystyle
1\cdot\frac{DG^{k-k_{1}+1}(W_{k_{1}-1})}{(\tau^{\ast})^{k-k_{1}+1}}\qquad\mbox{(by
Proposition \ref{prop_2-_2})}$ $\displaystyle\asymp$
$\displaystyle{\left(\frac{\tau}{\tau^{\ast}}\right)}^{k-k_{1}+1}\prod_{j=k_{1}-1}^{k-1}{\left(1+O({W_{j}}^{-2})\right)}\qquad\mbox{(as
\eqref{DGk=})}$ $\displaystyle\asymp$
$\displaystyle{\left(1+O(t^{2})\right)}^{k-k_{1}+1}\cdot\prod_{j=k_{1}-1}^{k-1}{\left(1+O({W_{j}}^{-2})\right)}\qquad\mbox{(since
$\displaystyle\frac{\tau}{\tau^{\ast}}=1+O(t^{2})$.)}$

Because $|W_{k_{1}-1}|\geq|B|-|W_{k_{1}-1}-B|>|B|-|B|/2=|B|/2=O(1/|t|)$ and
because $|W_{k_{1}-1}|\leq 1/(qd_{0}|t|)$, we have
$|W_{k_{1}-1}^{-2}|=O(|t|^{2})$.

By Proposition 9.2, we know

$|W_{k}-B|\geq|W_{k_{1}}-B|+(k-k_{1})d_{0}\geq(k-k_{1})d_{0}$

and thus $k-k_{1}\leq(|W_{k}|+|B|)/d_{0}=O(|t|^{-1})$. This implies that
$(1+O(t^{2}))^{k-k_{1}+2}=1+O(t)$.

It remains to obtain an estimate of
$\prod_{j=k_{1}}^{k-1}{\left(1+O(W_{j}^{-2})\right)}$. Again, it is enough to
obtain an estimate by showing that $\sum_{j=k_{1}}^{k-1}|W_{j}|^{-2}$ is
uniformly bounded in $k$ and $W_{0}$.

For the (ii-) case, since $|W_{j}|<1/(qd_{0}|t|)$, we obtain
$|W_{j}|\asymp|t|^{-1}$. Therefore,

$\sum_{j=k_{1}}^{k_{2}-1}|W_{j}|^{-2}=O(|t|)\qquad\mbox{(since
$k-k_{1}=O(|t|^{-1})$ so is $k_{2}-k_{1}$.)}$

The (iii-) case follows exactly as the (iii+) case.

The proof of Proposition 10.3 is complete. $\blacksquare$

### Lemma H in the branched coordinates.

To give estimates for the sums of the form

$\sum_{k=0}^{m-1}\frac{1}{|Df_{c}^{kpq}(z_{j})|}\quad\text{and}\quad\sum_{k=0}^{m-1}\frac{|c-{\hat{c}}|+|f_{c}^{kpq}(z_{j})-b_{j}(c)|}{|Df_{c}^{kpq}(z_{j})|}$

with $0\leq j\leq p-1$, we rewrite them in $W$-coordinates. It is enough to
consider the case of $j=0$, since we may apply the same argument as in the
proof of Lemma G.

Let $W=\Phi_{t}(z)=\Psi_{t}\circ\psi_{t}(z)$ for $z\in\hat{U}$ such that
$W_{k}:=\Phi_{t}(z_{kpq})$ for each $k$. Note that $|D\psi_{t}(z)|\asymp 1$
for any $z\in\hat{U}$, where the implicit constant is independent of
$t\in\Delta$. Since $D\Psi_{t}(w)=\lambda_{t}^{q^{2}}/w^{q+1}$, we have

$|D\Phi_{t}(z)|=|D\Psi_{t}(w)\cdot
D\psi_{t}(z)|\asymp|w|^{-(q+1)}\asymp|W|^{1+1/q}.$

By the chain rule, we obtain

$\displaystyle|Df_{c_{t}}^{kpq}(z_{0})|$ $\displaystyle=|D(\Phi_{t}^{-1}\circ
G^{k}\circ\Phi_{t})(z_{0})|=|D\Phi_{t}^{-1}(W_{k})\cdot DG^{k}(W_{0})\cdot
D\Phi_{t}(z_{0})|$
$\displaystyle\asymp|W_{k}|^{-1-1/q}\cdot|DG^{k}(W_{0})|\cdot|W_{0}|^{1+1/q}$
$\displaystyle={\left|\frac{W_{0}}{W_{k}}\right|}^{1+1/q}\cdot|DG^{k}(W_{0})|.$

We let

${\mathcal{S}}_{1}:=\sum_{k=0}^{m-1}{\left|\frac{W_{k}}{W_{0}}\right|}^{1+1/q}\cdot\frac{1}{|DG^{k}(W_{0})|}\asymp\sum_{k=0}^{m-1}\frac{1}{|Df_{c_{t}}^{kpq}(z_{0})|},$

and

${\mathcal{S}}_{2}:=\frac{1}{|W_{0}|^{1/q}}\sum_{k=0}^{m-1}{\left|\frac{W_{k}}{W_{0}}\right|}\cdot\frac{1}{|DG^{k}(W_{0})|}\asymp\sum_{k=0}^{m-1}\frac{|f_{c_{t}}^{kpq}(z_{0})-b_{t}|}{|Df_{c_{t}}^{kpq}(z_{0})|},$

where we used the fact that
$|f_{c}^{kpq}(z_{0})-b_{0}(c)|=|z_{kpq}-b_{t}|\asymp|W_{k}|^{-1/q}$. Now Lemma
H is reduced to the estimates

${\mathcal{S}}_{1}=O{\left(\frac{1}{|t|}\right)}\quad\text{and}\quad{\mathcal{S}}_{2}=O{\left(\frac{1}{|t|^{1-1/q}}\right)}.$

Indeed, Proposition 2.2 implies

$\sum_{k=0}^{m-1}\frac{1}{|Df_{c_{t}}^{kpq}(z_{0})|}\asymp{\mathcal{S}}_{1}=O{\left(\frac{1}{|t|}\right)}=O{\left(\frac{1}{\sqrt{|c_{t}-{\hat{c}}|}}\right)}$

in Case 1, and

$\displaystyle\sum_{k=0}^{m-1}\frac{|c_{t}-{\hat{c}}|+|f_{c_{t}}^{kpq}(z_{0})-b_{t}|}{|Df_{c_{t}}^{kpq}(z_{0})|}$
$\displaystyle\asymp|c_{t}-{\hat{c}}|{\mathcal{S}}_{1}+{\mathcal{S}}_{2}$
$\displaystyle=O{\left(|t|\cdot\frac{1}{|t|}+\frac{1}{|t|^{1-1/q}}\right)}=O{\left(\frac{1}{|c_{t}-{\hat{c}}|^{1-1/q}}\right)}$

in Case 2.

### Proof of Lemma H for Case $2^{-}$.

We will use tha fact that $\tau^{\ast}=1+qt+O(t^{2})$ and

$|\tau^{\ast}|=1+{\mathrm{Re}\,}(qt)+O(|t^{2}|)\geq 1+qd_{0}|t|$

for $t\in\Delta$ if we take a smaller $T_{0}$ if necessary.

First, suppose that $|W_{0}|\geq K_{\mathrm{I}}/|t|$. Since it must be
$|W_{k}|\leq 1/(qd_{0}|t|)$ for any $k=0,1,\ldots,m$ by Proposition 9.1
(otherwise $W_{k}$ is attracted by $\infty$), we have $|W_{0}|\asymp|t|^{-1}$
and $|W_{k}/W_{0}|=O(1)$. Hence by Proposition 10.3, we have

${\mathcal{S}}_{1}=\sum_{k=0}^{m-1}{\left|\frac{W_{k}}{W_{0}}\right|}^{1+1/q}\cdot\frac{1}{|DG^{k}(W_{0})|}=\sum_{k=0}^{m-1}O(1)\cdot\frac{1}{|\tau^{\ast}|^{k}}=O{\left(\sum_{k=0}^{m-1}\frac{1}{(1+qd_{0}|t|)^{k}}\right)}=O{\left(\frac{1}{|t|}\right)}$

and

${\mathcal{S}}_{2}=\frac{1}{|W_{0}|^{1/q}}\sum_{k=0}^{m-1}{\left|\frac{W_{k}}{W_{0}}\right|}\cdot\frac{1}{|DG^{k}(W_{0})|}=|t|^{1/q}\sum_{k=0}^{m-1}O(1)\cdot\frac{1}{|\tau^{\ast}|^{k}}=O{\left(\frac{1}{|t|^{1-1/q}}\right)}.$

Next, suppose that $|W_{0}|\leq K_{\mathrm{I}}/|t|$. By Proposition 10.1 and
Lemma I, we have $|\arg(-W_{k})|\leq\pi/4$ and thus
$|W_{k}|/\sqrt{2}\leq-{\mathrm{Re}\,}W_{k}\leq|W_{k}|$ for any
$k=0,1,\ldots,m$. By Proposition 10.1 again we have
${\mathrm{Re}\,}W_{0}+k/2\leq{\mathrm{Re}\,}W_{k}$. Hence

$0\leq|W_{k}|/\sqrt{2}\leq-{\mathrm{Re}\,}W_{k}\leq-{\mathrm{Re}\,}W_{0}-k/2\leq|W_{0}|-k/2$
(10.3)

and this implies

${\left|\frac{W_{k}}{W_{0}}\right|}\leq\sqrt{2}\cdot\frac{|W_{0}|-k/2}{|W_{0}|}=O(1).$

In particular, by letting $k=m$ in (10.3) we have $m\leq 2|W_{0}|=O(|W_{0}|)$.

Since $|W_{0}|\leq K_{\mathrm{I}}/|t|$, we have

${\mathcal{S}}_{1}=\sum_{k=0}^{m-1}O(1)\cdot\frac{1}{|\tau^{\ast}|^{k}}\leq\sum_{k=0}^{m-1}O(1)\cdot
1^{k}=O(m)=O(|W_{0}|)=O{\left(\frac{1}{|t|}\right)}$

and

$\displaystyle{\mathcal{S}}_{2}$
$\displaystyle=\frac{1}{|W_{0}|^{1/q}}\sum_{k=0}^{m-1}O(1)\cdot\frac{1}{|\tau^{\ast}|^{k}}\leq\frac{1}{|W_{0}|^{1/q}}\sum_{k=0}^{m-1}O(1)\cdot
1^{k}$
$\displaystyle=\frac{O(m)}{|W_{0}|^{1/q}}=O(|\,W_{0}|^{1-1/q}|)=O{\left(\frac{1}{|t|^{1-1/q}}\right)}.$

This completes the proof for Case 2-.

### Preliminary for Case 1 and Case $2^{+}$.

We will use the fact that $\tau^{-1}=1+qt+O(t^{2})$ and

$|\tau|^{-1}=1+{\mathrm{Re}\,}(qt)+O(|t^{2}|)\geq 1+qd_{0}|t|$

for $t\in\Delta$ if we take a smaller $T_{0}$ if necessary.

It is very convenient to use Ueda’s modulus defined as follows (see [U]): For
$W\neq\infty$,

$N(W):=|W-B|-|B|.$

Then one can easily check that

  * •

$|W|\geq N(W)$; and

  * •

$N(W)\geq|W|/3$ when $|W-B|\geq 4|B|$.

Indeed, $|W|\geq N(W)$ is just the triangle inequality; and when $|W-B|\geq
4|B|$, we have $|W|\geq 3|B|$ and thus $N(W)/|W|=|1-B/W|-|B/W|\geq 1/3$.

Here is another useful fact:

######  Proposition 10.4.

In Case 1 and Case 2+, we have

$N(G^{-1}(W))\geq|\tau|^{-1}N(W)+d_{0}$

by taking sufficiently small $\hat{R}$ and $T_{0}$ in the definitions of
$\hat{U}$ and $\Delta$.

### Proof.

Since $|G^{-1}(W)-B|=|\tau|^{-1}|W-B|+O(|W^{-1}|)$, we have

$N(G^{-1}(W))=|\tau|^{-1}N(W)+\frac{|\tau|^{-1}-1}{|1-\tau|}+O(|W^{-1}|).$

By taking a smaller $T_{0}$ if necessary, we have

$\frac{|\tau|^{-1}-1}{|1-\tau|}=\frac{{\mathrm{Re}\,}t}{|t|}(1+O(|t|))\geq\frac{\cos
A_{0}}{2}=2d_{0}$

for any $t\in\Delta$. By taking a smaller $\hat{R}$ (hence a larger $\hat{r}$)
if necessary, we have $O(|W^{-1}|)\leq(\cos A_{0})/4=d_{0}$ for any
$|W|\geq\hat{r}$. Hence we have the desired inequality. $\blacksquare$

######  Proposition 10.5.

In Case 1 and Case 2+, we have

${\left|\frac{W_{k}}{W_{0}}\right|}=O(|\tau|^{k})$

for any $k=0,1,\ldots,m$.

### Proof.

We adopt the argument of Proposition 10.3 and assume that

  1. (i+)

$|W_{j}-B|\geq 4|B|$ for $j=0,\ldots,k_{1}-1$,

  2. (ii+)

$|W_{j}-B|<4|B|$ and $|W_{j}|>K_{\mathrm{I}}/|t|$ for
$j=k_{1},\ldots,k_{2}-1$,

  3. (iii+)

$|W_{j}-B|<4|B|$ and $|W_{j}|\leq K_{\mathrm{I}}/|t|$ for $j=k_{2},\ldots,k$

for some $k_{1}$ and $k_{2}$. Now we consider the product

${\left|\frac{W_{k}}{W_{0}}\right|}={\left|\frac{W_{k_{1}-1}}{W_{0}}\right|}\cdot{\left|\frac{W_{k_{2}}}{W_{k_{1}-1}}\right|}\cdot{\left|\frac{W_{k}}{W_{k_{2}}}\right|}.$
(10.4)

In the case of (i+), we can apply Proposition 10.4 and it follows that

$|W_{0}|\geq
N(W_{0})\geq|\tau|^{-(k_{1}-1)}N(W_{k_{1}-1})\geq|\tau|^{-k_{1}+1}|W_{k_{1}-1}|/3.$

Hence $|W_{k_{1}-1}/W_{0}|\leq 3\,|\tau|^{k_{1}-1}$. Since $|W_{k_{1}-1}|\geq
3|B|=3/(q|t|)+O(1)$ and $|W_{k_{2}}|\leq K_{\mathrm{I}}/|t|$, we have
$|W_{k_{2}}/W_{k_{1}-1}|=O(1)$. Since $|W_{j}|$ is decreasing in the case of
(iii+) by Lemma I, we have $|W_{k}/W_{k_{2}}|\leq 1$. Hence by (10.4), we
obtain $|W_{k}/W_{0}|=O(|\tau|^{k_{1}-1})$.

To show the estimate $|W_{k}/W_{0}|=O(|\tau|^{k})$, it is enough to show that
$|\tau|^{k-(k_{1}-1)}=O(1)$. Because $|W_{k_{1}}-B|\leq 4|B|$ and hence
$|W_{k_{1}}|\leq 5|B|$, by (9.1) we have

$|W_{k_{1}}-B|\geq|W_{k}-B|+(k-k_{1})\,d_{0}\geq(k-k_{1})\,d_{0}$

and thus $k-k_{1}\leq(|W_{k_{1}}|+|B|)/d_{0}=O(|t|^{-1})$. Hence

$|\tau|^{k-(k_{1}-1)}=(1-{\mathrm{Re}\,}(qt)+O(t^{2}))^{k-k_{1}}\cdot|\tau|=O(1)$

and this completes the proof. $\blacksquare$

### Proof of Lemma H for Case 1 and Case $2^{+}$.

By Propositions 10.3 and 10.5, we have

${\mathcal{S}}_{1}=\sum_{k=0}^{m-1}{\left|\frac{W_{k}}{W_{0}}\right|}^{1+1/q}\cdot\frac{1}{|DG^{k}(W_{0})|}=\sum_{k=0}^{m-1}O(|\tau|^{k(1+1/q)})\cdot\frac{1}{|\tau|^{k}}=O{\left(\sum_{k=0}^{m-1}|\tau|^{k/q}\right)}.$

Since $|\tau|^{1/q}\leq 1-d_{0}|t|$ for $t\in\Delta$ (by taking a smaller
$T_{0}$ if necessary), we obtain ${\mathcal{S}}_{1}=O(|t|^{-1})$.

Similarly we have

${\mathcal{S}}_{2}=\frac{1}{|W_{0}|^{1/q}}\sum_{k=0}^{m-1}{\left|\frac{W_{k}}{W_{0}}\right|}\cdot\frac{1}{|DG^{k}(W_{0})|}=\frac{1}{|W_{0}|^{1/q}}\sum_{k=0}^{m-1}O(|\tau|^{k})\cdot\frac{1}{|\tau|^{k}}=\frac{O(m)}{|W_{0}|^{1/q}}.$

First we suppose that $|W_{0}-B|<4|B|$. Then we have
$|W_{0}|<5|B|=O(|t|^{-1})$. By (9.1),

$|W_{0}|\geq|W_{0}-B|-|B|\geq|W_{m}-B|-|B|+md_{0}\geq-|W_{m}|+md_{0}.$

By the assumption that $W_{m}\in\hat{A}$, we have $|W_{m}|<6\hat{r}$ and thus
$md_{0}\leq|W_{0}|+6\hat{r}$. We obtain $m=O(|W_{0}|)$ since
$5\hat{r}<|W_{0}|=O(|t|^{-1})$. Hence

${\mathcal{S}}_{2}=\frac{O(m)}{|W_{0}|^{1/q}}=O(|W_{0}|^{1-1/q})=O{\left(\frac{1}{|t|^{1-1/q}}\right)}.$

Next we suppose that $|W_{0}-B|\geq 4|B|$. There exists a $k\geq 0$ such that

  * •

$|W_{j}-B|\geq 4|B|$ for $0\leq j\leq k$; and

  * •

$|W_{j}-B|<4|B|$ for $k\leq j\leq m$.

(We may regard $k$ and $m$ as $k_{1}-1$ and $k$ in the proof of Proposition
10.5 respectively.) Since $|W_{k}/W_{0}|\leq 3|\tau|^{k}$ as in the proof of
Proposition 10.5, and $|W_{k}|\geq 3|B|>1/(q|t|)$ for $t\in\Delta$, we have
$3|\tau|^{k}\geq 1/(q|tW_{0}|)$. Hence

$1<|\tau|^{-k}\leq 3q|tW_{0}|,\quad\mbox{or equivalently}\quad
0<k\log|\tau|^{-1}\leq\log(3q|tW_{0}|).$

This implies that

$0<k({\mathrm{Re}\,}(qt)+O(t^{2}))\leq\log(3q|tW_{0}|).$

Since ${\mathrm{Re}\,}(qt)+O(t^{2})\geq(q|t|\cos A_{0})/2$ for $t\in\Delta$
(by taking a smaller $T_{0}$ if necessary), we conclude

$k=O{\left(\frac{\log(3q|tW_{0}|)}{|t|}\right)}.$

As in the proof of Proposition 10.5, we have $m-k=O(1/|t|)$. Since
$|W_{0}|\geq 3|B|\geq 2/(q|t|)$ (for $t\in\Delta$ by taking a smaller $T_{0}$
if necessary), we obtain $\log(3q|tW_{0}|)\geq\log 6>1$ and thus

$m=k+(m-k)=O{\left(\frac{\log(3q|tW_{0}|)}{|t|}\right)}+O{\left(\frac{1}{|t|}\right)}=O{\left(\frac{\log(3q|tW_{0}|)}{|t|}\right)}.$

Hence we conclude that

${\mathcal{S}}_{2}=\frac{O(m)}{|W_{0}|^{1/q}}=O{\left(\frac{\log(3q|tW_{0}|)}{|tW_{0}|^{1/q}}\right)}\cdot\frac{1}{|t|^{1-1/q}}=O{\left(\frac{1}{|t|^{1-1/q}}\right)},$

where we used the fact that $x^{-1/q}\log(3qx)$ is bounded if $3qx\geq 6$.
$\blacksquare$

##  11 Proof of Lemma D

In this section we give a proof of Lemma D. Since the proof heavily relies on
Lemma I, we restate Lemma D in terms of the parameter $t\in\Delta\cup\\{0\\}$.
(Note that we have ${\hat{c}}=c_{0}$).

######  Proposition 11.1.

There exist constants $K_{\mathrm{D}}>0$ and $\nu_{0}\in(0,R_{0})$ such that
for any $\nu\in(0,\nu_{0})$, there exists a constant
$T_{0}=T_{0}(\nu)\in(0,2\cos A_{0})$ such that for any
$t\in\Delta=\Delta(A_{0},T_{0})$, $\zeta\in P(f_{c_{0}})$, and $z\in
J(f_{c_{t}})-{\mathcal{V}}(c_{t})$,

$|\zeta-z|\geq K_{\mathrm{D}}\,\nu.$

Without loss of generality, we may assume that $K_{\mathrm{D}}\leq 1$.

### Proof.

We prove it by contradiction. Suppose that for any $K_{\mathrm{D}}>0$ and
$\nu_{0}\in(0,R_{0})$, there exists a $\nu\in(0,\nu_{0})$ such that for any
$T_{0}\in(0,2\cos A_{0})$, there exist $t\in\Delta=\Delta(A_{0},T_{0})$,
$\zeta\in P(f_{c_{0}})$, and $z\in J(f_{c_{t}})-{\mathcal{V}}(c_{t})$ such
that

$|\zeta-z|<K_{\mathrm{D}}\,\nu.$

For instance, let $K_{\mathrm{D}}:=1/n$, $\nu_{0}:=R_{0}/n$ for integer $n\geq
1$. Then for any sufficiently large $n$, we may assume that $R_{0}/n<1$ and
there exists a $\nu_{n}\in(0,R_{0}/n)$ such that for $T_{0}:=2\nu_{n}^{2q}\cos
A_{0}$ we can find sequences $t_{n}\in\Delta=\Delta(A_{0},T_{0})$,
$\zeta_{n}\in P(f_{c_{0}})$, and $z_{n}\in J(f_{c_{n}})-{\mathcal{V}}(c_{n})$
with $c_{n}=c_{t_{n}}$ such that

$|\zeta_{n}-z_{n}|<\frac{1}{n}\cdot\nu_{n}.$

(Now, $\mathcal{V}(c_{n})=\bigcup_{k=0}^{p-1}f_{c_{n}}^{-k}(V_{0})$ and
$V_{0}=\mathbb{D}({\hat{b}},\nu_{n})$.) We may assume that $\zeta_{n}$ and
$z_{n}$ tends to the same limit $\hat{\zeta}\in P(f_{c_{0}})$ by passing
through a subsequence, since $P(f_{c_{0}})$ is compact and the Julia sets
$J(f_{c_{n}})$ are uniformly bounded. Moreover, there exists a $j\geq 0$ such
that $f_{c_{0}}^{j}(\hat{\zeta})$ is contained in
$\mathbb{D}({\hat{b}},R_{0}/2)\subset U_{0}=\mathbb{D}({\hat{b}},R_{0})$. By
replacing $\zeta_{n}$ and $z_{n}$ by $f_{c_{0}}^{j}(\zeta_{n})$ and
$f_{c_{n}}^{j}(z_{n})$ respectively, we may assume that $\zeta_{n}\in U_{0}$
and $z_{n}\in U_{0}-V_{0}$ for sufficiently large $n$.

In $w$-coordinate $w=\psi_{0}(z)$ given in Proposition 2.3 for $t=0$, the
local dynamics of $f_{c_{0}}^{pq}$ in $U_{0}$ can be observed as $w\mapsto
w(1+w^{q}+O(w^{2q}))$ and the orbit of $\psi_{0}(\zeta_{n})$ by this map tends
to $0$ tangentially to the attracting direction (that is, the set of $w$’s
with ${\mathrm{Im}\,}(w^{q})=0$ and ${\mathrm{Re}\,}(w^{q})<0$). This implies
that (by taking sufficiently large $j$ if necessary) $\psi_{0}(\zeta_{n})$ is
contained in the set
$X:=\\{w\in\psi_{0}(U_{0})\,:\,|\arg(-w^{q})|\leq\pi/4\\}$.

Figure 9: The sets $X$ and $Y$ in $w$-coordinate for $q=3$.

Next we consider $z_{n}\in U_{0}-V_{0}$ in the local coordinate
$w=\psi_{t_{n}}(z)$. Since $\psi_{t_{n}}$ is univalent and
$\nu_{n}<|z_{n}-{\hat{b}}|<R_{0}$, there exists a constant $C_{7}>0$
independent of $n$ such that $w_{n}:=\psi_{t_{n}}(z_{n})$ satisfies
$|w_{n}|\geq C_{7}\nu_{n}$. In $W$-coordinate
$W=\Psi_{t_{n}}(w)=-\lambda_{t_{n}}^{q^{2}}/(qw^{q})$, we have
$\arg(-W)=-\arg(w^{q})+O(t_{n})$ since $\lambda_{t_{n}}^{q^{2}}=1+O(t_{n})$.
Now, for $W_{n}=\Psi_{t_{n}}(w_{n})=\Psi_{t_{n}}\circ\psi_{t_{n}}(z_{n})$ with
sufficiently large $n$, we obtain

$|W_{n}|\leq\left|\frac{1+O(t_{n})}{qC_{7}^{q}\nu_{n}^{q}}\right|\leq\left|\frac{\sqrt{2\cos
A_{0}}}{qC_{7}^{q}}\frac{1+O(|t_{n}|)}{\sqrt{|t_{n}|}}\right|\leq\frac{K_{\mathrm{I}}}{|t_{n}|}.$

Therefore, by Lemma I, we have $|\arg(-W_{n})|\leq\pi/4$ since $z_{n}\in
J(f_{c_{n}})\cap U_{0}$. Hence in $w$-coordinate we have
$|\arg(w_{n}^{q})|=|\arg(-W_{n})|+O(|t_{n}|)\leq\pi/4+O(\nu_{n}^{2q})$ for
$w_{n}=\psi_{t_{n}}(z_{n})$.

We also have
$|\psi_{t_{n}}(z_{n})-\psi_{0}(z_{n})|=O(|t_{n}|)=O(\nu_{n}^{2q})$, since the
map $(t,z)\mapsto\psi_{t}(z)$ is holomorphic in both $t$ and $z$. Hence
$\psi_{0}(z_{n})$ is contained in
$Y:=\\{w\in\psi_{0}(U_{0})\,:\,|\arg(w^{q})|\leq\pi/3,\,|w|\geq
C_{7}\nu_{n}\cdot(3/4)\\}$ for sufficiently large $n$. It follows that
$|\psi_{0}(\zeta_{n})-\psi_{0}(z_{n})|$ is larger than the distance between
the sets $X$ and $Y$, which is comparable with the radius $\nu_{n}$ of $V_{0}$
(see Figure 9). However, the univalence of $\psi_{0}$ implies that
$|\psi_{0}(\zeta_{n})-\psi_{0}(z_{n})|$ is not comparable with $\nu_{n}$ by
$|\psi_{0}(\zeta_{n})-\psi_{0}(z_{n})|\asymp|\zeta_{n}-z_{n}|<\nu_{n}/n$. This
is a contradiction. $\blacksquare$

##  12 Proof of Lemma C

We will show that $|Df_{c}^{M^{\prime}-M}(z_{M})|\geq\kappa_{\mathrm{C}}/\nu$
for some constant $\kappa_{\mathrm{C}}$ that does not depend on $c=c_{t}$ with
$t\in\Delta$. By choosing $\nu$ sufficiently small, we have
$\Lambda:=\kappa_{\mathrm{C}}/\nu>1$.

As in the proof of Lemma A, we assume that $M=0$ and $M^{\prime}=mpq+L$ for
which $z_{0}\in V_{0}$ (equivalently, $|z_{0}-{\hat{b}}|<\nu$),
$z_{(m-1)pq}\in U_{0}$, $z_{mpq}\notin U_{0}$ (equivalently,
$|z_{mpq}-{\hat{b}}|\geq R_{0}$), and $z_{M^{\prime}}\in V_{0}$. By the chain
rule we have

$|Df_{c}^{M^{\prime}}(z_{0})|=|Df_{c}^{mpq}(z_{0})|\cdot|Df_{c}^{L}(z_{mpq})|.$
(12.1)

First let us give an estimate of $|Df_{c}^{mpq}(z_{0})|$. For
$c=c_{t}~{}(t\in\Delta)$, let $b(c)$ be the periodic point $b_{t}$ of period
$p$ given in Proposition 2.3 with $b(c)\to{\hat{b}}$ as $c=c_{t}$ tends to
${\hat{c}}$. We may assume that $|b(c)-{\hat{b}}|\leq\nu$ for
$c\approx{\hat{c}}$, and by the Koebe distortion Theorem, we have

$|Df_{c}^{mpq}(z_{0})|\asymp\frac{|z_{mpq}-b(c)|}{|z_{0}-b(c)|}\geq\frac{|z_{mpq}-{\hat{b}}|-|b(c)-{\hat{b}}|}{|z_{0}-{\hat{b}}|+|b(c)-{\hat{b}}|}\geq\frac{R_{0}-\nu}{\nu+\nu}.$

This implies that $|Df_{c}^{mpq}(z_{0})|\geq C_{8}/\nu$ for some constant
$C_{8}>0$ independent of $c\approx{\hat{c}}$, $z_{0}$ and $\nu\ll 1$.

Next we give an estimate of the form $|Df_{c}^{L}(z_{mpq})|\geq C_{9}$, where
$C_{9}$ is a constant independent of $c\approx{\hat{c}}$, $\nu\ll 1$, and
$z_{0}\in J(f_{c})$. (Then by (12.1) the proof is done by setting
$\kappa_{\mathrm{C}}:=C_{8}C_{9}$.)

As in the proof of Lemma A, by shrinking $U_{0}$ if necessary, we may always
assume that $L>p$. Since $z_{M^{\prime}}\in V_{0}$, we have
$z_{M^{\prime}-p}\in f_{c}^{-p}(V_{0})$ and thus
$|Df_{c}^{p}(z_{M^{\prime}-p})|\geq\xi^{p}$. By Proposition 4.1 and inequality
(4.1) in Lemma F we obtain

$\displaystyle|Df_{c}^{L}(z_{mpq})|=$
$\displaystyle|Df_{c}^{L-p}(z_{mpq})||Df_{c}^{p}(z_{M^{\prime}-p})|$
$\displaystyle\geq$
$\displaystyle\frac{\rho(z_{mpq})}{\rho(z_{M^{\prime}-p})}A^{L-p}\cdot\xi^{p}$
$\displaystyle\geq$
$\displaystyle\rho(z_{mpq})\cdot\mathrm{dist}\,(z_{M^{\prime}-p},P(f_{\hat{c}}))\cdot\xi^{p}.$

Since $z_{mpq}\notin U_{0}$ and $z_{mpq}$ has a definite distance from the
parabolic cycle, the distance between $z_{mpq}$ and the postcritical set
$P(f_{\hat{c}})$ is larger than a positive constant independent of
$c\approx{\hat{c}}$, $\nu\ll 1$, and $z_{0}\in J(f_{c})$. (Indeed, we may
apply the proof of Lemma D to obtain
$\mathrm{dist}\,(z_{mpq},P(f_{\hat{c}}))\geq K_{\mathrm{D}}R_{0}$ by taking a
smaller $R_{0}$ if necessary.) Hence we always have $\rho(z_{mpq})\asymp 1$.
Finally we show that $\mathrm{dist}\,(z_{M^{\prime}-p},P(f_{\hat{c}}))$ has a
uniform distance away from zero. Since $z_{M^{\prime}}\in V_{0}$,
$\mathrm{dist}\,(z_{M^{\prime}-p},P(f_{\hat{c}}))$ can be close to $0$ only if
$z_{M^{\prime}-p}$ belongs to the connected component $V_{0}^{\prime}$ of
$f_{c}^{-p}(V_{0})$ contained in $U_{0}$. (See Figure 6.) However, the local
dynamics of $f_{c}^{p}$ on $U_{0}$ does not allow any point $z\in J(f_{c})$
satisfying both $z\in V_{0}^{\prime}-V_{0}$ and $f_{c}^{p}(z)\in V_{0}$. (More
precisely, we may apply Proposition 10.1 and Lemma I to exclude such a $z$ in
the Julia set.) Hence $z_{M^{\prime}-p}$ belongs to a connected component of
$f_{c}^{-p}(V_{0})$ that has a definite distance from $P(f_{\hat{c}})$ and
$\mathrm{dist}\,(z_{M^{\prime}-p},P(f_{\hat{c}}))$ is bounded below by a
positive constant independent of $c\approx{\hat{c}}$, $\nu\ll 1$, and
$z_{0}\in J(f_{c})$. In conclusion, $|Df_{c}^{L}(z_{mpq})|$ is bounded away
from zero by a constant independent of $c\approx{\hat{c}}$, $\nu\ll 1$, and
$z_{0}\in J(f_{c})$. $\blacksquare$

##  13 Proofs of Theorems 1.2 and 1.3

In this section we prove Theorem 1.2 and Theorem 1.3 together.

### Proofs of Theorems 1.2 and 1.3.

Let $\sigma_{0}$ be any point in the thick internal ray
${\mathcal{I}}(\theta,\delta)$ and $\ell:=|\sigma_{0}-{\hat{c}}|$. For each
$n=0,1,\ldots$, let

$E_{n}:={\left\\{c\in{\mathcal{I}}(\theta,\delta)\,:\,\frac{\ell}{2^{n+1}}\leq|c-{\hat{c}}|\leq\frac{\ell}{2^{n}}\right\\}}.$

Note that $\mathrm{diam}\,E_{n}\asymp\ell/2^{n}$ and that for any $c\in E_{n}$
we have $|c-{\hat{c}}|\asymp\ell/2^{n}$.

We take a $\sigma_{n}\in E_{n}$ for each $n\geq 1$. Then we can join
$\sigma_{n}$ and $\sigma_{n+1}$ by a piecewise smooth path $\gamma_{n}$ of
length compatible with $\ell/2^{n}$ contained in $E_{n}\cup E_{n+1}$. By the
Main Theorem we have

${\left|\frac{d}{dc}z(c)\right|}\leq\frac{K}{|c-{\hat{c}}|^{1-1/Q}}\asymp{\left(\frac{\ell}{2^{n}}\right)}^{-1+1/Q}$

for any $c\in\gamma_{n}$ and $n\geq 0$. Hence

$|z(\sigma_{n+1})-z(\sigma_{n})|={\left|\int_{\gamma_{n}}\frac{d}{dc}z(c)\,dc\right|}=O{\left({\left(\frac{\ell}{2^{n}}\right)}^{-1+1/Q}\cdot\frac{\ell}{2^{n}}\right)}=O{\left(\frac{1}{2^{n/Q}}\right)}\cdot\ell^{1/Q}.$

It follows that the sequence $\\{z(\sigma_{n})\\}_{n\geq 0}$ is Cauchy, and we
denote the limit by $z({\hat{c}})$. (One can easily check that the limit does
not depend on the choice of the sequences $\\{\sigma_{n}\\}$ and
$\\{\gamma_{n}\\}$.) Moreover, we have

$|z(\sigma_{0})-z(\sigma_{n})|\leq
O{\left(1+\frac{1}{2^{1/Q}}+\cdots+\frac{1}{2^{(n-1)/Q}}\right)}\cdot\ell^{1/Q}=O(\ell^{1/Q})$

and thus
$|z(\sigma_{0})-z({\hat{c}})|=O(\ell^{1/Q})=O(|\sigma_{0}-{\hat{c}}|^{1/Q})$,
where the implicit constants depend only on ${\hat{c}}$ and the thickness
$\delta$ of the thick internal ray. This proves (1.2) of Theorem 1.2.

### Semiconjugacy.

Next we show that $z({\hat{c}})$ belongs to the Julia set $J(f_{\hat{c}})$.
For each $z_{\ast}\in J(f_{\sigma})$ and its motion
$z(c)=h_{c}(z_{\ast})=H(c,z_{\ast})$ along the thick internal ray
${\mathcal{I}}(\theta,\delta)$, we define $h_{\hat{c}}(z_{\ast})$ by the limit
$z({\hat{c}})$ given as above. Since $h_{c}:J(f_{\sigma})\to J(f_{c})$ is
continuous and the convergence of $h_{c}$ to $h_{\hat{c}}$ as
$c\in{\mathcal{I}}(\theta,\delta)$ tends to ${\hat{c}}$ is uniform,
$h_{\hat{c}}:J(f_{\sigma})\to h_{\hat{c}}(J(f_{\sigma}))$ is continuous as
well. Hence by $f_{c}\circ h_{c}=h_{c}\circ f_{\sigma}$ we obtain
$f_{\hat{c}}\circ h_{\hat{c}}=h_{\hat{c}}\circ f_{\sigma}$. In particular,
this implies that $f_{\hat{c}}\circ
h_{\hat{c}}(J(f_{\sigma}))=h_{\hat{c}}(J(f_{\sigma}))$ and thus the image
$h_{\hat{c}}(J(f_{\sigma}))$ is a compact forward invariant set contained in
the filled Julia set of $f_{\hat{c}}$. Suppose that there exists a
$z_{\ast}\in J(f_{\sigma})$ such that $h_{\hat{c}}(z_{\ast})=z({\hat{c}})\in
h_{\hat{c}}(J(f_{\sigma}))$ belongs to the Fatou set of $f_{\hat{c}}$. It
actually belongs to the basin of attraction of $\hat{b}$, and thus there
exists an $l$ such that $f_{{\hat{c}}}^{l+kpq}(z({\hat{c}}))$ is contained in
$V_{0}$ for any $k\geq 0$. Since $h_{\hat{c}}$ is continuous, the image of any
nearby point of $z_{*}$ under $h_{\hat{c}}$ also belongs to the same basin. In
addition, since the points that are not eventually periodic in the dynamics of
$f_{\sigma}$ are dense in $J(f_{\sigma})$, we may assume that the orbit
$f_{\sigma}^{n}(z_{\ast})~{}(n\in\mathbb{N}_{0})$ of $z_{\ast}$ never lands on
the periodic points. By
$f_{{\hat{c}}}^{l+kpq}(z({\hat{c}}))=h_{\hat{c}}(f_{\sigma}^{l+kpq}(z_{\ast}))$
and uniform convergence of $h_{c}$ to $h_{\hat{c}}$, we have
$h_{c}(f_{\sigma}^{l+kpq}(z_{\ast}))\in V_{0}$ for any $k$ and
$c\approx{\hat{c}}$. Since $h_{c}(f_{\sigma}^{l+kpq}(z_{\ast}))\in J(f_{c})$,
Proposition 2.4 implies that $h_{c}(f_{\sigma}^{l+kpq}(z_{\ast}))$ is a
repelling periodic point contained in $U_{0}$. This is impossible because
$z_{\ast}$ is not eventually periodic. This completes the proof of Theorem
1.2.

To confirm that $h_{\hat{c}}$ is a semiconjugacy, we show surjectivity of
$h_{{\hat{c}}}:J(f_{\sigma})\to J(f_{\hat{c}})$: First we take any repelling
periodic point $\hat{x}\in J(f_{\hat{c}})$. Since there is a holomorphic
family $x(c)$ of repelling periodic points for $c$ sufficiently close to
${\hat{c}}$ such that $\hat{x}=x({\hat{c}})$, we have some $z_{0}\in
J(f_{\sigma})$ with $h_{c}(z_{0})=x(c)$ for any
$c\in{\mathcal{I}}(\theta,\delta)$. In particular, we have
$h_{\hat{c}}(z_{0})=\hat{x}$. Next we take any $w\in J(f_{{\hat{c}}})$ and a
sequence of repelling periodic points $\hat{x}_{n}$ of $f_{{\hat{c}}}$ that
converges to $w$ as $n\to\infty$. (Such a sequence exists since repelling
periodic points are dense in the Julia set.) Let $z_{n}\in J(f_{\sigma})$ be
the repelling periodic point with $h_{\hat{c}}(z_{n})=\hat{x}_{n}$. Then any
accumulation point $y$ of the sequence $z_{n}$ satisfies $h_{\hat{c}}(y)=w$ by
continuity.

Finally we check properties (1) – (3) of Theorem 1.3. Property (3) is an
immediate consequence of (1.2) in Theorem 1.2. (In particular, we obtain
Corollary 1.4.) To show (1) and (2), let $\eta_{c}:=h_{\hat{c}}\circ
h_{c}^{-1}:J(f_{c})\to J(f_{\hat{c}})$ for $c\approx{\hat{c}}$ such that
$\eta_{c}$ is a semiconjugacy between $f_{c}$ and $f_{\hat{c}}$ and satisfies
$|\eta_{c}(z)-z|\leq K^{\prime}|c-{\hat{c}}|^{1/Q}$ by (3). We assume that $c$
is sufficiently close to ${\hat{c}}$ such that
$K^{\prime}|c-{\hat{c}}|^{1/Q}<K_{\mathrm{D}}\,\nu/4$, where the constant
$K_{\mathrm{D}}\leq 1$ is given in Lemma D. Suppose that
$\eta_{c}(z)=\eta_{c}(z^{\prime})$ for some distinct points $z,\,z^{\prime}\in
J(f_{c})$. Let $z_{n}:=f_{c}^{n}(z)$ and
$z_{n}^{\prime}:=f_{c}^{n}(z^{\prime})$ for $n\in\mathbb{N}$. Then
$\eta_{c}\circ f_{c}=f_{\hat{c}}\circ\eta_{c}$ implies
$\eta_{c}(z_{n})=\eta_{c}(z_{n}^{\prime})$ for any $n\in\mathbb{N}$, and thus

$|z_{n}-z_{n}^{\prime}|\leq|\eta_{c}(z_{n})-z_{n}|+|\eta_{c}(z_{n}^{\prime})-z_{n}^{\prime}|=2K^{\prime}|c-{\hat{c}}|^{1/Q}<K_{\mathrm{D}}\nu/2<\nu/2.$
(13.1)

Suppose that the orbit $z_{n}~{}(n\in\mathbb{N})$ never lands on repelling
periodic points of $f_{c}$ in $\hat{U}=\mathbb{D}({\hat{b}},\hat{R})$
described in assertion (3) or (4) of Proposition 2.3. By Proposition 2.4, the
orbit $z_{n}^{\prime}~{}(n\in\mathbb{N})$ must behave in the same way. Let
$\\{n_{k}\\}_{k\in\mathbb{N}}$ be a subsequence such that
$z_{n},z_{n}^{\prime}\in J(f_{c})-{\mathcal{V}}(c)$ when $n=n_{k}$ for each
$k$. By Lemma D, $\mathbb{D}(z_{n},K_{\mathrm{D}}\nu)$ does not contain any
point in the postcritical set $P(f_{\hat{c}})$ when $n$ ranges over the
subsequence $\\{n_{k}\\}_{k\in\mathbb{N}}$. Hence we have a univalent branch
$g_{n}$ of $f_{c}^{-n}$ defined on $\mathbb{D}(z_{n},K_{\mathrm{D}}\nu)$ that
sends $z_{n}$ and $z_{n}^{\prime}$ to $z$ and $z^{\prime}$. However, since
$f_{c}$ is hyperbolic, the Koebe distortion theorem (see [A, Theorem 5-3])
implies

$|z-z^{\prime}|\leq\mathrm{diam}\,g_{n}(\mathbb{D}(z_{n},K_{\mathrm{D}}\nu/2))\asymp\frac{\nu}{|Df_{c}^{n}(z)|}\to
0$

as $n=n_{k}\to\infty$. This contradicts the assumption $z\neq z^{\prime}$.
Hence we may assume that the orbit $z_{n}~{}(n\in\mathbb{N})$ lands on a
repelling fixed point of $f_{c}^{pq}$ in $V_{0}=\mathbb{D}({\hat{b}},\nu)$.
Let $z_{N}$ be such a repelling fixed point.

### Case 1 or Case 2+.

By (13.1) and Proposition 2.4, $z_{N}^{\prime}$ must be the only repelling
fixed point of $f_{c}^{pq}$ in $V_{0}$ given in assertion (3) of Proposition
2.3, and thus $z_{N}=z_{N}^{\prime}$. Since $z\neq z^{\prime}$, we must have
$z_{n}\neq z_{n}^{\prime}=-z_{n}$ for some $n<N$. However, this is impossible
because the Julia set $J(f_{\hat{c}})$ and the critical point 0 have a
definite distance, and the same holds for $J(f_{c})$ by (1.3) for
$c\approx{\hat{c}}$. (Indeed, $J(f_{c})$ converges to $J(f_{\hat{c}})$ as $c$
tends to ${\hat{c}}$ along a thick internal ray. See Corollary 1.4.) Hence we
conclude that $\eta_{c}$ is injective. Since $h_{\hat{c}}=\eta_{c}\circ h_{c}$
and $h_{c}$ is a conjugacy, property (1) holds.

### Case 2-.

By the same argument as above, $z_{N}^{\prime}$ must be one of the $q$
repelling fixed points of $f_{c}^{pq}$ in $V_{0}$ given in assertion (4) of
Proposition 2.3. Hence we conclude that $\eta_{c}(z_{N})$ is a parabolic
periodic point ${\hat{b}}$ and property (2) holds by the relation
$h_{\hat{c}}=\eta_{c}\circ h_{c}$. $\blacksquare$

## Acknowledgments

Chen was partly supported by MOST 108-2115-M-001-005 and 109-2115-M-001-006.
Kawahira was partly supported by JSPS KAKENHI Grants numbers 16K05193 and
19K03535. They thank the hospitality of Academia Sinica, Nagoya University,
RIMS in Kyoto University, and Tokyo Institute of Technology where parts of
this research were carried out.

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Yi-Chiuan Chen
Institute of Mathematics
Academia Sinica
Taipei 106319, Taiwan
<EMAIL_ADDRESS>
Tomoki Kawahira
Graduate School of Economics
Hitotsubashi University
Tokyo 186-8602, Japan
<EMAIL_ADDRESS>