# Dynamics of superattracting skew products on the attracting basins: Böttcher
coordinates and plurisubharmonic functions

Kohei Ueno  Daido University, Nagoya 457-8530, Japan<EMAIL_ADDRESS>
###### Abstract.

We study the dynamics of a superattracting skew product $f$ on the attracting
basin. As the first strategy, we find out forward $f$-invariant wedge-shaped
regions in the basin, on some of which $f$ is conjugate to monomial maps, and
consider whether the unions of all the preimages of the regions coincide with
the basin. As the second strategy, we show the existence and properties of
several kinds of plurisubharmonic functions of $f$, the main functions of
which are induced from the Böttcher coordinates, and investigate the
asymptotic behavior of the functions toward the boundaries of the unions.
Consequently, we obtain a plurisubharmonic function on the complement of
specific fibers in the basin, which is continuous and pluriharmonic on open
and dense subsets of the complement and describes an certain weighted vertical
dynamics well.

###### Key words and phrases:

###### 2020 Mathematics Subject Classification:

Primary 32H50; Secondary 37F80

This work was supported by the Research Institute for Mathematical Sciences, a
Joint Usage/Research Center located in Kyoto University.

##  1\. Introduction

###  1.1. Background

The dynamics around superattracting fixed points of holomorphic functions is
completely understood in dimension one. Let $p(z)=az^{\delta}+O(z^{\delta+1})$
be a holomorphic function defined on $\mathbb{C}$, where $a\neq 0$ and
$\delta\geq 2$. Then it has a superattracting fixed point at the origin.
Böttcher’s theorem [4] indicates that the limit

$\varphi_{p}(z)=\lim_{n\to\infty}p_{0}^{-n}\circ p^{n}(z)$

exists on a small neighborhood of the origin, where $p^{n}$ is the $n$-th
iterate of $p$, $p_{0}(z)=az^{\delta}$ and the branch is taken such that
$p_{0}^{-n}\circ p_{0}^{n}=id$. This limit is called the Böttcher coordinate
for $p$ at the origin, and it conjugates $p$ to $p_{0}$. We refer [13] for
details. We remark that the order of $p^{n}$ at the origin is clearly
$\delta^{n}$, which is also called the attraction rate of $p^{n}$. Let $A_{p}$
be the attracting basin of $p$ at the origin, and $E_{p}$ the union of all the
preimages of the origin. We can extend $\varphi_{p}$ from the neighborhood to
$A_{p}$ until it meets the other critical points than the origin. On the other
hand, the limit

$G_{p}(z)=\lim_{n\to\infty}\dfrac{1}{\delta^{n}}\log|p^{n}(z)|$

exists on $A_{p}$. It is subharmonic and negative on $A_{p}$ and harmonic on
$A_{p}-E_{p}$. Note that $G_{p}=\log|\varphi_{p}|$ on the neighborhood of the
origin. We remark that similar statements hold even if $p$ is a holomorphic
germ.

Several studies have been made toward the generalization of Böttcher’s theorem
to higher dimensions. For example, Ushiki [29], Ueda [18] and Buff et al [5]
studied the case in which holomorphic germs, with superattracting fixed
points, have the Böttcher coordinates on neighborhoods of the points. The
germs in [29] are conjugate to monomial maps, whereas the germs in [18] and
[5] are conjugate to homogeneous and quasihomogeneous maps, respectively.

However, Böttcher’s theorem does not extend to higher dimensions entirely as
pointed out by Hubbard and Papadopol [10]. Let $f$ be a two dimensional
holomorphic germ with a superattracting fixed point at the origin. Unlike the
one dimensional case, in which a superattracting fixed point of a holomorphic
germ is a isolated critical point and forward invariant under the germ, the
origin is contained in the critical set of $f$, which may have a singularity
at the origin and is not forward invariant under $f$ in general. Hence it is
difficult or impossible to take a nice normal form of $f$ on a neighborhood of
the origin.

Rigidity is a keyword for the study of the local dynamics of superattracting
germs. We say that $f$ is rigid if the critical set is contained in a totally
invariant set with normal crossing singularities. Favre [7] introduced and
classified attracting rigid germs. Favre and Jonsson [9] proved that any two
dimensional superattracting germ can be blown up to a rigid germ, and hence
the original germ is rigid on an open set whose closure contains the fixed
point. Using the normal form on the open set, they proved that the asymptotic
attraction rate is a quadratic integer, gave an inequality on the attraction
rates, and constructed a dynamically nice plurisubharmonic function defined on
the neighborhood of the fixed point. Ruggiero [16] extended their results to
more general cases.

We restrict our attention to superattracting skew products in this paper. A
skew product is a map of the form $f(z,w)=(p(z),q(z,w))$. The fundamental
properties of the dynamics of polynomial skew products is well studied and
summarized in [11] and [8], in which the main topics are the Green functions,
currents and measures. Lilov [12] studied the local and semi-local dynamics of
holomorphic skew products near a superattracting invariant fiber. As natural
extensions of the one dimensional results, he obtained nice normal forms on
neighborhoods of periodic points which are geometrically attracting, parabolic
and Siegel on fiber direction, except the superattracting case. See also [15]
and [14] for the dynamics of skew products near an invariant fiber of
different types. Whereas the dynamics of skew products is mild in the sense
that it can be regarded as the intermediate between the one and two
dimensional dynamics, it has very complicated aspects and gives new phenomena
of the dynamics in dimension two. For example, Astorg et al [3] found
polynomial skew products with wandering domains, which is the first example of
non-invertible polynomial maps with wandering domains. See also [2] and [1]
for other types of polynomial skew products with wandering domains.
Furthermore, polynomial skew products are used in [6] and [17] to construct
robust bifurcations, i.e. open sets contained in a bifurcation locus.

###  1.2. Previous results: Böttcher coordinates

We have studied the dynamics of superattracting skew products and succeeded to
construct Böttcher coordinates on invariant wedge-shaped regions in [23],
[24], [25] and [26]. Let us briefly review main results in [26], which is a
consequence of the studies. Let $f(z,w)=(p(z),q(z,w))$ be a skew product with
a superattracting fixed point at the origin. More precisely, we assume that
the eigenvalues of $Df(0)$ are both zero; this assumption is called nilpotent
in [16], whereas the assumption $Df(0)=0$ is imposed in [9]. Hence we may
write $p(z)=az^{\delta}+O(z^{\delta+1})$ and $q(z,w)=bz+\sum_{i,j\geq
0,i+j\geq 2}b_{ij}z^{i}w^{j}$, where $a\neq 0$ and $\delta\geq 2$. Let
$bz=b_{10}z^{1}w^{0}$ and $q(z,w)=\sum b_{ij}z^{i}w^{j}$ for short.

It is clear that the dominant term of $p$ is $az^{\delta}$. We found in [26]
that the order $\delta$ of $p$ and the Newton polygon of $q$ assign a term
$b_{\gamma d}z^{\gamma}w^{d}$ of $q$ that is dominant on a wedge-shaped region
$U$, where $U=\\{|z|^{l_{1}+l_{2}}<r^{l_{2}}|w|,|w|<r|z|^{l_{1}}\\}$ for some
rational numbers $0\leq l_{1}<\infty$ and $0<l_{2}\leq\infty$, and that
preserves $U$. Let $f_{0}(z,w)=(az^{\delta},b_{\gamma d}z^{\gamma}w^{d})$.

######  Lemma 1.1 ([26]).

If $d\geq 2$ or if $d=1$ and $\delta\neq T_{k}$ for any $k$, then $f\sim
f_{0}$ on $U$ as $r\to 0$, and $f(U)\subset U$ for small $r>0$.

The notation $f\sim f_{0}$ on $U$ as $r\to 0$ means that the ratios of the
first and second components of $f$ and $f_{0}$ tends to $1$ on $U$ as $r\to
0$, and the definition of $T_{k}$ is given below. Moreover, we proved that the
limit

$\phi(z,w)=\lim_{n\to\infty}f_{0}^{-n}\circ f^{n}(z,w)$

exists on $U$, where the branch is taken such that $f_{0}^{-n}\circ
f_{0}^{n}=id$.

######  Theorem 1.2 ([26]).

If $d\geq 2$ or if $d=1$ and $\delta\neq T_{k}$ for any $k$, then $\phi$ is
biholomorphic on $U$ for small $r>0$, which conjugates $f$ to $f_{0}$.
Moreover, $\phi\sim id$ on $U$ as $r\to 0$.

We call $\phi$ the Böttcher coordinate for $f$ on $U$, although the second
component of $\phi$ looks like the Kœnigs coordinate if $d=1$.

Let us denote the precise definitions in the results. We define the Newton
polygon $N(q)$ of $q$ as the convex hull of the union of $D(i,j)$ with
$b_{ij}\neq 0$, where $D(i,j)=\\{(x,y)\ |\ x\geq i,y\geq j\\}$. Let
$(n_{1},m_{1})$, $(n_{2},m_{2})$, $\cdots,(n_{s},m_{s})$ be the vertices of
$N(q)$, where $n_{1}<n_{2}<\cdots<n_{s}$ and $m_{1}>m_{2}>\cdots>m_{s}$. Let
$T_{k}$ be the $y$-intercept of the line $L_{k}$ passing through the vertices
$(n_{k},m_{k})$ and $(n_{k+1},m_{k+1})$ for each $1\leq k\leq s-1$.

  * Case 1

If $s=1$, then $N(q)$ has the only one vertex, which is denoted by
$(\gamma,d)$.
We define $l_{1}=l_{2}^{-1}=0$ and so $U=\\{|z|<r,|w|<r\\}$.

The results for Case 1 are classical and the proofs are simpler than the other
cases. We remark that $f$ in Case 1 is rigid and belongs to class 6 or class 4
in [7] if $d\geq 2$ or $d=1$, respectively, and a description on Böttcher
coordinates is given in pp.498-499 in [7].

Therefore, we only deal with the case $s>1$ hereafter, which is divided into
the following three cases:

  * Case 2

If $s>1$ and $\delta\leq T_{s-1}$, then we define

$(\gamma,d)=(n_{s},m_{s}),\ l_{1}=\frac{n_{s}-n_{s-1}}{m_{s-1}-m_{s}}\text{
and }l_{2}^{-1}=0.$

Hence $U=\\{|z|<r,|w|<r|z|^{l_{1}}\\}$.

  * Case 3

If $s>1$ and $T_{1}\leq\delta$, then we define

$(\gamma,d)=(n_{1},m_{1}),\ l_{1}=0\text{ and
}l_{2}=\frac{n_{2}-n_{1}}{m_{1}-m_{2}}.$

Hence
$U=\\{|z|^{l_{2}}<r^{l_{2}}|w|,|w|<r\\}=\\{r^{-l_{2}}|z|^{l_{2}}<|w|<r\\}$.

  * Case 4

If $s>2$ and $T_{k}\leq\delta\leq T_{k-1}$ for some $2\leq k\leq s-1$, then we
define

$(\gamma,d)=(n_{k},m_{k}),\ l_{1}=\frac{n_{k}-n_{k-1}}{m_{k-1}-m_{k}}\text{
and }l_{1}+l_{2}=\frac{n_{k+1}-n_{k}}{m_{k}-m_{k+1}}.$

Hence
$U=\\{|z|^{l_{1}+l_{2}}<r^{l_{2}}|w|,|w|<r|z|^{l_{1}}\\}=\\{r^{-l_{2}}|z|^{l_{1}+l_{2}}<|w|<r|z|^{l_{1}}\\}$.

We remark that if $\delta>T_{1}$ and $d=1$, then $f$ is rigid and belongs to
class 4 in [7] and hence $f$ is conjugate to $f_{0}$ not only on $U$ but also
on a neighborhood of the origin; see [26, pp.44-45] for a detail.

Note that if $\delta=T_{k}$ for some $k$, then there are two different
dominant terms of $q$, whereas there is the only one dominant term if
$\delta\neq T_{k}$ for any $k$. Moreover, if the two dominant terms satisfy
the degree condition, then we have two disjoint invariant regions on which $f$
is conjugate to each of the two different monomial maps.

Using the same idea and arguments as in [26], we gave inequalities on the
attraction rates in [28], which imply that the asymptotic attraction rate is
$\delta$ if $\gamma>0$ or $\delta\leq d$, and is $d$ if $\gamma=0$ and
$\delta>d$.

###  1.3. New results: weighted plurisubharmonic functions

To consider the global dynamics, we assume that $f$ is defined on
$\mathbb{C}^{2}$ or on $\\{|z|<R\\}\times\mathbb{C}$ for large $R$ such that
$A_{p}$ is relatively compact in $\\{|z|<R\\}$. We give a summary of main
results for Cases 2, 3 and 4, respectively, and Tables 1, 2 and 3 as
comparison charts of the results for Cases 2, 3 and 4, respectively, in the
next section. Although those results are important, they are so much and
require to introduce the interval or rectangle $\mathcal{I}_{f}$ instead of
$l_{1}$ and $l_{2}$. Therefore, we only give three consequences of the main
results in this subsection, which are separated into the case $\delta<T_{s-1}$
and the case $\delta\geq T_{s-1}$.

Let us first state two consequences of the main results for the case
$\delta<T_{s-1}$. Let $A_{f}$ be the union of all the preimages of $U$ and
$A_{0}$ the attracting basin. If $\delta<T_{s-1}$, then $A_{f}$ almost
coincides with $A_{0}$. Let $E_{z}$ and $E_{w}$ be the unions of all the
preimages of $\\{z=0\\}$ and $\\{w=0\\}$, respectively, and $E=E_{z}\cup
E_{w}$.

######  Theorem 1.3.

If $\delta<T_{s-1}$, then $A_{f}=A_{0}-E_{z}$.

To describe the vertical dynamics, we introduce the following limits in Case
2:

$\displaystyle G_{z}^{\alpha}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{\alpha}}\right|\text{
if }\delta\neq d,\text{ and}$ $\displaystyle G_{z}^{\infty}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\dfrac{|w_{n}|}{|z_{n}|^{\frac{\gamma}{d}n}}\text{
if }\delta=d,$

where $\alpha=\gamma/(\delta-d)$ and $(z_{n},w_{n})=f^{n}(z,w)$. Note that

$G_{z}^{\alpha}(w)=\log\left|\dfrac{\phi_{2}(z,w)}{\phi_{1}(z)^{\alpha}}\right|\text{
and }G_{z}^{\infty}(w)=\log|\phi_{2}(z,w)|$

if the Böttcher coordinate $\phi$ exists, where $\phi=(\phi_{1},\phi_{2})$,
and that $G_{z}^{\alpha}\circ f=dG_{z}^{\alpha}$ and $G_{z}^{\infty}\circ
f=d^{n}G_{z}^{\infty}+n\gamma d^{n-1}G_{p}$. Hence the existence of $\phi$ on
$U$ induces the existence of the limits above on $A_{f}$, whose properties
depend on the magnitude relation between $\delta$ and $d$.

######  Theorem 1.4.

Let $\delta<T_{s-1}$. If $\delta<d$, then $G_{z}^{\alpha}$ is plurisubharmonic
and negative on $A_{0}$ and
$G_{z}^{\alpha}(w)=\log\left|z^{-\alpha}w\right|+o(1)$ on $U$ as $r\to 0$. If
$\delta=d$, then $G_{z}^{\infty}$ is plurisubharmonic on $A_{0}-E_{z}$ and
$G_{z}^{\infty}(w)=\log\left|w\right|+o(1)$ on $U$ as $r\to 0$. If
$\delta>d\geq 1$, then $G_{z}^{\alpha}$ is plurisubharmonic on $A_{0}-E_{z}$
and $G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U$ as $r\to 0$.
For all the cases, the functions are pluriharmonic on $A_{0}-E$.

Detailed versions of the two theorems above are given as Theorems 2.3 and 2.4,
respectively, in the next section. We remark that the notation $\alpha_{0}$ is
used instead of $\alpha$ in [26].

Let us next state a consequence of the main results for the case $\delta\geq
T_{s-1}$. Note that $d\geq 1$ for this case. The weight $\alpha$ is not
defined for the special case $\gamma=0$ and $\delta=d$, whereas $\delta>d$ and
so $\alpha\geq 0$ except the special case if $\delta\geq T_{s-1}$. By
redefining $\alpha$ for the case $\delta\geq T_{s-1}$, which coincides with
$\gamma/(\delta-d)$ except the special case, we can always define the limit
$G_{z}^{\alpha}$ for the case $\delta\geq T_{s-1}$, which exists on
$A_{f}-E_{z}$ if $\phi$ exists on $U$. Moreover, the limit

$G_{z}^{\alpha,+}(w)=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log^{+}\left|\dfrac{w_{n}}{z_{n}^{\alpha}}\right|$

exists and behaves well on the complement of specific fibers in the basin. Let
$E_{deg}$ be the union of all the preimages of the set of fibers in
$A_{p}\times\mathbb{C}$ that are degenerated to points by $f$. In the
following theorem we set $(\gamma,d)$ as $(n_{k},m_{k})$ if $f$ has two
dominant terms $(n_{k},m_{k})$ and $(n_{k+1},m_{k+1})$; we remark that
$\alpha$ depends only on $f$.

######  Theorem 1.5.

Let $\delta\geq T_{s-1}$ and let $d\geq 2$ or $\delta\neq T_{k}$ for any $k$.
If $\gamma>0$ or $\delta=d$, then $\alpha>0$, and $G_{z}^{\alpha,+}$ is well-
defined on $A_{0}-E_{z}$, plurisubharmonic and continuous on $A_{0}-E_{z}\cup
E_{deg}$, pluriharmonic outside the boundary of
$\\{G_{z}^{\alpha,+}>0\\}\cap(A_{0}-E_{z}\cup E_{deg})$ and
$G_{z}^{\alpha,+}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U-\\{z=0\\}$ as
$r\to 0$. If $\gamma=0$ and $\delta>d$, then $\alpha=0$, and
$G_{z}^{\alpha,+}$ is well-defined on $A_{0}$, plurisubharmonic and continuous
on $A_{0}-E_{deg}$, pluriharmonic outside the boundary of
$\\{G_{z}^{\alpha,+}>0\\}\cap(A_{0}-E_{deg})$ and
$G_{z}^{\alpha,+}(w)=\log\left|w\right|+o(1)$ on $U$ as $r\to 0$.

If $\alpha>0$ and $d\geq 2$, then the set
$\\{G_{z}^{\alpha,+}>0\\}\cap(A_{0}-E_{z})$ is expressed by the union of all
the preimages of another invariant wedge-shaped region, which is included in
$U$ and whose slope relates to $\alpha$. See Theorems 2.9 and 2.17 for
details.

The limits $G_{z}^{\alpha}$ and $G_{z}^{\alpha,+}$ are good tools for the
study of the dynamics of $w_{n}/z_{n}^{\alpha}$ on $A_{f}-E_{z}$ and
$A_{0}-E_{z}$, respectively. In particular, the set where $G_{z}^{\alpha,+}$
is not pluriharmonic seems to be a Julia set of the dynamics of
$w_{n}/z_{n}^{\alpha}$ and we expect that this Julia set is fractal when
$\delta=T_{k}$ for some $k$, which is true at least for Examples 6.17 and
6.18.

###  1.4. New results: typical plurisubharmonic functions

There are other limits such as

$G_{z}(w)=\lim_{n\to\infty}\dfrac{1}{\lambda^{n}}\log\left|w_{n}\right|\text{
and
}G_{f}(z,w)=\lim_{n\to\infty}\dfrac{1}{\lambda^{n}}\log\left|(z_{n},w_{n})\right|,$

where $\lambda=\max\\{\delta,d\\}$ and $|(z,w)|=\max\\{|z|,|w|\\}$. We remark
that $\lambda$ depends only on $f$. Although these limits are typical to study
the dynamics of a holomorphic map $f$, they are less useful than the previous
limits to discover the weighted vertical dynamics; roughly speaking, if
$\delta\geq d$, then either these limits are constant on the intersection of
each fiber and $A_{f}$, or $\gamma=0$ and so $f_{0}$ is just a product. Tables
4 and 5 in the next section help us to understand the statements below.

First, the existence of the Böttcher coordinate induces the following.

######  Theorem 1.6.

If $\delta<d$, then $f$ is in Case 2, $G_{z}=G_{z}^{\alpha}$ on $A_{0}$ and
$G_{f}=0$ on $A_{0}-E_{z}$. If $\gamma>0$ and $\delta=d$, then $f$ is in Case
2, $G_{z}=-\infty$ and $G_{f}=G_{p}$ on $A_{0}$. If $\gamma=0$ and $\delta=d$,
then $f$ is in Case 3, $G_{z}$ is pluriharmonic on $A_{f}$ and
$G_{z}(w)=\log\left|w\right|+o(1)$ on $U$ as $r\to 0$. If $\delta>d\geq 2$ or
if $\delta>d=1$ and $\delta\neq T_{k}$ for any $k$, then $G_{z}=\alpha G_{p}$
on $A_{0}-E_{w}$, $A_{f}-E_{w}$ or $A_{f}$. More precisely, $G_{z}=\alpha
G_{p}$ on $A_{0}-E_{w}$ if $\delta<T_{s-1}$ and $d\geq 1$, on $A_{f}-E_{w}$ if
$\delta=T_{s-1}$ and $d\geq 2$, and on $A_{f}$ if $\delta>T_{s-1}$, and if
$d\geq 2$ or if $d=1$ and $\delta\neq T_{k}$.

Although $G_{z}$ is useful for the special case $\gamma=0$ and $\delta=d$, the
monomial map $f_{0}$ is just a product and the dynamics is rather easy. We
remark that if $\gamma=0$ and $\delta>d$, then $f$ is in Case 3 and $G_{z}=0$
on $A_{f}$.

Even if $d=0$, we can prove that $G_{z}=\alpha G_{p}$ on $A_{0}$ unless
$\delta=T_{s-1}$, although the Böttcher coordinate does not exist on $U$.

######  Theorem 1.7.

If $T_{s-1}>\delta>d=0$, then $f$ is in Case 2 and $G_{z}=\alpha G_{p}$ on
$A_{0}$.

If $\delta=T_{s-1}$, then the dominant term $(m_{s-1},n_{s-1})$ belongs to
Case 3 or Case 4, and the other dominant term $(m_{s},n_{s})$ belongs to Case
2. Since $\delta>m_{s-1}$ and $m_{s}\geq 2$, the equality $G_{z}=\alpha G_{p}$
holds on the two disjoint invariant regions as stated in Theorem 1.6. Hence it
follows from Corollary 2.15 that $G_{z}=\alpha G_{p}$ on $A_{0}-E_{w}$.

######  Corollary 1.8.

If $T_{s-1}=\delta>m_{s-1}>m_{s}\geq 2$, then $G_{z}=\alpha G_{p}$ on
$A_{0}-E_{w}$.

The limit

$G_{f}^{\alpha}(z,w)=\lim_{n\to\infty}\dfrac{1}{\lambda^{n}}\log\left|(z_{n}^{\alpha},w_{n})\right|$

is useful to hide the set where we do not know whether the limit $G_{z}$
exists, and it summarizes the statements for some cases in the theorems above
as follows.

######  Theorem 1.9.

If $\gamma=0$ and $\delta=d$, then $G_{f}^{\alpha}$ is plurisubharmonic on
$A_{0}$, $G_{f}^{\alpha}>\alpha G_{p}$ on $A_{f}$ and $G_{f}^{\alpha}=\alpha
G_{p}$ on $A_{0}-A_{f}$. If $\delta>d$, then $G_{f}^{\alpha}=\alpha G_{p}$ on
$A_{0}$ unless $d\leq 1$ and $\delta=T_{k}$ for some $k$.

We remark that if $\gamma=0$ and $\delta>d$, then $G_{f}^{\alpha}=0$ on
$A_{0}$ by defining that $z^{0}=1$. The three theorems above are consequences
of Theorems 2.5, 2.10 and 2.19.

######  Corollary 1.10.

At least one of the limits $G_{z}$, $G_{f}$ and $G_{f}^{\alpha}$ is
plurisubharmonic on $A_{0}$ for any case unless $d\leq 1$ and $\delta=T_{k}$
for some $k$.

The plurisubharmonic function $G_{FJ}$ constructed in [9] has to satisfy the
equation $G_{FJ}\circ f=c_{\infty}G_{FJ}$, where $c_{\infty}$ is the
asymptotic attraction rate for $f$. If $\gamma>0$ or $\delta\leq d$, then
$G_{p}$ is a candidate of $G_{FJ}$ since $c_{\infty}=\delta$ and, moreover,
$G_{f}^{\alpha}$ is another candidate when $\alpha>0$. However,
$G_{z}^{\alpha}$, $G_{z}^{\infty}$ and $G_{z}^{\alpha,+}$ do not satisfy the
equation above. On the other hand, if $\gamma=0$ and $\delta>d$, then
$G_{z}^{\alpha}$ is a candidate of $G_{FJ}$ since $c_{\infty}=d$, whereas
$G_{p}$ is not. However, $\alpha=0$ and $f_{0}$ is just a product for this
case.

###  1.5. Organization and comments

We give main results for each case in Section 2. To state our results more
precisely, we introduce the interval or rectangle $\mathcal{I}_{f}$: our
previous results, Lemma 1.1 and Theorem 1.2, hold not only for $l_{1}$ and
$l_{2}$ but also for any real number in the interval or numbers in the
rectangle, which numbers are called weights in the previous papers. In this
sense, there are infinitely many invariant wedges if $\delta\neq T_{k}$ for
any $k$. For Cases 2 and 4, we find out new invariant wedges, on which
$b_{\gamma d}z^{\gamma}w^{d}$ may not be the dominant term in $q$, and claim
that the unions of all the preimages of the wedges coincide with $A_{0}-E_{z}$
for almost all the cases. These results imply Theorems 1.3 and are important
to induce Theorems 1.5 and 1.7 and Corollary 1.8. Tables are given in each
subsection as comparison charts of the main results.

We review our previous study in [26] more precisely in Section 3. We give the
definition of $\mathcal{I}_{f}$, calculate it, and illustrate detailed
versions of Lemma 1.1 and Theorem 1.2 in terms of blow-ups.

The proofs of the results for Case 2 are given in Sections 4 and 5. With an
illustration in terms of blow-ups, we first prove the results on invariant
wedges and the unions of all the preimages of the wedges in Section 4. We next
show the existence and properties of plurisubharmonic functions, and
investigate the asymptotic behavior of the functions toward the boundaries of
the unions in Section 5, which makes clear the magnitude relation of the
unions. The proofs of the results for Cases 3 and 4 are given in Sections 6
and 7, respectively. Moreover, we provide examples of polynomial skew products
that are semiconjugate to polynomial products in Section 6, which demonstrate
that $G_{z}^{\alpha}$ and $G_{z}^{\alpha,+}$ are good tools for the study of
the weighted vertical dynamics.

Similar studies have been made for polynomial skew products in [20], [22] and
[23] to understand the dynamics near infinity. Although the idea and the
statements are similar, and we apply the same arguments to investigate the
asymptotic behavior of plurisubharmonic functions, the study in this paper is
more valuable in the sense that we only dealt with the case when the specific
term in $q$ is dominant on a wedge-shaped region, and with the specific weight
in the previous papers above.

##  2\. Summary of main results for each case

Main results for each case are given in this section. We deal with Cases 2, 3
and 4 in Sections 2.1, 2.2 and 2.3, respectively. Moreover, we provide tables
for each case, which help us to recognize the differences of properties of
plurisubharmonic functions depending on the magnitude relation of $\delta$ and
$d$, of $\delta$ and $T_{j}$ and of $\gamma$ and $0$. Furthermore, we provide
tables of properties of the typical plurisubharmonic functions in Section 2.4.

###  2.1. Main results for Case 2

Let $s>1$, $\delta\leq T_{s-1}$ and $(\gamma,d)=(n_{s},m_{s})$ in this
subsection. Then $\gamma>0$. Table 1 is given below as a comparison chart of
main results for Case 2. We remark that the descriptions on the asymptotic
behavior of $G_{z}^{\alpha}$ and $G_{z}^{\infty}$ in the table are not
precise. Actually, we have to remove $E_{z}$ and $E_{deg}$, the sets of
fibers, from the boundary $\partial A_{f}^{l}$ of $A_{f}^{l}$; see
Propositions 5.15, 5.27, 5.34 and 5.43 in Section 5 for complete descriptions.

Table 1. Comparison chart of main results for Case 2 | $\mathcal{I}_{f}$ |  $G_{z}^{\alpha}$ or $G_{z}^{\infty}$ | limsup | $G_{z}$
---|---|---|---|---
$T>d>\delta$ | $[l_{1},\infty)$ |  $G_{z}^{\alpha}$ psh on $A_{0}$ | $G_{z}^{\alpha}\to 0$ | $G_{z}=G_{z}^{\alpha}$
| $\alpha<0$ | ph on $A_{0}-E$ | as $\to\partial A_{f}^{l}$ | on $A_{0}$
|  |  $\fallingdotseq\log|z^{-\alpha}w|$ on $U$ | for any $l$ in $\mathcal{I}_{f}$ |
$T>\delta=d$ | $[l_{1},\infty)$ |  $G_{z}^{\infty}$ psh on $A_{0}-E_{z}$ | $G_{z}^{\infty}\to\infty$ | $G_{z}=-\infty$
|  | ph on $A_{0}-E$ | as $\to\partial A_{f}^{l}$ | on $A_{0}$
|  |  $\fallingdotseq\log|w|$ on $U$ | for any $l$ in $\mathcal{I}_{f}$ |
$T>\delta>d$ | $[l_{1},\alpha]$ |  $G_{z}^{\alpha}$ psh on $A_{0}-E_{z}$ |  $G_{z}^{\alpha}\to\infty$ if $l<\alpha$ | $G_{z}=\alpha G_{p}$
$d\geq 1$ | $\alpha>0$ | ph on $A_{0}-E$ |  $G_{z}^{\alpha}\to 0$ or | on $A_{0}-E_{w}$
|  |  $\fallingdotseq\log\left|w/z^{\alpha}\right|$ on $U$ | bdd if $l=\alpha$ |
$T>\delta>d$ | $[l_{1},\alpha]$ | $\nexists\phi$ |  | $G_{z}=\alpha G_{p}$
$d=0$ | $\alpha>0$ |  |  | on $A_{0}$
$T=\delta>d$ | $\\{l_{1}\\}$ |  $G_{z}^{\alpha}$ psh on $A_{f}$ | $G_{z}^{\alpha}\to 0$ | $G_{z}=\alpha G_{p}$
$d\geq 2$ | $\alpha=l_{1}$ | ph on $A_{f}-E$ | as $\to\partial A_{f}$ | on $A_{f}-E_{w}$

We first exhibit a result on invariant wedges, and give an affirmative answer
to the question whether the unions of all the preimages of the wedges coincide
with the basin. Let $U^{l}=\\{|z|<r,|w|<r|z|^{l}\\}$ and
$U_{r_{1},r_{2}}^{l}=\\{|z|<r_{1},|w|<r_{2}|z|^{l}\\}$. Note that
$U=U^{l_{1}}$. The following is a detailed version of Lemma 1.1, which is
stated also in the next section as Lemma 3.1.

######  Lemma 2.1 ([26]).

The following holds for any $l$ in $\mathcal{I}_{f}$ if $d\geq 2$ and for any
$l$ in $\mathcal{I}_{f}-\\{\alpha\\}$ if $d=1$ and $\delta<T_{s-1}$, where
$\mathcal{I}_{f}=[l_{1},\infty)$ if $\delta\leq d$ and
$\mathcal{I}_{f}=[l_{1},\alpha]$ if $\delta>d$: $f\sim f_{0}$ on $U^{l}$ as
$r\to 0$, and $f(U^{l})\subset U^{l}$ for small $r>0$.

The interval $\mathcal{I}_{f}$ plays an important role to study the dynamics
of $f$ as explained in the next section. Note that $\min\mathcal{I}_{f}=l_{1}$
and $U^{l_{1}}$ is the largest wedge among $U^{l}$ for any $l$ in
$\mathcal{I}_{f}$. We have succeeded to extend the second statement of this
lemma to the following.

######  Theorem 2.2.

The following holds for any $l>0$ if $\delta\leq d$ and for any $0<l<\alpha$
if $\delta>d$: $f(U_{r_{1},r_{2}}^{l})\subset U_{r_{1},r_{2}}^{l}$ for
suitable small $r_{1}$ and $r_{2}$.

Whereas we used the property $f\sim f_{0}$ on $U^{l}$ as $r\to 0$ to show that
$f(U^{l})\subset U^{l}$ in the lemma above, we apply different arguments to
prove this theorem; the property above does not hold since $l$ does not belong
to $\mathcal{I}_{f}$ if $0<l<l_{1}$. Note that $U_{r_{1},r_{2}}^{l}$ is larger
than $U_{r_{1},r_{2}}^{l_{1}}$ for any $0<l<l_{1}$ and that there is no
condition on the degree $d$ in the theorem.

Let $A_{f}^{l}$ be the union of all the preimages of $U^{l}$. Note that
$A_{f}=A_{f}^{l_{1}}$. We have also succeeded to give the following
affirmative answer to the question, which implies Theorem 1.3 since
$l_{1}<\alpha$ if $\delta<T_{s-1}$.

######  Theorem 2.3.

It follows that $A_{f}^{l}=A_{0}-E_{z}$ for any $l>0$ if $\delta\leq d$ and
for any $0<l<\alpha$ if $\delta>d$ and $(n_{1},m_{1})\neq(0,\delta)$.

We next exhibit main results on plurisubharmonic functions. If the Böttcher
coordinate exists on $U$, then one can show that the limits $G_{z}^{\alpha}$
and $G_{z}$ exist and are plurisubharmonic on $A_{f}-E_{z}$ and $A_{f}$,
respectively. Moreover, thanks to Theorem 1.3, we obtain the following
theorems.

######  Theorem 2.4.

Let $\delta<T_{s-1}$. If $\delta<d$, then $G_{z}^{\alpha}=G_{z}$ on $A_{0}$,
which is plurisubharmonic and negative on $A_{0}$ and
$G_{z}^{\alpha}(w)=\log\left|z^{-\alpha}w\right|+o(1)$ on $U$ as $r\to 0$. If
$\delta=d$, then $G_{z}^{\infty}$ is plurisubharmonic on $A_{0}-E_{z}$ and
$G_{z}^{\infty}(w)=\log\left|w\right|+o(1)$ on $U$ as $r\to 0$. If
$\delta>d\geq 1$, then $G_{z}^{\alpha}$ is plurisubharmonic on $A_{0}-E_{z}$
and $G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U$ as $r\to 0$.
For all the cases, the functions are pluriharmonic on $A_{0}-E$.

On the other hand, if $T_{s-1}=\delta>d\geq 2$, then $G_{z}^{\alpha}$ is
plurisubharmonic on $A_{f}$, pluriharmonic on $A_{f}-E_{w}$ and
$G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U$ as $r\to 0$.

######  Theorem 2.5.

Let $\delta<T_{s-1}$. If $\delta<d$, then $G_{z}=G_{z}^{\alpha}$ and $G_{f}=0$
on $A_{0}$. If $\delta=d$, then $G_{z}=-\infty$ and $G_{f}=G_{p}$ on $A_{0}$.
If $\delta>d\geq 1$, then $G_{z}=\alpha G_{p}$ on $A_{0}-E_{w}$ and $-\infty$
on $E_{w}$. If $d=0$, then $G_{z}=\alpha G_{p}$ on $A_{0}$. In particular, if
$\delta>d$, then $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$.

On the other hand, if $T_{s-1}=\delta>d\geq 2$, then $G_{z}=\alpha G_{p}$ on
$A_{f}-E_{w}$ and $-\infty$ on $E_{w}$. Hence $G_{f}^{\alpha}=\alpha G_{p}$ on
$A_{f}$. Moreover, if $n_{s-1}>0$, then $G_{z}=\alpha G_{p}$ on $A_{0}-E_{w}$
and $-\infty$ on $E_{w}$. Hence $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$.

By investigating the asymptotic behavior of $G_{z}^{\alpha}$ toward the
boundary $\partial A_{f}^{l}$, we can show that $A_{f}^{\alpha}$ is smaller
than $A_{f}^{l_{1}}$.

######  Proposition 2.6.

If $T_{s-1}>\delta>d\geq 1$, then $A_{f}^{\alpha}\subsetneq A_{f}^{l_{1}}$.

Moreover, we give a sufficient condition for $A_{f}^{\alpha}$ and
$A_{f}^{l_{1}}$ being unbounded.

######  Proposition 2.7.

Let $T_{s-1}>\delta>d\geq 1$. If $q$ is polynomial and
$f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then both $A_{f}^{\alpha}$ and
$A_{f}^{l_{1}}$ are unbounded.

###  2.2. Main results for Case 3

Let $s>1$, $\delta\geq T_{1}$ and $(\gamma,d)=(n_{1},m_{1})$ in this
subsection. Then $\delta\geq T_{1}\geq d\geq 1$, and $\delta>d$ if $\gamma>0$.
Table 2 is given below as a comparison chart of main results for Case 3; see
Proposition 6.4 for a complete description on the asymptotic behavior of
$G_{z}^{\alpha}$ toward $\partial A_{f}^{l}\cap A_{0}$.

Table 2. Comparison chart of main results for Case 3 |  $\gamma>0$ & $\delta>T_{1}$ |  $\gamma>0$ & $\delta=T_{1}$ |  $\gamma=0$ & $\delta>T_{1}$ |  $\gamma=0$ & $\delta=T_{1}$
---|---|---|---|---
$\mathcal{I}_{f}$ | $[\alpha,l_{2}]$ | $\\{\alpha\\}=\\{l_{2}\\}$ | $(0,l_{2}]$ | $(0,l_{2}]$
$\alpha$ | $\alpha>0$ | $\alpha>0$ | $\alpha=0$ | $\alpha=l_{2}>0$
$d$ | $\delta>d\geq 1$ | $\delta>d\geq 1$ | $\delta>d\geq 2$ | $\delta=d\geq 2$
$f_{0}$ | $(z^{\delta},z^{\gamma}w^{d})$ | $(z^{\delta},z^{\gamma}w^{d})$ | $(z^{\delta},w^{d})$ | $(z^{\delta},w^{\delta})$
$G_{z}^{\alpha}$ |  ph on $A_{f}-E_{z}$ |  ph on $A_{f}-E_{z}$ |  ph on $A_{f}$ |  ph on $A_{f}-E_{z}$
|  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$ |  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$ |  $\fallingdotseq\log|w|$ on $U$ |  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$
|  |  if $d\geq 2$ |  |
lim |  $G_{z}^{\alpha}\to 0$ or | $G_{z}^{\alpha}\to 0$ | $G_{z}^{\alpha}\to-\infty$ | $G_{z}^{\alpha}\to(l-\alpha)G_{p}$
|  bdd if $l=\alpha$ |  as $\to\partial A_{f}^{\alpha}\cap A_{0}$ |  as $\to\partial A_{f}^{l}\cap A_{0}$ |  for any $l$ in $\mathcal{I}_{f}$
| $G_{z}^{\alpha}\to-\infty$ |  |  for any $l$ in $\mathcal{I}_{f}$ | In particular,
|  for any $l>\alpha$ |  |  |  $G_{z}^{\alpha}\to 0$ if $l=\alpha$
$G_{z}^{\alpha,+}$ | psh & conti on | psh & conti on | psh & conti on | psh & conti on
| $A_{0}-E_{z}\cup E_{deg}$ | $A_{0}-E_{z}\cup E_{deg}$ | $A_{0}-E_{deg}$ | $A_{0}-E_{z}\cup E_{deg}$
|  $G_{z}^{\alpha,+}>0$ on $A_{f}^{\alpha}$ |  $G_{z}^{\alpha,+}>0$ on $A_{f}^{\alpha}$ |  |  $G_{z}^{\alpha,+}>0$ on $A_{f}^{\alpha}$
|  $=0$ on $A_{0}-A_{f}^{\alpha}$ |  $=0$ on $A_{0}-A_{f}^{\alpha}$ |  |  $=0$ on $A_{0}-A_{f}^{\alpha}$
|  if $d\geq 2$ |  |  |
$A_{f}^{l}$ | $A_{f}^{\alpha}\subsetneq A_{f}^{l}=A_{f}^{l_{2}}$ |  | $A_{f}^{l}=A_{f}^{l_{2}}$ | $A_{f}^{l^{\prime}}\subsetneq A_{f}^{l^{\prime\prime}}$
|  for any $l>\alpha$ |  |  for any $l$ in $\mathcal{I}_{f}$ |  for any $l^{\prime}<l^{\prime\prime}$
$G_{z}$ |  $G_{z}=\alpha G_{p}$ on $A_{f}$ |  $G_{z}=\alpha G_{p}$ on $A_{f}$ |  $G_{z}=0$ on $A_{f}$ |  $\fallingdotseq\log|w|$ on $U$
|  |  |  |  ph on $A_{f}$
$G_{f}^{\alpha}$ |  $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$ |  $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$ |  $G_{f}^{\alpha}=0$ on $A_{0}$ |  psh on $A_{0}-E_{deg}$
|  (psh on $A_{0}$) |  (psh on $A_{0}$) |  (psh on $A_{0}$) |  $G_{f}^{\alpha}>\alpha G_{p}$ on $A_{f}^{\alpha}$
|  |  |  |  $=\alpha G_{p}$ on $A_{0}-A_{f}^{\alpha}$

Although we want to define $\alpha$ as $\gamma/(\delta-d)$, it is not defined
if $\gamma=0$ and $\delta=d$, which corresponds to the exceptional case
$(n_{1},m_{1})=(0,\delta)$ in Theorem 2.3. To overcome this problem, we
redefine $\alpha$ as the minimum of $l\geq 0$ such that a line
$x+ly-l\delta=0$ intersects $N(q)$; this definition works not only when
$\delta\geq T_{1}$ but also when $\delta>m_{s}$. Then the line that takes the
minimum passes through $(0,\delta)$ and $(\gamma,d)$, and $\alpha$ coincides
with $\gamma/(\delta-d)$ unless $\gamma=0$ and $\delta=d$. Moreover, $\alpha$
is well-defined even if $\gamma=0$ and $\delta=d$, which coincides with
$l_{2}=n_{2}/(\delta-m_{2})$.

We exhibit results only on plurisubharmonic functions; we have no results on
invariant wedges and do not know whether the unions of all the preimages of
the wedges coincide with the basin.

The existence of the Böttcher coordinate induces the following.

######  Proposition 2.8.

Let $d\geq 2$ or $\delta>T_{1}$. If $\gamma>0$ or $\delta=d$, then $\alpha>0$
and $G_{z}^{\alpha}$ is pluriharmonic on $A_{f}-E_{z}$. Moreover,
$G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U-\\{z=0\\}$ as
$r\to 0$. If $\gamma=0$ and $\delta>d$, then $\alpha=0$ and $G_{z}^{\alpha}$
is pluriharmonic on $A_{f}$. Moreover,
$G_{z}^{\alpha}(w)=\log\left|w\right|+o(1)$ on $U$ as $r\to 0$.

Let $U^{l}=\\{|z|^{l}<r^{l}|w|,|w|<r\\}$ and $A_{f}^{l}$ be the union of all
the preimages of $U^{l}$. Note that $U=U^{l_{2}}$ and $A_{f}=A_{f}^{l_{2}}$.
Although the interval $\mathcal{I}_{f}$ plays an important role as the same as
Case 2, where $\mathcal{I}_{f}=[\alpha,l_{2}]$ if $\gamma>0$ and
$\mathcal{I}_{f}=(0,l_{2}]$ if $\gamma=0$, we avoid to use this notation in
this subsection. By investigating the asymptotic behavior of $G_{z}^{\alpha}$
and $G_{z}$ toward the boundary $\partial A_{f}^{l}$, we obtain the theorems
and proposition below.

######  Theorem 2.9.

Let $d\geq 2$ or $\delta>T_{1}$. If $\gamma>0$ or $\delta=d$, then
$G_{z}^{\alpha,+}$ is well-defined on $A_{0}-E_{z}$, plurisubharmonic and
continuous on $A_{0}-E_{z}\cup E_{deg}$ and pluriharmonic outside the boundary
of $\\{G_{z}^{\alpha,+}>0\\}\cap(A_{0}-E_{z}\cup E_{deg})$. More precisely, if
$d\geq 2$, then $G_{z}^{\alpha,+}>0$ on $A_{f}^{\alpha}$ and
$G_{z}^{\alpha,+}=0$ on $A_{0}-A_{f}^{\alpha}$, which is pluriharmonic on
$A_{0}-\partial A_{f}^{\alpha}\cup E_{z}\cup E_{deg}$. If $\gamma=0$ and
$\delta>d$, then $G_{z}^{\alpha,+}$ is well-defined on $A_{0}$,
plurisubharmonic and continuous on $A_{0}-E_{deg}$ and pluriharmonic outside
the boundary of $\\{G_{z}^{\alpha,+}>0\\}\cap(A_{0}-E_{deg})$.

######  Theorem 2.10.

If $\delta>d\geq 2$ or if $\delta>d=1$ and $\delta>T_{1}$, then $G_{z}=\alpha
G_{p}$ on $A_{f}$ and $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$. If $\gamma=0$
and $\delta=d$, then $G_{z}$ is pluriharmonic and continuous on $A_{f}$,
$G_{z}(w)=\log\left|w\right|+o(1)$ on $U$ as $r\to 0$, $G_{f}^{\alpha}$ is
plurisubharmonic and continuous on $A_{0}-E_{deg}$, $G_{f}^{\alpha}>\alpha
G_{p}$ on $A_{f}$ and $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}-A_{f}$.

We remark that if $\gamma=0$ and $\delta>d$, then $G_{z}=0$ on $A_{f}$ and
$G_{f}^{\alpha}=0$ on $A_{0}$ and that the equality $G_{f}^{\alpha}=\alpha
G_{p}$ on $A_{0}$ holds even if $d=1$ and $\delta=T_{1}$ as stated in Remark
6.8.

######  Proposition 2.11.

If $\delta>T_{1}$, then $A_{f}^{l}=A_{f}^{l_{2}}$ for any $\alpha<l\leq
l_{2}$. Moreover, if $\gamma>0$, then $A_{f}^{\alpha}\subsetneq
A_{f}^{l_{2}}$. If $\gamma=0$ and $\delta=d$, then $\delta=T_{1}$ and
$A_{f}^{l^{\prime}}\subsetneq A_{f}^{l^{\prime\prime}}$ for any
$0<l^{\prime}<l^{\prime\prime}\leq l_{2}$.

###  2.3. Main results for Case 4

Let $s>2$, $T_{k}\leq\delta\leq T_{k-1}$ and $(\gamma,d)=(n_{k},m_{k})$ for
some $2\leq k\leq s-1$ in this subsection. Then $\gamma>0$ and $\delta>d\geq
1$. Table 3 is given below as a comparison chart of main results for Case 4;
see Propositions 7.13, 7.16 and 7.19 for complete descriptions on the
asymptotic behavior of $G_{z}^{\alpha}$. In the table the notation $A$ means
the attracting set corresponding to the other dominant term and we denote
$A_{f}^{l_{1},\alpha-l_{1}}$ as $A_{f}^{\cdots}$ for short.

Table 3. Comparison chart of main results for Case 4 | $T_{k}<\delta<T_{k-1}$ | $T_{k}=\delta<T_{k-1}$ | $T_{k}<\delta=T_{k-1}$ | $T_{k}<\delta=T_{k-1}$
---|---|---|---|---
|  |  |  & $n_{k-1}>0$ |  & $n_{k-1}=0$
$\mathcal{I}_{f}$ | $[l_{1},\alpha]\times[\alpha,l_{1}+l_{2}]$ | $[l_{1},\alpha)\times\\{\alpha\\}$ | $\\{\alpha\\}\times(\alpha,l_{1}+l_{2}]$ | $\\{\alpha\\}\times(\alpha,l_{1}+l_{2}]$
| $-\\{(\alpha,\alpha)\\}$ |  |  |
$U_{r_{1},r_{2}}^{l,+}$ | $f(U_{r_{1},r_{2}}^{l,+})\subset U_{r_{1},r_{2}}^{l,+}$ | $f(U_{r_{1},r_{2}}^{l,+})\subset U_{r_{1},r_{2}}^{l,+}$ | $f(U_{r_{1},r_{2}}^{l,+})\subset U_{r_{1},r_{2}}^{l,+}$ | $f(U_{r_{1},r_{2}}^{l,+})\subset U_{r_{1},r_{2}}^{l,+}$
|  for any $0<l<\alpha$ |  for any $0<l<\alpha$ |  for any $0<l<\alpha$ |  for any $0<l<\alpha$
$A_{f}^{l,+}$ | $A_{f}^{l,+}=A_{0}-E_{z}$ | $A_{f}^{l,+}=A_{0}-E_{z}$ | $A_{f}^{l,+}=A_{0}-E_{z}$ |
|  for any $0<l<\alpha$ |  for any $0<l<\alpha$ |  for any $0<l<\alpha$ |
$\phi$ |  $\exists\phi$ on $U$ |  $\exists\phi$ on $U$ if $d\geq 2$ |  $\exists\phi$ on $U$ if $d\geq 2$ |  $\exists\phi$ on $U$ if $d\geq 2$
$G_{z}^{\alpha}$ |  ph on $A_{f}$ |  ph on $A_{f}$ |  ph on $A_{f}$ |  ph on $A_{f}$
|  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$ |  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$ |  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$ |  $\fallingdotseq\log|w/z^{\alpha}|$ on $U$
lim |  $G_{z}^{\alpha}\to\infty$ if $l<\alpha$ | $G_{z}^{\alpha}\to\infty$ | $G_{z}^{\alpha}\to 0$ | $G_{z}^{\alpha}\to 0$
sup |  $G_{z}^{\alpha}\to 0$ or bdd if |  as $\to\partial A_{f}^{l,+}$ |  as $\to\partial A_{f}^{\alpha,+}$ |  as $\to\partial A_{f}^{\alpha,+}$
|  $l=\alpha$ as $\to\partial A_{f}^{l,+}$ |  for any $l$ in $\mathcal{I}_{f}^{1}$ |  |
lim |  $G_{z}^{\alpha}\to 0$ or bdd if |  $G_{z}^{\alpha}\to 0$ as |  $G_{z}^{\alpha}\to-\infty$ as |  $G_{z}^{\alpha}\to-\infty$ as
|  $l=\alpha$, $G_{z}^{\alpha}\to-\infty$ | $\to\partial A_{f}^{l_{1},\alpha-l_{1}}\cap A_{0}$ | $\to\partial A_{f}^{\alpha,l-\alpha}\cap A_{0}$ | $\to\partial A_{f}^{\alpha,l-\alpha}\cap A_{0}$
|  if $l>\alpha$ as $\to$ |  |  for any $l$ in $\mathcal{I}_{f}^{2}$ |  for any $l$ in $\mathcal{I}_{f}^{2}$
| $\partial A_{f}^{l_{1},l-l_{1}}-\partial A_{f}^{l_{1},+}$ |  |  |
$A_{f}^{l_{\scalebox{0.5}{(1)}},l_{\scalebox{0.5}{(2)}}}$ |  $A_{f}^{l_{\scalebox{0.5}{(1)}},l_{\scalebox{0.5}{(2)}}}=A_{f}$ for | $A_{f}^{l,\alpha-l}=A_{f}$ | $A_{f}^{\alpha,l-\alpha}=A_{f}$ | $A_{f}^{\alpha,l-\alpha}=A_{f}$
|  most $l_{\scalebox{0.5}{(1)}}$ and $l_{\scalebox{0.5}{(2)}}$ |  for any $l$ in $\mathcal{I}_{f}^{1}$ |  for any $l$ in $\mathcal{I}_{f}^{2}$ |  for any $l$ in $\mathcal{I}_{f}^{2}$
| $A_{f}^{l_{1},\alpha-l_{1}}\subsetneq A_{f}$ |  |  |
| $A_{f}^{\alpha,l_{1}+l_{2}-\alpha}\subsetneq A_{f}$ |  |  |
other | $\nexists$ | $(n_{k+1},m_{k+1})$ | $(n_{k-1},m_{k-1})\Rightarrow A$ | $(n_{1},m_{1})\Rightarrow A$
term |  | Case 2 or Case 4 | Case 3 or Case 4 | Case 3
$G_{z}^{\alpha,+}$ | psh & conti on | psh & conti on | psh & conti on | psh & conti on
| $A_{0}-E_{z}\cup E_{deg}$ | $A_{0}-E_{z}\cup E_{deg}$ | $A_{0}-E_{z}\cup E_{deg}$ | $A_{0}-E_{z}\cup E_{deg}$
|  $>0$ on $A_{f}^{l_{1},\alpha-l_{1}}$ |  $>0$ on $A_{f}^{l_{1},\alpha-l_{1}}$ |  $>0$ on $A$ |  $>0$ on $A$
|  $=0$ on $A_{0}-A_{f}^{\cdots}$ |  $=0$ on $A_{0}-A_{f}^{\cdots}$ |  $=0$ on $A_{0}-A$ |  $=0$ on $A_{0}-A$
|  if $d\geq 2$ |  |  |
$G_{z}$ |  $G_{z}=\alpha G_{p}$ on $A_{f}$ |  $G_{z}=\alpha G_{p}$ on $A_{f}$ |  $G_{z}=\alpha G_{p}$ on $A_{f}$ |  $G_{z}=\alpha G_{p}$ on $A_{f}$
$G_{f}^{\alpha}$ |  $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$ |  $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$ |  $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$ |  psh on $A_{0}-E_{deg}$
|  |  |  |  $G_{f}^{\alpha}>\alpha G_{p}$ on $A$
|  |  |  |  $=\alpha G_{p}$ on $A_{0}-A$

We first exhibit a result on invariant wedges, and give a partial affirmative
answer to the question whether the unions of all the preimages of the wedges
coincide with the basin, which are similar to Case 2. Let
$U^{l_{(1)},l_{(2)}}=\\{|z|^{l_{(1)}+l_{(2)}}<r^{l_{(2)}}|w|,|w|<r|z|^{l_{(1)}}\\}$.
Note that $U=U^{l_{1},l_{2}}$. A detailed version of Lemma 1.1 is given in the
next section as Lemma 3.10, in which the wedge $U^{l_{(1)},l_{(2)}}$ and
intervals $\mathcal{I}_{f}^{1}$ and $\mathcal{I}_{f}^{2}$ are used. Let
$U^{l,+}=\\{|z|<r,|w|<r|z|^{l}\\}$. Although it might be more precise to
define $U^{l,+}$ with the inequality $|z|<r^{1+1/l_{2}}$, we define it by the
inequality $|z|<r$ for simplicity. Lemma 3.10 in the next section induces the
following.

######  Lemma 2.12.

The following holds for any $l$ in $\mathcal{I}_{f}^{1}$ if $d\geq 2$ and for
any $l$ in $\mathcal{I}_{f}^{1}-\\{\alpha\\}$ if $d=1$ and
$T_{k}<\delta<T_{k-1}$, where $\mathcal{I}_{f}^{1}=[l_{1},\alpha]$ if
$T_{k}<\delta<T_{k-1}$, $\mathcal{I}_{f}^{1}=[l_{1},\alpha)$ if
$\delta=T_{k}$, and $\mathcal{I}_{f}^{1}=\\{l_{1}\\}=\\{\alpha\\}$ if
$\delta=T_{k-1}$: $f(U^{l,+})\subset U^{l,+}$ for small $r>0$.

Note that $\min\mathcal{I}_{f}^{1}=l_{1}$ and $U^{l_{1},+}$ is the largest
wedge among $U^{l,+}$ for any $l$ in $\mathcal{I}_{f}^{1}$. Let
$U_{r_{1},r_{2}}^{l,+}=\\{|z|<r_{1},|w|<r_{2}|z|^{l}\\}$. We can extend this
lemma to the following theorem by the same arguments as Case 2.

######  Theorem 2.13.

The following holds for any $0<l<\alpha$: $f(U_{r_{1},r_{2}}^{l,+})\subset
U_{r_{1},r_{2}}^{l,+}$ for suitable small $r_{1}$ and $r_{2}$.

Note that $U_{r_{1},r_{2}}^{l}$ is larger than $U_{r_{1},r_{2}}^{l_{1}}$ for
any $0<l<l_{1}$ and that there is no condition on the degree $d$ in this
theorem.

Let $A_{f}^{l,+}$ be the union of all the preimages of $U^{l,+}$. We obtain
the following partial affirmative answer to the question as the same as Case
2.

######  Theorem 2.14.

If $(n_{1},m_{1})\neq(0,\delta)$, then $A_{f}^{l,+}=A_{0}-E_{z}$ for any
$0<l<\alpha$.

This theorem implies the following since $l_{1}<\alpha$ if $\delta<T_{k-1}$.

######  Corollary 2.15.

If $\delta<T_{k-1}$, then $A_{f}^{l_{1},+}=A_{0}-E_{z}$.

We next exhibit main results on plurisubharmonic functions. The existence of
the Böttcher coordinate induces the following.

######  Proposition 2.16.

If $d\geq 2$ or $T_{k}<\delta<T_{k-1}$, then $G_{z}^{\alpha}$ is pluriharmonic
on $A_{f}$ and $G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U$
as $r\to 0$.

Let $A_{f}^{l_{(1)},l_{(2)}}$ be the union of all the preimages of
$U^{l_{(1)},l_{(2)}}$. Thanks to Corollary 2.15, we obtain the following
theorem by investigating the asymptotic behavior of $G_{z}^{\alpha}$ toward
the boundary $\partial A_{f}^{l_{(1)},l_{(2)}}$.

######  Theorem 2.17.

Let $\delta<T_{k-1}$. If $d\geq 2$ or $\delta>T_{k}$, then $G_{z}^{\alpha,+}$
is well-defined on $A_{0}-E_{z}$, plurisubharmonic and continuous on
$A_{0}-E_{z}\cup E_{deg}$ and pluriharmonic outside the boundary of
$\\{G_{z}^{\alpha,+}>0\\}\cap(A_{0}-E_{z}\cup E_{deg})$. More precisely, if
$d\geq 2$, then $G_{z}^{\alpha,+}>0$ on $A_{f}^{l_{1},\alpha-l_{1}}$ and
$G_{z}^{\alpha,+}=0$ on $A_{0}-A_{f}^{l_{1},\alpha-l_{1}}$, which is
pluriharmonic on $A_{0}-\partial A_{f}^{l_{1},\alpha-l_{1}}\cup E_{z}\cup
E_{deg}$.

The investigation of the asymptotic behavior of $G_{z}^{\alpha}$ toward
$\partial A_{f}^{l_{(1)},l_{(2)}}$ also shows us the magnitude relation of
$A_{f}^{l_{(1)},l_{(2)}}$ among any $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ and
$l_{(2)}$ in $\mathcal{I}_{f}^{2}$.

######  Proposition 2.18.

If $T_{k}<\delta<T_{k-1}$, then $A_{f}^{l_{(1)},l_{(2)}}=A_{f}^{l_{1},l_{2}}$
for any $l_{1}\leq l_{(1)}<\alpha$ and $\alpha<l_{(1)}+l_{(2)}\leq
l_{1}+l_{2}$. Moreover, $A_{f}^{l_{1},\alpha-l_{1}}\subsetneq
A_{f}^{l_{1},l_{2}}$, $A_{f}^{\alpha,l_{1}+l_{2}-\alpha}\subsetneq
A_{f}^{l_{1},l_{2}}$ and $A_{f}^{l_{1},\alpha-l_{1}}\cap
A_{f}^{\alpha,l_{1}+l_{2}-\alpha}=\emptyset$. If $\delta=T_{k}$ and $d\geq 2$,
then $A_{f}^{l,\alpha-l}=A_{f}^{l_{1},\alpha-l_{1}}$ for any $l_{1}\leq
l<\alpha$. If $\delta=T_{k-1}$ and $d\geq 2$, then
$A_{f}^{\alpha,l-\alpha}=A_{f}^{\alpha,l_{2}-\alpha}$ for any $0<l\leq l_{2}$.

The existence of the Böttcher coordinate and Corollary 2.15 induce the
following.

######  Theorem 2.19.

If $d\geq 2$ or $T_{k}<\delta<T_{k-1}$, then $G_{z}=\alpha G_{p}$ on $A_{f}$.
If $(n_{1},m_{1})\neq(0,\delta)$, then $G_{f}^{\alpha}=\alpha G_{p}$ on
$A_{0}$.

###  2.4. Tables for typical plurisubharmonic functions

We end this section with the following two tables for the typical
plurisubharmonic functions.

Table 4. Comparison chart of typical psh functions | $G_{z}$ | $G_{f}$ | $G_{f}^{\alpha}$
---|---|---|---
$\delta<d$ |  $G_{z}=G_{z}^{\alpha}$ on $A_{0}$ |  $G_{f}=0$ on $A_{0}-E_{z}$ |  $G_{f}^{\alpha}=0$ on $A_{0}-E_{z}$
| (psh on $A_{0}$) |  $=-\infty$ on $E_{z}$ |  $=\infty$ on $E_{z}$
$\delta=d$ and |  $G_{z}=-\infty$ on $A_{0}$ |  $G_{f}=G_{p}$ on $A_{0}$ |
$\gamma>0$ |  | (psh on $A_{0}$) |
$\delta=d$ and | ph on $A_{f}$ | ph on $A_{f}$ | psh on $A_{0}$
$\gamma=0$ |  $G_{z}\fallingdotseq\log|w|$ on $U$ |  |  $G_{f}^{\alpha}\fallingdotseq\log|w|$ on $U$
$\delta>d$, and |  $G_{z}=\alpha G_{p}$ on … | psh on … |  $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$
$d\neq 1$ or $\delta\neq T_{k}$ | (see Table 5) | (see Table 5) | (psh on $A_{0}$)
Table 5. Comparison chart of $G_{z}$ and $G_{f}$ when $\delta>d$ on … | if …
---|---
$A_{0}-E_{w}$ | if $\delta<T_{s-1}$ and $d\geq 1$
$A_{0}$ | if $\delta<T_{s-1}$ and $d=0$
$A_{f}-E_{w}$ | if $\delta=T_{s-1}$ and $d\geq 2$
$A_{f}$ | if $\delta>T_{s-1}$, and if $d\geq 2$
| or if $d=1$ and $\delta\neq T_{k}$

##  3\. Intervals of weights and Blow-ups

More precise review of the study in the previous paper [26] is given in this
section. We deal with Cases 2, 3 and 4 in Sections 3.1, 3.2 and 3.3,
respectively. For each case, we give the definition of the interval or
rectangle $\mathcal{I}_{f}$, calculate it, restate Lemma 1.1 and Theorem 1.2
by using $\mathcal{I}_{f}$, and illustrate them in terms of blow-ups.

###  3.1. Intervals of weights and Blow-ups for Case 2

Let $s>1$,

$\delta\leq T_{s-1},\ (\gamma,d)=(n_{s},m_{s})\text{ and
}l_{1}=\frac{n_{s}-n_{s-1}}{m_{s-1}-m_{s}}.$

Note that $\gamma>0$ and $d\geq 0$ by the setting.

####  3.1.1. Intervals and Böttcher coordinates

We define the interval $\mathcal{I}_{f}$ as

$\mathcal{I}_{f}=\left\\{\ l>0\ |\begin{array}[]{lcr}l\delta\leq\gamma+ld\leq
i+lj\text{ for any $i$ and $j$ s.t. }b_{ij}\neq 0\end{array}\right\\}.$

It follows that $\min\mathcal{I}_{f}=l_{1}$. In fact, if $\delta>d$, then

$\mathcal{I}_{f}=\left[\max\left\\{\dfrac{\gamma-i}{j-d}\right\\},\dfrac{\gamma}{\delta-d}\right]=\left[\frac{\gamma-
n_{s-1}}{m_{s-1}-d},\dfrac{\gamma}{\delta-d}\right]=\left[l_{1},\alpha\right],$

which is mapped to $[\delta,T_{s-1}]$ by the transformation $l\to
l^{-1}\gamma+d$. If $\delta\leq d$, then the inequality $l\delta\leq\gamma+ld$
is trivial and so $\mathcal{I}_{f}=[l_{1},\infty)$. We remark that

$\mathcal{I}_{f}=\left\\{\ l>0\ |\begin{array}[]{lcr}l\delta\leq\gamma+ld\leq
n_{j}+lm_{j}\text{ for any }1\leq j\leq s-1\end{array}\right\\},$

which expression is used in Case 4.

Let $U^{l}=\\{|z|<r,|w|<r|z|^{l}\\}$. Note that $U=U^{l_{1}}$.

######  Lemma 3.1 ([26]).

The following hold for any $l$ in $\mathcal{I}_{f}$ if $d\geq 2$ and for any
$l$ in $\mathcal{I}_{f}-\\{\alpha\\}$ if $d=1$ and $\delta<T_{s-1}$: $f\sim
f_{0}$ on $U^{l}$ as $r\to 0$, and $f(U^{l})\subset U^{l}$ for small $r>0$.

This lemma induces the Böttcher coordinate for $f$ on $U^{l}$.

######  Theorem 3.2 ([26]).

The following holds for any $l$ in the previous lemma: $\phi$ is biholomorphic
on $U^{l}$ and conjugates $f$ to $f_{0}$ for small $r>0$. Moreover, $\phi\sim
id$ on $U^{l}$ as $r\to 0$.

Note that $U^{l_{1}}$ is the largest open set among $U^{l}$ for any $l$ in
$\mathcal{I}_{f}$.

####  3.1.2. Blow-ups

Assuming that $l$ in $\mathcal{I}_{f}$ is integer, we illustrate our previous
results in terms of blow-ups. We also assume that $p(z)=z^{\delta}$ and
$b_{\gamma d}=1$ for simplicity. Let $\pi_{1}(z,c)=(z,z^{l}c)$ and
$\tilde{f}=\pi_{1}^{-1}\circ f\circ\pi_{1}$. Note that $\pi_{1}$ is the $l$-th
compositions of the blow-up $(z,c)\to(z,zc)$. Then

$\displaystyle\tilde{f}(z,c)$
$\displaystyle=(\tilde{p}(z),\tilde{q}(z,c))=\left(p(z),\
\dfrac{q(z,z^{l}c)}{p(z)^{l}}\right)$ $\displaystyle=\left(z^{\delta},\
z^{\gamma+ld-l\delta}c^{d}+\sum b_{ij}z^{i+lj-l\delta}c^{j}\right)$
$\displaystyle=\left(z^{\delta},\ z^{\gamma+ld-l\delta}c^{d}\left\\{1+\sum
b_{ij}z^{(i+lj)-(\gamma+ld)}c^{j-d}\right\\}\right).$

Let $\tilde{\gamma}=\gamma+ld-l\delta$ and $\tilde{i}=i+lj-l\delta$. Then
$\tilde{q}(z,c)=z^{\tilde{\gamma}}c^{d}+\sum b_{ij}z^{\tilde{i}}c^{j}$ and
$0\leq\tilde{\gamma}\leq\tilde{i}$ for any $(i,j)$ such that $b_{ij}\neq 0$.
Therefore, the Newton polygon of $\tilde{q}$ has just one vertex
$(\tilde{\gamma},d)$, $N(\tilde{q})=D(\tilde{\gamma},d)$, and we have the
following.

######  Proposition 3.3 ([26]).

If $l$ in $\mathcal{I}_{f}$ is integer, then $\tilde{f}$ is holomorphic and
skew product on a neighborhood of the origin. More precisely,

$\tilde{f}(z,c)=\left(z^{\delta},\
z^{\tilde{\gamma}}c^{d}\left\\{1+\eta(z,c)\right\\}\right),$

where $\eta\to 0$ as $z$ and $c\to 0$. Moreover, it has a superattracting
fixed point at the origin if $d\geq 2$ or if $d=1$ and $l\delta<\gamma+ld$.

######  Remark 3.4.

Even if $l$ is rational, we can lift $f$ to a holomorphic skew product similar
to $\tilde{f}$ as stated in Section 5.1 in [26]. In particular, $l_{1}$ is
rational.

Because $\tilde{f}$ is a holomorphic skew product in Case 1, it is easy to
construct the Böttcher coordinate for $\tilde{f}$ on a neighborhood of the
origin, which induces the Böttcher coordinate for $f$ on $U^{l}$.

######  Remark 3.5.

The affine transformation

$A_{1}\begin{pmatrix}i\\\ j\end{pmatrix}=\begin{pmatrix}i+lj-l\delta\\\
j\end{pmatrix}=\begin{pmatrix}1&l\\\ 0&1\end{pmatrix}\begin{pmatrix}i\\\
j\end{pmatrix}-\begin{pmatrix}l\delta\\\ 0\end{pmatrix}$

maps the basis $\\{(1,0),(-l,1)\\}$ to $\\{(1,0),(0,1)\\}$. In other words, it
maps a horizontal line and a line with slope $-l^{-1}$ to the same horizontal
line and a vertical line. More precisely, it maps the line with slope
$-l^{-1}$ that passes through $(i,\delta)$ to the vertical line $\\{x=i\\}$.
In particular, if $l=l_{1}$, then it maps the line $L_{s-1}$ with slope
$-l_{1}^{-1}$ and the horizontal line that intersect at $(\gamma,d)$, to the
vertical and horizontal lines that intersect at $(\tilde{\gamma},d)$.

###  3.2. Intervals of weights and Blow-ups for Case 3

Let $s>1$,

$T_{1}\leq\delta,\ (\gamma,d)=(n_{1},m_{1})\text{ and
}l_{2}=\frac{n_{2}-n_{1}}{m_{1}-m_{2}}.$

Note that $\gamma\geq 0$ and $\delta\geq T_{1}\geq d\geq 1$ by the setting.

####  3.2.1. Intervals and Böttcher coordinates

We define the interval $\mathcal{I}_{f}$ as

$\mathcal{I}_{f}=\left\\{\ l>0\ |\begin{array}[]{lcr}\gamma+ld\leq
l\delta\text{ and }\gamma+ld\leq i+lj\text{ for any $i$ and $j$ s.t.
}b_{ij}\neq 0\end{array}\right\\}.$

It follows that $\max\mathcal{I}_{f}=l_{2}$. In fact, if $\gamma>0$, then
$\delta>d$ and

$\mathcal{I}_{f}=\left[\dfrac{\gamma}{\delta-d},\min\left\\{\dfrac{i-\gamma}{d-j}\right\\}\right]=\left[\dfrac{\gamma}{\delta-d},\frac{n_{2}-\gamma}{d-m_{2}}\right]=\left[\alpha,l_{2}\right],$

which is mapped to $[T_{1},\delta]$ by the transformation $l\to
l^{-1}\gamma+d$. If $\gamma=0$, then the inequality $\gamma+ld\leq l\delta$ is
trivial since $d\leq\delta$, and so $\mathcal{I}_{f}=(0,l_{2}]$.

Let $U^{l}=\\{|z|^{l}<r^{l}|w|,|w|<r\\}$. Note that $U=U^{l_{2}}$.

######  Lemma 3.6 ([26]).

The following hold for any $l$ in $\mathcal{I}_{f}$ if $d\geq 2$ and for any
$l$ in $\mathcal{I}_{f}-\\{\alpha\\}$ if $d=1$ and $\delta>T_{1}$: $f\sim
f_{0}$ on $U^{l}$ as $r\to 0$, and $f(U^{l})\subset U^{l}$ for small $r>0$.

######  Theorem 3.7 ([26]).

The following holds for any $l$ in the previous lemma: $\phi$ is biholomorphic
on $U^{l}$ and conjugates $f$ to $f_{0}$ for small $r>0$. Moreover, $\phi\sim
id$ on $U^{l}$ as $r\to 0$.

Note that $U^{l_{2}}$ is the largest wedge among $U^{l}$ for any $l$ in
$\mathcal{I}_{f}$.

####  3.2.2. Blow-ups

Assuming that $l^{-1}$ is integer for $l$ in $\mathcal{I}_{f}$, we illustrate
our previous results in terms of blow-ups. Let $\pi_{2}(t,w)=(tw^{l^{-1}},w)$
and $\tilde{f}=\pi_{2}^{-1}\circ f\circ\pi_{2}$. Note that $\pi_{2}$ is the
$l^{-1}$-th compositions of the blow-up $(t,w)\to(tw,w)$. Let $b_{\gamma d}=1$
for simplicity. Then

$\tilde{f}(t,w)=(\tilde{p}(t,w),\tilde{q}(t,w))=\left(\dfrac{p(tw^{l^{-1}})}{q(tw^{l^{-1}},w)^{l^{-1}}},\
q(tw^{l^{-1}},w)\right)\text{ and }$ $\displaystyle\tilde{q}(t,w)$
$\displaystyle=q(tw^{l^{-1}},w)=t^{\gamma}w^{l^{-1}\gamma+d}+\sum
b_{ij}t^{i}w^{l^{-1}i+j}$
$\displaystyle=t^{\gamma}w^{l^{-1}\gamma+d}\left\\{1+\sum
b_{ij}t^{i-\gamma}w^{(l^{-1}i+j)-(l^{-1}\gamma+d)}\right\\}.$

Let $\tilde{d}=l^{-1}\gamma+d$ and $\tilde{j}=l^{-1}i+j$. Then
$\tilde{q}(t,w)=t^{\gamma}w^{\tilde{d}}+\sum b_{ij}t^{i}w^{\tilde{j}}$,
$d\leq\tilde{d}\leq\delta$ and $\tilde{d}\leq\tilde{j}$ for any $(i,j)$ such
that $b_{ij}\neq 0$. Therefore, the Newton polygon of $\tilde{q}$ has just one
vertex $(\gamma,\tilde{d})$, $N(\tilde{q})=D(\gamma,\tilde{d})$, and we have
the following.

######  Proposition 3.8 ([26]).

If $l^{-1}$ is integer for $l$ in $\mathcal{I}_{f}$, then $\tilde{f}$ is
holomorphic on a neighborhood of the origin. More precisely,

$\tilde{f}(t,w)=\left(t^{\delta-l^{-1}\gamma}w^{l^{-1}(\delta-\tilde{d})}\\{1+\zeta(t,w)\\},\
t^{\gamma}w^{\tilde{d}}\\{1+\eta(t,w)\\}\right),$

where $\zeta$ and $\eta\to 0$ as $t$ and $w\to 0$. Moreover, it has a
superattracting fixed point at the origin if $d\geq 2$ or if $d=1$ and
$\gamma+ld<l\delta$.

Although $\tilde{f}$ is not skew product, it is a perturbation of a monomial
map near the origin. Hence we can construct the Böttcher coordinate for
$\tilde{f}$ on a neighborhood of the origin by similar arguments as in Section
6 in [26], or one may refer to pp.498-499 in [7], which induces the Böttcher
coordinate for $f$ on $U^{l}$.

######  Remark 3.9.

The linear transformation

$A_{2}\begin{pmatrix}i\\\ j\end{pmatrix}=\begin{pmatrix}i\\\
l^{-1}i+j\end{pmatrix}=\begin{pmatrix}1&0\\\
l^{-1}&1\end{pmatrix}\begin{pmatrix}i\\\ j\end{pmatrix}$

maps the basis $\\{(1,-l^{-1}),(0,1)\\}$ to $\\{(1,0),(0,1)\\}$. In other
words, it maps a line with slope $-l^{-1}$ and a vertical line to a horizontal
line and the same vertical line. In particular, if $l=l_{2}$, then it maps the
line $L_{1}$ with slope $-l_{2}^{-1}$ and the vertical line that intersect at
$(\gamma,d)$, to the horizontal and vertical lines that intersect at
$(\gamma,\tilde{d})$.

###  3.3. Intervals, Rectangles and Blow-ups for Case 4

Let $s>2$,

$T_{k}\leq\delta\leq T_{k-1}\text{ for some }2\leq k\leq s-1,$
$(\gamma,d)=(n_{k},m_{k}),\,l_{1}=\frac{n_{k}-n_{k-1}}{m_{k-1}-m_{k}}\text{
and }l_{1}+l_{2}=\frac{n_{k+1}-n_{k}}{m_{k}-m_{k+1}}.$

Note that $\gamma>0$ and $\delta>d\geq 1$ by the setting.

####  3.3.1. Intervals, Rectangles and Böttcher coordinates

We define the interval $\mathcal{I}_{f}^{1}$ as

$\mathcal{I}_{f}^{1}=\left\\{\ l_{(1)}>0\
\Bigg{|}\begin{array}[]{lcr}l_{(1)}\delta\leq\gamma+l_{(1)}d\\\
\gamma+l_{(1)}d\leq n_{j}+l_{(1)}m_{j}\text{ for }1\leq j\leq k-1\\\
\gamma+l_{(1)}d<n_{j}+l_{(1)}m_{j}\text{ for }k+1\leq j\leq
s\end{array}\right\\},$

the interval $\mathcal{I}_{f}^{2}$ associated with $l_{(1)}$ in
$\mathcal{I}_{f}^{1}$ as

$\mathcal{I}_{f}^{2}=\mathcal{I}_{f}^{2}(l_{(1)})=\left\\{\ l_{(2)}>0\
\Big{|}\begin{array}[]{lcr}\tilde{\gamma}+l_{(2)}d\leq l_{(2)}\delta\text{ and
}\tilde{\gamma}+l_{(2)}d\leq\tilde{i}+l_{(2)}j\\\ \text{for any $i$ and $j$
s.t. }b_{ij}\neq 0\end{array}\right\\},$

where $\tilde{\gamma}=\gamma+l_{(1)}d-l_{(1)}\delta$ and
$\tilde{i}=i+l_{(1)}j-l_{(1)}\delta$, and the rectangle $\mathcal{I}_{f}$ as

$\mathcal{I}_{f}=\\{(l_{(1)},l_{(1)}+l_{(2)})\ |\
l_{(1)}\in\mathcal{I}_{f}^{1},l_{(2)}\in\mathcal{I}_{f}^{2}\\}.$

Let us calculate the intervals and rectangle more practically. Note that
$\alpha>0$ since $\delta>d$ and $\gamma>0$. Since $n_{j}<\gamma$ and $m_{j}>d$
for any $1\leq j\leq k-1$, and $n_{j}>\gamma$ and $m_{j}<d$ for any $k+1\leq
j\leq s$,

$\displaystyle\mathcal{I}_{f}^{1}$ $\displaystyle=$
$\displaystyle\left[\max_{1\leq j\leq k-1}\left\\{\dfrac{\gamma-
n_{j}}{m_{j}-d}\right\\},\min_{k+1\leq j\leq
s}\left\\{\dfrac{n_{j}-\gamma}{d-m_{j}}\right\\}\right)\cap\left(0,\dfrac{\gamma}{\delta-d}\right]$
$\displaystyle=$ $\displaystyle\left[\dfrac{\gamma-
n_{k-1}}{m_{k-1}-d},\dfrac{n_{k+1}-\gamma}{d-m_{k+1}}\right)\cap\left(0,\dfrac{\gamma}{\delta-d}\right]=[l_{1},l_{1}+l_{2})\cap(0,\alpha].$

On the other hand,

$\displaystyle\mathcal{I}_{f}^{2}$ $\displaystyle=$
$\displaystyle\left[\dfrac{\tilde{\gamma}}{\delta-d},\dfrac{\tilde{n}_{k+1}-\tilde{\gamma}}{d-m_{k+1}}\right]\cap\mathbb{R}_{>0}=\left[\dfrac{\gamma}{\delta-d}-l_{(1)},\dfrac{n_{k+1}-\gamma}{d-m_{k+1}}-l_{(1)}\right]\cap\mathbb{R}_{>0}$
$\displaystyle=$ $\displaystyle[\alpha-
l_{(1)},l_{1}+l_{2}-l_{(1)}]\cap\mathbb{R}_{>0}.$

If $T_{k}<\delta<T_{k-1}$, then it follows from the inequality
$l_{1}<\alpha<l_{1}+l_{2}$ that

$\mathcal{I}_{f}^{1}=[l_{1},\alpha],\ \mathcal{I}_{f}^{2}=[\alpha-
l_{(1)},l_{1}+l_{2}-l_{(1)}]\cap\mathbb{R}_{>0}\text{ and so }$
$\mathcal{I}_{f}=[l_{1},\alpha]\times[\alpha,l_{1}+l_{2}]-\\{(\alpha,\alpha)\\}.$

If $T_{k}=\delta<T_{k-1}$, then it follows from the inequality
$l_{1}<\alpha=l_{1}+l_{2}$ that

$\mathcal{I}_{f}^{1}=[l_{1},\alpha),\ \mathcal{I}_{f}^{2}=\\{\alpha-
l_{(1)}\\}\text{ and so }\mathcal{I}_{f}=[l_{1},\alpha)\times\\{\alpha\\}.$

If $T_{k}<\delta=T_{k-1}$, then it follows from the inequality
$l_{1}=\alpha<l_{1}+l_{2}$ that

$\mathcal{I}_{f}^{1}=\\{\alpha\\},\ \mathcal{I}_{f}^{2}=(0,l_{2}]\text{ and so
}\mathcal{I}_{f}=\\{\alpha\\}\times(\alpha,l_{1}+l_{2}].$

In particular, $\min\mathcal{I}_{f}^{1}=l_{1}$ and $\max\\{l_{(1)}+l_{(2)}\ |\
l_{(1)}\in\mathcal{I}_{f}^{1},l_{(2)}\in\mathcal{I}_{f}^{2}\\}=l_{1}+l_{2}$.

Let
$U^{l_{(1)},l_{(2)}}=\\{|z|^{l_{(1)}+l_{(2)}}<r^{l_{(2)}}|w|,|w|<r|z|^{l_{(1)}}\\}$.
Note that $U=U^{l_{1},l_{2}}$.

######  Lemma 3.10 ([26]).

The following hold for any $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in
$\mathcal{I}_{f}^{2}$ if $d\geq 2$, and for any $l_{(1)}$ in
$\mathcal{I}_{f}^{1}-\\{\alpha\\}$ and $l_{(2)}$ in
$\mathcal{I}_{f}^{2}-\\{\alpha-l_{(1)}\\}$ if $d=1$ and
$T_{k}<\delta<T_{k-1}$: $f\sim f_{0}$ on $U^{l_{(1)},l_{(2)}}$ as $r\to 0$,
and $f(U^{l_{(1)},l_{(2)}})\subset U^{l_{(1)},l_{(2)}}$ for small $r>0$.

######  Theorem 3.11 ([26]).

The following holds for any $l_{(1)}$ and $l_{(2)}$ in the previous lemma:
$\phi$ is biholomorphic on $U^{l}$ and conjugates $f$ to $f_{0}$ for small
$r>0$. Moreover, $\phi\sim id$ on $U^{l_{(1)},l_{(2)}}$ as $r\to 0$.

Note that $U^{l_{1},l_{2}}$ is the largest wedge among $U^{l_{(1)},l_{(2)}}$
for any $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in
$\mathcal{I}_{f}^{2}$.

####  3.3.2. Blow-ups

Assuming that $l_{(1)}$ and $l_{(2)}^{-1}$ are integer for $l_{(1)}$ in
$\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in $\mathcal{I}_{f}^{2}$, we illustrate
our previous results in terms of blow-ups. The strategy is to combine the
blow-ups in Cases 2 and 3. We first blow-up $f$ to $\tilde{f}_{1}$ by
$\pi_{1}$ as in Case 2. It then turns out that $\tilde{f}_{1}$ is a
holomorphic skew product in Case 3. We next blow-up $\tilde{f}_{1}$ to
$\tilde{f}_{2}$ by $\pi_{2}$ as in Case 3. The map $\tilde{f}_{2}$ is a
perturbation of a monomial map near the origin, and we obtain the Böttcher
coordinate.

Let us restate the first blow-up. Let $\pi_{1}(z,c)=(z,z^{l_{(1)}}c)$ and
$\tilde{f}_{1}=\pi_{1}^{-1}\circ f\circ\pi_{1}$ as in Case 2, and let
$\tilde{f}_{1}(z,c)=(\tilde{p}_{1}(z),\tilde{q}_{1}(z,c))$. Let
$p(z)=z^{\delta}$ and $b_{\gamma d}=1$ for simplicity. Then

$\tilde{f}_{1}(z,c)=\left(z^{\delta},z^{\gamma+l_{(1)}d-l_{(1)}\delta}c^{d}+\sum
b_{ij}z^{i+l_{(1)}j-l_{(1)}\delta}c^{j}\right).$

Let $\tilde{\gamma}=\gamma+l_{(1)}d-l_{(1)}\delta$ and
$\tilde{i}=i+l_{(1)}j-l_{(1)}\delta$ as in Case 2. Then
$0\leq\tilde{\gamma}\leq\tilde{i}$ for any $(i,j)$ such that $b_{ij}\neq 0$,
and we have the following.

######  Proposition 3.12 ([26]).

If $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ is integer, then $\tilde{f}_{1}$ is
holomorphic and skew product on a neighborhood of the origin. More precisely,

$\tilde{f}_{1}(z,c)=\left(z^{\delta},z^{\tilde{\gamma}}c^{d}+\sum
b_{ij}z^{\tilde{i}}c^{j}\right).$

Moreover, it has a superattracting fixed point at the origin if $d\geq 2$ or
if $d=1$ and $l_{(1)}\delta<\gamma+l_{(1)}d$.

Note that $(\tilde{\gamma},d)$ is the vertex of the Newton polygon
$N(\tilde{q}_{1})$ whose $x$-coordinate is minimum, and that
$N(\tilde{q}_{1})$ has other vertices such as $(\tilde{n}_{k+1},m_{k+1})$.
Hence the situation resembles that of Case 3.

We illustrate that $\tilde{f}_{1}$ is actually in Case 3. Recall that $L_{k}$
is the line passing through the vertices $(\gamma,d)$ and $(n_{k+1},m_{k+1})$,
and $T_{k}$ is the $y$-intercept of $L_{k}$. The slope of $L_{k}$ is
$-(l_{1}+l_{2})^{-1}$ and so $T_{k}=(l_{1}+l_{2})^{-1}\gamma+d$. Let
$\tilde{L}_{k}$ be the line passing through the vertices $(\tilde{\gamma},d)$
and $(\tilde{n}_{k+1},m_{k+1})$, and $\tilde{T}_{k}$ the $y$-intercept of
$\tilde{L}_{k}$, where $\tilde{n}_{k+1}=n_{k+1}+l_{(1)}m_{k+1}-l_{(1)}\delta$.
Then the slope of $\tilde{L}_{k}$ is $-(l_{1}+l_{2}-l_{(1)})^{-1}$ and so
$\tilde{T}_{k}=(l_{1}+l_{2}-l_{(1)})^{-1}\tilde{\gamma}+d$. Therefore, the
inequality $T_{k}\leq\delta$ implies the inequality $\tilde{T}_{k}\leq\delta$.
More precisely, $\tilde{T}_{k}<\delta$ if $T_{k}<\delta$, and
$\tilde{T}_{k}=\delta$ if $T_{k}=\delta$. In particular,

######  Proposition 3.13 ([26]).

If $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ is integer, then $\tilde{f}_{1}$ is a
holomorphic skew product in Case 3.

Let us restate the second blow-up. Let $\pi_{2}(t,c)=(tc^{l_{(2)}^{-1}},c)$
and $\tilde{f}_{2}=\pi_{2}^{-1}\circ\tilde{f}_{1}\circ\pi_{2}$ as in Case 3,
and let $\tilde{f}_{2}(t,c)=(\tilde{p}_{2}(t,c),\tilde{q}_{2}(t,c))$. Let
$\tilde{d}=l_{(2)}^{-1}\tilde{\gamma}+d$ and
$\tilde{j}=l_{(2)}^{-1}\tilde{i}+j$ as in Case 3. Then
$\tilde{d}\leq\tilde{j}$ and $\tilde{d}\leq\delta$ for any $(i,j)$ such that
$b_{ij}\neq 0$. Note that $(\tilde{\gamma},\tilde{d})$ is the minimum in the
sense that $0\leq\tilde{\gamma}\leq\tilde{i}$ and
$d\leq\tilde{d}\leq\tilde{j}$ for any $(i,j)$ such that $b_{ij}\neq 0$.
Therefore, the Newton polygon of $\tilde{q}_{2}$ has just one vertex
$(\tilde{\gamma},\tilde{d})$: $N(\tilde{q}_{2})=D(\tilde{\gamma},\tilde{d})$,
and we have the following.

######  Proposition 3.14 ([26]).

If $l_{(1)}$ and $l_{(2)}^{-1}$ are integer for $l_{(1)}$ in
$\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in $\mathcal{I}_{f}^{2}$, then
$\tilde{f}_{2}$ is holomorphic on a neighborhood of the origin. More
precisely,

$\tilde{f}_{2}(t,c)=\left(t^{\delta-
l_{(2)}^{-1}\tilde{\gamma}}c^{l_{(2)}^{-1}(\delta-\tilde{d})}\\{1+\zeta_{2}(t,c)\\},\
t^{\tilde{\gamma}}c^{\tilde{d}}\\{1+\eta_{2}(t,c)\\}\right),$

where $\zeta_{2}$ and $\eta_{2}\to 0$ as $t$ and $c\to 0$. Moreover, it has a
superattracting fixed point at the origin if $d\geq 2$ or if $d=1$,
$l_{(1)}\delta<\gamma+l_{(1)}d$ and $\tilde{\gamma}+l_{(2)}d<l_{(2)}\delta$.

Consequently, we can construct the Böttcher coordinate for $\tilde{f}_{2}$ on
$\\{|t|<r,|c|<r\\}$, which induces that for $\tilde{f}_{1}$ on
$\\{|z|<r|c|^{l_{(2)}^{-1}},|c|<r\\}$ and that for $f$ on
$U^{l_{(1)},l_{(2)}}$.

######  Remark 3.15.

The affine transformation

$A\begin{pmatrix}i\\\ j\end{pmatrix}=\begin{pmatrix}1&0\\\
l_{2}^{-1}&1\end{pmatrix}\left\\{\begin{pmatrix}1&l_{1}\\\
0&1\end{pmatrix}\begin{pmatrix}i\\\
j\end{pmatrix}-\begin{pmatrix}l_{1}\delta\\\ 0\end{pmatrix}\right\\}$

is the composition of the two affine transformations

$A_{1}\begin{pmatrix}i\\\
j\end{pmatrix}=\begin{pmatrix}i+l_{1}j-l_{1}\delta\\\ j\end{pmatrix}\text{ and
}A_{2}\begin{pmatrix}i\\\ j\end{pmatrix}=\begin{pmatrix}i\\\
l_{2}^{-1}i+j\end{pmatrix}.$

Although statements similar to below hold for any $l_{(1)}$ in
$\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in $\mathcal{I}_{f}^{2}$, instead of
$l_{1}$ and $l_{2}$, we illustrate only the case $l_{(1)}=l_{1}$ and
$l_{(2)}=l_{2}$ for simplicity. The transformation $A_{1}$ maps the basis
$\\{(1,-(l_{1}+l_{2})^{-1}),(-l_{1},1)\\}$ to $\\{(1,-l_{2}^{-1}),(0,1)\\}$.
In other words, it maps the line $L_{k}$ with slope $-(l_{1}+l_{2})^{-1}$ and
the line $L_{k-1}$ with slope $-l_{1}^{-1}$, which intersect at $(\gamma,d)$,
to the line $\tilde{L}_{k}$ with slope $-l_{2}^{-1}$ and the vertical line
that intersect at $(\tilde{\gamma},d)$. The transformation $A_{2}$ maps the
basis $\\{(1,-l_{2}^{-1}),(0,1)\\}$ to $\\{(1,0),(0,1)\\}$. In other words, it
maps the line $\tilde{L}_{k}$ and the vertical line that intersect at
$(\tilde{\gamma},d)$, to the horizontal and vertical lines that intersect at
$(\tilde{\gamma},\tilde{d})$.

##  4\. Case 2: Invariant wedges and Attracting sets

We deal with Case 2 in this and the next sections: we prove Theorems 2.2 and
2.3, the main results on invariant wedges and the unions of all the preimages
of the wedges, in this section and prove Theorems 2.4 and 2.5 and Propositions
2.6 and 2.7, the main results on plurisubharmonic functions, in the next
section. Let $s>1$,

$\delta\leq T_{s-1}\text{ and }(\gamma,d)=(n_{s},m_{s}).$

Note that $\gamma>0$ and $d\geq 0$ by the setting.

Before we give proofs of Theorems 2.2 and 2.3 in Section 4.2, we illustrate
the theorems in terms of blow-ups in Section 4.1. Actually, we check the
following.

######  Proposition 4.1.

Let $\delta\leq d$. Then $\tilde{f}$ is superattracting at the origin and
degenerates the $c$-axis for any $l>0$. Let $\delta>d$. Then $\tilde{f}$ is
superattracting at the origin for any $0<l<\alpha$, even for $l=\alpha$ if
$d\geq 2$, and degenerates the $c$-axis for any $0<l<\alpha$ unless
$(n_{1},m_{1})=(0,\delta)$.

Roughly speaking, if $\tilde{f}$ is superattracting at the origin, then the
open set $U_{r_{1},r_{2}}^{l}$ should be invariant under $f$, and if
$\tilde{f}$ degenerates the $c$-axis, then one may expect that
$A_{f}^{l}=A_{0}-E_{z}$.

######  Remark 4.2 (Another illustration of Theorem 2.3).

One may expect that the vertical dynamics eventually converges to the dynamics
on the $w$-axis since $p^{n}(z)\to 0$ as $n\to\infty$. If $n_{1}>0$, then $f$
degenerates the $w$-axis to the origin and so one may expect that Theorem 2.3
holds. If $n_{1}=0$, then $q(0,w)=b_{n_{1}m_{1}}w^{m_{1}}(1+o(1))$ and so one
may expect that Theorem 2.3 holds if $\delta<m_{1}$ since $p(z)\sim
az^{\delta}$.

###  4.1. Illustration of Theorems 2.2 and 2.3 in terms of blow-ups

Recall that $\tilde{f}=\pi_{1}^{-1}\circ f\circ\pi_{1}$, where
$\pi_{1}(z,c)=(z,z^{l}c)$. Let $\tilde{f}(z,c)=(\tilde{p}(z),\tilde{q}(z,c))$
and we assume that $p(z)=z^{\delta}$ for simplicity. Then
$\tilde{p}(z)=z^{\delta}$ and $\tilde{q}(z,c)=\sum
b_{ij}z^{i+lj-l\delta}c^{j}$. Let $\tilde{\gamma}=\gamma+ld-l\delta$ and
$\tilde{n}_{j}=n_{j}+lm_{j}-l\delta$. The shape of the Newton polygon
$N(\tilde{q})$ of $\tilde{q}$ is useful to check whether $\tilde{f}$ is
superattracting at the origin, and whether $\tilde{f}$ degenerates the
$c$-axis. Recall that $A_{1}$ maps the line with slope $-l^{-1}$ that passes
through the point $(i,\delta)$ to the vertical line $\\{x=i\\}$, whereas it
preserves all the horizontal lines.

Let us describe the shape of the Newton polygon of $\tilde{q}$ and properties
of $\tilde{f}$ for the cases $\delta\leq d$, $\delta>d$ and $\delta<T_{s-1}$,
and $\delta>d$ and $\delta=T_{s-1}$ in Sections 4.1.1, 4.1.2 and 4.1.3,
respectively, which implies Proposition 4.1.

####  4.1.1. $\delta\leq d$

Recall that $\mathcal{I}_{f}=[l_{1},\infty)$ if $\delta\leq d$. See Table 6
below for a summary of the shape of $N(\tilde{q})$ and properties of
$\tilde{f}$, where the notation NA means that $\tilde{f}$ is superattracting
at the origin, and the notation Deg means that $\tilde{f}$ degenerates the
$c$-axis.

Table 6. Shape of $N(\tilde{q})$ when $\delta\leq d$ $l$ | $0<l<l_{1}$ | $l_{1}$ | $l_{1}<l$
---|---|---|---
$\tilde{\gamma}$ | $0<\tilde{n}_{s-1}<\tilde{\gamma}$ | $0<\tilde{\gamma}=\tilde{n}_{s-1}<\tilde{n}_{j}$ | $0<\tilde{\gamma}<\tilde{n}_{j}$
|  $0<\tilde{n}_{j}$ for any $j$ | for $j\leq s-2$ | for $j\leq s-1$
$N(\tilde{q})$ | has other | has the only | has the only
| vertices | one vertex | one vertex
| than $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)$
$\tilde{f}$ | SA and Deg | SA and Deg | SA and Deg

For any $l$ in $\mathcal{I}_{f}$, $N(\tilde{q})=(\tilde{\gamma},d)$ and
$\tilde{\gamma}>0$. Hence

$\tilde{q}(z,c)=z^{\tilde{\gamma}}c^{d}\left\\{1+\eta(z,c)\right\\},$

where $\eta\to 0$ as $z$ and $c\to 0$, and $\tilde{f}$ is superattracting at
the origin and degenerates the $c$-axis. More precisely,
$\tilde{\gamma}=\tilde{n}_{s-1}<\tilde{n}_{j}$ for any $1\leq j\leq s-2$ if
$l=l_{1}$, and $\tilde{\gamma}<\tilde{n}_{j}$ for any $1\leq j\leq s-1$ if
$l>l_{1}$.

On the other hand, for any $0<l<l_{1}$, $N(\tilde{q})$ has other vertices than
$(\tilde{\gamma},d)$, whereas $(\tilde{\gamma},d)$ is the vertex of
$N(\tilde{q})$ whose $y$-coordinate is the minimum and $\tilde{\gamma}>0$.
More precisely, the candidates of vertices of $N(\tilde{q})$ are
$(\tilde{n}_{j},m_{j})$’s, and $(\tilde{n}_{s-1},m_{s-1})$ is actually one of
the vertices of $N(\tilde{q})$. However, since $\tilde{n}_{j}>0$ for any $j$,
$\tilde{f}$ is superattracting at the origin and degenerates the $c$-axis.

####  4.1.2. $\delta>d$ and $\delta<T_{s-1}$

Recall that $\mathcal{I}_{f}=[l_{1},\alpha]$ if $\delta>d$ and
$\delta<T_{s-1}$. See Table 7 below for a summary of the shape of
$N(\tilde{q})$ and properties of $\tilde{f}$.

Table 7. Shape of $N(\tilde{q})$ when $\delta>d$ and $\delta<T_{s-1}$ $l$ | $0<l<l_{1}$ | $l_{1}$ | $l_{1}<l<\alpha$ | $\alpha$
---|---|---|---|---
$\tilde{\gamma}$ | $0<\tilde{n}_{s-1}<\tilde{\gamma}$ | $0<\tilde{\gamma}=\tilde{n}_{s-1}<\tilde{n}_{j}$ | $0<\tilde{\gamma}<\tilde{n}_{j}$ | $0=\tilde{\gamma}<\tilde{n}_{j}$
|  $0<\tilde{n}_{j}$ for any $j$ | for $j\leq s-2$ | for $j\leq s-1$ | for $j\leq s-1$
$N(\tilde{q})$ | has other | has the only | has the only | has the only
| vertices | one vertex | one vertex | one vertex
| than $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)=(0,d)$
$\tilde{f}$ | SA and Deg | SA and Deg | SA and Deg | SA if $d\geq 2$

For any $l$ in $\mathcal{I}_{f}$, $N(\tilde{q})=(\tilde{\gamma},d)$. Moreover,
if $l<\alpha$, then $\tilde{\gamma}>0$ and so $\tilde{f}$ is superattracting
at the origin and degenerates the $c$-axis. On the other hand, if $l=\alpha$,
then $\tilde{\gamma}=0$,

$\tilde{q}(z,c)=c^{d}\left\\{1+\eta(z,c)\right\\},$

and so $\tilde{f}$ is superattracting at the origin only if $d\geq 2$, and
$\tilde{f}$ does not degenerate the $c$-axis. For any $0<l<l_{1}$, the same
things hold as the previous case. We remark that if $l>\alpha$, then
$\tilde{\gamma}<0$ and so $\tilde{f}$ is not holomorphic.

####  4.1.3. $\delta>d$ and $\delta=T_{s-1}$

If $\delta=T_{s-1}$, then $\delta>d$ and
$\mathcal{I}_{f}=\\{l_{1}\\}=\\{\alpha\\}$. The shape of $N(\tilde{q})$ and
properties of $\tilde{f}$ differ whether $(n_{1},m_{1})=(0,\delta)$ or not.

If $(n_{1},m_{1})\neq(0,\delta)$, then the situation is somehow similar to the
previous case. If $l=\alpha$, then
$\tilde{\gamma}=\tilde{n}_{s-1}=0<\tilde{n}_{j}$ for any $1\leq j\leq s-2$ and
so $\tilde{f}$ is superattracting at the origin only if $d\geq 2$, and it does
not degenerate the $c$-axis. For any $0<l<l_{1}$, the same things hold as the
previous case. See Table 8 below for a summary.

Table 8. Shape of $N(\tilde{q})$ when $\delta=T_{s-1}$ and $(n_{1},m_{1})\neq(0,\delta)$ $l$ | $0<l<l_{1}=\alpha$ | $l_{1}=\alpha$
---|---|---
$\tilde{\gamma}$ | $0<\tilde{n}_{s-1}<\tilde{\gamma}$ | $0=\tilde{\gamma}=\tilde{n}_{s-1}<\tilde{n}_{j}$
|  $0<\tilde{n}_{j}$ for any $j$ | for $j\leq s-2$
$N(\tilde{q})$ | has other | has the only
| vertices | one vertex
| than $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)=(0,d)$
$\tilde{f}$ | SA and Deg | SA if $d\geq 2$

On the other hand, if $(n_{1},m_{1})=(0,\delta)$, then the situation differs
from the previous case. Note that $s=2$ for this case. If $l=\alpha$, then
$\tilde{\gamma}=\tilde{n}_{1}=0$,

$\tilde{q}(z,c)=b_{0\delta}c^{\delta}+\cdots+b_{\gamma
d}c^{d}+\sum_{i+lj>l\delta}b_{ij}z^{i+lj-l\delta}c^{j}=c^{d}\left\\{1+\eta(z,c)\right\\}$

and so $\tilde{f}$ is superattracting at the origin only if $d\geq 2$, and
$\tilde{f}$ does not degenerate the $c$-axis. For any $0<l<l_{1}$,
$\tilde{\gamma}>\tilde{n}_{1}=0$,

$\tilde{q}(z,c)=b_{0\delta}c^{\delta}+\sum_{i+lj>l\delta}b_{ij}z^{i+lj-l\delta}c^{j}$

and so $\tilde{f}$ is superattracting at the origin, but it does not
degenerate the $c$-axis. See Table 9 below for a summary.

Table 9. Shape of $N(\tilde{q})$ when $(n_{1},m_{1})=(0,\delta)$ $l$ | $0<l<l_{1}=\alpha$ | $l_{1}=\alpha$
---|---|---
$\tilde{\gamma}$ | $0=\tilde{n}_{1}<\tilde{\gamma}$ | $0=\tilde{n}_{1}=\tilde{\gamma}$
$N(\tilde{q})$ | has other | has the only
| vertices | one vertex
| than $(\tilde{\gamma},d)$ | $(\tilde{\gamma},d)=(0,d)$
$\tilde{f}$ | SA | SA if $d\geq 2$

###  4.2. Proof of Theorems 2.2 and 2.3

The following quantity plays an important role in the proofs:

$D=D_{l}=\min\left\\{l^{-1}i+j\ |\ b_{ij}\neq 0\right\\}.$

By definition, $D>d$ for any $l>0$ since $D\geq l^{-1}\gamma+d$ and
$\gamma>0$. Note that $l^{-1}i+j$ is the $y$-intercept of the line with slope
$-l^{-1}$ that passes through $(i,j)$. Hence, for any $l>0$, the minimum is
attached by a vertex $(n_{j},m_{j})$. Let $(N,M)$ be such a vertex:
$D=l^{-1}M+N$. Then we have the following.

######  Lemma 4.3.

It follows that $(N,M)=(\gamma,d)$ and $T_{s-1}\geq
D=l^{-1}\gamma+d\geq\delta$ for any $l$ in $\mathcal{I}_{f}$. Moreover,
$D=T_{s-1}$ for $l=l_{1}$, and $D=\delta$ for $l=\alpha$ if $\delta>d$. On the
other hand, $(N,M)\neq(\gamma,d)$ and $D=l^{-1}M+N\geq T_{s-1}\geq\delta$ for
any $0<l<l_{1}$. More precisely, $(M,N)=(n_{1},m_{1})$ or
$(M,N)=(n_{k},m_{k})$ for some $2\leq k\leq s-1$

$\text{ if }0<l\leq\dfrac{n_{2}-n_{1}}{m_{1}-m_{2}}\text{ or
}\dfrac{n_{k}-n_{k-1}}{m_{k-1}-m_{k}}\leq
l\leq\dfrac{n_{k+1}-n_{k}}{m_{k}-m_{k+1}}.$

Consequently, the inequality $D\geq\delta$ holds for any $l>0$ if $\delta\leq
d$, and for any $0<l\leq\alpha$ if $\delta>d$. Let us make clear when the
inequality $D>\delta$ or the equality $D=\delta$ holds. If $\delta\leq d$,
then $D>d\geq\delta$ for any $l>0$. If $\delta>d$, then $D\geq\delta$ for any
$0<l<\alpha$, and $D=\delta$ for $l=\alpha$. More precisely, if $\delta>d$,
then the inequality $D>\delta$ holds for any $0<l<\alpha$ except the special
case; if $(n_{1},m_{1})=(0,\delta)$, then $s=2$ and $D=T_{1}=T_{s-1}=\delta$
for any $0<l<\alpha$.

######  Lemma 4.4.

The inequality $D>\delta$ holds for any $l>0$ if $\delta\leq d$ and for any
$0<l<\alpha$ if $\delta>d$ and $(n_{1},m_{1})\neq(0,\delta)$. On the other
hand, the equality $D=\delta$ holds for $l=\alpha$ if $\delta>d$ and for any
$0<l\leq\alpha$ if $\delta>d$ and $(n_{1},m_{1})=(0,\delta)$.

####  4.2.1. Proof of Theorem 2.2

Theorem 2.2 follows from the inequality $D\geq\delta$. We use the following
for the case $D>\delta$.

######  Lemma 4.5.

For any $l>0$ and for any small $r_{1}$ and $r_{2}$, there exists a constant
$C>0$ such that $|q(z,w)|\leq C|z|^{lD}$ on $U^{l}_{r_{1},r_{2}}$.

###### Proof.

Since $|w|<r_{2}|z|^{l}$ on $U^{l}_{r_{1},r_{2}}$ and $i+lj\geq lD$ for any
$(i,j)$ such that $b_{ij}\neq 0$,

$\displaystyle|q(z,w)|$
$\displaystyle\leq\sum|b_{ij}||z|^{i}|w|^{j}<\sum|b_{ij}|r_{2}^{j}|z|^{i+lj}$
$\displaystyle=\sum_{i+lj=lD}|b_{ij}|r_{2}^{j}|z|^{lD}+\sum_{i+lj>lD}|b_{ij}|r_{2}^{j}|z|^{i+lj}$
$\displaystyle=|z|^{lD}\Big{\\{}\sum_{i+lj=lD}|b_{ij}|r_{2}^{j}+\sum_{i+lj>lD}|b_{ij}|r_{2}^{j}|z|^{i+lj-
lD}\Big{\\}}$
$\displaystyle\leq|z|^{lD}\Big{\\{}\sum_{i+lj=lD}|b_{ij}|r_{2}^{j}+\sum_{i+lj>lD}|b_{ij}|r_{2}^{j}r_{1}^{i+lj-
lD}\Big{\\}}$

on $U^{l}_{r_{1},r_{2}}$. The first sum in the last expression is bounded from
above since the set of the pairs $(i,j)$ such that $i+lj=lD$ and $b_{ij}\neq
0$ is finite, and the second sun tends to $0$ as $r_{1}\to 0$ since $i+lj-
lD>0$. ∎

The following is a precise version of Theorem 2.2.

######  Theorem 4.6.

The following holds for any $l>0$ if $\delta\leq d$ and for any $0<l<\alpha$
if $\delta>d$: $f(U_{r_{1},r_{2}}^{l})\subset U_{r_{1},r_{2}}^{l}$ for
suitable small $r_{1}$ and $r_{2}$. More precisely, $r_{1}$ and $r_{2}$ have
to satisfy the inequality $Cr_{1}^{l(D-\delta)}<r_{2}$ or
$2Cr_{1}^{l(D^{*}-\delta)}<r_{2}$ for some constants $C$ and $D^{*}$ if
$D>\delta$ or $D=\delta$, where $D^{*}>\delta$.

###### Proof.

Let $U=U^{l}_{r_{1},r_{2}}$ in the proof. Since the inequality $|p(z)|<r_{1}$
is trivial, it is enough to show the existence of $r_{1}$ and $r_{2}$ such
that $|q(z,w)|<r_{2}|p(z)|^{l}$ for any $(z,w)$ in $U$. We have the inequality
$D\geq\delta$ by Lemma 4.4. The proof differs whether $D>\delta$ or
$D=\delta$.

We first consider the case $D>\delta$. It follows from Lemma 4.5 that

$|p(z)|>C_{1}|z|^{\delta}\text{ and }|q(z,w)|\leq C_{2}|z|^{lD}\text{ on }U$

for some constants $C_{1}$ and $C_{2}$. Hence

$\left|\dfrac{q(z,w)}{p(z)^{l}}\right|<\dfrac{C_{2}|z|^{lD}}{(C_{1}|z|^{\delta})^{l}}=\dfrac{C_{2}}{C_{1}^{l}}|z|^{l(D-\delta)}<Cr_{1}^{l(D-\delta)}\text{
on }U,$

where $C=C_{2}\cdot C_{1}^{-l}$. Since $D>\delta$, the map $f$ preserves $U$
if $Cr_{1}^{l(D-\delta)}<r_{2}$.

We next consider the case $D=\delta$; that is, the special case
$(n_{1},m_{1})=(0,\delta)$. Recall that $\delta>d$ and $(N,M)=(n_{1},m_{1})$
for any $0<l<\alpha$ if $(n_{1},m_{1})=(0,\delta)$. Let us assume that
$p(z)=z^{\delta}$ for simplicity. Then

$\displaystyle\left|\dfrac{q(z,w)}{p(z)^{l}}\right|$
$\displaystyle\leq\dfrac{\sum|b_{ij}||z|^{i}|w|^{j}}{|z|^{l\delta}}\leq\sum|b_{ij}|r_{2}^{j}|z|^{i+lj-l\delta}$
$\displaystyle=\sum_{i+lj=l\delta}|b_{ij}|r_{2}^{j}+\sum_{i+lj>l\delta}|b_{ij}|r_{2}^{j}|z|^{i+lj-l\delta}$
$\displaystyle=|b_{NM}|r_{2}^{M}+\sum_{i+lj>l\delta}|b_{ij}|r_{2}^{j}|z|^{i+lj-l\delta}\text{
on }U.$

Since $M=m_{1}=\delta\geq 2$, $|b_{NM}|r_{2}^{M}\leq|b_{NM}|r_{2}^{2}<r_{2}/2$
if $r_{2}<(2|b_{NM}|)^{-1}$. Let

$D^{*}=\min\left\\{l^{-1}i+j\ |\ (i,j)\neq(0,\delta)\text{ and }b_{ij}\neq
0\right\\}.$

Then $D^{*}>D=\delta$ and the following inequality holds on $U$ for some
constant $C$:

$\sum_{i+lj>l\delta}|b_{ij}|r_{2}^{j}|z|^{i+lj-l\delta}=\Big{\\{}\sum_{i+lj\geq
lD^{*}}|b_{ij}|r_{2}^{j}|z|^{i+lj-lD^{*}}\Big{\\}}|z|^{l(D^{*}-\delta)}$
$<\Big{\\{}\sum_{i+lj\geq lD^{*}}|b_{ij}|r_{2}^{j}r_{1}^{i+lj-
lD^{*}}\Big{\\}}r_{1}^{l(D^{*}-\delta)}\leq Cr_{1}^{l(D^{*}-\delta)}.$

Therefore, if $2Cr_{1}^{l(D^{*}-\delta)}<r_{2}<(2|b_{NM}|)^{-1}$, then

$\left|\dfrac{q(z,w)}{p(z)^{l}}\right|<\dfrac{r_{2}}{2}+\dfrac{r_{2}}{2}=r_{2}.$

∎

######  Remark 4.7.

As the same strategy, we can show that the statement in the theorem above
holds even for $l=\alpha$ if $\delta>d\geq 2$, without using the fact $f\sim
f_{0}$ on $U$ as in Lemmas 2.1 and 3.1. Let us give a short proof. As in the
proof above,

$\left|\dfrac{q(z,w)}{p(z)^{l}}\right|\leq\sum_{i+\alpha
j=\alpha\delta}|b_{ij}|r_{2}^{j}+\sum_{i+\alpha
j>\alpha\delta}|b_{ij}|r_{2}^{j}|z|^{i+\alpha j-\alpha\delta}\text{ on
}U_{r_{1},r_{2}}^{\alpha}.$

Note that the set of the pairs $(i,j)$ such that $i+\alpha j=\alpha\delta$ and
$b_{ij}\neq 0$ is finite, which includes $(\gamma,d)$, and the minimum $j$
such that $i+\alpha j=\alpha\delta$ is $d$. Since $d\geq 2$, the first sum is
less than $r_{2}/2$ for small $r_{2}$. Let

$\displaystyle D^{*}$ $\displaystyle=\min\left\\{{\alpha}^{-1}i+j\ |\
(i,j)\neq(\gamma,d)\text{ and }b_{ij}\neq 0\right\\}$ $\displaystyle\text{ if
}\delta<T_{s-1},\text{ and}$ $\displaystyle D^{*}$
$\displaystyle=\min\left\\{{\alpha}^{-1}i+j\ |\ i+\alpha j>\alpha\delta\text{
and }b_{ij}\neq 0\right\\}$ $\displaystyle\text{ if }\delta=T_{s-1}.$

Then $D^{*}>D=\delta$ and the second sum is less than or equal to
$Cr_{1}^{l(D^{*}-\delta)}$ for some constant $C$, which is less than $r_{2}/2$
for small $r_{1}$.

######  Remark 4.8.

As in the proof above, if $D>\delta$, then $r_{1}$ has to satisfy the
inequality

$r_{1}\leq\left(\dfrac{r_{2}}{C}\right)^{\frac{1}{l(D-\delta)}}.$

Note that $l(D-\delta)=N+lM-l\delta\to N$ as $l\to 0$, and $N=n_{1}$ for small
enough $l$ by Lemma 4.3. Hence, for small enough $l$, it is enough to satisfy
the inequality

$r_{1}\leq\left(\dfrac{r_{2}}{C}\right)^{\frac{1}{n_{1}}}.$

Therefore, $r_{1}$ needs not converge to $0$ as $l\to 0$.

####  4.2.2. Proof of Theorem 2.3

Let $V^{l}=\\{0<|z|<r,|w|\geq r|z|^{l},|w|<r_{3}\\}$. Then the union
$U^{l}\cup V^{l}\cup\\{z=0\\}$ is a neighborhood of the origin, where
$U^{l}=\\{|z|<r,|w|<r|z|^{l}\\}$.

######  Lemma 4.9.

For any $l>0$ and for any small $r$ and $r_{3}$, there exists a constant $C>0$
such that $|q(z,w)|\leq C|w|^{D}$ on $V^{l}$.

###### Proof.

Since $|z|\leq(r^{-1}|w|)^{l^{-1}}$ on $V^{l}$,

$\displaystyle|q(z,w)|$
$\displaystyle\leq\sum|b_{ij}z^{i}w^{j}|\leq\sum|b_{ij}|r^{-l^{-1}i}|w|^{l^{-1}i+j}$
$\displaystyle=\sum_{i+lj=lD}|b_{ij}|r^{-l^{-1}i}|w|^{D}+\sum_{i+lj>lD}|b_{ij}|r^{-l^{-1}i}|w|^{l^{-1}i+j}$
$\displaystyle\leq|w|^{D}\Big{\\{}\sum_{i+lj=lD}|b_{ij}|r^{-l^{-1}i}+\sum_{i+lj>lD}|b_{ij}|r^{-l^{-1}i}r_{3}^{(l^{-1}i+j)-D}\Big{\\}}$

on $V^{l}$. The first sum in the last expression is bounded from above since
the set of the pairs $(i,j)$ such that $i+lj=lD$ and $b_{ij}\neq 0$ is finite,
and the second sun tends to $0$ as $r_{3}\to 0$ since $l^{-1}i+j-D>0$. ∎

The proposition below follows from the inequality $D>\delta$ and the lemma
above, which implies Theorem 2.3.

######  Proposition 4.10.

The following holds for any $l>0$ if $\delta\leq d$ and for any $0<l<\alpha$
if $\delta>d$ and $(n_{1},m_{1})\neq(0,\delta)$: for any $(z,w)$ in $V^{l}$
with suitable small $r$ and $r_{3}$, there exists an integer $n$ such that
$f^{n}(z,w)$ belongs to $U^{l}$.

###### Proof.

It follows from Lemma 4.9 that

$|p(z)|>(C_{1}|z|)^{\delta}\text{ and }|q(z,w)|<(C_{2}|w|)^{D}\text{ on
}V^{l}$

for some constants $C_{1}$ and $C_{2}$. We derive a contradiction by assuming
that $f^{n}(z,w)$ belongs to $V^{l}$ for all $n>0$. Let
$f^{n}(z,w)=(z_{n},w_{n})$. Then $r|z_{n}|^{l}\leq|w_{n}|$ by the assumption.
Since

$|z_{n}|>C_{1}^{\frac{\delta^{n}-1}{\delta-1}}|z|^{\delta^{n}}>C_{1}^{\frac{\delta^{n}}{\delta-1}}|z|^{\delta^{n}}\text{
and
}|w_{n}|<C_{2}^{\frac{D^{n}-1}{D-1}}|w|^{D^{n}}<C_{2}^{\frac{D^{n}}{D-1}}|w|^{D^{n}},$

we have the inequality

$r(C_{1}^{\frac{1}{\delta-1}}|z|)^{l\delta^{n}}<r|z_{n}|^{l}\leq|w_{n}|<(C_{2}^{\frac{1}{D-1}}|w|)^{D^{n}}.$

Recall that $D>\delta$ by Lemma 4.4. By taking $r$ and $r_{3}$ small enough,
we may assume that $C_{1}^{\frac{1}{\delta-1}}r<1$ and
$C_{2}^{\frac{1}{D-1}}r_{3}<1$. Therefore, the convergence speed of
$(C_{2}^{\frac{1}{D-1}}|w|)^{D^{n}}$ is faster than that of
$\\{(C_{1}^{\frac{1}{\delta-1}}|z|)^{l}\\}^{\delta^{n}}$, which contradict the
inequality above. ∎

##  5\. Case 2: Plurisubharmonic functions

We continue the study of the dynamics for Case 2. We show the existence and
properties of plurisubharmonic functions and prove Theorems 2.4 and 2.5 and
Propositions 2.6 and 2.7 in this section.

We first deal with the monomial skew products in Section 5.1. Although they
are not exactly in Case 2, the investigation in this subsection is helpful to
understand the other cases. We then deal with the cases $\delta<d$ and
$\delta=d$ in Sections 5.2 and 5.3, respectively. The case $T_{s-1}>\delta>d$
is dealt with in Section 5.4, which is separated into the subcases $d\geq 1$
and $d=0$. We finally deal with the case $\delta=T_{s-1}$ in Section 5.5,
which is separated into the subcases $n_{s-1}>0$ and $n_{s-1}=0$. One may
refer Table 1 in Section 2.1 for a comparison chart.

Theorem 2.4 follows from Propositions 5.5, 5.7, 5.22, 5.30 and 5.42. Theorem
2.5 follows from Propositions 5.5, 5.24, Corollary 5.25, Proposition 5.31,
Corollary 5.32, Propositions 5.41, 5.44 and 5.45. Proposition 2.6 follows from
Corollary 5.35. Proposition 2.7 follows from Corollary 5.38.

###  5.1. Monomial maps

Let $f_{0}(z,w)=(z^{\delta},z^{\gamma}w^{d})$, $\delta\geq 2$, $\gamma\geq 0$,
$d\geq 1$ and $\gamma+d\geq 2$. Then $f_{0}$ has a superattracting fixed point
at the origin. Let

$\alpha=\dfrac{\gamma}{\delta-d}$

if $\delta\neq d$. Note that $\alpha<0$ if $\delta<d$, and $\alpha>0$ if
$\delta>d$.

Let $f_{0}^{n}(z,w)=(z_{n},w_{n})$. Then $z_{n}=z^{\delta^{n}}$ and
$w_{n}=z^{\gamma_{n}}w^{d^{n}}$, where

$\gamma_{n}=\alpha(\delta^{n}-d^{n})\text{ if }\delta\neq d,\text{ and
}\gamma_{n}=n\gamma d^{n-1}\text{ if }\delta=d.$

It is clear that $G_{p}(z)=\log|z|$.

We first consider the case $\delta\neq d$. If $\delta\neq d$ and $z\neq 0$,
then

$w_{n}=z^{\alpha(\delta^{n}-d^{n})}w^{d^{n}}=z^{\alpha\delta^{n}}\left(\dfrac{w}{z^{\alpha}}\right)^{d^{n}}.$

Hence

$\dfrac{w_{n}}{z_{n}^{\alpha}}=\left(\dfrac{w}{z^{\alpha}}\right)^{d^{n}}\text{
and so }G_{z}^{\alpha}(w)=\log\left|\dfrac{w}{z^{\alpha}}\right|.$

######  Lemma 5.1.

If $\delta<d$, then $G_{z}^{\alpha}(w)=\log\left|z^{-\alpha}w\right|$ on
$\mathbb{C}^{2}$. If $\delta>d$ and $\gamma>0$, then
$G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|$ on $\\{z\neq 0\\}$. If
$\delta\neq d$ and $\gamma=0$, then $G_{z}^{\alpha}(w)=\log|w|$ on
$\mathbb{C}^{2}$.

We also have the following.

######  Lemma 5.2.

If $\delta<d$, then $G_{z}(w)=\log\left|z^{-\alpha}w\right|$ on
$\mathbb{C}^{2}$. If $\delta>d$ and $\gamma>0$, then $G_{z}(w)=\alpha\log|z|$
on $\\{w\neq 0\\}$ and $G_{z}(w)=-\infty$ on $\\{w=0\\}$. If $\delta>d$ and
$\gamma=0$, then $G_{z}(w)=0$ on $\\{w\neq 0\\}$ and $G_{z}(w)=-\infty$ on
$\\{w=0\\}$.

###### Proof.

If $\delta<d$, then

$\displaystyle G_{z}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|z^{\alpha\delta^{n}}\left(\dfrac{w}{z^{\alpha}}\right)^{d^{n}}\right|=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|\dfrac{(z^{-\alpha}w)^{d^{n}}}{(z^{-\alpha})^{\delta^{n}}}\right|$
$\displaystyle=-\lim_{n\to\infty}(-\alpha)\left(\dfrac{\delta}{d}\right)^{n}\log|z|+\log\left|z^{-\alpha}w\right|=\log\left|z^{-\alpha}w\right|$

on $\\{z\neq 0\\}$. It is clear that $G_{z}(w)=-\infty$ on $\\{z=0\\}$.

If $\delta>d$ and $\gamma>0$, then

$\displaystyle G_{z}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{\delta^{n}}\log\left|z^{\alpha\delta^{n}}\left(\dfrac{w}{z^{\alpha}}\right)^{d^{n}}\right|$
$\displaystyle=\alpha\log|z|+\lim_{n\to\infty}\left(\dfrac{d}{\delta}\right)^{n}\log\left|\dfrac{w}{z^{\alpha}}\right|=\alpha\log|z|$

on $\\{zw\neq 0\\}$. It is clear that $G_{z}(w)=-\infty$ on $\\{zw=0\\}$. ∎

######  Remark 5.3.

If $\delta\neq d$, then $f_{0}$ is semiconjugate to the product
$(z,c)\to(z^{\delta},c^{d})$ by $\pi(z,c)=(z^{\delta-d},z^{\gamma}c)$. We can
replace $\pi$ by $\pi(z,c)=(z,z^{\alpha}c)$ if $\alpha$ is integer.

We next consider the case $\delta=d$. If $\delta=d$ and $z\neq 0$, then

$w_{n}=z^{n\gamma d^{n-1}}w^{d^{n}}\text{ and so }G_{z}^{\infty}(w)=\log|w|.$

If $\delta=d$, $\gamma>0$ and $|z|<1$, then

$\displaystyle G_{z}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|z^{n\gamma
d^{n-1}}w^{d^{n}}\right|$
$\displaystyle=\lim_{n\to\infty}\left\\{n\dfrac{\gamma}{d}\log|z|+\log|w|\right\\}=-\infty.$

######  Lemma 5.4.

If $\delta=d$ and $\gamma>0$, then $G_{z}^{\infty}(w)=\log|w|$ on $\\{z\neq
0\\}$ and $G_{z}(w)=-\infty$ on $\\{|z|<1\\}$. If $\delta=d$ and $\gamma=0$,
then $G_{z}^{\infty}(w)=G_{z}(w)=\log|w|$ on $\mathbb{C}^{2}$.

These lemmas can be summarized into the following two tables.

Table 10. Psh functions for monomial maps when $\gamma>0$ |  $G_{z}^{\alpha}$ or $G_{z}^{\infty}$ | $G_{z}$ | $A_{0}$
---|---|---|---
$\delta<d$ |  $\log\left|z^{-\alpha}w\right|$ on $\mathbb{C}^{2}$ |  $\log\left|z^{-\alpha}w\right|$ on $\mathbb{C}^{2}$ | $\\{|z|<1,|z^{-\alpha}w|<1\\}$
$\delta>d$ |  $\log\left|w/z^{\alpha}\right|$ on $\\{z\neq 0\\}$ |  $\alpha\log\left|z\right|$ on $\\{w\neq 0\\}$ | $\\{|z|<1\\}$
$\delta=d$ |  $\log\left|w\right|$ on $\\{z\neq 0\\}$ |  $-\infty$ on $\\{|z|<1\\}$ | $\\{|z|<1\\}$
Table 11. Psh functions for monomial maps when $\gamma=0$ |  $G_{z}^{\alpha}$ or $G_{z}^{\infty}$ | $G_{z}$ | $A_{0}$
---|---|---|---
$\delta<d$ |  $\log\left|w\right|$ on $\mathbb{C}^{2}$ |  $\log\left|w\right|$ on $\mathbb{C}^{2}$ | $\\{|z|<1,|w|<1\\}$
$\delta>d$ |  $\log\left|w\right|$ on $\mathbb{C}^{2}$ |  $0$ on $\\{w\neq 0\\}$ | $\\{|z|<1,|w|<1\\}$
$\delta=d$ |  $\log\left|w\right|$ on $\mathbb{C}^{2}$ |  $\log\left|w\right|$ on $\mathbb{C}^{2}$ | $\\{|z|<1,|w|<1\\}$

###  5.2. $\delta<d$

Recall that $\alpha<0$ and $\mathcal{I}_{f}=[l_{1},\infty)$ if $\delta<d$.

We first show the existence and properties of plurisubharmonic functions on
$A_{f}^{l}$ for any $l$ in $\mathcal{I}_{f}$ in Section 5.2.1, and then
evaluate the limit supremum and inferior of $G_{z}^{\alpha}$ toward the
boundary $\partial A_{f}^{l}$ in Sections 5.2.2 and 5.2.3, respectively.

####  5.2.1. Existence and basic properties

Because $f$ is conjugate to $f_{0}$ on $U^{l}$ for any $l$ in
$\mathcal{I}_{f}$, the limits $G_{z}^{\alpha}$ and $G_{z}$ exist on $U^{l}$.
More precisely, as shown below,

$G_{z}^{\alpha}(w)=G_{z}(w)=\log\left|\phi_{1}(z)^{-\alpha}\phi_{2}(z,w)\right|=\log\left|z^{-\alpha}w\right|+o(1)\text{
on }U^{l}\text{ as }r\to 0,$

where $\phi=(\phi_{1},\phi_{2})$ is the Böttcher coordinate for $f$ on
$U^{l}$. It extends to a plurisubharmonic function on $A_{f}^{l}$ via the
equation $G^{\alpha}\circ f=dG^{\alpha}$. We remark that
$\left|\phi_{1}^{-\alpha}\phi_{2}\right|$ extends to a continuous function
from $A_{f}^{l}$ to $[0,1)$. Theorem 2.3 indicates that
$A_{f}^{l}=A_{0}-E_{z}$ for any $l$ in $\mathcal{I}_{f}$. Consequently, we
obtain the following.

######  Proposition 5.5.

If $\delta<d$, then $G_{z}^{\alpha}=G_{z}$ on $A_{0}$. It is plurisubharmonic
and negative on $A_{0}-E_{z}$, pluriharmonic on $A_{0}-E$ and $-\infty$ on
$E\cap A_{0}$. Moreover,
$G_{z}^{\alpha}(w)=\log\left|z^{-\alpha}w\right|+o(1)$ on $U^{l_{1}}$ as $r\to
0$.

###### Proof.

Let us show the equality $G_{z}^{\alpha}=G_{z}$ on $A_{0}$. It follows that

$\displaystyle G_{z}^{\alpha}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{\alpha}}\right|=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log|w_{n}|-\lim_{n\to\infty}\dfrac{\alpha}{d^{n}}\log|z_{n}|$
$\displaystyle=G_{z}(w)-0=G_{z}(w)\text{ on }U^{l_{1}},$

which extends to $A_{0}-E_{z}$. The equality $G_{z}^{\alpha}=-\infty$ on
$E_{z}$ follows from the definition. On the other hand, the equality
$G_{z}=-\infty$ on $E_{z}\cap A_{0}$ follows from the dynamics of $f$ on the
$w$-axis. In fact, $q(0,w)$ degenerates the $w$-axis to the origin if
$n_{1}>0$, or it has a superattracting fixed point at $w=0$ with the order
$m_{1}$ if $n_{1}=0$, where $m_{1}>d$.

The last estimate in the statement follows from the lemma below. ∎

######  Lemma 5.6.

If $\delta<d$, then $G_{p}=\log|\phi_{1}|$ and
$G_{z}^{\alpha}=\log\left|\phi_{1}^{-\alpha}\phi_{2}\right|$ on $U^{l_{1}}$.

###### Proof.

Since $f_{0}^{n}(z,w)=(z^{\delta^{n}},z^{\gamma_{n}}w^{d^{n}})$ and
$\gamma_{n}=\alpha(\delta^{n}-d^{n})$,

$f_{0}^{-n}(Z,W)=\left(Z^{1/\delta^{n}},\left(\dfrac{W}{Z^{\gamma_{n}/\delta^{n}}}\right)^{1/d^{n}}\right)=\left(Z^{1/\delta^{n}},Z^{\alpha/\delta^{n}}\left(\dfrac{W}{Z^{\alpha}}\right)^{1/d^{n}}\right).$

Let $(z_{n},w_{n})=f^{n}(z,w)$. Since
$\displaystyle\phi(z,w)=\lim_{n\to\infty}f_{0}^{-n}\circ f^{n}(z,w)$,

$\phi(z,w)=\lim_{n\to\infty}\left(z_{n}^{1/\delta^{n}},z_{n}^{\alpha/\delta^{n}}\left(\dfrac{w_{n}}{z_{n}^{\alpha}}\right)^{1/d^{n}}\right).$

Hence
$\displaystyle\log|\phi_{1}(z)|=\lim_{n\to\infty}\delta^{-n}\log|z_{n}|=G_{p}(z)$
and

$\log|\phi_{2}(z,w)|=\lim_{n\to\infty}\dfrac{\alpha}{\delta^{n}}\log|z_{n}|+\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{\alpha}}\right|=\alpha
G_{p}(z)+G_{z}^{\alpha}(w).$

Therefore,
$G_{z}^{\alpha}(w)=\log|\phi_{2}(z,w)|-\alpha\log|\phi_{1}(z)|=\log\left|\phi_{1}(z)^{-\alpha}\phi_{2}(z,w)\right|$
on $U^{l_{1}}$. ∎

Moreover, $G_{z}^{\alpha}$ is plurisubharmonic on $A_{0}$.

######  Proposition 5.7.

If $\delta<d$, then $G_{z}^{\alpha}$ is plurisubharmonic on $A_{0}$.

###### Proof.

There is a constant $C>0$ such that $|q(z,w)|\leq C|w|^{d}$ on a neighborhood
of the origin, because $d\leq j$ for any $j$ such that $b_{ij}\neq 0$. Hence
we have the inequality

$A|q(z,w)|\leq(A|w|)^{d}$

on the neighborhood, where $A=|C|^{1/(d-1)}$. Let
$g_{n}(z,w)=d^{-n}\log|Aw_{n}|$. Then $g_{n+1}\leq g_{n}$ and so $\\{g_{n}\\}$
is a decreasing sequence of plurisubharmonic functions. Therefore, $g_{n}$
converges to a plurisubharmonic function $g_{\infty}$ on the neighborhood.
Since $g_{n}(z,w)=d^{-n}\log|w_{n}|+d^{-n}\log|A|$, it follows that
$g_{\infty}(z,w)=G_{z}^{\alpha}(w)$. ∎

####  5.2.2. Asymptotic behavior of $G_{z}^{\alpha}$ toward $\partial
A_{f}^{l}$: Limit supremum

We omit to denote the condition $\delta<d$ in the statements in this and the
next subsections.

Let $S^{l}=\\{|z|<r,|w|=r|z|^{l}\\}=\partial U^{l}\cap\\{|z|<r\\}$ and
$S_{n}^{l}=f^{-n}(S^{l})$.

######  Lemma 5.8.

It follows for any $l$ in $\mathcal{I}_{f}$ and for any $z$ in $A_{p}-E_{p}$
that $S_{n}^{l}\cap\mathbb{C}_{z}\neq\emptyset$ for any large $n$, and
$G_{z}^{\alpha}|_{S_{n}^{l}}\to 0$ as $n\to\infty$.

The proof is similar to that of Theorem 4.5 in [23], but not the same; we deal
with any $l$ in $\mathcal{I}_{f}$ not only $\alpha$. We also remark that one
can prove the lemma by a similar fashion to Theorem 5.4 in [23] since
$G_{z}^{\alpha}=G_{z}$.

###### Proof.

By Proposition 5.5, there is a constant $C>0$ such that

$\left|G_{z}^{\alpha}(w)-\log\left|z^{-\alpha}w\right|\right|<C\text{ on
}U^{l}.$

Since $\left|z^{-\alpha}w\right|=r|z|^{l-\alpha}$ on $S^{l}$, where
$l-\alpha>l\geq l_{1}>0$,

$\left|\ G_{p^{n}(z)}^{\alpha}(Q_{z}^{n}(w))-\log r|p^{n}(z)|^{l-\alpha}\
\right|\leq C\ \text{ on }S_{n}^{l}.$

Since $G_{p^{n}(z)}^{\alpha}(Q_{z}^{n}(w))=d^{n}G_{z}^{\alpha}(w)$,

$\left|\ G_{z}^{\alpha}(w)-d^{-n}\log r|p^{n}(z)|^{l-\alpha}\ \right|\leq
d^{-n}C\ \text{ on }S_{n}^{l}.$

Since $\deg p=\delta<d$ and $G_{p}(z)\neq-\infty$ by the setting,

$\dfrac{1}{d^{n}}\log|p^{n}(z)|^{l-\alpha}=\left(\dfrac{\delta}{d}\right)^{n}\cdot\dfrac{l-\alpha}{\delta^{n}}\log|p^{n}(z)|\to
0\cdot(l-\alpha)G_{p}(z)=0$

and $d^{-n}C\to 0$ as $n\to\infty$. ∎

We want to show that $S_{n}^{l}$ converges to the boundary $\partial
A_{f}^{l}$ in $A_{p}\times\mathbb{P}^{1}$ as $n\to\infty$. Let
$S_{\infty}^{l}$ be the set of the accumulation points of $S_{n}^{l}$ in
$\overline{A_{p}}\times\mathbb{P}^{1}$ as $n\to\infty$:

$S_{\infty}^{l}=\\{x\in\overline{A_{p}}\times\mathbb{P}^{1}\,|\,\exists\
x_{k}\in S_{n_{k}}^{l}\text{ such that }x_{k}\to x\text{ and
}n_{k}\to\infty\text{ as }k\to\infty\\}.$

Since the sequence $f^{-n}(U^{l})$ is monotonically increasing,

$S_{\infty}^{l}=\\{x\in\overline{A_{p}}\times\mathbb{P}^{1}\,|\,\exists\
x_{n}\in S_{n}^{l}\text{ such that }x_{n}\to x\text{ as }n\to\infty\\}.$

To show the convergence of $S_{n}^{l}$ to $\partial A_{f}^{l}$, we impose a
non-degenerate condition. We say that $f$ is non-degenerate if $f$ does not
degenerate any fiber to a point. We first prepare a lemme and a remark,
assuming that $f$ is non-degenerate.

######  Lemma 5.9.

If $f$ is non-degenerate, then for any $l$ in $\mathcal{I}_{f}$ and for any
$n\geq 0$,

  1. (1)

$f^{-n}(\partial U^{l})=\partial f^{-n}(U^{l})$,

  2. (2)

$f^{n}(f^{-n}(\partial U^{l}))\subset\partial U^{l}$, and
$f^{n}(f^{-n}(\partial U^{l}))=\partial U^{l}$ if $f$ is polynomial,

  3. (3)

$f^{-n}(S^{l})=f^{-n}(\partial U^{l})\cap\\{(z,w)\,|\,|p^{n}(z)|<r_{1}\\}$,

  4. (4)

$f^{n}(f^{-n}(S^{l}))\subset S^{l}$, and $f^{n}(f^{-n}(S^{l}))=S^{l}$ if $f$
is polynomial,

  5. (5)

$f^{-1}(\partial A_{f}^{l})=\partial A_{f}^{l}$, $f(\partial
A_{f}^{l})\subset\partial A_{f}^{l}$, and $f(\partial A_{f}^{l})=\partial
A_{f}^{l}$ if $f$ is polynomial,

  6. (6)

$f^{-1}(S_{\infty}^{l})=S_{\infty}^{l}$, $f(S_{\infty}^{l})\subset
S_{\infty}^{l}$, and $f(S_{\infty}^{l})=S_{\infty}^{l}$ if $f$ is polynomial.

We only give an outline of the proof of (1). The inclusion $f^{-n}(\partial
U)\supset\partial f^{-n}(U)$ follows since $f$ is continuous. The inverse
inclusion $f^{-n}(\partial U)\subset\partial f^{-n}(U)$ follows since $f$ is
open map, which is guaranteed by the non-degenerate assumption. The condition
of $f$ being polynomial can be replaced by $f$ being surjective on a suitable
global region. For example, if $f(\mathbb{C}^{2})=\mathbb{C}^{2}$, then the
equalities in the lemma hold.

######  Remark 5.10.

We proved in [26] that $f\sim f_{0}$ on $U^{l}$ as $r\to 0$, and
$f(U^{l})\subset U^{l}$ for any $l$ in $\mathcal{I}_{f}$. With a slight change
of the proof, one can show for any $l$ in $\mathcal{I}_{f}$ that

$\overline{f(U^{l})}-\\{0\\}\subset U^{l}\text{ and
}\overline{f(U^{l})}\cap\partial U^{l}=\\{0\\}.$

Moreover, if $f$ is non-degenerate, then for any $l$ in $\mathcal{I}_{f}$ and
for any $n\geq 0$,

$\overline{f^{-n}(U^{l})}-f^{-n}(0)\subset f^{-(n+1)}(U^{l})\text{ and
}\overline{f^{-n}(U^{l})}\cap\partial f^{-(n+1)}(U^{l})=f^{-n}(0).$

Let $\lim_{n\to\infty}f^{-n}(\partial U^{l})$ be the set of the accumulation
points of $f^{-n}(\partial U^{l})$ in $\overline{A_{p}}\times\mathbb{P}^{1}$
as $n\to\infty$. The following follows from the monotonicity of
$f^{-n}(U^{l})$ and Lemma 5.9.

######  Proposition 5.11.

If $f$ is non-degenerate, then $\displaystyle\lim_{n\to\infty}f^{-n}(\partial
U^{l})=\partial A_{f}^{l}$ for any $l$ in $\mathcal{I}_{f}$.

###### Proof.

Let $U=U^{l}$ and let us denote $\lim_{n\to\infty}$ by $\lim$ in the proof for
simplicity.

We first show the inclusion $\lim f^{-n}(\partial U)\subset\partial A_{f}$.
Let $x\in\lim f^{-n}(\partial U)$. Then we may assume that there exist
$x_{n}\in f^{-n}(\partial U)$ such that $x_{n}\to x$ as $n\to\infty$. Lemma
5.9 implies that $f^{-n}(\partial U)=\partial
f^{-n}(U)\subset\overline{f^{-n}(U)}\subset\overline{A_{f}}$. Since
$\overline{A_{f}}$ is closed, $x\in\overline{A_{f}}$. Let us show that
$x\in\partial A_{f}$ by contradiction. If $x\in A_{f}$, then there exists $N$
such that $f^{N}(x)\in U$. Hence $x\in f^{-N}(U)$ and so $x_{n}\in f^{-N}(U)$
for any large $n$, which contradicts the fact that $\partial f^{-n}(U)\cap
f^{-N}(U)=\emptyset$ for any $n>N$.

We next show the opposite direction. Let $x\in\partial A_{f}$. Then
$x\in\overline{A_{f}}$ and so we may assume that there exists $x_{n}\in
f^{-n}(U)$ such that $x_{n}\to x$ as $n\to\infty$. Any path connecting $x$ and
$x_{n}$ intersects with $\partial f^{-n}(U)$. Let $y_{n}$ be the intersection
of $\partial f^{-n}(U)$ and the segment connecting $x$ and $x_{n}$. Then
$y_{n}\to x$ as $n\to\infty$ and so $x\in\lim\partial f^{-n}(U)$, which
coincides with $\lim f^{-n}(\partial U)$ by Lemma 5.9. ∎

Since $S^{l}=\partial U^{l}\cap\\{|z|<r\\}$ and
$\lim_{n\to\infty}f^{-n}(\partial U^{l}-S^{l})\subset\partial
A_{p}\times\mathbb{C}$, the proposition above implies the following.

######  Corollary 5.12.

If $f$ is non-degenerate, then $S_{\infty}^{l}\subset\partial A_{f}^{l}$ and
$S_{\infty}^{l}\cap(A_{p}\times\mathbb{P}^{1})=\partial
A_{f}^{l}\cap(A_{p}\times\mathbb{P}^{1})$ for any $l$ in $\mathcal{I}_{f}$.

We obtain the following from Lemma 5.8 and the corollary above.

######  Proposition 5.13.

If $f$ is non-degenerate, then for any $l$ in $\mathcal{I}_{f}$ and for any
$(z_{0},w_{0})$ in $\partial
A_{f}^{l}\cap\\{(A_{p}-E_{p})\times\mathbb{P}^{1}\\}$,

$\limsup_{(z,w)\in A_{f}^{l}\to(z_{0},w_{0})}G_{z}^{\alpha}(w)=0.$

Let $E_{deg}$ be the union of all the preimages of the set of vertical lines
in $A_{p}\times\mathbb{C}$ that are degenerated to points by $f$:

$E_{deg}=\bigcup_{L}\Big{\\{}\bigcup_{n\geq 0}f^{-n}(L)\Big{\\}},$

where $L$ is a fiber in $A_{p}\times\mathbb{C}$ that is degenerated to a point
by $f$. Let $E_{deg}^{*}=\pi_{z}(E_{deg})$ where $\pi_{z}$ is the projection
to the $z$-coordinate.

######  Remark 5.14.

If $\delta<d$, then $f$ maps $\\{w=0\\}$ to itself. Hence
$f(E_{deg})\subset\\{w=0\\}$ and so $E_{deg}\subset A_{f}\cup E_{z}$.

Because $f$ is non-degenerate on $A_{p}\times\mathbb{C}-E_{z}\cup E_{deg}$, we
obtain the following by similar arguments as above.

######  Proposition 5.15.

For any $l$ in $\mathcal{I}_{f}$ and for any $(z_{0},w_{0})$ in $\partial
A_{f}^{l}\cap\\{(A_{p}-E_{p}\cup E_{deg}^{*})\times\mathbb{P}^{1}\\}$,

$\limsup_{(z,w)\in A_{f}^{l}\to(z_{0},w_{0})}G_{z}^{\alpha}(w)=0.$

This proposition implies the following, which also follows from Theorem 2.3.

######  Corollary 5.16.

It follows that $A_{f}^{l}=A_{f}^{l_{1}}$ for any $l$ in $\mathcal{I}_{f}$.

###### Proof.

By definition, $A_{f}^{l}\subset A_{f}^{l_{1}}$ for any $l>l_{1}$. It is
enough to show that

$A_{f}^{l}-E_{deg}=A_{f}^{l_{1}}-E_{deg}$

for any $l>l_{1}$ since it implies that $A_{f}^{l}\cap
E_{deg}=A_{f}^{l_{1}}\cap E_{deg}$. We derive the contradiction, assuming that
the equality above does not hold. Let $G=G_{z}^{\alpha}=G_{z}$ and $\partial
A=\partial A_{f}^{l}\cap A_{f}^{l_{1}}\cap(A_{p}\times\mathbb{C}-E_{deg})$. If
the equality above does not hold, then $\partial A\neq\emptyset$. Since $G$ is
plurisubharmonic on $A_{f}^{l_{1}}$ and since $A_{f}^{l}\cap E_{z}=\emptyset$,
it follows from the proposition above that $G\geq 0$ on $\partial A$. On the
other hand, $G\leq 0$ on $A_{f}^{l_{1}}$ since
$G=\lim_{n\to\infty}d^{-n}\log|w_{n}|$ and $|w_{n}|\to 0$ as $n\to\infty$ on
$A_{f}^{l_{1}}$. Hence $G=0$ on $\partial A$, which contradicts to the maximum
principle for $G$. ∎

######  Question 5.17.

Can we remove the non-degenerate condition?

######  Question 5.18.

Note that $\partial A_{f}\cap E_{z}\neq\emptyset$ and, moreover, $\partial
A_{f}\cap E_{z}$ have an interior point in $E_{z}$ since $A_{f}=A_{0}-E_{z}$
by Theorem 2.3. Can $\partial A_{f}\cap\mathbb{C}_{z}$ have an interior point
in $\mathbb{C}_{z}$ for some $z$ in $A_{p}-E_{p}$?

####  5.2.3. Asymptotic behavior of $G_{z}^{\alpha}$ toward $\partial
A_{f}^{l}$: Limit inferior

It follows from the assumption $m_{j}\geq d>\delta\geq 2$ that $f$ maps
$\\{w=0\\}$ to itself. If $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then we expect
that the limit infimum of $G_{z}^{\alpha}$ toward $\partial A_{f}^{l}$ is
$-\infty$ since $G_{z}^{\alpha}=-\infty$ on $E_{w}-E_{z}$. Actually, we obtain
the following with the assumption that $q$ is polynomial, which guarantees the
surjectivity of the function.

######  Proposition 5.19.

If $q$ is polynomial and $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then for any
$l$ in $\mathcal{I}_{f}$ and for any $z$ in $A_{p}-E_{p}\cup E_{deg}^{*}$,

$\liminf_{w\in A_{f}^{l}\cap\mathbb{C}_{z}\to\partial
A_{f}^{l}\cap\mathbb{P}_{z}^{1}}G_{z}^{\alpha}(w)=-\infty,$

where $\pi_{z}$ is the projection to the $z$-coordinate and
$\mathbb{P}_{z}^{1}=\\{z\\}\times\mathbb{P}^{1}$.

###### Proof.

Fix any $l$ in $\mathcal{I}_{f}$ and any $z$ in $A_{p}-E_{p}\cup E_{deg}^{*}$.
It is enough to show that $E_{w}\cap\mathbb{C}_{z}$ accumulates to $\partial
A_{f}^{l}\cap\mathbb{P}_{z}^{1}$ since $G_{z}^{\alpha}=-\infty$ on
$E_{w}-E_{z}$.

Let $f^{n}(z,w)=(p^{n}(z),Q_{z}^{n}(w))$. In other words, let

$Q_{z}^{n}(w)=q_{z_{n-1}}\circ\cdots\circ q_{z_{1}}\circ q_{z}(w),$

where $z_{j}=p^{j}(z)$. Hence $Q_{z}^{n}$ is the composition of n holomorphic
functions on $w$, which are not constant since $z\notin E_{deg}^{*}$.

We first check that the number of $(Q_{z}^{n})^{-1}(0)$ tends to infinity as
$n\to\infty$. Let $q(z,w)=w^{d}h(z,w)$ and $H=\\{z\in\mathbb{C}\,|\,\exists
w\neq 0\text{ such that }h(z,w)=0\\}$. Then
$H=\\{z\in\mathbb{C}\,|\,q_{z}^{-1}(0)\supsetneq\\{0\\}\\}$ and it is enough
to show that $H$ is the complement of only finitely many points. Because $h$
is polynomial, $\\{z\in\mathbb{C}\,|\,\exists w\text{ such that
}h(z,w)=0\\}=\mathbb{C}$. On the other hand,
$\\{z\in\mathbb{C}\,|\,h(z,0)=0\\}$ is a finite set. If not, it follows from
the identity theorem that $h(z,0)\equiv 0$, which contradicts the minimality
of $d$. Therefore, $H$ is the complement of only finitely many points.

We next show that $(Q_{z}^{n})^{-1}(0)-\\{0\\}$ converges to $\partial
A_{f}^{l}\cap\mathbb{P}_{z}^{1}$ as $n\to\infty$, which completes the proof.
Let $w_{n}$ be a point in $(Q_{z}^{n})^{-1}(0)-\\{0\\}$ and $\\{w_{n_{k}}\\}$
a subsequence of $\\{w_{n}\\}$ that converges to a point $w_{\infty}$.
Assuming that $w_{\infty}$ belongs to $A_{f}^{l}\cap\mathbb{C}_{z}$, we derive
the contradiction. Since $(z,w_{\infty})$ belongs to $A_{f}^{l}$, there is an
integer $N$ such that $f^{N}$ maps a neighborhood of $(z,w_{\infty})$ to
$U^{l}$. Because $f\sim f_{0}$ on $U^{l}$, it follows for each sufficient
large $k$ that $Q_{z_{N}}^{n}(Q_{z}^{N}(w_{n_{k}}))=Q_{z}^{n+N}(w_{n_{k}})\neq
0$ for any $n\geq 0$. Hence
$Q_{z}^{n_{k}}(w_{n_{k}})=Q_{z}^{n+N}(w_{n_{k}})\neq 0$ for $n=n_{k}-N$, which
contradicts the definition of $w_{n_{k}}$. ∎

######  Question 5.20.

It is known that if $p$ is polynomial and $p^{-1}(0)\supsetneq\\{0\\}$, then
$p^{-n}(0)-\\{0\\}$ converges to $J_{p}$ densely. Does the same or similar
thing hold for skew products? More precisely, if $f$ is polynomial and
$f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then does
$[f^{-n}\\{w=0\\}-\\{w=0\\}]\cap(A_{p}\times\mathbb{C})$ converge to $\partial
A_{f}\cap(A_{p}\times\mathbb{P}^{1})$ densely? If this holds, then we can
revise the proposition above as follows: for any $l$ in $\mathcal{I}_{f}$ and
for any $(z_{0},w_{0})$ in $\partial A_{f}^{l}\cap\\{(A_{p}-E_{p}\cup
E_{deg}^{*})\times\mathbb{P}^{1}\\}$,

$\liminf_{(z,w)\in A_{f}^{l}\to(z_{0},w_{0})}G_{z}^{\alpha}(w)=-\infty.$

######  Question 5.21.

Can we remove the condition that the map is polynomial in the previous
proposition and question?

###  5.3. $\delta=d$

Recall that $\mathcal{I}_{f}=[l_{1},\infty)$ if $\delta=d$.

We first show the existence and properties of plurisubharmonic functions on
$A_{f}^{l}$ for any $l$ in $\mathcal{I}_{f}$. Because $f$ is conjugate to
$f_{0}$ on $U^{l}$ for any $l$ in $\mathcal{I}_{f}$, the limit
$G_{z}^{\infty}$ exists on $U^{l}$. More precisely, as shown below,

$G_{z}^{\infty}(w)=\log\left|\phi_{2}(z,w)\right|=\log\left|w\right|+o(1)\text{
on }U^{l}\text{ as }r\to 0.$

It extends to $A_{f}^{l}$ via the equation $G_{z}^{\infty}\circ
f^{n}=d^{n}G_{z}^{\infty}+n\gamma d^{n-1}G_{p}$, and $A_{f}^{l}=A_{0}-E_{z}$
for any $l$ in $\mathcal{I}_{f}$ by Theorem 2.3. Consequently, we obtain the
following.

######  Proposition 5.22.

If $\delta=d$, then $G_{z}^{\infty}$ is plurisubharmonic on $A_{0}-E_{z}$. It
is pluriharmonic on $A_{0}-E$, $-\infty$ on $E_{w}-E_{z}$, $\infty$ on
$E_{z}-E_{w}$ and not defined on $E_{z}\cap E_{w}$. Moreover,
$G_{z}^{\infty}(w)=\log\left|w\right|+o(1)$ on $U^{l_{1}}$ as $r\to 0$.

The last estimate in the proposition follows from the lemma below.

######  Lemma 5.23.

If $\delta=d$, then $G_{z}^{\infty}=\log\left|\phi_{2}\right|$ on $U^{l_{1}}$.

###### Proof.

Since $\gamma_{n}=n\gamma d^{n-1}$,

$\phi_{2}(z,w)=\lim_{n\to\infty}\left(\dfrac{w_{n}}{z_{n}^{\gamma_{n}/d^{n}}}\right)^{1/d^{n}}=\lim_{n\to\infty}\left(\dfrac{w_{n}}{z_{n}^{n\gamma/d}}\right)^{1/d^{n}}$

and so

$\log|\phi_{2}(z,w)|=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{n\gamma/d}}\right|=G_{z}^{\infty}(w)\text{
on }U^{l_{1}}.$

∎

The limits $G_{z}$ and $G_{f}$ also exist on $U^{l_{1}}$, which extend to
$A_{0}$.

######  Proposition 5.24.

If $\delta=d$, then $G_{z}=-\infty$ on $A_{0}$.

###### Proof.

It follows that

$\displaystyle G_{z}^{\infty}(w)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{n\gamma/d}}\right|=\lim_{n\to\infty}\dfrac{1}{d^{n}}\log|w_{n}|-\lim_{n\to\infty}\dfrac{n\gamma}{d}\cdot\dfrac{1}{d^{n}}\log|z_{n}|$
$\displaystyle=G_{z}(w)-\infty\cdot G_{p}(z)=G_{z}(w)+\infty.$

Since $G_{z}^{\infty}<\infty$ on $A_{0}-E_{z}$, $G_{z}=-\infty$ on
$A_{0}-E_{z}$. The equality $G_{z}=-\infty$ on $E_{z}\cap A_{0}$ follows from
the dynamics on the $w$-axis as the same as the proof of Proposition 5.5. ∎

######  Corollary 5.25.

If $\delta=d$, then $G_{f}=G_{p}$ on $A_{0}$.

We next investigate the asymptotic behavior of $G_{z}^{\infty}$ as $(z,w)$ in
$A_{f}$ tends to the boundary $\partial
A_{f}^{l}\cap(A_{p}\times\mathbb{P}^{1})$. Since the arguments are similar to
the previous case $\delta<d$, we omit the proofs except the following lemma.

######  Lemma 5.26.

Let $\delta=d$. It follows for any $l$ in $\mathcal{I}_{f}$ and for any $z$ in
$A_{p}-E_{p}$ that $S_{n}^{l}\cap\mathbb{C}_{z}\neq\emptyset$ for any large
$n$, and $G_{z}^{\infty}|_{S_{n}}\to+\infty$ as $n\to\infty$.

###### Proof.

By Proposition 5.22, there is a constant $C>0$ such that

$\left|G_{z}^{\infty}(w)-\log|w|\right|<C\text{ on }U^{l}.$

Since $|w|=r|z|^{l}$ on $S_{n}^{l}$,

$\left|G_{p^{n}(z)}^{\infty}(Q_{z}^{n}(w))-\log r|p^{n}(z)|^{l}\right|<C\text{
on }S_{n}^{l}.$

Since $G_{z}^{\infty}\circ f^{n}=d^{n}G_{z}^{\infty}+n\gamma d^{n-1}G_{p}$,

$\left|G_{z}^{\infty}(w)+n\dfrac{\gamma}{d}G_{p}(z)-\dfrac{\log
r}{d^{n}}-\dfrac{l}{d^{n}}\log|p^{n}(z)|\right|\leq\dfrac{C}{d^{n}}\text{ on
}S_{n}^{l}.$

Since $\deg p=\delta=d$ and $G_{p}(z)\neq-\infty$ by the setting,

$-n\dfrac{\gamma}{d}G_{p}(z)+\dfrac{l}{d^{n}}\log|p^{n}(z)|\to+\infty-
lG_{p}(z)=+\infty$

as $n\to\infty$. ∎

This lemma and Corollary 5.12 induce the following.

######  Proposition 5.27.

Let $\delta=d$. For any $l$ in $\mathcal{I}_{f}$ and for any $(z_{0},w_{0})$
in $\partial A_{f}^{l}\cap\\{(A_{p}-E_{p}\cup
E_{deg}^{*})\times\mathbb{P}^{1}\\}$,

$\limsup_{(z,w)\in A_{f}^{l}\to(z_{0},w_{0})}G_{z}^{\infty}(w)=\infty.$

This proposition implies the following, which also follows from Theorem 2.3.

######  Corollary 5.28.

If $\delta=d$, then $A_{f}^{l}=A_{f}^{l_{1}}$ for any $l$ in
$\mathcal{I}_{f}$.

Note that $f$ maps $\\{w=0\\}$ to itself since $m_{j}\geq d=\delta\geq 2$.

######  Proposition 5.29.

Let $\delta=d$. If $q$ is polynomial and $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$,
then for any $l$ in $\mathcal{I}_{f}$ and for any $z$ in $A_{p}-E_{p}\cup
E_{deg}^{*}$,

$\liminf_{w\in A_{f}^{l}\cap\mathbb{C}_{z}\to\partial
A_{f}^{l}\cap\mathbb{P}_{z}^{1}}G_{z}^{\infty}(w)=-\infty.$

###  5.4. $T_{s-1}>\delta>d$

Recall that $\alpha>0$ and $\mathcal{I}_{f}=[l_{1},\alpha]$ if
$T_{s-1}>\delta>d$.

The dynamics differs whether $d\geq 1$ or $d=0$; the Böttcher coordinate
exists only if $d\geq 1$. We deal with the case $d\geq 1$ in Section 5.4.1 and
the case $d=0$ in Section 5.4.2, respectively.

####  5.4.1. $d\geq 1$

We first show the existence and properties of plurisubharmonic functions on
$A_{f}^{l}$ for any $l$ in $\mathcal{I}_{f}$. Because $f$ is conjugate to
$f_{0}$ on $U^{l}$ for any $l$ in $\mathcal{I}_{f}$ if $d\geq 2$, and for any
$l$ in $\mathcal{I}_{f}-\\{\alpha\\}$ if $d=1$, the limit $G_{z}^{\alpha}$
exists on $U^{l}$. More precisely,

$G_{z}^{\alpha}(w)=\log\left|\dfrac{\phi_{2}(z,w)}{\phi_{1}(z)^{\alpha}}\right|=\log\left|\dfrac{w}{z^{\alpha}}\right|+o(1)\text{
on }U^{l}\text{ as }r\to 0.$

It extends to $A_{f}^{l}$, and $A_{f}^{l}=A_{0}-E_{z}$ for any $l$ in
$\mathcal{I}_{f}-\\{\alpha\\}$ by Theorem 2.3. Consequently, we obtain the
following.

######  Proposition 5.30.

If $T_{s-1}>\delta>d\geq 1$, then $G_{z}^{\alpha}$ is plurisubharmonic on
$A_{0}-E_{z}$. It is pluriharmonic on $A_{0}-E$, $-\infty$ on $E_{w}-E_{z}$,
$\infty$ on $E_{z}-E_{w}$ and not defined on $E_{z}\cap E_{w}$. Moreover,
$G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on $U_{r}^{l_{1}}$ as
$r\to 0$.

The limit $G_{z}$ also exists on $U^{l_{1}}$. More precisely,

$G_{z}(w)=\alpha G_{p}(z)\text{ on }U^{l_{1}}-\\{w=0\\}$

and $G_{z}(w)=-\infty$ on $\\{w=0\\}$, which extends to $A_{0}$.

######  Proposition 5.31.

If $T_{s-1}>\delta>d\geq 1$, then $G_{z}=\alpha G_{p}$ on $A_{0}-E_{w}$ and
$-\infty$ on $E_{w}$. It is plurisubharmonic on $A_{0}-(E_{w}-E_{z})$,
pluriharmonic on $A_{0}-E$ and $-\infty$ on $E\cap A_{0}$.

###### Proof.

It follows that

$\displaystyle G_{z}(w)-\alpha G_{p}(z)$
$\displaystyle=\lim_{n\to\infty}\dfrac{1}{\delta^{n}}\log|w_{n}|-\lim_{n\to\infty}\dfrac{\alpha}{\delta^{n}}\log|z_{n}|=\lim_{n\to\infty}\dfrac{1}{\delta^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{\alpha}}\right|$
$\displaystyle=\lim_{n\to\infty}\left(\dfrac{d}{\delta}\right)^{n}\cdot\dfrac{1}{d^{n}}\log\left|\dfrac{w_{n}}{z_{n}^{\alpha}}\right|=0\cdot
G_{z}^{\alpha}(w)=0$

on $U^{l_{1}}-\\{w=0\\}$, which extends to $A_{0}-E$. The equality
$G_{z}=-\infty$ on $E_{w}$ follows from the definition, and the equality
$G_{z}=-\infty$ on $E_{z}\cap A_{0}$ follows from the dynamics on the $w$-axis
as the same as the proof of Proposition 5.5. ∎

######  Corollary 5.32.

If $T_{s-1}>\delta>d\geq 1$, then $G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$.

We next investigate the asymptotic behavior of $G_{z}^{\alpha}$ as $(z,w)$ in
$A_{f}^{l}$ tends to the boundary $\partial
A_{f}^{l}\cap(A_{p}\times\mathbb{P}^{1})$. Since the arguments are similar to
the case $\delta<d$, we omit the proofs except the following lemma. We also
omit to denote the assumption $T_{s-1}>\delta>d\geq 1$ for all the statements
below.

######  Lemma 5.33.

Fix any $z$ in $A_{p}-E_{p}$. Then $S_{n}^{l}\cap\mathbb{C}_{z}\neq\emptyset$
for any large $n$, and

$G_{z}^{\alpha}|_{S_{n}^{l}}\begin{cases}\ \to\infty\text{ as
}n\to\infty&\text{ if }l_{1}\leq l<\alpha,\\\ \ \to 0\text{ as
}n\to\infty&\text{ if }l=\alpha\text{ and }d\geq 2,\\\ \ \text{is bounded as
}n\to\infty&\text{ if }l=\alpha\text{ and }d=1.\end{cases}$

###### Proof.

The proof is similar to that of Lemma 5.8 except the assumptions of $\delta$
and $d$, and thus $\alpha$. By Proposition 5.30, there is a constant $C>0$
such that

$\left|G_{z}^{\alpha}(w)-\log\left|\dfrac{w}{z^{\alpha}}\right|\right|<C\text{
on }U^{l}.$

Since $\left|z^{-\alpha}w\right|=r|z|^{l-\alpha}$ on $S^{l}$, where
$l-\alpha\leq 0$,

$\left|\ G_{p^{n}(z)}^{\alpha}(Q_{z}^{n}(w))-\log r|p^{n}(z)|^{l-\alpha}\
\right|\leq C\ \text{ on }S_{n}^{l}.$

Since $G_{p^{n}(z)}^{\alpha}(Q_{z}^{n}(w))=d^{n}G_{z}^{\alpha}(w)$,

$\left|\ G_{z}^{\alpha}(w)-d^{-n}\log r|p^{n}(z)|^{l-\alpha}\ \right|\leq
d^{-n}C\ \text{ on }S_{n}^{l}.$

If $l<\alpha$ and $d\geq 2$, then

$\dfrac{1}{d^{n}}\log|p^{n}(z)|^{l-\alpha}=\left(\dfrac{\delta}{d}\right)^{n}\cdot\dfrac{l-\alpha}{\delta^{n}}\log|p^{n}(z)|\to\infty\cdot(l-\alpha)G_{p}(z)=\infty$

as $n\to\infty$ since $\deg p=\delta>d$ and $G_{p}(z)\neq-\infty$. If
$l<\alpha$ and $d=1$, then

$\log|p^{n}(z)|^{l-\alpha}=(l-\alpha)\log|p^{n}(z)|\to(l-\alpha)\cdot(-\infty)=\infty$

as $n\to\infty$. On the other hand, if $l=\alpha$ and $d\geq 2$, then

$d^{-n}\log r|p^{n}(z)|^{l-\alpha}=d^{-n}\log r\to 0$

and $d^{-n}C\to 0$ as $n\to\infty$. If $l=\alpha$ and $d=1$, then

$\left|\ G_{z}^{\alpha}(w)-\log r\ \right|\leq C\ \text{ on }S_{n}^{l}.$

∎

This lemma and Corollary 5.12 induce the following.

######  Proposition 5.34.

For any $(z_{0},w_{0})$ in $\partial A_{f}^{l}\cap\\{(A_{p}-E_{p}\cup
E_{deg}^{*})\times\mathbb{P}^{1}\\}$,

$\limsup_{(z,w)\in A_{f}^{l}\to(z_{0},w_{0})}G_{z}^{\alpha}(w)=\begin{cases}\
\infty&\text{ if }l_{1}\leq l<\alpha,\\\ \ 0&\text{ if }l=\alpha\text{ and
}d\geq 2,\end{cases}$

and $G_{z}^{\alpha}$ is bounded as $(z,w)$ in $A_{f}^{l}$ tends to
$(z_{0},w_{0})$ if $l=\alpha$ and $d=1$.

######  Corollary 5.35.

It follows that $A_{f}^{\alpha}\subsetneq A_{f}^{l}=A_{f}^{l_{1}}$ for any $l$
in $\mathcal{I}_{f}-\\{\alpha\\}$.

Note that $f$ maps $\\{w=0\\}$ to itself since $m_{j}\geq d\geq 1$.

######  Proposition 5.36.

If $q$ is polynomial and $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then for any
$l$ in $\mathcal{I}_{f}$ and for any $z$ in $A_{p}-E_{p}\cup E_{deg}^{*}$,

$\liminf_{w\in A_{f}^{l}\cap\mathbb{C}_{z}\to\partial
A_{f}^{l}\cap\mathbb{P}_{z}^{1}}G_{z}^{\alpha}(w)=-\infty.$

Let $E_{w}^{\infty}$ be the set of the accumulation points of
$f^{-n}\\{w=0\\}-\\{w=0\\}$ in $\mathbb{C}\times\mathbb{P}^{1}$ as
$n\to\infty$. The proof of Proposition 5.19 implies the following.

######  Proposition 5.37.

If $q$ is polynomial and $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then

$E_{w}^{\infty}\cap\\{(A_{p}-E_{p}\cup
E_{deg}^{*})\times\mathbb{P}^{1}\\}\subset\bigcap_{l_{1}\leq
l\leq\alpha}\partial A_{f}^{l}.$

Consequently, $E_{w}^{\infty}$ is unbounded if $f$ satisfies the assumptions
since $A_{f}^{\alpha}\subsetneq A_{f}^{l_{1}}$. Since
$E_{w}^{\infty}\cap(A_{p}\times\mathbb{P}^{1})$ is included in the closure of
$A_{f}^{l}$ in $\mathbb{C}\times\mathbb{P}^{1}$, we obtain the following.

######  Corollary 5.38.

If $q$ is polynomial and $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then
$A_{f}^{l}$ is unbounded for any $l$ in $\mathcal{I}_{f}$.

####  5.4.2. $d=0$

If $d=0$, then $f$ can not be conjugate to $f_{0}$ on any open set, since
$f_{0}$ is not dominant, where $f_{0}(z,w)=(z^{\delta},z^{\gamma})$. However,
the same statements as Lemma 3.1 hold for any $l$ in
$\mathcal{I}_{f}-\\{\alpha\\}$, and we can show that $G_{z}=\alpha G_{p}$ on
$A_{0}$.

######  Lemma 5.39.

Let $T_{s-1}>\delta>d=0$. The following hold for any $l$ in
$\mathcal{I}_{f}-\\{\alpha\\}$: $f\sim f_{0}$ on $U_{r_{1},r_{2}}^{l}$ as
$r_{1}$ and $r_{2}\to 0$, and $f(U_{r_{1},r_{2}}^{l})\subset
U_{r_{1},r_{2}}^{l}$ for suitable small $r_{1}$ and $r_{2}$.

The first statement follows from the same argument as in [26], and it holds
even for $l=\alpha$. The second statement follows from Theorem 2.2.

######  Remark 5.40 (Illustration of the second statement in terms of blow-
ups).

Let $\tilde{f}$ be a formal blow-up of $f$ by $\pi(z,c)=(z,z^{l}c)$. Then
$\tilde{f}(z,c)=\left(z^{\delta},\
z^{\tilde{\gamma}}\left\\{1+o(1)\right\\}\right)$, where
$\tilde{\gamma}=\gamma-l\delta\geq 0$ since $d=0$. If
$l\in\mathcal{I}_{f}-\\{\alpha\\}$, then $\tilde{\gamma}>0$ and so the origin
is a superattracting fixed point of $\tilde{f}$. In particular, $\tilde{f}$
preserves a neighborhood of the origin, and so $f$ preserves
$U_{r_{1},r_{2}}^{l}$. On the other hand, if $l=\alpha$, then
$\tilde{\gamma}=0$ and so the origin is neither superattracting nor a fixed
point of $\tilde{f}$.

######  Proposition 5.41.

If $T_{s-1}>\delta>d=0$, then $G_{z}=\alpha G_{p}$ on $A_{0}$.

###### Proof.

Fix any $l$ in $\mathcal{I}_{f}-\\{\alpha\\}$. Let $U=U_{r_{1},r_{2}}^{l}$ and
let $(z_{n},w_{n})=f^{n}(z,w)$ for any $(z,w)$ in $U$. The second statement in
the lemma above implies that $(z_{n},w_{n})$ also belongs to $U$. The first
statement in the lemma implies that

$C_{1}|(z_{n})^{\gamma}|<|w_{n}|<C_{2}|(z_{n})^{\gamma}|$

for some constants $C_{1}<1$ and $C_{2}>1$. Hence $\delta^{-n}\log|w_{n}|$ is
bounded by

$\dfrac{1}{\delta^{n}}\log
C_{j}+\dfrac{\gamma}{\delta}\cdot\dfrac{1}{\delta^{n-1}}\log|z_{n}|$

from below and above for $j=1$ and $j=2$, respectively. Therefore,
$G_{z}=\alpha G_{p}$ on $U$ and so on $A_{f}^{l}$. Thanks to Theorem 2.3, the
equality holds on $A_{0}$. ∎

###  5.5. $\delta=T_{s-1}$

Recall that $\delta>d$ and $\mathcal{I}_{f}=\\{l_{1}\\}=\\{\alpha\\}$ if
$\delta=T_{s-1}$.

Because $f$ is conjugate to $f_{0}$ on $U^{\alpha}$ if $d\geq 2$, the limit
$G_{z}^{\alpha}$ exists on $U^{\alpha}$, which extends to $A_{f}^{\alpha}$.

######  Proposition 5.42.

If $T_{s-1}=\delta>d\geq 2$, then $G_{z}^{\alpha}$ is plurisubharmonic on
$A_{f}^{\alpha}$, pluriharmonic on $A_{f}^{\alpha}-E_{w}$ and $-\infty$ on
$E_{w}$. Moreover, $G_{z}^{\alpha}(w)=\log\left|w/z^{\alpha}\right|+o(1)$ on
$U_{r}^{\alpha}$ as $r\to 0$.

The asymptotic behavior of $G_{z}^{\alpha}$ toward $\partial A_{f}^{\alpha}$
is as follows.

######  Proposition 5.43.

If $T_{s-1}=\delta>d\geq 2$, then for any $(z_{0},w_{0})$ in $\partial
A_{f}^{\alpha}\cap\\{(A_{p}-E_{p}\cup E_{deg}^{*})\times\mathbb{P}^{1}\\}$,

$\limsup_{(z,w)\in A_{f}^{\alpha}\to(z_{0},w_{0})}G_{z}^{\alpha}(w)=0.$

Moreover, if $q$ is polynomial and $f^{-1}\\{w=0\\}\supsetneq\\{w=0\\}$, then
for any $z$ in $A_{p}-E_{p}\cup E_{deg}^{*}$,

$\liminf_{w\in A_{f}^{\alpha}\cap\mathbb{C}_{z}\to\partial
A_{f}^{\alpha}\cap\mathbb{P}_{z}^{1}}G_{z}^{\alpha}(w)=-\infty.$

The dynamics of $f$ differs whether $n_{s-1}>0$ or $n_{s-1}=0$. If
$n_{s-1}>0$, then Theorem 2.3 implies the following inequality and equalities
on $A_{0}$.

######  Proposition 5.44.

Let $\delta=T_{s-1}$. If $n_{s-1}>0$, then $G_{z}^{*}\leq\alpha G_{p}$ on
$A_{0}$ and $G_{z}=\alpha G_{p}$ on $A_{0}-A_{f}^{\alpha}$. In particular,
$G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$. Moreover, if $d\geq 2$, then
$G_{z}=\alpha G_{p}$ on $A_{0}-E_{w}$ and $-\infty$ on $E_{w}$.

On the other hand, if $n_{s-1}=0$, then we have the same inequality and almost
the same equalities only on $A_{f}^{\alpha}$.

######  Proposition 5.45.

Let $\delta=T_{s-1}$. If $n_{s-1}=0$, then $G_{z}^{*}\leq\alpha G_{p}$ on
$A_{f}^{\alpha}$. In particular, $G_{f}^{\alpha}=\alpha G_{p}$ on
$A_{f}^{\alpha}$. Moreover, if $d\geq 2$, then $G_{z}=\alpha G_{p}$ on
$A_{f}^{\alpha}-E_{w}$ and $-\infty$ on $E_{w}$.

######  Remark 5.46.

If $\delta=T_{s-1}$, then $f$ has the other dominant term $(n_{s-1},m_{s-1})$
than $(n_{s},m_{s})$, which belongs to Case 3 or Case 4. Moreover, if
$n_{s-1}=0$, then $s=2$ and $(n_{1},m_{1})=(n_{s-1},m_{s-1})=(0,\delta)$,
which belongs to Case 3. Therefore, we can apply the results in Case 3 or Case
4. For example, variants of Corollary 6.7 and Corollary 7.9 imply that
$G_{f}^{\alpha}=\alpha G_{p}$ on $A_{0}$ if $n_{s-1}>0$ and that
$G_{f}^{\alpha}$ is plurisubharmonic and continuous on $A_{0}-E_{deg}$ if
$n_{s-1}=0$.

##  6\. Case 3

We deal with Case 3 in this section. Let $s>1$,

$T_{1}\leq\delta\text{ and }(\gamma,d)=(n_{1},m_{1}).$

Note that $\gamma\geq 0$ and $\delta\geq T_{1}\geq d\geq 1$ by the setting.

Although we observe that the blow-up $\tilde{f}$ has the required properties
in Section 6.1, there are no results on invariant wedges and the unions of all
the preimages of wedges for Case 3. We exhibit results on the existence and
properties of plurisubharmonic functions and on the asymptotic behavior of the
functions toward the boundaries of the unions in Section 6.2, which imply all
the main results stated in Section 2.2. Although we omit to prove the results
on the existence and properties since they are similar to Case 2, we prove the
results on the asymptotic behavior in Section 6.3. One may refer Table 2 in
Section 2.2 for a comparison chart of the results.

Two examples are given in Section 6.4. For any polynomial $h$, we can
construct a polynomial skew product that is semiconjugate to a polynomial
product of the form $(z^{\delta},h(w))$. Dynamics of such a map is rather easy
to understand. More precisely, all the limits of the map are expresses by the
counterparts of $h$ and, in particular, the boundary of of the set where
$G_{z}^{\alpha,+}$ is not pluriharmonic is expresses by the Julia set of $h$.

###  6.1. Observation of Blow-ups

Let us observe that the blow-up $\tilde{f}$ is superattracting at the origin
and degenerates the $t$-axis for a suitable weight $l$. If $l$ belongs to
$\mathcal{I}_{f}$, then $\tilde{q}(t,w)\sim t^{\gamma}w^{\tilde{d}}$ on a
neighborhood of the origin and so

$\tilde{f}(t,w)=\left(t^{\delta-l^{-1}\gamma}w^{l^{-1}(\delta-\tilde{d})}\\{1+\zeta(t,w)\\},\
t^{\gamma}w^{\tilde{d}}\\{1+\eta(t,w)\\}\right),$

where $\tilde{d}=l^{-1}\gamma+d$. Moreover, if $\delta>T_{1}$ and $l>\alpha$,
then $\delta-\tilde{d}>0$ and so we have the following.

######  Proposition 6.1.

If $\delta>T_{1}$, then $\tilde{f}$ is superattracting at the origin and
degenerates the $t$-axis for any $l$ in $\mathcal{I}_{f}-\\{\alpha\\}$.

Although $\tilde{f}$ degenerates the $t$-axis, we do not know whether
$A_{f}^{l}$ contains $A_{0}-E_{w}$.

###  6.2. Plurisubharmonic functions

Let us first review $\mathcal{I}_{f}$ and related quantities, which depends on
whether $\delta>T_{1}$ and whether $\gamma>0$; see the table below for a
summary. If $\delta>T_{1}$ and $\gamma>0$, then $\delta>d$, $\alpha>0$ and
$\mathcal{I}_{f}=[\alpha,l_{2}]$. If $\delta>T_{1}$ and $\gamma=0$, then
$\delta>d$, $\alpha=0$ and $\mathcal{I}_{f}=(0,l_{2}]$. Moreover, $d\geq 2$
because of the assumption for $f$ being superattracting at the origin. If
$\delta=T_{1}$ and $\gamma>0$, then $\delta>d$, $\alpha>0$ and
$\mathcal{I}_{f}=\\{\alpha\\}=\\{l_{2}\\}$. If $\delta=T_{1}$ and $\gamma=0$,
then $\delta=d$, $\alpha=l_{2}$ and $\mathcal{I}_{f}=(0,l_{2}]$; recall that
we redefined $\alpha$ to be $l_{2}$ in Section 2.2, whereas
$\gamma/(\delta-d)$ is not defined for this case.

Table 12. Informations on $\mathcal{I}_{f}$ and related quantities for Case 3 | $\gamma>0$ | $\gamma=0$
---|---|---
$\delta>T_{1}$ | $\mathcal{I}_{f}=[\alpha,l_{2}]$ | $\mathcal{I}_{f}=(0,l_{2}]$
|  $\delta>d\geq 1$ and $\alpha>0$ |  $\delta>d\geq 2$ and $\alpha=0$
$\delta=T_{1}$ | $\mathcal{I}_{f}=\\{\alpha\\}=\\{l_{2}\\}$ | $\mathcal{I}_{f}=(0,l_{2}]$
|  $\delta>d\geq 1$ and $\alpha>0$ |  $\delta=d\geq 2$ and $\alpha=l_{2}>0$

In particular, the dynamics differs more or less whether $\gamma>0$ or
$\gamma=0$. If $\gamma=0$, then $f_{0}$ is just a product. More precisely, if
$\gamma=0$ and $\delta>d$, then $\alpha=0$ and so the properties of
$G_{z}^{\alpha}$ are different from the other cases. On the other hand, if
$\gamma=0$ and $\delta=d$, then $f_{0}(z,w)=(z^{\delta},w^{\delta})$ and so
the properties of $G_{z}$ and the magnitude relation of $A_{f}^{l}$ for any
$l$ in $\mathcal{I}_{f}$ are different from the other cases.

Recall that $f$ is conjugate to the monomial map $f_{0}$ on $U^{l}$ for any
$l$ in $\mathcal{I}_{f}$ if $d\geq 2$ and for any $l$ in
$\mathcal{I}_{f}-\\{\alpha\\}$ if $d=1$ and $\delta>T_{1}$ as stated in Lemma
3.6. Therefore, $G_{z}^{\alpha}$ exists on $U^{l}-\\{z=0\\}$ for any $l$ as
above if $\gamma>0$ or $\delta=d$. More precisely,

$G_{z}^{\alpha}(w)=\log\left|\dfrac{\phi_{2}}{\phi_{1}^{\alpha}}\right|=\log\left|\dfrac{w}{z^{\alpha}}\right|+o(1)\text{
on }U^{l}-\\{z=0\\}\text{ as }r\to 0,$

which extends to a pluriharmonic function on $A_{f}^{l}-E_{z}$. On the other
hand, if $\gamma=0$ and $\delta>d$, then $\alpha=0$ and $G_{z}^{\alpha}$
exists on $U^{l}$ for any $l$ in $\mathcal{I}_{f}$. More precisely,

$G_{z}^{\alpha}(w)=\log\left|\phi_{2}\right|=\log\left|w\right|+o(1)\text{ on
}U^{l}\text{ as }r\to 0,$

which extends to a pluriharmonic function on $A_{f}^{l}$.

######  Proposition 6.2.

The following hold for any $l$ in $\mathcal{I}_{f}$ if $d\geq 2$ and for any
$l$ in $\mathcal{I}_{f}-\\{\alpha\\}$ if $d=1$ and $\delta>T_{1}$: If
$\gamma>0$ or $\delta=d$, then $\alpha>0$ and $G_{z}^{\alpha}$ is