# Attraction rates for iterates of a superattracting skew product

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

Let $f(z,w)=(p(z),q(z,w))$ be a holomorphic skew product with a
superattracting fixed point at the origin. In the previous paper we have
succeeded to specify a dominant term of $q$ by the order of $p$ and the Newton
polygon of $q$, and to construct a Böttcher coordinate on an invariant wedge.
By using the same idea and terminologies, we give inequalities of attraction
rates for the vertical dynamics of $f$ in this paper. The results hold not
only for the superattracting case, but for all the other cases.

###### 2010 Mathematics Subject Classification:

Primary 32H50; Secondary 37C25

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

##  1\. Introduction

Let $f:(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ be a holomorphic germ with a
superattracting fixed point at the origin, and let $f^{n}$ be the $n$-th
iterate of $f$. We define $c(f)$ as the smallest degree of any term in the
Taylor expansion of $f$ in local coordinates, which is independent of the
choice of coordinates, and we call $c(f)$ the attraction rate of $f$. From the
viewpoint of complex dynamics, it is important to study the behavior of the
attraction rates for iterates of $f$, because it gives a measure of the rate
at which nearby points are attracted to the origin under iteration. Moreover,
the limit $c_{\infty}=\lim_{n\to\infty}\sqrt[n]{c(f^{n})}$, which we call the
asymptotic attraction rate of $f$, measures the growth of the sequence
$\\{c(f^{n})\\}_{n\geq 1}$ of attraction rates. Favre and Jonsson [1] proved
that $c_{\infty}$ is a quadratic integer, and there exists $D\in(0,1]$ such
that $Dc_{\infty}^{n}\leq c(f^{n})\leq c_{\infty}^{n}$ for any $n\geq 1$. This
result is derived from their result on normal forms of $f$, which is obtained
by blow-ups, and applied to construct a pluriharmonic function with the
adequate invariance property. By using valuative techniques similar to those
developed by Favre and Jonsson, Gignac and Ruggiero [2] proved that the
sequence $\\{c(f^{n})\\}_{n\geq 1}$ eventually satisfies an integral linear
recursion relation, which, up to replacing $f$ by an iterate, can be taken to
have order at most two.

In this paper we are concerned with skew products, and provide equalities and
inequalities of the attraction rates for the vertical dynamics. A holomorphic
germ of the form $f(z,w)=(p(z),q(z,w))$ is called a skew product. We assume
that it has a fixed point at the origin; thus we have the Taylor expansions

$p(z)=a_{\delta}z^{\delta}+O(z^{\delta+1})\text{ and }q(z,w)=\sum_{i+j\geq
1}b_{ij}z^{i}w^{j},$

where $a_{\delta}\neq 0$ and $\delta\geq 1$, in local coordinates. We define
$c(f)$, $c(p)$ and $c(q)$ as the smallest degrees of any term in this Taylor
expansion of $f$, $p$ and $q$, respectively. Then $c(f)=\min\\{c(p),c(q)\\}$,
$c(p)=\delta$ and

$c(q)=\min\\{i+j:b_{ij}\neq 0\\}.$

Let $f^{n}(z,w)=(p^{n}(z),Q^{n}(z,w))$. Then
$c(f^{n})=\min\\{c(p^{n}),c(Q^{n})\\}$ and $c(p^{n})=\delta^{n}$. Our aim is
to provide an estimate on the attraction rate $c(Q^{n})$, which is deeply
related to the Newton polygon of $Q^{n}$ as stated later. We remark that,
although an estimate on $c(Q^{n})$ implies that on $c(f^{n})$, the opposite
direction is not true in general.

We first recall our previous result in [3]. Assuming that the origin is
superattracting, we have succeeded to show the existence of a dominant term
$b_{\gamma d}z^{\gamma}w^{d}$ of $q$ and to construct a Böttcher coordinate
that conjugates $f$ to the monomial map $f_{0}$ on the wedge $U$, where
$f_{0}(z,w)=(a_{\delta}z^{\delta},b_{\gamma d}z^{\gamma}w^{d})$ and
$U=\\{|z|^{l_{1}+l_{2}}<r^{l_{2}}|w|,|w|<r|z|^{l_{1}}\\}$ for some rational
numbers $0\leq l_{1}<\infty$ and $0<l_{2}\leq\infty$ and for small $r>0$. The
bidegree $(\gamma,d)$ and the rational numbers $l_{1}$ and $l_{2}$ are
determined by the order of $p$ and the Newton polygon of $q$. 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 unique vertex, which is denoted by $(\gamma,d)$,
and we define $l_{1}=l_{2}^{-1}=0$. Hence $U=\\{|z|<r,|w|<r\\}$.

If $s=1$, then $b_{\gamma d}z^{\gamma}w^{d}$ is clearly the dominant term of
$q$ and the result is classical. Difficulties appear when $s>1$, which is
divided into the following three cases.

  * Case 2

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

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

  * Case 3

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

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

  * 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

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

Note that $\gamma>0$ for Case 2, $d>0$ for Case 3, and $\gamma d>0$ for Case 4
by the setting. The rational numbers $-l_{1}^{-1}$ and $-(l_{1}+l_{2})^{-1}$
are the slopes of the lines $L_{k-1}$ and $L_{k}$ for Case 4, and the same
correspondence holds for all the cases if we define $L_{0}=\\{x=n_{1}\\}$ and
$L_{s}=\\{y=m_{s}\\}$.

A precise statement of our previous result is the following.

######  Theorem 1.1 (Lemmas 1.1 and 1.7 and Theorems 1.2 and 1.8 in [3]).

If $\delta\geq 2$ and $d\geq 2$, or if $\delta\geq 2$, $d=1$ and $\delta\neq
T_{k}$ for any $k$, then $f$ preserves $U$ and there is a biholomorphic map
defined on $U$ that conjugates $f$ to $f_{0}$ for small $r>0$.

The map $\phi$ was called the Böttcher coordinate for $f$ on $U$ and
constructed as the limit of the compositions of $f_{0}^{-n}$ and $f^{n}$ in
[3]. As detailed versions using intervals of weights, we also exhibit Theorems
2.4, 2.9 and 2.15 for Cases 2, 3 and 4, respectively, in Section 2.

We next state our results on the attraction rates. We have found that the same
idea as in [3] can be applied to study the attraction rate of $Q^{n}$, not
only for the superattracting case but for all the other cases. For Case 1, it
is clear that $Q^{n}$ contains the dominant term $z^{\gamma_{n}}w^{d^{n}}$ and
$c(Q^{n})=\gamma_{n}+d^{n}$, where
$\gamma_{n}=\gamma(\delta^{n-1}+\delta^{n-2}d+\cdots+d^{n-1})$. Hence
$c(f^{n})=\min\\{\delta^{n},\gamma_{n}+d^{n}\\}$ and so
$c_{\infty}=\min\\{\delta,d\\}$. Difficulties appear for Cases 2, 3 and 4, and
the situation differs whether $d>0$ or not for Case 2. To overcome the
difficulties, it is useful to consider the following quantity:

$w(q)=w_{l}(q)=\min\\{i+lj:b_{ij}\neq 0\\}.$

Actually, we obtain the following equalities of $w(Q^{n})$.

######  Theorem 1.2.

It follows for any $n\geq 1$ that $Q^{n}$ contains the term
$z^{\gamma_{n}}w^{d^{n}}$ and

  1. (1)

$w_{l_{1}}(Q^{n})=\gamma_{n}+l_{1}d^{n}$ if $d>0$ for Case 2,

  2. (2)

$w_{l_{2}}(Q^{n})=\gamma_{n}+l_{2}d^{n}$ for Case 3,

  3. (3)

$w_{l_{1}}(Q^{n})=\gamma_{n}+l_{1}d^{n}$ and
$w_{l_{1}+l_{2}}(Q^{n})=\gamma_{n}+(l_{1}+l_{2})d^{n}$ for Case 4.

Moreover, for any $n\geq 1$,

  1. $(4)$

$w_{l_{1}}(Q^{n})=\gamma_{n}$ if $d=0$ for Case 2.

Although we omit the coefficient of the term $z^{\gamma_{n}}w^{d^{n}}$ in the
statement for simplicity, it is equal to
$a_{\delta}^{\gamma_{n-1}+\gamma_{n-2}+\cdots+\gamma_{2}+\gamma_{1}}b_{\gamma
d}^{d^{n-1}+d^{n-2}+\cdots+d+1}$, where $\gamma_{1}=\gamma$. As detailed
versions using intervals of weights, we also exhibit Theorems 3.1, 4.1 5.1 and
6.1 for each cases. Because $w(Q^{n})$ coincides with the minimum
$x$-intercept of the lines with slope $-l^{-1}$ that intersect the Newton
polygon $N(Q^{n})$, this theorem shows us the shape of $N(Q^{n})$. More
precisely, $N(Q^{n})$ is included in the upper-right region that is surrounded
by the two lines with slopes $-l_{1}^{-1}$ and $-(l_{1}+l_{2})^{-1}$, which
intersect at $(\gamma_{n},d^{n})$. Because $c(Q^{n})$ coincides with the
minimum $x$-intercept of the lines with slope $-1$ that intersect $N(Q^{n})$,
we obtain the following inequalities of the attraction rate $c(Q^{n})$.

######  Theorem 1.3.

It follows for any $n\geq 1$ that $c(Q^{n})\leq\gamma_{n}+d^{n}$ and

  1. (1)

$\min\\{l_{1}^{-1},1\\}\gamma_{n}+d^{n}\leq c(Q^{n})$ if $d>0$ for Case 2,

  2. (2)

$\gamma_{n}+\min\\{l_{2},1\\}d^{n}\leq c(Q^{n})$ for Case 3,

  3. (3)

$\min\\{l_{1}^{-1},1\\}\gamma_{n}+\min\\{l_{1}+l_{2},1\\}d^{n}\leq c(Q^{n})$
for Case 4.

On the other hand, for any $n\geq 1$,

  1. $(4)$

$\min\\{l_{1}^{-1},1\\}\gamma_{n}\leq
c(Q^{n})\leq\max\\{l_{1}^{-1},1\\}\gamma_{n}$ if $d=0$ for Case 2.

This theorem is restated as Corollaries 3.2, 4.2, 5.2 and 6.2 for each cases.
Moreover, we can improve these inequalities by investigating the vertices of
$N(Q^{n})$ that are previous and/or next to $(\gamma_{n},d^{n})$; see Theorems
3.3, 4.3, 4.4, 5.3 and 6.3 for improved versions. We remark that, whereas the
term $z^{\gamma_{n}}w^{d^{n}}$ may vanish for some $n_{0}$ if $d=0$ for Case
2, the equalities (4) in Theorems 1.2 and 1.3 follow from the existence of the
other vertex of $N(Q^{n})$ that should be previous to $(\gamma_{n},d^{n})$ if
the term $z^{\gamma_{n}}w^{d^{n}}$ did not vanish for any $n\geq n_{0}$.

Let $\alpha=\gamma/(\delta-d)$ when $\delta\neq d$. As a corollary of Theorem
1.3, we obtain the following inequalities of the attraction rate $c(f^{n})$.

######  Corollary 1.4.

Let $\gamma d>0$. Then $c_{\infty}=\delta$ and

  1. (1)

$\alpha\delta^{n}\leq c(f^{n})<\delta^{n}$ for any $n\geq 1$ if $\delta>d$ and
$\alpha<1$, or

  2. (2)

$c(f^{n})=\delta^{n}$ for any $n\geq 1$ if $\delta>d$ and $\alpha\geq 1$ or if
$\delta\leq d$

for Cases 2, 3 and 4. On the other hand,

  1. $(3)$

$c_{\infty}=\delta$ and $D\delta^{n}\leq c(f^{n})\leq\delta^{n}$ for any
$n\geq 1$, where $D=1,\gamma/\delta$ or $l_{1}^{-1}\gamma/\delta$,
if $d=0$ for Case 2, and

  2. $(4)$

$c_{\infty}=\min\\{\delta,d\\}$ and $Dc_{\infty}^{n}\leq c(f^{n})\leq
c_{\infty}^{n}$ for any $n\geq 1$, where $D=1$ or $l_{2}$,
if $\gamma=0$ for Case 3.

The organization of this paper is as follows. In Section 2 we review the
related definitions and results in [3]. More precisely, we recall the
definitions of the intervals of weights, and illustrate Theorems 2.4, 2.9 and
2.15, detailed versions of Theorem 1.1, by blow-ups. Although we do not use
blow-ups to prove the theorems, they are useful to explain the theorems when
the weights are integer. We then prove Theorems 1.2 and 1.3 or, more
precisely, detailed versions of Theorem 1.2 and improved versions of Theorem
1.3, for Case 2 when $d>0$, Case 2 when $d=0$, Case 3 and Case 4 in Sections
3, ,4, 5 and 6, respectively. Finally, we induce Corollary 1.4 from Theorem
1.3 in Section 7.

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

In this section we give a summary of our previous results in [3]: we introduce
intervals of weights, explain benefits of the intervals in terms of blow-ups,
and state the results on Böttcher coordinates by means of the intervals. We
deal with Cases 2, 3 and 4 in Sections 2.1, 2.2 and 2.3, respectively. The
intervals for Cases 2 and 3 are also used to describe equalities of $w(Q^{n})$
in Sections 3, 4 and 5. For Case 4 we use another interval to describe
equalities of $w(Q^{n})$ in Section 6, but it is closely related to the
intervals and the rectangle introduced here. Although we do not use blow-ups
in the proofs of our main theorems, they are useful to explain our results in
both the previous and this papers. Let

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

if $\delta\neq d$, and assume that $a_{\delta}=1$ and $b_{\gamma d}=1$ for
simplicity.

###  2.1. Interval 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 $j\geq d$ for any $(i,j)$ such that $b_{ij}\neq 0$ by
the setting.

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 $j$ such that }b_{ij}\neq 0\end{array}\right\\}.$

If $\delta>d$, then

$\mathcal{I}_{f}=\left[\max_{i,j}\left\\{\dfrac{\gamma-i}{j-d}\ \Big{|}\
b_{ij}\neq 0\text{ and }j>d\right\\},\dfrac{\gamma}{\delta-d}\right].$

Since the ratio $(\gamma-i)/(d-j)$ is the slope of the line passing through
$(\gamma,d)$ and $(i,j)$, we can take the maximum over $(i,j)$ at
$(n_{s-1},m_{s-1})$:

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

which is mapped to $[\delta,T_{s-1}]$ by the transformation $l\to
l^{-1}\gamma+d$. Therefore, $\mathcal{I}_{f}$ can be identified with the set
of the lines passing through $(\gamma,d)$, whose slopes are in
$[-\alpha,-l_{1}]$ or, equivalently, whose $y$-intercepts are in
$[\delta,T_{s-1}]$. If $\delta\leq d$, then the inequality
$l\delta\leq\gamma+ld$ is trivial and so $\mathcal{I}_{f}=[l_{1},\infty)$. In
particular, $\min\mathcal{I}_{f}=l_{1}$.

Assuming that $l$ in $\mathcal{I}_{f}$ is integer, we explain benefits of the
weights in $\mathcal{I}_{f}$ and our previous results in terms of blow-ups.
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 we have

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

Let $\tilde{i}=i+lj-l\delta$, $\tilde{\gamma}=\gamma+ld-l\delta$ and
$\tilde{n}_{j}=n_{j}+lm_{j}-l\delta$. Then the Newton polygon $N(\tilde{q})$
of $\tilde{q}$ has just one vertex $(\tilde{\gamma},d)$:
$N(\tilde{q})=D(\tilde{\gamma},d)$.

######  Lemma 2.1.

It follows that $0\leq\tilde{\gamma}\leq\tilde{i}$ for any $l$ in
$\mathcal{I}_{f}$ and for any $(i,j)$ such that $b_{ij}\neq 0$. In particular,
$0\leq\tilde{\gamma}\leq\tilde{n}_{j}$ for any $l$ in $\mathcal{I}_{f}$ and
for any $j$. More precisely, $\tilde{\gamma}=\tilde{n}_{s-1}$ and
$\tilde{\gamma}<\tilde{n}_{j}$ for any $j\neq s-1$, $s$ if $l=l_{1}$, and
$\tilde{\gamma}<\tilde{n}_{j}$ for any $j\neq s$ if $l_{1}<l$. Moreover,
$\tilde{\gamma}>0$ if $l<\alpha$, and $\tilde{\gamma}=0$ if $l=\alpha$.

######  Remark 2.2.

The blow-up of $f$ can be transfered to the affine transformation of $N(q)$.
The affine transformation

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

maps the basis $\\{(1,0),(-l_{1},1)\\}$ to $\\{(1,0),(0,1)\\}$. In other
words, $A_{1}$ maps a horizontal line and the line $L_{s-1}$ with slope
$-l_{1}^{-1}$ to the same horizontal line and a vertical line.

######  Proposition 2.3.

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

$\tilde{f}(z,c)=\big{(}z^{\delta}(1+o(z)),\
z^{\tilde{\gamma}}c^{d}(1+o(z,c))\big{)},$

and it has a fixed point at the origin if $d>0$. Moreover, if $\delta\geq 2$
and $d\geq 2$, or if $\delta\geq 2$, $d=1$ and $\delta<T_{s-1}$, then the
fixed point is superattracting.

Therefore, if $l$ is integer and $\tilde{f}$ is superattracting, then it is
easy to construct the Böttcher coordinate for $\tilde{f}$ on a neighborhood of
the origin, that conjugates $\tilde{f}$ to
$(z,c)\to(z^{\delta},z^{\tilde{\gamma}}c^{d})$, because $\tilde{f}$ is a
holomorphic skew product in Case 1. Consequently, we obtain the Böttcher
coordinate for $f$ on $U_{r}^{l}$ that conjugates $f$ to $f_{0}$, where
$f_{0}(z,w)=(z^{\delta},z^{\gamma}w^{d})$ and
$U_{r}^{l}=\\{|z|<r,|w|<r|z|^{l}\\}$. Actually, we can construct the Böttcher
coordinate for $f$ on $U_{r}^{l}$ directly even if $l$ in $\mathcal{I}_{f}$ is
not integer nor rational number.

######  Theorem 2.4 ([3]).

If $\delta\geq 2$ and $d\geq 2$, or if $\delta\geq 2$, $d=1$ and
$\delta<T_{s-1}$, then for any $l$ in $\mathcal{I}_{f}$, $f$ preserves
$U_{r}^{l}$ and there is a biholomorphic map defined on $U_{r}^{l}$ that
conjugates $f$ to $f_{0}$ for small $r$.

Note that $U_{r}^{l_{1}}$ is the largest region among $U_{r}^{l}$ for any $l$
in $\mathcal{I}_{f}$.

######  Remark 2.5.

Even if $l$ is rational, we can lift $f$ to a holomorphic skew product. In
fact, let $\pi_{1}(\mathsf{z},c)=(\mathsf{z}^{r},\mathsf{z}^{s}c)$ and
$\tilde{f}=\pi_{1}^{-1}\circ f\circ\pi_{1}$, where $s/r=l$. Then

$\tilde{f}(\mathsf{z},c)=\left(\mathsf{z}^{\delta}(1+o(\mathsf{z})),\ \sum
b_{ij}\mathsf{z}^{ri+sj-s\delta}c^{j}(1+o(\mathsf{z}))\right)$

and it is well-defined, holomorphic and skew product.

###  2.2. Interval 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 $\delta\geq d>0$ and $i\geq\gamma$ for any $(i,j)$ such that
$b_{ij}\neq 0$ by the setting.

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

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

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

$\displaystyle\mathcal{I}_{f}$
$\displaystyle=\left[\dfrac{\gamma}{\delta-d},\min_{i,j}\left\\{\dfrac{i-\gamma}{d-j}:b_{ij}\neq
0\text{ and }j<d\right\\}\right]$
$\displaystyle=\left[\dfrac{\gamma}{\delta-d},\min_{1<j\leq
s}\left\\{\dfrac{n_{j}-\gamma}{d-m_{j}}\right\\}\right]=\left[\dfrac{\gamma}{\delta-d},\frac{n_{2}-\gamma}{d-m_{2}}\right]=\left[\alpha,l_{2}\right],$

which is mapped to $[T_{1},\delta]$ by the transformation $l\to
l^{-1}\gamma+d$. Therefore, $\mathcal{I}_{f}$ can be identified with the set
of the lines passing through $(\gamma,d)$, whose slopes are in
$[-l_{2},-\alpha]$ or, equivalently, whose $y$-intercepts are in
$[\delta,T_{1}]$. If $\gamma=0$, then the inequality $\gamma+ld\leq l\delta$
is trivial since $d\leq\delta$, and so $\mathcal{I}_{f}=(0,l_{2}]$. In
particular, $\max\mathcal{I}_{f}=l_{2}$.

Assuming that $l^{-1}$ is integer for $l$ in $\mathcal{I}_{f}$, we explain
benefits of the weights in $\mathcal{I}_{f}$ and 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 $(t,w)\to(tw,w)$. Then we have

$\tilde{f}(t,w)=(\tilde{p}(t,w),\tilde{q}(t,w))=\left(\dfrac{p(tw^{l^{-1}})}{q(tw^{l^{-1}},w)^{l^{-1}}},\
q(tw^{l^{-1}},w)\right).$

Let $\tilde{j}=l^{-1}i+j$. Then $\tilde{q}(t,w)=\sum
b_{ij}t^{i}w^{l^{-1}i+j}=\sum b_{ij}t^{i}w^{\tilde{j}}$. Let
$\tilde{d}=l^{-1}\gamma+d$. Then the Newton polygon $N(\tilde{q})$ of
$\tilde{q}$ has just one vertex $(\gamma,\tilde{d})$:
$N(\tilde{q})=D(\gamma,\tilde{d})$.

######  Lemma 2.6.

It follows that $d<\tilde{d}\leq\tilde{j}$ for any $l$ in $\mathcal{I}_{f}$
and for any $(i,j)$ such that $b_{ij}\neq 0$.

######  Remark 2.7.

The linear transformation

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

maps the basis $\\{(1,-l_{2}^{-1}),(0,1)\\}$ to $\\{(1,0),(0,1)\\}$. In other
words, $A_{2}$ maps the line $L_{1}$ with slope $-l_{2}^{-1}$ and a vertical
line to a horizontal line and the same vertical line.

Hence
$\tilde{q}(t,w)=t^{\gamma}w^{l^{-1}\gamma+d}(1+o(t,w))=t^{\gamma}w^{\tilde{d}}(1+o(t,w))$
and so

$\tilde{p}(t,w)=\dfrac{\\{tw^{l^{-1}}\\}^{\delta}(1+o(t,w))}{\\{t^{\gamma}w^{\tilde{d}}(1+o(t,w))\\}^{l^{-1}}}=t^{\delta-l^{-1}\gamma}w^{l^{-1}(\delta-\tilde{d})}(1+o(t,w)).$

Note that $\tilde{d}\leq\delta$ since $\gamma+ld\leq l\delta$, and so
$\delta-l^{-1}\gamma\geq d>0$ and $\delta-\tilde{d}\geq 0$.

######  Proposition 2.8.

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

$\tilde{f}(t,w)=\big{(}t^{\delta-l^{-1}\gamma}w^{l^{-1}(\delta-\tilde{d})}(1+o(t,w)),\
t^{\gamma}w^{\tilde{d}}(1+o(t,w))\big{)},$

and it has a fixed point at the origin. Moreover, if $\delta\geq 2$ and $d\geq
2$, or if $\delta\geq 2$, $d=1$ and $\delta>T_{1}$, then the fixed point is
superattracting.

Although $\tilde{f}$ is not skew product, it is a perturbation of a monomial
map near the origin. Therefore, if $l^{-1}$ is integer and $\tilde{f}$ is
superattracting, then we can construct the Böttcher coordinate for $\tilde{f}$
by similar arguments in Section 6 of the previous paper [3], which induces the
Böttcher coordinate for $f$ on $U_{r}^{l}$, where
$U_{r}^{l}=\\{|z|<r|w|^{l},|w|<r\\}$. Actually, we can construct the Böttcher
coordinate for $f$ on $U_{r}^{l}$ directly for any $l$ in $\mathcal{I}_{f}$.

######  Theorem 2.9 ([3]).

If $\delta\geq 2$ and $d\geq 2$, or if $\delta\geq 2$, $d=1$ and
$\delta>T_{1}$, then for any $l$ in $\mathcal{I}_{f}$, $f$ preserves
$U_{r}^{l}$ and there is a biholomorphic map defined on $U_{r}^{l}$ that
conjugates $f$ to $f_{0}$ for small $r$.

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

######  Remark 2.10.

If $l^{-1}$ is rational, then a similar lift is well-defined only for limited
cases. Let $\pi_{2}(t,\mathsf{w})=(t\mathsf{w}^{r},\mathsf{w}^{s})$ and
$\tilde{f}=\pi_{2}^{-1}\circ f\circ\pi_{2}$, where $s/r=l$. Then $\tilde{f}$
is well-defined if $\gamma/s$ is integer.

###  2.3. Intervals of weights and Blow-ups for Case 4

Let $s>2$,

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

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

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

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\tilde{i}+l_{(2)}j\text{
and }\tilde{\gamma}+l_{(2)}d\leq l_{(2)}\delta\\\ \text{for any $i$ and $j$
such that }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

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

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

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

In particular, $\min\mathcal{I}_{f}^{1}=l_{1}$ and, as a remark,

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

On the other hand,

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

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

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

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

$\mathcal{I}_{f}^{1}=[l_{1},\alpha],\
\mathcal{I}_{f}^{2}=\begin{cases}[\alpha-l_{(1)},l_{1}+l_{2}-l_{(1)}]&\text{
if }l_{(1)}<\alpha\\\ (0,l_{1}+l_{2}-\alpha]&\text{ if
}l_{(1)}=\alpha\end{cases}$ $\text{and so
}\mathcal{I}_{f}=[l_{1},\alpha]\times[\alpha,l_{1}+l_{2}]-\\{(\alpha,\alpha)\\}.$

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

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

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

Assuming that $l_{1}$ and $l_{2}^{-1}$ are integer, we explain our previous
results in terms of blow-ups. Although the same arguments hold for any
$l_{(1)}$ in $\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in $\mathcal{I}_{f}^{2}$, the
case $l_{(1)}=l_{1}$ and $l_{(2)}=l_{2}$ is most important for the study in
the previous and this papers, and so we fix $l_{1}$ and $l_{2}$ for
simplicity. 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 for $f$ on a wedge $U$ if the origin is
superattracting.

Let us explain the first blow-ups. Let $\tilde{f}_{1}=\pi_{1}^{-1}\circ
f\circ\pi_{1}$, where $\pi_{1}(z,c)=(z,z^{l_{1}}c)$, and
$\tilde{i}=i+l_{1}j-l_{1}\delta$ as in Case 2. Then

$\displaystyle\tilde{f}_{1}(z,c)$
$\displaystyle=(\tilde{p}_{1}(z),\tilde{q}_{1}(z,c))=\left(p(z),\
\dfrac{q(z,z^{l_{1}}c)}{p(z)^{l_{1}}}\right)$
$\displaystyle=\left(z^{\delta}(1+o(z)),\ \sum
b_{ij}z^{\tilde{i}}c^{j}(1+o(z,c))\right).$

######  Proposition 2.11.

If $l_{1}$ is integer, then $\tilde{f}_{1}$ is well-defined, holomorphic and
skew product on a neighborhood of the origin, which has a fixed point at the
origin. Moreover, if $\delta\geq 2$ and $d\geq 2$, or if $\delta\geq 2$, $d=1$
and $T_{k}<\delta<T_{k-1}$, then the fixed point is superattracting.

Let $\tilde{\gamma}=\gamma+l_{1}d-l_{1}\delta$,
$\tilde{i}=i+l_{1}j-l_{1}\delta$ and
$\tilde{n}_{j}=n_{j}+l_{1}m_{j}-l_{1}\delta$ as in Case 2. Then
$0\leq\tilde{\gamma}\leq\tilde{n}_{j}$ for any $j$. In particular,
$(\tilde{\gamma},d)$ is minimum in the sense that
$\tilde{\gamma}\leq\tilde{i}$, and $d\leq j$ if $\tilde{\gamma}=\tilde{i}$.
Hence $(\tilde{\gamma},d)$ is the vertex of the Newton polygon
$N(\tilde{q}_{1})$ whose $x$-coordinate is minimum. However,
$N(\tilde{q}_{1})$ 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 let $\tilde{T}_{k}$ be the $y$-intercept
of $\tilde{L}_{k}$. Then the slope of $\tilde{L}_{k}$ is $-l_{2}^{-1}$ and so
$\tilde{T}_{k}=l_{2}^{-1}\tilde{\gamma}+d$ because the affine transformation
$A_{1}$ in Remark 2.2 maps the basis $\\{(1,0),(-(l_{1}+l_{2}),1)\\}$ to
$\\{(1,0),(-l_{2},1)\\}$. Moreover, $\tilde{T}_{k}\leq\delta$ since
$T_{k}\leq\delta$. More precisely, $\tilde{T}_{k}<\delta$ if $T_{k}<\delta$,
and $\tilde{T}_{k}=\delta$ if $T_{k}=\delta$.

######  Proposition 2.12.

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

We next explain the second blow-ups. 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.
Then

$\tilde{f}_{2}(t,c)=(\tilde{p}_{2}(t,c),\tilde{q}_{2}(t,c))=\left(\dfrac{\tilde{p}_{1}(tc^{l_{2}^{-1}})}{\tilde{q}_{1}(tc^{l_{2}^{-1}},c)^{l_{2}^{-1}}},\
\tilde{q}_{1}(tc^{l_{2}^{-1}},c)\right).$

Let $\tilde{d}=l_{2}^{-1}\tilde{\gamma}+d$ and
$\tilde{j}=l_{2}^{-1}\tilde{i}+j$ as in Case 3. Then $\tilde{d}\leq\tilde{j}$
for any $(i,j)$ such that $b_{ij}\neq 0$, and $\tilde{d}\leq\delta$. In
particular, the minimality of $(\tilde{\gamma},\tilde{d})$ follows.

######  Lemma 2.13.

It follows that $0\leq\tilde{\gamma}\leq\tilde{i}$ and
$d\leq\tilde{d}\leq\tilde{j}$ for any $(i,j)$ such that $b_{ij}\neq 0$.

Hence the Newton polygon of $\tilde{q}_{2}$ has just one vertex
$(\tilde{\gamma},\tilde{d})$: $N(\tilde{q}_{2})=D(\tilde{\gamma},\tilde{d})$.

######  Proposition 2.14.

If $l_{1}$ and $l_{2}^{-1}$ are integer, then $\tilde{f}_{2}$ is well-defined
and holomorphic on a neighborhood of the origin. More precisely,

$\tilde{f}_{2}(t,c)=\big{(}t^{\delta-
l_{2}^{-1}\tilde{\gamma}}c^{l_{2}^{-1}(\delta-\tilde{d})}(1+o(t,c)),\
t^{\tilde{\gamma}}c^{\tilde{d}}(1+o(t,c))\big{)},$

and it has a fixed point at the origin. Moreover, if $\delta\geq 2$ and $d\geq
2$, or if $\delta\geq 2$, $d=1$ and $T_{k}<\delta<T_{k-1}$, then the fixed
point is superattracting.

Therefore, if $l_{1}$ and $l_{2}^{-1}$ are integer and if $\tilde{f}_{2}$ is
superattracting, then we can construct the Böttcher coordinate for
$\tilde{f}_{2}$ on a neighborhood of the origin, which induces that for
$\tilde{f}_{1}$ on an open wedge and that for $f$ on $U$, where

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

Actually, we can construct the Böttcher coordinate for $f$ on $U$ even if
$l_{1}$ and $l_{2}^{-1}$ are not integer and, moreover, we can construct it
directly for any $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ and $l_{(2)}$ in
$\mathcal{I}_{f}^{2}$.

######  Theorem 2.15 ([3]).

If $\delta\geq 2$ and $d\geq 2$, or if $\delta\geq 2$, $d=1$ and
$T_{k}<\delta<T_{k-1}$, then for any $l_{(1)}$ in $\mathcal{I}_{f}^{1}$ and
$l_{(2)}$ in $\mathcal{I}_{f}^{2}$, $f$ preserves $U$ and there is a
biholomorphic map defined on $U$ that conjugates $f$ to $f_{0}$ for small $r$.

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

######  Remark 2.16.

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 two affine transformations

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

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,
which 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, which intersect at
$(\tilde{\gamma},d)$, to the horizontal line and the vertical line, which
intersect at $(\tilde{\gamma},\tilde{d})$. Therefore, $A$ maps the lines
$L_{k}$ and $L_{k-1}$ to the horizontal and vertical lines.

##  3\. Shape of Newton polygon of $Q^{n}$ for Case 2

In this section we deal with Case 2 under the condition $d>0$. 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 $\gamma>0$ by the setting. We first give a summary of the results in
Section 3.1, and illustrate some of them in terms of blow-ups in Section 3.2.
Preparing two lemmas in Section 3.3, we prove Theorem 3.1 below in Section
3.4. Finally, in Section 3.5 we show the existence of the vertex of $N(Q^{n})$
that is previous to $(\gamma_{n},d^{n})$, and specify it, which induces
Theorem 3.3 below.

###  3.1. Summary of results

We first show the following equalities.

######  Theorem 3.1.

Let $d>0$. Then $Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$ and

$w(Q^{n})=w(z^{\gamma_{n}}w^{d^{n}})=\gamma_{n}+ld^{n}$

for any $n\geq 1$ and for any $l$ in $\mathcal{I}_{f}$, where
$\mathcal{I}_{f}=[l_{1},\alpha]$ or $\mathcal{I}_{f}=[l_{1},\infty)$ if
$\delta>d$ or $\delta\leq d$.

Moreover, $(\gamma_{n},d^{n})$ is the vertex of $N(Q^{n})$ whose
$y$-coordinate is minimum. Hence the theorem above implies the following
corollary.

######  Corollary 3.2.

Let $d>0$. Then

  1. (1)

$c(Q^{n})=\gamma_{n}+d^{n}$ for any $n\geq 1$ if $l_{1}\leq 1$, or

  2. (2)

$l_{1}^{-1}\gamma_{n}+d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ for any $n\geq
1$ if $l_{1}>1$.

Furthermore, we can specify the vertex of $N(Q^{n})$ that is previous to
$(\gamma_{n},d^{n})$, which exists, and improve the corollary above as
follows.

######  Theorem 3.3.

Let $d>0$. Then

  1. (1)

$c(Q^{n})=\gamma_{n}+d^{n}$ for any $n\geq 1$ if $l_{1}\leq 1$, or

  2. (2)

$l_{1}^{-1}\gamma_{n}+d^{n}\leq c(Q^{n})<\gamma_{n}+d^{n}$ for any $n\geq 1$
if $l_{1}>1$.

More precisely,

  1. $(3)$

$l_{1}^{-1}\gamma_{n}+d^{n}<c(Q^{n})$ for any $n\geq 1$ if $l_{1}>1$ and if
$n_{1}>0$ or $s>2$.

Let $d>0$, $l_{1}>1$, $n_{1}=0$ and $s=2$. Then

  1. $(4)$

$l_{1}^{-1}\gamma+d=c(q)$ and $l_{1}^{-1}\gamma_{n}+d^{n}<c(Q^{n})$ for any
$n\geq 2$ if $\delta<T_{s-1}$, or

  2. $(5)$

$l_{1}^{-1}\gamma_{n}+d^{n}=c(Q^{n})$ for any $n\geq 1$ if $\delta=T_{s-1}$.

###  3.2. Illustration of results in terms of blow-ups

Assuming that $l_{1}$ is integer, we illustrate Theorem 3.1 and Corollary 3.2
by blow-ups. Let $\pi_{1}(z,c)=(z,z^{l_{1}}c)$ and
$\tilde{f}=\pi_{1}^{-1}\circ f\circ\pi_{1}$ as in Section 2.1. By Lemma 2.1,
$N(\tilde{q})=D(\tilde{\gamma},d)$, where
$\tilde{\gamma}=\gamma+l_{1}d-l_{1}\delta$. Hence $\tilde{f}$ is a skew
product in Case 1. If $d>0$, then $\tilde{Q}^{n}$ contains the term
$z^{\tilde{\gamma}_{n}}w^{d^{n}}$ and
$N(\tilde{Q}^{n})=D(\tilde{\gamma}_{n},d^{n})$, where
$\tilde{\gamma}_{n}=\gamma_{n}+l_{1}d^{n}-l_{1}\delta^{n}$. It implies that
$Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$, and that $N(Q^{n})$ is
included in the upper-right region that is surrounded by the line with slope
$-l_{1}^{-1}$ and the horizontal line, which intersect at
$(\gamma_{n},d^{n})$. Therefore, we obtain Theorem 3.1 and Corollary 3.2 when
$l_{1}$ is integer, because $w(Q^{n})$ or $c(Q^{n})$ is the minimum
$x$-intercept of the lines with slope $-l^{-1}$ or $-1$ that intersect
$N(Q^{n})$.

###  3.3. Preliminary lemmas: dynamics on the interval

The following affine function $R$ on the real line plays an important role:

$R(l)=\dfrac{\gamma+ld}{\delta}.$

If $\delta\neq d>0$, then $R$ is a contracting or expanding function around
the fixed point $\alpha$:

$R(l)=\dfrac{d}{\delta}(l-\alpha)+\alpha\text{ and so
}R^{n}(l)=\left(\dfrac{d}{\delta}\right)^{n}(l-\alpha)+\alpha.$

More precisely, $\alpha$ is attracting and $\alpha>0$ if $\delta>d$, and
$\alpha$ is repelling and $\alpha<0$ if $\delta<d$. On the other hand, if
$\delta=d$, then $R$ is a translation:

$R(l)=l+\dfrac{\gamma}{\delta}\text{ and so
}R^{n}(l)=l+\dfrac{\gamma}{\delta}n.$

Since $\mathcal{I}_{f}=[l_{1},\alpha]$ or $\mathcal{I}_{f}=[l_{1},\infty)$ if
$\delta>d$ or $\delta\leq d$, we can conclude as follows.

######  Lemma 3.4.

For any $l$ in $\mathcal{I}_{f}$, it follows that $R^{n}(l)$ belongs to
$\mathcal{I}_{f}$ for any $n\geq 1$ and the sequence $\\{R^{n}(l)\\}_{n\geq
1}$ is increasing. More precisely, $R^{n}(l)\to\alpha$ or $R^{n}(l)\to\infty$
as $n\to\infty$ if $\delta>d>0$ or $\delta\leq d$.

We can express $R^{n}$ by $\delta^{n}$, $\gamma_{n}$ and $d^{n}$ as follows.

######  Lemma 3.5.

For any $n\geq 1$,

$R^{n}(l)=\dfrac{\gamma_{n}+ld^{n}}{\delta^{n}}.$

###### Proof.

We show by induction. The equality for the case $n=1$ is trivial. Assume that
the equality holds for an integer $k>1$:

$R^{k}(l)=\dfrac{\gamma_{k}+ld^{k}}{\delta^{k}}.$

Note that $\delta^{k}\gamma+d\gamma_{k}=\gamma_{k+1}$ since

$\delta^{k}\gamma+d\gamma_{k}=\\{\delta^{k}+(\delta^{k-1}d+\delta^{k-2}d^{2}+\cdots+d^{k})\\}\gamma=\gamma_{k+1}.$

Therefore,

$R^{k+1}(l)=R(R^{k}(l))=\dfrac{\gamma+R^{k}(l)d}{\delta}=\dfrac{\delta^{k}\gamma+(\gamma_{k}+ld^{k})d}{\delta^{k+1}}=\dfrac{\gamma_{k+1}+ld^{k+1}}{\delta^{k+1}}.$

∎

######  Remark 3.6.

Let $f_{0}(z,w)=(z^{\delta},z^{\gamma}w^{d})$. Then
$f_{0}(z,cz^{l})=(z^{\delta},c^{d}z^{\gamma+ld})$ and

$f_{0}^{n}(z,cz^{l})=(z^{\delta^{n}},c^{d^{n}}z^{\gamma_{n}+ld^{n}})$

since $f_{0}^{n}(z,w)=(z^{\delta^{n}},z^{\gamma_{n}}w^{d^{n}})$. Note that the
ratio of the degrees with respect to $z$ of the second and first components of
$f_{0}^{n}(z,cz^{l})$ coincides with $R^{n}(l)$. In particular, $f_{0}$ maps
$\\{w=cz^{l}\\}$ to $\\{w=c^{d}z^{R(l)}\\}$, and $f_{0}^{n}$ maps
$\\{w=cz^{l}\\}$ to $\\{w=c^{d^{n}}z^{R^{n}(l)}\\}$.

###  3.4. Proof of Theorem 3.1

By definition, $w(q)=\gamma+ld$ for any $l$ in $\mathcal{I}_{f}$. Moreover, we
prove that $w(Q^{n})=\gamma_{n}+ld^{n}$ for any $l$ in $\mathcal{I}_{f}$ and
for any $n\geq 1$.

###### Proof of Theorem 3.1.

Fix any $l$ in $\mathcal{I}_{f}$, and let $w=w_{l}$. For a monomial term
$z^{i}w^{j}$, we call $i+lj$ the weight of $z^{i}w^{j}$. As the same as the
description of the theorem, we omit the coefficient of the term
$z^{\gamma_{n}}w^{d^{n}}$ for simplicity.

We first show the equality $w(Q^{2})=\gamma_{2}+ld^{2}$. Note that

$Q^{2}(z,w)=\sum b_{ij}\big{(}p(z)\big{)}^{i}\big{(}q(z,w)\big{)}^{j}.$

Since $w(p)=\delta$ and $w(q)=\gamma+ld$,

$w(Q^{2})\geq\min\\{\delta i+(\gamma+ld)j:b_{ij}\neq 0\\}.$

It follows from Lemma 3.4 that $i+R(l)j\geq\gamma+R(l)d$, and so

$\delta i+(\gamma+ld)j\geq\delta\gamma+(\gamma+ld)d$

by Lemma 3.5. Therefore,

$w(Q^{2})\geq\delta\gamma+(\gamma+ld)d=(\delta+d)\gamma+ld^{2}=\gamma_{2}+ld^{2}.$

On the other hand, the term

$\big{(}z^{\delta}\big{)}^{\gamma}\big{(}z^{\gamma}w^{d}\big{)}^{d}=z^{\gamma_{2}}w^{d^{2}}$

has weight $\gamma_{2}+ld^{2}$ and, moreover, has the smallest degree $d^{2}$
with respect to $w$ among all the terms in $Q^{2}$ of weight
$\gamma_{2}+ld^{2}$. Because $d>0$, there is no other construction than
$(z^{\delta})^{\gamma}(z^{\gamma}w^{d})^{d}$ that generates the term of
bidegree $(\gamma_{2},d^{2})$. Therefore, $Q^{2}$ contains the term
$z^{\gamma_{2}}w^{d^{2}}$ and so
$w(Q^{2})=w(z^{\gamma_{2}}w^{d^{2}})=\gamma_{2}+ld^{2}$.

We next show the equality $w(Q^{3})=\gamma_{3}+ld^{3}$ by the same strategy.
Note that

$Q^{3}(z,w)=\sum
b_{ij}\big{(}p^{2}(z)\big{)}^{i}\big{(}Q^{2}(z,w)\big{)}^{j}.$

Since $w(p^{2})=\delta^{2}$ and $w(Q^{2})=\gamma_{2}+ld^{2}$,

$w(Q^{3})\geq\min\\{\delta^{2}i+(\gamma_{2}+ld^{2})j:b_{ij}\neq 0\\}.$

It follows from Lemma 3.4 that $i+R^{2}(l)j\geq\gamma+R^{2}(l)d$, and so

$\delta^{2}i+(\gamma_{2}+ld^{2})j\geq\delta^{2}\gamma+(\gamma_{2}+ld^{2})d$

by Lemma 3.5. Therefore,

$w(Q^{3})\geq\delta^{2}\gamma+(\gamma_{2}+ld^{2})d=(\delta^{2}\gamma+d\gamma_{2})+ld^{3}=\gamma_{3}+ld^{3}.$

On the other hand, the term

$\big{(}z^{\delta^{2}}\big{)}^{\gamma}\big{(}z^{\gamma_{2}}w^{d^{2}}\big{)}^{d}=z^{\gamma_{3}}w^{d^{3}}$

has weight $\gamma_{3}+ld^{3}$ and, moreover, has the smallest degree $d^{3}$
with respect to $w$ among all the terms in $Q^{3}$ of weight
$\gamma_{3}+ld^{3}$. Because $d>0$, there is no other construction than
$(z^{\delta^{2}})^{\gamma}(z^{\gamma_{2}}w^{d^{2}})^{d}$ that generates the
term of bidegree $(\gamma_{3},d^{3})$. Therefore, $Q^{3}$ contains the term
$z^{\gamma_{3}}w^{d^{3}}$ and so
$w(Q^{3})=w(z^{\gamma_{3}}w^{d^{3}})=\gamma_{3}+ld^{3}$.

By repeating this process, it follows that $w(Q^{n})\geq\gamma_{n}+ld^{n}$ and
that $Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$ for any $n\geq 1$.
Therefore, $w(Q^{n})=w(z^{\gamma_{n}}w^{d^{n}})=\gamma_{n}+ld^{n}$. ∎

We remark that one can show that the coefficient of the term
$z^{\gamma_{n}}w^{d^{n}}$ is

$a_{\delta}^{\gamma_{n-1}+\gamma_{n-2}+\cdots+\gamma_{2}+\gamma_{1}}b_{\gamma
d}^{d^{n-1}+d^{n-2}+\cdots+d+1}$

from the construction of the term and the equality
$\delta^{k}\gamma+d\gamma_{k}=\gamma_{k+1}$ in Lemma 3.5.

Because $w(Q^{n})$ is the minimum $x$-intercept of the lines with slope
$-l^{-1}$ that intersect $N(Q^{n})$, Theorem 3.1 implies that
$(\gamma_{n},d^{n})$ belongs to the boundary of $N(Q^{n})$. Moreover, we
obtain the following corollary because the smallest degree with respect to $w$
of all terms in $Q^{n}$ is $d^{n}$.

######  Corollary 3.7.

The bidegree $(\gamma_{n},d^{n})$ is the vertex of $N(Q^{n})$ whose
$y$-coordinate is minimum for any $n\geq 1$.

Therefore, $N(Q^{n})$ is included in the upper-right region that is surrounded
by the line with slope $-l_{1}^{-1}$ and the horizontal line, which intersect
at $(\gamma_{n},d^{n})$.

Let $-M_{n}$ be the slope of the line passing through $(\gamma_{n},d^{n})$ and
the previous vertex if it exists, let $M_{n}=\infty$ if
$N(Q^{n})=D(\gamma_{n},d^{n})$, and let $M=M_{1}$. Then $M=l_{1}^{-1}$, and
Theorem 3.1 implies the following corollary and Corollary 3.2 because
$w(Q^{n})$ or $c(Q^{n})$ is the minimum $x$-intercept of the lines with slope
$-l^{-1}$ or $-1$ that intersect $N(Q^{n})$.

######  Corollary 3.8.

Let $d>0$. Then $M_{n}\geq M$ for any $n\geq 1$.

###  3.5. Previous vertices and proof of Theorem 3.3

Furthermore, we can show the existence of the vertex of $N(Q^{n})$ that is
previous to $(\gamma_{n},d^{n})$, and specify it, which induces the equality
$M_{n}=M$. Let

$(A,B)=(n_{s-1},m_{s-1}),$ $\displaystyle(A_{n},B_{n})$
$\displaystyle=((\delta^{n-1}+\delta^{n-2}d+\cdots+\delta
d^{n-2})\gamma+d^{n-1}A,Bd^{n-1})$
$\displaystyle=(\gamma_{n}-(\gamma-A)d^{n-1},Bd^{n-1})\text{ and}$
$\displaystyle(A_{n}^{*},B_{n}^{*})$
$\displaystyle=((\delta^{n-1}+\delta^{n-2}B+\cdots+\delta
B^{n-2}+B^{n-1})A,B^{n}).$

######  Proposition 3.9.

Let $d>0$. If $\delta<T_{s-1}$, then $(A_{n},B_{n})$ is the vertex of
$N(Q^{n})$ that is previous to $(\gamma_{n},d^{n})$ for any $n\geq 1$.
Moreover, $M_{n}=M$, and $\delta^{n}$ is smaller than the y-intercept of the
line passing through $(\gamma_{n},d^{n})$ and $(A_{n},B_{n})$. If
$\delta=T_{s-1}$, then $(A_{n}^{*},B_{n}^{*})$ is the vertex of $N(Q^{n})$
that is previous to $(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover,
$M_{n}=M$, and $\delta^{n}$ coincides with the y-intercept of the line passing
through $(\gamma_{n},d^{n})$ and $(A_{n}^{*},B_{n}^{*})$.

Before go into the proof, we show an idea how to construct $(A_{2},B_{2})$ and
$(A_{2}^{*},B_{2}^{*})$. Let us pick up a term
$(z^{\delta})^{I}(z^{i}w^{j})^{J}$ in $Q^{2}$, where $(i,j)\neq(\gamma,d)$. As
the same as the proof of Theorem 3.1,

$w((z^{\delta})^{I}(z^{i}w^{j})^{J})=\delta I+(i+lj)J\geq\delta
I+(\gamma+ld)J\geq\delta\gamma+(\gamma+ld)d=\gamma_{2}+ld^{2}$

for any $l$ in $\mathcal{I}_{f}$. The equality in the first inequality holds
if and only if $i+lj=\gamma+ld$ and $l=l_{1}$. Let $l=l_{1}$ hereafter. Note
that the set

$\\{(i,j):i+l_{1}j=\gamma+l_{1}d\\}$

lies on the side of $N(q)$ whose ends ares $(n_{s-1},m_{s-1})$ and
$(\gamma,d)$. If $\delta<T_{s-1}$, then the equality in the second inequality
holds if and only if $(I,J)=(\gamma,d)$, since
$\mathcal{I}_{f}\supsetneq\\{l_{1}\\}$ and $R(l_{1})>l_{1}$. Therefore,

$\big{(}z^{\delta}\big{)}^{\gamma}\big{(}z^{n_{s-1}}w^{m_{s-1}}\big{)}^{d}$

is the desired term in $Q^{2}$ and so
$(A_{2},B_{2})=(\delta\gamma+n_{s-1}d,m_{s-1}d)$. On the other hand, if
$\delta=T_{s-1}$, then the equality in the second inequality holds if and only
if $I+l_{1}J=\gamma+l_{1}d$, since $\mathcal{I}_{f}=\\{l_{1}\\}$ and
$R(l_{1})=l_{1}$. Therefore,

$\big{(}z^{\delta}\big{)}^{n_{s-1}}\big{(}z^{n_{s-1}}w^{m_{s-1}}\big{)}^{m_{s-1}}$

is the desired term in $Q^{2}$ and so $(A_{2}^{*},B_{2}^{*})=(\delta
n_{s-1}+n_{s-1}m_{s-1},m_{s-1}^{2})$.

To prove Proposition 3.9, we use the following lemma, which follows
immediately from the definition of $\gamma_{n}$.

######  Lemma 3.10.

The slopes of the line passing through $(0,\delta^{n})$ and
$(\gamma_{n},d^{n})$ are all the same for any $n\geq 1$.

As a consequence of Proposition 3.9 and Lemma 3.10, it follows that

$\displaystyle\dfrac{B_{n}-d^{n}}{\gamma_{n}-A_{n}}$
$\displaystyle=\dfrac{B-d}{\gamma-A}>\dfrac{\delta^{n}-d^{n}}{\gamma_{n}}=\dfrac{\delta-d}{\gamma}\text{
if $\delta<T_{s-1}$ or},$
$\displaystyle\dfrac{B_{n}^{*}-d^{n}}{\gamma_{n}-A_{n}^{*}}$
$\displaystyle=\dfrac{B-d}{\gamma-A}=\dfrac{\delta^{n}-d^{n}}{\gamma_{n}}=\dfrac{\delta-d}{\gamma}\text{
if $\delta=T_{s-1}$}.$

Now we are ready to prove Proposition 3.9.

###### Proof of Proposition 3.9.

As we saw in the explanation above, we have to choice $l_{1}$ as $l$ in
$\mathcal{I}_{f}$ to find out the vertex of $N(Q^{n})$ that is previous to
$(\gamma_{n},d^{n})$. Then $w_{l_{1}}(Q^{n})=\gamma_{n}+l_{1}d^{n}$ by Theorem
3.1.

If $\delta<T_{s-1}$, then it follows that the weight of any term of

$\displaystyle h(z,w)$ $\displaystyle:=b_{\gamma
d}\left(a_{\delta}z^{\delta}\right)^{\gamma}\bigg{(}\sum_{i+l_{1}j=\gamma+l_{1}d}b_{ij}z^{i}w^{j}\bigg{)}^{d}$
$\displaystyle=b_{\gamma
d}\left(a_{\delta}z^{\delta}\right)^{\gamma}\left(b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\right)^{d}$

is $\gamma_{2}+l_{1}d^{2}$, and all the weights of the other terms in
$Q^{2}(z,w)$ are bigger than $\gamma_{2}+l_{1}d^{2}$. The germ $h$ is a part
of $Q^{2}$ and, moreover,

$\big{(}z^{\delta}\big{)}^{\gamma}\big{(}z^{A}w^{B}\big{)}^{d}$

is the term of $h$ with the biggest degree with respect to $w$ in $h$. Let

$(A_{2},B_{2})=(\delta\gamma+Ad,Bd).$

Since there is no other construction that generates the term of bidegree
$(A_{2},B_{2})$, it is the vertex of $N(Q^{2})$ that is previous to
$(\gamma_{2},d^{2})$. Since the weight of $z^{\gamma_{2}}w^{d^{2}}$ and
$z^{A_{2}}w^{B_{2}}$ are the same, $M_{2}=M$. Consequently,

$\dfrac{B_{2}-d^{2}}{\gamma_{2}-A_{2}}=\dfrac{B-d}{\gamma-A}>\dfrac{\delta-d}{\gamma}=\dfrac{\delta^{2}-d^{2}}{\gamma_{2}}$

and so $\delta^{2}$ is smaller than the $y$-intercept of the line passing
through $(\gamma_{2},d^{2})$ and $(A_{2},B_{2})$.

It follows from the same arguments that the term

$\big{(}z^{\delta^{2}}\big{)}^{\gamma}\big{(}z^{A_{2}}w^{B_{2}}\big{)}^{d}=z^{\delta^{2}\gamma+A_{2}d}w^{B_{2}d}$

in $Q^{3}$ has weight $\gamma_{3}+l_{1}d^{3}$ and, moreover, has the biggest
degree with respect to $w$ among all the terms in $Q^{3}$ of the smallest
weight. Let

$(A_{3},B_{3})=(\delta^{2}\gamma+\delta
d\gamma+Ad^{2},Bd^{2})=(\gamma_{2}-(\gamma-A)d,Bd^{2}).$

Then it follows from the same arguments that $(A_{3},B_{3})$ is the vertex of
$N(Q^{3})$ which is previous to $(\gamma_{3},d^{3})$, $M_{3}=M$, and
$\delta^{3}$ is smaller than the $y$-intercept of the line passing through
$(0,\delta^{3})$ and $(\gamma_{3},d^{3})$. Repeating this process, we obtain
the required vertices and properties.

If $\delta=T_{s-1}$, then it follows that the weight of any term of

$\displaystyle h(z,w)$
$\displaystyle:=\sum_{I+l_{1}J=\gamma+l_{1}d}b_{IJ}\left(a_{\delta}z^{\delta}\right)^{I}\bigg{(}\sum_{i+l_{1}j=\gamma+l_{1}d}b_{ij}z^{i}w^{j}\bigg{)}^{J}$
$\displaystyle=b_{\gamma
d}\left(a_{\delta}z^{\delta}\right)^{\gamma}\left(b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\right)^{d}+\cdots$ $\displaystyle\ \
\ \ +b_{AB}\left(a_{\delta}z^{\delta}\right)^{A}\left(b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\right)^{B}$

is $\gamma_{2}+l_{1}d^{2}$, and all the weights of the other terms in
$Q^{2}(z,w)$ are bigger than $\gamma_{2}+l_{1}d^{2}$. The germ $h$ is a part
of $Q^{2}$ and, moreover,

$\big{(}z^{\delta}\big{)}^{A}\big{(}z^{A}w^{B}\big{)}^{B}$

is the term of $h$ with the biggest degree with respect to $w$ in $h$. Let

$(A_{2}^{*},B_{2}^{*})=(\delta A+AB,B^{2})=((\delta+B)A,B^{2}).$

Since there is no other construction that generates the term of bidegree
$(A_{2}^{*},B_{2}^{*})$, it is the vertex of $N(Q^{2})$ that is previous to
$(\gamma_{2},d^{2})$. Moreover, $M_{2}=M$ and so $\delta^{2}$ coincides with
the $y$-intercept of the line passing through $(\gamma_{2},d^{2})$ and
$(A_{2}^{*},B_{2}^{*})$.

It follows from the same arguments that the term

$\big{(}z^{\delta^{2}}\big{)}^{A}\big{(}z^{A_{2}}w^{B^{2}}\big{)}^{B}=z^{\delta^{2}A+A_{2}B}w^{B^{3}}$

in $Q^{3}$ has weight $\gamma_{3}+l_{1}d^{3}$ and, moreover, has the biggest
degree with respect to $w$ among all the terms in $Q^{3}$ of the smallest
weight. Let

$(A_{3}^{*},B_{3}^{*})=(\delta^{2}A+(\delta+B)AB,B^{3})=((\delta^{2}+\delta
B+B^{2})A,B^{3}).$

Then it follows from the same arguments that $(A_{3}^{*},B_{3}^{*})$ is the
vertex of $N(Q^{3})$ which is previous to $(\gamma_{3},d^{3})$, $M_{3}=M$, and
$\delta^{3}$ coincides with the $y$-intercept of the line passing through
$(0,\delta^{3})$ and $(\gamma_{3},d^{3})$. Repeating this process, we obtain
the required vertices and properties. ∎

The polynomial $h$ in the proof plays an important role also for Cases 3 and
4. We remark that $h(z,w)=b_{\gamma d}z^{\gamma}w^{d}$ for any $l$ in
$\mathcal{I}_{f}\setminus\\{l_{1}\\}$. We can illustrate the difference of $h$
depending on $l$ by the Newton polygon $N(Q^{n})$: the line passing through
$(\gamma,d)$, with slope $-l_{1}^{-1}$, intersects $N(Q^{n})$ with the side
whose ends are $(n_{s-1},m_{s-1})$ and $(\gamma,d)$, whereas the line passing
through $(\gamma,d)$, with slope $-l^{-1}$ for any $l$ in
$\mathcal{I}_{f}\setminus\\{l_{1}\\}$, intersects $N(Q^{n})$ only at the point
$(\gamma,d)$.

Proposition 3.9 induces Theorem 3.3, which improves Corollary 3.2.

###### Proof of Theorem 3.3.

We only show the improved parts. Recall that $N(Q^{n})$ is included in the
upper-right region that is surrounded by the line with slope $-l_{1}^{-1}$ and
the horizontal line, which intersect at $(\gamma_{n},d^{n})$, and that
$c(Q^{n})$ is the minimum $x$-intercept of the lines with slope $-1$ that
intersect $N(Q^{n})$.
(2) If $\delta<T_{s-1}$, then $c(Q^{n})\leq A_{n}+B_{n}<\gamma_{n}+d^{n}$. If
$\delta=T_{s-1}$, then $c(Q^{n})\leq A_{n}^{*}+B_{n}^{*}<\gamma_{n}+d^{n}$.
(3) If $n_{1}>0$ or $s>2$, then $A=n_{s-1}>0$. Hence $A_{n}>0$ or
$A_{n}^{*}>0$ since $\delta$, $\gamma$, $d$, $A$ and $B$ are all positive.
Hence $N(Q^{n})$ does not contain $(0,l_{1}^{-1}\gamma_{n}+d^{n})$, the
$y$-intercept of the line passing through $(\gamma_{n},d^{n})$ with slope
$-l_{1}^{-1}$. Therefore, $c(Q^{n})>l_{1}^{-1}\gamma_{n}+d^{n}$.
(4) Let $n_{1}=0$ and $s=2$. Then $A=n_{1}=0$. If $\delta<T_{s-1}$, then
$A_{1}=A=0$ and $A_{n}>0$ for any $n\geq 2$. Hence
$c(Q^{n})>l_{1}^{-1}\gamma_{n}+d^{n}$ for any $n\geq 2$.
(5) If $\delta=T_{s-1}$, then $A_{n}^{*}=0$ and
$c(Q^{n})=B_{n}^{*}=B^{n}=\delta^{n}=l_{1}^{-1}\gamma_{n}+d^{n}$. ∎

######  Remark 3.11.

If $d>0$, then $\gamma_{n}\to\infty$, and $A_{n}$ or $A_{n}^{*}\to\infty$ as
$n\to\infty$ since $\delta>0$ and $\gamma>0$.

######  Remark 3.12.

Recall that $f$ preserves the open wedge $U_{r}^{l}$ for any $l$ in
$\mathcal{I}_{f}$. Hence $f^{n}$ preserves $U_{r}^{l}$ for any $l$ in
$\mathcal{I}_{f}$. Therefore, one may expect that
$\mathcal{I}_{f}\subset\mathcal{I}_{f^{n}}$, where $\mathcal{I}_{f^{n}}$ is
the interval of the weights for $f^{n}$. In fact, Proposition 3.9 and Lemma
3.10 implies that the equality $\mathcal{I}_{f}=\mathcal{I}_{f^{n}}$ holds for
any $n\geq 1$.

##  4\. Shape of Newton polygon of $Q^{n}$ when $d=0$

We complete the investigation of $w(Q^{n})$ and $c(Q^{n})$ for Case 2 in this
section, assuming that $d=0$. We first give a rough explanation of the
differences between the cases $d>0$ and $d=0$, and a summary of the results in
Section 4.1. Some of the results are illustrated in terms of blow-ups in
Section 4.2. Preparing a lemma in Section 4.3, we give more detailed
explanations of the results for the cases $\delta<T_{s-1}$ and
$\delta=T_{s-1}$ in Sections 4.4 and 4.5, respectively, although we omit the
precise proofs of the results because they are more or less similar to the
case $d>0$.

###  4.1. Explanation of differences and Summary of results

The situation for the case $d=0$ is different from that for the previous case
$d>0$. Let us give here a rough explanation. If $d=0$, then
$(\gamma_{n},d^{n})=(\delta^{n-1}\gamma,0)$. Recall that, for the case $d>0$,
the bidegree $(A_{n},B_{n})$ or $(A_{n}^{*},B_{n}^{*})$ is the vertex of
$N(Q^{n})$ that is previous to $(\gamma_{n},d^{n})$ if $\delta<T_{s-1}$ or
$\delta=T_{s-1}$. If $d=0$ and $\delta<T_{s-1}$, then $(A_{n},B_{n})$
coincides with $(\gamma_{n},0)$, and the equality $M_{n}=M$ does not hold; in
fact, the inequality $M_{n}>M$ holds for any $n\geq 2$. However, the term
$z^{\gamma_{n}}$ remains forever, and the equality
$w(Q^{n})=w(z^{\gamma_{n}})=\gamma_{n}$ holds for any $n\geq 1$ and for any
$l$ in $\mathcal{I}_{f}$. On the other hand, if $d=0$ and $\delta=T_{s-1}$,
then $(A_{n},B_{n})$ coincides with $(\gamma_{n},0)$ and, moreover, the term
$z^{\gamma_{n}}$ may vanish. However, the term $z^{A_{n}^{*}}w^{B_{n}^{*}}$
remains forever, and the equality
$w_{l_{1}}(Q^{n})=w_{l_{1}}(z^{A_{n}^{*}}w^{B_{n}^{*}})=\gamma_{n}$ holds for
any $n\geq 1$, where $\mathcal{I}_{f}=\\{l_{1}\\}$. Consequently, even if
$d=0$, we have the same equality on $w(Q^{n})$ as the case $d>0$.

######  Theorem 4.1.

Let $d=0$. Then

$w(Q^{n})=\gamma_{n}$

for any $n\geq 1$ and for any $l$ in $\mathcal{I}_{f}$, where
$\mathcal{I}_{f}=[l_{1},\alpha]$.

Induced inequalities of $c(Q^{n})$ are similar to the case $d>0$, but
$c(Q^{n})$ can be bigger than $\gamma_{n}$ since the term $z^{\gamma_{n}}$ may
vanish.

######  Corollary 4.2.

Let $d=0$. Then

  1. (1)

$\gamma_{n}\leq c(Q^{n})\leq l_{1}^{-1}\gamma$ for any $n\geq 1$ if $l_{1}\leq
1$, or

  2. (2)

$l_{1}^{-1}\gamma_{n}\leq c(Q^{n})\leq\gamma_{n}$ for any $n\geq 1$ if
$l_{1}>1$.

Moreover, we have the following improved estimates on $c(Q^{n})$.

######  Theorem 4.3.

Let $d=0$ and $\delta<T_{s-1}$. Then

  1. (1)

$c(Q^{n})=\gamma_{n}$ for any $n\geq 1$ if $l_{1}\leq 1$, or

  2. (2)

$l_{1}^{-1}\gamma\leq c(q)<\gamma$ and
$l_{1}^{-1}\gamma_{n}<c(Q^{n})\leq\gamma_{n}$ for any $n\geq 2$ if $l_{1}>1$.

######  Theorem 4.4.

Let $d=0$ and $\delta=T_{s-1}$. Then

  1. (1)

$\gamma_{n}\leq c(Q^{n})\leq l_{1}^{-1}\gamma_{n}$ for any $n\geq 1$ if
$l_{1}\leq 1$, or

  2. (2)

$l_{1}^{-1}\gamma_{n}\leq c(Q^{n})<\gamma_{n}$ for any $n\geq 1$ if $l_{1}>1$.

More precisely, if $l_{1}<1$ and the term $z^{\gamma_{n}}$ vanish for some
$n_{0}$, then

  1. $(3)$

$\gamma_{n}<c(Q^{n})$ for any $n\geq n_{0}$.

###  4.2. Illustration of results in terms of blow-ups

Assuming that $l_{1}$ is integer, we illustrate Theorem 4.1 and Corollary 4.2
by blow-ups. Let $\pi_{1}(z,c)=(z,z^{l_{1}}c)$ and
$\tilde{f}=\pi_{1}^{-1}\circ f\circ\pi_{1}$. By Lemma 2.1,
$N(\tilde{q})=D(\tilde{\gamma},0)$. Although the term $z^{\tilde{\gamma}_{n}}$
may vanish if $d=0$, the inclusion $N(\tilde{Q}^{n})\subset
D(\tilde{\gamma}_{n},0)$ still holds. Hence $N(Q^{n})$ is included in the
upper-right region that is surrounded by the line with slope $-l_{1}^{-1}$ and
the $x$-axis, which intersect at $(\gamma_{n},0)$. Therefore, we obtain
Theorem 4.1 and Corollary 4.2 when $l_{1}$ is integer, because $w(Q^{n})$ or
$c(Q^{n})$ is the minimum $x$-intercept of the lines with slope $-l^{-1}$ or
$-1$ that intersect $N(Q^{n})$.

###  4.3. Preliminary lemma: dynamics on the interval

If $d=0$, then $R\equiv\alpha$. Hence $R$ collapses any point to $\alpha$, and
$\mathcal{I}_{f}=[l_{1},\alpha]$ since $\delta>d$.

######  Lemma 4.5.

For any $l$ in $\mathcal{I}_{f}$, it follows that $R^{n}(l)$ belongs to
$\mathcal{I}_{f}$ for any $n\geq 1$. Actually, $R\equiv\alpha$ and
$\mathcal{I}_{f}=[l_{1},\alpha]$.

###  4.4. The case $d=0$ and $\delta<T_{s-1}$

We showed in the proof of Theorem 3.3 that, if $d>0$ and $\delta<T_{s-1}$,
then the major part $h$ of $Q^{2}$,

$b_{\gamma d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{d},$

generates the important terms $z^{\gamma_{2}}w^{d^{2}}$ and
$z^{A_{2}}w^{B_{2}}$. However, if $d=0$ and $\delta<T_{s-1}$, then these terms
coincide:

$b_{\gamma 0}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
0}z^{\gamma}w^{0}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{0}=a_{\delta}^{\gamma}b_{\gamma
0}z^{\gamma_{2}}.$

Actually, it follows from the same arguments as the case $d>0$ that, if $d=0$
and $\delta<T_{s-1}$, then the term $z^{\gamma_{n}}$ remains forever as the
unique term of the smallest weight $\gamma_{n}$. Hence we obtain the following
three propositions, which implies Theorem 4.1 for the case $\delta<T_{s-1}$
and Theorem 4.3.

######  Proposition 4.6.

Let $d=0$ and $\delta<T_{s-1}$. Then $Q^{n}$ contains the term
$z^{\gamma_{n}}$ and

$w(Q^{n})=w(z^{\gamma_{n}})=\gamma_{n}$

for any $n\geq 1$ and for any $l$ in $\mathcal{I}_{f}$, where
$\mathcal{I}_{f}=[l_{1},\alpha]$.

######  Proposition 4.7.

Let $d=0$ and $\delta<T_{s-1}$. Then $M_{n}>M$ for any $n\geq 2$.

######  Proposition 4.8.

Let $d=0$ and $\delta<T_{s-1}$. Then

  1. (1)

$c(Q^{n})=\gamma_{n}$ for any $n\geq 1$ if $l_{1}\leq 1$, or

  2. (2)

$l_{1}^{-1}\gamma\leq c(q)\leq A+B<\gamma$ and
$l_{1}^{-1}\gamma_{n}<c(Q^{n})\leq\gamma_{n}$ for any $n\geq 2$ if $l_{1}>1$.

######  Remark 4.9.

Let $d=0$ and $\delta<T_{s-1}$. Then

$M<M_{n}\leq\dfrac{B_{n}^{*}}{\gamma_{n}-A_{n}^{*}}$

for any $n\geq 2$. If $A=0$, then $(A_{n}^{*},B_{n}^{*})=(0,B^{n})$. Hence
$B_{n}^{*}/\gamma_{n}\to\infty$ as $n\to\infty$. Moreover, if
$(A_{n}^{*},B_{n}^{*})$ is the vertex of $N(Q^{n})$ that is previous to
$(\gamma_{n},d^{n})$, then $M_{n}=B_{n}^{*}/\gamma_{n}\to\infty$ as
$n\to\infty$. If $A>0$, then $A_{n}^{*}>\gamma_{n}$ for any large $n$. Hence
it may happen that $M_{n}=\infty$ for any large $n$.

###  4.5. The case $d=0$ and $\delta=T_{s-1}$

We showed in the proof of Theorem 3.3 that, if $d>0$ and $\delta=T_{s-1}$,
then the major part $h$ of $Q^{2}$,

$\displaystyle h(z,w)$
$\displaystyle=\sum_{I+l_{1}J=\gamma+l_{1}d}b_{IJ}\left(a_{\delta}z^{\delta}\right)^{I}\bigg{(}\sum_{i+l_{1}j=\gamma+l_{1}d}b_{ij}z^{i}w^{j}\bigg{)}^{J}$
$\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{d}+\cdots$ $\displaystyle\ \
\ \ +b_{AB}\big{(}a_{\delta}z^{\delta}\big{)}^{A}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{B},$

generates the important terms $z^{\gamma_{2}}w^{d^{2}}$ and
$z^{A_{2}^{*}}w^{B_{2}^{*}}$. However, if $d=0$ and $\delta=T_{s-1}$, then $h$
has the terms that consist only of $z$ other than $b_{\gamma
0}z^{\gamma_{2}}$:

$\displaystyle h(z,w)$ $\displaystyle=b_{\gamma
0}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
0}z^{\gamma}w^{0}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{0}+\cdots$ $\displaystyle\ \
\ \ +b_{AB}\big{(}a_{\delta}z^{\delta}\big{)}^{A}\big{(}b_{\gamma
0}z^{\gamma}w^{0}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{B}$ $\displaystyle=b_{\gamma
0}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}+\cdots+b_{AB}\big{(}a_{\delta}z^{\delta}\big{)}^{A}\Big{\\{}\big{(}b_{\gamma
0}z^{\gamma}\big{)}^{B}+\cdots+\big{(}b_{AB}z^{A}w^{B}\big{)}^{B}\Big{\\}}$
$\displaystyle=a_{\delta}^{\gamma}b_{\gamma
0}z^{\delta\gamma}+\cdots+a_{\delta}^{A}b_{AB}b_{\gamma 0}^{B}z^{\delta
A+\gamma B}+\cdots+a_{\delta}^{A}b_{AB}^{B+1}z^{\delta A+AB}w^{B^{2}}.$

Because the weights of all the terms in $h$ are the same $\gamma_{2}$, it
follows that

$\delta\gamma=\delta A+\gamma B=\delta I+\gamma J$

for any $(I,J)$ such that $I+l_{1}J=\gamma+l_{1}d$. We remark that this
equality also follows directly from the condition $\delta=T_{s-1}$ since

$T_{s-1}=\dfrac{\gamma B}{\gamma-A}=\dfrac{\gamma J}{\gamma-I}.$

Hence

$\displaystyle h(z,w)$
$\displaystyle=\bigg{\\{}\sum_{I+l_{1}J=\gamma}a_{\delta}^{I}b_{IJ}b_{\gamma
0}^{J}\bigg{\\}}z^{\delta\gamma}+\cdots+a_{\delta}^{A}b_{AB}^{B+1}z^{A_{2}^{*}}w^{B^{2}}$
$\displaystyle=\Big{\\{}a_{\delta}^{\gamma}b_{\gamma
0}+\cdots+a_{\delta}^{A}b_{AB}b_{\gamma
0}^{B}\Big{\\}}z^{\delta\gamma}+\cdots+a_{\delta}^{A}b_{AB}^{B+1}z^{A_{2}^{*}}w^{B^{2}}.$

Therefore, the term $z^{\gamma_{2}}$ vanishes if

$\sum_{I+l_{1}J=\gamma}a_{\delta}^{I}b_{IJ}b_{\gamma 0}^{J}=0.$

If the term $z^{\gamma_{n}}$ remains forever, then the same equality on
$w(Q^{n})$ and inequalities on $c(Q^{n})$ hold as the case $d>0$. Note that,
for example, if $a_{\delta}$ and all the non-zero coefficients of $q$ are
positive, then the term $z^{\gamma_{n}}$ remains forever. Even if
$z^{\gamma_{n}}$ vanish for some $n=n_{0}$, the term
$z^{A_{n}^{*}}w^{B_{n}^{*}}$ remains forever and
$w_{l_{1}}(Q^{n})=w_{l_{1}}(z^{A_{n}^{*}}w^{B_{n}^{*}})=\gamma_{n}$ for any
$n\geq n_{0}$. In particular, we obtain the following two propositions, which
implies Theorem 4.1 for the case $\delta=T_{s-1}$ and Theorem 4.4.

######  Proposition 4.10.

Let $d=0$ and $\delta=T_{s-1}$. Then $w_{l_{1}}(Q^{n})=\gamma_{n}$ for any
$n\geq 1$ and $\mathcal{I}_{f}=\\{l_{1}\\}$.

######  Proposition 4.11.

Let $d=0$ and $\delta=T_{s-1}$. Then

  1. (1)

$\gamma_{n}\leq c(Q^{n})\leq A_{n}^{*}+B_{n}^{*}\leq l_{1}^{-1}\gamma_{n}$ for
any $n\geq 1$ if $l_{1}\leq 1$, or

  2. (2)

$l_{1}^{-1}\gamma_{n}\leq c(Q^{n})\leq A_{n}^{*}+B_{n}^{*}<\gamma_{n}$ for any
$n\geq 1$ if $l_{1}>1$.

More precisely, if $l_{1}<1$ and the term $z^{\gamma_{n}}$ vanish for some
$n_{0}$, then

  1. $(3)$

$\gamma_{n}<c(Q^{n})$ for any $n\geq n_{0}$.

##  5\. Shape of Newton polygon of $Q^{n}$ for Case 3

In this section we deal with 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 $\delta\geq d>0$ by the setting. We first give a summary of the
results in Section 5.1, and illustrate some of them in terms of blow-ups in
Section 5.2. Preparing a lemma in Section 5.3, we prove Theorem 5.1 below in
Section 5.4. Finally, in Section 5.5 we show the existence of the vertex of
$N(Q^{n})$ that is next to $(\gamma_{n},d^{n})$ in most cases, and specify it,
which induces Theorem 5.3 below.

###  5.1. Summary of results

######  Theorem 5.1.

It follows that $Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$ and

$w(Q^{n})=w(z^{\gamma_{n}}w^{d^{n}})=\gamma_{n}+ld^{n}$

for any $n\geq 1$ and for any $l$ in $\mathcal{I}_{f}$, where
$\mathcal{I}_{f}=[\alpha,l_{2}]$ or $\mathcal{I}_{f}=(0,l_{2}]$ if $\gamma>0$
or $\gamma=0$.

Moreover, it follows that $(\gamma_{n},d^{n})$ is the vertex of $N(Q^{n})$
whose $x$-coordinate is minimum. Hence the theorem above implies the following
corollary.

######  Corollary 5.2.

It follows that

  1. (1)

$c(Q^{n})=\gamma_{n}+d^{n}$ for any $n\geq 1$ if $l_{2}\geq 1$, or

  2. (2)

$\gamma_{n}+l_{2}d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ for any $n\geq 1$ if
$l_{2}<1$.

Let $(C,D)=(n_{2},m_{2})$, and we define $(C_{n},D_{n})$ and
$(C_{n}^{*},D_{n}^{*})$ as the same as Case 2. If $\delta>T_{1}$, then the
term $z^{C_{n}}w^{D_{n}}$ remains forever and $(C_{n},D_{n})$ is the vertex of
$N(Q^{n})$ that is next to $(\gamma_{n},d^{n})$. Similarly, if $\delta=T_{1}$
and $m_{2}>0$, then the term $z^{C_{n}^{*}}w^{D_{n}^{*}}$ remains forever and
$(C_{n}^{*},D_{n}^{*})$ is the vertex of $N(Q^{n})$ that is next to
$(\gamma_{n},d^{n})$. On the other hand, if $\delta=T_{1}$ and $m_{2}=0$, then
the term $z^{C_{n}^{*}}w^{D_{n}^{*}}$ may vanish. By using these vertices, we
can improve the inequalities for the case $l_{1}<1$ in Corollary 5.2 as
follows.

######  Theorem 5.3.

It follows that

  1. (1)

$c(Q^{n})=\gamma_{n}+d^{n}$ for any $n\geq 1$ if $l_{2}\geq 1$, or

  2. (2)

$\gamma_{n}+l_{2}d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ for any $n\geq 1$ if
$l_{2}<1$.

Let $l_{2}<1$ and $m_{2}>0$. Then

  1. $(3)$

$\gamma_{n}+l_{2}d^{n}<c(Q^{n})<\gamma_{n}+d^{n}$ for any $n\geq 1$.

Let $l_{2}<1$ and $m_{2}=0$. Then $s=2$ and

  1. $(4)$

$\gamma_{n}+l_{2}d^{n}=c(Q^{n})$ for any $n\geq 1$ if $\delta>T_{1}$,

  2. $(5)$

$\gamma_{n}+l_{2}d^{n}=c(Q^{n})$ for any $n\geq 1$ if $\delta=T_{1}$ and the
term $z^{C_{n}^{*}}w^{D_{n}^{*}}$ remain forever,

  3. $(6)$

$\gamma_{n}+l_{2}d^{n}<c(Q^{n})$ for any $n\geq n_{0}$ if $\delta=T_{1}$ and
the term $z^{C_{n}^{*}}w^{D_{n}^{*}}$ vanish for some $n_{0}$.

###  5.2. Illustration of results in terms of blow-ups

Assuming that $l_{2}^{-1}$ is integer, we illustrate Theorem 5.1 and Corollary
5.2 by blow-ups. Let $\pi_{2}(t,w)=(tw^{l_{2}^{-1}},w)$ and
$\tilde{f}=\pi_{2}^{-1}\circ f\circ\pi_{2}$ as in Section 2.2. By Lemma 2.6,
$N(\tilde{q})=D(\gamma,\tilde{d})$, where $\tilde{d}=l_{2}^{-1}\gamma+d$.
Although $\tilde{f}$ may not be skew product, it is close to a monomial map by
Proposition 2.8 and so $N(\tilde{Q}^{n})$ has the unique vertex, that should
be $(\gamma_{n},\tilde{d}^{n})$. Hence
$N(\tilde{Q}^{n})=D(\gamma_{n},\tilde{d}^{n})$. It implies that $Q^{n}$
contains the term $z^{\gamma_{n}}w^{d^{n}}$, and that $N(Q^{n})$ is included
in the upper-right region that is surrounded by the line with slope
$-l_{2}^{-1}$ and the vertical line, which intersect at $(\gamma_{n},d^{n})$.
Therefore, we obtain Theorem 5.1 and Corollary 5.2 when $l_{2}^{-1}$ is
integer, because $w(Q^{n})$ or $c(Q^{n})$ is the minimum $x$-intercept of the
lines with slope $-l^{-1}$ or $-1$ that intersect $N(Q^{n})$.

###  5.3. Preliminary lemma: dynamics on the interval

Recall that

$R(l)=\dfrac{d}{\delta}(l-\alpha)+\alpha,\text{ where
}\alpha=\dfrac{\gamma}{\delta-d}.$

If $\delta>d$, then $\gamma\geq 0$ and $R$ is a contracting function around
the fixed point $\alpha$, where $\alpha\geq 0$. If $\delta=d$, then $\gamma=0$
and so $R(l)=l$. Since $\mathcal{I}_{f}=[\alpha,l_{2}]$ or
$\mathcal{I}_{f}=(0,l_{2}]$ if $\gamma>0$ or $\gamma=0$, we can conclude as
follows.

######  Lemma 5.4.

For any $l$ in $\mathcal{I}_{f}$, it follows that $R^{n}(l)$ belongs to
$\mathcal{I}_{f}$ for any $n\geq 1$ and the sequence $\\{R^{n}(l)\\}_{n\geq
1}$ is decreasing or $R\equiv id$. More precisely, $R^{n}(l)\to\alpha$ as
$n\to\infty$ or $R\equiv id$ if $\delta>d$ or $\delta=d$.

###  5.4. Proof of Theorem 5.1

The proof of Theorem 5.1 is similar to the proof of Theorem 3.1 for Case 2.
However, to show that $Q^{2}$ contains the term $z^{\gamma_{2}}w^{d^{2}}$, we
use the polynomial $h$, which was first appeared in the proof of Proposition
3.9 for Case 2.

###### Proof of Theorem 5.1.

Fix any $l$ in $\mathcal{I}_{f}$. The inequality
$w(Q^{n})\geq\gamma_{n}+ld^{n}$ follows from Lemma 5.4 and the same arguments
as the proof of Theorem 3.1 for Case 2.

Here we only show that the term $z^{\gamma_{2}}w^{d^{2}}$ has the biggest
degree $d^{2}$ with respect to $w$ among all the terms in $Q^{2}$ of weight
$\gamma_{2}+ld^{2}$; hence $Q^{2}$ contains the term
$z^{\gamma_{2}}w^{d^{2}}$.

If $\delta>T_{1}$, then it follows that the weight of any term of

$h(z,w):=b_{\gamma
d}\left(a_{\delta}z^{\delta}\right)^{\gamma}\bigg{(}\sum_{i+lj=\gamma+ld}b_{ij}z^{i}w^{j}\bigg{)}^{d}$

is $\gamma_{2}+ld^{2}$, and all the weights of the other terms in $Q^{2}(z,w)$
are bigger than $\gamma_{2}+ld^{2}$. The germ $h$ is a part of $Q^{2}$ and,
moreover, the term
$(z^{\delta})^{\gamma}(z^{\gamma}w^{d})^{d}=z^{\gamma_{2}}w^{d^{2}}$ has the
biggest degree $d^{2}$ with respect to $w$ in $h$. We remark that

$h(z,w)=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{d}$

if $l=l_{2}$, and $h(z,w)=a_{\delta}^{\gamma}b_{\gamma
d}^{d+1}z^{\gamma_{2}}w^{d^{2}}$ if $l<l_{2}$.

If $\delta=T_{1}$, then $\mathcal{I}_{f}=\\{l_{2}\\}$ and it follows that the
weight of any term of

$\displaystyle h(z,w)$
$\displaystyle:=\sum_{I+l_{2}J=\gamma+l_{2}d}b_{IJ}\left(a_{\delta}z^{\delta}\right)^{I}\bigg{(}\sum_{i+l_{2}j=\gamma+l_{2}d}b_{ij}z^{i}w^{j}\bigg{)}^{J}$
$\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{d}+\cdots$ $\displaystyle\ \
\ \ +b_{CD}\big{(}a_{\delta}z^{\delta}\big{)}^{C}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{D}$

is $\gamma_{2}+l_{2}d^{2}$, and all the weights of the other terms in
$Q^{2}(z,w)$ are bigger than $\gamma_{2}+l_{2}d^{2}$. The germ $h$ is a part
of $Q^{2}$ and, moreover, the term
$(z^{\delta})^{\gamma}(z^{\gamma}w^{d})^{d}=z^{\gamma_{2}}w^{d^{2}}$ has the
biggest degree $d^{2}$ with respect to $w$ in $h$.

It follows from the same arguments that the term
$(z^{\delta^{2}})^{\gamma}(z^{\gamma_{2}}w^{d^{2}})^{d}=z^{\gamma_{3}}w^{d^{3}}$
has the biggest degree with respect to $w$ among all the terms in $Q^{3}$ of
the smallest weight $\gamma_{3}+ld^{3}$. Repeating this process, one can show
that the term $z^{\gamma_{n}}w^{d^{n}}$ has the biggest degree $d^{n}$ with
respect to $w$ among all the terms in $Q^{n}$ of weight $\gamma_{n}+ld^{n}$
inductively; hence the term $z^{\gamma_{n}}w^{d^{n}}$ remains forever. ∎

The proof of Theorem 5.1 also shows that $(\gamma_{n},d^{n})$ is the vertex of
$N(Q^{n})$. Moreover, we can prove that $(\gamma_{n},d^{n})$ is the vertex of
$N(Q^{n})$ whose $x$-coordinate is minimum. In fact, we prove that the order
of $Q^{n}$ with respect to $z$ is $\gamma_{n}$, whereas we showed for Case 2
that the order of $Q^{n}$ with respect to $w$ is $d^{n}$ in the proof of
Theorem 3.1.

######  Proposition 5.5.

The order of $Q^{n}$ with respect to $z$ is $\gamma_{n}$ for any $n\geq 1$.

###### Proof.

The equality for $n=1$ follows from the setting.

We first show the equality for $n=2$. Although

$Q^{2}(z,w)=\sum b_{ij}\big{(}p(z)\big{)}^{i}\big{(}q(z,w)\big{)}^{j},$

it is enough to consider the part

$\sum b_{ij}\big{(}z^{\delta}\big{)}^{i}\big{(}z^{\gamma}w^{d}\big{)}^{j}$

of $Q^{2}$ and show that

$\delta i+\gamma j\geq\delta\gamma+\gamma d=\gamma_{2}$

for any $(i,j)$ such that $b_{ij}\neq 0$ since we only interested in the order
with respect to $z$. It follows from Lemma 5.4 that $i+R(l)j\geq\gamma+R(l)d$,
and so

$\delta i+\gamma j+ldj\geq\delta\gamma+\gamma d+ld^{2}$

by Lemma 3.5. Let $(i,j)\neq(\gamma,d)$. If $j<d$ then $\delta i+\gamma
j>\delta\gamma+\gamma d$ since $ldj<ld^{2}$. If $j=d$ then $\delta i+\gamma
j>\delta\gamma+\gamma d$ since $i>\gamma$. If $j>d$ then $\delta i+\gamma
j>\delta\gamma+\gamma d$ since $i\geq\gamma$. Therefore, the order of $Q^{2}$
with respect to $z$ is $\gamma_{2}$.

We next show the equality for $n=3$ by the same strategy. Although

$Q^{3}(z,w)=\sum
b_{ij}\big{(}p^{2}(z)\big{)}^{i}\big{(}Q^{2}(z,w)\big{)}^{j},$

it is enough to consider the part

$\sum
b_{ij}\big{(}z^{\delta^{2}}\big{)}^{i}\big{(}z^{\gamma_{2}}w^{d^{2}}\big{)}^{j}$

of $Q^{3}$ and show that

$\delta^{2}i+\gamma_{2}j\geq\delta^{2}\gamma+\gamma_{2}d=\gamma_{3}$

for any $(i,j)$ such that $b_{ij}\neq 0$. It follows from Lemma 5.4 that
$i+R^{2}(l)j\geq\gamma+R^{2}(l)d$, and so

$\delta^{2}i+\gamma_{2}j+ld^{2}j\geq\delta^{2}\gamma+\gamma_{2}d+ld^{3}$

by Lemma 3.5. Let $(i,j)\neq(\gamma,d)$. Then
$\delta^{2}i+\gamma_{2}j>\delta^{2}\gamma+\gamma_{2}d$ as the same as above.
Therefore, the order of $Q^{3}$ with respect to $z$ is $\gamma_{3}$.

Repeating this process, we obtain the required equalities. ∎

######  Corollary 5.6.

The bidegree $(\gamma_{n},d^{n})$ is the vertex of $N(Q^{n})$ whose
$x$-coordinate is minimum for any $n\geq 1$.

Therefore, $N(Q^{n})$ is included in the upper-right region that is surrounded
by the line with slope $-l_{2}^{-1}$ and the vertical line, which intersect at
$(\gamma_{n},d^{n})$.

Let $-M_{n}$ be the slope of the line passing through $(\gamma_{n},d^{n})$ and
the next vertex if it exist, let $M_{n}=\infty$ if
$N(Q^{n})=D(\gamma_{n},d^{n})$, and let $M=M_{1}$. Then $M=l_{2}^{-1}$, and
Proposition 5.5 implies the following corollary and Corollary 5.2.

######  Corollary 5.7.

It follows that $M_{n}\leq M$ for any $n\geq 1$.

######  Remark 5.8.

If $\gamma>0$, then $\gamma_{n}\to\infty$ since $\delta>0$ and $d>0$. Hence,
if $\gamma>0$ or $d>1$, then $c(Q^{n})\to\infty$ as $n\to\infty$. On the other
hand, $\min\\{1,l_{2}\\}d^{n}\leq c(Q^{n})\leq d^{n}$ if $\gamma=0$, and so
$c(Q^{n})=1$ if $\gamma=0$ and $d=1$.

###  5.5. Next vertices and proof of Theorem 5.3

Furthermore, we can show the existence of the vertex of $N(Q^{n})$ that is
next to $(\gamma_{n},d^{n})$ in most cases, and specify it. Let

$(C,D)=(n_{2},m_{2}),$ $\displaystyle(C_{n},D_{n})$
$\displaystyle=((\delta^{n-1}+\delta^{n-2}d+\cdots+\delta
d^{n-2})\gamma+d^{n-1}C,Dd^{n-1})$
$\displaystyle=(\gamma_{n}-(\gamma-C)d^{n-1},Dd^{n-1})\text{ and}$
$\displaystyle(C_{n}^{*},D_{n}^{*})$
$\displaystyle=((\delta^{n-1}+\delta^{n-2}D+\cdots+\delta
D^{n-2}+D^{n-1})C,D^{n}).$

######  Proposition 5.9.

If $\delta>T_{1}$, then $(C_{n},D_{n})$ is the vertex of $N(Q^{n})$ that is
next to $(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover, $M_{n}=M$, and
$\delta^{n}$ is bigger than the $y$-intercept of the line passing through
$(\gamma_{n},d^{n})$ and $(C_{n},D_{n})$. If $\delta=T_{1}$ and $m_{2}>0$,
then $(C_{n}^{*},D_{n}^{*})$ is the vertex of $N(Q^{n})$ that is next to
$(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover, $M_{n}=M$, and $\delta^{n}$
coincides with the $y$-intercept of the line passing through
$(\gamma_{n},d^{n})$ and $(C_{n}^{*},D_{n}^{*})$.

###### Proof.

The proof is similar to that of Proposition 3.9 for Case 2. Using the
polynomial $h$ in the proof of Theorem 5.1, we only show how to construct
$(C_{2},D_{2})$ and $(C_{2}^{*},D_{2}^{*})$. Let $l=l_{1}$.

If $\delta>T_{1}$, then

$h(z,w)=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{d}.$

The desired term is $b_{\gamma
d}(a_{\delta}z^{\delta})^{\gamma}(b_{CD}z^{C}w^{D})^{d}$ and so
$(C_{2},D_{2})=(\delta\gamma+Cd,Dd)$.

If $\delta=T_{1}$, then

$\displaystyle h(z,w)$ $\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{d}+\cdots$ $\displaystyle\ \
\ \ +b_{CD}\big{(}a_{\delta}z^{\delta}\big{)}^{C}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{D}.$

The desired term is $b_{CD}(a_{\delta}z^{\delta})^{C}(b_{CD}z^{C}w^{D})^{D}$
and so $(C_{2}^{*},D_{2}^{*})=(\delta C+CD,D^{2})$. ∎

This proposition induces Theorem 5.3.

###### Proof of Theorem 5.3.

We only show the improved parts.
(3) Since $D=m_{2}>0$, $D_{n}=Dd^{n-1}>0$ if $\delta>T_{1}$, and
$D_{n}^{*}=D^{n}>0$ if $\delta=T_{1}$. Hence $c(Q^{n})>\gamma_{n}+l_{2}d^{n}$.
On the other hand, $c(Q^{n})\leq C_{n}+D_{n}<\gamma_{n}+d^{n}$ if
$\delta>T_{1}$, and $c(Q^{n})\leq C_{n}^{*}+D_{n}^{*}<\gamma_{n}+d^{n}$ if
$\delta=T_{1}$.
(4) If $\delta>T_{1}$, then $c(Q^{n})=C_{n}=\gamma_{n}+l_{2}d^{n}$.
(5) If $\delta=T_{1}$ and $z^{C_{n}^{*}}w^{D_{n}^{*}}$ remain forever, then
$c(Q^{n})=C_{n}^{*}=\gamma_{n}+l_{2}d^{n}$.
(6) If $\delta=T_{1}$ and $z^{C_{n}^{*}}w^{D_{n}^{*}}$ vanish for $n=n_{0}$,
then $c(Q^{n})>C_{n}^{*}=\gamma_{n}+l_{2}d^{n}$ for any $n\geq n_{0}$. ∎

##  6\. Shape of Newton polygon of $Q^{n}$ for Case 4

We deal with 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 $\delta>d>0$ and $\gamma>0$ by the setting. We first give another
interval of weights which is closely related to those in Section 2.3, and a
summary of the results in Section 6.1. Some of the results are illustrated in
terms of blow-ups in Section 6.2. Preparing a lemma in Section 6.3, we prove
Theorem 6.1 below in Section 6.4. Finally, in Section 6.5 we state the claims
on the existence of the vertices of $N(Q^{n})$ that are previous and next to
$(\gamma_{n},d^{n})$ in most cases, and specify it, which induces Theorem 6.3
below.

###  6.1. Another interval and Summary of results

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

$\mathcal{I}_{f}^{AR}=\left\\{\ l>0\ \Bigg{|}\begin{array}[]{lcr}\gamma+ld\leq
n_{j}+lm_{j}\text{ for }j\leq k-1\\\ \gamma+ld\leq n_{j}+lm_{j}\text{ for
}j\geq k+1\end{array}\right\\}.$

Then

$\displaystyle\mathcal{I}_{f}^{AR}$ $\displaystyle=\left\\{\ l>0\
\Bigg{|}\begin{array}[]{lcr}\gamma+ld\leq n_{k-1}+lm_{k-1}\\\ \gamma+ld\leq
n_{k+1}+lm_{k+1}\end{array}\right\\}$ $\displaystyle=\left[\dfrac{\gamma-
n_{k-1}}{m_{k-1}-d},\dfrac{n_{k+1}-\gamma}{d-m_{k+1}}\right]=[l_{1},l_{1}+l_{2}].$

As shown in the theorem below, this interval is suitable for describing the
result on $w(Q^{n})$ rather than the intervals $\mathcal{I}_{f}^{1}$ and
$\mathcal{I}_{f}^{2}$ and the rectangle $\mathcal{I}_{f}$ in Section 2.3.

######  Theorem 6.1.

It follows that $Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$ and

$w(Q^{n})=w(z^{\gamma_{n}}w^{d^{n}})=\gamma_{n}+ld^{n}$

for any $n\geq 1$ and for any $l\in\mathcal{I}_{f}^{AR}$, where
$\mathcal{I}_{f}^{AR}=[l_{1},l_{1}+l_{2}]$.

Moreover, $(\gamma_{n},d^{n})$ is a vertex of $N(Q^{n})$ for any $n\geq 1$.
Hence the theorem above implies the following corollary.

######  Corollary 6.2.

It follows that

  1. (1)

$c(Q^{n})=\gamma_{n}+d^{n}$ for any $n\geq 1$ if $l_{1}\leq 1\leq
l_{1}+l_{2}$,

  2. (2)

$l_{1}^{-1}\gamma_{n}+d^{n}\leq c(Q^{n})<\gamma_{n}+d^{n}$ for any $n\geq 1$
if $l_{1}>1$, or

  3. (3)

$\gamma_{n}+(l_{1}+l_{2})d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ for any
$n\geq 1$ if $l_{1}+l_{2}<1$.

As the same as Cases 2 and 3, by investigating the vertices of $N(Q^{n})$ that
are previous and next to $(\gamma_{n},d^{n})$, we can improve the corollary as
follows.

######  Theorem 6.3.

It follows that

  1. (1)

$c(Q^{n})=\gamma_{n}+d^{n}$ for any $n\geq 1$ if $l_{1}\leq 1\leq
l_{1}+l_{2}$,

  2. (2)

$l_{1}^{-1}\gamma_{n}+d^{n}\leq c(Q^{n})<\gamma_{n}+d^{n}$ for any $n\geq 1$
if $l_{1}>1$, or

  3. (3)

$\gamma_{n}+(l_{1}+l_{2})d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ for any
$n\geq 1$ if $l_{1}+l_{2}<1$.

More precisely, for any $n\geq 1$,

  1. $(4)$

$l_{1}^{-1}\gamma_{n}+d^{n}<c(Q^{n})$ if $l_{1}>1$ and $n_{k-1}>0$, and

  2. $(5)$

$\gamma_{n}+(l_{1}+l_{2})d^{n}<c(Q^{n})<\gamma_{n}+d^{n}$ if $l_{1}+l_{2}<1$
and $m_{k+1}>0$.

Let $l_{1}+l_{2}<1$ and $m_{k+1}=0$. Then

  1. $(6)$

$\gamma_{n}+(l_{1}+l_{2})d^{n}=c(Q^{n})$ for any $n\geq 1$ if $\delta>T_{k}$,

  2. $(7)$

$\gamma_{n}+(l_{1}+l_{2})d^{n}=c(Q^{n})$ for any $n\geq 1$ if $\delta=T_{k}$
and the term $z^{C_{n}^{*}}w^{D_{n}^{*}}$ remain forever, or

  3. $(8)$

$\gamma_{n}+(l_{1}+l_{2})d^{n}<c(Q^{n})$ for any $n\geq n_{0}$ if
$\delta=T_{k}$ and the term $z^{C_{n}^{*}}w^{D_{n}^{*}}$ vanish for some
$n_{0}$.

###  6.2. Illustration of results in terms of blow-ups

Assuming that $l_{1}$ and $l_{2}^{-1}$ are integer, we illustrate Theorem 6.1
and Corollary 6.2 by blow-ups. Let $\tilde{f}_{1}=\pi_{1}^{-1}\circ
f\circ\pi_{1}$, where $\pi_{1}(z,c)=(z,z^{l_{1}}c)$, as in Case 2 and let
$\tilde{f}_{2}=\pi_{2}^{-1}\circ\tilde{f}_{1}\circ\pi_{2}$, where
$\pi_{2}(t,w)=(tw^{l_{2}^{-1}},w)$, as in Case 3. By Lemma 2.13,
$N(\tilde{q}_{2})=D(\tilde{\gamma},\tilde{d})$. Although $\tilde{f}_{2}$ may
not be skew product, it is close to a monomial map by Proposition 2.14 and so
$N(\tilde{Q}_{2}^{n})$ has the unique vertex, that should be
$(\tilde{\gamma}_{n},\tilde{d}^{n})$. Hence
$N(\tilde{Q}_{2}^{n})=D(\tilde{\gamma}_{n},\tilde{d}^{n})$. It implies that
$Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$, and that $N(Q^{n})$ is
included in the upper-right region that is surrounded by the two lines with
slopes $-l_{1}^{-1}$ and $-(l_{1}+l_{2})^{-1}$, which intersect at
$(\gamma_{n},d^{n})$. Therefore, we obtain Theorem 6.1 and Corollary 6.2 when
$l_{1}$ and $l_{2}^{-1}$ are integer, because $w(Q^{n})$ or $c(Q^{n})$ is the
minimum $x$-intercept of the lines with slope $-l^{-1}$ or $-1$ that intersect
$N(Q^{n})$.

###  6.3. Preliminary lemma: dynamics on the interval

Recall that

$R(l)=\dfrac{d}{\delta}(l-\alpha)+\alpha$

and so $R$ is a contracting function around the fixed point $\alpha$ since
$\delta<d$. Since $\mathcal{I}_{f}^{AR}$ contains $\alpha$, we have the
following lemma.

######  Lemma 6.4.

For any $l$ in $\mathcal{I}_{f}^{AR}$, it follows that $R^{n}(l)$ belongs to
$\mathcal{I}_{f}^{AR}$ for any $n\geq 1$ and $R^{n}(l)\to\alpha$ as
$n\to\infty$. More precisely, the sequence $\\{R^{n}(l)\\}_{n\geq 1}$ is
increasing if $l<\alpha$ and decreasing if $l>\alpha$.

###  6.4. Proof of Theorem 6.1

The proof of Theorem 6.1 is almost the same as the proof of Theorem 5.1 for
Case 3; we use the polynomial $h$ to show that $Q^{2}$ contains the term
$z^{\gamma_{2}}w^{d^{2}}$.

###### Proof of Theorem 6.1.

Fix any $l$ in $\mathcal{I}_{f}^{AR}$. The inequality
$w(Q^{n})\geq\gamma_{n}+ld^{n}$ follows from Lemma 6.4 and the same arguments
as the proof of Theorem 3.1 for Case 2.

Here we only give an explanation that $Q^{2}$ contains the term
$z^{\gamma_{2}}w^{d^{2}}$. Let $h$ be the part of $Q^{2}$ that consists of the
terms of the smallest weight $\gamma_{2}+ld^{2}$. Although $h$ depends on $l$,
we can show the claim for any $l$ in $\mathcal{I}_{f}$ as follows.

First, let $l=l_{1}$. Then

$\displaystyle h(z,w)$ $\displaystyle=b_{\gamma
d}\left(a_{\delta}z^{\delta}\right)^{\gamma}\bigg{(}\sum_{i+l_{1}j=\gamma+l_{1}d}b_{ij}z^{i}w^{j}\bigg{)}^{d}$
$\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{d}\text{ if
$\delta<T_{k-1}$, and}$ $\displaystyle h(z,w)$
$\displaystyle=\sum_{I+l_{1}J=\gamma+l_{1}d}b_{IJ}\left(a_{\delta}z^{\delta}\right)^{I}\bigg{(}\sum_{i+l_{1}j=\gamma+l_{1}d}b_{ij}z^{i}w^{j}\bigg{)}^{J}$
$\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{d}+\cdots$ $\displaystyle\ \
\ \ +b_{AB}\big{(}a_{\delta}z^{\delta}\big{)}^{A}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{AB}z^{A}w^{B}\big{)}^{B}\text{ if
$\delta=T_{k-1}$}.$

Hence the term $z^{\gamma_{2}}w^{d^{2}}$ has the smallest degree with respect
to $w$ in $h$.

Next, let $l=l_{1}+l_{2}$. Then

$\displaystyle h(z,w)$ $\displaystyle=b_{\gamma
d}\left(a_{\delta}z^{\delta}\right)^{\gamma}\bigg{(}\sum_{i+(l_{1}+l_{2})j=\gamma+(l_{1}+l_{2})d}b_{ij}z^{i}w^{j}\bigg{)}^{d}$
$\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{d}\text{ if
$\delta>T_{k+1}$, and}$ $\displaystyle h(z,w)$
$\displaystyle=\sum_{I+(l_{1}+l_{2})J=\gamma+(l_{1}+l_{2})d}b_{IJ}\left(a_{\delta}z^{\delta}\right)^{I}\bigg{(}\sum_{i+(l_{1}+l_{2})j=\gamma+(l_{1}+l_{2})d}b_{ij}z^{i}w^{j}\bigg{)}^{J}$
$\displaystyle=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{d}+\cdots$ $\displaystyle\ \
\ \ +b_{CD}\big{(}a_{\delta}z^{\delta}\big{)}^{C}\big{(}b_{\gamma
d}z^{\gamma}w^{d}+\cdots+b_{CD}z^{C}w^{D}\big{)}^{D}\text{ if
$\delta=T_{k+1}$}.$

Hence the term $z^{\gamma_{2}}w^{d^{2}}$ has the biggest degree with respect
to $w$ in $h$.

Finally, let $l_{1}<l<l_{1}+l_{2}$. Then

$h(z,w)=b_{\gamma
d}\big{(}a_{\delta}z^{\delta}\big{)}^{\gamma}\big{(}b_{\gamma
d}z^{\gamma}w^{d}\big{)}^{d}=a_{\delta}^{\gamma}b_{\gamma
d}^{d+1}z^{\gamma_{2}}w^{d^{2}}$

since $\\{(i,j):i+lj=\gamma+ld\text{ and }b_{ij}\neq 0\\}=\\{(\gamma,d)\\}$.

Applying the same argument inductively, one can show that the term
$z^{\gamma_{n}}w^{d^{n}}$ has the biggest degree with respect to $w$ among all
the terms in $Q^{n}$ of the smallest weight $\gamma_{n}+ld^{n}$, and so
$Q^{n}$ contains the term $z^{\gamma_{n}}w^{d^{n}}$, for any $n\geq 1$. ∎

The proof above implies the following.

######  Corollary 6.5.

The bidegree $(\gamma_{n},d^{n})$ is a vertex of $N(Q^{n})$ for any $n\geq 1$.

Therefore, $N(Q^{n})$ is included in the upper-right region that is surrounded
by the two lines with slopes $-l_{1}^{-1}$ and $-(l_{1}+l_{2})^{-1}$, which
intersect at $(\gamma_{n},d^{n})$.

Let $-M_{n}(l_{1})$ and $-M_{n}(l_{1}+l_{2})$ be the slopes of the line
passing through $(\gamma_{n},d^{n})$ and the previous vertex, and the line
passing through $(\gamma_{n},d^{n})$ and the next vertex, respectively. Let
$M(l_{1})=M_{1}(l_{1})$ and $M(l_{1}+l_{2})=M_{1}(l_{1}+l_{2})$. Then
$M(l_{1})=l_{1}^{-1}$ and $M(l_{1}+l_{2})=(l_{1}+l_{2})^{-1}$. Theorem 6.1 and
Corollary 6.5 imply the following corollary and Corollary 6.2.

######  Corollary 6.6.

It follows that $M_{n}(l_{1})\geq M(l_{1})$ and $M_{n}(l_{1}+l_{2})\leq
M(l_{1}+l_{2})$ for any $n\geq 1$.

###  6.5. Claims on previous and next vertices

As in Cases 2 and 3, we can show the existence of the vertices of $N(Q^{n})$
that are previous and next to $(\gamma_{n},d^{n})$ in most cases, and specify
them. Let

$(A,B)=(n_{k-1},m_{k-1})\text{ and }(C,D)=(n_{k+1},m_{k+1}).$

Let $(A_{n},B_{n})$ and $(A_{n}^{*},B_{n}^{*})$ be the same as Case 2, and let
$(C_{n},D_{n})$ and $(C_{n}^{*},D_{n}^{*})$ be the same as Case 3.

######  Proposition 6.7.

If $\delta<T_{k-1}$, then $(A_{n},B_{n})$ is the vertex of $N(Q^{n})$ that is
previous to $(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover,
$M_{n}(l_{1})=M(l_{1})$, and $\delta^{n}$ is smaller than the $y$-intercept of
the line passing through $(\gamma_{n},d^{n})$ and $(A_{n},B_{n})$. If
$\delta=T_{k-1}$, then $(A_{n}^{*},B_{n}^{*})$ is the vertex of $N(Q^{n})$
that is previous to $(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover,
$M_{n}(l_{1})=M(l_{1})$, and $\delta^{n}$ coincides with the $y$-intercept of
the line passing through $(\gamma_{n},d^{n})$ and $(A_{n}^{*},B_{n}^{*})$.

######  Proposition 6.8.

If $\delta>T_{k}$, then $(C_{n},D_{n})$ is the vertex of $N(Q^{n})$ that is
next to $(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover,
$M_{n}(l_{1}+l_{2})=M(l_{1}+l_{2})$, and $\delta^{n}$ is bigger than the
$y$-intercept of the line passing through $(\gamma_{n},d^{n})$ and
$(C_{n},D_{n})$. If $\delta=T_{k}$ and $m_{k+1}>0$, then
$(C_{n}^{*},D_{n}^{*})$ is the vertex of $N(Q^{n})$ that is next to
$(\gamma_{n},d^{n})$ for any $n\geq 1$. Moreover,
$M_{n}(l_{1}+l_{2})=M(l_{1}+l_{2})$, and $\delta^{n}$ coincides with the
$y$-intercept of the line passing through $(\gamma_{n},d^{n})$ and
$(C_{n}^{*},D_{n}^{*})$.

The proofs of these propositions are the same as those of Propositions 3.9 and
5.9 for Cases 2 and 3, and these propositions induces Theorem 6.3 as Cases 2
and 3.

##  7\. A corollary: Estimates on attraction rate of $f^{n}$

In the last section we derive inequalities of $c(f^{n})$ from the inequalities
of $c(Q^{n})$ in Theorem 1.3. We first give a summary of our results on the
inequalities of $c(f^{n})$ in Section 7.1. Detailed estimates and explanations
for Cases 1, Case 2 when $d>0$, Case 2 when $d=0$, Case 3 and Case 4 are given
in Sections 7.2, 7.3, 7.4, 7.5 and 7.6, respectively.

###  7.1. A summary

It is clear that $c(f^{n})\leq\delta^{n}$ since

$c(f^{n})=\min\\{c(p^{n}),c(Q^{n})\\}=\min\\{\delta^{n},c(Q^{n})\\}\leq\delta^{n}.$

Theorem 1.3 induces the following inequalities of $c(f^{n})$.

######  Theorem 7.1.

Let $\gamma d>0$. Then, for any $n\geq 1$,

  1. (1)

$\alpha\delta^{n}\leq c(f^{n})<\delta^{n}$ if $\delta>d$ and $\alpha<1$, or

  2. (2)

$c(f^{n})=\delta^{n}$ if $\delta>d$ and $\alpha\geq 1$ or if $\delta\leq d$.

On the other hand, for any $n\geq 1$,

  1. $(3)$

$c(f^{n})=\min\\{1,\gamma/\delta\\}\delta^{n}$ if $d=0$ and $s=1$,

  2. $(4)$

$c(f^{n})=\min\\{\delta^{n},d^{n}\\}$ if $\gamma=0$ and $s=1$,

  3. $(5)$

$\min\\{1,\gamma/\delta,l_{1}^{-1}\gamma/\delta\\}\delta^{n}\leq
c(f^{n})\leq\delta^{n}$ if $d=0$ and $s>1$,

  4. $(6)$

$l_{2}d^{n}\leq c(f^{n})\leq d^{n}$ if $\gamma=0$ and $s>1$.

In particular, $c_{\infty}=\delta$ if $\gamma>0$, and
$c_{\infty}=\min\\{\delta,d\\}$ if $\gamma=0$.

Note that the cases (3) and (4) occur only for Case 1, the case (5) occurs
only for Case 2, and the case (6) occur only for Case 3. We give improved
versions of (5) in Section 7.4.

###  7.2. Estimates for Case 1

For Case 1, we may assume that $f$ is a monomial map because $c(p^{n})$ and
$c(Q^{n})$ are the same. Let $f(z,w)=(z^{\delta},z^{\gamma}w^{d})$, where
$\delta\geq 1$, $\gamma\geq 0$, $d\geq 0$ and $\gamma+d\geq 1$. Then
$f^{n}(z,w)=(z^{\delta^{n}},z^{\gamma_{n}}w^{d^{n}})$ and so
$c(p^{n})=\delta^{n}$ and $c(Q^{n})=\gamma_{n}+d^{n}$. Since
$\gamma_{n}=\gamma(\delta^{n-1}+\delta^{n-2}d+\delta^{n-3}d^{2}+\cdots+d^{n-1})$,

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

We first consider the case $\gamma d>0$. If $\delta>d$, then
$\gamma_{n}+d^{n}=\alpha\delta^{n}+(1-\alpha)d^{n}\sim\alpha\delta^{n}$ and

$\begin{cases}\
\alpha\delta^{n}<\gamma_{n}+d^{n}<\alpha\delta^{n}+(1-\alpha)\delta^{n}=\delta^{n}&\text{
if }\alpha>1,\\\ \
\delta^{n}=\alpha\delta^{n}+(1-\alpha)\delta^{n}<\gamma_{n}+d^{n}<\alpha\delta^{n}&\text{
if }\alpha<1,\\\ \
\gamma_{n}+d^{n}=(\delta^{n}-d^{n})+\delta^{n}=\delta^{n}&\text{ if
}\alpha=1.\end{cases}$

Here the notation $A_{n}\sim B_{n}$ means that the ratio of $A_{n}$ and
$B_{n}$ tends to $1$ as $n\to\infty$. Since
$c(f^{n})=\min\\{\delta^{n},c(Q^{n})\\}$, we can summarize estimates on
$c(Q^{n})$ and $c(f^{n})$ as follows.

Table 1. Estimates for Case 1 when $\gamma d>0$ and $\delta>d$ $\alpha<1$ | $\alpha\delta^{n}<c(Q^{n})<\delta^{n}$ | $\alpha\delta^{n}<c(f^{n})<\delta^{n}$
---|---|---
$\alpha>1$ | $\delta^{n}<c(Q^{n})<\alpha\delta^{n}$ | $c(f^{n})=\delta^{n}$
$\alpha=1$ | $c(Q^{n})=\delta^{n}$ | $c(f^{n})=\delta^{n}$

If $\delta<d$, then
$\gamma_{n}+d^{n}=(-\alpha+1)d^{n}-(-\alpha)\delta^{n}\sim(-\alpha+1)d^{n}$
and

$\delta^{n}<d^{n}<\gamma_{n}+d^{n}<(-\alpha+1)d^{n}.$

If $\delta=d$, then $\gamma_{n}+d^{n}=n\gamma\delta^{n-1}+\delta^{n}\sim
n\gamma\delta^{n-1}$ and $\delta^{n}=d^{n}<\gamma_{n}+d^{n}$. Consequently, we
can summarize estimates on $c(Q^{n})$ for Case 1 when $\gamma d>0$ as follows.

Table 2. Estimates on $c(Q^{n})$ for Case 1 when $\gamma d>0$ $\delta>d$ | $c(Q^{n})\sim\alpha\delta^{n}$ | $\min\\{\alpha,1\\}\delta^{n}\leq c(Q^{n})\leq\max\\{\alpha,1\\}\delta^{n}$
---|---|---
$\delta<d$ | $c(Q^{n})\sim(-\alpha+1)d^{n}$ | $\delta^{n}<c(Q^{n})<(-\alpha+1)d^{n}$
$\delta=d$ | $c(Q^{n})\sim n\gamma\delta^{n-1}$ | $\delta^{n}<c(Q^{n})=n\gamma\delta^{n-1}+\delta^{n}$

Since $c(f^{n})=\min\\{\delta^{n},c(Q^{n})\\}$, we obtain the following
estimates on $c(f^{n})$.

Table 3. Estimates on $c(f^{n})$ for Case 1 when $\gamma d>0$ $\delta>d$ | $\min\\{\alpha,1\\}\delta^{n}\leq c(f^{n})\leq\delta^{n}$
---|---
$\delta<d$ | $c(f^{n})=\delta^{n}$
$\delta=d$ | $c(f^{n})=\delta^{n}$

We next consider the case $\gamma d=0$. If $\gamma=0$, then $c(Q^{n})=d^{n}$
and so $c(f^{n})=\min\\{\delta^{n},d^{n}\\}$. If $d=0$, then
$c(Q^{n})=\gamma_{n}=\gamma\delta^{n-1}$ and so
$c(f^{n})=\min\\{\delta,\gamma\\}\delta^{n-1}$.

Table 4. Estimates for Case 1 when $\gamma d=0$ $\gamma=0$ | $c(Q^{n})=d^{n}$ | $c(f^{n})=\min\\{\delta^{n},d^{n}\\}$
---|---|---
$d=0$ | $c(Q^{n})=\gamma\delta^{n-1}$ | $c(f^{n})=\min\\{\delta,\gamma\\}\delta^{n-1}$

###  7.3. Estimates for Case 2 when $d>0$

Recall that $\gamma>0$ by the setting. If $d>0$ and $l_{1}\leq 1$, then
$c(Q^{n})=\gamma_{n}+d^{n}$ by Theorem 1.3 and so we have the same
inequalities as Case 1. Let $d>0$ and $l_{1}>1$. Then
$l_{1}^{-1}\gamma_{n}+d^{n}\leq c(Q^{n})<\gamma_{n}+d^{n}$ by Theorem 1.3, and

$\l_{1}^{-1}\gamma_{n}+d^{n}=l_{1}^{-1}\alpha\delta^{n}+\left(1-l_{1}^{-1}\alpha\right)d^{n}.$

If $\delta>d$, then $\mathcal{I}_{f}=[l_{1},\alpha]$ and $\alpha\geq l_{1}>1$.
Hence $1-l_{1}^{-1}\alpha\leq 0$ and so

$\delta^{n}=l_{1}^{-1}\alpha\delta^{n}+\left(1-l_{1}^{-1}\alpha\right)\delta^{n}\leq
l_{1}^{-1}\gamma_{n}+d^{n}\leq c(Q^{n})<\gamma_{n}+d^{n}<\alpha\delta^{n}.$

If $\delta<d$, then

$\delta^{n}<d^{n}<l_{1}^{-1}\gamma_{n}+d^{n}\leq
c(Q^{n})<\gamma_{n}+d^{n}<(-\alpha+1)d^{n}.$

If $\delta=d$, then

$\delta^{n}=d^{n}<l_{1}^{-1}\gamma_{n}+d^{n}\leq
c(Q^{n})<\gamma_{n}+d^{n}=n\gamma\delta^{n-1}+\delta^{n}.$

Therefore, we can summarize estimates on $c(Q^{n})$ and $c(f^{n})$ for Case 2
when $d>0$ and $l_{1}>1$ as follows, which are almost the same as Case 1.

Table 5. Estimates for Case 2 when $d>0$ and $l_{1}>1$ $\delta>d$ | $\delta^{n}\leq c(Q^{n})<\alpha\delta^{n}$ | $c(f^{n})=\delta^{n}$
---|---|---
$\delta<d$ | $\delta^{n}<c(Q^{n})<(-\alpha+1)d^{n}$ | $c(f^{n})=\delta^{n}$
$\delta=d$ | $\delta^{n}<c(Q^{n})=n\gamma\delta^{n-1}+\delta^{n}$ | $c(f^{n})=\delta^{n}$

In particular, $\delta^{n}\leq c(Q^{n})$ and $c(f^{n})=\delta^{n}$ if $d>0$
and $l_{1}>1$.

###  7.4. Estimates for Case 2 when $d=0$

Recall that $\gamma>0$. Let $d=0$. If $\delta<T_{s-1}$, then

$\begin{cases}\ c(Q^{n})=\gamma_{n}&\text{ if }l_{1}\leq 1\\\ \
l_{1}^{-1}\gamma_{n}\leq c(Q^{n})\leq\gamma_{n}&\text{ if }l_{1}>1\end{cases}$

by Theorem 4.3. If $\delta=T_{s-1}$, then

$\begin{cases}\ \gamma_{n}\leq c(Q^{n})\leq l_{1}^{-1}\gamma_{n}&\text{ if
}l_{1}\leq 1\\\ \ l_{1}^{-1}\gamma_{n}\leq c(Q^{n})\leq\gamma_{n}&\text{ if
}l_{1}>1\end{cases}$

by Theorem 4.4 or Theorem 1.3. Since $\gamma_{n}=\gamma\delta^{n-1}$ if $d=0$,
we can summarize estimates on $c(Q^{n})$ and $c(f^{n})$ for Case 2 when $d=0$
as follows.

Table 6. Estimates for Case 2 when $d=0$

$\delta<T_{s-1}$ | $c(Q^{n})$ | $c(f^{n})$
---|---|---
$l_{1}\leq 1$ | $c(Q^{n})=\gamma\delta^{n-1}$ | $c(f^{n})=\min\\{\delta,\gamma\\}\delta^{n-1}$
$l_{1}>1$ | $l_{1}^{-1}\gamma\delta^{n-1}\leq c(Q^{n})\leq\gamma\delta^{n-1}$ | $\min\\{\delta,l_{1}^{-1}\gamma\\}\delta^{n-1}\leq c(f^{n})\leq\min\\{\delta,\gamma\\}\delta^{n-1}$

$\delta=T_{s-1}$ | $c(Q^{n})$ | $c(f^{n})$
---|---|---
$l_{1}\leq 1$ | $\gamma\delta^{n-1}\leq c(Q^{n})\leq l_{1}^{-1}\gamma\delta^{n-1}$ | $\min\\{\delta,\gamma\\}\delta^{n-1}\leq c(f^{n})\leq\min\\{\delta,l_{1}^{-1}\gamma\\}\delta^{n-1}$
$l_{1}>1$ | $l_{1}^{-1}\gamma\delta^{n-1}\leq c(Q^{n})\leq\gamma\delta^{n-1}$ | $\min\\{\delta,l_{1}^{-1}\gamma\\}\delta^{n-1}\leq c(f^{n})\leq\min\\{\delta,\gamma\\}\delta^{n-1}$

In particular,
$\min\\{1,\gamma/\delta,l_{1}^{-1}\gamma/\delta\\}\delta^{n}\leq
c(f^{n})\leq\delta^{n}$ if $d=0$. We remark that this rough inequality also
follows from Theorem 1.3.

###  7.5. Estimates for Case 3

Recall that $\delta>d>0$ if $\gamma>0$, and $\delta\geq d>0$ if $\gamma=0$ by
the setting. If $l_{2}\geq 1$, then $c(Q^{n})=\gamma_{n}+d^{n}$ by Theorem 1.3
and so we have the same inequalities as Case 1. Let $l_{2}<1$. Then
$\gamma_{n}+l_{2}d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ by Theorem 1.3. If
$\gamma>0$, then $\mathcal{I}_{f}=[\alpha,l_{2}]$ and $\alpha\leq l_{2}<1$.
Hence

$\alpha\delta^{n}\leq\alpha\delta^{n}+(l_{2}-\alpha)d^{n}=\gamma_{n}+l_{2}d^{n}\leq
c(Q^{n})\leq\gamma_{n}+d^{n}<\delta^{n}.$

If $\gamma=0$, then $l_{2}d^{n}\leq c(Q^{n})\leq d^{n}$ and so $l_{2}d^{n}\leq
c(f^{n})\leq d^{n}$. Therefore, we can summarize estimates on $c(Q^{n})$ and
$c(f^{n})$ for Case 3 when $l_{2}<1$ as follows.

Table 7. Estimates for Case 3 when $l_{2}<1$ $\gamma>0$ (and $\delta>d$) | $\alpha\delta^{n}\leq c(Q^{n})<\delta^{n}$ | $\alpha\delta^{n}\leq c(f^{n})<\delta^{n}$
---|---|---
$\gamma=0$ (and $\delta\geq d$) | $l_{2}d^{n}\leq c(Q^{n})\leq d^{n}$ | $l_{2}d^{n}\leq c(f^{n})\leq d^{n}$

###  7.6. Estimates for Case 4

Recall that $\delta>d>0$ and $\gamma>0$ by the setting and that
$l_{1}\leq\alpha\leq l_{1}+l_{2}$. If $l_{1}\leq 1\leq l_{1}+l_{2}$, then
$c(Q^{n})=\gamma_{n}+d^{n}$ by Theorem 1.3 and so we have the same
inequalities as Case 1. If $l_{1}>1$, then $l_{1}^{-1}\gamma_{n}+d^{n}\leq
c(Q^{n})<\gamma_{n}+d^{n}$ by Theorem 1.3 and so $\delta^{n}\leq
c(Q^{n})<\alpha\delta^{n}$ since $\alpha\geq l_{1}>1$. If $l_{1}+l_{2}<1$,
then $\gamma_{n}+(l_{1}+l_{2})d^{n}\leq c(Q^{n})\leq\gamma_{n}+d^{n}$ by
Theorem 1.3 and so

$\alpha\delta^{n}\leq\alpha\delta^{n}+(l_{1}+l_{2}-\alpha)d^{n}=\gamma_{n}+(l_{1}+l_{2})d^{n}\leq
c(Q^{n})\leq\gamma_{n}+d^{n}<\delta^{n}$

since $\alpha\leq l_{1}+l_{2}<1$. We can classify the estimates on $c(Q^{n})$
and $c(f^{n})$ for Case 4 in terms of $\alpha$, and obtain the following
summary, which is almost the same as Case 1 when $\delta>d>0$ and $\gamma>0$.

Table 8. Estimates for Case 4 $\alpha<1$ | $\alpha\delta^{n}\leq c(Q^{n})<\delta^{n}$ | $\alpha\delta^{n}\leq c(f^{n})<\delta^{n}$
---|---|---
$\alpha>1$ | $\delta^{n}\leq c(Q^{n})<\alpha\delta^{n}$ | $c(f^{n})=\delta^{n}$
$\alpha=1$ | $c(Q^{n})=\delta^{n}$ | $c(f^{n})=\delta^{n}$

## References

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