# Primitive tuning for non-hyperbolic polynomials

Yimin Wang  Shanghai Center for Mathematical Sciences, Fudan University, No.
2005 Songhu Road, Shanghai 200438, China<EMAIL_ADDRESS>
###### Abstract.

Let $f_{0}$ be a polynomial of degree $d_{1}+d_{2}$ $(d_{1}\geq 2,d_{2}\geq
1)$with a periodic critical point $0$ of multiplicity $d_{1}-1$ and a Julia
critical point of multiplicity $d_{2}$. We show that if $f_{0}$ is primitive,
free of neutral periodic points and non-renormalizable at the Julia critical
point, then the straightening map
$\chi_{f_{0}}:\mathcal{C}(\lambda_{f_{0}})\to\mathcal{C}_{d_{1}}$ is a
bijection. More precisely, $f^{m_{0}}$ has a polynomial-like restriction which
is hybrid equivalent to some polynomial in $\mathcal{C}_{d_{1}}$ for each map
$f\in\mathcal{C}(\lambda_{f_{0}})$, where $m_{0}$ is the period of $0$ under
$f_{0}$. On the other hand, $f_{0}$ can be tuned with any polynomial
$g\in\mathcal{C}_{d_{1}}$. As a consequence, we conclude that the
straightening map $\chi_{f_{0}}$ is a homeomorphism from
$\mathcal{C}(\lambda_{f_{0}})$ onto the Mandelbrot set when $d_{1}=2$. This
together with the main result in [24] solve the problem for primitive tuning
for cubic polynomials with connected Julia sets thoroughly.

##  1\. Introduction

Tuning and straightening were first introduced by Douady-Hubbard [6] to prove
the existence of baby Mandelbrot sets in the quadratic family. Roughly
speaking, tuning is a procedure to replace the bounded superattracting Fatou
components with the copies of a filled Julia set of another polynomial and
respect some combinatorial properties. Douady-Hubbard proved any quadratic
polynomial which has a periodic critical point can be tuned with any quadratic
polynomials in the Mandelbrot set $\mathcal{M}$.

In [8], Inou-Kiwi generalized the definition of the straightening maps for
higher degree polynomials. They used some combinatorial argument to show that
any primitive postcritically finite hyperbolic polynomial can be tuned with
any hyperbolic generalized polynomial with fiber-wise connected Julia set. In
[24], Shen and the author used quasiconformal surgery and ran Thurston’s
Algorithm to prove that any primitive postcritically finite hyperbolic
polynomial can be tuned with any generalized polynomial with fiber-wise
connected Julia set. Therefore, the problem for primitive tuning had been
solved thoroughly for hyperbolic polynomials.

The question which remains is whether primitive non-hyperbolic polynomials can
be tuned with generalized polynomials or not. Let us consider the family
$\mathcal{P}_{d}$ of polynomials of degree $d$ $(d\geq 2)$ normalized as the
following form

$f:z\mapsto z^{d}+a^{d-1}z^{d-1}+\cdots+a_{2}z^{2}+a_{0}.$

Note that each map $f\in\mathcal{P}_{d}$ is monic and has a critical point
$0$. The set

$\mathcal{C}:=\\{f\in\mathcal{P}_{d}\mid J_{f}~{}\text{is connected}\\}$

is called the connectedness locus of degree $d$. Let
$f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ be a polynomial with a periodic critical
point $0$ of multiplicity $d_{1}-1$ and a Julia critical point $c_{0}$ of
multiplicity $d_{2}$. Let $\lambda_{f_{0}}$ denote the rational lamination of
$f_{0}$. We believe that if $f_{0}$ is renormalizable at the free critical
point $c_{0}$, then there is a hyperbolic post-critically finite polynomial
$f_{*}\in\mathcal{C}_{d_{1}+d_{2}}$ with non-trivial rational lamination such
that $\lambda_{f_{*}}\subset\lambda_{f_{0}}$. The primitive tuning for such a
hyperbolic post-critically finite polynomial $f_{*}$ had been studied
thoroughly in [24]. So we will restrict our attention to the case that $f_{0}$
is non-renormalizable at the free critical point $c_{0}$. There is no loss of
generality in assuming $f_{0}$ is free of neutral periodic points by [8,
Theorem 5.17]. Let $\mathcal{C}(\lambda_{f_{0}})$ denote the set of
combinatorially $\lambda_{f_{0}}$\- renormalizable maps. For the precise
definition of $\mathcal{C}(\lambda_{f_{0}})$, see §2. Set
$g\in\mathcal{C}_{d_{1}}$. We say $f_{0}$ can be tuned with $g$ if there
exists $f\in\mathcal{C}(\lambda_{f_{0}})$ such that $f^{m_{0}}$ has a
polynomial-like restriction which is hybrid equivalent to $g$ (and the hybrid
conjugacy respects the corresponding external markings). Such an $f$ is called
a tuning of $f_{0}$ by $g$. We say $f_{0}$ is primitive if any two Fatou
components of $f_{0}$ have disjoint closures.

The aim of this paper is to show:

######  Theorem A.

Let $f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ be a primitive polynomial with a
periodic critical point $0$ of multiplicity $d_{1}-1$ and a Julia critical
point $c_{0}$ of multiplicity $d_{2}$. Fix an internal angle system $\alpha$
for $f_{0}$. If $f_{0}$ has no neutral periodic points and $f_{0}$ is non-
renormalizable at $c_{0}$, then the straightening map
$\chi_{f_{0}}:\mathcal{C}(\lambda_{f_{0}})\to\mathcal{C}_{d_{1}}$ induced by
$f_{0}$ and $\alpha$ is a bijection.

The proof falls naturally into two parts: surjectivity and injectivity. To
show the surjectivity of $\chi_{f_{0}}$, it suffices to show:

######  Theorem B.

Under the assumptions in the Theorem A, $f_{0}$ can be tuned with any
polynomial $g\in\mathcal{C}_{d_{1}}$.

The basic idea is to use qc surgery and apply [24, Theorem 5.1]. The main
difficulty is that we cannot construct a desired tuning by using qc surgery
and Thurston’s Algorithm directly. In [24], one can construct a quasiregular
map with certain dynamical properties so that all the ramification points lie
in the union of all the small filled Julia sets. Thus the postcritical set
will not affect the combinatorics (the landing relation for external rays).
However, $f_{0}$ has a free Julia critical point in our case. This does affect
the combinatorics when we do qc surgery. To overcome this difficulty, we
construct the tuning by approximation instead of constructing it directly.
More precisely, we first control the orbit of the free critical point
carefully when we do qc surgery to show there is a sequence $\\{f_{k}\\}$ of
polynomials such that $f^{m}_{k}$ is hybrid equivalent to a given
$g\in\mathcal{C}_{d_{1}}$ and $f_{k}$ has the same combinatorics as $f_{0}$ up
to depth $k$ for all $k\in\mathbb{N}$. Then we show the limit $f$ of
$\\{f_{k}\\}$ is our desired tuning.

It is worth pointing out that Douady-Hubbard’s method does not work since
$\mathcal{C}_{d_{1}}$ may contain multi-critical maps. Even for the case
$d_{1}=2$, one cannot apply Douady-Hubbard’s Theorem directly since
$\mathcal{C}(\lambda_{f_{0}})$ is not an analytic family in general.

The proof of the injectivity of $\chi_{f_{0}}$ is based on the arguments
developed in [1]. For any $f,\widetilde{f}\in\mathcal{C}(\lambda_{f_{0}})$
with $\chi_{f_{0}}(f)=\chi_{f_{0}}(\widetilde{f})$, we show that $f$ and
$\widetilde{f}$ are conformally conjugate. To this end, We first construct a
sequence $\\{\Phi_{k}\\}$ of qc maps such that $\Phi_{k}$ is a weak pseudo-
conjugacy between $f$ and $\widetilde{f}$ up to depth $k$ for all
$k\in\mathbb{N}$. Then we prove some technical lemmas to show that there
exists a sequence $\\{k_{n}\\}$ so that we can make $\Phi_{k_{n}}$ be a
pseudo-conjugacy between $f$ and $\widetilde{f}$ up to depth $k_{n}$ by
adjustment. Moreover, $\\{\Phi_{k_{n}}\\}$ can be made uniformly qc. Finally,
we show the limit $\Phi$ of $\\{\Phi_{k_{n}}\\}$ is a conformal conjugacy
between $f$ and $\widetilde{f}$.

Combining Theorem A, [20, proposition 4.7] and [8, Theorem 8.1], we conclude:

######  Theorem C.

Under the assumptions in Theorem A, if $d_{1}=2$ then the straightening map
$\chi_{f_{0}}:\mathcal{C}(\lambda_{f_{0}})\to\mathcal{C}_{2}=\mathcal{M}_{2}$
is a homeomorphism, where $\mathcal{M}_{2}$ is the Mandelbrot set.

Theorem C together with the main results in [24] solve the problem for
primitive tuning for cubic maps with connected Julia sets thoroughly. For more
related topics on qc surgery, we refer the readers to [2, 4, 23].

The paper is organized as follows. In §2, we review some known facts on
rational laminations and give the precise definition of tuning. In §3 and
§LABEL:sec:Kahn, we follow the ideas in [24] to construct a specific Yoccoz
puzzle which satisfies some certain dynamical properties and prove a useful
lemma on qc distortion. Our case is slightly different from that of [24] by
virtue of the existence of the free Julia critical point. But it is fair to
say the proofs are similar in spirit. The readers who are familiar with the
results in [24] may skip the proofs in these two section. In § 5, we build up
qc weak pseudo-conjugacies between $f$ and maps in
$\mathcal{C}(\lambda_{f_{0}})$, which turns out to be one of the most powerful
tool in the proof of Theorem A. The aim of section 5 is to prove Theorem B. To
this end, we first use qc surgery and apply [24, Theorem 5.1] to construct a
sequence $\\{f_{k}\\}$ from $f_{0}$ and a given polynomial
$g\in\mathcal{C}_{d_{1}}$ such that $f_{k}$ has the same combinatorics as
$f_{0}$ up to depth $k$ and $f^{m_{0}}_{k}$ has a polynomial-like restriction
hybrid equivalent to $g$ for all $k\in\mathbb{N}$. Then we show the limit of
such a sequence is a tuning of $f_{0}$ by $g$. In §6, we prove some technical
lemmas and then prove Theorem A.

Acknowledgment. The author would like to thank Hiroyuki Inou and Weixiao Shen
for helpful discussions and comments. This work was supported by CSC.

##  2\. Laminations and Tuning

Throughout this paper we will fix a primitive non-hyperbolic polynomial
$f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ with a periodic critical point $0$ of
multiplicity $d_{1}-1$ and a free Julia critical point $c_{0}$ of multiplicity
$d_{2}$.

In this section, we will recall some background about laminations and give the
definition of tuning.

###  2.1. Rational laminations

For each polynomial $f\in\mathcal{P}_{d}$, the Green function is defined as

$G_{f}(z)=\lim_{n\to\infty}\frac{1}{{d}^{n}}\log^{+}|f^{n}(z)|,$

where $\log^{+}=\max(\log,0).$

The set

$D_{f}(\infty):=\\{z\mid f^{n}(z)\to\infty\\}$

is called the basin of infinity of $f$. The complement
$K(f)=K_{f}=\mathbb{C}\setminus D_{f}(\infty)$ of the basin of infinity is
called the filled Julia set of $f$. The common boundary of $K(f)$ and
$D_{f}(\infty)$ is called the Julia set of $f$ and denoted by $J(f)=J_{f}$.

There exists a unique conformal map

$\phi_{f}:D_{f}(\infty)\setminus\\{z\in\mathbb{C}\mid G_{f}(z)\leq
r_{f}\\}\to\\{z\mid|z|>r_{f}\\}$

such that $\phi(z)/z\to 1$ as $z\to\infty$ and such that $\phi_{f}\circ
f(z)=(\phi_{f}(z))^{d}$ on $D_{f}(\infty)\setminus\\{z\in\mathbb{C}\mid
G_{f}(z)\leq r_{f}\\}$, where $r_{f}=\max\\{G_{f}(c)\mid c~{}\text{is a
critical point of}~{}f\\}$. Moreover, $G_{f}(z)=\log|\phi_{f}(z)|$.

The equipotential curve of height $h$ of $f$ is defined as

$E_{f}(h)=\\{z\mid G_{f}(z)=h\\},$

and the external ray of angle $t\in\mathbb{R}/\mathbb{Z}$ as

$\mathcal{R}_{f}(t)=\\{\phi_{f}^{-1}(re^{i2\pi t})\mid r_{f}<r<\infty\\}.$

An external ray $R_{f}(t)$ which is not bifurcated has a smooth extension:
$\mathcal{R}_{f}(t)=\\{\phi_{f}^{-1}(re^{i2\pi t})\mid 1<r<\infty\\}$. We say
the external ray $\mathcal{R}_{f}(t)$ lands at some point $z_{0}$ if it is not
bifurcated and $\lim_{r\to 1}\phi_{f}^{-1}(re^{i2\pi t})=z_{0}$.

######  Definition 2.1 (Rational lamination).

The rational lamination $\lambda_{f}$ of $f\in\mathcal{C}_{d}$ is the
equivalence relation on $\mathbb{Q}/\mathbb{Z}$ so that $s\sim t$ if and only
if $\mathcal{R}_{f}(s)$ and $\mathcal{R}_{f}(t)$ land at a common point.

We say $f\in\mathcal{C}_{d}$ is combinatorially
$\lambda_{f_{0}}$-renormalizable if $\lambda_{f}\supset\lambda_{f_{0}}$. Let
$\mathcal{C}(\lambda_{f_{0}})$ denote the set of all the combinatorially
$\lambda_{f_{0}}$-renormalizable maps. By [8, Theorem 8.1],
$\mathcal{C}(\lambda_{f_{0}})$ is compact.

###  2.2. Renormalization and Tuning

Recall that a polynomial-like map is a holomorphic proper map $g:U\to V$,
where $U\Subset V$ are quasidisks in $\mathbb{C}$. We call the set
$K(g):=\\{z\in U\mid f^{k}(z)\in U~{}\text{for all}~{}k\in\mathbb{N}\\}$ the
filled Julia set of $g$. Two polynomial-like maps $g$ and $\widetilde{g}$ are
said to be qc conjugate if there exists a qc map
$\varphi:\mathbb{C}\to\mathbb{C}$ such that $\varphi\circ
g=\widetilde{g}\circ\varphi$ near $K(g)$. We say $g$ and $\widetilde{g}$ are
hybrid equivalent if they are qc conjugate and there exists a corresponding qc
conjugacy $\varphi$ between them such that $\partial\bar{\varphi}=0$ a.e. on
$K(g)$. For a more detailed treatment, we refer the reader to [6, 13].

Let $f\in\mathcal{P}_{d}$ and let $\mathrm{Crit}(f)$ denote the set of
critical points of $f$. We say $f$ is renormalizable at $c$ for some
$c\in\mathrm{Crit}(f)$ if there exist quasidisks
$c\in\Omega\Subset\Omega^{\prime}$ and $p\geq 1$ such that

  * •

$f^{p}|_{\Omega}:\Omega\to\Omega^{\prime}$ is a polynomial-like map with
connected Julia set;

  * •

for any $c^{\prime}\in\mathrm{Crit}(f)$, there is at most one $0\leq j<p-1$
such that $c^{\prime}\in f^{j}(\Omega)$;

  * •

$p>1$ or $\mathrm{Crit}(f)\not\subset\Omega$.

The Douady-Hubbard Straightening Theorem says that each polynomial-like map
$g$ with connected Julia set is hybrid to a unique polynomial up to an affine
conjugacy. To determine the straightening uniquely, it is convenient to
introduce an external marking for $g$.

######  Definition 2.2 (Access and external marking).

Let $g:U\to V$ be a polynomial-like map connected filled Julia set. A path to
$K(g)$ is a continuous map $\gamma:[0,1]\to U$ such that $\gamma((0,1])\subset
U\setminus K(g)$ and $\gamma(0)\in J(g)$. We say two paths $\gamma_{0}$ and
$\gamma_{1}$ to $K(g)$ are homotopic if there exists a continuous map
$\tilde{\gamma}:[0,1]\times[0,1]\to U$ such that

  1. (1)

$t\mapsto\tilde{\gamma}(s,t)$ is a path to $K(g)$ for all $s\in[0,1]$;

  2. (2)

$\tilde{\gamma}(0,t)=\gamma_{0}(t)$ and $\tilde{\gamma}(1,t)=\gamma_{1}(t)$
for all $t\in[0,1]$;

  3. (3)

$\tilde{\gamma}(s,0)=\gamma_{0}(0)$ for all $s\in[0,1]$.

An access to $K(g)$ is a homotopy class of paths to $K(g)$.

An external marking of $f$ is an access $\Gamma$ to $K(g)$ which is forward
invariant in the following sense. For every representative $\gamma$ of
$\Gamma$, the connected component of $g(\gamma)\cap U$ which intersects $J(f)$
is also a representative of $\Gamma$.

The standard external marking for a polynomial $f$ with connected Julia set is
defined as the homotopy class of $\mathcal{R}_{f}(0)$ in the sense of paths to
$K(f)$.

Recall that $f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ is a primitive polynomial with
a periodic critical point $0$ of multiplicity $d_{1}-1$ and a free Julia
critical point $c_{0}$ of multiplicity $d_{2}$. Let $U_{0}$ be the Fatou
component of $f_{0}$ which contains the superattracting periodic point $0$. It
is well-known that $\partial U_{0}$ is a Jordan curve (see [22]). Let $m_{0}$
be the period of $0$ under $f_{0}$. Note that $f^{m_{0}}|_{U_{0}}$ has a
polynomial-like extension $F_{0}:U\to V$ such that
$K(F_{0})=\overline{U_{0}}$.

A polynomial $f\in\mathcal{C}(\lambda_{f_{0}})$ is said to be
$\lambda_{f_{0}}$-renormalizable if there exist quasidisks $U^{\prime}\Subset
V^{\prime}$ such that $F=f^{m_{0}}|_{U^{\prime}}:U^{\prime}\to V^{\prime}$ is
a polynomial-like map with the filled Julia set

$K(F)=K(U_{0},f):=\bigcap\limits_{\theta\sim_{\lambda_{f_{0}}}\theta^{\prime}}S(\theta,\theta^{\prime},U_{0})\cap
K(f),$

where $S(\theta,\theta^{\prime},U_{0})$ is the component of
$\mathbb{C}\setminus\overline{\mathcal{R}_{f}(\theta)\cup\mathcal{R}_{f}(\theta^{\prime})}$
which contains external rays landing on $\partial U_{0}$. Such a polynomial-
like $F$ is called a $\lambda_{f_{0}}$-renormalization for $f$. Note that a
$\lambda_{f_{0}}$-renormalization $F$ for $f$ is unique up to the hybrid
conjugacy.

To define a canonical external marking for the
$\lambda_{f_{0}}$-renormalization $F$, we need the definition of internal
angle system which was introduced by Inou-Kiwi [8]. An internal angle system
for $f_{0}$ is a homeomorphism $\alpha:\partial U_{0}\to\mathbb{R}/\mathbb{Z}$
such that $\alpha\circ f^{m_{0}}=m_{d_{1}}\circ\alpha\mod 1$ on $\partial
U_{0}$. Such a homeomorphism is determined by $\alpha^{-1}(0)$ which is a
$f^{m_{0}}_{0}$-fixed point. Choose an external angle $\theta_{0}$ such that
$\mathcal{R}_{f_{0}}(\theta_{0})$ land at $\alpha^{-1}(0)$. The external
marking for $F$ induced by $\alpha$ is defined as the homotopy class of the
component of $\overline{\mathcal{R}_{f}}(\theta)\cap U$ which intersects
$K(F)$ in the sense of paths to $K(F)$. The homotopy class does not depend on
the choice of $\theta_{0}$ (see [8, Remark 3.16]), so it is well defined.

######  Definition 2.3 (Tuning).

Let $f_{0}$ and an internal angle system $\alpha$ be given. We say $f_{0}$ can
be tuned with a polynomial $g\in\mathcal{C}_{d_{1}}$ if there exists a
polynomial $f\in\mathcal{C}(\lambda_{f_{0}})$ and quasidisks
$U^{\prime}\Subset V^{\prime}$ with the following properties:

  * •

$F=f^{m_{0}}|_{U^{\prime}}:U^{\prime}\to V^{\prime}$ is a
$\lambda_{f_{0}}$-renormalization for $f$ which is hybrid equivalent to $g$;

  * •

there exists a corresponding hybrid conjugacy $\varphi$ between $F$ and $g$
which respects the external markings, that is, $\varphi$ maps the external
marking for $F$ induced by $\alpha$ to the standard external marking for $g$.

Such an $f$ is called a tuning of $f_{0}$ by $g$.

Clearly, $f_{0}$ can be tuned with $z\mapsto z^{d_{1}}$ and $f_{0}$ itself is
a tuning of $f_{0}$ by $z\mapsto z^{d_{1}}$. In this paper, we show that
$f_{0}$ can be tuned with any $g\in\mathcal{C}_{d_{1}}$:

######  Theorem B.

Let $f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ be a primitive polynomial with a
periodic critical point $0$ of multiplicity $d_{1}-1$ and a Julia critical
point $c_{0}$ of multiplicity $d_{2}$. Fix an internal angle system $\alpha$
for $f_{0}$. If $f_{0}$ has no neutral periodic points and $f_{0}$ is non-
renormalizable at $c_{0}$, then $f_{0}$ can be tuned with any
$g\in\mathcal{C}_{d_{1}}$.

The proof of Theorem B will be given in section 5.

##  3\. Yoccoz puzzle

From this section to the end of this paper, we assume $f_{0}$ is not
renormalizable at $c_{0}$ and has no neutral periodic points. A finite set $Z$
is said to be $f_{0}$-admissible if the following hold:

  * •

$f_{0}(Z)\subset Z$,

  * •

each periodic point in $Z$ is repelling and does not lie in the orbit of the
critical point $c_{0}$;

  * •

for each $z\in Z$, there exist at least two external rays landing at $z$.

Note that the definition differs slightly from that of [24] as we require that
the periodic points in $Z$ do not lie in the orbit of the free critical point.
This condition is used to guarantee the free critical point is not contained
in the boundary of any Yoccoz puzzle pieces which are defined as following.

Let $\Gamma_{0}=\Gamma^{Z}_{0}$ denote the union of all the external rays
landing on $Z$, the set $Z$ itself and the equipotential curve of height $1$.
For each $n\geq 1$, define $\Gamma^{Z}_{n}=f_{0}^{-n}(\Gamma^{Z}_{0})$. A
bounded component of $\mathbb{C}\setminus\Gamma^{Z}_{n}$ is called a
$Z$-puzzle piece of depth $n$. Let $Y^{(n)}_{Z}=Y^{(n)}_{Z}(0)$ and
$X^{(n)}_{Z}=Y^{(n)}_{Z}(c_{0})$ denote the puzzle pieces of depth $n$ which
contains $0$ and $c_{0}$ respectively.

The aim of this section is to prove the following:

######  Theorem 3.1.

Let $f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ be a primitive polynomial with a
periodic critical point $0$ of multiplicity $d_{1}-1$ and a Julia critical
point $c_{0}$ of multiplicity $d_{2}$. If $f_{0}$ has no neutral cycles and
$f_{0}$ is non-renormalizable at $c_{0}$, then there exists a buried
biaccessible repelling periodic point $\tau_{0}$ with the following
properties:

  1. (1)

$Z_{0}:=\mathrm{orb}(\tau_{0})$ is an $f_{0}$-admissible finite set;

  2. (2)

$\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{0}}=\overline{U_{0}}$, where $U_{0}$ is
Fatou component of $f_{0}$ which contains $0$;

  3. (3)

$\bigcap_{n=0}^{\infty}X^{(n)}_{Z_{0}}=\\{c_{0}\\}$;

The proof is slightly different from that of [24, Theorem 3.1] since $f_{0}$
has a free Julia critical point in this case. However, the methods of the
proofs are similar in spirit. To make this section self-contained, we give a
proof of Theorem 3.1 here. The readers who are familiar with the results and
proofs in [24] may skip the proof of Theorem 3.1.

We say a point $w\in J_{f_{0}}$ is bi-accessible if it is the common landing
points of two distinct external rays. A point in $J_{f_{0}}$ is called buried
if it is not in the boundary of any bounded Fatou component.

######  Lemma 3.1 ([24, Lemma 3.1]).

Let $f$ be a polynomial with connected Julia set. Then any bi-accessible point
in the boundary of a bounded Fatou component is eventually periodic.

###### Proof.

See the proof of [24, Lemma 3.1]. ∎

######  Lemma 3.2.

Under the assumption in Theorem 3.1, $f_{0}$ has a bi-accessible repelling
periodic point on $\partial U_{0}$.

###### Proof.

Since all the periodic point of $f_{0}$ are non-neutral and $f_{0}$ is not
renormalizable at $c_{0}$, the Julia set $J_{f_{0}}$ of $f_{0}$ is locally
connected ([11, Theorem 1.1]). Then the lemma follows by the same method as in
the proof of [24, Lemma 3.2].

∎

######  Lemma 3.3.

Let $f\in\mathcal{C}_{d^{\prime}}$ for some $d^{\prime}\geq 2$ and let $U$ be
a fixed Fatou component of $f$. If $p$ is a fixed point lying on $\partial U$
and $\mathcal{R}_{f}(t)$ lands at $p$, then $d^{\prime}t=t\mod 1$.

###### Proof.

Let $\Theta$ denote the set of the arguments $\theta$ for which
$\mathcal{R}_{f}(\theta)$ lands at $p$. Note that $m_{d^{\prime}}$ maps
$\Theta$ onto itself and preserves cyclic order, where
$m_{d^{\prime}}:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$, $x\mapsto
d^{\prime}x\mod 1$. Thus all the arguments $\theta\in\Theta$ have the same
period under $m_{d}^{\prime}$. It suffices to show there exists
$\theta_{0}\in\Theta$ such that $m_{d^{\prime}}(\theta_{0})=\theta_{0}$. If
$\Theta$ is a singleton, then $d^{\prime}t=t\mod 1$ since both $t$ and
$m_{d^{\prime}}(t)$ belong to $\Theta$. Now we assume $\\#\Theta\geq 2$. Let
$\Gamma$ be the closure of the union of all the external rays with arguments
$\theta\in\Theta$. Let $\Omega$ be the component of
$\mathbb{C}\setminus\Gamma$ which contains $U$ and let $\Omega^{\prime}$ be
the component of $\mathbb{C}\setminus f^{-1}(\Gamma)$ which contains $U$. Then
$\partial\Omega\subset\partial\Omega^{\prime}$ and
$f:\Omega^{\prime}\to\Omega$ is a holomorphic proper map. It follows that $f$
must fix the boundary of $\Omega$. In other words, the external rays in
$\partial V$ are fixed under $f$.

∎

######  Lemma 3.4.

Under the assumption in Theorem 3.1, $f_{0}$ has a buried biaccessible
repelling periodic point which does not lie in the orbit of the critical point
$c_{0}$.

###### Proof.

The proof is based on the pigeonhole principle. By Lemma 3.2, there is a bi-
accessible repelling periodic point $p$ on $\partial U_{0}$. Since $f_{0}$ is
primitive, the period of $p$ must be a multiple of the period of $U_{0}$.
Without loss of generality, we assume $U_{0}$ is fixed by $f_{0}$. By Lemma
3.3, the ray period of $p$ is the same as the period of $p$. In other words,
if the period of $p$ is $\ell$, then each external ray landing at $p$ has
external angle $\theta$ with $d^{\ell}\theta=\theta\mod 1$, where
$d=d_{1}+d_{2}$.

Note that the polynomial $f^{\ell}_{0}$ has $d^{\ell}$ fixed point and $0$ is
its unique attracting fixed point. As $f_{0}$ has no neutral periodic points,
we know all the fixed points of $f^{\ell}_{0}$ are repelling except $0$. Since
$p$ is bi-accessible, by Lemma 3.3, there are at least two
$f^{\ell}_{0}$-fixed external rays landing at $p$. Thus there are at most
$(d^{\ell}-3)$ $f^{\ell}_{0}$-fixed external rays landing at other
$d^{\ell}-2$ repelling fixed points. Since each $f^{\ell}_{0}$-fixed external
ray must land at a repelling $f^{\ell}_{0}$-fixed point, there exists a
repelling fixed point $\widehat{\alpha}$ of $f^{\ell}_{0}$ at which there are
no $f^{\ell}_{0}$-fixed external rays landing. Thus, there are at least two
external rays (which are not $f^{\ell}_{0}$-fixed) landing at
$\widehat{\alpha}$. It follows $\widehat{\alpha}\notin\partial U_{0}$ from
Lemma 3.3. Since $f_{0}$ is primitive, and so is $f^{\ell}_{0}$,
$\widehat{\alpha}$ cannot lie on the boundary of any preimage of $U_{0}$ under
$f^{\ell}_{0}$. Note that all the Fatou component of $f^{\ell}_{0}$ is a
preimage of $U_{0}$ under $f^{\ell}_{0}$. We conclude that $\widehat{\alpha}$
is a buried repelling fixed point of $f^{\ell}_{0}$. If $\widehat{\alpha}$
does not lie in the orbit of $c_{0}$, then we are done.

Now assume $\widehat{\alpha}\in\mathrm{orb}(c_{0})$. Let $\ell^{\prime}>\ell$
denote the ray period of $\widehat{\alpha}$. Let $P(\ell^{\prime})$ denote the
set of all the repelling periodic points with period $\ell^{\prime}$. For any
$z\in P(\ell^{\prime})$, there is at least one external ray with period
divisible by $\ell^{\prime}$ landing at $z$. As there are at least two
external rays of period $\ell^{\prime}$ landing at $\widehat{\alpha}\notin
P(\ell^{\prime})$, there are at most $d^{\ell^{\prime}}-d^{\ell^{\prime}-1}-1$
external rays with period $\ell^{\prime}$ possible to land on
$P(\ell^{\prime})$. Since
$\\#P(\ell^{\prime})=d^{\ell^{\prime}}-d^{\ell^{\prime}-1}$, there exists an
$\widehat{\alpha}^{\prime}\in P(\ell^{\prime})$ such that there are no
$f^{\ell^{\prime}}_{0}$-fixed external rays landing at
$\widehat{\alpha}^{\prime}$. So $\widehat{\alpha}^{\prime}$ is a biaccessible
repelling $f^{\ell^{\prime}}_{0}$-fixed point. Again by Lemma 3.3 and the
primitive property of $f_{0}$, we can conclude that
$\widehat{\alpha}^{\prime}$ is buried. Since
$\widehat{\alpha}\in\mathrm{orb}(c_{0})$ and $\ell^{\prime}>\ell$,
$\widehat{\alpha}^{\prime}$ cannot lie in the critical orbit
$\mathrm{orb}(c_{0})$.

∎

###### Proof of (1) and (2) of Theorem 3.1.

Recall that $m_{0}$ is the period of $0$. By Lemma 3.4, $f_{0}$ has a buried
biaccessible repelling periodic point $\xi_{1}$ such that
$\xi_{1}\notin\mathrm{orb}(c_{0})$. We use $Z_{1}=\mathrm{orb}(\xi_{1})$ to
make Yoccoz puzzle. Let $s_{1}\leq m_{0}$ be the smallest positive integer
such that $f^{s_{1}}_{0}(0)\in\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{1}}$. Then
there exist positive integers $q_{1}$ and $u$ such that
$f^{s_{1}}_{0}:Y^{(n+s_{1})}_{Z_{1}}\to Y^{(n)}_{Z_{1}}$ is a holomorphic
proper map of degree $u$ for all $n\geq q_{1}$. We claim $u=d_{1}$. Indeed, if
$u>d_{1}$ then there exists $0\leq t_{1}<s_{1}$ such that $c_{0}\in
f^{t_{1}}_{0}(\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{1}})$. This is impossible
since $f_{0}$ is non-renormalizable at $c_{0}$.

By the thickening technique ([21]),
$f^{s_{1}}_{0}:Y^{(q_{1}+s_{1})}_{Z_{1}}\to Y^{(q_{1})}_{Z_{1}}$ can extend to
a polynomial-like map $g_{1}$ with filled Julia set
$K(g_{1})=\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{1}}$. By Douady-Hubbard
Straightening Theorem, $g_{1}$ is hybrid to a primitive postcritically finite
hyperbolic polynomial $Q_{1}$ of degree $d_{1}$. If $s_{1}=m_{0}$, then
$Q_{1}=z^{d_{1}}$ and so
$\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{1}}=\overline{U_{0}}$. Then we take
$\tau_{0}=\xi_{1}$ and we are done. If $s_{1}\neq m_{0}$, then $Q_{1}\neq
z^{d_{1}}$. It follows [24, Lemma 3.3] that $Q_{1}$ has a buried biaccessible
repelling periodic point $\zeta_{1}$. Let $h_{1}$ denote the hybrid conjugacy
between $g_{1}$ and $Q_{1}$. Then $\xi_{2}=h^{-1}_{1}(\zeta_{1})$ is
accessible along at least two curves (the preimage of two external rays of
$Q_{1}$ under $h^{-1}_{1}$) which are not homotopy equivalent relative to
$\mathbb{C}\setminus K(f_{0})$. By [18, Theorem 2], there are at least two
$f_{0}$-external rays landing at $\xi_{2}$. So $\xi_{2}$ is a buried
biaccessible repelling periodic point of $f_{0}$. By using a similar argument
in the proof of Lemma 3.4, we can further assume
$\xi_{2}\notin\mathrm{orb}(c_{0})$.

Now let $Z_{2}=\mathrm{orb}(\xi_{2})$ and construct Yoccoz puzzle. Let $s_{2}$
be the smallest positive integer such that
$f^{s_{2}}_{0}(0)\in\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{2}}$. Note that
$s_{2}>s_{1}$. Indeed, the external rays landing at $\xi$ separate $0$ from
$f_{0}(\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{1}}),\ldots,f_{0}^{s_{1}-1}(\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{1}})$
and $f^{s_{1}}_{0}(0)$.

Inductively, we can obtain a strictly increasing sequence $\\{s_{k}\\}$ and a
sequence $\\{Z_{k}=\mathrm{orb}(\xi_{k})\\}$ with the following properties:

  * •

$\xi_{k}$ is a buried biaccessible repelling periodic point of $f_{0}$ which
lies outside the orbit of $c_{0}$;

  * •

$s_{k}\leq m_{0}$ is the smallest positive integer such that
$f^{s_{k}}_{0}(0)\in\bigcap_{n=0}^{\infty}Y^{(n)}_{Z_{k}}$.

Since $s_{k}$ is a strictly increasing sequence of positive integers bounded
by $m_{0}$, there is a first moment $k_{0}$ such that $s_{k}=m_{0}$. Take
$\tau_{0}=\xi_{k}$ and we are done. Indeed,
$f^{m_{0}}_{0}:Y^{(m_{0}+n)}_{Z_{k}}\to Y^{(n)}_{Z_{k}}$ has a polynomial-like
extension which is hybrid to $z\mapsto z^{d_{1}}$ for all sufficiently large
$n$. Thus the filled Julia set of $f^{m_{0}}_{0}|_{Y^{(m_{0}+n)}}$ must equal
to $\overline{U_{0}}$.

∎

From now on, we will fix an $f_{0}$-admissible set
$Z_{0}=\mathrm{orb}({\tau_{0}})$ which is given by the above proof. For each
$n\in\mathbb{N}$, let $Y^{(n)}$ and $X^{(n)}$ denote the critical puzzle
pieces $Y^{(n)}_{Z_{0}}(0)$ and $Y^{(n)}_{Z_{0}}(c_{0})$ respectively. We
complete the proof of Theorem 3.1 by showing that $X^{(n)}$ shrinks to a
point:

######  Lemma 3.5.

$\bigcap_{n\in\mathbb{N}}X^{(n)}=\\{c_{0}\\}$.

We say $c_{0}$ is combinatorially recurrent if for any $n\in\mathbb{N}$, there
exists $j\geq 1$ such that $f^{j}_{0}(c_{0})\in X^{(n)}$. The proof for the
non-combinatorially recurrent case is simple, and is treated in the same way
as in the quadratic case. See [21, Lemma 1.8,Theorem 1.9] for an example. To
deal with the combinatorially recurrent case, we need the following
definition.

######  Definition 3.1.

A generalized polynomial-like map is a holomorphic proper map $g:U\to V$ with
the following properties:

  * •

$V$ is a quasidisk;

  * •

each component of $U$ is a quasidisk contained compactly in $V$;

  * •

$g$ has finitely many critical points.

For more details for generalized polynomial-like map, we refer the readers to
[9]. Now we use the Yoccoz puzzle induced by $Z_{0}$ to associate a
generalized polynomial-like map to $f_{0}$.

######  Definition 3.2.

An APL  map $g:U\to V$ is a holomorphic proper map which satisfies the
following : $U\subset V$ and $K(g):=\bigcap_{n}g^{-n}(U)$ is compactly
contained in $U$.

It follows from [16, Lemma 2.4] that every APL map has a polynomial-like
restriction with the filled Julia set equal to $K(g)$.

######  Lemma 3.6.

If $c_{0}$ is combinatorially recurrent, then there exists a positive integer
$\nu_{0}$ such that the component of the domain of the first return map to
$X^{(\nu_{0})}$ which contains $c_{0}$ is compactly contained $X^{(\nu_{0})}$.

###### Proof.

By the proof of (1) and (2) of Theorem 3.1, there exists $\kappa_{0}>0$ large
enough so that the following holds:

  * •

$\overline{Y^{(\kappa_{0})}},\overline{f_{0}(Y^{(\kappa_{0})})},\ldots,\overline{f^{m_{0}-1}_{0}(Y^{(\kappa_{0})})}$
are disjoint;

  * •

$f^{m_{0}}_{0}:Y^{(\kappa_{0})}\to Y^{(\kappa_{0}-m_{0})}$ is an APL map with
filled Julia set $\overline{U_{0}}$.

Let $P_{0}=X^{(\kappa_{0})}$. For $n\geq 1$, we define inductively $P_{n}$ as
the component of the domain of first return map to $P_{n-1}$ which contains
$c_{0}$. For all $n\in\mathbb{N}$, the first return map
$R_{n}|_{P_{n+1}}:P_{n+1}\to P_{n}$ is $d_{2}$-to-$1$. Note that there exists
a first moment $s_{0}$ such that $P_{s_{0}+1}\Subset P_{s_{0}}$. It is well
known since $f_{0}$ is non-renormalizable at $c_{0}$, see the proof of [11,
Lemma 2.2] for an example. Let $X^{(\nu_{0})}=P_{s_{0}}$ and we are done. ∎

###### Proof of Lemma 3.5.

Assume $c_{0}$ is combinatorially recurrent and let $X^{(\nu_{0})}=P_{s_{0}}$,
$P_{n}~{}(n\in\mathbb{N})$ be as in the proof Lemma 3.6. It suffices to show
$P_{n}$ shrinks to $c_{0}$. Let $V=P_{s_{0}+1}$ and let $R:U\to V$ be the
first return map to $V$ under $f_{0}$. Note that $R:U\to V$ is a generalized
polynomial map with a unique critical point $c_{0}$. Since $f_{0}$ is non-
renormalizable at $-c_{0}$, so is $R:U\to V$. This implies the diameter of the
puzzle piece $X^{(n)}$ shrinks to $0$ and $J_{f_{0}}$ is locally connected at
$c_{0}$, see [9] and [11, Theorem 1.5]. ∎

Every puzzle piece which attaches $Z_{0}$ also shrinks to a point:

######  Proposition 3.1.

Let $z\in\bigcup_{j=0}^{\infty}f_{0}^{-j}(Z_{0})$. Then

$\sup\\{\mathrm{diam}(Y)\mid Y\text{ is a puzzle piece of depth
}n~{}\text{such that}~{}z\in\overline{Y}\\}\to 0\text{ as }n\to\infty.$

###### Proof.

According to Theorem 3.1, there exists $n_{0}$ large such that
$Y\cap(X^{(n_{0})}\cup Y^{(n_{0})})=\emptyset$. Then the proposition follows
by [21, Lemma 1.8]. ∎

For a more general result by Kozlovski and van Strien, see [11, Theorem 1.1].
The following corollary is a combinatorial version for Proposition 3.1:

######  Corollary 3.1.

Let $z\in\bigcup_{j\geq 0}f^{-j}_{0}(Z_{0})$ and let $\Theta(z)$ be the set of
angles $\theta$ for which $\mathcal{R}_{f_{0}}(\theta)$ lands at $z$. Then for
each component $(\theta_{1},\theta_{2})$ of
$\mathbb{R}/\mathbb{Z}\setminus\Theta(z)$, there exist two monotone sequences
$\\{a_{n}\\}$ and $\\{b_{n}\\}$ in $(\theta_{1},\theta_{2})$ such that
$a_{n}\sim_{\lambda_{f_{0}}}b_{n}$ for all $n>0$ and $a_{n}\to\theta_{1}$ and
$b_{n}\to\theta_{2}$ as $n$ tends to infinity.

As an application, we prove:

######  Lemma 3.7.

Suppose two external rays $\mathcal{R}_{f_{0}}(t)$ and
$\mathcal{R}_{f_{0}}(s)$ of $f_{0}$ land at $z_{s}$ and $z_{t}$ respectively.
If $z_{t},z_{s}\in\bigcup\limits_{n=0}^{\infty}f^{-n}_{0}(Z_{0})$ and
$z_{t}\neq z_{s}$, then $\mathcal{R}_{f}(t)$ and $\mathcal{R}_{f}(s)$ land at
two different points for every $f\in\mathcal{C}(\lambda_{f_{0}})$.

###### Proof.

Let $\Theta(z_{t})$ and $\Theta(z_{s})\subset\mathbb{R}/\mathbb{Z}$ be the set
of landing angles for $z_{t}$ and $z_{s}$ under $f_{0}$ respectively. There
are two component $C$ and $D$ of
$\mathbb{R}/\mathbb{Z}\setminus(\Theta(z_{t})\cup\Theta(z_{s}))$ such that
$C=(\theta_{\max}(s),\theta_{\min}(t))$ and
$D=(\theta_{\max}(t),\theta_{\min}(s))$ where
$\theta_{\min}(t),\theta_{\max}(t)\in\Theta(z_{t})$ and
$\theta_{\min}(s),\theta_{\max}(s)\in\Theta(z_{s})$. By Corollary 3.1, there
exist $\\{a_{n}\\}\subset C$ and $\\{b_{n}\\}\subset D$ such that
$a_{n}\sim_{\lambda_{f_{0}}}b_{n}$ and $a_{n}\to\theta_{\min}(t)$,
$b_{n}\to\theta_{\max}(t)$ as $n\to\infty$. Now assume $\mathcal{R}_{f}(t)$
and $\mathcal{R}_{f}(s)$ land at a common point $w$. Since
$\lambda_{f_{0}}\subset\lambda_{f}$, $\mathcal{R}_{f}(a_{n})$ and
$\mathcal{R}_{f}(b_{n})$ land at a common point. Moreover, this point must be
$w$ by virtue of the construction. Hence, there are infinitely many external
rays of $f$ landing at $w$, which is a contradiction. ∎

The above lemma implies all the $f\in\mathcal{C}(\lambda_{f_{0}})$ have the
same puzzle structure with respect to $Z_{0}$.

${\beginpicture\setcoordinatesystem units<.8cm,.8cm>\par\circulararc
360degreesfrom33centerat00\plot-20-33/\plot-20-3-3/\plot-20-4.250/\plot-10-1.93.8/\par\par\par\par\plot
2033/\plot 203-3/\plot
204.250/\plot-10-1.9-3.8/\par\put(0.0,0.0){$}whichisgivenbyTheorem~{}\ref{thm:puzzle}.\par
Inthissection,wewillmodifytheargumentin~{}\cite[cite]{[\@@bibref{}{SW}{}{}]}toobtainaK-
qcextensionprinciple.ThisenablesustousethecriterionforconvergenceofThurston^{\prime}salgorithmwhichwasestablishedin\cite[cite]{[\@@bibref{}{SW}{}{}]}.\par\par
Let$L_n$denotethedomainofthefirstlandingmapto$Y^{(n)}$:\begin{equation}L_{n}=\left\\{z\in\mathbb{C}:\exists
k\geq 0\text{ such that }f_{0}^{k}(z)\in
Y^{(n)}\right\\}.\end{equation}Ourmaingoalinthesectionistoshow:\begin{Theorem}Foranypuzzlepiece$Y$,thereexistsapositiveinteger$N$suchthat$$\mathcal{QD}(L_{N}\cap
Y,Y)<\infty.$$\end{Theorem}Asin\cite[cite]{[\@@bibref{}{SW}{}{}]},thedifficultyinprovingtheabovetheoremisthatthelandingdomains$L_{N}$maycomearbitrarilyclosetotheboundaryof$Y$.Todealwiththesituation,weshallneedatoymodeldevelopedbyKahn(\cite[cite]{[\@@bibref{}{Kahn}{}{}]}).WefirstconstructanexpandingmapwhichisquasiconformallyconjugatetoKahn^{\prime}srecursivelynotchedsquaremodel,whichwillimplyTheorem~{}\ref{thm:Kqcpuzzle'}holdsforallthepuzzlepiecesofdepth$0$.Tocompletetheproof,weuseapullbackargument.Theproofisadaptedfrom~{}\cite[cite]{[\@@bibref{}{SW}{}{},
Section~{}4]}.Thereaderswhoarefamiliarwiththatof\cite[cite]{[\@@bibref{}{SW}{}{}]}mayskiptheproof.Themaindifferenceisthefollowing.In\cite[cite]{[\@@bibref{}{SW}{}{}]},$\partial
Y'$isdisjointfromthepost-criticalsetforanypuzzlepiece$Y'$since$f_0$ispost-
criticallyfinite.Thusforeach$z\in J_{f_0} \cap
Y'$,wecanpullbackadefinitesmallneighborhoodof$z$univalentlytoanarbitrarydepth.Inourcase,$f_0$hasafreeJuliacriticalpoint$c_0$whichmayaccumulateontheboundariesofallthepuzzlepieces.Thuswemustchooseaneighborhoodof$\tau_0$carefullysothatwecanpullbackitunivalentlytosomegivendepth.Tomakeitself-
contained,wegiveanindependentproofinthissection.\par\par\@@numbered@section{subsection}{toc}{Quasiconformal
Distortion Bounds} \par\par\par Let us first recall some terminology from
\cite[cite]{[\@@bibref{}{Kahn}{}{}]}. Let $U\subset\mathbb{C}$ be a Jordan
domain and $A$ be a measurable subset of $U$. We say that $(A,U)$ has {\em
bounded qc distortion} if there exists a constant $K\geq 1$ with the following
property: if $\varphi:U\to\varphi(U)$ is a quasiconformal map and
$\bar{\partial}\varphi=0$ a.e. outside $A$, then there is a $K$-q.c map
$\tilde{\varphi}:U\to\varphi(U)$ such that $\tilde{\varphi}=\varphi$ on
$\partial U$. Let $\mathcal{QD}(A,U)$ denote the smallest $K$ satisfying the
property. \par\par\par\par\par We shall need the following easy facts.
\begin{Lemma}\cite[cite]{[\@@bibref{}{Kahn}{}{}, Fact~{}1.3.6]} If $A\subset
U$ is compact, then $\mathcal{QD}(A,U)<\infty$. \end{Lemma}
\par\begin{Lemma}\cite[cite]{[\@@bibref{}{Kahn}{}{}, Fact~{}1.3.4]} Let $U$
and $V$ be Jordan domains in $\mathbb{C}$ and let $A$ be a measurable subset
of $U$. If there exists a $L$-qc map $g:U\to V$ and
$\mathcal{QD}(A,U)<\infty$, then $$\mathcal{QD}(g(A),V)\leq
L^{2}\mathcal{QD}(A,U).$$ \end{Lemma} \par\begin{Lemma} The following
statements are equivalent: \begin{enumerate} \par\enumerate@item@[(i)]
$\mathcal{QD}(A,U)=C<\infty$; \par\enumerate@item@[(ii)] For any qc map
$\varphi:U\to\varphi(U)$, if $\mathrm{Dil}(\varphi)\leq K$ for some $K\geq 1$
a.e. outside $A$, then there is a $KC$-qc map
$\widetilde{\varphi}:U\to\varphi(U)$ such that $\widetilde{\varphi}=\varphi$
on $\partial U$. \par\end{enumerate} \end{Lemma}
See~{}\cite[cite]{[\@@bibref{}{SW}{}{}, Lemma~{}4.3]} for a proof.
\begin{Lemma} Let $U$ be a Jordan domain and let $A$ be a measurable subset of
$U$. Assume there is an open subset $W$ of $U$ with the following properties:
\begin{itemize} \par\itemize@item@$W$ is a disjoint union of finitely many
Jordan domains; \par\itemize@item@for each component $T$ of $W$, $\partial
T\cap A=\emptyset$ and $\mathcal{QD}(A\cap T,T)<\infty$;
\par\itemize@item@$A\setminus W$ is compactly contained in $U$.
\par\end{itemize} Then $\mathcal{QD}(A,U)<\infty$. \end{Lemma} \par\@proof Let
$\phi:U\to\phi(U)$ be a quasiconformal map with $\bar{\partial}\phi=0$ a.e.
outside $A$. For each component $T$ of $W$, there exists a $K_{T}$-qc map
$\phi_{T}:T\to\phi(T)$ such that $\phi_{T}|_{\partial T}=\phi|_{\partial T}$
since $\mathcal{QD}(A\cap T,T)<\infty$. Let $K=\max\limits_{T}K_{T}$. Replace
$\phi$ by $\phi_{T}$ on each component $T$, we get a qc map $\varphi$ which is
$K$-qc a.e. outside $A\setminus W$ and $\varphi|_{\partial U}=\phi|_{\partial
U}$. Since $A\setminus W$ is compactly contained in $U$,
$\mathcal{QD}(A\setminus W,U)=K^{\prime}<\infty$. By Lemma~{}\ref{qc2}, there
exists a $KK^{\prime}$-qc map with the same boundary value as $\phi$.
\qed\end@proof \par We shall now recall the {\em recursively notched square
model} developed in~{}\cite[cite]{[\@@bibref{}{Kahn}{}{}]}. Let
$S=(0,1)\times(-1/2,1/2)$. Let $\mathcal{I}$ denote the collection of the
components of $(0,1)\setminus\mathcal{C}$, where $\mathcal{C}$ is the ternary
Cantor set. Let
\begin{equation}\mathcal{N}=\bigcup_{I\in\mathcal{I}}\overline{I}\times[-|I|/2,|I|/2]\end{equation}
which is a countable disjoint union of closed squares. The following is
~{}\cite[cite]{[\@@bibref{}{Kahn}{}{}, Lemma 2.1.1]}: \begin{Theorem}
$\mathcal{QD}(\mathcal{N},S)<\infty.$ \end{Theorem}
\par\par\par\par\par\par\par\par\par\@@numbered@section{subsection}{toc}{Reduce
to the toy model} In the following we write $\mathcal{R}(\theta)$ for
$\mathcal{R}_{f_{0}}(\theta)$. A {\em combinatorial ray-pair} is, by
definition, a pair $(\theta,\tilde{\theta})$ such that
$\theta\neq\tilde{\theta}$ and $\theta\sim\tilde{\theta}$. The union
$\overline{\mathcal{R}(\theta)}\cup\overline{\mathcal{R}(\tilde{\theta})}$ is
called the {\em geometric ray-pair} with respect to $(\theta,\tilde{\theta})$
and we denote it by $\mathcal{R}(\theta,\tilde{\theta})$. By a {\em slice} we
mean an open set
$S(\theta_{1},\theta_{2},\tilde{\theta}_{2},\tilde{\theta}_{1})$ bounded by
two disjoint geometric ray-pairs $\mathcal{R}(\theta_{i},\tilde{\theta}_{i})$,
$i=1,2$ such that
$\theta_{1},\theta_{2},\tilde{\theta}_{2},\tilde{\theta}_{1}$ lie in
$\mathbb{R}/\mathbb{Z}$ in the anticlockwise order. \par\par Let $\tau_{0}$ be
given by Theorem~{}\ref{thm:puzzle}, the external rays landing at $\tau_{0}$
cut the complex plane $\mathbb{C}$ into finitely many sectors
$S_{0},\cdots,S_{q-1}$ where $q$ is the number of external rays landing at
$\tau_{0}$. Without loss of generality, we assume $0\in S_{0}$. The boundary
of $S_{0}$ is a geometric ray-pair: there exists
$\theta^{-},\theta^{+}\in\mathbb{R}/\mathbb{Z}$ such that $\partial
S_{0}=\mathcal{R}(\theta^{-},\theta^{+}).$ By
Proposition~{}\ref{prop:noncrpiece}, for $n$ sufficiently large, there exists
a unique puzzle piece $V_{n}$ lying in $S_{0}$ such that $\tau_{0}\in\partial
V_{n}$ and $0\notin V_{n}$. The boundary of $V_{n}$ consists of portions of an
equipotential curve and a finite set of geometric ray-pairs, cut off at that
potential height. Among these geometric ray-pairs, there exists a unique ray-
pair $\mathcal{R}(t_{n},s_{n})$ which separates $\tau_{0}$ from $0$. Let
${S}(n)$ be the slice bounded by $\mathcal{R}(t_{n},s_{n})$ and $\partial
S_{0}$. Let $\widehat{S}(n)=\\{z\in{S}(n):G_{f_{0}}(z)<(d_{1}+d_{2})^{-n}\\}$.
Without loss of generality, we assume $\theta^{-},t_{n},s_{n},\theta^{+}$ lie
in $\mathbb{R}/\mathbb{Z}$ in the anticlockwise order. The following lemma is
a consequence of Proposition~{}\ref{prop:noncrpiece}.
\begin{Lemma}$\text{diam}(\widehat{S}(n))\to 0$ as $n\to\infty$. \end{Lemma}
\par\par\par\par Let $p$ be the period of $\theta^{+}$. By
Lemma~{}\ref{lem:slice}, we can choose $N_{1}$ sufficiently large so that the
closure of $\widehat{S}:=\widehat{S}(N_{1})$ is disjoint from the set
$\bigcup_{j=0}^{p}f^{j}_{0}(\\{0,c_{0}\\})$. Thus we can define a single-
valued univalent branch $g$ of $f^{-p}_{0}$ on a neighborhood of
$\widehat{S}(N_{1})$ such that $g(\tau_{0})=\tau_{0}$. \par Let
$\tau^{\prime}_{0}$ be the landing point of the external ray
$\mathcal{R}(t_{N_{1}})$ and let $r$ be the first moment for which
$f^{r}_{0}(R(t_{N_{1}},s_{N_{1}}))=\partial S_{0}$. Note that
$g^{k}(\widehat{S}(N_{1}))\subset\widehat{S}(N_{1}+kp)$ for all
$k\in\mathbb{N}$. By Lemma~{}\ref{lem:slice}, the closure of
$\widehat{S}(N_{1}+kp)$ is disjoint from the set
$\bigcup_{j=0}^{r}f^{j}_{0}(\\{0,c_{0}\\}$ for all $k$ large. Thus we can
define a single-valued univalent branch $h_{k}$ of $f^{-r}_{0}$ on a
neighborhood of $\widehat{S}(N_{1}+kp)$ such that
$h_{k}(\tau_{0})=\tau^{\prime}_{0}$ for all $k$ large. Let
$\widehat{A}=g(\widehat{S}(N_{1}))$ and $\widehat{B}_{k}=h_{k}\circ
g^{kp}(\widehat{S}(N_{1}))$. Note that $A$ and $B_{k}$ are quasidisks. It
follows from Lemma~{}\ref{lem:slice} that there exists $k_{0}$ such that
$\overline{\widehat{A}}\cap\overline{\widehat{B}_{k}}=\emptyset$ for all
$k\geq k_{0}$. Let $\widehat{B}=\widehat{B}_{k_{0}}$ and $h=h_{k_{0}}$. Define
a map $\widehat{F}:\widehat{A}\cup\widehat{B}\to\widehat{S}$ with
$\widehat{F}|_{\widehat{A}}=f^{p}_{0}$ and
$\widehat{F}|_{\widehat{B}}=f^{r+k_{0}p}_{0}$. \par\par\par Without loss
generality, we assume $\widehat{A}$ and $\widehat{B}$ are truncations of
slices $S(\theta^{-},t,s,\theta^{+})$ and
$S(t_{N_{1}},t^{\prime},s^{\prime},s_{N_{1}})$ by equipotential curves
respectively for some $t,s,t^{\prime},s^{\prime}\in\mathbb{R}/\mathbb{Z}$.
Moreover, we may assume
$f^{r}_{0}(\mathcal{R}(t_{N_{1}}))=\mathcal{R}(\theta^{-})$. For any
$\eta\in\mathbb{R}/\mathbb{Z}$ and $\kappa>0$, let $\xi(\eta,\kappa)$ denote
the intersection point of the external ray $\mathcal{R}(\eta)$ and the
equipotential curve of height $\kappa$. Let $\Phi:\mathbb{H}_{+}\cap\\{z\mid
0<\Im z<1\\}\to\mathbb{C}\setminus K(f_{0})$ be a map defined as
$\Phi(z)=\phi^{-1}_{f_{0}}(e^{2\pi iz})$ where $\mathbb{H}_{+}$ is the right
half-plane and $\phi_{f_{0}}$ is the B\"{o}ttcher coordinate of $f_{0}$. Let
$\gamma_{-}$ and $\gamma_{+}$ be the line segment with endpoints
$\\{\Phi^{-1}(\xi(t,d^{-N_{1}-p})),\Phi^{-1}(\xi(t^{\prime},d^{-N_{1}-k_{0}p-r}))\\}$
and
$\\{\Phi^{-1}(\xi(s,d^{-N_{1}-p})),\Phi^{-1}(\xi(s^{\prime},d^{-N_{1}-k_{0}p-r}))\\}$
respectively, where $d=d_{1}+d_{2}$. Let $\widehat{C}$ be the closure of the
bounded component of
$\mathbb{C}\setminus(\overline{A}\cup\overline{B}\cup\Phi(\gamma_{-})\cup\Phi(\gamma_{+}))$.
\par\par\par\par Let $A=(0,1/3)\times(-1/6,1/6)$, $B=(2/3,1)\times(-1/6,1/6)$,
$C=[1/3,2/3]\times[-1/6,1/6]$ and $S=(0,1)\times(-1/2,1/2)$ and define a map
$$F:A\cup B\to S,$$ as follows: $$F(z)=\left\\{\begin{array}[]{ll}3z,&\mbox{
if }z\in A;\\\ 3(1-z),&\mbox{ if }z\in B.\end{array}\right.$$
\par\par\par\par\par\par\begin{Lemma} There is a qc homeomorphism
$H:S\to\widehat{S}$ such that \begin{enumerate} \par\enumerate@item@[(i)]
$H(A)=\widehat{A},H(B)=\widehat{B},H(C)=\widehat{C},$
\par\enumerate@item@[(ii)] $H\circ F=\widehat{F}\circ H$ holds on
$\overline{A\cup B}$. \par\end{enumerate} \end{Lemma} \@proof\par We claim
that there is qc map $H_{0}:S\to\widehat{S}$ such that (i) holds and (ii)
holds on $\partial A\cup\partial B$ (with $H$ replaced by $H_{0}$). Indeed,
one can easiy construct a homeomorphism with the desired properties by using
B\"{o}ttcher coordinate. It can be made global qc by Beurling-Ahlfors
extension since $\partial S,\partial A,\partial B,\partial C$ are all quasi-
circles. \par Now we can construct inductively a sequence
$\\{H_{n}\\}_{n=0}^{\infty}$ of qc maps by pull-back which has the following
properties: \begin{itemize} \par\itemize@item@$H_{n+1}=H_{n}$ on
$S\setminus(A\cup B);$ \par\itemize@item@$H_{n}\circ F=\widehat{F}\circ
H_{n+1}$ holds on $\overline{A\cup B}$. \par\end{itemize} These maps $H_{n}$
have the same maximal dilatation as $H_{0}$, and they eventually stablize for
any point in the set $X=\\{z\in S:F^{n}(z)\not\in\bigcup A\cup B\mbox{ for
some }n\\}$. Since $F$ is uniformly expanding, the set $X$ is dense in $S$, it
follows that $H_{n}$ converges to a qc map $H$ which satisfies the
requirements. \par\par\par\qed\end@proof \par Let
$\widehat{\mathcal{N}}=\\{z\in\widehat{S}:\exists n\geq 1\text{ such that
}\widehat{F}^{n}(z)\text{ is well-defined and belongs
to}~{}~{}\widehat{C}\\}.$ Note that $H(\mathcal{N})=\widehat{\mathcal{N}}$,
where $\mathcal{N}$ is as in (\ref{eqn:setN}). The following Lemma follows
immediately by Lemma~{}\ref{lem:qc homeo}, Theorem~{}\ref{thm:Kahn} and
Lemma~{}\ref{qc1}.
\begin{Lemma}$\mathcal{QD}(\widehat{\mathcal{N}},\widehat{S})<\infty.$
\end{Lemma} \par\par\par\par\@proof[Proof of Theorem~\ref{thm:Kqcpuzzle'}] For
each $z\in\partial Y\cap J(f_{0})$, there exists a unique geometric ray-pair
$\Gamma(z)\ni z$ such that each componet of $\Gamma(z)\setminus\\{z\\}$
intesects $\overline{Y}$. Let $m(z)$ be the first moment for which
$f^{m(z)}_{0}(\Gamma(z))=\partial S_{0}=\mathcal{R}(\theta^{-},\theta^{+})$.
Recall that $g$ is a single-valued univalent branch of $f^{-p}_{0}$ on a
neighborhood of $\widehat{S}(N_{1})$ such that $g(\tau_{0})=\tau_{0}$ and
$h=h_{k_{0}}$ is a single-valued univalent branch of $f^{-r}_{0}$ on a
neighborhood of $\widehat{S}(N_{1}+k_{0}p)$ such that
$h(\tau_{0})=\tau^{\prime}_{0}$. As $g^{k}(\widehat{S}(N_{1}))$ shrinks to
$\tau_{0}$ as $k$ tends to infinity, there exists $k_{1}=k_{1}(z)>k_{0}$ large
enough so that $\overline{g^{k}(\widehat{S}(N_{1}))}$ is disjoint from
$\bigcup_{j=0}^{m}f^{j}_{0}(\\{0,c_{0}\\})$. So we can define a single-valued
univalent branch $\psi$ of $f^{-m(z)}_{0}$ on a neighborhood of
$\overline{g^{k_{1}}(\widehat{S}(N_{1}))}$ such that $\psi(\tau_{0})=z$.
\par\par Take $N(z)=3(m(z)+N_{1}+k_{1}p)$ and let $W=W(z):=\psi\circ
g^{k_{1}}(\widehat{S}(N_{1}))$. Then any landing domain of $Y^{(N(z))}$ does
not intersect $\partial W$ and $\partial A\cup\partial B$. Therefore,
$L_{N(z)}\cap\widehat{S}\subset\widehat{\mathcal{N}}$, which implies
$$\mathcal{QD}(L_{N(z)}\cap\widehat{S},\widehat{S})\leq\mathcal{QD}(\widehat{\mathcal{N}},\widehat{S})=Q<\infty.$$
Note that $f_{0}^{m(z)+k_{1}p}$ maps a neighborhood of $W$ conformally onto a
neighborhood of $\widehat{S}$ and $f_{0}^{m(z)+k_{1}p}(L_{N(z)}\cap W)\subset
L_{N(z)}\cap\widehat{S}$. For otherwise there exists a moment $0\leq
T<m(z)+k_{1}p$ such that $f^{T}_{0}(W)$ contains a critical point. This
contradicts that $f^{m(z)+k_{1}p}_{0}|_{W}$ is a conformal map. Therefore, by
Lemma~{}\ref{qc1}, $$\mathcal{QD}(L_{N(z)}\cap
W,W)\leq\mathcal{QD}(L_{N(z)}\cap\widehat{S},\widehat{S})<Q.$$ \par Let
$E:=\\{z\mid z\in\partial Y\cap J(f_{0})\\}$. Take $N=\max\\{N(z)\mid z\in
E\\}$. Then $$\sup\limits_{z\in E}\mathcal{QD}(L_{N}\cap
W(z),W(z))\leq\sup\limits_{z\in E}\mathcal{QD}(L_{N(z)}\cap W(z),W(z)<Q.$$
Since $\min\\{k_{1}(z)\mid z\in E\\}$ can be taken arbitrarily large, we may
assume $W(z)$'s are mutually disjoint. Note that $L_{N}\setminus\bigcup_{z\in
E}W(z)$ is compactly contained in $Y$. By Lemma~{}\ref{lem:qc3},
$$\mathcal{QD}(L_{N}\cap Y,Y)<\infty.$$ \par\par\par\qed\end@proof
\par\par\par\par\par\par\@@numbered@section{section}{toc}{Maps in $\mathcal
C(\lambda_{f_0})$} Recall that $Z_{0}=\mathrm{orb}(\tau_{0})$ is an
$f_{0}$-admissible given by Theorem~{}\ref{thm:puzzle}. In this section, we
study the combinatorial properties for the maps in
$\mathcal{C}(\lambda_{f_{0}})$. More precisely, we show all the maps in
$\mathcal{C}(\lambda_{f_{0}})$ have the same puzzle structure as $f_{0}$.
Moreover, for any $f\in\mathcal{C}(\lambda_{f_{0}})$ there is a qc weak
pseudo-conjugacy between $f_{0}$ and $f$ up to an arbitrarily given depth.
\par\par\@@numbered@section{subsection}{toc}{Weak pseudo-conjugacy and pseudo-
conjugacy} \par We say a polynomial $\widetilde{f}$ {\em has the same
combinatorics as $f_{0}$ up to depth $n$} if there exists a homemorphism
$H:\mathbb{C}\to\mathbb{C}$ which satisfies the following: \begin{itemize}
\par\itemize@item@$H$ is identity in the B\"{o}ttcher coordinate for $f_{0}$
and $\widetilde{f}$ near $\infty$,i.e.,
$H=\phi^{-1}_{\widetilde{f}}\circ\phi_{f_{0}}$ in a neighborhood of infinity,
where $\phi_{f_{0}}$ and $\phi_{\widetilde{f}}$ are B\"{o}ttcher map for
$f_{0}$ and $\widetilde{f}$ respectively; \par\itemize@item@$H\circ
f_{0}=\widetilde{f}\circ H$ outside the union of all the $f_{0}$-puzzle pieces
of depth $n$. (Note that all the puzzle pieces are open in our definition.)
\par\end{itemize} Such a homeomorphism $H$ is called a {\em weak pseudo-
conjugacy up to depth $n$} between $f_{0}$ and $\widetilde{f}$. If $H\circ
f_{0}=\widetilde{f}\circ H$ holds everywhere outside all the critical puzzle
pieces of depth $n$, then such an $H$ is called a {\em pseudo-conjugacy up to
depth $n$ }. \par\par\begin{Lemma}For any $f\in\mathcal{C}(\lambda_{f_{0}})$,
$f$ has the same combinatorics as $f_{0}$ up to an arbitrary depth.
\end{Lemma} \@proof By definition, it suffices to show that for any $k$, there
exists a homeomorphism $H_{k}:\mathbb{C}\to\mathbb{C}$ with the following
properties: \begin{itemize} \par\itemize@item@$H_{k}$ is identity in the
B\"{o}ttcher coordinate for $f_{0}$ and $f$ near $\infty$, i.e.,
$H_{k}=\phi^{-1}_{f}\circ\phi_{f_{0}}$ in a neighborhood of infinity, where
$\phi_{f_{0}}$ and $\phi_{f}$ are B\"{o}ttcher map for $f_{0}$ and $f$
respectively; \par\itemize@item@$H_{k}\circ f_{0}=f\circ H_{k}$ holds
everywhere outside any puzzle piece of depth $k$. \par\end{itemize} Indeed,
such a homeomorphism can be constructed as following. First let
$H_{k}=\phi^{-1}_{f}\circ\phi_{f_{0}}$ outside the union of puzzle pieces of
depth $k$. By Lemma~{}\ref{lem:nondegenerate}, such an $H_{k}$ is well
defined. Then for each puzzle piece $Y$ of depth $k$, we extend
$H_{k}|_{\partial Y}$ into $Y$ to be a homeomorphism in an arbitrary way.
\qed\end@proof \par\par\par\par\par\par\par\begin{Lemma} If an
$f_{0}$-external ray $\mathcal{R}_{f_{0}}(t)$ lands at $\tau_{0}$, then the
$f$-external ray $\mathcal{R}_{f}(t)$ lands at a repelling periodic point of
$f$ for any $f\in\mathcal{C}(\lambda_{f_{0}})$. \end{Lemma} \par\par\@proof To
obtain a contradiction, we suppose that the landing point $\tau_{f}$ of
$\mathcal{R}_{f}(t)$ is parabolic. Then $\tau_{f}$ must attract at least one
of the critical points of $f$. Let $\mathrm{Crit}(f)$ denote the set of the
critical points of $f$. By Proposition~{}\ref{prop:noncrpiece} and
Theorem~{}\ref{thm:puzzle}, there exists $k_{0}>0$ such that $\partial
Y^{(k-m_{0})}\cap\mathrm{orb}(\tau_{0})=\emptyset$ and
$f^{m_{0}}_{0}:Y^{(k)}\to Y^{(k-m_{0})}$ is an APL map for all $k\geq k_{0}$.
For any $k\geq k_{0}$, let $H_{k}$ be a weak pseudo-conjugacy between $f_{0}$
and $f$. We claim $\mathrm{Crit}(f)\subset H_{k}(Y^{(k)}\cup X^{(k)})$.
Indeed, $f_{0}:Y\to f_{0}(Y)$ is a conformal map for any puzzle piece
$Y\notin\\{Y^{(k)},X^{(k)}\\}$ of depth $k$. Thus $f_{0}|_{\partial
Y}:\partial Y\to\partial f_{0}(Y)$ is a homeomorphism, and so is $f:\partial
H_{k}(Y)\to\partial H_{k}(f_{0}(Y))=\partial f(H_{k}(Y))$ since $H_{k}$ is a
conjugacy between $f_{0}$ and $f$ on $\partial Y$. This implies $f:H_{k}(Y)\to
f(H_{k}(Y))$ is a conformal map. Hence $H_{k}(Y)$ contains none of the
critical points of $f$. \par By a similar argument, $f:H_{k}(Y^{(k_{0})})\to
H_{k}(f(Y^{(k_{0})}))$ is a holomorphic proper map of degree $d_{1}$.
Moreover, the orbit of any $c\in\mathrm{Crit}(f)\cap H_{k}(Y^{(k_{0})})$ lies
in
$\bigcup_{j=1}^{m_{0}}f^{j}(H_{k}(Y^{(k_{0})}))=\bigcup_{j=1}^{m_{0}}H_{k}(f^{j}_{0}(Y^{(k_{0})})$.
Since $\partial Y^{(k-m_{0})}\cap\mathrm{orb}(\tau_{0})=\emptyset$ and
$\tau_{f}=H_{k}(\tau_{0})$, one can conclude $\mathrm{orb}(\tau_{f})$ cannot
attract any critical point of $f$ which lies in $H_{k}(Y^{(k)})$. \par On the
other hand, it can be proved that $\bigcap_{k\geq k_{0}}H_{k}(X^{(k)})$ is a
singleton by the same method as in the proof of Lemma~{}\ref{lem:X shrinking}.
Note that such a singleton is a Julia critical point of $f$, which cannot be
attracted by $\mathrm{orb}(\tau_{f})$ either. \par As we have showed that none
of the critical points of $f$ can be attracted to the orbit of $\tau_{f}$, a
contradiction. \par\par\par\par\par\qed\end@proof
\par\par\par\par\par\par\begin{Lemma}If $f\in\mathcal{C}(\lambda_{f_{0}})$,
then for each $k\in\mathbb{N}$, there exists a qc weak pseudo-conjugacy
between $f_{0}$ to $f$ up to depth $k$. \end{Lemma} \par\@proof Let $\Theta$
denote the angles $\theta$ for which $\mathcal{R}_{f_{0}}(\theta)$ lands at
$\tau_{0}$. By Lemma~{}\ref{lem:tau repelling}, for each $\theta\in\Theta$,
the landing point of $\mathcal{R}_{f}(\theta)$ is a repelling periodic point
of $f$. \par For each $k\in\mathbb{N}$, we can construct such a qc weak
pseudo-conjugacy $H_{k}$ between $f_{0}$ and $f$ up to depth $k$ by hand.
First we define $H_{k}$ as identity in the B\"{o}ttcher coordinate for $f_{0}$
and $f$ outside the union of the puzzle pieces of depth $k$. It is well-known
that for each puzzle piece $Y$ of depth $k$, $H_{k}|_{\partial Y}$ can extend
to a qc homeomorphism form $\overline{Y}$ onto $\overline{H_{k}(Y)}$ since
$J_{f}\cap\partial Y$ are repelling (pre)-periodic points (see
\cite[cite]{[\@@bibref{}{KSS}{}{}, Lemma~{}5.3]}). \par\qed\end@proof
\par\par\par\par\par\par\par\begin{Lemma} Let $n_{0}>m_{0}$ be an integer such
that $Y^{(n_{0})},f_{0}(Y^{(n_{0})}),\ldots,f_{0}^{m_{0}-1}(Y^{(n_{0})})$ are
disjoint and $F_{0}=f^{m_{0}}_{0}|_{Y^{(n_{0})}}:Y^{(n_{0})}\to
Y^{(n_{0}-m_{0})}$ is an APL map. Suppose
$\widetilde{f}\in\mathcal{P}_{d_{1}+d_{2}}$ has the same combinatorics as
$f_{0}$ up to depth $n_{0}$. Assume there exists a qc weak pseudo-conjugacy
between $f_{0}$ and $\widetilde{f}$ up to depth $n_{0}$ such that
$$\widetilde{F}=\widetilde{f}^{m_{0}}|_{\Phi(Y^{(n_{0})}}:\Phi(Y^{(n_{0})})\to\Phi(Y^{(n_{0}-m_{0})})$$
is an APL map of degree $d_{1}$ with connected filled Julia set
$K(\tilde{F})$. Then $\mathcal{R}_{\widetilde{f}}(s)$ and
$\mathcal{R}_{\widetilde{f}}(t)$ land at a common point if
$\mathcal{R}_{f_{0}}(s)$ and $\mathcal{R}_{f_{0}}(t)$ land at a common point
which belongs to $\partial U_{0}$. \end{Lemma} \par\@proof We define a new qc
map $H:\mathbb{C}\setminus K(F)\to\mathbb{C}\to\setminus K(\widetilde{F})$
such that $H=\Phi$ on $\mathbb{C}\setminus Y^{(n_{0})}$ and $H\circ
F=\widetilde{F}\circ H$ on $Y^{(n_{0})}\setminus K(F)$. The construction of
$H$ is the following. \par Let $U=Y^{(n_{0})}$, $V=Y^{(n_{0}-m_{0})}$,
$\widetilde{U}=\Phi(Y^{(n_{0})})$ and $\widetilde{V}=\Phi(Y^{(n_{0}-m_{0})})$.
First choose quasidisks $W,W^{\prime}$ and
$\widetilde{W},\widetilde{W}^{\prime}$ such that $K(F)\Subset W\Subset
W^{\prime}\Subset U$ and
$K(\widetilde{F})\Subset\widetilde{W}\Subset\widetilde{W}^{\prime}\Subset\widetilde{U}$
respectively. Choose a qc homeomorphism $\Psi:W\setminus
K(F)\to\widetilde{W}\setminus K(\widetilde{F})$. Define $H_{0}=\Phi$ on
$\mathbb{C}\setminus W^{\prime}$ and $H_{0}=\Psi$ on $W\setminus K(F)$. Then
interpolate quasiconformally to obtain a qc homeomorphism
$H_{0}:\mathbb{C}\setminus K(F)\to\mathbb{C}\setminus K(\widetilde{F})$. Set
$\widehat{H}_{0}=H_{0}|_{V\setminus K(F)}$. Since $F:U\setminus K(F)\to
V\setminus K(F)$ and $\widetilde{F}:\widetilde{U}\to K(\widetilde{F})$ are
coverings of the same degree , we lift $\widehat{H}_{0}$ to a $K$-qc map
$\widehat{H}_{1}:U\setminus K(F)\to\widetilde{U}\setminus K(\widetilde{F})$
such that $\widehat{H}_{1}|_{\partial U}=\widehat{H}_{0}=H_{0}=\Phi$. Let
$H_{1}:\mathbb{C}\setminus K(F)\to\mathbb{C}\setminus K(\tilde{F})$ be a
$K$-qc map which is obtained by glueing $H_{0}|_{\mathbb{C}\setminus U}$ and
$\widehat{H}_{1}$ together. By definition, $\widetilde{F}\circ
H_{1}=H_{0}\circ F$ on $U\setminus K(F)$. Inductively, we can obtain a
sequence of $K$-qc maps $H_{n}:\mathbb{C}\setminus K(F)\to\mathbb{C}\setminus
K(\tilde{F})$ such that $\widetilde{F}\circ H_{n+1}=H_{n}\circ F$ on
$U\setminus K(F)$. Since $H_{n}$ are $K$-qc and eventually stable on
$\mathbb{C}\setminus K(F)$, $H_{n}$ converges to a $K$-qc map
$H:\mathbb{C}\setminus K(F)\to\mathbb{C}\setminus K(\tilde{F})$ which is
desired. \par Note that $H$ maps $\mathcal{R}_{f_{0}}(s)$ and
$\mathcal{R}_{f_{0}}(t)$ onto $\mathcal{R}_{\widetilde{f}}(s)$ and
$\mathcal{R}_{\widetilde{f}}(t)$ respectively. Let
$\varphi:\mathbb{C}\setminus\overline{\mathbb{D}}\to\mathbb{C}\setminus
K(\widetilde{F})$ denote a Riemann mapping. Since $\partial K(F)=\partial
U_{0}$ is a Jordan curve, $\varphi^{-1}\circ H$ extends to a homeomorphism
from $\mathbb{C}\setminus K(F)$ onto $\mathbb{C}\setminus\mathbb{D}$. Let
$\zeta(\theta)$ and $\widetilde{\zeta}(\theta)$ denote the landing point of
the external ray $\mathcal{R}_{f_{0}}(\theta)$ and
$\mathcal{R}_{\widetilde{f}}(\theta)$ of angle
$\theta\in\mathbb{Q}/\mathbb{Z}$ respectively. Note that
$\varphi^{-1}(\mathcal{R}_{\widetilde{f}}(s))=\varphi^{-1}\circ
H(\mathcal{R}_{f_{0}}(s))$ lands at $x=\varphi^{-1}\circ
H(\zeta(s))=\varphi^{-1}\circ H(\zeta(t))$ and
$\mathcal{R}_{\widetilde{f}}(s)$ lands at $\widetilde{\zeta}(s)$. By
Lindel\"{o}f Theorem (see \cite[cite]{[\@@bibref{}{McM1}{}{},
Theorem~{}6.3]}), the hyperbolic geodesic $\varphi([0,x])$ lands at
$\widetilde{\zeta}(s)$. In the same manner we can see that $\varphi([0,x])$
also lands at $\widetilde{\zeta}(t)$. Thus
$\widetilde{\zeta}(s)=\widetilde{\zeta}(t)$, in other words,
$\mathcal{R}_{\widetilde{f}}(s)$ and $\mathcal{R}_{\widetilde{f}}(t)$ land at
a common point. \qed\end@proof
\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\@@numbered@section{section}{toc}{Quasiconformal
surgery} Recall $f_{0}$ is a primitive polynomial with a periodic critical
point $0$ of period $m_{0}$ and a Julia critical critical point $c_{0}$ at
which $f_{0}$ is non-renormalizable. Let $Z_{0}=\mathrm{orb}(\tau_{0})$ be
given by Theorem~{}\ref{thm:puzzle}. Fix $\kappa_{0}$ large enough so that the
following holds: \par\begin{itemize}
\par\itemize@item@$\overline{Y^{(\kappa_{0})}},\overline{f_{0}(Y^{(\kappa_{0})})},\ldots,\overline{f^{m_{0}-1}_{0}(Y^{(\kappa_{0})})}$
are disjoint; \par\itemize@item@$f^{m_{0}}_{0}:Y^{(\kappa_{0})}\to
Y^{(\kappa_{0}-m_{0})}$ is an APL map with filled Julia set
$\overline{U_{0}}$. \par\end{itemize} The existence of $\kappa_{0}$ follows
from Theorem~{}\ref{thm:puzzle} and the primitive property of $f_{0}$. \par
The aim of this section is to prove Theorem~{}B.
In~{}\cite[cite]{[\@@bibref{}{SW}{}{}]}, qc surgery and Thurston Algorithm was
applied successfully to construct the desired tuning directly. Unfortunately,
we cannot construct our desired tuning in our case directly. As the orbit of
the free critical point may accumulate on the boundary $\partial U_{0}$ of the
Fatou component $U_{0}$, it is impossible to respect all the combinatorics
when one does qc surgery. Instead, we first show that for an arbitrary given
depth there is a polynomial $P$ which has the same combinatorics as $f_{0}$ up
to this depth and has some other dynamical properties. More precisely, we
prove: \begin{Theorem}Let $f_{0}\in\mathcal{C}_{d_{1}+d_{2}}$ be a primitive
polynomial with a periodic critical point $0$ of multiplicity $d_{1}-1$ and a
Julia critical point $c_{0}$ of multiplicity $d_{2}$. Assume $f_{0}$ has no
neutral cycle and is not renormalizable at $c_{0}$. Fix an internal angle
system $\alpha$ for $f_{0}$ and choose an external angle $\theta_{0}$ such
that $\mathcal{R}_{f_{0}}(\theta_{0})$ lands at $\alpha^{-1}(0)$. Then for any
$k>\kappa_{0}$ and $g\in\mathcal{C}_{d_{1}}$, there exists a polynomial
$f_{k}\in\mathcal{P}_{d_{1}+d_{2}}$ with the following properties:
\begin{enumerate} \par\enumerate@item@there exists a qc weak pseudo-conjugacy
$\Phi_{k}$ between $f_{0}$ and $f_{k}$ up to depth $k$;
\par\enumerate@item@$\max\\{G_{f_{k}}(c^{\prime})\mid c^{\prime}~{}\text{is a
critical point of}~{}f_{k}\\}\leq d^{-k}$, where $d=d_{1}+d_{2}$;
\par\enumerate@item@$f_{k}^{m_{0}}:\Phi_{k}(Y^{(\kappa_{0})})\to\Phi_{k}(Y^{(\kappa_{0}-m_{0})})$
is an APL map with a polynomial-like restriction which is hybrid equivalent to
$g$ and the hybrid conjugacy $\varphi_{k}$ can be chosen so that $\varphi_{k}$
sends $\mathcal{R}_{f_{k}}(\theta_{0})$ to $\mathcal{R}_{g}(0)$.
\par\end{enumerate} \end{Theorem} \begin{Remark}In $(3)$ of
Theorem~{}\ref{thm:sequence}, the statement $\varphi_{k}$ sends
$\mathcal{R}_{f_{k}}(\theta_{0})$ to $\mathcal{R}_{g}(0)$ means that there
exist neighborhoods $W_{k}$ and $W$ of the landing points of
$\mathcal{R}_{f_{k}}(\theta_{0})$ and $\mathcal{R}_{g}(0)$ respectively such
that $\varphi_{k}$ maps the connected component of
$\overline{\mathcal{R}_{f_{k}}(\theta_{0})}\cap W_{k}$ which intersects
$J_{f_{k}}$ onto the connected component of $\overline{\mathcal{R}_{g}(0)}\cap
W$ which intersects $J_{g}$. \end{Remark} \par\par The proof is based on
quasiconformal surgery and techniques developed in~{}
\cite[cite]{[\@@bibref{}{SW}{}{}]}. Then we take a suitable limit and use some
combinatorial facts to show that: \par\begin{Theorem B}Under the assumptions
in the Theorem~{}A, $f_{0}$ can be tuned with any polynomial
$g\in\mathcal{C}_{d_{1}}$. \end{Theorem B}
\par\par\par\@@numbered@section{subsection}{toc}{Constructions of quasiregular
maps}Let us recall some terminologies and notations in
\cite[cite]{[\@@bibref{}{SW}{}{}]}, we say a quasiregular map $\widetilde{f}$
is a {\em quasi-polynomial} if $\widetilde{f}^{-1}(\infty)=\infty$ and
$\widetilde{f}$ is holomorphic in a neighborhood of $\infty$. An open set
$\mathcal{U}$ is called {\em nice with respect to $\widetilde{f}$} if each
component of $\mathcal{U}$ is a Jordan domain and
$\widetilde{f}^{k}(\partial\mathcal{U})\cap\mathcal{U}=\emptyset$ for each
$k\geq 1$. For a nice open set $U$, let
$$D(\mathcal{U},\widetilde{f})=\\{z\in\mathbb{C}:\exists n\geq 1\text{ such
that }\widetilde{f}^{n}(z)\in\mathcal{U}\\}$$ denote the domain of the first
entry map to $\mathcal{U}$ under $\widetilde{f}$. \par A nice open set
$\mathcal{U}$ is said to be {\em$\widetilde{f}$-free} if $\mathcal{U}\cap
P(\widetilde{f})=\emptyset$, where $P(\widetilde{f})$ is the closure of the
orbit of the ramification points of $P(\widetilde{f})$. \par\par Let
$\mathcal{B}$ be a nice open set with respect to $\widetilde{f}$. We say
$\mathcal{B}$ is {\em$M$-nice with respect to $\widetilde{f}$} if
$\mathcal{B}$ is nice with respect to $\widetilde{f}$ and for each component
$B$ of $\mathcal{B}$, the following three conditions hold:
\begin{equation}\mathrm{diam}(B)^{2}\leq M\mathrm{area}(B);\end{equation}
\begin{equation}\frac{\mathrm{area}(B\setminus
D(\mathcal{B},\widetilde{f}))}{\mathrm{area}(B)}>M^{-1};\end{equation}
\begin{equation}\mathcal{QD}(D(\mathcal{B},\widetilde{f})\cap B,B)\leq
M,\end{equation} where $\mathcal{QD}$ is as defined in \S\ref{sec:Kahn}.
\par\par\par\par\begin{Lemma} Let $g\in\mathcal{C}_{d_{1}}$. Then for any
positive integer $k>\kappa_{0}$, there is a quasi-polynomial
$\widetilde{f}_{k}$ of degree $d_{1}+d_{2}$ and an open set
$\mathcal{B}_{k}\Subset Y^{(k)}$ satisfying the following properties:
\begin{itemize} \par\itemize@item@$\widetilde{f}_{k}=f_{0}$ on
$\mathbb{C}\setminus Y^{(k)}$; \par\itemize@item@$\mathcal{B}_{k}$ is
$\widetilde{f}_{k}$-free and $M_{k}$-nice for some $M_{k}>0$;
\par\itemize@item@there exists a positive integer $T_{k}$ such that for every
$z$ and $n\geq 1$, if $\widetilde{f}_{k}^{j}(z)\not\in\mathcal{B}_{k}$ for
each $0\leq j<n$, then $\\#\\{0\leq
j<n:\widetilde{f}^{j}_{k}(z)\in\mathcal{A}_{k}\\}\leq T_{k}$, where
$\mathcal{A}_{k}=\\{z\mid\bar{\partial}\widetilde{f}_{k}(z)\neq 0\\}$;
\par\itemize@item@$\widetilde{F}_{k}=\widetilde{f}^{m_{0}}_{k}|_{Y^{(\kappa_{0})}}:Y^{(\kappa_{0})}\to
Y^{(\kappa_{0}-m_{0})}$ is a quasiregular map of degree $d_{1}$ with a
polynomial-like restriction $P_{k}$ which is hybrid equivalent to $g$.
Furthermore,
$K(\widetilde{F}_{k}):=\bigcap_{n=0}^{\infty}\widetilde{F}^{-n}_{k}(Y^{(\kappa_{0})})$
is disjoint from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k}\cup\mathcal{A}_{k})$
and each orbit of $\widetilde{F}_{k}$ passes through $\mathcal{A}_{k}$ at most
$T_{k}$ times. \par\itemize@item@there exists a
$\widetilde{f}_{k}^{m_{0}}$-invariant ray $\gamma_{k}$ landing on
$K(\widetilde{F}_{k})$ such that $\gamma_{k}\cap(\mathbb{C}\setminus
Y^{(\kappa_{0})})=\mathcal{R}_{f_{0}}(\theta_{0})$ and there exists a hybrid
conjugacy $\psi_{k}$ between $P_{k}$ and $g$ which sends $\gamma_{k}$ to
$\mathcal{R}_{g}(0)$. \par\end{itemize} \end{Lemma} We postpone the proof of
this lemma to the end of this subsection and show now how it implies
Theorem~{}\ref{thm:sequence}. \@proof[Proof of Theorem~\ref{thm:sequence}]
Note that $\widetilde{f}_{k}$ satisfies the assumption
of~{}\cite[cite]{[\@@bibref{}{SW}{}{}, Theorem~{}5]}. Thus it follows there
exists a polynomial $f_{k}\in\mathcal{P}_{d_{1}+d_{2}}$ and a continuous
surjection $h_{k}:\mathbb{C}\to\mathbb{C}$ such that $f_{k}\circ
h_{k}=h_{k}\circ\widetilde{f}_{k}$. Moreover, $h_{k}$ coincides with some qc
map $\lambda_{k}$ on
$\mathbb{C}\setminus\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k})$
and $h_{k}$ is comformal in a neighborhood of $\infty$. Since
$\widetilde{f}_{k}=f_{0}$ outside $Y^{(k)}$ and $h_{k}$ is a conformal
conjugacy between $f_{k}$ and $\widetilde{f}_{k}$ near infinity, $f_{k}\circ
h_{k}=h_{k}\circ f_{0}$ in a neighborhood of $\infty$. Thus
$h_{k}=\lambda_{k}$ is identity in the B\"{o}ttcher coordinate for $f_{0}$ and
$f_{k}$ near infinity. This implies the $f_{k}^{m_{0}}$-invariant ray
$h_{k}(\gamma_{k})$ is the external ray $\mathcal{R}_{f_{k}}(\theta_{0})$.
\par First we show that $\lambda_{k}$ is a qc weak pseudo-conjugacy between
$f_{0}$ and $f_{k}$ up to depth $k$. Let $\mathcal{Y}_{k}$ denote the set of
puzzle pieces of depth $k$ and let
$\mathcal{V}_{k}=\mathbb{C}\setminus\bigcup_{Y\in\mathcal{Y}_{k}}Y$. It
remains to show that $\lambda_{k}\circ f_{0}=f_{k}\circ\lambda_{k}$ on
$\mathcal{V}_{k}$. It suffices to show $\lambda_{k}=h_{k}$ on
$\mathcal{V}_{k}$, which follows easily from the fact that $\mathcal{V}_{k}$
is disjoint from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k})$. Indeed,
$\mathcal{V}_{k}$ is disjiont from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(Y^{(k)})$ since
$\widetilde{f}_{k}=f_{0}$ outside $Y^{(k)}$. \par Now we turn to prove that
$\max\\{G_{f_{k}}(c^{\prime})\mid c^{\prime}~{}\text{is a critical point
of}~{}f_{k}\\}\leq 1/d^{k}$, where $d=d_{1}+d_{2}$. Note that
$\phi:=\lambda^{-1}_{k}$ is qc and identity in the B\"{o}ttcher coordinate for
$f_{k}$ and $f_{0}$ on $\\{z\mid G_{f_{k}}(z)>1/d^{k}\\}$. Suppose, contrary
to our claim, that there is a critical point $c_{k}$ of $f_{k}$ with
$G_{f_{k}}(c_{k})>1/d^{k}$. Then there exists a small disk $D_{k}$ around
$c_{k}$ such that $G_{f_{k}}(z)>1/d^{k}$ for all $z\in D_{k}$. Then $\phi\circ
f_{k}=f_{0}\circ\phi$ on $D_{k}$ since $D_{k}$ is disjoint from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k})$. It follows
$\phi(c_{k})$ is a critical point of $f_{0}$ and
$G_{f_{0}}(\phi(c_{k}))=G_{f_{k}}(c_{k})>1/d^{k}>0$. Thus $\phi(c_{k})$ is in
the basin of infinity of $f_{0}$. This is absurd since $J_{f_{0}}$ is
connected. \par\par We proceed to show that
$F_{k}:=f^{m_{0}}_{k}:h_{k}(Y^{(\kappa_{0})})\to
h_{k}(Y^{(\kappa_{0}-m_{0})})$ is an APL map with a polynomial-like
restriction hybrid equivalent to $g$. Since $Y^{(\kappa_{0})}$ and
$Y^{(\kappa_{0}-m_{0})}$ are nice w.r.t. $\widetilde{f}_{k}$ and
$\mathcal{B}_{k}\Subset Y^{(\kappa_{0})}\subset Y^{(\kappa_{0}-m_{0})}$,
$\partial Y^{(\kappa_{0})}$ and $\partial Y^{(\kappa_{0}-m_{0})}$ lie outside
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k})$. Thus
$h_{k}=\lambda_{k}$ on $\partial Y^{(\kappa_{0})}\cup\partial
Y^{(\kappa_{0}-m_{0})}$. Moreover, $h_{k}=\lambda_{k}$ and
$\bar{\partial}\lambda_{k}=0$ a.e. on $K(\widetilde{F}_{k})$ since
$K(\widetilde{F}_{k})$ is disjoint from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k}\cup\mathcal{A}_{k})$.
Note that $\widetilde{f}_{k}^{-n}(\mathcal{B}_{k})$ is a countable union of
disjoint Jordan domains which are nice w.r.t $\widetilde{f}_{k}$ and
$h_{k}=\lambda_{k}$ on
$\mathbb{C}\setminus\widetilde{f}_{k}^{-n}(\mathcal{B}_{k})$. It follows
$\lambda_{k}$ is homotopic to $h_{k}$ rel $K(\widetilde{F}_{k})$. Therefore,
by homotopy lifting, there is a sequence of qc maps
$H_{j}:Y^{(\kappa_{0}-m_{0})}\to\lambda_{k}(Y^{(\kappa_{0}-m_{0})})$ homotopic
to $h_{k}$ rel $K(\widetilde{F}_{k})$ such that $H_{j}=\lambda_{k}$ on
$Y^{(\kappa_{0}-m_{0})}\setminus\overline{Y^{(\kappa_{0})}}$ and $F_{k}\circ
H_{j+1}=H_{j}\circ\widetilde{F}_{k}$. Since $F_{k}$ is holomorphic and each
orbit of $\widetilde{F}_{k}$ can pass through $\mathcal{A}_{k}$ at most
$T_{k}$ times, the maximal dilatation of $H_{j}$ is uniformly bounded. As
$H_{j}(z)$ eventually stablizes for all $z\in Y^{(\kappa_{0}-m_{0})}$, $H_{j}$
converges to a qc map $H$. Clearly, $\bar{\partial}H=0$ holds a.e. on
$K(\widetilde{F}_{k})$ since so does $H_{j}$ for all $j$. Assume
$P_{k}=\widetilde{F}_{k}|_{\Omega_{k}}:\Omega_{k}\to\Omega^{\prime}_{k}$ is a
polynomial-like restriction of $\widetilde{F}_{k}$ which is hybrid equivalent
to $g$ and $\psi_{k}$ is a hybrid conjugacy between $P_{k}$ and $g$ which
sends $\gamma_{k}$ to $\mathcal{R}_{g}(0)$ . Then
$F_{k}|_{H(\Omega_{k})}:H(\Omega_{k})\to H(\Omega^{\prime}_{k})$ is also a
polynomial-like map hybrid equivalent to $g$. Note that $H(\gamma_{k}\cap
Y^{(\kappa_{0})})=h_{k}(\gamma_{k})\cap H(Y^{(\kappa_{0})})$, which is a part
of the external ray $\mathcal{R}_{f_{k}}(\theta_{0})$. Thus $\psi_{k}\circ
H^{-1}$ sends the external ray $\mathcal{R}_{f_{k}}(\theta_{0})$ to
$\mathcal{R}_{g}(0)$. \qed\end@proof \par Now we prove
Lemma~{}\ref{lem:surgery} to finish this subsection. \par\@proof[Proof of
Lemma~\ref{lem:surgery}] Since $f_{0}$ is primitive and non-renormalizable at
$c_{0}$, the critical orbit
$\mathrm{orb}(c_{0})\cap\overline{U_{0}}=\emptyset$. By
Theorem~{}\ref{thm:puzzle}, $Y^{(n)}$ shrinks to $\overline{U_{0}}$ as $n$
tends to infinity. Thus for any positive integer $k>\kappa_{0}$, we can choose
a positive integer $\ell=\ell(k)>k$ with the following properties:
\begin{itemize}
\par\itemize@item@$\overline{Y^{(\ell)}},\overline{f_{0}(Y^{(\ell)})},\ldots,\overline{f^{m_{0}-1}_{0}(Y^{(\ell)})}$
are disjoint; \par\itemize@item@$f^{m_{0}}_{0}:Y^{(\ell+m_{0})}\to Y^{(\ell)}$
is an APL map with filled Julia set $\overline{U_{0}}$. \par\end{itemize} By
Theorem~{}\ref{thm:Kqcpuzzle'}, there exists a positive integer
$N_{0}=N_{0}(\ell)>\ell+m_{0}$ such that $$\max\\{\mathcal{QD}(L_{N_{0}}\cap
Y^{(\ell)},Y^{(\ell)}),\mathcal{QD}(L_{N_{0}}\cap
X^{(\ell)},X^{(\ell)}\\}<\infty.$$ It follows from Theorem~{}\ref{thm:puzzle}
that there exists a positive integer $i_{0}$ such that
$Y^{(N_{0}+i_{0}m_{0})}\Subset Y^{(N_{0})}$. Let $A$ denote the annuli
$Y^{(N_{0})}\setminus\overline{Y^{(N_{0}+i_{0}m_{0})}}$. \par\par{\bf Case
1.}There exists a first moment $s_{0}$ such that
$x_{0}:=f^{s_{0}}_{0}(c_{0})\in A$. \par We define a quasiregular map
$\widetilde{f}_{0}:A\to f_{0}(A)$ such that
$d^{-N_{0}}<G_{f_{0}}(\widetilde{f}_{0}(x_{0}))<{d^{-N_{0}+1}}$,
$d=d_{1}+d_{2}$ and $\widetilde{f}_{0}=f_{0}$ on
$\mathcal{R}_{f_{0}}(\theta_{0})\cap A$ as following. Fix $y_{0}\in
f_{0}(A^{\prime})$ with $G_{f_{0}}(y_{0})>d^{-N_{0}}$. Define
$\widetilde{f}_{0}=f_{0}$ on $\partial
A\cup(\mathcal{R}_{f_{0}}(\theta_{0})\cap A)$ and interpolate quasiregularly
on $A$ so that $\widetilde{f}_{0}(x_{0})=y_{0}$. \par Now choose $r_{0}$ large
enough so that
$\\{c_{0},f_{0}(c_{0}),\ldots,f^{s_{0}}_{0}(c_{0})\\}\cap\overline{Y^{(r_{0})}}=\emptyset$.
Note that $f^{m_{0}}_{0}:Y^{(r_{0}+m_{0})}\to Y^{(r_{0})}$ is an APL map with
filled Julia set $\overline{U_{0}}$. Since $\overline{U_{0}}\Subset
Y^{(r_{0}+m_{0})}$, by \cite[cite]{[\@@bibref{}{LY}{}{}, Lemma~{}2.4]}, there
exist quasidisks $U\Subset V\subset Y^{(r_{0}+m_{0})}$ such that
$f^{m_{0}}_{0}:U\to V$ is a polynomial-like map with filled Julia set
$\overline{U_{0}}$. Then take a polynomial-like restriction
$g|_{\widehat{W}}:\widehat{W}\to\widehat{U}$ of $g$. The quasidisk $U$ can be
chosen so that $\mathcal{R}_{f_{0}}(\theta_{0})$ intersects $\partial U$
transversally (at a unique point). Let $\xi$ denote the corresponding
intersecting point. Similarly, we can assume $\mathcal{R}_{g}(0)$ intersects
$\partial\widehat{U}$ at a unique point $\zeta$. Let
$\xi^{\prime}\in\mathcal{R}_{f_{0}}(\theta_{0})$ and
$\zeta^{\prime}\in\mathcal{R}_{g}(0)$ so that
$f^{m_{0}}_{0}(\xi^{\prime})=\xi$ and $g(\zeta^{\prime})=\zeta$. Let
$\varphi:U\to\widehat{U}$ be the unique conformal map from $U$ onto
$\widehat{U}$ with $\varphi(\xi^{\prime})=\zeta^{\prime}$ and
$\varphi(\xi)=\zeta$. Set $W=\varphi^{-1}(\widehat{W})$. \par\par\par\par\par
Let us define $\widehat{F}_{k}=\varphi^{-1}\circ g\circ\varphi$ on
$\overline{W}$ and $\widehat{F}_{k}=f^{m_{0}}_{0}$ on $\partial U$. Then we
interpolate quasiregularly on $U\setminus\overline{W}$ so that
$\widehat{F}_{k}(\varphi^{-1}(\mathcal{R}_{g}(0)\cap(\widehat{U}\setminus\widehat{W})))=f^{m_{0}}_{0}(\mathcal{R}_{f_{0}}(\theta_{0}))$.
Let $\mathcal{I}$ denote the inverse map of the first entry map
$f^{m_{0}-1}_{0}:f_{0}(U)\to V$ which is conformal. Then we define
$\widetilde{f}_{k}$ as following:
$$\widetilde{f}_{k}(z)=\begin{cases}\widetilde{f}_{0}(z),&z\in A\\\
\mathcal{I}\circ\widehat{F}_{k}(z),&z\in U\\\
f_{0}(z),&z\in\mathbb{C}\setminus(A\cup U).\end{cases}$$ Note that
$\widetilde{f}_{k}(\varphi^{-1}(\mathcal{R}_{g}(0)\cap(\widehat{U}\setminus\widehat{W})))=f_{0}(\mathcal{R}_{f_{0}}(\theta_{0}))$.
\par In this case, $\bar{\partial}\widetilde{f}_{k}=0$ holds almost everywhere
outside $\mathcal{A}_{k}={A}\cup U\setminus\overline{W}$. Let
$R(\mathcal{A}_{k})$ denote the return domian to $\mathcal{A}_{k}$ under
$\widetilde{f}_{k}$. For any $z\in\mathcal{A}_{k}$, there exists a smallest
positive integer $\iota(z)$ such that $\widetilde{f}_{k}^{\iota(z)}(z)\in
Y^{(\ell)}\setminus\overline{Y^{(\ell+m_{0})}}$. It is easy to see
$T:=\sup_{z\in\mathcal{A}_{k}}\iota(z)<\infty$. Set $$\Omega:=\\{z\in
Y^{(\ell)}\setminus\overline{Y^{(\ell+m_{0})}}\mid\exists j>1~{}\text{such
that}~{}\widetilde{f}_{k}^{j}(z)\in Y^{(\ell)}\cup X^{(\ell)}\\}.$$ Let
${\mathcal{B}}_{k}$ be the union of the component of $D(\Omega)$ which
intersects $R(\mathcal{A}_{k})$, where $D(\Omega)$ is the domian of the first
entry map to $\Omega$ under $\widetilde{f}_{k}$. \par\vskip 6.0pt plus 2.0pt
minus 2.0pt{\bf Fact 1.} {\em$\mathcal{B}_{k}$ is $\widetilde{f}_{k}$-free}.
\par Clearly, $\mathcal{B}_{k}\subset Y^{(N_{0})}$ since $A\subset
Y^{(N_{0})}$ and $\partial Y^{(N_{0})}$ is nice with respect to
$\widetilde{f}$. Note that
$\widetilde{f}_{k}^{j}(c_{0})=f^{j-s_{0}-1}_{0}\circ\widetilde{f}_{0}(x_{0})$
for all $j>s_{0}$ by the definition of $\widetilde{f}_{k}$. Thus
$G_{f_{0}}(\widetilde{f}_{k}^{j}(c_{0}))>d^{-N_{0}}$ for all $j>s_{0}$, which
implies $\widetilde{f}_{k}^{j}(c_{0})\notin\overline{Y^{(N_{0})}}$ for all
$j>s_{0}$. So
$Y^{(N_{0})}\cap\\{\widetilde{f}^{j}(c_{0})\\}_{j\in\mathbb{N}}=\\{x_{0}\\}$
by the definition of $s_{0}$ and $\widetilde{f}_{k}$. It suffices to show
$x_{0}\notin\overline{\mathcal{B}_{k}}$. If
$x_{0}\in\overline{\mathcal{B}_{k}}$, then there exists a sequence
$\\{z_{j}\\}\subset\mathcal{B}_{k}$ such that $z_{j}\to x_{0}$. So
$\widetilde{f}_{k}(z_{j})\to f_{0}(y_{0})$. Since
$d^{-N_{0}}<G_{f_{0}}(y_{0})<d^{N_{0}+1}$,
$d^{-N_{0}}<G_{f_{0}}(\widetilde{f}_{k}(z_{j}))<d^{-N_{0}+1}$ for all $j$
large. Thus the orbit of $\widetilde{f}_{k}(z_{j})$ under $\widetilde{f}_{k}$
can never pass through $Y^{\ell}$ after it enters into
$Y^{(\ell)}\setminus\overline{Y^{(\ell+m_{0})}}$. This implies
$\widetilde{f}_{k}(z_{j})\notin D(\Omega)$ and so does $z_{j}$ for $j$ large,
a contradiction. \par\vskip 6.0pt plus 2.0pt minus 2.0pt\par{\bf Fact 2.}
{\em$\mathcal{B}_{k}$ is $M_{k}$-nice with respect to $\widetilde{f}_{k}$ for
some $M_{k}>0$}. \par We first observe that $\Omega$ and hence
$\mathcal{B}_{k}$ is nice with respect to $\widetilde{f}_{k}$. Fix a component
$B$ of $\mathcal{B}_{k}$, let $s^{\prime}$ be the first entry time of $B$ into
$\Omega$, and let $t$ denote the return time of
$\widetilde{f}_{k}^{s^{\prime}}(B)$ into $Y^{(\ell)}\cup X^{(\ell)}$. Then
$\widetilde{f}_{k}^{s^{\prime}+t}$ maps $B$ homeomorphically onto a component
of $Y^{(\ell)}\cup X^{(\ell)}$ and the map
$\widetilde{f}_{k}^{s^{\prime}+t}|_{B}$ is $K^{T}$-qc, where $K$ is the
maximal dilatation of $\widetilde{f}_{k}$. Indeed, $s^{\prime}\leq T$ since
$B$ intersect $\mathcal{A}_{k}$. Note that
$\widetilde{f}_{k}^{s^{\prime}+t}(D(\mathcal{B}_{k})\cap B)\subset L_{N_{0}}$
since $\mathcal{B}_{k}\subset Y^{(N_{0})}$. It follows from Lemma~{}\ref{qc1}
and Theorem~{}\ref{thm:Kqcpuzzle'} that $$\mathcal{QD}(D(\mathcal{B})\cap
B,B)\leq(K^{T})^{2}\max\\{\mathcal{QD}(L_{N_{0}}\cap
Y^{(\ell)},Y^{(\ell)}),\mathcal{QD}(L_{N_{0}}\cap
X^{(\ell)},X^{(\ell)})\\}=:M^{\prime}_{k}<\infty.$$ Note that there is $M>0$
such that $$M^{2}\mathrm{area}(Y^{(\ell)}\setminus L_{N_{0}})\geq
M\mathrm{area}(Y^{(\ell)})\geq\mathrm{diam}(Y^{(\ell)})^{2}$$ and
$$M^{2}\mathrm{area}(X^{(\ell)}\setminus L_{N_{0}})\geq
M\mathrm{area}(X^{(\ell)})\geq\mathrm{diam}(X^{(\ell)})^{2}$$ since
$Y^{(\ell)}$ and $X^{(\ell)}$ are quasidisks and $(Y^{(\ell)}\setminus
L_{N_{0}})\cap D_{f_{0}}(\infty)\neq\emptyset$, $(X^{(\ell)}\setminus
L_{N_{0}})\cap D_{f_{0}}(\infty)\neq\emptyset$. As
$\widetilde{f}^{s^{\prime}+t}_{k}|_{B}$ has a $K^{T}$-qc extension from $B$
onto a neighborhor of $\widetilde{f}^{s^{\prime}+t}_{k}(B)$, there is
$M^{\prime}>0$ (only depends on $M$ and $K^{T}$) such that $$M^{\prime
2}\mathrm{area}(B)\setminus D(\mathcal{B}_{k}))\geq
M^{\prime}\mathrm{area}(B)\geq\mathrm{diam}(B)^{2}$$ by area distortion of
$K^{T}$-qc maps. Indeed, $\partial Y^{(\ell)}$ and $\partial X^{(\ell)}$ does
not intersect the orbit of the ramification points of $\widetilde{f}_{k}$.
Thus there are definite neighborhoods $V(0)$ and $V(c_{0})$ of $\partial
Y^{(\ell)}$ and $\partial X^{(\ell)}$ respectively so that
$P(\widetilde{f}_{k})\cap(V(0)\setminus Y^{(\ell)})=\emptyset$ and
$P(\widetilde{f}_{k})\cap(V(c_{0})\setminus X^{(\ell)})=\emptyset$. Hence
$\widetilde{f}^{s^{\prime}+t}_{k}|_{B}$ has a $K^{T}$-qc extension from $B$
onto a definite neighborhood of $Y^{(\ell)}$ or $X^{(\ell)}$. Let
$M_{k}=\max\\{M^{\prime}_{k},M^{\prime}\\}$. Then $\mathcal{B}_{k}$ is
$M_{k}$-nice. \par\vskip 6.0pt plus 2.0pt minus 2.0pt\par{\bf Fact 3.} {\em
For every $z$ and $n\geq 1$, if
$\widetilde{f}_{k}^{j}(z)\not\in\mathcal{B}_{k}$ for each $0\leq j<n$, then
$\\#\\{0\leq j<n:\widetilde{f}^{j}_{k}(z)\in\mathcal{A}_{k}\\}\leq 1$.}
\par\par By the definition of $\mathcal{B}_{k}$,
$R(\mathcal{A}_{k})\subset\mathcal{B}_{k}$. So if
$\widetilde{f}_{k}^{j}(z)\not\in\mathcal{B}_{k}$ for each $0\leq j<n$, then
$\widetilde{f}_{k}^{j}(z)\not\in R(\mathcal{A}_{k})$ for each $0\leq j<n$.
This implies $\widetilde{f}_{k}^{j}(z)\not\in\mathcal{A}_{k}$ for each $1\leq
j<n$. \par\par\vskip 6.0pt plus 2.0pt minus 2.0pt\par\par\par{\bf Fact 4.}
{\em$\widetilde{F}_{k}:=\widetilde{f}^{m_{0}}_{k}:Y^{(\kappa_{0})}\to
Y^{(\kappa_{0}-m_{0})}$ is a quasiregular map of degree $d_{1}$ with a
polynomial-like restriction which is hybrid equivalent to $g$. Moreover,
$K(\widetilde{F}_{k}):=\bigcap_{n=0}^{\infty}\widetilde{F}^{-n}_{k}(W_{k})$ is
disjoint from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k}\cup\mathcal{A}_{k})$
and each orbit of $\widetilde{F}_{k}$ passes through $\mathcal{A}_{k}$ at most
$T+1$ times.} \par Since $\widetilde{f}_{k}:Y^{(\kappa_{0})}\to
f_{0}(Y^{(\kappa_{0})})$ is a qusiregular map of degree $d_{1}$ and
$f^{m_{0}-1}_{0}:f_{0}(Y^{(\kappa_{0})})\to Y^{(\kappa_{0}-m_{0})}$ is a
univalent map, $\widetilde{f}^{m_{0}}_{k}:Y^{(\kappa_{0})}\to
Y^{(\kappa_{0}-m_{0})}$ is a quasiregular map of degree $d_{1}$. Note that
$\widetilde{f}^{m_{0}}_{k}|_{W}=\widehat{F}_{k}|_{W}=\varphi^{-1}\circ
g\circ\varphi|_{W}$ is conformally conjugate to $g$ and
$K(\widetilde{F}_{k})=K(\widehat{F}_{k}|_{W})=\varphi^{-1}(K(g))$. Clearly,
$\varphi^{-1}(K(g))$ is disjoint from
$\bigcup_{n=0}^{\infty}\widetilde{f}_{k}^{-n}(\mathcal{B}_{k}\cup\mathcal{A}_{k})$.
For each point $z\in Y^{(\kappa_{0})}\setminus K(\widetilde{F}_{k})$, its
orbit will first pass through $Y^{(\ell)}\setminus\overline{Y^{(\ell+m_{0})}}$
and eventually pass through
$Y^{(\kappa_{0}-m_{0})}\setminus\overline{Y^{(\kappa_{0})}}$ under
$\widetilde{F}_{k}$. Let $t^{\prime}$ be the first moment so that
$\widetilde{F}_{k}^{t^{\prime}}(z)\in\mathcal{A}_{k}$. Then the orbit of
$\widetilde{F}_{k}^{t^{\prime}}(z)$ under $\widetilde{F}_{k}$ can pass through
$\mathcal{A}_{k}$ at most $\iota(F_{k}^{t^{\prime}}(z))\leq T$ times. \par\par
Let $\gamma_{k}$ denote the union of $\mathcal{R}_{f_{0}}(\theta_{0})\setminus
U$ and $\varphi^{-1}(\mathcal{R}_{g}(0))\cap U$. Clearly, $\varphi$ is a
hybrid conjugacy between $\widehat{F}_{k}|_{W}$ to $g$ which sends
$\gamma_{k}$ to $\mathcal{R}_{g}(0)$. \vskip 6.0pt plus 2.0pt minus
2.0pt\par\par{\bf Case 2.} The critical orbit $\mathrm{orb}(c_{0})$ does not
pass through the annulus $A$. Then there exists $\ell^{\prime}$ large such
that
$Y^{(\ell^{\prime})}\cap\overline{\\{c_{0},f_{0}(c_{0}),\ldots\\}}=\emptyset$.
By Theorem~{}\ref{thm:Kqcpuzzle'}, there exists a positive integer
$N^{\prime}_{0}=N^{\prime}_{0}(\ell^{\prime})>\ell^{\prime}+m_{0}$ such that
$\mathcal{QD}(L_{N^{\prime}_{0}}\cap
Y^{(\ell^{\prime})},Y^{(\ell^{\prime})})<\infty$. \par Now choose
$r_{0}>N^{\prime}_{0}$. Without loss of generality, we assume $r_{0}$ is the
same as in Case~{}1. Let $W,U,\widehat{W},\widehat{U}$, $\widehat{F}_{k}$ and
$\mathcal{I}$ be as in Case~{}1. This time, we define
$$\widetilde{f}_{k}(z)=\begin{cases}\mathcal{I}\circ\widehat{F}_{k}(z),&z\in
U\\\ f_{0}(z),&z\in\mathbb{C}\setminus U.\end{cases}$$ In this case,
$\bar{\partial}\widetilde{f}_{k}=0$ holds almost everywhere outside
$\mathcal{A}_{k}=U\setminus\overline{W}$. Again let $R(\mathcal{A}_{k})$
denote the return domian to $\mathcal{A}_{k}$ under $\widetilde{f}_{k}$. Let
$$\Omega^{\prime}:=\\{z\in
Y^{(\ell^{\prime})}\setminus\overline{Y^{(\ell^{\prime}+m_{0})}}\mid\exists
j>1~{}\text{such that}~{}\widetilde{f}_{k}^{j}(z)\in Y^{(\ell^{\prime})}\\}$$
and let $\mathcal{B}_{k}$ be the union of the component of
$D(\Omega^{\prime})$ which intersects $R(\mathcal{A}_{k})$, where
$D(\Omega^{\prime})$ is the domian of the first entry map to $\Omega^{\prime}$
under $\widetilde{f}_{k}$. Similar to Case~{}1, one can check
$\widetilde{f}_{k}$ is a desired map in Lemma~{}\ref{lem:surgery}.
\qed\end@proof
\par\par\par\par\par\par\par\par\par\par\@@numbered@section{subsection}{toc}{Proof
of Theorem~B} Fix $g\in\mathcal{C}_{d_{1}}$. It follows from
Theorem~{}\ref{thm:sequence} that there exists a sequence $\\{f_{k}\\}$ of
polynomials and a sequence of $\\{\Phi_{k}\\}$ of qc maps satisfies the
following: \begin{enumerate} \par\enumerate@item@$\Phi_{k}$ is a qc weak
pseudo-conjugacy between $f_{0}$ and $f_{k}$ up to depth $k$;
\par\enumerate@item@$\max\\{G_{f_{k}}(c^{\prime})\mid c^{\prime}~{}\text{is a
critical point of}~{}f_{k}\\}\leq d^{-k}$;
\par\enumerate@item@$f_{k}^{m_{0}}:\Phi_{k}(Y^{(\kappa_{0})})\to\Phi_{k}(Y^{(\kappa_{0}-m_{0})})$
is an APL map with a polynomial-like restriction which is hybrid equivalent to
$g$. \par\end{enumerate} By~{}\cite[cite]{[\@@bibref{}{BH}{}{},
Proposition~{}3.6]}, $\\{f_{k}\\}$ lies in a compact set. Thus $\\{f_{k}\\}$
has a convergent subsequence. Without loss of generality, we assume $f_{k}\to
f$ as $k\to\infty$. We now finish the proof of Theorem~{}B by claiming that
$f$ is our desired tuning. \par\@proof[Proof of Theorem~B] Note that
$G_{f}(c)=0$ for any critical point $c$ of $f$ since
$$\max\\{G_{f_{k}}(c^{\prime})\mid c^{\prime}~{}\text{is a critical point
of}~{}f_{k}\\}\leq d^{-k}$$ holds for all $k\in\mathbb{N}$. Thus,
$f\in\mathcal{C}_{d_{1}+d_{2}}$. Let
$\mathcal{E}=\\{E_{j}\\}_{j\in\mathbb{N}}$ be the set of all the equivalence
classes of $\lambda_{f_{0}}$. We say $E\in\mathcal{E}$ is admissible if
$\mathcal{R}_{f}(t)$ does not land at the grand orbit of a parabolic periodic
point or a critical point of $f_{0}$ or $f$ for all $t\in E$. Let
$\mathcal{E}^{\prime}:=\\{E\in\mathcal{E}\mid E~{}\text{is admissible}\\}$. We
claim that for each $E\in\mathcal{E}^{\prime}$, $E$ is contained in an
equivalence class of $\lambda_{f}$. \par Assume $t,s\in
E\in\mathcal{E}^{\prime}$, we prove that $\mathcal{R}_{f}(t)$ and
$\mathcal{R}_{f}(s)$ land at a common point. Let $\zeta$ and $\xi$ be the
landing point of $\mathcal{R}_{f}(t)$ and $\mathcal{R}_{f}(s)$ respectively.
By the definition of $\mathcal{E}^{\prime}$, $\zeta$ and $\xi$ are repelling
periodic point of $f$. By holomorphic motion, we know there exists
$\zeta_{k}\to\zeta$ and $\xi_{k}\to\xi$ so that $\mathcal{R}_{f_{k}}(t)$ and
$\mathcal{R}_{f_{k}}(s)$ land at $\zeta_{k}$ and $\xi_{k}$ respectively for
all $k$ larege (see also ~{}\cite[cite]{[\@@bibref{}{GJ}{}{}, Lemma~{}B.1]}).
\par{\bf Case 1.} $\mathcal{R}_{f_{0}}(t)$ does not land on
$\bigcup\limits_{n=0}^{\infty}f^{-n}_{0}(\partial U_{0})$. Recall that $U_{0}$
is the Fatou component of $f_{0}$ which contains $0$. In this case, there
exists $k$ large such that
$f^{n}_{0}(\mathcal{R}_{f_{0}}(t)\cup\mathcal{R}_{f_{0}}(s))$ lies outside
$Y^{(k)}\cup X^{(k)}$ for all $n\in\mathbb{N}$. It is well known that we can
adjust the weak pseudo-conjugacy $\Phi_{k}$ to be a pseudo conjugacy. See
\cite[cite]{[\@@bibref{}{AKLS}{}{}, Lemma~{}4.3]} and
\cite[cite]{[\@@bibref{}{KSS}{}{}, section~{}5]} for examples, see also
Lemma~{}\ref{lem:pseudo}. Thus there is no loss of generality in assuming
$\Phi_{k}$ is pseudo-conjugacy between to $f_{0}$ and $f_{k}$ up to depth $k$.
This implies that $\mathcal{R}_{f}(t)=\Phi^{-1}_{k}(\mathcal{R}_{f_{0}}(t))$
and $\mathcal{R}_{f}(s)=\Phi^{-1}_{k}(\mathcal{R}_{f_{0}}(s))$ land at a
common point for all $k$ large. So $\zeta_{k}=\xi_{k}$ for $k$ large. Hence
$\zeta=\xi$. \par\par{\bf Case 2.} $\mathcal{R}_{f_{0}}(t)$ lands on $\partial
U_{0}$. It follows from Lemma~{}\ref{lem: landing on boundary} that
$\zeta_{k}=\xi_{k}$ for all $k$, and so $\zeta=\xi$. \par{\bf Case 3.}
$\mathcal{R}_{f_{0}}(t)$ lands on $f^{-k_{0}}_{0}(\partial U_{0})$ for some
$k_{0}\geq 1$. Without loss of generality, we assume $k_{0}$ is the first
moment so that $f^{k_{0}}_{0}(\mathcal{R}_{f_{0}}(t))$ lands on $\partial
U_{0}$. Let $k_{1}$ be the smallest positive integer such
$f^{j}_{0}(\mathcal{R}_{f_{0}}(t)\cup\mathcal{R}_{f_{0}}(s))$ lies outside
$Y^{(k_{1})}\cup X^{(k_{1})}$ for $0\leq j<k_{0}-1$. Such an integer $k_{1}$
exists since $Y^{(n)}$ shrinks to $\overline{U_{0}}$ and $X^{(n)}$ shrinks to
$\\{c_{0}\\}$. Let $X$ be the puzzle piece of depth $k_{0}+k_{1}$ which
contains the landing point of $\mathcal{R}_{f_{0}}(t)$. Then
$f^{k_{0}}_{0}:X\to Y^{(k_{1})}$ is a conformal map. Note that
$f^{k_{0}}_{k}:\Phi^{-1}_{k}(X)\to\Phi^{-1}_{k}(Y^{(k_{1})})$ is a holomorphic
proper map for all $k$ large since $\Phi_{k}$ is a weak pseudo conjugacy
between $f_{0}$ and $f_{k}$ up to depth $k$. Since $\Phi_{k}\circ
f^{k_{0}}_{m}=f^{k_{0}}_{k}\circ\Phi_{k}$ on $\partial X$,
$f^{k_{0}}_{k}|_{\partial\Phi^{-1}_{k}(X)}$ is a homeomorphism for $k$ large.
Thus $f^{k_{0}}_{k}:\Phi^{-1}_{k}(X)\to\Phi^{-1}_{k}(Y^{(k_{0})})$ is
conformal. Moreover, $f^{k_{0}}_{k}|_{\Phi^{-1}_{k}(X)}$ maps the landing
points of $\mathcal{R}_{f_{k}}(t)$ and $\mathcal{R}_{f_{k}}(s)$ to a common
point which is a common landing point of
$f^{k_{0}}_{k}(\mathcal{R}_{f_{k}}(t))$ and
$f^{k_{0}}_{k}(\mathcal{R}_{f_{k}}(s))$ (see Case~{}2). This implies
$\mathcal{R}_{f_{k}}(t)$ and $\mathcal{R}_{f_{k}}(s)$ land at a common point
for all $k$ large, and so do $\mathcal{R}_{f}(t)$ and $\mathcal{R}_{f}(s)$,
since $f^{k_{0}}_{k}|_{\Phi^{-1}_{k}(X)}$ is conformal for $k$ large. \par\par
By~{}\cite[cite]{[\@@bibref{}{IK}{}{}, Lemma~{}8.3]}, $\lambda_{f_{0}}$ is the
smallest equivalence relation in $\mathbb{Q}/\mathbb{Z}$ contains
$(\bigcup_{E\in\mathcal{E}^{\prime}}E)\times(\bigcup_{E\in\mathcal{E}^{\prime}}E)$.
Thus $\lambda_{f_{0}}\subset\lambda_{f}$. \par It remains to show there exists
quasidisks $U\Subset V$ and a qc map $\varphi:\mathbb{C}\to\mathbb{C}$ such
that $F:=f^{m_{0}}|_{U}:U\to V$ is a polynomial-like map with filled Julia set
$K(F)=K(U_{0},f)$ which is hybrid equivalent to $g$ and $\varphi$ is a hybrid
conjugacy between $F$ and $g$ respecting the external markings. \par Let
$\Theta$ denote the angles $\eta$ for which $\mathcal{R}_{f_{0}}(\eta)$ lands
at $\tau_{0}$. By Lemma~{}\ref{lem:tau repelling}, for each $\eta\in\Theta$,
the landing point of $\mathcal{R}_{f}(\eta)$ is a repelling periodic point of
$f$. By Lemma~{}\ref{lem:weak pseudo}, there is a qc weak pseudo-conjugacy
$\Psi$ between $f_{0}$ and $f$ up to depth $\kappa_{0}$. Note that
$f^{m_{0}}_{0}:Y^{(\kappa_{0})}\to Y^{(\kappa_{0}-m_{0})}$ is holomorphic
proper map of degree $d_{1}$, so is
$F:=f^{m_{0}}:\Psi(Y^{(\kappa_{0})})\to\Psi(Y^{(\kappa_{0}-m_{0})})$. Since
$J_{f}\cap\partial\Psi(Y^{(\kappa_{0})})$ are repelling (pre-)periodic points
of $f$, by thickening technique~{}\cite[cite]{[\@@bibref{}{Mil4}{}{}]}, there
are quasidisks $\Psi(Y^{(\kappa_{0})})\subset
U,\Psi(Y^{(\kappa_{0}-m_{0})})\subset V$ such that $f^{m_{0}}:U\to V$ is a
polynomial-like map of degree $d_{1}$. Similarly, there are quasidisks
$\Phi_{k}(Y^{(\kappa_{0})})\subset
U_{k},\Phi_{k}(Y^{(\kappa_{0}-m_{0})})\subset V_{k}$ such that
$F_{k}:=f^{m_{0}}_{k}:U_{k}\to V_{k}$ is a polynomial-like map of degree
$d_{1}$. Moreover, we can choose $U_{k}$, $V_{k}$ carefully so that
$\mathrm{mod}(V_{k}\setminus\overline{U_{k}})\geq\mathrm{mod}(V\setminus\overline{U})/2$
for $k$ large since $f_{k}\to f$. By Theorem~{}\ref{thm:sequence}, there
exists qc maps $\psi_{k}:\mathbb{C}\to\mathbb{C}$ such that $\psi_{k}$ is a
hybrid conjugacy between $F_{k}$ and $g$ and $\psi_{k}$ sends the homotopy
class of $\mathcal{R}_{f_{k}}(\theta_{0})$ to the homotopy class of
$\mathcal{R}_{g}(0)$. Note that we can assume $\psi_{k}$ are uniformly $K$-qc
for some $K\geq 1$ since
$\mathrm{mod}(V_{k}\setminus\overline{U_{k}})\geq\mathrm{mod}(V\setminus\overline{U})/2$
for all $k\in\mathbb{N}$. By the compactness of normalized $K$-qc maps,
without loss of generality, we may assume $\psi_{k}$ converges uniformly to
some $K$-qc map $\varphi$. Note that
$F_{k}\circ\psi^{-1}_{k}=\psi^{-1}_{k}\circ g$ near $K(g)$ and
$\bar{\partial}\psi^{-1}_{k}=0$ a.e. on $K(g)$. Thus
$F\circ\varphi^{-1}=\varphi^{-1}\circ g$ near $K(g)$ and
$\bar{\partial}\varphi^{-1}=0$ a.e. on $K(g)$. This implies $\varphi$ is a
hybrid conjugacy between $F$ and $g$. \par We now proceed to show that
$\varphi$ sends the homotopy class of $\mathcal{R}_{f}(\theta)$ to the
homotopy class of $\mathcal{R}_{g}(0)$. One can easily check that
$\overline{\mathcal{R}_{f_{k}}(\theta_{0})}$ converges uniformly to
$\overline{\mathcal{R}_{f}(\theta_{0})}$ in the sense of Hausdorff topology by
using the fact that $\psi_{k}$ converges uniformly to $\varphi$ and the local
dynamical property near the landing point of $\mathcal{R}_{g}(0)$. Thus
$\varphi^{-1}\circ\psi_{k}(\mathcal{R}_{f_{k}}(\theta_{0}))$ is homotopic to
$\mathcal{R}_{f}(\theta_{0})$ in the sense of paths to $K(F)$ for $k$
sufficiently large. On the other hand,
$\varphi^{-1}\circ\psi_{k}(\mathcal{R}_{f_{k}}(\theta_{0}))$ is homotopic to
$\varphi^{-1}(\mathcal{R}_{g}(0))$ since
$\psi_{k}(\mathcal{R}_{f_{k}}(\theta_{0}))$ is homotopic to
$\mathcal{R}_{g}(0)$ by the definition of $\psi_{k}$. Thus
$\mathcal{R}_{f}(\theta_{0})$ is homotopic to
$\varphi^{-1}(\mathcal{R}_{g}(0))$. \par The proof is complete by showing
$K(F)=K(U_{0},f)$. This is easy by using the weak pesudo-conjugacies between
$f_{0}$ and $f$. The details are left to the readers. \par\par\qed\end@proof
\par\begin{Remark}There is another way to prove Theorem~{}B (suggested by
Hiroyuki Inou). One can first find a sequence
$\\{F_{k}\\}\subset\mathcal{C}_{d_{1}+d_{2}}$ of post-critically finite
polynomials such that $F_{k}$ has the same combinatorics as $f_{0}$ up to
depth $k$ for all $k\in\mathbb{N}$. See \cite[cite]{[\@@bibref{}{Kiwi}{}{},
Section~{}7]} for a combinatorial argument and this can also been done by qc
surgery. Then we use qc surgery to construct quasiregular maps
$\widetilde{F}_{k}$ satisfies all the conditions in Lemma~{}\ref{lem:surgery}
with $f_{0}$ and $\widetilde{f}_{k}$ replaced by $F_{k}$ and
$\widetilde{F}_{k}$ respectively. The construction is essentially the same as
that in Case~{}2 in the proof of Lemma~{}\ref{lem:surgery}.
By~{}\cite[cite]{[\@@bibref{}{SW}{}{}, Theorem~{}5.1]}, there exists
$f_{k}\in\mathcal{C}_{d_{1}+d_{2}}$ such that $f_{k}$ has the same
combinatorics as $f_{0}$ up to depth $k$ and $f_{k}^{m_{0}}$ has a polynomial-
like restriction which is hybrid equivalent to $g\in\mathcal{C}_{d_{1}}$.
Again, the limit $f$ of $f_{k}$ is our desired tuning. We mention here that
using this method one can prove Theorem~{}B more easily when $d_{1}=2$.
Indeed, $\mathcal{C}(\lambda_{F_{k}})$ is an analytic family for all
$k\in\mathbb{N}$. So one may apply Douady-Hubbard's Theorem to obtain $f_{k}$
when $d_{1}=2$. \end{Remark} \par\begin{Remark}To fix the idea and not to make
the notations too complicate, we only deal with the case that $f_{0}$ has one
periodic critical point and one Julia critical point. Actually the method of
the proof of Theorem~{}B also works for the following case. Let
$f\in\mathcal{C}_{d}$ be a polynomial with only periodic critical points and
non-renormalizable Julia critical points.
Following~{}\cite[cite]{[\@@bibref{}{IK}{}{}]}, one can define the reduced
mapping scheme $T(f)$ for $f$. It can be proved by the same method as in proof
of Theorem~{}B that $f$ can be tuned with any generalized polynomial
$g\in\mathcal{C}(T(f))$. The proof is left to the readers who are interested
in it. \end{Remark}
\par\par\par\par\@@numbered@section{section}{toc}{Bijectivity for the
Straightening map} Let $f_{0}$ and $\alpha$ be given as in the assumption of
Theorem~{}A. \par For any $f\in\mathcal{C}(\lambda_{f_{0}})$, by
Lemma~{}\ref{lem:weak pseudo}, there is a qc weak pseudo-conjugacy $\Psi_{f}$
between $f$ and $f_{0}$ up to depth $\kappa_{0}$. Note that
$f^{m_{0}}:Y^{(\kappa_{0})}\to Y^{(\kappa_{0}-m_{0})}$ is an APL map of degree
$d_{1}$, and so is
$F:=f^{m_{0}}:\Psi_{f}(Y^{(\kappa_{0})})\to\Psi_{f}(Y^{(\kappa_{0}-m_{0})})$.
By \cite[cite]{[\@@bibref{}{LY}{}{}, Lemma~{}2.4]}, $F$ has a polynomial-like
restriction with connected filled Julia set $K(F)$. By the discussions in
section~{}2, there is a unique external marking $\Gamma$ for $F$ induced by
the internal angle system $\alpha$. It follows Douady-Hubbard Straightening
Theorem that there is a unique polynomial $\chi(f)\in\mathcal{C}_{d_{1}}$ such
that the polynomial-like restriction of $F$ is hybrid equivalent to $\chi(f)$
if the external markings are respected (the conjugacy sends $\Gamma$ to the
standard marking for $\chi(f)$). Therefore, we have a well defined map:
$$\chi:\mathcal{C}(\lambda_{f_{0}})\to\mathcal{M},f\mapsto\chi(f).$$ We call
$\chi$ {\em the straightening map induced by $f_{0}$}. Clearly, $f$ is a
tuning of $f_{0}$ by $\chi(f)$. Theorem~{}B implies that: \begin{Theorem}The
staightening map $\chi$ is a surjection. \end{Theorem} \par\par\par Now we use
the arguments developed in~{}\cite[cite]{[\@@bibref{}{AKLS}{}{}]} to show the
injectivity of $\chi$: \par\begin{Theorem} $\chi$ is an injection.
\end{Theorem} We postpone the proof to the end of this section and first prove
some useful lemmas. \begin{Lemma}Let
$f,\widetilde{f}\in\mathcal{C}(\lambda_{f_{0}})$. For any $k\geq\kappa_{0}$,
let $H_{k}$ $(\text{resp.}~{}\widetilde{H}_{k})$ be a qc weak pseudo-conjugacy
between $f_{0}$ and $f$ $(\text{resp.}~{}\widetilde{f})$ up to depth $k$ and
let $\Lambda_{k}=\widetilde{H}_{k}\circ H^{-1}_{k}$. If
$\chi(f)=\chi(\widetilde{f})$, then there exists $K=K(f)>0$ such that
$\Lambda_{k}|_{\partial{H_{k}(Y^{(k)})}}$ has a $K$-qc extension
$\widehat{\Lambda}_{k}$ from $H_{k}(Y^{(k)})$ onto
$\widetilde{H}_{k}(Y^{(k)})$ for all $k\geq\kappa_{0}+m_{0}$. \end{Lemma}
\@proof It is easy to see there exists $K>0$ such that
$\Lambda_{\kappa_{0}+j}|_{\partial{H_{\kappa_{0}+j}(Y^{(\kappa_{0}+j)})}}$ has
a desired $K$-qc extension $\widehat{\Lambda}_{\kappa_{0}+j}$ for
$j=1,\dots,m_{0}$. Indeed, one can choose polynomial-like restrictions
$F=f^{m_{0}}:U\to V$ and
$\widetilde{F}=\widetilde{f}^{m_{0}}:\widetilde{U}\to\widetilde{V}$ such that
$U\Subset H_{(\kappa_{0}+m_{0})}(Y^{(\kappa_{0}+m_{0})})$ and
$\widetilde{U}\Subset\widetilde{H}_{\kappa_{0}+m_{0}}(Y^{(\kappa_{0}+m_{0})})$.
Let $\psi:U\to\widetilde{U}$ be a hybrid conjugacy between $f$ and
$\widetilde{f}$ respecting the canonical external marking. Define
$\widehat{\Lambda}_{\kappa_{0}+j}=\psi$ on $U$ and
$\widehat{\Lambda}_{\kappa_{0}+j}=\Lambda_{\kappa_{0}+j}$ on
$\partial\Lambda_{\kappa_{0}+j}(Y^{(\kappa_{0}+j)})$. Then interpolate
quasiconformally. Let $K$ be the maxium of the maximal dilatation of
$\widehat{H}_{\kappa_{0}+j}$. \par Now the Lemma follows easily by induction.
Note that $f^{k-\kappa_{0}-j}:H_{k}(Y^{(k)})\setminus K(F)\to
H_{k}(Y^{\kappa_{0}+j})\setminus K(F)$ and
$\widetilde{f}^{k-\kappa_{0}-j}:\widetilde{H}_{k}(Y^{(k)})\setminus
K(\widetilde{F})\to\widetilde{H}_{k}(Y^{\kappa_{0}+j})\setminus
K(\widetilde{F})$ are coverings of the same degree for all such $k$ that
$m_{0}\mid k-\kappa_{0}-j$. We now define
$\widehat{\Lambda}_{k}|_{H_{k}(Y^{(k)})\setminus K(F)}$ as the lift of
$\widehat{\Lambda}_{\kappa_{0}+j}|_{H_{k}(Y^{(\kappa_{0}+j})\setminus K(F)}$
(this part is $K$-qc) and $\widehat{\Lambda}_{k}|_{K(F)}=\psi$ (this part is
conformal). It follows ~{}\cite[cite]{[\@@bibref{}{L2}{}{}, Lemma~{}10.4]} and
~{}\cite[cite]{[\@@bibref{}{DH2}{}{}, Lemma~{}2, page 303]} that
$\widehat{\Lambda}_{k}$ is a $K$-qc map. \qed\end@proof \par\begin{Remark}In
the proof of the above lemma, we know that the $K$-qc extension
$\widehat{\Lambda}_{k}$ of $\Lambda_{k}$ can be made conformal a.e. on $K(F)$.
Moreover, $\widehat{\Lambda}_{k}\circ
f^{m_{0}}=\widetilde{f}^{m_{0}}\circ\widehat{\Lambda}_{k}$. \end{Remark}
\par\par\par\par\par\begin{Lemma}Let
$\Omega,\Omega_{*},\widetilde{\Omega},\widetilde{\Omega}_{*}$ be quasidisks.
Let $H_{*}:\Omega_{*}\to\widetilde{\Omega}_{*}$ be a qc map and let
$H:\partial\Omega\to\partial\widetilde{\Omega}$ be a homeomorphism. Assume $A$
and $\widetilde{A}$ are compact subset of $\Omega_{*}$ and
$\widetilde{\Omega}_{*}$ respectively. If there exist holomorphic proper maps
$g:\Omega\to\Omega_{*}$ and
$\widetilde{g}:\widetilde{\Omega}\to\widetilde{\Omega}_{*}$ of the same degree
with the following properties: \begin{itemize} \par\itemize@item@the critical
values of $g$ and $\widetilde{g}$ lie in $A$ and $\widetilde{A}$ respectively;
\par\itemize@item@$H\circ g=\widetilde{g}\circ\widetilde{H}$ on
$\partial\Omega$. \par\end{itemize} Then $H$ has a $K$-qc extension from
$\Omega$ onto $\widetilde{\Omega}$ where $K$ only depends on $H$, $A$ and
$A^{\prime}$. \end{Lemma} \par\@proof Fix some $x\in\Omega_{*}$. Let
$\phi_{*}:\Omega_{*}\to\mathbb{D}$ and
$\widetilde{\phi}_{*}:\widetilde{\Omega}_{*}\to\mathbb{D}$ be conformal maps
such that $\phi_{*}(x)=0$ and $\widetilde{\phi}_{*}(H_{*}(x))=0$. Choose $y\in
g^{-1}(x)$ and $\widetilde{y}\in\widetilde{g}^{-1}(H_{*}(x))$. Let
$\phi:\Omega\to\mathbb{D}$ and
$\widetilde{\phi}:\widetilde{\Omega}\to\mathbb{D}$ be conformal maps such that
$\phi(y)=0$ and $\widetilde{\phi}(\widetilde{y})=0$. Let
$\widehat{H}_{*}={\widetilde{\phi}_{*}}\circ
H\circ{\phi_{*}}^{-1}:\mathbb{D}\to\mathbb{D}$ and
$\widehat{H}={\widetilde{\phi}}\circ
H\circ{\phi}^{-1}:\partial\mathbb{D}\to\partial\mathbb{D}$. Since $A$ and
$\widetilde{A}$ lie compactly in $\Omega_{*}$ and $\widetilde{\Omega}_{*}$
respectively, there exists $0<r=r(A,A^{\prime},x,H_{*}(x))<1$ such that
$\phi_{*}(A),\widetilde{\phi}_{*}(\widetilde{A})\subset\mathbb{D}_{r^{2}}$. By
\cite[cite]{[\@@bibref{}{AKLS}{}{}, Lemma~{}3.2]}, $\widehat{H}$ has a $K$-qc
extension $\widehat{H}:\mathbb{D}\to\mathbb{D}$. Thus $H$ has a $K$-qc
extension from $\Omega$ onto $\widetilde{\Omega}$. \qed\end@proof
\par\begin{Lemma}Let $f,\widetilde{f}$ and $\\{\Lambda_{k}\\}$ be as in
Lemma~{}\ref{lem:Fatou}. Then there exists $K>0$ and a sequence $\\{k_{n}\\}$
such that $\Lambda_{k_{n}}|_{X^{(k_{n})}}$ has a $K$-qc extension from
$H_{k_{n}}(X^{(k_{n})})$ onto $\widetilde{H}_{k_{n}}(X^{(k_{n})})$.
\end{Lemma} \@proof Recall that there exists $\kappa_{0}>0$ with the following
properties. \begin{itemize}
\par\itemize@item@$\overline{Y^{(\kappa_{0})}},\overline{f_{0}(Y^{(\kappa_{0})})},\ldots,\overline{f^{m_{0}-1}_{0}(Y^{(\kappa_{0})})}$
are disjoint; \par\itemize@item@$f^{m_{0}}_{0}:Y^{(\kappa_{0})}\to
Y^{(\kappa_{0}-m_{0})}$ is an APL map with filled Julia set
$\overline{U_{0}}$. \par\end{itemize} We can further assume $\kappa_{0}$ is
large enough so that
$\tau_{0}\notin\bigcup_{j=1}^{m_{0}}f^{j}_{0}(\overline{Y^{(\kappa_{0})}})$.
\par If $c_{0}$ is combinatorially recurrent, i.e., for any $n\in\mathbb{N}$
there exists $\ell>0$ such that $f^{\ell}_{0}(c_{0})\in X^{(n)}$, then the
lemma follows from~{}\cite[cite]{[\@@bibref{}{AKLS}{}{}, Theorem~{}4.4]}. \par
Now we assume $c_{0}$ is combinatorially non-recurrent. Then there exists
$\ell_{0}>\kappa_{0}$ such that $f^{j}_{0}(c_{0})\notin X^{(\ell_{0})}$. For
each $j\in\mathbb{N}$, let us denote $f^{j}_{0}(c_{0})$ by $c_{j}$. \par\vskip
6.0pt plus 2.0pt minus 2.0pt\par{\bf Case 1.} $\partial
U_{0}\cap\omega(c_{0})\neq\emptyset$. Then for any $n>\ell_{0}$, there exists
a smallest non-negative integer $k^{\prime}_{n}$ such that
$c_{k^{\prime}_{n}}\in Y^{(n)}$. Let $k_{n}=k^{\prime}_{n}+\ell_{0}$. Consider
the map $f^{k^{\prime}_{n}}_{0}:Y^{(k_{n})}(c_{0})\to
Y^{(\ell_{0})}(c_{k^{\prime}_{n}})=Y^{(\ell_{0})}$. Note that this map is
$d_{1}$-to-$1$. For otherwise, there exists a smallest positive integer
$0<s<k^{\prime}_{n}$ such that
$f^{s}_{0}(Y^{(k_{n})}(c_{0}))=Y^{(k_{n}-s)}(c_{s})$ contains $0$. Thus
$c_{s}\in Y^{(k_{n}-s)}$. By the definition of $k^{\prime}_{n}$, $c_{s}\in
Y^{(k_{n}-s)}\setminus Y^{(n)}$. Note that
$F_{0}:=f^{m_{0}}_{0}:Y^{(\ell_{0}+m_{0})}\to Y^{(\ell_{0})}$ is a first
return map. Since $f^{k^{\prime}_{n}-s}_{0}(Y^{(k_{n}-s)})=Y^{(\ell_{0})}$,
there exists $u$ such that $F^{u}_{0}(Y^{(k_{n}-s)})=Y^{(\ell_{0})}$. So
$c_{k^{\prime}_{n}}=f^{k^{\prime}_{n}-s}_{0}(c_{s})=F^{u}_{0}(c_{s})\notin
Y^{(n)}$ since $F_{0}^{-1}(Y^{(n)})\subset Y^{(n)}$ and $c_{s}\notin Y^{(n)}$.
\par Now take $n_{0}$ large so that $Y^{(n)}\Subset Y^{(\ell_{0})}$ for all
$n\geq n_{0}$. Since $f,\widetilde{f}\in\mathcal{C}(\lambda_{f_{0}})$, take a
sufficiently large positive integer $k$, we know that \begin{itemize}
\par\itemize@item@$H_{k}(Y^{(n)})\subset H_{n_{0}}(Y^{(n_{0})})\Subset
H_{n_{0}}(Y^{(\ell_{0})})$;
\par\itemize@item@$f^{k_{n}-\ell_{0}}:H_{k_{n}}(Y^{(k_{n})}(c_{0}))\to
H_{k_{n}}(Y^{(\ell_{0})})$ is $d_{1}$-to-$1$ and its critical value lies in
$H_{k}(Y^{(n)})\subset H_{n_{0}}(Y^{(n_{0})})\Subset
H_{n_{0}}(Y^{(\ell_{0})})$. \par\end{itemize} This statement also holds for
the corresponding objects with tildes. Since
$\widetilde{f}^{k^{\prime}_{n}}\Lambda_{\ell_{0}}=\Lambda_{k^{\prime}_{n}}\circ
f^{k^{\prime}_{n}}$ on $\partial H_{k_{n}}(Y^{(k_{n})}(c_{0}))$, by
Lemma~{}\ref{lem:pullback}, $H_{k_{n}}$ has a $K$-qc extension where $K$ only
depends on $\Lambda_{\ell_{0}},H_{n_{0}}(Y^{(n_{0})})$ and
$\widetilde{H}_{n_{0}}(Y^{(n_{0})})$. \par\vskip 6.0pt plus 2.0pt minus
2.0pt\par{\bf Case 2.} $\tau_{0}\in\omega(c_{0})$. Let $q$ be the ray period
of $\tau_{0}$. For each $n\in\mathbb{N}$, let $Y^{n}_{*}$ be the interior of
the closure of the union of all the puzzle pieces of depth $n$ which attaches
$\tau_{0}$. Since $\tau_{0}$ is a repelling fixed point of $f^{q}_{0}$, there
exist small disks $\tau_{0}\in D\Subset D^{\prime}\Subset Y^{\ell_{0}}_{*}$
such that $f^{q}_{0}:D\to D^{\prime}$ is a conformal map. Let $g$ denote the
inverse map of $f^{q}_{0}:D\to D^{\prime}$. Since $Y^{n}_{*}$ shrinks to
$\tau_{0}$, there exists $n_{1}>\ell_{0}$ large such that
$Y^{n_{1}}_{*}\subset D^{\prime}$. Then there exists $s_{1}$ and $n_{2}$ such
that $Y^{n_{2}}_{*}=g^{s_{1}}(Y^{n_{1}}_{*})\Subset Y^{n_{1}}_{*}$ as $g$ is
contracting. Since $\tau_{0}\in\omega(c_{0})$, for any $n$, there exists
$t>n+n_{2}$ such that $c_{t}\in Y^{n_{2}}_{*}$. Then there exists
$t^{\prime}>t$ so that $c_{t^{\prime}}\in Y^{n_{1}}_{*}\setminus
Y^{n_{2}}_{*}$. Note that $\tau_{0}$ lies on the boundary of
$Y^{(\ell_{0})}(c_{t^{\prime}})$ since $n_{1}>\ell_{0}$. Let
$k_{n}=t^{\prime}+\ell_{0}$. We consider the map
$f^{t^{\prime}}_{0}:Y^{(k_{n})}(c_{0})\to Y^{(\ell_{0})}(c_{t^{\prime}})$. We
claim this map is $(d_{2}+1)$-to-$1$. For otherwise, there exists
$0<s^{\prime}<t^{\prime}$ such that $c_{s^{\prime}}\in
Y^{(t^{\prime}+\ell_{0}-s^{\prime})}\subset Y^{(\kappa_{0})}$. Since
$\tau_{0}\notin\bigcup_{j=1}^{m_{0}}f^{j}_{0}(\overline{Y^{(\kappa_{0})}})$,
we conclude that $\tau_{0}$ does not lie on the boundary of
$Y^{(\ell_{0})}(c_{t^{\prime}})=f^{t^{\prime}-s^{\prime}}_{0}(Y^{(t^{\prime}+\ell_{0}-s^{\prime})})$.
This is a contradiction. Again take a sufficiently large positive integer $k$,
we know that \begin{itemize} \par\itemize@item@$H_{k}(Y^{n_{1}}_{*}\setminus
Y^{n_{2}}_{*})=H_{n_{2}}(Y^{n_{1}}_{*}\setminus Y^{n_{2}}_{*})\Subset
H_{n_{2}}(Y^{(\ell_{0})}_{*})$;
\par\itemize@item@$f^{k_{n}-\ell_{0}}:H_{k_{n}}(Y^{(k_{n})}(c_{0}))\to
H_{k_{n}}(Y^{(\ell_{0})}(c_{k_{n}-\ell_{0}}))$ is $(d_{2}+1)$-to-$1$ and its
critical value lies in $H_{n_{2}}(Y^{n_{1}}_{*}\setminus Y^{n_{2}}_{*})$.
\par\end{itemize} We claim that the critical value $v_{n}$ of
$f^{k_{n}-\ell_{0}}|_{H_{k_{n}}(Y^{(k_{n})}(c_{0}))}$ lies in a definite
compact subset of $H_{\ell_{0}}(Y^{(\ell_{0})}(c_{k_{n}-\ell_{0}}))$. For
otherwise, $\\{v_{n}\\}$ accumulates on the boundary of
$H_{\ell_{0}}(Y^{(\ell_{0})}(c_{k_{n}-\ell_{0}}))$. Since $v_{n}\in
H_{n_{2}}(Y^{n_{1}}_{*}\setminus Y^{n_{2}}_{*})$, the only possibility is that
$\\{v_{n}\\}$ accumulates on the external rays truncated by height
${(d_{1}+d_{2})}^{-n_{1}}$ and $(d_{1}+d_{2})^{-n_{2}}$. This implies $v_{n}$
lies in the Fatou set of $f$ for infinitely many $n$, which is a
contradiction. The above statements also hold for the corresponding objects
with tildes. Since
$\widetilde{f}^{k_{n}-\ell_{0}}\Lambda_{\ell_{0}}=\Lambda_{k_{n}-\ell_{0}}\circ
f^{k_{n}-\ell_{0}}$ on $\partial H_{k_{n}}(Y^{(k_{n})}(c_{0}))$, by
Lemma~{}\ref{lem:pullback}, $H_{k_{n}}$ has a $K$-qc extension. \par\par\vskip
6.0pt plus 2.0pt minus 2.0pt\par\par{\bf Case 3.}
$\tau_{0}\notin\omega(c_{0})$ and $\partial U_{0}\cap\omega(c_{0})=\emptyset$.
In this case we can choose $\ell^{\prime}>\ell_{0}$ large enough so that
$c_{j}\notin Y^{(\ell^{\prime})}$ for all $j\in\mathbb{N}$. For any
$n\in\mathbb{N}$, let $k_{n}=n+\ell^{\prime}$. Let us consider
$f^{k_{n}-\ell^{\prime}}_{0}:Y^{(k_{n})}(c_{0})\to Y^{\ell^{\prime}}(c_{n})$.
Clearly, it is unicritical and $(d_{2}+1)$-to-$1$. Since
$\tau_{0}\notin\omega(c_{0})$, the critical vaule $c_{n}$ lies in a definite
compact subset of $Y^{\ell^{\prime}}(c_{n})$ (by a similar argument in
Case~{}2). Since $f\in\mathcal{C}(\lambda_{f_{0}})$, we know
$f^{k_{n}-\ell^{\prime}}:H_{k_{n}}(Y^{(k_{n})}(c_{0}))\to
H_{\ell^{\prime}}(Y^{\ell^{\prime}}(c_{n}))$ is also $(d_{2}+1)$-to-$1$ and
its critical value lies in a definite compact subset
$H_{\ell^{\prime}}(Y^{\ell^{\prime}}(c_{n}))$ (by a similar argument in
Case~{}2). This statement also holds for the corresponding objects with
tildes. Since
$\widetilde{f}^{k_{n}-\ell^{\prime}}\Lambda_{\ell^{\prime}}=\Lambda_{k_{n}-\ell^{\prime}}\circ
f^{k_{n}-\ell^{\prime}}$ on $\partial H_{k_{n}}(Y^{(k_{n})}(c_{0}))$, by
Lemma~{}\ref{lem:pullback}, $H_{k_{n}}$ has a $K$-qc extension. \qed\end@proof
\par\begin{Lemma}Let $f,\widetilde{f}\in\mathcal{C}(\lambda_{f_{0}})$. Assume
$H$ $(\text{resp.}~{}\widetilde{H})$ is weak pseudo-conjugacy up to depth $n$
between $f_{0}$ and $f$ $(\text{resp.}~{}\widetilde{f})$ for some $n>0$. Let
$\Lambda=\widetilde{H}\circ H^{-1}$. If $\Lambda|_{\partial
H(Y^{(n)}\cup\partial X^{(n)})}$ admits a $K$-qc extension to
$\widetilde{H}(Y^{(n)}\cup X^{(n)})$, then there exists a $K$-qc map
$\widehat{\Lambda}$ such that $\widehat{\Lambda}$ is identity in the
B\"{o}ttcher coordinate for $f$ and $\widetilde{f}$ near $\infty$ and
$\widehat{\Lambda}\circ f=\widetilde{f}\circ\widehat{\Lambda}$ on
$\mathbb{C}\setminus H(Y^{(n)}\cup\partial X^{(n)})$. \par\end{Lemma} \@proof
Without loss of generality, we assume $\Lambda$ is $K$-qc on $H(Y^{(n)}\cup
X^{(n)})$. Let $L_{n}$ denote the domain of the first landing map to
$H(Y^{(n)}\cup X^{(n)})$: $$L_{n}=\\{z\mid\exists k\geq 0~{}\text{such
that}~{}f^{k}(z)\in H(Y^{(n)}\cup X^{(n)})\\}.$$ For each component $W$ of
$L_{n}$, we define $\phi_{W}$ be identity in the B\"{o}ttcher coordinate for
$f$ and $\widetilde{f}$ on $\partial W$ and interpolate homemorphicly. Note
that $\phi_{W}(W)$ is a first landing domain of $H(Y^{(n)}\cup X^{(n)})$ since
$\lambda_{f_{0}}\subset\lambda_{f}\cap\lambda_{\widetilde{f}}$. Thus we can
define $\widehat{\Lambda}|_{W}:W\to\phi_{W}(W)$ as the univalent pullback of
$H$. Then define $\widehat{\Lambda}$ on $\mathbb{C}\setminus(K(f)\cup L_{n})$
as identity in the B\"{o}ttcher coordinate for $f$ and $\widetilde{f}$. These
two maps match on the common boundary since $\Lambda|_{\partial H(Y^{(n)}\cup
X^{(n)})}$ is identity in the B\"{o}ttcher coordinate for $f$ and
$\widetilde{f}$. Note that $\widehat{\Lambda}$ is $K$-qc. By
\cite[cite]{[\@@bibref{}{KSS}{}{}, Fact~{}5.1]}, the residual set
$$E(H(Y^{(n)}\cup X^{(n)}):=\\{z\in K(f)\mid f^{j}(z)\notin H(Y^{(n)}\cup
X^{(n)})~{}\text{for all}~{}j\in\mathbb{N}\\}$$ is a nowhere dense compact set
with Lebesgue measure $0$. Thus $\widehat{\Lambda}$ admits a $K$-qc extension
to the whole complex plane $\mathbb{C}$. \qed\end@proof \par Let
$f\in\mathcal{C}(\lambda_{f_{0}})$ For any $n\in\mathbb{N}$, let $H_{n}$ be a
weak pseudo-conjugacy between $f_{0}$ and $f$ up to depth $n$. Define
$\mathcal{K}(f):=\bigcap_{n\in\mathbb{N}}H_{n}(Y^{(n)})$ and
$\mathcal{S}(f):=\bigcap_{n\in\mathbb{N}}H_{n}(X^{(n)})$. By using the same
argument as in the proof of Lemma~{}\ref{lem:X shrinking}, one can show
$\mathcal{S}(f)$ is a singleton which is a Julia critical point
$c^{\prime}_{f}$ of $f$. \par\par\begin{Lemma} The set
$E(f):=\\{z\in\mathbb{C}\setminus\mathcal{K}(f)\mid\omega(z)\cap(\mathcal{K}(f)\cup\mathcal{S}(f))\neq\emptyset\\}$
has zero Lebesgue measure for all $f\in\mathcal{C}(\lambda_{f_{0}})$.
\end{Lemma} \@proof\par{\bf Case 1.} $c_{0}$ is combinatorially recurrent.
\par We first show that for
$E^{\prime}(f)=\\{z\in\mathbb{C}\mid\omega(z)\cap\mathcal{S}(f)\neq\emptyset\\}$
has zero Lebesgue measure. To this end, we construct a generalized polynomial
like map (complex box map) as in the proof of Lemma~{}\ref{lem:X shrinking}.
Let $P_{0}=X^{(\kappa_{0})}$. For $n\geq 1$, we define $P_{n}$ as the
component of the first return map to $P_{n-1}$ which contains $c_{0}$
inductively. It is well known this is a moment $n_{0}$ so that
$P_{n_{0}+1}\Subset P_{n_{0}}$ since $f_{0}$ is not renormalizable at $c_{0}$
(see~{}\cite[cite]{[\@@bibref{}{KS}{}{}, Lemma~{}2.2]} for an example). Let
$H_{n_{0}+1}$ be a weak pseudo-conjugacy between $f_{0}$ and $f$ up to depth
$n_{0}+1$. Let $V_{f}:=H_{n_{0}+1}(P_{n_{0}+1})$ and let $U_{f}$ be the domain
of the first return to $V_{f}$ under $f$. The first return map $R_{f}:U_{f}\to
V_{f}$ is our desired generalized polynomial-like map. Similarly, let
$V_{0}:=P_{n_{0}+1}$ and $R_{0}:U_{0}\to V_{0}$ be the first return map under
$f_{0}$. Since $f_{0}$ is non-renormalizable at $c_{0}$, so is $R_{0}:U_{0}\to
V_{0}$. As $f\in\mathcal{C}(\lambda_{f_{0}})$, $R_{f}:U_{f}\to V_{f}$ is also
non-renormalizable. By~{}\cite[cite]{[\@@bibref{}{KS}{}{}, Theorem~{}1.5]}
(see also \cite[cite]{[\@@bibref{}{L3}{}{}]}), $K(R_{f})$ has Lebesgue measure
$0$ where $K(R_{f}):=\\{z\in U_{f}\mid R^{n}_{f}(z)\in U_{f}~{}\text{for
all}~{}n\in\mathbb{N}\\}$. Note that $E^{\prime}(f)\subset K(R_{f})$, so
$E^{\prime}(f)$ has zero Lebesgue measure. \par Now we show $E(f)\setminus
E^{\prime}(f)$ has zero Lebesgue measure. It suffices to show that for any
$z\in E(f)\setminus E^{\prime}(f)$, $z$ is not a density point of
$E(f)\setminus E^{\prime}(f)$. Let $z_{k}:=f^{k}(z)$ for all $k\in\mathbb{N}$.
Since $z\notin\mathcal{S}(f)$, there exists $\ell>0$ such that
$\mathrm{orb}(z)\cap H_{\ell}(X^{(\ell)}))=\emptyset$. Now consider the APL
map $f^{m_{0}}:H_{\ell+m_{0}}(Y^{(\ell+m_{0})})\to H_{\ell}(Y^{(\ell)})$.
Choose $n_{1}$ large so that $H_{n_{1}}(Y^{(n_{1})})\Subset
H_{\ell}(Y^{(\ell)})$. Let $\delta$ denote the Hausdorff distance between
$\overline{H}_{n_{1}}(Y^{(n_{1})})\setminus
H_{n_{1}+m_{0}}(Y^{(n_{1}+m_{0})})$ and $\partial
H_{\ell}(Y^{(\ell)})\cup\mathcal{K}(f)$. As they are disjoint closed set,
$\delta>0$. Since $\omega(z)\cap\mathcal{K}(f)\neq\emptyset$, there is a
strictly increasing sequence $\\{k_{n}\\}_{n\in\mathbb{N}}$ such that
$z_{k_{n}}\in
H_{n_{1}}(Y^{(n_{1})})\setminus\overline{H_{n_{1}+m_{0}}(Y^{(n_{1}+m_{0})})}$.
Let $D_{n}$ be the component of $f^{-k_{n}}(\mathbb{D}(z_{k_{n}},\delta))$
which contains $z$ for all $n\in\mathbb{N}$. Note that
$f^{k_{n}}:D_{n}\to\mathbb{D}(z_{k_{n}},\delta)$ is conformal. Indeed, any
pullback of $\mathbb{D}(z_{k_{n}},\delta)$ cannot intersect
$H_{\ell}(X^{(\ell)}))$ since $\mathrm{orb}(z)\cap
H_{\ell}(X^{(\ell)}))=\emptyset$ and $\mathbb{D}(z_{k_{n}},\delta)\subset
H_{\ell}(Y^{(\ell)})$. By the Koebe Distortion Theorem, $z$ is not a density
point of $K(f)$. \par\par{\bf Case 2.} $c_{0}$ is combinatorially non-
recurrent. Fix $\ell>\kappa_{0}$ large so that $f^{j}_{0}(c_{0})\notin
X^{(\ell)}$ for all $j\geq 1$. \par In this case, we first show
$E^{\prime\prime}(f)=\\{z\in
E(f)\mid\omega(z)\cap\mathcal{K}(f)\neq\emptyset\\}$ has zero Lebesgue
measure. The proof is similar to the second part of Case~{}1. Take $z\in
E^{\prime\prime}(f)$ and let $z_{k}:=f^{k}(z)$ for all $k\in\mathbb{N}$.
Choose $n_{1}$ large so that $H_{n_{1}}(Y^{(n_{1})})\Subset
H_{\ell}(Y^{(\ell)})$. Let $\delta$ denote the Hausdorff distance between
$\overline{H}_{n_{1}}(Y^{(n_{1})})\setminus
H_{n_{1}+m_{0}}(Y^{(n_{1}+m_{0})})$ and $\partial
H_{\ell}(Y^{(\ell)})\cup\mathcal{K}(f)$. As they are disjoint closed set,
$\delta>0$. Since $\omega(z)\cap\mathcal{K}(f)\neq\emptyset$, there is a
strictly increasing sequence $\\{k_{n}\\}_{n\in\mathbb{N}}$ such that
$z_{k_{n}}\in
H_{n_{1}}(Y^{(n_{1})})\setminus\overline{H_{n_{1}+m_{0}}(Y^{(n_{1}+m_{0})})}$.
Let $D_{n}$ and $D^{\prime}_{n}$ be the component of
$f^{-k_{n}}(\mathbb{D}(z_{k_{n}},\delta))$ and
$f^{-k_{n}}(\mathbb{D}(z_{k_{n}},\delta/2))$ which contains $z$ for all
$n\in\mathbb{N}$ respectively. Note that
$f^{k_{n}}:D_{n}\to\mathbb{D}(z_{k_{n}},\delta)$ is a holomorphic proper map
of degree at most $d_{2}+1$ (since $c_{0}$ is combinatorially non-recurrent).
Thus $f^{k_{n}}:D^{\prime}_{n}\to\mathbb{D}(z_{k_{n}},\delta/2)$ can be
written as $\psi_{n}\circ Q_{0}\circ\phi_{n}$ where $Q_{0}(z)=z^{d_{2}+1}$,
$\psi_{n}$ and $\phi_{n}$ are conformal maps with uniformly bounded
distortion. Then it follows $z$ is not a density point of
$E^{\prime\prime}(f)$ by a standard argument. \par It remains to show that
$E(f)\setminus E^{\prime\prime}(f)$ has zero Lebesgue measure. Take $z\in
E(f)\setminus E^{\prime\prime}(f)$ and let $z_{k}:=f^{k}(z)$ for all
$k\in\mathbb{N}$. Fix $n_{2}$ large so that $H_{n_{2}}(X^{(n_{2})})\Subset
H_{\ell}(X^{(\ell)})$. Let $\delta_{1}$ denote the Hausdorff distance between
$\omega(z)$ and $\mathcal{K}(f)$ and let $\delta_{2}$ denote the Hausdorff
distance between $\partial H_{n_{2}}(X^{(n_{2})})$ and $\partial
H_{\ell}(X^{(\ell)})$. Take $\delta<\min(\delta_{1},\delta)$. Since
$\omega(z)\cap\mathcal{S}(f)\neq\emptyset$, there exists a strictly increasing
sequence $\\{k_{n}\\}$ such that $z_{k_{n}}\in H_{n_{2}}(X^{(n_{2})})$. Let
$U_{n}$ be the component of $f^{-k_{n}}(\mathbb{D}(z_{k_{n}},\delta))$ which
contains $z$ for all $n\in\mathbb{N}$. Clearly,
$f^{k_{n}}:U_{n}\to\mathbb{D}(z_{k_{n}},\delta)$ is conformal by the
construction. It follows from Koebe Distortion Theorem that $z$ is not a
density point of $E(f)\setminus E^{\prime\prime}(f)$. \qed\end@proof \par\par
We are now in a position to show Theorem~{}\ref{thm:injection}. \@proof[Proof
of Theorem~\ref{thm:injection}] Assume there are
$f\in\mathcal{C}(\lambda_{f_{0}})$ and
$\widetilde{f}\in\mathcal{C}(\lambda_{f_{0}})$ such that
$\chi(f)=\chi(\widetilde{f})$. We prove that $f=\widetilde{f}$. \par For any
$n\in\mathbb{N}$, let $H_{n}$ be a qc weak pesudo-conjugacy between $f_{0}$
and $f$ and let $\widetilde{H}_{n}$ be a qc weak pesudo-conjugacy between
$f_{0}$ and $\widetilde{f}$. Let $\Lambda_{n}=\widetilde{H}_{n}\circ
H^{-1}_{n}$ for all $n\in\mathbb{N}$. It follow there exists a sequence
$\\{k_{n}\\}$ and $K>0$ with the following properties: \begin{itemize}
\par\itemize@item@$\Lambda_{k_{n}}$ has a $K$-qc extension from
$H_{k_{n}}(Y^{(k_{n})}\cup X^{(k_{n})})$ onto
$\widetilde{H}_{k_{n}}(Y^{(k_{n})}\cup X^{(k_{n})})$;
\par\itemize@item@$\bar{\partial}\Lambda_{k_{n}}=0$ a.e. on $\mathcal{K}(f)$
\par\end{itemize} by Lemma~{}\ref{lem:Fatou}, Lemma~{}\ref{lem:Julia} and
Remark~{}\ref{re:filled}. Then by Lemma~{}\ref{lem:pseudo}, we can further
assume $\Lambda_{k_{n}}$ is global $K$-qc and $\Lambda_{k_{n}}\circ
f=\widetilde{f}\circ\Lambda_{k_{n}}$ on $\mathbb{C}\setminus
H(Y^{(n)}\cup\partial X^{(n)})$. By the compactness of normalized $K$-qc maps,
$\\{\Lambda_{k_{n}}\\}$ has a convergent subsequence. Without loss of
generality, we assume $\Lambda_{k_{n}}$ converges to a $K$-qc map
$\Phi:\mathbb{C}\to\mathbb{C}$. Clearly $\Phi\circ f=\widetilde{f}\circ\Phi$
on $\mathbb{C}\setminus(\mathcal{K}(f)\cup\mathcal{S}(f))$. By the
construction in the proof of Lemma~{}\ref{lem:pseudo}, we know that
$\bar{\partial}\Lambda_{k_{n}}$ only supports on
$\bigcup_{j=0}^{\infty}f^{-j}(H_{k_{n}}(Y^{(k_{n})}\cup
X^{(k_{n})}\setminus\mathcal{K}(f)))$. Since these sets shrink to $E(f)$ which
has zero Lebesgue measure by Lemma~{}\ref{lem:measure}, we conclude $\Phi$ is
conformal. By the proof of Lemma~{}\ref{lem:Fatou} and
Remark~{}\ref{re:filled}, we know $\Phi\circ
f^{m_{0}}=\widetilde{f}^{m_{0}}\circ\Phi$ on $\mathcal{K}(f)$. Since
$\Phi\circ f=\widetilde{f}\circ\Phi$ on
$\bigcup_{j=1}^{m_{0}-1}f^{j}(\mathcal{K}(f))$, we have
$$\widetilde{f}^{m_{0}-1}\circ\widetilde{f}_{0}\circ\Phi=\Phi\circ
f^{m_{0}}=\Phi\circ f^{m_{0}-1}\circ f=\widetilde{f}^{m_{0}-1}\circ\Phi\circ
f$$ on $\mathcal{K}(f)$. As
$\widetilde{f}^{m_{0}-1}|_{\widetilde{f}(\mathcal{K}(\widetilde{f}))}$ is one-
to-one, we conclue $\Phi\circ f=\widetilde{f}\circ\Phi$ on $\mathcal{K}(f)$.
Thus, $\Phi\circ f=\widetilde{f}\circ\Phi$ on the whole complex plane
$\mathbb{C}$. Since $\Phi$ is conformal and is tangent to $id$ at $\infty$,
$\Phi$ is identity. \qed\end@proof \par Theorem~{}A follows immediately from
Theorem~{}\ref{thm:surjection} and Theorem~{}\ref{thm:injection}. Theorem~{}C
is a consequence of Theorem~{}A, \cite[cite]{[\@@bibref{}{McM2}{}{},
Propsotion~{}4.7]} and the compactness of $\mathcal{C}(\lambda_{f_{0}})$
(\cite[cite]{[\@@bibref{}{IK}{}{}, Theorem~{}8.1]}).
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