# Skew Products on the Berkovich Projective Line

Richard A. P. Birkett

###### Abstract.

In this article, we develop a dynamical theory for what shall be called a
_skew product_ on the Berkovich projective line,
$\phi_{*}:\mathbb{P}^{1}_{\text{an}}(K)\to\mathbb{P}^{1}_{\text{an}}(K)$ over
a non-Archimedean field $K$. These functions are defined algebraically yet
strictly generalise the notion of a _rational map_ on
$\mathbb{P}^{1}_{\text{an}}$. We describe the analytical, algebraic, and
dynamical properties of skew products, including a study of periodic points,
and a Fatou/Julia dichotomy. The article culminates with the classification of
the connected components of the Fatou set.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Skew Products on the Berkovich Projective Line
    3. 1.3 Properties of Skew Products
    4. 1.4 Dynamics of a Skew Product
    5. 1.5 Fatou and Julia
    6. 1.6 Classification of Fatou Components
    7. 1.7 Organisation
    8. 1.8 Acknowledgements
  2. 2 Background
    1. 2.1 Non-Archimedean Metrics
    2. 2.2 Disks
    3. 2.3 Projective Line and Affinoids
    4. 2.4 Power Series and Constructing Seminorms
    5. 2.5 Taylor Series
    6. 2.6 Laurent Series
    7. 2.7 Seminorms of Power Series
    8. 2.8 The Berkovich Projective Line
    9. 2.9 The Berkovich Affine Line
    10. 2.10 The Berkovich Projective Line
    11. 2.11 Berkovich Disks, Affinoids, and Directions
    12. 2.12 Paths and Hyperbolic Metric
    13. 2.13 Rational Maps
    14. 2.14 Reduction
  3. 3 Skew Products on the Berkovich Projective Line
    1. 3.1 Motivation
    2. 3.2 The Problem
    3. 3.3 Skew Products
    4. 3.4 Properties of Skew Products
    5. 3.5 Local Degrees in Directions
    6. 3.6 Local Degrees at Points
    7. 3.7 Reduction and Computation of Local Degrees
    8. 3.8 The Injectivity and Ramification Loci
    9. 3.9 Geometric and Topological Ideas
  4. 4 Periodic Points
  5. 5 Fatou and Julia
  6. 6 Fatou Components of Skew Products
    1. 6.1 Berkovich Fatou Components
    2. 6.2 Attracting Components
    3. 6.3 Indifferent Components
    4. 6.4 Classification
    5. 6.5 Wandering Domains

##  1\. Introduction

A rational map
$\phi:\mathbb{P}^{1}_{\text{an}}(K)\to\mathbb{P}^{1}_{\text{an}}(K)$ on the
Berkovich projective line can be thought of as a $K$-algebra endomorphism
$\phi^{*}:K(z)\to K(z)$. This article is dedicated to developing an analytical
and dynamical theory for a map called a _skew product_. In this more general
context, we relax the condition that $\phi^{*}|_{K}$ is the identity, but ask
that that it respects the non-Archimedean metric.

After unraveling the algebraic structure of a skew product, we will see that
the map is still piecewise linear on $\mathbb{P}^{1}_{\text{an}}$, however the
slopes are not necessarily integers, nor at least $1$. The possibility of
contraction leads to more interesting dynamics, especially for fixed points
and the Fatou-Julia dichotomy. In the ‘simple’ case (where the slopes are
integers) we generalise the Rivera-Letelier style classification of Fatou
components, and certain facts about wandering domains. These latter results
will be fundamental for applications to dynamics in two dimensions.

The results in the present article and the forthcoming applications feature in
the author’s PhD thesis.

###  1.1. Motivation

The dynamical theory of a skew product will be essential for applications to
the dynamics of rational maps on a complex surface in a later article. For the
purposes of understanding the dynamical degrees (algebraic entropy) and
algebraic stability of a rational map $f:X\dashrightarrow X$ on a surface, one
often needs to understand the potential dynamical behaviour of $f$ for all
possible birational changes of coordinates. Favre and Jonsson [FJ11]
considered the case of a polynomial map
$f:\mathbb{P}^{2}_{\mathbb{C}}\dashrightarrow\mathbb{P}^{2}_{\mathbb{C}}$
which is invariant over the line at infinity. They translate the dynamics of
$f$ to that of a universal ‘dual graph’ of all possible exceptional curves
over this line. The ‘valuative tree’ [FJ04] they obtain is equivalent to a one
dimensional Berkovich space over the Puiseux series, and the corresponding
function turns out to essentially be a contraction mapping. Typically, such an
induced function will not be a contraction mapping. The later application of
the present article will be to the case of a skew product
$\phi:X\dashrightarrow X$ on a complex ruled surface, and hence lends its name
to the non-Archimedean version. Classically, a skew product is a map of the
form $\phi(x,y)=(\phi_{1}(x),\phi_{2}(x,y))$ on a product space. When
$X=\mathbb{P}^{1}\times\mathbb{P}^{1}$, the rational map $\phi$ corresponds to
a $\mathbb{C}$-algebra endomorphism
$\phi^{*}:\mathbb{C}(x)(y)\to\mathbb{C}(x)(y)$. Furthermore if
$\phi_{1}=\operatorname{id}$ then this is a $\mathbb{C}(x)$-algebra
endomorphism, and so it extends to a rational map
$\phi_{2}:\mathbb{P}^{1}_{\text{an}}(\mathbb{K})\to\mathbb{P}^{1}_{\text{an}}(\mathbb{K})$
over the field $\mathbb{K}$ of complex Puiseux series in $x$. DeMarco and
Faber [DF14, DF16] used this perspective when $\phi_{1}=\operatorname{id}$ to
find a (possibly singular) surface $\hat{X}$ where $\phi$ is algebraically
stable. The core of their argument was to use the Rivera-Letelier
classification of Fatou components [RL03a, RL03b] for the corresponding
Berkovich rational map. Inspired by this result, we will deal with the case of
complex skew products where $\phi_{1}$ is not necessarily trivial. In this
more general setting, $\phi^{*}:\mathbb{C}(x)(y)\to\mathbb{C}(x)(y)$ will not
be a $\mathbb{C}(x)$-algebra endomorphism; whence the induced function
$\phi_{*}:\mathbb{P}^{1}_{\text{an}}(\mathbb{K})\to\mathbb{P}^{1}_{\text{an}}(\mathbb{K})$
cannot correspond to a rational map, but to a different kind of map - the non-
Archimedean _skew product_.

###  1.2. Skew Products on the Berkovich Projective Line

Aside from examples, this theory will be discussed for a general non-
Archimedean field $K$; further study of the specialisation to the Puiseux
series $K=\mathbb{K}$ and applications to complex/algebraic dynamics will be
deferred to a sequel. Skew products on the Berkovich projective line are of
independent interest as a generalisation of Berkovich _rational maps_ , for
instance because they also encompass a class of field automorphisms.

linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,todo:
linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,Mention Vladimir
Berkovich somewhere.

Emerging out of the study of the dynamics of rational maps over the p-adic
numbers there has been substantial interest in the notions of non-Archimedean
Fatou and Julia sets [Dre03, Hsi96, Hsi00, Béz01, Béz04]. The landmark theses
of Robert L. Benedetto [Ben98, Ben00, Ben01a, Ben01b] and Juan Rivera-Letelier
[RL03a] together established the essential theory for the dynamics of non-
Archimedean Fatou components. Rivera-Letelier, in particular, brought the
insight of extending a rational map from $\mathbb{P}^{1}(K)$ to a _rational
map_ on a non-Archimedean analytic space $\mathbb{P}^{1}_{\text{an}}(K)$ which
compactifies the former. He gave the classification of Fatou components over
the $p$-adics. Following this there were numerous papers on the structure of
the Fatou and Julia sets [Ben02a, Ben02b, Ben05, Ben06, BBP07, RL03b, RL04,
RL05]linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,todo:
linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,Add Faber?. The
field exploded and many fruitful connections appeared between complex and non-
Archimedean dynamics. For example by Kiwi [Kiw06, Kiw14, Kiw15], by Baker and
DeMarco [BD11, BDM13], by Favre and Gauthier [FG18], by Dujardin and Favre
[DF17], by DeMarco and Faber [DF14, DF16], and by Favre [Fav20] – to name just
a selection. Following this progress Fatou and Julia theory, three independent
groups of mathematicians developed potential theory and proved
equidistribution theorems in the non-Archimedean setting, mimicking the
complex dynamical case; namely by Chambert-Loir and Thuillier [CL06, Thu05],
by Baker and Rumely [BR06], and by Favre and Rivera-Letelier [FRL04, FRL06,
FRL07]. For a fuller account of this history we refer the reader to the
excellent survey by Benedetto [Ben22]. For theoretical background the author
recommends the books by Benedetto, and also Baker and Rumely [Ben19, BR10].

In particular, however, the reader is advised to compare with [Ben19] as they
read section 2. In this work we will use the notation built up by Benedetto,
and follow his development when possible for building up the theory of non-
Archimedean skew products. The author hopes this will allow for an easy
adjustment to the reader already familiar with Berkovich rational maps.
linecolor=red,backgroundcolor=red!25,bordercolor=red,linecolor=red,backgroundcolor=red!25,bordercolor=red,todo:
linecolor=red,backgroundcolor=red!25,bordercolor=red,Move to section 2?

For this introduction, we briefly recall the structure of the Berkovich
projective line $\mathbb{P}^{1}_{\text{an}}(K)$. Unless otherwise stated, we
consider $(K,|\cdot|)$ to be an arbitrary non-Archimedean algebraically closed
field, and $\mathbb{\mathbb{C}}_{v}$ its completion. Algebraically,
$\mathbb{P}^{1}_{\text{an}}$ is the set of seminorms on the ring $K[y]$ which
extend the norm $\left|\,\cdot\,\right|$ on $K$, and also a point at infinity.
Topologically, it is uniquely path connected, meaning it is a tree, however
not in the finite combinatorial sense because it has a dense set of vertices
on any interval. These (interior) vertices, called _Type II_ points, have one
branch for every element of $\mathbb{P}^{1}(k)$, where $k$ is the residue
field of $K$. All other points on an open interval are _Type III_ points (edge
points). Every Type II and III point $\zeta=\zeta(a,r)$ corresponds to a
closed disk $\overline{D}(a,r)\subset\mathbb{P}^{1}(K)$, with $r$ in the value
group $\left|K^{\times}\right|$ or not respectively. The points in the
classical projective line
$\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$ form a set of
endpoints called _Type I_ points, which alone would be a totally disconnected
and non-compact set. There are other endpoints called _Type IV_ points,
corresponding empty intersections of nested disks. All of these naturally
correspond to seminorms through the geometric data and Berkovich showed
[Ber90] that the four types listed constitute the whole space. The _hyperbolic
plane_ $\mathbb{H}=\mathbb{P}^{1}_{\text{an}}(K)\setminus\mathbb{P}^{1}(K)$ is
defined as the set of Type II, III, and IV points; it is endowed with a
_hyperbolic metric_ $d_{\mathbb{H}}$. See subsection 2.8 for more.

Much like a rational map, the starting point for a skew product is a
homomorphism $\phi^{*}:K(y)\to K(y)$. To define a Berkovich _rational map_ we
would equivalently require that $\phi^{*}$ is a $K$-algebra homomorphism, i.e.
the identity on $K$. Unfortunately this is too restrictive for the application
to complex skew products $\phi(x,y)=(\phi_{1}(x),\phi_{2}(x,y))$ because
$\left.\phi^{*}\right|_{\mathbb{K}}:x\mapsto\phi_{1}(x)$ is not the identity
map on $K=\mathbb{K}$. On the other hand, if we allow the homomorphism
$\phi^{*}$ to be completely arbitrary, the induced map on the Berkovich
projective line becomes unwieldy. Instead we impose on $\phi^{*}$ the
condition that that $\phi^{*}$ uniformly scales the norm on $K$. This is
easily satisfied in the geometric/complex case, for example

$\left|\phi^{*}(x)\right|=\left|\phi_{1}(x)\right|=\left|c_{n}x^{n}+c_{n+1}x^{n+1}+\cdots\right|=\left|x\right|^{n}.$

In general, if $\left|\phi^{*}(a)\right|=\left|a\right|^{1/q}$ for every $a\in
K$ then we call $\phi^{*}$ an _equivariant skew endomorphism_ , and define the
_skew product_ $\phi_{*}$ by

$\displaystyle\phi_{*}:\mathbb{P}^{1}_{\text{an}}(K)$
$\displaystyle\longrightarrow\mathbb{P}^{1}_{\text{an}}(K)$
$\displaystyle\zeta$ $\displaystyle\longmapsto\phi_{*}(\zeta)$
$\displaystyle\text{where }\left\|\psi\right\|_{\phi_{*}(\zeta)}$
$\displaystyle=\left\|\phi^{*}(\psi)\right\|_{\zeta}^{q}$

We call $q$ the _scale factor_ , and for general $K$ this can be any positive
real number; in the geometric/complex case, $q=1/n$.

###  1.3. Properties of Skew Products

Fundamental to understanding the structure of skew products is the
decomposition result of Theorem 3.3. We define
$\phi_{1}^{*}=\left.\phi^{*}\right|_{K}$ but extended trivially to $K(y)$, and
secondly we define $\phi_{2}^{*}$ to capture only the action of $\phi^{*}$ on
$y$. However, perhaps unintuitively, $\phi_{1*}$ acts as $(\phi_{1}^{*})^{-1}$
on classical points. Every skew endomorphism has this decomposition
$\phi^{*}=\phi_{2}^{*}\circ\phi_{1}^{*}$, and it descends to the skew product
$\phi_{*}=\phi_{1*}\circ\phi_{2*}$. Most important are the facts that
$\phi_{2*}$ is a non-Archimedean rational map on $\mathbb{P}^{1}_{\text{an}}$
and $\phi_{1*}$ uniformly scales the metric in the hyperbolic plane
$\mathbb{H}\subset\mathbb{P}^{1}_{\text{an}}$ by a factor of $q$, see Theorem
3.4. The term _scale factor_ for $q$ was chosen with this in mind.

In Theorem 3.7 and preceding results we demonstrate that any non-constant skew
product $\phi_{*}$ is a continuous, open mapping that is precisely the unique
continuous extension of $(\phi_{1}^{*})^{-1}\circ\phi_{2}$ on
$\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$. As one would hope, it
preserves the types of each Berkovich point. Furthermore, Theorem 3.8 states
that a skew product maps connected affinoids to connected affinoids, and by
subsection 3.6 it will map each component of $\phi_{*}^{-1}(U)$ to $U$ in a
$d$-to-$1$ manner.

We then proceed to consider the map on tangent directions
$\phi_{\\#}:T_{\zeta}\mathbb{P}^{1}_{\text{an}}\to
T_{\phi_{*}(\zeta)}\mathbb{P}^{1}_{\text{an}}$ and degrees (local and in
tangent directions) for a skew product; these degrees are the _relative
degree_ $\operatorname{rdeg}(\phi_{*})$, _local degree_ $\deg_{\zeta}(\phi)$
and _local degree in a direction_ $\deg_{\phi,{\bf v}}(\phi)$. subsection 3.5,
Theorem 3.18 and Theorem 3.20 show that the previously established relations
between these quantities for rational maps still hold in the new setting.
After reintroducing reduction for skew products we obtain a generalisation
Theorem 3.24 of further consequential results [Ben19, Theorem 7.34, Lemma
7.35]. These results state that the tangent map $\phi_{\\#}$ can be understood
through the reduction $\overline{\phi}$ and that the tangent map disagrees
with $\phi_{*}$ itself when and only when a _bad direction_ ${\bf v}$ contains
a preimage of the entire projective line. Theorem 3.13 extends the idea that
on a small interval with local degree $m$, $\phi_{*}$ is a linear map with
gradient $mq$.

We reuse the definitions of ramification locus
$\operatorname{Ram}(\phi)=\left\\{\zeta\in\mathbb{P}^{1}_{\text{an}}:\,\deg_{\zeta}(\phi)\geq
2\right\\}$, and injectivity locus
$\operatorname{Inj}(\phi)=\mathbb{P}^{1}_{\text{an}}\setminus\operatorname{Ram}(\phi)$.
We also say a skew product $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ is _tame_ iff
the underlying rational map $\phi_{2}$ is tame. Note that if
$\operatorname{char}K=p$ then we can always at the least find a decomposition
where $\phi_{2}$ is separable, by transferring (inverse) Frobenius
automorphisms onto $\phi_{1}^{*}$. We discuss the the separability and
uniqueness of the decomposition of skew products. Again, thanks to
decomposition, we obtain Theorem 3.28 as a direct corollary of the theorem by
Xander Faber for rational maps [Fab13a]. This says that
$\operatorname{Ram}(\phi)$ is a subset of
$\operatorname{Hull}(\operatorname{Crit}(\phi_{2}))$ whose endpoints lie in
$\operatorname{Crit}(\phi_{2})=\operatorname{Ram}(\phi)\cap\mathbb{P}^{1}$.
linecolor=red,backgroundcolor=red!25,bordercolor=red,linecolor=red,backgroundcolor=red!25,bordercolor=red,todo:
linecolor=red,backgroundcolor=red!25,bordercolor=red,Should I write cor or
thm?As a further corollary, we find that if $\phi_{*}$ is tame then
$\deg_{\zeta}(\phi)=1$ for every Type IV point of
$\mathbb{P}^{1}_{\text{an}}$.

Subsection 3.9 contains some possibly new proofs and restatements of
established topological ideas, and an upgrade for Theorem 3.13 as Theorem
3.34. We highlight the Extreme Value Theorem Theorem 3.30 and the
Hyperbolicity Theorem (Theorem 3.35). Subsection 3 contains more results that
we leave out of this introduction.

###  1.4. Dynamics of a Skew Product

Having established the basic analytical properties of a skew product, which
largely appear to be the same, it is natural to ask: which dynamical
properties of rational maps still hold for skew products? Despite the
decomposition theorem, a skew product is _nowhere_ analytic, and under
iteration this means various algebraic techniques used to prove dynamical
results for rational maps will fail in the new setting – we are required to
think more topologically.

At first, the situation may seem discouraging, because whilst a rational map
always has finitely many fixed points in $K$, a skew product may have
uncountably many or zero classical fixed points, see Example 4.2, Example 4.4.

Skew products do end up behaving like rational maps in many important ways,
but there are marked and fascinating differences. For instance, consider
periodic points on the hyperbolic space, $\mathbb{H}$. We see that when $q<1$,
a skew product can have interior _attracting_ points $\zeta\in\mathbb{H}$ with
all directions attracting, except perhaps finitely many indifferent ones.
Moreover a Type II point could have both attracting and repelling directions -
we call these _saddle_ points. For skew products, a Type III or IV periodic
point may be repelling or attracting; see Example 4.5 for an example of a
repelling Type III point. Under a reasonable hypothesis about the value group,
we prove that Type III points are indifferent Theorem 4.8. We also resolve the
issue with Type IV points for a skew product $\phi_{*}$ with scale factor
$q\geq 1$, Theorem 4.7. More is discussed in section 4.

###  1.5. Fatou and Julia

######  Definition 1.1.

Let $\phi_{*}$ be a skew product. We say an open set
$U\subseteq\mathbb{P}^{1}_{\text{an}}$ is _dynamically stable_ under
$\phi_{*}$ iff $\displaystyle\bigcup_{n\geq 0}\phi_{*}^{n}(U)$ omits
infinitely many points of $\mathbb{P}^{1}_{\text{an}}$. The _(Berkovich) Fatou
set_ of $\phi_{*}$, denoted $\mathcal{F}_{\phi,\text{an}}$, is the subset of
$\mathbb{P}^{1}_{\text{an}}$ consisting of all points
$\zeta\in\mathbb{P}^{1}_{\text{an}}$ having a dynamically stable
neighbourhood. The _(Berkovich) Julia set_ of $\phi_{*}$ is the complement
$\mathcal{J}_{\phi,\text{an}}=\mathbb{P}^{1}_{\text{an}}\setminus\mathcal{F}_{\phi,\text{an}}$
of the Berkovich Fatou set.

We first seek to classify the periodic points of a skew product as Fatou or
Julia, generalising [Ben01a, Theorem 8.7] for rational maps which asserts that
all Type III and IV points are Fatou, and that Type II points are Julia if
they repel or have non-periodic bad directions. Unfortunately this is rather
troublesome; to begin with the above discussion, we know that Type III or IV
points could be repelling, and there might exist attracting directions and
saddle points. Fortunately, the new theorem below takes on very clean form,
perhaps even more so than the original version for rational maps, but albeit
of different flavour. It turns out that the main deciding factor is the
multiplier, and that saddle points are always Julia because they are
numerically repelling. We say a direction ${\bf v}\in
T_{\zeta}\mathbb{P}^{1}_{\text{an}}$ is _exceptional_ iff it has a finite
backward orbit when restricted to the orbit of the periodic point $\zeta$.

######  Theorem A.

Let $\phi_{*}$ be a skew product of scale factor $q$, and let
$\zeta\in\mathbb{P}^{1}_{\text{an}}$ be a periodic point of Type II, III, or
IV of period $n\geq 1$. Then $\zeta\in\mathcal{J}_{\phi,\text{an}}$ is Julia
if and only if either

  1. (i)

$\zeta$ is numerically repelling i.e. $\deg_{\zeta}(\phi)q>1$; or

  2. (ii)

$\zeta$ is Type II and either none of the directions ${\bf v}\in
T_{\zeta}\mathbb{P}^{1}_{\text{an}}$ are exceptional or any of the bad
directions ${\bf v}\in T_{\zeta}\mathbb{P}^{1}_{\text{an}}$ is not
exceptional.

Moreover if $\zeta\in\mathcal{F}_{\phi,\text{an}}$ is Fatou then every
direction intersecting $\mathcal{J}_{\phi,\text{an}}$ is exceptional and is
$\phi_{\\#}^{j}({\bf v})$ for some bad direction ${\bf v}$.

Unfortunately, $q<1$ means that fixed classical points will often be
‘superrepelling’, and worse also Fatou according to the definition of
$\mathcal{F}_{\phi,\text{an}}$; see Example 5.1. However a (non-
super-)repelling point is always Julia, as stated in section 5.

One can also extend the results of Benedetto about wandering domains of
rational maps [Ben05] to skew products. In particular, Theorem 6.14 says that
if $\phi_{*}$ is a skew product of relative degree $d\geq 2$ and scale factor
$q\geq 1$ then any wandering domain $U\subseteq\mathcal{F}_{\phi,\text{an}}$
eventually becomes a disk under iteration. If additionally our base field
$K=\mathbb{K}$ the Puiseux series and $q=1$, then the boundary points of $U$
are Type II Julia preperiodic points.

###  1.6. Classification of Fatou Components

We also take the same definitions of attracting, indifferent and wandering
Fatou component as with rational maps, see [Ben19]. We occasionally specialise
to simple ($q=1$) skew products which are defined over a discretely valued
subfield. For instance, these results will apply to simple _$k$ -rational_
skew products, meaning they are defined over the field $k((x))$ of Laurent
series with coefficients in some algebraically closed characteristic $0$ field
$k$ (the residue field).

The following theorems describing the attracting and indifferent components
are a generalisation to skew products of those due to Rivera-Letelier [RL03b,
RL03a, RL05].

######  Theorem B.

Let $\phi_{*}$ be a skew product of relative degree $d\geq 2$, let
$a\in\mathbb{P}^{1}(K)$ be an attracting periodic point of minimal period
$m\geq 1$, and let $U\subseteq\mathbb{P}^{1}_{\text{an}}$ be the immediate
attracting basin of $\phi_{*}$. Then the following hold.

  1. (i)

$U$ is a periodic Fatou component of $\mathcal{F}_{\phi,\text{an}}$, of period
$m$.

  2. (ii)

$U$ is either an open Berkovich disk or a domain of Cantor type.

  3. (iii)

If $U$ is a disk, then its unique boundary point is a Type II repelling
periodic point, of minimal period dividing $m$.

  4. (iv)

If $U$ is of Cantor type, then its boundary is homeomorphic to the Cantor set
and is contained in the Berkovich Julia set.

  5. (v)

The mapping $\phi_{*}^{m}:U\to U$ is $l$-to-$1$, for some integer $l\geq 2$.

######  Theorem C.

Let $L$ be a discretely valued subfield of $K$, and let $\phi_{*}$ be a simple
skew product defined over $L$ of relative degree $d\geq 2$. Let
$U\subseteq\mathbb{P}^{1}_{\text{an}}$ be an indifferent component for
$\phi_{*}$. Then the following hold.

  1. (a)

$U$ is a periodic connected component of $\mathcal{F}_{\phi,\text{an}}$, of
some minimal period $m\geq 1$.

  2. (b)

$U$ is a rational open connected Berkovich affinoid.

  3. (c)

Each of the finitely many points of the boundary $\partial U$ is a Type II
point lying in the Berkovich Julia set.

  4. (d)

$\phi_{*}^{m}$ permutes the boundary points of $U$; in particular, each is
periodic.

  5. (e)

The mapping $\phi_{*}^{m}:U\to U$ is bijective.

We end this section of the introduction with the most important theorem in
section 6 – we recover the classification of Fatou components, which was
originally proved for rational maps by Rivera-Letelier [RL03a, RL03b].

######  Theorem D (Classification of Fatou Components).

Let $L$ be a discretely valued subfield of $K$. Let
$\phi_{*}:\mathbb{P}^{1}_{\text{an}}(K)\to\mathbb{P}^{1}_{\text{an}}(K)$ be a
simple skew product defined over $L$ of relative degree $d\geq 2$ with
Berkovich Fatou set $\mathcal{F}_{\phi,\text{an}}$, and let
$U\subset\mathcal{F}_{\phi,\text{an}}$ be a periodic Fatou component. Then $U$
is either an indifferent component or an attracting component, but not both.

In closing the introduction, we mention that H. Nie and S. Zhao have developed
a noteworthy alternative approach to these problems and have an independent
proof of the classification of Fatou components. They also plan to release a
proof of equidistribution for skew products.

###  1.7. Organisation

We start with section 2, an overview of non-Archimedean analysis and algebra
that will be useful in practice. Next, section 3 provides a lengthy
development of the non-Archimedean skew product, from motivation to local
degrees, and other geometric results. In the final sections we will explore
the dynamics of a skew product, again comparing to its rational cousin. In
section 4 we give an elementary study of periodic points. Section 5 is
dedicated to defining and understanding the basic properties of the Fatou and
Julia set of a skew product; we examine how periodic points are related to the
Fatou-Julia dichotomy, proving Theorem A. The ultimate goal is to prove the
generalisation, Theorem D, of the Rivera-Letelier classification of Fatou
components, which will be the focus of section 6.

###  1.8. Acknowledgements

First and foremost I would like to thank my doctoral advisor, Jeffrey Diller,
for suggesting this avenue of research, for his time, encouragement, and for
sharing his remarkable editorial expertise.

Along with my advisor, I thank my dissertation committee, Nicole Looper, Eric
Riedl, and Roland Roeder, for their patience, insight, and helpful comments
which lead to a more effective exposition of new concepts.

I am grateful to Xander Faber for various helpful discussions about non-
Archimedean dynamics.

I would like to show my appreciation to Robert Benedetto for his exceptional
book which imparted a great deal of intuition and insight. His mathematical
framework for rational maps positively guided the development of skew
products.

Lastly, I thank Hongming Nie and Shengyuan Zhao for intriguing discussions
about their ‘twisted rational maps’, which are equivalent to the skew products
introduced here.

In addition, I am grateful to the NSF for its support of my work through grant
DMS-1954335.

##  2\. Background

###  2.1. Non-Archimedean Metrics

######  Definition 2.1.

A metric space $(X,d)$ is _non-Archimedean_ iff

$d(x,z)\leq\max\left\\{d(x,y),d(y,z)\right\\}\quad\forall x,y,z\in X.$

Such a $d$ is often called an _ultrametric_ , due to this _strong triangle
inequality_. In an algebraic context, we would like this to derive from a norm
which respects addition and multiplication. Furthermore, we will want to
consider something called a _seminorm_ , informally this is a norm where we
relax the condition $\left\|a\right\|=0\implies a=0$. The following notions
will be fundamentally important in this work.

######  Definition 2.2.

Let $G$ be a group. A _seminorm_ is a function
$\left\|\cdot\right\|:G\to\mathbb{R}_{\geq 0}$ such that $\left\|0\right\|=0$,
$\left\|a\right\|=\left\|-a\right\|\quad\forall a\in G$, and
$\left\|a+b\right\|\leq\left\|a\right\|+\left\|b\right\|\quad\forall a,b\in
G$.

  * •

This is a _norm_ iff additionally

$\left\|a\right\|=0\implies a=0.$

  * •

A seminorm $\left\|\cdot\right\|$ on $G$ is said to be _non-Archimedean_ iff

$\left\|a+b\right\|\leq\max\left\\{\left\|a\right\|,\left\|b\right\|\right\\}\quad\forall
a,b\in G.$

  * •

A seminorm $\left\|\cdot\right\|$ on a ring $R$ is _multiplicative_ iff

$\left\|a\cdot b\right\|=\left\|a\right\|\cdot\left\|b\right\|\quad\forall
a,b\in R.$

  * •

We say a field $(K,\left|\cdot\right|)$ is _non-Archimedean_ iff
$\left|\cdot\right|$ is a multiplicative non-Archimedean norm on $K$. In this
case we refer to $\left|\cdot\right|$ as an _absolute value_.

It is clear that any non-Archimedean norm induces a non-Archimedean metric.

######  Remark 2.1.

A _(semi)valuation_ can always be obtained by taking $\log$ of a (semi)norm,
$\log_{\varepsilon}\left\|\cdot\right\|$, for any $\varepsilon\in(0,1)$. We
apply the other adjectives of the previous definition respectively.

######  Remark 2.2.

Note that $\left\|a\right\|=\left\|-a\right\|\quad\forall a\in R$ on a ring is
implied by the multiplicative condition.

######  Example 2.1 (Trivial Norm).

Let $G$ be any group. Then there is a non-Archimedean norm
$\left\|\cdot\right\|_{\text{triv}}$ such that
$\left\|0\right\|_{\text{triv}}=0$ and $\left\|a\right\|_{\text{triv}}=1$ for
any $a\in G\setminus\left\\{0\right\\}$. If $G$ is an integral domain (e.g. a
field), this is also multiplicative.

######  Definition 2.3.

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field.

  1. (i)

The _ring of integers_ of $K$ is given by

$\mathcal{O}_{K}=\left\\{a\in K:\,\left|a\right|\leq 1\right\\}.$

  2. (ii)

This $\mathcal{O}_{K}$ has a unique maximal ideal,

$\mathcal{M}_{K}=\left\\{a\in K:\,\left|a\right|<1\right\\}.$

  3. (iii)

The _residue field_ of $K$ is the quotient field
$k=\mathcal{O}_{K}/\mathcal{M}_{K}$.

  4. (iv)

The _value group_ , $\left|K^{\times}\right|\leq(0,\infty)$ is the range of
$\left|\cdot\right|$ on $K^{\times}=K\setminus\left\\{0\right\\}$.

It is possible that $K$ has _mixed characteristic_ meaning
$\operatorname{char}K=0$ and $\operatorname{char}k=p$. Otherwise they have
_equal characteristic_ $\operatorname{char}K=\operatorname{char}k$. The value
group is a (multiplicative) subgroup of reals; if the value group is dense in
$(0,\infty)$ then we shall say it is _densely valued_ ; if this is non-trivial
but not dense in $(0,\infty)$ then it must be cyclic, i.e.
$|K^{\times}|\cong\mathbb{Z}$, in which case we say $\left|\cdot\right|$ is
_discretely valued_.

######  Proposition 2.1.

Let $(G,\left\|\cdot\right\|)$ be a non-Archimedean group. An infinite series
$\sum_{n=1}^{N}a_{n}$ is Cauchy if and only if $a_{n}\to 0$; in particular
when $(G,\left\|\cdot\right\|)$ is complete an infinite series converges if
and only if its terms tend to $0$.

######  Proposition 2.2 (Completion).

Given a non-Archimedean group $(G,\left\|\cdot\right\|)$, we can form it’s
completion $\hat{G}$. This can be thought of as all convergent series over
$G$. The induced norm is naturally given by

$\left\|\lim_{n\to\infty}a_{n}\right\|=\lim_{n\to\infty}\left\|a_{n}\right\|.$

######  Example 2.2 ($p$-adic Numbers).

Consider $\mathbb{Q}=(\mathbb{Q},|\cdot|_{p})$, the rational numbers with
$|\cdot|_{p}$ the $p$-adic norm. This is defined such that
$\left|\frac{a}{b}p^{n}\right|=p^{-n}$ if $a,b\in\mathbb{Z}$ are coprime to
$p$. One can easily check that this norm is non-Archimedean and
multiplicative. This norm also gives rational numbers a natural $p$-adic
expansion. For example when $p=5$, we can write

$\frac{42}{5}=2\cdot 5^{-1}+3\cdot 1+1\cdot 5^{1}.$

Observe that $\left|\frac{42}{5}\right|_{5}=5^{1}$ is given by

$\left|\frac{42}{5}\right|_{5}=\max\left\\{|2\cdot 5^{-1}|_{5},|3\cdot
1|_{5},|1\cdot 5^{1}|_{5}\right\\}$
$=\max\left\\{|2|_{5}|5^{-1}|_{5},|3|_{5}|1|_{5},|1|_{5}|5^{1}|_{5}\right\\}$
$=\max\left\\{|5^{-1}|_{5},|1|_{5},|5^{1}|_{5}\right\\}$
$=\max\left\\{5^{1},1,5^{-1}\right\\}.$

This is a very typical way we use the non-Archimedean property of a
(semi)norm.

The completion of $(\mathbb{Q},|\cdot|_{p})$ is
$(\mathbb{Q}_{p},|\cdot|_{p})$, the $p$_-adic numbers_. One should think of
$\mathbb{Q}_{p}$ as all the half-infinite $p$-adic expansions, i.e. ‘Laurent
series’ in the ‘variable’ $p$. The ring of integers of $\mathbb{Q}_{p}$ is the
$p$_-adic integers_ , $\mathbb{Z}_{p}$ whose elements are the ‘Maclaurin
series’ in $p$. One can check that
$\mathcal{M}_{\mathbb{Q}_{p}}=p\mathbb{Z}_{p}$. The residue field for both
$\mathbb{Q}_{p}$ and $\mathbb{Q}$ is $\mathbb{Z}/p$, meaning they have mixed
characteristic. Finally, we define $\mathbb{\mathbb{C}}_{p}$ as the completion
of $\mathbb{Q}_{p}$.

This shows that a power series structure provides a simple way to define a
non-Archimedean seminorm, using the “lowest order term”, or the lowest index.
Here is another.

######  Example 2.3 (Formal Laurent series).

Consider $k((x))=\operatorname{Frac}k[[x]]$, the field of formal Laurent
series over some base field $k$. Note that these have finite principle part.
We can define the order of vanishing norm $|\cdot|$ as follows, beginning with
a fixed $\varepsilon\in(0,1)$. For any $m_{0}\in\mathbb{Z}$ and coefficients
$(c_{m})\subset k$, let

$f(x)=\sum_{m=m_{0}}^{\infty}c_{m}x^{k}$

where $c_{m_{0}}\neq 0$. Then $\left|f\right|=\varepsilon^{m_{0}}$. We see
that the constants $k$ are all given the trivial norm and a series $f$ has its
norm determined by its lowest order term,
$\left|f\right|=\left|c_{m}x^{m}\right|=\left|x^{m}\right|$. Here the residue
field is $k$ itself, which naturally lives inside $k((x))$, hence these are of
equal characteristic. This field is discretely valued, with value group
$\left\\{\varepsilon^{n}:\,n\in\mathbb{Z}\right\\}$.

The following lemmata spell out the idea we have already used in the above
examples, that the norm of an element is always the norm of its dominant
summand.

######  Proposition 2.3 (Strong triangle equality).

Let $\left\|\cdot\right\|$ be a non-Archimedean seminorm on $G$. Then

$\left\|a\right\|\neq\left\|b\right\|\implies\left\|a+b\right\|=\max\left\\{\left\|a\right\|,\left\|b\right\|\right\\}.$

###### Proof.

$\left\|a+b\right\|\leq\max\left\\{\left\|a\right\|,\left\|b\right\|\right\\}$
by the non-Archimedean definition, hence
$\left\|a+b\right\|\leq\left\|a\right\|$. Also we have
$\left\|a\right\|=\left\|a+b+-b\right\|\leq\max\left\\{\left\|a+b\right\|,\left\|-b\right\|\right\\}=\max\left\\{\left\|a+b\right\|,\left\|b\right\|\right\\}$.
If WLOG $\left\|a\right\|>\left\|b\right\|$, it must be that
$\max\left\\{\left\|a+b\right\|,\left\|b\right\|\right\\}=\left\|a+b\right\|$.
Therefore $\left\|a+b\right\|=\left\|a\right\|$. ∎

######  Proposition 2.4 (Extended strong triangle (in)equality).

Suppose that $\sum_{j=1}^{\infty}a_{j}$ converges in
$(G,\left\|\cdot\right\|)$. Then

$\left\|\sum_{j=1}^{\infty}a_{j}\right\|\leq\max_{j}\left\|a_{j}\right\|,$

and moreover if $\left\|a_{N}\right\|>\left\|a_{j}\right\|\forall j\neq N$,
then

$\left\|\sum_{j=1}^{\infty}a_{j}\right\|=\left\|a_{N}\right\|.$

The proof follows from the previous two results. The following definition
provides not only a key example, but also one of the main fields of interest
in sequels and the author’s thesis.

######  Example 2.4 (Puiseux series).

Let $k$ be a field. We shall define $\mathbb{K}(k)=\mathbb{K}$, the field of
_Puiseux 111or Newton-Puiseux series_ over $k$ with variable $x$. For
$a=a(x)\in\mathbb{K}$ there is an $n\in\mathbb{N}$, $m_{0}\in\mathbb{Z}$ and
coefficients $(c_{m})\subset k$ such that we write

$a=\sum_{m=m_{0}}^{\infty}c_{m}x^{\frac{m}{n}}.$

Addition and multiplication works as with any power series. Moreover, these
series converge from their partial sums under the non-Archimedean metric
defined next. We fix a value of $0<|x|=\varepsilon<1$ and give $k$ the trivial
norm by setting $|c|=1\quad~{}\forall c\in k^{\times}$. This information
uniquely determines a norm on $\mathbb{K}$ in the sense of subsection 2.1:
assuming $c_{m_{0}}\neq 0$,

$|a|=|x|^{\frac{m_{0}}{n}}.$

This dependence of the power series on the denominator $n$ is the only thing
preventing $\mathbb{K}$ from being complete with respect to
$\left|\cdot\right|$. The completion of $\mathbb{K}$, the _Levi-Civita field_
$\mathbb{\hat{K}}$, is the field with elements of the form

$\gamma=\sum_{j=0}^{\infty}c_{j}x^{r_{j}},\text{ where
}(r_{j})\subset\mathbb{Q},\ r_{j}\to\infty.$

######  Theorem 2.5 (Puiseux’s Theorem).

The Puiseux series, $\mathbb{K}(\mathbb{C})$, is the algebraic closure of the
formal Laurent series $\mathbb{C}((x))$.

This is a useful field to use when working with germs of algebraic curves in
$\mathbb{C}^{2}$. _Puiseux’s Theorem_ says that any irreducible curve
$P(x,y)=0$ crossing $\left\\{x=0\right\\}$ (except the line itself) can be
given locally by branches of the form $y=\gamma(x)$ where $\gamma$ is a
Puiseux series.
linecolor=olive,backgroundcolor=olive!25,bordercolor=olive,linecolor=olive,backgroundcolor=olive!25,bordercolor=olive,todo:
linecolor=olive,backgroundcolor=olive!25,bordercolor=olive,I think the $n$ is
the degree of ramification at $x=0$. Note that $\mathbb{K}$ is the direct
limit $\varinjlim\mathbb{C}((x^{\frac{1}{n}}))$. The Levi-Civita field
$\mathbb{\hat{K}}$ is both algebraically closed and complete. When
$\operatorname{char}k=0$, the Puiseux series over $\overline{k}$,
$\mathbb{K}(k)$ is the algebraic closure of $k((x))$. However
$\overline{k((x))}$ is larger in positive characteristic.

Throughout this article, when we use Puiseux series, we will not declare an
$\varepsilon\in(0,1)$ for which $\left|x\right|=\varepsilon$ as in the
definition above. Instead, we will simply refer to the quantity
$\left|x\right|$ which intrinsically provides the same information. This also
serves as a visual reminder to the order of vanishing of a series, for
instance if $\left|a(x)\right|=\left|x\right|^{3/2}$ then the first non-zero
term of the Puiseux series $a(x)$ must be $cx^{3/2}$.

###  2.2. Disks

######  Definition 2.4.

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field222although similar can
be said for any non-Archimedean metric space. We define the open and closed
disks of radius $r$ centred at $a\in K$, respectively below.

$D(a,r):=\left\\{b\in K:\,|b-a|<r\right\\}$ $\overline{D}(a,r):=\left\\{b\in
K:\,|b-a|\leq r\right\\}$

If the radius $r$ of a disk is in the value group $|K^{\times}|$, we say this
disk (and its radius $r$) is _rational_ , otherwise, we say it is
_irrational_.

By convention we will allow notationally that
$\overline{D}(a,0)=\left\\{a\right\\}$, but not formally refer to this as a
‘disk’. The terminology of rationality follows from the notion in fields like
the Puiseux series, where the value group is
$\left|\mathbb{K}^{\times}\right|=\left|x\right|^{\mathbb{Q}}\cong\mathbb{Q}$,
however we will use these adjectives even if the value group is not isomorphic
to $\mathbb{Q}$. It follows immediately from the definition that for an
irrational radius $r$, open and closed disks coincide,
$D(a,r)=\overline{D}(a,r)$. The non-Archimedean metric results in some weird
quirks for disks. For example, consider $a,b\in K$ such that
$\left|a-b\right|=r>0$. Then in an Archimedean space, the overlap of the disks
$\overline{D}(a,r)$, $\overline{D}(b,r)$ would be a non-trivial proper subset,
much similar to the overlap of $D(a,r)$ and $D(b,r)$. In this non-Archimedean
setting we have that $\overline{D}(a,r)=\overline{D}(b,r)$ but $D(a,r)\cap
D(b,r)=\emptyset$.
linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,todo:
linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,Draw nice
picture Moreover any two disks are either disjoint or nested (or equal).
Perhaps confusingly, the rational closed disk $\overline{D}(a,r)$ is never the
closure of the rational open disk $D(a,r)$. The following proposition, lifted
from [Ben19, Proposition 2.4], details these differences.

######  Proposition 2.6.

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field.

  1. (i)

Let $a,b\in K$, and $R\geq r>0$ such that $a\in D(b,R)$. Then

$D(a,r)\subseteq D(b,R)D(a,R)=D(b,R).$

  2. (ii)

Let $a,b\in K$, and $R\geq r>0$ such that $a\in\overline{D}(b,R)$. Then

$\overline{D}(a,r)\subseteq\overline{D}(b,R)\overline{D}(a,R)=\overline{D}(b,R).$

  3. (iii)

Let $D_{1},D_{2}$ be disks such that $D_{1}\cap D_{2}\neq\emptyset$. Then
either

$D_{1}\subseteq D_{2}\text{ or }D_{1}\supseteq D_{2}$

  4. (iv)

All disks in $K$ are clopen.

  5. (v)

$K$ is totally disconnected.

######  Remark 2.3.

If $D$ is an open (resp. closed) disk then the smallest possible radius $r$
for which $D=D(a,r)$ (resp. $\overline{D}(a,r)$) is also its _diameter_ ,
$\operatorname{diam}(D)=\sup_{a,b\in D}\left|a-b\right|$. If $K$ is densely
valued, then there is a unique radius. See [Ben19, Proposition 2.5]. In
particular, whenever $K$ is algebraically closed, then
$\left|K^{\times}\right|$ is a $\mathbb{Q}$-vector space dense in
$(0,\infty)$.

It is common for non-Archimedean fields to not be _spherically complete_ \-
that is, there may be a sequence of nested closed disks

$\overline{D}(a_{1},r_{1})\supset\overline{D}(a_{2},r_{2})\supset\overline{D}(a_{3},r_{3})\supset\overline{D}(a_{4},r_{4})\supset\cdots$

with empty intersection

$\bigcap_{n=1}^{\infty}\overline{D}(a_{n},r_{n})=\emptyset.$

Of course, in a complete field, if $r_{n}\to 0$, then this intersection is
always a singleton. As an example, consider the following sequence in the
complete non-Archimedean field $\mathbb{\hat{K}}$.

$\overline{D}\left(1,\left|x^{1-1/2}\right|\right)\supset\overline{D}\left(1+x^{1-1/2},\left|x^{1-1/3}\right|\right)\supset\overline{D}\left(1+x^{1-1/2}+x^{1-1/3},\left|x^{1-1/4}\right|\right)\supset\cdots$
$\cdots\supset\overline{D}\left(\sum_{n=1}^{N-1}x^{1-1/n},\left|x^{1-1/N}\right|\right)\supset\cdots$

One can check it has a non-empty intersection if and only if the infinite
series $\sum x^{1-1/n}$ converges in $\mathbb{\hat{K}}$. The limit does not
exist! We will return to this idea when we discuss Type IV norms in the
Berkovich projective line. In any case, if the intersection of nested closed
disks is non-empty, containing say $a\in K$, then

$\bigcap_{n=1}^{\infty}\overline{D}(a_{n},r_{n})=\overline{D}(a,r),$

where $r=\lim r_{n}$ is possibly zero.

###  2.3. Projective Line and Affinoids

So far we have discussed a non-Archimedean field $K=\mathbb{A}^{1}(K)$, which
is the affine line, but in general we shall work on the projective line
$\mathbb{P}^{1}(K)=\mathbb{A}^{1}(K)\cup\left\\{\infty\right\\}$ with its
usual definition. It is natural to extend the definition of disks and their
types to one that is invariant of the fractional linear transformations
$\operatorname{PGL}(2,K)$. We shall also recall (from [Ben19, §3.5]) important
topological objects called affinoids, which are merely disks subtracted from
disks.

######  Definition 2.5.

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field. A _disk_ is any one
of the following.

  * •

A _rational closed disk_ $D\subset\mathbb{P}^{1}(K)$ is either a rational
closed disk $D\subset K$ or $D=\mathbb{P}^{1}(K)\setminus E$ where $E\subset
K$ is a rational open disk.

  * •

A _rational open disk_ $D\subset\mathbb{P}^{1}(K)$ is either a rational open
disk $D\subset K$ or $D=\mathbb{P}^{1}(K)\setminus E$ where $E\subset K$ is a
rational closed disk.

  * •

An _irrational disk_ $D\subset\mathbb{P}^{1}(K)$ is either an irrational disk
$D\subset K$ or $D=\mathbb{P}^{1}(K)\setminus E$ where $E\subset K$ is an
irrational disk.

The generalisation of subsection 2.2 (iii) would include the possibility that
two disks cover the whole space.

######  Definition 2.6.

A _connected affinoid_ is a nonempty intersection of finitely many disks
$D_{1},\dots,D_{n}$. If all of the disks $D_{1},\dots,D_{n}$ are closed, open,
rational, or irrational, then the connected affinoid $D_{1}\cap\cdots\cap
D_{n}$ is respectively said to be closed, open, rational, or irrational. The
connected open affinoid of the form
$\left\\{r<\left|z-a\right|<R\right\\}=D(a,R)\setminus\overline{D}(a,r)$ is
called an _open annulus_.

If two connected affinoids $U$ and $V$ and non-empty intersection, then both
$U\cap V$ and $U\cup V$ are connected affinoids.

###  2.4. Power Series and Constructing Seminorms

To study rational functions on a non-Archimedean field $K$, we will want to
understand them as analytic functions in neighbourhoods. Here, ‘analytic’
means given by Taylor or Laurent series, rather than some notion of
holomorphy, however it remains true that any rational function is locally
analytic. In this section we will define and recall some notions of convergent
power series for non-Archimedean fields. See [Ben19, §3].

###  2.5. Taylor Series

Following subsection 2.1 a Taylor series

$f(y)=\sum_{n=0}^{\infty}c_{n}(y-a)^{n}\in K[[y-a]]$

will converge at $y=b$ if and only if $\left|c_{n}(b-a)^{n}\right|\to 0$. Let
$\left|b-a\right|=r$, then the series converges at $b$ if and only if it
converges at every $b^{\prime}\in\overline{D}(a,r)$. This behaviour is a
little nicer than the Archimedean situation in complex analysis.

######  Definition 2.7.

The _radius of convergence_ , $R\in[0,\infty]$, for a Taylor series $f(y)$ as
above is

$R=\sup\left\\{r\in\mathbb{R}:\,\left|c_{n}\right|r^{n}\to 0\right\\}.$

The _domain of convergence_ of a Taylor series $f(y)$, is defined to be

$\operatorname{dom}(f)=\left\\{a\in K:\,f(a)\text{ converges}\right\\}.$

######  Proposition 2.7.

Let $(K,\left|\cdot\right|)$ is non-trivially and densely valued non-
Archimedean field. Let $a\in K$ and $f(y)\in K[[y-a]]$ be a Taylor series with
radius of convergence $R$. Then

$\operatorname{dom}(f)=\begin{cases}D(a,R)=\overline{D}(a,R),&\text{if
}R\notin\left|K^{\times}\right|,\text{ or otherwise}\\\ D(a,R),&\text{if
}\left|c_{n}\right|R^{n}\nrightarrow 0,\text{ and}\\\
\overline{D}(a,R),&\text{if }\left|c_{n}\right|R^{n}\to 0.\end{cases}$

######  Definition 2.8.

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field. We define the power
series rings

$\mathcal{A}(a,r)=\left\\{f\in{K[[y-a]]}:\,D(a,r)\subseteq\operatorname{dom}(f)\right\\}$
$\overline{\mathcal{A}}(a,r)=\left\\{f\in{K[[y-a]]}:\,\overline{D}(a,r)\subseteq\operatorname{dom}(f)\right\\}$

Note that the polynomials live in every power series ring, moreover $K\subset
K[y]\subset\overline{\mathcal{A}}(a,r)\subset\mathcal{A}(a,r)$.

######  Definition 2.9 (Weierstrass Degree).

Let $a\in K$,

$f(y)=\sum_{n=0}^{\infty}c_{n}(y-a)^{n}$

be a non-zero power series with radius of convergence $r>0$, and let
$D=\operatorname{dom}(f)$. The _Weierstrass degree_
$\operatorname{wdeg}_{D}(f)$ of $f$ is defined according to two cases as
follows.

  1. (i)

If $D=\overline{D}(a,r)$ is a rational closed disk, then

$\operatorname{wdeg}_{D}(f)=\max\left\\{d\in\mathbb{N}:\,\left|c_{d}\right|r^{d}=\max_{n}\left|c_{n}\right|r^{n}\right\\}.$

  2. (ii)

If $D=D(a,r)$ is an irrational or a rational open disk, then

$\operatorname{wdeg}_{D}(f)=\min\left\\{d\in\mathbb{N}:\,\left|c_{d}\right|r^{d}=\sup_{n}\left|c_{n}\right|r^{n}\right\\}\cup\left\\{\infty\right\\}.$

The Weierstrass degree plays a crucial role in understanding the zeros images
of analytic functions on a non-Archimedean field. One can easily check that
$\operatorname{wdeg}_{D}(g)+\operatorname{wdeg}_{D}(h)$ =
$\operatorname{wdeg}_{D}(gh)$ and hence any power series with a multiplicative
inverse (a unit) has Weierstrass degree $0$. The Weierstrass preparation
theorem shows that this quantity really is a ‘degree’ on $D$.

######  Theorem 2.8 (Weierstrass Preparation Theorem).

Let $(K,\left|\cdot\right|)$ be a complete non-Archimedean field, $a\in K$,
$r\in\left|K^{\times}\right|$, and $f\in\overline{\mathcal{A}}(a,r)$ be a non-
zero power series. Then there exists a monic polynomial $g\in K[y]$ of degree
$d=\operatorname{wdeg}_{\overline{D}(a,r)}(f)$ and a unit power series
$h\in\overline{\mathcal{A}}^{\times}(a,r)$ such that $f=gh$ and all the zeroes
of $g$ lie in $\overline{D}(a,r)$.

This has several consequences; immediately, we see that in an algebraically
closed complete field, such a power series $f$ has $d$ zeroes (counting
multiplicity) in the disk; secondly, moreover $f$ is a $d$-to-$1$ mapping on
$\overline{D}(0,r)$.

######  Theorem 2.9.

Let $(K,\left|\cdot\right|)$ be a complete and algebraically closed non-
Archimedean field, $a\in K$, and

$f(y)=\sum_{n=0}^{\infty}c_{n}(y-a)^{n}$

be a non-zero power series converging on a disk $D$ of radius $r>0$. Suppose
that $d=\operatorname{wdeg}_{D}(f-c_{0})<\infty$, then

  1. (i)

$f(D)$ is a disk of the same type as $D$ (rational closed, rational open, or
irrational), centred at $f(a)=c_{0}$, of radius $\left|c_{d}\right|r^{d}$; and

  2. (ii)

$f:D\to f(D)$ is a $d$-to-$1$ mapping, counting multiplicity.

###  2.6. Laurent Series

The _formal_ Laurent series $K((y-a))=\operatorname{Frac}K[[y-a]]$ represent
the set of power series in $(y-a)$ with infinitely many positive powers but
only _finitely many negative_ ones. It is not hard to see that most of the
results in the previous subsection apply to formal Laurent series on punctured
domains. To be precise, if

$f(y)=\sum_{n=-n_{0}}^{\infty}c_{n}(y-a)^{n}$

is a formal Laurent series ($n_{0}>0$) and

$f_{+}(y)=\sum_{n=0}^{\infty}c_{n}(y-a)^{n}$

converges on a disk $\overline{D}(a,r)$, then $f$ converges on
$\overline{D}^{*}(a,r)=\overline{D}(a,r)\setminus\left\\{0\right\\}$. However,
it will be useful to consider bi-infinite Laurent series when we inspect
rational maps.

For example, suppose $0<\left|a\right|<\left|b\right|$ and we want to consider
the rational map

$f(y)=\frac{1}{y-a}-\frac{1}{y-b}$

over the annulus
$U=\left\\{\left|a\right|<\left|z\right|<\left|b\right|\right\\}$. We may
expand both as series in powers of $y$ using the usual binomial trick

$(1-t)^{-1}=1+t+t^{2}+\cdots,$

but this converges when and only when $\left|t\right|<1$. Therefore on the
annulus $U$ we must expand $f$ as

$f(y)=\frac{1}{y}\frac{1}{1-\frac{a}{y}}+\frac{1}{b}\frac{1}{1-\frac{y}{b}}=\frac{1}{y}\sum_{n=0}^{\infty}\left(\frac{a}{y}\right)^{n}+\frac{1}{b}\sum_{n=0}^{\infty}\left(\frac{y}{b}\right)^{n}.$

In general, we can study rational maps through Laurent series converging on
annuli.

######  Definition 2.10.

Let $K$ be a densely valued non-Archimedean field and
$(c_{n})_{n=-\infty}^{\infty}\subset K$. A _Laurent series_ $f(y)$ about $a\in
K$ is a series of the form

$f(y)=\sum_{n\in\mathbb{Z}}c_{n}(y-a)^{n}\in K[[y-a,(y-a)^{-1}]].$

The _inner_ and _outer radii of convergence_ for $f(y)$, $r,R\in[0,\infty]$,
are defined respectively (if they exist) as

$r=\inf\left\\{s\in\mathbb{R}:\,\left|c_{n}\right|s^{n}\to 0\right\\}$
$R=\sup\left\\{s\in\mathbb{R}:\,\left|c_{n}\right|s^{n}\to 0\right\\}.$

The _domain of convergence_ of a Laurent series $f(y)$, is defined to be

$\operatorname{dom}(f)=\left\\{a\in K:\,f(a)\text{ converges}\right\\}.$

######  Proposition 2.10.

Let $(K,\left|\cdot\right|)$ be a densely valued non-Archimedean field. Let
$a\in K$ consider a Laurent series

$f(y)=\sum_{n\in\mathbb{Z}}c_{n}(y-a)^{n}\in K[[y-a,(y-a)^{-1}]].$

Then $f(y)$ will converge at $y=b$ if and only if
$\left|c_{n}(b-a)^{n}\right|\to 0$ both as $n\to\infty$ and as $n\to-\infty$.
Hence $f(y)$ converges for some $y=b$ if and only if the inner and outer radii
of convergence $r$ and $R$ both exist (with $r\leq\left|b-a\right|\leq R$). In
this case moreover the domain of convergence, $\operatorname{dom}(f)$ is one
of the following annuli

$\left\\{r<\left|z-a\right|<R\right\\}=D(a,R)\setminus\overline{D}(a,r),$
$\left\\{r\leq\left|z-a\right|<R\right\\}=D(a,R)\setminus D(a,r),$
$\left\\{r<\left|z-a\right|\leq
R\right\\}=\overline{D}(a,R)\setminus\overline{D}(a,r),$
$\left\\{r\leq\left|z-a\right|\leq R\right\\}=\overline{D}(a,R)\setminus
D(a,r),$

depending only on the boundary cases, whether $\left|c_{n}\right|r^{n}\to 0$
and/or $\left|c_{n}\right|R^{n}\to 0$.

######  Definition 2.11.

Let

$f(y)=\sum_{n\in\mathbb{Z}}c_{n}(y-a)^{n}$

be a Laurent series about $a\in K$. On any open annulus
$U=\left\\{r<\left|z-a\right|<R\right\\}\subset\operatorname{dom}(f)$ we
define

  1. (i)

the _inner Weierstrass degree_ $\overline{\operatorname{wdeg}}_{a,r}(f)$ of
$f$ to be the largest integer $M\in\mathbb{Z}$ such that
$\left|c_{M}\right|r^{M}=\sup_{n\in\mathbb{Z}}\left|c_{n}\right|r^{n}$, or
$-\infty$ if there is no such integer; and

  2. (ii)

the _outer Weierstrass degree_ $\operatorname{wdeg}_{a,R}(f)$ of $f$ to be the
smallest integer $N\in\mathbb{Z}$ such that
$\left|c_{N}\right|R^{N}=\sup_{n\in\mathbb{Z}}\left|c_{n}\right|R^{n}$, or
$\infty$ if there is no such integer.

Note that for Taylor series,
$\overline{\operatorname{wdeg}}_{a,r}(f)=\operatorname{wdeg}_{\overline{D}(a,r)}$
and $\operatorname{wdeg}_{a,r}(f)=\operatorname{wdeg}_{D(a,r)}$.

Despite the hypothesis of the definition, one can think of the inner and outer
Weierstrass degrees as a function of the radii and coefficients

$\operatorname{wdeg}_{a,R}(f)=\min\left\\{d\in\mathbb{N}:\,\left|c_{d}\right|R^{d}=\sup_{n}\left|c_{n}\right|R^{n}\right\\}\cup\left\\{\infty\right\\},$

ignorant of domains or annuli. As the radius $R$ changes for the annulus
$U=\left\\{r<\left|z-a\right|<R\right\\}$, $U$ may absorb zeroes of $f(y)$;
for each new zero (counted with multiplicity) the outer Weierstrass degree
will _increase_ by one, and on annuli without zeroes the Weierstrass degree
remains constant. This is made explicit in the proposition below, see [Ben19,
Proposition 3.32].
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result about local degrees As suggested by the example above, all rational
functions have Laurent series expansions on annuli away from their poles. If
one considers the Weierstrass degree only as a function of the rational map
and the radius, independent of the Laurent series representation. Then one can
interpret this number a count of poles and zeroes. Indeed, we can see that the
Weierstrass degree _decreases_ by one every time the radius crosses a pole of
$f(y)$.

######  Proposition 2.11.

Let $(K,\left|\cdot\right|)$ be an algebraically closed, complete non-
Archimedean field and let $f(y)\in K(y)$ be a rational function.

  1. (i)

If $f$ has no poles in $U=\left\\{r<\left|z-a\right|<R\right\\}$, an open
annulus, then $f$ has a unique Laurent series expansion converging on $U$.

  2. (ii)

Hence we may write $\overline{\operatorname{wdeg}}_{a,r}(f)$ and
$\operatorname{wdeg}_{a,R}(f)$ for the inner and outer Weierstrass degrees of
this unique series at radius $R$ about $a$. Hence these quantities are well
defined and finite for any $r,R>0$.

  3. (iii)

If $f$ has no poles or zeros in $U=\left\\{r<\left|z-a\right|<R\right\\}$,
then the inner and outer Weierstrass degrees of $f$ on $U$ coincide. i.e.
$\overline{\operatorname{wdeg}}_{r}(f)=\operatorname{wdeg}_{R}(f)$.

  4. (iv)

If $f$ has $N_{\infty}$ poles and $N_{0}$ zeros in the open disk $D(a,R)$,
then

$\operatorname{wdeg}_{a,R}(f)=N_{0}-N_{\infty}.$

  5. (v)

If $f$ has $N_{\infty}$ poles and $N_{0}$ zeros in the closed disk
$\overline{D}(a,R)$, then

$\overline{\operatorname{wdeg}}_{a,R}(f)=N_{0}-N_{\infty}.$

  6. (vi)

If $f$ has $N_{\infty}$ poles and $N_{0}$ zeros in
$U=\left\\{r<\left|z-a\right|<R\right\\}$, then

$\operatorname{wdeg}_{a,R}(f)-\overline{\operatorname{wdeg}}_{a,r}(f)=N_{0}-N_{\infty}.$

  7. (vii)

If $f$ has $N_{\infty}$ poles and $N_{0}$ zeros in the circle
$\overline{D}(a,R)\setminus D(a,R)$, then

$\overline{\operatorname{wdeg}}_{a,R}(f)-\operatorname{wdeg}_{a,R}(f)=N_{0}-N_{\infty}.$

One can extend Theorem 2.9 to the case of Laurent series and annuli as follows
(see [Ben19, Theorem 3.33]).

######  Theorem 2.12.

Let $(K,\left|\cdot\right|)$ be an algebraically closed, complete non-
Archimedean field. Let $0<r<R$, let $U=\left\\{r<\left|z-a\right|<R\right\\}$
be an open annulus, and let $f(y)$ be a non-constant convergent Laurent series
on $U\subset\operatorname{dom}(f)$. Write

$f(y)=\sum_{n\in\mathbb{Z}}c_{n}(y-a)^{n}$

and suppose that $f-c_{0}$ has finite inner and outer Weierstrass degrees
$M\leq N\in\mathbb{Z}$, respectively. Let

$s=\left|c_{M}\right|r^{M}t=\left|c_{N}\right|R^{N}.$

  1. (i)

If $M<N$, then $f(U)=D\left(c_{0},\max\left\\{s,t\right\\}\right)$.

  2. (ii)

If $M=N\geq 1$, then $f(U)=\left\\{s<\left|z-c_{0}\right|<t\right\\}$, and the
mapping $f:U\to f(U)$ is $M$-to-$1$.

  3. (iii)

If $M=N\leq-1$, then $f(U)=\left\\{t<\left|z-c_{0}\right|<s\right\\}$, and the
mapping $f:U\to f(U)$ is $(-M)$-to-$1$.

It follows that in the last two cases,
$\left|f(z)-c_{0}\right|=\left|c_{M}(z-a)^{M}\right|$, for any $z\in U$.

Finally, we recall a description of how a rational map acts on affinoids, this
is lifted from [Ben19, Proposition 3.29].

######  Theorem 2.13.

Let $(K,\left|\cdot\right|)$ be an algebraically closed, complete non-
Archimedean field, and $U\subseteq\mathbb{P}^{1}(K)$ be a connected affinoid.
Let $f(y)\in K(y)$ be a rational function of degree $d\geq 1$. Then

  1. (i)

$f(U)$ is either $\mathbb{P}^{1}(K)$ or a connected affinoid of the same type,
if any, as $U$, and

  2. (ii)

$f^{-1}(U)$ is a disjoint union $V_{1}\cup\cdots\cup V_{m}$ of connected
affinoids, each of the same type, if any, as $U$, and with $1\leq m\leq d$.
Moreover, for each $i=1,\dots,m$, there is an integer $1\leq d_{i}\leq d$ such
that every point in $U$ has exactly $d_{i}$ preimages in $V_{i}$, counting
multiplicity, and such that $d_{1}+\cdots+d_{m}=d$.

###  2.7. Seminorms of Power Series

Power series rings $K[[y-a]]$ can be equipped with many different non-
Archimedean seminorms, but we shall focus on those which agree with the
absolute value on $K$. These shall constitute the seminorms of the Berkovich
affine line. Whenever we define a seminorm $\left\|\cdot\right\|_{\zeta}$,
notationally we will use $\left\|\cdot\right\|_{\zeta}$ and $\zeta$
interchangeably. This will make more sense after defining the Berkovich affine
line which contains $K$ in the form of Type I points.

######  Definition 2.12 (Type I Seminorm).

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field and $a\in K$. We
define a function called a _Type I seminorm_
$\left\|\cdot\right\|_{a}:K[[y-a]]\to[0,\infty)$ by

$\left\|f\right\|_{a}=\left|f(a)\right|.$

######  Proposition 2.14.

Let $a\in K$, then $\left\|\cdot\right\|_{a}$ is a well defined non-
Archimedean, multiplicative seminorm on $K[[y-a]]\supset K[y]$, which extends
the norm $\left|\cdot\right|$ on $K$, i.e.
$\left\|c\right\|_{a}=\left|c\right|\quad\forall c\in K$. However it is never
a norm since $\left\|y-a\right\|_{a}=0$.

######  Definition 2.13 (Type II/III Norm).

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field, $a\in K$, $0<R$. We
define the ring

$\mathcal{L}(a,R)=\left\\{f\in{K[[y-a,(y-a)^{-1}]]}:\,\operatorname{wdeg}_{a,R}(f)<\infty\right\\}.$

We also define a function
$\left\|\cdot\right\|_{\zeta(a,R)}:\mathcal{L}(a,r)\to[0,\infty)$ by

$\left\|f\right\|_{\zeta(a,R)}=\left|c_{d}\right|R^{d}$

where

$f(y)=\sum_{n=0}^{\infty}c_{n}(y-a)^{n}\in\mathcal{L}(a,R),\quad\operatorname{wdeg}_{a,R}(f)=d.$

We say $\zeta(a,R)$ is a Type II or Type III norm if $R$ is rational or not,
respectively.

######  Proposition 2.15.

Let $a\in K$ and $r>0$. Then $\left\|\cdot\right\|_{\zeta(a,r)}$ is a well
defined non-Archimedean, multiplicative norm on
$\mathcal{L}(a,r)\supset\overline{\mathcal{A}}(a,r)\supset K[y]$ which extends
the norm $\left|\cdot\right|$ on $K$. Hence, for any disk
$D(b,R)\supsetneq\overline{D}(a,r)$, $\left\|\cdot\right\|_{\zeta(a,r)}$ is a
norm on $\overline{\mathcal{A}}(b,R)$ and $\mathcal{A}(b,R)$, moreover

$\left\|f\right\|_{\zeta(a,r)}\leq\left\|f\right\|_{\zeta(b,R)}\quad\forall
f\in\overline{\mathcal{A}}(b,R).$

If $f\in\overline{\mathcal{A}}(a,R)$ or more generally $f$ is a Laurent series
converging on a closed annulus
$E=\left\\{R-\varepsilon\leq\left|z-a\right|\leq R\right\\}$, i.e.
$\left|c_{n}\right|R^{n}\to 0$ as $n\to\pm\infty$, then the supremum
$\sup_{n}\left|c_{n}\right|R^{n}$ is attained at some $d\in\mathbb{Z}$ and
thus the outer Weierstrass degree of $f$ is finite. Therefore
$\mathcal{L}(a,r)\supset\overline{\mathcal{A}}(a,r)$. However if $f$ converges
on $D(a,r)$ but not $\overline{D}(a,r)$ then the sequence
$\left|c_{n}\right|r^{n}$ may not attain its supremum or be bounded.

By subsection 2.6, any rational function $f(y)$ for any radius $R$ has a
Laurent expansion on $U=\left\\{R-\varepsilon<\left|z-a\right|<R\right\\}$,
hence the above definition works for all rational functions $f\in
K(y)\subset\mathcal{L}(a,r)$. One could also consider the opposite annulus
$\left\\{R<\left|z-a\right|<R+\varepsilon\right\\}$ and try make a similar
definition using the inner Weierstrass degree; or one could pick a different
centre $b\in\overline{D}(a,R)$. These all turn out to be equal. Further, the
Type II/III norm $\left\|\cdot\right\|_{\zeta(a,r)}$ is actually a ‘sup-norm’
on $\overline{\mathcal{A}}(a,r)$. This is showcased in [Ben19] but we shall
state and prove a little more.
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but doubt this is really
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######  Proposition 2.16.

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field, $a\in K$, $r>0$, and
$f(y)$ be a non-constant rational function. Pick $\varepsilon>0$ such that $f$
has no poles on $V=\left\\{R<\left|z-a\right|<R+\varepsilon\right\\}$ and
write

$f(y)=\sum_{n\in\mathbb{Z}}c_{n}(y-a)^{n},\quad\overline{\operatorname{wdeg}}_{a,R}(f)=d.$

Then

$\left\|f\right\|_{\zeta(a,R)}=\left|c_{d}\right|R^{d}=\lim_{\left|b\right|\to
R}\left|f(b)\right|.$

Where the limit excludes $\left|b\right|=R$. Moreover,
$\left\|f\right\|_{\zeta(a,R)}=\left|f(b)\right|$ for every $b$ in all but
finitely many residue classes of $\overline{D}(a,R)$, i.e. avoiding any open
disks $D(b,R)\subset\overline{D}(a,R)$ containing zeroes or poles of $f$.
Furthermore, $\zeta(a,R)$ depends only on the choice of closed disk
$\overline{D}(a,R)$. That is for any $b\in\overline{D}(a,R)$ we have that

$\zeta(a,R)=\zeta(b,R).$

This is mostly remarkable because the values
$\overline{\operatorname{wdeg}}_{a,R}(f)$ and $\operatorname{wdeg}_{a,R}(f)$
could be different, deriving from distinct Laurent series. The result derives
from the following:

######  Proposition 2.17.

Let $(K,\left|\cdot\right|)$ be a densely valued non-Archimedean field, $a\in
K$, $r>0$, and consider the non-zero Taylor series

$f(y)=\sum_{n=0}^{\infty}c_{n}(y-a)^{n}.$

  1. (i)

If $f$ converges on $D=D(a,r)$ with Weierstrass degree
$\operatorname{wdeg}_{a,r}(f)=d$, then

$\left\|f\right\|_{\zeta(a,r)}=\left|c_{d}\right|r^{d}=\sup_{b\in
D}\left|f(b)\right|=\sup_{b\in
D}\left\|f\right\|_{b}=\lim_{\left|b\right|\nearrow r}\left\|f\right\|_{b}.$

  2. (ii)

If $f$ converges on $E=\overline{D}(a,r)$ with Weierstrass degree
$\overline{\operatorname{wdeg}}_{a,r}(f)=\overline{d}$, then

$\left\|f\right\|_{\zeta(a,r)}=\left|c_{\overline{d}}\right|r^{\overline{d}}=\sup_{b\in
E}\left|f(b)\right|=\sup_{b\in E}\left\|f\right\|_{b}.$

  3. (iii)

If $f$ converges on $D(a,R)\supsetneq\overline{D}(a,r)$, then

$\left\|f\right\|_{\zeta(a,r)}=\left|c_{\overline{d}}\right|r^{\overline{d}}=\lim_{\left|b\right|\searrow
r}\left\|f\right\|_{b}.$

  4. (iv)

Moreover, $\left\|f\right\|_{\zeta(a,r)}=\left|f(b)\right|$ for every $b$ in
all but finitely many residue classes of $\overline{D}(a,r)$, to be precise we
could pick any $b\in\overline{D}(a,r)\setminus(D(a_{1},r)\cup\cdots\cup
D(a_{\overline{d}},r))$, where $(a_{j})$ are the finitely many solutions to
$f(y)=0$.

Furthermore, $\zeta(a,R)$ depends only on the choice of closed disk
$\overline{D}(a,R)$. That is for any $b\in\overline{D}(a,R)$ we have that

$\zeta(a,R)=\zeta(b,R).$

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move to a later part. Be careful about $z^{p}-z$. Degree less than res char
should be enough. Add an equivalence using
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###### Proof.

Since $f\in\mathcal{L}(a,r)$, we have by definition
$\left\|f\right\|_{\zeta(a,r)}=\left|c_{d}\right|r^{d}$. Since $f$ has
Weierstrass degree $d=\operatorname{wdeg}_{a,r}(f)$, for $n>d$ we have that
$\left|c_{n}\right|r^{n}\leq\left|c_{d}\right|r^{d}$ and for $n<d$ we have
$\left|c_{n}\right|r^{n}<\left|c_{d}\right|r^{d}$. By rearranging the former
we get that $\left|c_{n}\right|r^{n-d}\leq\left|c_{d}\right|$ and hence for
any $s<r$ we have $\left|c_{n}\right|s^{n-d}<\left|c_{d}\right|$ which implies
$\left|c_{n}\right|s^{n}<\left|c_{d}\right|s^{d}$ for every $n>d$. On the
other hand, for the finitely many $0\leq n<d$ we can use continuity to find an
$s_{0}<r$ large enough such that for every $s\in(s_{0},r)$ the latter
inequality remains true $\left|c_{n}\right|s^{n}<\left|c_{d}\right|s^{d}$.
Suppose that $b\in D(a,r)$, specifically with $\left|b-a\right|=s<r$. Then by
subsection 2.1

$\left|f(b)\right|=\left|\sum_{n=0}^{\infty}c_{n}(b-a)^{n}\right|\leq\max_{n}\left|c_{n}\right|\left|b-a\right|^{n}=\max_{n}\left|c_{n}\right|s^{n}\leq\max_{n}\left|c_{n}\right|r^{n}=\left|c_{d}\right|r^{d},$

and moreover if $s_{0}<s<r$, then
$\left|f(b)\right|=\left|c_{d}\right|\left|b-a\right|^{d}=\left|c_{d}\right|s^{d}$,
so as $s\nearrow r$, we have $\left|f(b)\right|\to\left|c_{d}\right|r^{d}$.
Items (ii), (iii) can be proven similarly.

It is enough to show (iv) in the completion of the algebraic closure of $K$,
since disregarding finitely many disks in a field extension will do the same
in $K$. If $\overline{d}=0$, then by subsection 2.1
$\left|f(b)\right|=\left|c_{0}\right|$ for every $b\in\overline{D}(a,r)$.
Otherwise, $\overline{d}=\overline{\operatorname{wdeg}}_{a,r}(f-c_{0})$ and
hence $\left|c_{0}\right|\leq\left\|f\right\|_{\zeta(a,r)}$. By Theorem 2.9 we
have that
$f(\overline{D}(a,r))=\overline{D}\left(c_{0},\left\|f\right\|_{\zeta(a,r)}\right)=\overline{D}\left(0,\left\|f\right\|_{\zeta(a,r)}\right)$,
$f(D(b,r))=D\left(f(b),\left\|f\right\|_{\zeta(a,r)}\right)$, and there are at
most $\overline{d}$ such open disks $D(a_{1},r),\dots,D(a_{\overline{d}},r)$
whose image contains $0$, i.e.
$f(D(a_{j},r))=D\left(0,\left\|f\right\|_{\zeta(a,r)}\right)$. Hence for any
$b$ _not_ in such a disk, we find
$f(b)\in\overline{D}\left(0,\left\|f\right\|_{\zeta(a,r)}\right)\setminus
D\left(0,\left\|f\right\|_{\zeta(a,r)}\right)$ and so
$\left|f(b)\right|=\left\|f\right\|_{\zeta(a,r)}$. ∎

Observe that if one considers the definition of Type II/III norm $\zeta(a,R$)
with $R=0$ we recover the Type I _semi_ norm. We make one last definition of a
seminorm; later we will see this is necessary to complete the Berkovich line.

######  Definition 2.14 (Type IV Norm).

Let $(K,\left|\cdot\right|)$ be a non-Archimedean field, and suppose the
following nested sequence of disks has empty intersection.

$\overline{D}(a_{1},r_{1})\supset\overline{D}(a_{2},r_{2})\supset\overline{D}(a_{3},r_{3})\supset\overline{D}(a_{4},r_{4})\supset\cdots$

Also let

$\mathcal{A}(\zeta)=\bigcup_{n}\overline{\mathcal{A}}(a_{n},r_{n}).$

We define a function
$\left\|\cdot\right\|_{\zeta}:\mathcal{A}(\zeta)\to[0,\infty)$ by

$\left\|f\right\|_{\zeta}=\inf_{n\geq N}\left\|f\right\|_{\zeta(a_{n},r_{n})}$

where $f\in\overline{\mathcal{A}}(a_{N},r_{N})\subset\mathcal{A}(\zeta)$. We
call $\zeta$ the _Type IV norm_ associated to the sequence
$(\overline{D}(a_{n},r_{n}))$.

Note that subsection 2.7 says that the sequence
$\left\|f\right\|_{\zeta(a_{n},r_{n})}$ above is decreasing, so this infimum
is a limit. Moreover for a fixed $f$, one can show that for large enough $n$,
$f\in\overline{\mathcal{A}}^{\times}(a_{n},r_{n})$ is a unit, and so the
sequence $\left\|f\right\|_{\zeta(a_{n},r_{n})}$ is eventually constant.
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propConsider a rational function $f(y)$. We know that $f$ has finitely many
poles, so for some large $N$, these poles lie outside of
$\overline{D}(a_{n},r_{n})$, hence $f\in\overline{\mathcal{A}}(a_{n},r_{n})$.
Therefore $K(y)\subset\mathcal{A}(\zeta)$ for every Type IV $\zeta$.

######  Definition 2.15.

We say the sequences

$\overline{D}(a_{1},r_{1})\supset\overline{D}(a_{2},r_{2})\supset\overline{D}(a_{3},r_{3})\supset\overline{D}(a_{4},r_{4})\supset\cdots$

and

$\overline{D}(b_{1},s_{1})\supset\overline{D}(b_{2},s_{2})\supset\overline{D}(b_{3},s_{3})\supset\overline{D}(b_{4},s_{4})\supset\cdots$

are _equivalent_ if for any $n\in\mathbb{N}$ we can find an $N\in\mathbb{N}$
such that $\overline{D}(a_{n},r_{n})\supset\overline{D}(b_{N},s_{N})$ and vice
versa.

######  Proposition 2.18.

Type IV norms are non-Archimedean multiplicative norms on
$\mathcal{A}(\zeta)\supset K(y)$. If $\zeta$ and $\zeta^{\prime}$ are Type IV
norms associated to equivalent nested sequences of disks, then
$\zeta=\zeta^{\prime}$.

linecolor=red,backgroundcolor=red!25,bordercolor=red,linecolor=red,backgroundcolor=red!25,bordercolor=red,todo:
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inverses? Maybe need to think about including trivial residue field.

###  2.8. The Berkovich Projective Line

It is an easy exercise to show that given any ring $A$, a non-Archimedean
multiplicative norm $\left\|\cdot\right\|$ on $A$ extends to a non-Archimedean
norm on $\operatorname{Frac}A$ by
$\left\|a/b\right\|=\left\|a\right\|/\left\|b\right\|$. This provides a
simpler proof without annuli of domains of convergence, as in the last
section, that the Type II, III, and IV norms automatically extend to $K(y)$.
Hence the set of non-Archimedean multiplicative norms on $K[y]$ is the same as
those on $K(y)$, and it would be natural to define the Berkovich line simply
as the set of such norms on $K(y)$. It would be easy to define the action of a
rational map $\phi$ on this space of norms by
$\left\|f(y)\right\|_{\phi_{*}(\zeta)}=\left\|f\circ\phi(y)\right\|_{\zeta}$,
because $f\in K(y)\implies f\circ\phi\in K(y)$. Unfortunately, this
construction fails in the case of Type I _seminorms_ , since
$\left\|f/g\right\|_{a}$ is infinite when $g(a)=0$. One of the important
reasons to define the Berkovich projective line
$\mathbb{P}^{1}_{\text{an}}(K)$ is that we would like a complete and compact
space (indeed to compactify $\mathbb{P}^{1}(K)$), and the Type I points, seen
as $\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$ will play a key
role as (limiting) endpoints in the resulting tree. Since the Type I points
are ill-defined on $K(y)$, we shall define the Berkovich affine line as
seminorms over $K[y]$. However the reader is encouraged to bear in mind that
the Type II, III, and IV points in the space are norms on $K(y)$, and Type I
norms can be treated as a special case.

Throughout the remainder of this section let $K$ denote an algebraically
closed field and $\mathbb{\mathbb{C}}_{v}$ a complete and algebraically closed
field, which in most cases will be the completion of the former
$\mathbb{\mathbb{C}}_{v}=\widehat{K}$.

###  2.9. The Berkovich Affine Line

######  Definition 2.16.

The Berkovich affine line
$\mathbb{A}^{1}_{\text{an}}=\mathbb{A}^{1}_{\text{an}}(K)$ is the set of non-
Archimedean multiplicative seminorms on $K[y]$ extending
$(K,\left|\cdot\right|)$. (Meaning
$\left\|a\right\|=\left|a\right|\quad\forall a\in K$.) A topology is given to
$\mathbb{A}^{1}_{\text{an}}(K)$ by taking the coarsest topology for which
$\left\|\cdot\right\|_{\zeta}\mapsto\left\|f\right\|_{\zeta}$ is continuous
for every $f\in K[y]$.

We often refer to the elements
$\zeta\in\mathbb{A}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})$ (or
$\mathbb{P}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})$) as _points_ , but we
shall write $\left\|\cdot\right\|_{\zeta}$ when we think of $\zeta$ as a
seminorm. It turns out that $\mathbb{A}^{1}_{\text{an}}$ is a tree; one way to
see this is through its poset structure which we define now and expand upon
later.

######  Definition 2.17.

We define a partial order $\preceq$ on $\mathbb{A}^{1}_{\text{an}}$ by

$\zeta\preceq\xi\iff\left\|f\right\|_{\zeta}\leq\left\|f\right\|_{\xi}\quad\forall
f\in\mathbb{\mathbb{C}}_{v}[y].$

######  Definition 2.18.

Let $\zeta\in\mathbb{A}^{1}_{\text{an}}$. Its _absolute value_ is

$\left|\zeta\right|=\left\|y\right\|_{\zeta}.$

Its _diameter_ is

$\operatorname{diam}(\zeta)=\inf_{a\in\mathbb{\mathbb{C}}_{v}}\left\|y-a\right\|_{\zeta}.$

It is clear that diameter exists and
$\operatorname{diam}(\zeta)\leq\left|\zeta\right|$. Some key examples here are
that $\operatorname{diam}(\zeta(a,r))=r$,
$\left|\zeta(a,r)\right|=\min\left\\{\left|a\right|,r\right\\}$,
$\operatorname{diam}(a)=0$, and $\left|a\right|=\left|a\right|$.

######  Definition 2.19.

We define the _open_ and _closed Berkovich disks_ of radius $r>0$ centred at
$a\in K$, respectively below.

$D_{\text{an}}(a,r):=\left\\{\zeta\in\mathbb{A}_{\text{an}}(K):\,\left\|y-a\right\|_{\zeta}<r\right\\}$
$\overline{D}_{\text{an}}(a,r):=\left\\{\zeta\in\mathbb{A}_{\text{an}}(K):\,\left\|y-a\right\|_{\zeta}\leq
r\right\\}$

If $r\in|K^{\times}|$, we say this disk is _rational_ , otherwise, we say it
is _irrational_.

There is a natural inclusion of
$K=\mathbb{A}^{1}(K)\subset\mathbb{A}^{1}_{\text{an}}(K)$; recall that for
every $a\in K$ we have the Type I seminorm $\left\|\cdot\right\|_{a}$. We call
these the _classical points_ of $\mathbb{A}^{1}_{\text{an}}$, again referring
to $a$ as a ‘point’ of $\mathbb{A}^{1}_{\text{an}}$. Furthermore we can
consider the classical disk $D(a,r)$ as a subset of
$\mathbb{A}^{1}_{\text{an}}$. In the following proposition and throughout, we
allow for radii to be zero where $D_{\text{an}}(a,r)=\emptyset$ and
$\overline{D}_{\text{an}}(a,r)=\left\\{a\right\\}$, although these are not
‘disks’.

######  Proposition 2.19.

The following hold.

  * •

$D_{\text{an}}(a,r)$ is open and $\overline{D}_{\text{an}}(a,r)$ is closed.

  * •

$\displaystyle D(a,r)=\mathbb{A}^{1}(K)\cap D_{\text{an}}(a,r)$

  * •

$\displaystyle\overline{D}(a,r)=\mathbb{A}^{1}(K)\cap\overline{D}_{\text{an}}(a,r)$

  * •

$\displaystyle D_{\text{an}}(a,r)\subseteq D_{\text{an}}(b,s)\iff
D(a,r)\subseteq D(b,s)$

  * •

$\displaystyle\overline{D}_{\text{an}}(a,r)\subseteq\overline{D}_{\text{an}}(b,s)\iff\overline{D}(a,r)\subseteq\overline{D}(b,s)$

  * •

$\displaystyle\zeta(a,r)\in D_{\text{an}}(b,s)\iff\overline{D}(a,r)\subseteq
D(b,s)$

  * •

$\displaystyle\zeta(a,r)\in\overline{D}_{\text{an}}(b,s)\iff\overline{D}(a,r)\subseteq\overline{D}(b,s)$

  * •

$\displaystyle\zeta(a,r)\succeq\xi\quad\iff\xi\in\overline{D}_{\text{an}}(a,r)$

  * •

$\displaystyle\overline{D}_{\text{an}}(a,r)=\left\\{\zeta(a,r)\right\\}\
\cup\bigcup_{b\in\overline{D}(a,r)}D_{\text{an}}(b,r)$.
Hence $\zeta(a,r)$ is the unique boundary point of
$\overline{D}_{\text{an}}(a,r)$.

  * •

$\displaystyle\operatorname{diam}(\xi)\leq
r\quad\forall\xi\in\overline{D}_{\text{an}}(a,r)$ with equality if and only if
$\xi=\zeta(a,r)$.

  * •

$\displaystyle\zeta\in\overline{D}_{\text{an}}(a,r)\implies\left\|y-b\right\|_{\zeta}=\left|a-b\right|$
for every $b\notin\overline{D}(a,r)$.
Hence if $\zeta,\xi\in\overline{D}_{\text{an}}(a,r)$ then $\zeta=\xi$ if and
only if $\left\|y-b\right\|_{\zeta}=\left\|y-b\right\|_{\xi}$ for every
$b\in\overline{D}(a,r)$.

We see some clear differences compared to classical disks. For irrational
$r\notin\left|K^{\times}\right|$, whereas we had $\overline{D}(a,r)=D(a,r)$,
now there is a distinction
$\overline{D}_{\text{an}}(a,r)=\left\\{\zeta(a,r)\right\\}\cup
D_{\text{an}}(a,r)$. Actually it is for these reasons that
$\mathbb{A}^{1}_{\text{an}}$ is connected, but $\mathbb{A}^{1}(K)$ was not.
Indeed for any $a,r$, $D_{\text{an}}(a,r)$ is never closed and
$\overline{D}_{\text{an}}(a,r)$ is never open.

In the previous section we laid out four types of seminorm, with some
equivalent definitions. The following theorem of Berkovich [Ber90] says these
are the only four.

######  Theorem 2.20 (Berkovich’s Classification).

Let $\zeta\in\mathbb{A}^{1}(\mathbb{\mathbb{C}}_{v})$. Then $\zeta$ is of
exactly one of the following four
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  1. I:

$\left\|\cdot\right\|_{\zeta}=\left\|\cdot\right\|_{a}$ for some unique
$a\in\mathbb{\mathbb{C}}_{v}$.

  2. II:

$\left\|\cdot\right\|_{\zeta}=\left\|\cdot\right\|_{\zeta(a,r)}$ corresponding
to a unique rational closed disk
$\overline{D}(a,r)\subset\mathbb{\mathbb{C}}_{v}$.

  3. III:

$\left\|\cdot\right\|_{\zeta}=\left\|\cdot\right\|_{\zeta(a,r)}$ corresponding
to a unique irrational disk $\overline{D}(a,r)\subset\mathbb{\mathbb{C}}_{v}$.

  4. IV:

$\left\|\cdot\right\|_{\zeta}=\lim_{n\to\infty}\left\|\cdot\right\|_{\zeta_{n}}$
where $\zeta_{n}=\zeta(a_{n},r_{n})$ corresponds to a decreasing nested
sequence of closed disks $\overline{D}(a_{n},r_{n})$ with empty intersection
in $\mathbb{\mathbb{C}}_{v}$. The sequence is unique up to equivalence as in
Definition 2.15.

###### Sketch Proof.

Following subsection 2.7 and subsection 2.7 there is little more to say about
uniqueness. The trick to classification is the following:

  * •

Any $\zeta$ is contained in a nested sequence of closed Berkovich disks
$\overline{D}_{\text{an}}(a_{n},r_{n})$ with
$r_{n}\to\operatorname{diam}(\zeta)$; starting with
$\zeta\in\overline{D}_{\text{an}}(0,r_{0})$ for
$r_{0}=\left|\zeta\right|=\left\|y\right\|_{\zeta}$.

  * •

$\left\|\cdot\right\|_{\zeta}\leq\lim_{n\to\infty}\left\|\cdot\right\|_{\zeta_{n}}$
where $\zeta_{n}=\zeta(a_{n},r_{n})$. We will prove equality.

  * •

If the limit of disks is non-empty then
$\bigcap_{n}\overline{D}(a_{n},r_{n})=\overline{D}(a,r)$ (with $r$ possibly
$0$) and therefore $\zeta=\zeta(a,r)$ because
$\zeta\in\overline{D}_{\text{an}}(a,r)$ and has diameter $r$ (by subsection
2.9).

  * •

If $r_{n}\to 0$ then the limit $\bigcap_{n}\overline{D}(a_{n},r_{n})$ is the
single point $\left\\{a\right\\}$ and $\zeta=a$ is Type I.

  * •

Otherwise $\zeta=\zeta(a,r)$ is Type II iff
$r\in\left|\mathbb{\mathbb{C}}_{v}^{\times}\right|$ and Type III iff
$r\notin\left|\mathbb{\mathbb{C}}_{v}^{\times}\right|$.

  * •

If $\bigcap_{n}\overline{D}(a_{n},r_{n})=\emptyset$, then $\zeta$ is the Type
IV point $\lim_{n\to\infty}\zeta_{n}$ associated with this sequence. Indeed,
both are in $\overline{D}_{\text{an}}(a_{n},r_{n})$ for every $n$ so they
agree on $y-b$ for every $b$ outside of
$\bigcap_{n}\overline{D}(a_{n},r_{n})$, by the last part of subsection 2.9.

∎

###  2.10. The Berkovich Projective Line

The projective line of a field $\mathbb{P}^{1}$ is often defined as
$\mathbb{A}^{1}\cup\left\\{\infty\right\\}$, or more rigorously as two
$\mathbb{A}^{1}$ affine lines glued over
$\mathbb{A}^{1}\setminus\left\\{0\right\\}$ using the transition map $z\mapsto
1/z$. Here we can do either for the Berkovich projective line. In order for
this transition map to work need to extend seminorms on $K[y]$ to
$K\left[y,y^{-1}\right]$. Fortunately, this is not a big problem. For any
$f\in K[y,y^{-1}]$ we can write $f(y)=g(y)/y^{d}$ with $g\in K[y]$ and some
$d\in\mathbb{N}$. Every seminorm
$\zeta\in\mathbb{A}^{1}_{\text{an}}\setminus\left\\{0\right\\}$ can be defined
on $f$ by

$\left\|f(y)\right\|_{\zeta}=\left\|g(y)/y^{d}\right\|_{\zeta}=\left\|g(y)\right\|_{\zeta}/\left\|y^{d}\right\|_{\zeta}$

since $\left\|y\right\|_{\zeta}\neq 0$. Conversely, any seminorm on
$K[y,y^{-1}]$ is in $\mathbb{A}^{1}_{\text{an}}\setminus\left\\{0\right\\}$.

######  Definition 2.20.

We define the _Berkovich projective line_
$\mathbb{P}^{1}_{\text{an}}=\mathbb{P}^{1}_{\text{an}}(K)$ with two charts
given by $\mathbb{A}^{1}_{\text{an}}(K)$ using the homeomorphism

$\mathbb{A}^{1}_{\text{an}}(K)\setminus\left\\{0\right\\}\to\mathbb{A}^{1}_{\text{an}}(K)\setminus\left\\{0\right\\}$

given by

$\left\|f(y)\right\|_{\frac{1}{\zeta}}=\left\|f\left(\frac{1}{y}\right)\right\|_{\zeta}.$

######  Remark 2.4.

We will see later that this transition map naturally extends to an involution
$\zeta\mapsto 1/\zeta$ of $\mathbb{P}^{1}_{\text{an}}$ with the point $\infty$
defined as $1/0$. We also refer to $\infty$ as a (Type I) classical point and
part of the natural inclusion
$\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$. With this hindsight,
we can expand the definition of the classical points $a\in\mathbb{P}^{1}$ to
evaluate elements of $K(y)$ as follows: for any $f\in K(y)$ we define
$\left\|f\right\|_{a}=\left|f(a)\right|$, allowing for
$\left\|f\right\|_{a}=\infty$ if and only if $f(a)=\infty$. Finally, we can
make sense of $\infty=1/0$:

$\left|f(\infty)\right|=\left\|f(y)\right\|_{\infty}=\left\|f(y)\right\|_{\frac{1}{0}}=\left\|f\left(\frac{1}{y}\right)\right\|_{0}.$

For more, see Benedetto [Ben19, §6].

The most ‘central’ point in $\mathbb{P}^{1}_{\text{an}}$ is the Type II point
$\zeta(0,1)$, we call this the _Gauss point_. This is mostly because the ring
of integers $\mathcal{O}_{K}$ corresponds to $\overline{D}(0,1)$ and the
residue classes $\overline{b}\in k=\mathcal{O}_{K}/\mathcal{M}_{K}$ correspond
to the open disks $D(b,1)\subset\overline{D}(0,1)$. The corresponding
Berkovich open disks will play a significant role in analysing maps etc. Often
when considering a Type II point $\zeta(a,r)$, it will be useful to find a
$\operatorname{PGL}(2,\mathbb{\mathbb{C}}_{v})$ transformation, changing
coordinates in $\mathbb{P}^{1}_{\text{an}}$, such that $\zeta(a,r)$ is moved
to $\zeta(0,1)$.

###  2.11. Berkovich Disks, Affinoids, and Directions

We further extend our definitions of Berkovich disks and affinoids.

######  Definition 2.21.

A _Berkovich disk_ in $\mathbb{P}^{1}_{\text{an}}$ is any one of the
following.

  * •

A _closed Berkovich disk_ $D\subset\mathbb{P}^{1}_{\text{an}}$ is either
$\overline{D}_{\text{an}}(a,r)\subset\mathbb{A}^{1}_{\text{an}}$ or
$\mathbb{P}^{1}_{\text{an}}\setminus D_{\text{an}}(a,r)$.

  * •

A _open Berkovich disk_ $D\subset\mathbb{P}^{1}_{\text{an}}$ is either
$D_{\text{an}}(a,r)\subset\mathbb{A}^{1}_{\text{an}}$ or
$D=\mathbb{P}^{1}_{\text{an}}\setminus\overline{D}_{\text{an}}(a,r)$.

  * •

A disk is _rational_ if $r\in\left|\mathbb{\mathbb{C}}_{v}\right|$ and
_irrational_ otherwise.

######  Definition 2.22.

A _connected Berkovich affinoid_ is a nonempty intersection of finitely many
Berkovich disks $D_{1},\dots,D_{n}$. If all of the disks $D_{1},\dots,D_{n}$
are closed, open, rational, or irrational, then the connected affinoid
$D_{1}\cap\cdots\cap D_{n}$ is respectively said to be closed, open, rational,
or irrational.

The connected open affinoid of the form

$\left\\{\zeta\in\mathbb{A}^{1}_{\text{an}}:\,r<\left\|y-a\right\|_{\zeta}<R\right\\}=D_{\text{an}}(a,R)\setminus\overline{D}_{\text{an}}(a,r)$

is called an _open annulus_. We will often abuse notation and write this as

$\left\\{r<\left|\zeta-a\right|<R\right\\},$

distinguished from the classical annulus by the use of the Greek $\zeta$
instead of the Roman $z$.

A _Berkovich affinoid_ is a finite union of connected Berkovich affinoids. We
may apply the usual adjectives as appropriate.

######  Theorem 2.21.

  * •

The set of open connected Berkovich affinoids in $\mathbb{P}^{1}_{\text{an}}$
forms a basis for the weak topology.

  * •

In particular, for any Type II point $\zeta(a,r)$ and open set $U$ containing
$\zeta(a,r)$, $U$ contains $D_{\text{an}}(a,R)\setminus(D_{1}\cup\cdots\cup
D_{n})$ where each $D_{j}$ is a closed Berkovich disk of the form
$\overline{D}_{\text{an}}(b,s)\subsetneq\overline{D}_{\text{an}}(a,r)$ and
$R>r$.

  * •

The Berkovich projective line
$\mathbb{P}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})$ is a connected,
complete, compact, Hausdorff space. Moreover, every connected affinoid is
connected.

From subsection 2.9 it is clear that for any Type II or III point,
$\mathbb{P}^{1}_{\text{an}}\setminus\zeta(a,r)$ is a union of open Berkovich
disks. To be precise

$\mathbb{P}^{1}_{\text{an}}\setminus\zeta(a,r)=\left(\mathbb{P}^{1}_{\text{an}}\setminus\overline{D}_{\text{an}}(a,r)\right)\cup\bigcup_{b\in\overline{D}(a,r)}D_{\text{an}}(b,r).$

However we should really write this as a disjoint union. In the special case
of the Gauss point, it is somewhat easier to see that we should have one
distinct open disk in $\mathbb{P}^{1}_{\text{an}}\setminus\zeta(a,r)$ for
every element of $\mathbb{P}^{1}(k)$, where $k$ is the residue field.

######  Definition 2.23.

Let $\zeta\in\mathbb{P}^{1}_{\text{an}}$. The connected components of
$\mathbb{P}^{1}_{\text{an}}\setminus\left\\{\zeta\right\\}$ are called the
_residue classes_ , or _directions_ , or _tangent vectors_ at $\zeta$. The set
of directions at $\zeta$ is denoted $T_{\zeta}\mathbb{P}^{1}_{\text{an}}$. For
any $\xi\in\mathbb{P}^{1}_{\text{an}}\setminus\left\\{\zeta\right\\}$ we
define $\vec{v}(\xi)$ to be the (unique) direction at $\zeta$ containing
$\xi$.

######  Proposition 2.22.

Let $\zeta\in\mathbb{P}^{1}_{\text{an}}$. Then
$T_{\zeta}\mathbb{P}^{1}_{\text{an}}$ can be described by

  1. (i)

If $\zeta$ is Type I or IV, then there is only one direction at $\zeta$,
meaning $\zeta$ is an endpoint of the tree.

  2. (ii)

If $\zeta=\zeta(a,r)$ is Type II, then
$T_{\zeta}\mathbb{P}^{1}_{\text{an}}\cong\mathbb{P}^{1}(k)$, meaning $\zeta$
has one direction for each distinct open disk $D_{\text{an}}(b,r)$ for
$b\in\overline{D}(a,r)$, and also
$\mathbb{P}^{1}_{\text{an}}\setminus\overline{D}(a,r)$, the residue class
associated with $\infty$.

  3. (iii)

If $\zeta=\zeta(a,r)$ is Type III, then the two directions are
$D_{\text{an}}(a,r)$ and
$\mathbb{P}^{1}_{\text{an}}\setminus\overline{D}(a,r)$.

###  2.12. Paths and Hyperbolic Metric

Recall from subsection 2.9 the definition of the partial order $\preceq$ on
$\mathbb{A}^{1}_{\text{an}}$; we naturally extend this to
$\mathbb{P}^{1}_{\text{an}}$ by asserting that $\zeta\preceq\infty$ for every
$\zeta\in\mathbb{A}^{1}_{\text{an}}$.

######  Proposition 2.23.

  1. (i)

The relation $\preceq$ defines a partial order on
$\mathbb{P}^{1}_{\text{an}}$.

  2. (ii)

All Type I and IV points are minimal with respect to $\preceq$.

  3. (iii)

$\infty$ is a maximum point.

  4. (iv)

For any $\zeta,\xi\in\mathbb{A}^{1}_{\text{an}}$ with $\xi\preceq\zeta$, we
have $\operatorname{diam}(\xi)\leq\operatorname{diam}(\zeta)$, with equality
if and only if $\xi=\zeta$.

  5. (v)

For any two $\zeta_{0},\zeta_{1}\in\mathbb{P}^{1}_{\text{an}}$ there is a
unique least upper bound, $\zeta_{0}\vee\zeta_{1}$, defined below.

######  Definition 2.24.

Let $\zeta_{0},\zeta_{1}\in\mathbb{P}^{1}_{\text{an}}$. The _least upper
bound_ or _join_ of $\zeta_{0}$ and $\zeta_{1}$, denoted
$\zeta_{0}\vee\zeta_{1}$, is the unique element of
$\mathbb{P}^{1}_{\text{an}}$ such that:

  1. (i)

$\zeta_{0},\zeta_{1}\preceq\zeta_{0}\vee\zeta_{1}$; and

  2. (ii)

if $\xi\in\mathbb{A}^{1}_{\text{an}}$ and $\zeta_{0},\zeta_{1}\preceq\xi$,
then $\zeta_{0}\vee\zeta_{1}\preceq\xi$.

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######  Definition 2.25.

Let $X$ be a topological space. We say $X$ is uniquely path-connected iff for
any two distinct points $x_{0},x_{1}\in X$, there is a unique subset $I\subset
X$ containing $x_{0}$ and $x_{1}$ such that $I$ is homeomorphic to the real
closed interval $[0,1]$, with the homeomorphism mapping $x_{0}$ to $0$ and
$x_{1}$ to $1$. We call $[x_{0},x_{1}]=I$ the _closed interval_ between
$x_{0}$ and $x_{1}$. We call,
$(x_{0},x_{1})=[x_{0},x_{1}]\setminus\left\\{x_{0},x_{1}\right\\}$ an _open
interval_. We can similarly define the half-open intervals $[x_{0},x_{1})$,
and $(x_{0},x_{1}]$.

######  Theorem 2.24.

Let $U\subseteq\mathbb{P}^{1}_{\text{an}}$ be a connected Berkovich affinoid,
then $U$ is uniquely path-connected. Hence $\mathbb{P}^{1}_{\text{an}}$ is
locally connected. Moreover, for any $\zeta_{0},\zeta_{1}\in U$, all points on
$(\zeta_{0},\zeta_{1})$, are of Type II or III. A path $[\zeta_{0},\zeta_{1}]$
is always of the form

$[\zeta_{0},\zeta_{0}\vee\zeta_{1}]\cup[\zeta_{0}\vee\zeta_{1},\zeta_{1}]=\left\\{\zeta_{0}\preceq\xi\preceq\zeta_{0}\vee\zeta_{1}\right\\}\cup\left\\{\zeta_{0}\vee\zeta_{1}\preceq\xi\preceq\zeta_{1}\right\\}.$

######  Definition 2.26.

Let $S\subseteq\mathbb{P}^{1}_{\text{an}}$ be any subset of the Berkovich
projective line. The _convex hull_ of $S$ is the set

$\operatorname{Hull}(S)=\left\\{\xi\in\mathbb{P}^{1}_{\text{an}}:\,\exists\zeta_{0},\zeta_{1}\in
S\text{ such that }\xi\in[\zeta_{0},\zeta_{1}]\right\\}.$

######  Definition 2.27.

The set $\mathbb{H}=\mathbb{P}^{1}_{\text{an}}(K)\setminus\mathbb{P}^{1}(K)$
is the _hyperbolic space_ over $K$. We define a _hyperbolic metric_
$d_{\mathbb{H}}:\mathbb{H}\times\mathbb{H}\to[0,\infty)$ given by

$d_{\mathbb{H}}(\zeta,\xi)=2\log\left(\operatorname{diam}(\zeta\vee\xi)\right)-\log\left(\operatorname{diam}(\zeta)\right)-\log\left(\operatorname{diam}(\xi)\right).$

######  Remark 2.5.

The hyperbolic metric measures distances by the logarithm of diameter along
the lines of the poset structure. Observe that when $\zeta\preceq\xi$ then
$\zeta\vee\xi=\xi$, so

$d_{\mathbb{H}}(\zeta,\xi)=\log\left(\operatorname{diam}(\zeta)\right)-\log\left(\operatorname{diam}(\xi)\right).$

Then in general, since
$[\zeta,\xi]=[\zeta,\zeta\vee\xi]\cup[\zeta\vee\xi,\xi]$ by Theorem 2.24, it
is natural to see that

$d_{\mathbb{H}}(\zeta,\xi)=d_{\mathbb{H}}(\zeta,\zeta\vee\xi)+d_{\mathbb{H}}(\zeta\vee\xi,\xi).$

The topology given by $d_{\mathbb{H}}$ is much stronger than the weak
topology, even though they agree on intervals. For instance, a hyperbolic ball
around a Type II point does not contain a single direction, but every (weak)
open neighbourhood contains all but finitely many.

###  2.13. Rational Maps

Recall that the aim of this part of this article is to generalise the notion
of _rational maps_ on the Berkovich projective line and also its dynamical
theory. When we define _skew products_ on $\mathbb{P}^{1}_{\text{an}}$ it will
not only be useful to refer and compare them to rational maps, but we will
want to understand skew products through their associated rational maps.

The purpose of this section is to recall some of the definitions and basic
theory of _rational maps_ on the Berkovich projective line. However much of
this theory will be deferred to the following sections (and referenced) as we
generalise these results to skew products, since the reader can always recover
the original theorem as a special case of the newly printed one.

_Notation._ Typically authors write $\phi$ for both a rational function in
$\mathbb{\mathbb{C}}_{v}(y)$ and also the induced function
$\phi:\mathbb{P}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})\to\mathbb{P}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})$.
In this case we will instead distinguish $\phi_{*}$ as the induced function on
the Berkovich line and $\phi^{*}$ as the homomorphism induced on the function
field $\mathbb{\mathbb{C}}_{v}(y)$. By now the reader will also have noticed
the propensity to write rational functions as $f(y)$ rather than $f(x)$,
$f(t)$, or $f(z)$; whilst the latter would have been fine, we preserve the
variable $x$ for the variable in the field of Puiseux series $\mathbb{K}$,
giving a natural extension from the field $\mathbb{C}(x,y)$ to
$\mathbb{K}(y)$.

For the following definition, recall Remark 2.4 about considering a Type I
point as an honorary $[0,\infty]$-valued seminorm on $K(y)$. Indeed, for any
$a\in\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$ one can check that
$\phi_{*}(a)=\phi(a)$.
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######  Definition 2.28.

Let $\phi(y)\in\mathbb{\mathbb{C}}_{v}(y)$ be a rational function. Then we
define the associated _rational map_ on the Berkovich projective line by

$\displaystyle\phi_{*}:\mathbb{P}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})$
$\displaystyle\longrightarrow\mathbb{P}^{1}_{\text{an}}(\mathbb{\mathbb{C}}_{v})$
$\displaystyle\zeta$ $\displaystyle\longmapsto\phi_{*}(\zeta)$
$\displaystyle\text{where }\left\|f(y)\right\|_{\phi_{*}(\zeta)}$
$\displaystyle=\left\|f\circ\phi(y)\right\|_{\zeta}.$

Importantly, this function is well defined because either
$\left\|\cdot\right\|_{\phi_{*}(\zeta)}$ is a seminorm on $K[y]$ which extends
the one on $K$. To see the latter, consider
$\left\|a\right\|_{\phi_{*}(\zeta)}$ for $a\in K$. Applying $\phi$ to a
constant does nothing i.e. $a\circ\phi(y)=a$ and so
$\left\|a\right\|_{\phi_{*}(\zeta)}=\left\|a\circ\phi\right\|_{\zeta}=\left\|a\right\|_{\zeta}=\left|a\right|$.
This will be the main challenge in making a more general definition later.

######  Theorem 2.25.

Let $\phi\in\mathbb{\mathbb{C}}_{v}(y)$ be a rational function. Then the
function $\phi_{*}:\mathbb{P}^{1}_{\text{an}}\to\mathbb{P}^{1}_{\text{an}}$ of
Definition 2.28 is the unique continuous extension of the rational function
$\phi:\mathbb{P}^{1}\to\mathbb{P}^{1}$ to $\mathbb{P}^{1}_{\text{an}}$.

######  Proposition 2.26.

Let $\phi,\psi\in\mathbb{\mathbb{C}}_{v}(y)$ be rational functions, then
$(\psi\circ\phi)_{*}=\psi_{*}\circ\phi_{*}$.

######  Theorem 2.27.

Let $\phi\in\mathbb{\mathbb{C}}_{v}(y)$ be a non-constant rational function.

  1. (i)

Suppose $D(a,r)$ contains no poles of $\phi$ and $\phi(D(a,r))=D(b,s)$, then
$\phi_{*}(\zeta(a,r))=\zeta(b,s)$.

  2. (ii)

$\phi$ preserves the types of points.

  3. (iii)

$\phi$ is an open mapping.

###  2.14. Reduction

As suggested by Definition 2.23 and subsection 2.11, the most natural way to
think about the directions at a Type II point $\zeta$ is by identifying each
one with a residue in $\mathbb{P}^{1}(k)$. In this subsection we shall discuss
reduction of elements and maps from $K$ or $K(y)$ to $k$ or $k(y)$, and what
we learn about local degrees. This will generalise the content of [Ben19,
§7.5].
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Recall that the residue field of $K$ is the quotient field
$k=\mathcal{O}_{K}/\mathcal{M}_{K}$. The quotient map $\mathcal{O}_{K}\to k$
is called the _reduction map_. We denote the reduction of
$a\in\mathcal{O}_{K}$ by $\overline{a}$. This has a more useful extension to
the projective line $\mathbb{P}^{1}(K)$ since every element can be written as
$[a_{0}:a_{1}]$ with $a_{0},a_{1}\in\mathcal{O}_{K}$.

$\displaystyle\mathbb{P}^{1}(K)$
$\displaystyle\longrightarrow\mathbb{P}^{1}(k)$ $\displaystyle[a_{0}:a_{1}]$
$\displaystyle\longmapsto[\overline{a}_{0}:\overline{a}_{1}]$

Furthermore, this induces a reduction map on $\mathcal{O}_{K}[y]\to k[y]$ by
reducing each coefficient in the polynomial. Reduction of rational functions
of $K$ is a little troublesome. One can write any $f\in\mathbb{P}^{1}(K(y))$
as a fraction $f=g/h$ or ratio $[g:h]$ of polynomials in $\mathcal{O}_{K}[y]$,
with at least one coefficient in $g$ or $h$ having absolute value $1$. We then
define the reduction $\overline{f}\in\mathbb{P}^{1}(k(y))$ as follows:

$\displaystyle\mathbb{P}^{1}(K(y))$
$\displaystyle\longrightarrow\mathbb{P}^{1}(k(y))$ $\displaystyle f=[g:h]$
$\displaystyle\longmapsto[\overline{g}:\overline{h}]=\overline{f}$

_Warning_ : this definition is sensitive to the choice of $f$ and $g$ \- if
one allows both $g,h\in\mathcal{M}_{K}$ then the reduction will be ill defined
as $[0:0]$.

Unfortunately, reduction can change the basic properties of a polynomial.

######  Example 2.5.

Three examples of reduction over the complex numbers.

  1. (i)

Let $g(x,y)\in\mathbb{C}(x)[y]$ be defined by $g=xy^{2}+y-1$. Then
$\overline{g}=y-1$ so $\deg(g)\neq\deg(\overline{g})$.

  2. (ii)

Let $g(x,y)=(y-x)y$, then $\overline{g}=y^{2}$. We see that $g$ had two
distinct roots, but its reduction has one despite having the same degree.

  3. (iii)

Let $f=(y-x)/y$ then $\overline{f}=1$.

A rational function $\phi(y)\in K(y)$ induces a rational map.

$\displaystyle\overline{\phi}:\mathbb{P}^{1}(k)$
$\displaystyle\longrightarrow\mathbb{P}^{1}(k)$ $\displaystyle\overline{a}$
$\displaystyle\longmapsto\overline{\phi}(\overline{a})$

######  Definition 2.29.

Suppose $K$ is a non-Archimedean field which might not be algebraically
closed, and let $\phi\in K(y)$. We say that $\phi(y)=\frac{g(y)}{h(y)}$ has
_explicit good reduction_ iff $\deg(\overline{\phi})=\deg(\phi)$, otherwise
$\phi$ has _bad reduction_. If there is a fractional linear transformation
$\eta\in\operatorname{PGL}(2,K)$ such that $\eta\circ\phi\circ\eta^{-1}$ has
explicit good reduction then we say $\phi$ has _good reduction_. If instead
there is such an $\eta\in\operatorname{PGL}(2,\overline{K})$ then $\phi$ has
_potentially good reduction_.

If $\phi$ has explicit good reduction then
$\overline{a}\mapsto\overline{\phi(a)}$ is well defined and equal to
$\overline{\phi}(\overline{a})$. Conversely, if the degree drops then we can
find $a,b\in K$ with $\phi(a)=0$, $\phi(b)=\infty$ and the same reduction
$\overline{a}=\overline{b}$, thus
$\overline{0}=\overline{\phi(a)}\neq\overline{\phi(b)}=\overline{\infty}$;
moreover both may be distinct from $\overline{\phi}(\overline{a})$.

For a much more thorough discussion of reduction, see [Ben19, §4.3].

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include more in this section.

##  3\. Skew Products on the Berkovich Projective Line

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section title?

The aim of this section is to define a _skew product_ on the Berkovich
projective line and compare it to a rational map. Whilst these maps have new
and unusual quirks and are a strict generalisation of Berkovich rational maps,
most of the properties we are used to will be recovered.

######  Definition 3.1.

Let $\Psi$ be an endomorphism of $K(y)$ extending an automorphism of $K$, i.e.
the following diagram commutes:

${K(y)}$${K(y)}$${K}$${K}$$\scriptstyle{\Psi}$$\scriptstyle{\Psi_{1}}$

In this case we will call $\Psi:K(y)\to K(y)$ a _skew endomorphism_ of $K(y)$.
We will typically denote the restriction $\left.\Psi\right|_{K}$ by
$\Psi_{1}$.

###  3.1. Motivation

Often, we shall think of $\Psi$ coming from a rational map of ruled surfaces.
We will give detail on this construction in a sequel, but describe a special
case of the situation now to give examples and motivation. Classically, a skew
product (in analysis and geometry) is one of the form

$\phi(x,y)=(\phi_{1}(x),\phi_{2}(x,y))$

defined on some product space $A\times B$. Let us focus on the simple case of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ over the field $k$. One may think of the
following diagram commuting with a first projection map $h(x,y)=x$; this will
help to generalise the concept later.

${\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{h}$$\scriptstyle{\phi}$$\scriptstyle{h}$$\scriptstyle{\phi_{1}}$

The information given by $\phi$ is equivalent to a $k$-algebra homomorphism of
function fields.

$\displaystyle\phi^{*}:k(x,y)$ $\displaystyle\longrightarrow k(x,y)$
$\displaystyle x$ $\displaystyle\longmapsto\phi_{1}(x)$ $\displaystyle y$
$\displaystyle\longmapsto\phi_{2}(x,y)$

After changing coordinates we may assume that $\phi_{1}(0)=0$ and look in a
neighbourhood of $x=0$ then we obtain a $k$-algebra map
$\phi_{1}^{*}:k[[x]]\to k[[x]]$ which extends to one of the local function
field $\phi^{*}:k((x))(y)\to k((x))(y)$. In more algebraic terminology, we
took the completion of the local ring $k[x]_{(x)}$.

${k((x))(y)}$${k((x))(y)}$${k((x))}$${k((x))}$$\scriptstyle{\phi^{*}}$$\scriptstyle{h^{*}}$$\scriptstyle{h^{*}}$$\scriptstyle{\phi_{1}^{*}}$

After taking the algebraic closure of $k((x))$ to obtain the Puiseux series
$\mathbb{K}(k)$, this map can be extended to a $k$-algebra endomorphism
$\phi^{*}:\mathbb{K}(y)\to\mathbb{K}(y)$. We can write
$\phi_{1}^{*}(x)=\phi_{1}(x)\in k[[x]]$ with $\phi_{1}(x)=\lambda
x^{n}+\mathcal{O}(x^{n+1})$ and $\lambda\in k^{\times}$ then this extends to
an ‘equivariant’ skew endomorphism over $\mathbb{K}$. With $\phi_{2}\in
k((x))(y)$ we call such a map a _$k$ -rational skew endomorphism_.

If $\phi_{1}(x)$ were the identity, then $\phi$ would represent the rational
map $y\mapsto\phi_{2}(y)$ over $k((x))$ and naturally induce a Berkovich
rational map on $\mathbb{P}^{1}_{\text{an}}(\mathbb{K})$. Unfortunately,
$\phi_{1}$ is rarely trivial, and $\phi$ will not translate to a Berkovich
rational map. A different mapping on the Berkovich projective line is needed -
_the skew product._

###  3.2. The Problem

If we had that $\Psi_{1}=\operatorname{id}$ then $\Psi$ would be a
$K$_-algebra endomorphism_ of $K(y)$, indeed it would be the _rational map_
$y\mapsto\Psi(y)$ over $\mathbb{P}^{1}(K)$. We could then define a Berkovich
rational map
$\Psi_{*}:\mathbb{P}^{1}_{\text{an}}\to\mathbb{P}^{1}_{\text{an}}$, as in
subsection 2.13, by

$\left\|f\right\|_{\Psi_{*}(\zeta)}=\left\|\Psi(f)\right\|_{\zeta}.$

The crucial calculation for the skew product to be well-defined was that
$\Psi_{*}(\zeta)$ preserves the norm on $K$, meaning that for every $a\in K$,
$\left\|a\right\|_{\Psi_{*}(\zeta)}=\left|a\right|$. We might naïvely try this
definition with an arbitrary skew endomorphism; let $a\in K$, then by the
expected definition we have

$\left\|a\right\|_{\Psi_{*}(\zeta)}=\left\|\Psi(a)\right\|_{\zeta}=\left|\Psi(a)\right|=\left|\Psi_{1}(a)\right|.$

In general, unlike the rational case, $\Psi_{1}$ is an arbitrary field
automorphism of $K$ and could do anything to the absolute value. In the above
we need $\left|\Psi_{1}(a)\right|=\left|a\right|$. Requiring this is
reasonable, but for the definition of skew product below, we only ask that
$\left|\Psi_{1}(a)\right|=\left|a\right|^{\frac{1}{q}}$ uniformly for some
$q>0$. The special cases where $1/q\in\mathbb{N}$, especially $q=1$, will be
of great interest in applications of this theory.

The construction of an arbitrary algebraically defined map on a Berkovich
space is not new; see for instance [FJ04]. Specifically, one can always
normalise $\left\|\Psi(f)\right\|_{\zeta}$ as necessary depending on both
$\Psi$ and $f$, to ensure a well-defined function. In any case, this will be a
geometrically natural definition because, roughly speaking, the corresponding
prime ideal of $\zeta$ i.e.
$p_{\zeta}=\left\\{f:\left\|f\right\|_{\zeta}<1\right\\}$ is mapped to the
corresponding prime of its image

$\Psi^{-1}(p_{\zeta})=\left\\{g:\Psi(g)\in
p_{\zeta}\right\\}=\left\\{g:\left\|\Psi(g)\right\|_{\zeta}<1\right\\}=\left\\{g:\left\|g\right\|_{\Psi_{*}(\zeta)}<1\right\\}.$

Our construction is arbitrary enough to be applicable to a broader class of
examples in complex dynamics, but the uniform normalisation factor allows for
the dynamical behaviour of $\Psi_{*}$ to be better understood.

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constructions e.g. Favre-Jonsson

###  3.3. Skew Products

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this is geometrically natural

######  Definition 3.2 (Skew Product).

Suppose that $\Psi:K(y)\to K(y)$ is a skew endomorphism of $K(y)$ and there is
a $q$ such that
$\left|\Psi(a)\right|=\left|\Psi_{1}(a)\right|=\left|a\right|^{\frac{1}{q}}$
for every $a\in K$. Then we say $\Psi$ is _equivariant_ with _scale factor_
$q$. Given such a $\Psi$, we define $\Psi_{*}$, a _skew product over $K$_, as
follows.

$\displaystyle\Psi_{*}:\mathbb{P}^{1}_{\text{an}}(K)$
$\displaystyle\longrightarrow\mathbb{P}^{1}_{\text{an}}(K)$
$\displaystyle\zeta$ $\displaystyle\longmapsto\Psi_{*}(\zeta)$
$\displaystyle\text{where }\left\|f\right\|_{\Psi_{*}(\zeta)}$
$\displaystyle=\left\|\Psi(f)\right\|_{\zeta}^{q}$

If $q=1$ then we call $\Psi_{*}$ a _simple_ skew product. Otherwise, if $q<1$
we say it is _superattracting_ , and if $q>1$ we may say it is
_superrepelling_.

Consider the case deriving from a skew product
$\phi(x,y)=(\phi_{1}(x),\phi_{2}(x,y))$ on a surface, with
$\phi_{1}(x)=\lambda x^{n}+\mathcal{O}(x^{n+1})$, and recall the above
discussion that we have a $k$-rational skew endomorphism $\phi^{*}$. Then the
induced skew product on the Berkovich projective line,
$\phi_{*}:\mathbb{P}^{1}_{\text{an}}(\mathbb{K})\to\mathbb{P}^{1}_{\text{an}}(\mathbb{K})$
will be called a _$k$ -rational skew product_ and has scale factor
$q=\frac{1}{n}$. In particular, if $n=1$ then $\phi_{*}$ is a simple
$k$-rational skew product; this name takes after the fact $x=0$ is a simple
zero of $\phi_{1}(x)$. Furthermore, note that $0$ is a superattracting fixed
point of $\phi_{1}(x)$ when $n>1$, and hence why we call $\phi_{*}$
‘superattracting’. A deeper discussion of $k$-rational skew products will be
provided in the sequel, but we will continue to use them for examples and
explain basic constructions.

######  Theorem 3.1.

Suppose that $\Psi$ is an equivariant skew endomorphism. Then the skew product
$\Psi_{*}:\mathbb{P}^{1}_{\text{an}}\to\mathbb{P}^{1}_{\text{an}}$ is a well
defined map on the Berkovich projective line.

###### Proof.

It is clear that $\Psi_{*}(\zeta)$ is a multiplicative seminorm because $\Psi$
is a ring homomorphism.
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to check that $\Psi_{*}(\zeta)$ extends the norm on $K$. Indeed given $a\in
K$,

$\left\|a\right\|_{\Psi_{*}(\zeta)}=\left\|\Psi(a)\right\|_{\zeta}^{q}=\left|\Psi(a)\right|^{q}=\left(\left|a\right|^{\frac{1}{q}}\right)^{q}=\left|a\right|.$

∎

######  Proposition 3.2.

If $\Phi,\Psi$ are equivariant skew endomorphisms of $K(y)$ then
$(\Phi\circ\Psi)_{*}=\Psi_{*}\circ\Phi_{*}$ i.e. $(\cdot)_{*}$ is a
contravariant functor.

###### Proof.

$\left\|f\right\|_{\Psi_{*}\circ\Phi_{*}(\zeta)}=\left\|\Psi(f)\right\|_{\Phi_{*}(\zeta)}=\left\|\Phi(\Psi(f))\right\|_{\zeta}=\left\|f\right\|_{(\Phi\circ\Psi)_{*}(\zeta)}$

∎

The following definition and theorem is fundamental to working with skew
products - it says we can always decompose a skew product into a field
automorphism and a rational map. On $\mathbb{P}^{1}_{\text{an}}$ this will
help because the former induces a bijection with some nice geometric
properties and the latter induces a Berkovich rational map which is well
understood.

######  Definition 3.3.

Let $\Psi$ be a skew endomorphism. We define the following two homomorphisms.
Firstly $\Psi_{1}=\left.\Psi\right|_{K}$ but extended trivially to $K(y)$, and
secondly with $\Psi_{2}$ we distill the action of $\Psi$ on $y$.

$\displaystyle\Psi_{1}:K(y)$ $\displaystyle\longrightarrow K(y)$
$\displaystyle a$ $\displaystyle\longmapsto\Psi(a)\quad\forall a\in K$
$\displaystyle y$ $\displaystyle\longmapsto y$ $\displaystyle\Psi_{2}:K(y)$
$\displaystyle\longrightarrow K(y)$ $\displaystyle a$
$\displaystyle\longmapsto a\qquad\ \ \forall a\in K$ $\displaystyle y$
$\displaystyle\longmapsto\Psi(y)$

######  Theorem 3.3.

Let $\Psi$ be an equivariant skew endomorphism. Then

  * •

$\Psi=\Psi_{2}\circ\Psi_{1}$;

  * •

$\Psi_{*}=\Psi_{1*}\circ\Psi_{2*}$;

  * •

$\Psi_{2*}$ is a rational map on $\mathbb{P}^{1}_{\text{an}}$.

###### Proof.

Clearly $\Psi_{1},\Psi_{2}$ are ring homomorphisms, so it is enough to show
$\Psi=\Psi_{2}\circ\Psi_{1}$ on generators of $K(y)$, or more simply on an
arbitrary $a\in K$ and on $y$.

$\Psi_{2}\circ\Psi_{1}(y)=\Psi_{2}(y)=\Psi(y)$

The last equality is by the definition of $\Psi_{2}$.

$\Psi_{2}\circ\Psi_{1}(a)=\Psi_{2}(\Psi(a))=\Psi(a)$

The last equality is because $\Psi_{2}(b)=b$ for any $b\in K$, such as
$b=\Psi(a)$. We finish the proof using subsection 3.3. ∎

Our notation is inspired by the case of a $k$-rational skew product.

$\displaystyle\phi:\mathbb{P}^{1}\times\mathbb{P}^{1}$
$\displaystyle\dashrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$
$\displaystyle(x,y)$ $\displaystyle\longmapsto(\phi_{1}(x),\phi_{2}(x,y))$

In this case $\Psi=\phi^{*}$ and we consider the induced skew product. From
the original geometric perspective, $\phi_{1}(x)=\phi^{*}(x)$ and
$\phi_{2}(x,y)=\phi^{*}(y)$. Then the decomposition of $\Psi=\phi^{*}$ is very
natural since it separetes into its into its actions on $x$ and $y$:

$\displaystyle\Psi_{1}:k(x,y)$ $\displaystyle\longrightarrow k(x,y)$
$\displaystyle\Psi_{2}:k(x,y)$ $\displaystyle\longrightarrow k(x,y)$
$\displaystyle x$ $\displaystyle\longmapsto\phi_{1}(x)$ $\displaystyle x$
$\displaystyle\longmapsto x$ $\displaystyle y$ $\displaystyle\longmapsto y$
$\displaystyle y$ $\displaystyle\longmapsto\phi_{2}(x,y)$

We write $\phi_{1}^{*}=\Psi_{1}$ and $\phi_{2}^{*}=\Psi_{2}$. One may verify
that $\phi^{*}=\phi_{2}^{*}\circ\phi_{1}^{*}$ and
$\phi_{*}=\phi_{1*}\circ\phi_{2*}$, but antecedent to Theorem 3.3 is the (set
theoretic) composition

$\phi(x,y)=(\phi_{1}(x),\phi_{2}(x,y))=(\phi_{1}(x),y)\circ(x,\phi_{2}(x,y)).$

From now on, we may denote an equivariant skew endomorphism by $\phi^{*}$ even
if it is not derived from a geometric skew product on a surface. To be clear,
to say $\phi_{*}$ is a skew product still will mean it derives from an
equivariant skew endomorphism $\phi^{*}$, but it may not be $k$-rational.
Furthermore, we may write $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ as a cue to the
splitting from Definition 3.3.

######  Remark 3.1.

We can see now that not only does a skew product generalise the definition of
a rational map, but every skew product is a composition of a rational map and
the action of a field automorphism. This will be most useful for our
understanding of how a skew product acts in one iterate. It will be much
harder to understand multiple iterations (its dynamics), however the
decomposition is still valuable.
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it not being rational or conjugate to one.

###  3.4. Properties of Skew Products

The following theorem say that given $\phi_{*}=\phi_{1*}\circ\phi_{2*}$, the
field automorphism part, $\phi_{*}$ is a geometrically nice map on
$\mathbb{P}^{1}_{\text{an}}$. However caution is warranted: a non-trivial
field automorphism will induce a highly non-analytic map.

######  Theorem 3.4.

Let $\Psi$ be an equivariant automorphism of $K$ extended trivially to $K(y)$
with scale factor $q$, i.e. $\Psi_{2}=\operatorname{id}$. Then the induced
skew product
$\Psi_{*}:\mathbb{P}^{1}_{\text{an}}\to\mathbb{P}^{1}_{\text{an}}$

  1. (i)

is a homeomorphism on $\mathbb{P}^{1}_{\text{an}}(K)$;

  2. (ii)

scales hyperbolic distances by a factor of $q$;

  3. (iii)

is the unique continuous extension of $\Psi^{-1}$ on
$\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$;

  4. (iv)

maps Berkovich points to those of the same type; and

  5. (v)

is order preserving on the poset $(\mathbb{P}^{1}_{\text{an}},\preceq)$.

In
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$\displaystyle\Psi_{*}(a)$ $\displaystyle=\Psi^{-1}(a)$
$\displaystyle\Psi_{*}(D(a,r))$ $\displaystyle=D(\Psi^{-1}(a),r^{q})$
$\displaystyle\Psi_{*}\left(\overline{D}(a,r)\right)$
$\displaystyle=\overline{D}(\Psi^{-1}(a),r^{q})$
$\displaystyle\Psi_{*}(\zeta(a,r))$ $\displaystyle=\zeta(\Psi^{-1}(a),r^{q})$
$\displaystyle\Psi_{*}(D_{\text{an}}(a,r))$
$\displaystyle=D_{\text{an}}(\Psi^{-1}(a),r^{q})$
$\displaystyle\Psi_{*}\left(\overline{D}_{\text{an}}(a,r)\right)$
$\displaystyle=\overline{D}_{\text{an}}(\Psi^{-1}(a),r^{q})$

######  Remark 3.2.

The fact that this skew product $\phi_{1*}$ extends the _inverse_ of the
homomorphism $\phi_{1}^{*}$ on $\mathbb{P}^{1}(K)$ is somewhat more natural in
the geometric setting over the Puiseux series. Here we see it as a ‘pre-
composition’ of functions of $x$, where
$\gamma(x)\mapsto\gamma(\phi_{1}^{-1}(x))$. This is very different from how
homomorphisms on $K$ acting only on $y$ (‘rational maps’) generate a post-
composition-like function on $\mathbb{P}^{1}(K)$.

Consider a germ of a curve through $x=0$, namely $x\mapsto(x,\gamma(x))$. Then
$\phi=(\phi_{1},\phi_{2})$ applied to this gives
$x\mapsto(\phi_{1}(x),\phi_{2}(x,\gamma(x))$. To rewrite this in the form
$(x,\tilde{\gamma}(x))$ we must _precompose_ with $\phi_{1}^{-1}(x)$ to get

$x\mapsto(x,\phi_{2}(\phi_{1}^{-1}(x),\gamma(\phi_{1}^{-1}(x)))).$

######  Proposition 3.5.

Let $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ be a simple $k$-rational skew product.
Then $\phi_{*}:\mathbb{P}^{1}(\mathbb{K})\to\mathbb{P}^{1}(\mathbb{K})$ acts
as follows

$\displaystyle\phi_{*}:\mathbb{P}^{1}(\mathbb{K})$
$\displaystyle\longrightarrow\mathbb{P}^{1}(\mathbb{K})$ $\displaystyle a(x)$
$\displaystyle\longmapsto\phi_{2}(\phi_{1}^{-1}(x),a(\phi_{1}^{-1}(x)))=\phi_{2}\circ(\operatorname{id}\times
a)\circ\phi_{1}^{-1}(x)$

###### Proof of Theorem 3.4.

Since $\Psi$ is an isomorphism,
$\left\|f\right\|_{(\Psi^{-1})_{*}(\zeta)}=\left\|\Psi^{-1}(f)\right\|_{\zeta}$
provides an inverse to $\Psi_{*}$.

First we prove that the restriction of $\Psi_{*}$ to classical points is equal
to $\Psi^{-1}$ on $\mathbb{P}^{1}(K)$. Let $\zeta=a\in K$ be Type I, then

$\left\|y-b\right\|_{\Psi_{*}(\zeta)}=\left\|y-\Psi(b)\right\|_{\zeta}^{q}=\left|a-\Psi(b)\right|^{q}=\left|\Psi(\Psi^{-1}(a)-b)\right|^{q}$
$=\left(\left|\Psi^{-1}(a)-b\right|^{\frac{1}{q}}\right)^{q}=\left|\Psi^{-1}(a)-b\right|=\left\|y-b\right\|_{\Psi^{-1}(a)}.$

It is a similar exercise in the definitions to prove that
$\psi_{*}(\infty)=\infty$.

Now observe that since $|\Psi_{*}(a)-\Psi_{*}(b)|=|\Psi^{-1}(a-b)|=|a-b|^{q}$
we have:

$\displaystyle\Psi_{*}(D(a,r))$ $\displaystyle=D(\Psi^{-1}(a),r^{q})$
$\displaystyle\Psi_{*}(\overline{D}(a,r))$
$\displaystyle=\overline{D}(\Psi^{-1}(a),r^{q})$

It follows that

$\Psi_{*}(\zeta(a,r))=\zeta(\Psi^{-1}(a),r^{q})$

since for $f\in K[y]$

$\left\|\Psi(f)\right\|_{\zeta(a,r)}=\sup_{b\in
D(a,r)}\left\|\Psi(f)\right\|_{b}=\sup_{b\in
D(a,r)}\left\|f\right\|_{\Psi_{*}(b)}=\sup_{b^{\prime}\in
D(\Psi^{-1}(a),r^{q})}\\!\\!\\!\left\|f\right\|_{b^{\prime}}=\left\|f\right\|_{\zeta(\Psi^{-1}(a),r^{q})}$

Similarly, since
$\left\|y-\Psi^{-1}(a)\right\|_{\Psi_{*}(\zeta)}=\left\|y-a\right\|_{\zeta}^{q}$
we have

$\displaystyle\Psi_{*}(D_{\text{an}}(a,r))$
$\displaystyle=D_{\text{an}}(\Psi^{-1}(a),r^{q})$
$\displaystyle\Psi_{*}(\overline{D}_{\text{an}}(a,r))$
$\displaystyle=\overline{D}_{\text{an}}(\Psi^{-1}(a),r^{q})$

This determines the images of Type II/III points. Note that
$\left|a\right|=r\iff\left|\Psi^{-1}(a)\right|=r^{q}$, hence
$r\in\left|K^{\times}\right|\iff r^{q}\in\left|K^{\times}\right|$. Thus Type
II and III points are individually preserved.

We have also implicitly shown that for disks $D$ and $E$ we have

$D\subseteq E\iff\Psi_{*}(D)\subseteq\Psi_{*}(E),$

and similar for their Berkovich versions $D_{\text{an}}$ and $E_{\text{an}}$
this is equivalent to

$D_{\text{an}}\subseteq
E_{\text{an}}\iff\Psi_{*}(D_{\text{an}})\subseteq\Psi_{*}(E_{\text{an}}).$

This shows that a nested sequence of disks $D_{1}\supsetneq D_{2}\supsetneq
D_{3}\supsetneq\cdots$ remains nested; it is also clear that this has empty
intersection if and only if the sequence of images do. Therefore Type IV
points are preserved since the images of Type IV points can be described by
the image of such a sequence of disks.

To see that $\Psi_{*}$ is order preserving, recall that $\zeta\prec\xi$
implies that $\xi=\zeta(a,R)$ or $\xi=\infty$. Since $\infty=\Psi_{*}(\infty)$
is a maximum, the latter case is trivial for all $\zeta$. In the case that
$\xi=\zeta(a,R)$, we know that $\zeta\in\overline{D}_{\text{an}}(a,R)$.

$\Psi_{*}(\zeta)\in\Psi_{*}(\overline{D}_{\text{an}}(a,R))=\overline{D}_{\text{an}}(\Psi^{-1}(a),R^{q}),$

therefore

$\zeta\preceq\zeta(a,R)\iff\Psi_{*}(\zeta)\preceq\Psi_{*}(\zeta(a,R)).$

We have just shown that $\Psi_{*}$ preserves types, the ordering $\preceq$,
and also that

$\operatorname{diam}(\Psi_{*}(\zeta))=\operatorname{diam}(\zeta)^{q}.$

Since the basis of the topology if given by open affinoids, and these are
finite intersections of disks, to show continuity, we only need to look at
preimages of disks. Hence we know from the above that $\Psi_{*}$ is an open
map. Since $\Psi_{*}$ has an inverse given by $(\Psi^{-1})_{*}$, we also have

$\Psi_{*}^{-1}(D_{\text{an}}(\gamma,r))=D_{\text{an}}(\Psi(\gamma),r^{\frac{1}{q}}),$
$\Psi_{*}^{-1}(\bar{D}_{\text{an}}(\gamma,r))=\bar{D}_{\text{an}}(\Psi(\gamma),r^{\frac{1}{q}}),$

proving the continuity. Thus $\Psi_{*}$ is a homeomorphism.

Since $\Psi_{*}$ is order preserving, to show the scaling or isometry on the
hyperbolic metric we begin with pairs of related points. Suppose that
$\zeta\prec\xi$ have diameter $r$ and $R$ respectively. Then
$\Psi_{*}(\zeta)\prec\Psi_{*}(\xi)$ and hence

$d_{\mathbb{H}}(\Psi_{*}(\zeta),\Psi_{*}(\xi))=\log(R^{q})-\log(r^{q})=q\left(\log(R)-\log(r)\right)=q\,d_{\mathbb{H}}(\zeta,\xi).$

Otherwise, we have $\zeta$ and $\xi$ unrelated with diameters $r,s$ and join
$\zeta(a,R)$. By $\Psi_{*}$ and $\Psi_{*}^{-1}$ preserving order one can check
that $\Psi_{*}(\zeta\vee\xi)=\Psi_{*}(\zeta)\vee\Psi_{*}(\xi)$. Therefore the
shortest path from $\Psi_{*}(\zeta)$ to $\Psi_{*}(\xi)$ is through
$\zeta(\Psi^{-1}(a),R^{q})$, which is the homeomorphic image of $[\zeta,\xi]$.
Since $\operatorname{diam}(\Psi_{*}(\zeta))=\operatorname{diam}(\zeta)^{q}$,
the length of this path will be

$d_{\mathbb{H}}(\Psi_{*}(\zeta),\Psi_{*}(\xi))=2\log(R^{q})-\log(r^{q})-\log(s^{q})=q\,d_{\mathbb{H}}(\zeta,\xi).$

∎

######  Theorem 3.6.

Let $\Psi$ be an equivariant automorphism of $K$ extended trivially to $K(y)$
with scale factor $q$. For any Berkovich affinoid
$W\subseteq\mathbb{P}^{1}_{\text{an}}$, let
$W_{\text{I}}=W\cap\mathbb{P}^{1}(K)$. Then $\Psi_{*}(W)$ is the Berkovich
affinoid of the same type (if any) corresponding to
$\Psi_{*}(W_{\text{I}})=\Psi^{-1}(W_{\text{I}})$, and $\Psi_{*}^{-1}(W)$ is
the Berkovich affinoid of the same type (if any) corresponding to
$\Psi_{*}^{-1}(W_{\text{I}})=\Psi(W_{\text{I}})$. Boundaries are mapped
bijectively to boundaries.

###### Proof.

By Theorem 3.4, $\Psi_{*}$ bijectively maps $D(a,r)$ to $D(\Psi^{-1}(a),r)$ if
and only if it maps $D_{\text{an}}(a,r)$ to $D_{\text{an}}(\Psi^{-1}(a),r)$.
The same goes for closed disks on the affine line, and similarly for disks in
the projective line. Since $\Psi_{*}$ is a bijection, an affinoid
$W=D_{1}\cap\cdots\cap D_{n}$ is mapped to
$\Psi_{*}(D_{1})\cap\cdots\cap\Psi_{*}(D_{n})$ etc. In fact, $\Psi_{*}(W)$ has
the same number of ‘holes’ as $W$. ∎

######  Theorem 3.7.

Let $\phi_{*}$ be a non-constant skew product over $K$. Then $\phi_{*}$

  1. (i)

is a continuous function;

  2. (ii)

is an open mapping;

  3. (iii)

is the unique continuous extension of $(\phi_{1}^{*})^{-1}\circ\phi_{2}$ on
$\mathbb{P}^{1}(K)\subset\mathbb{P}^{1}_{\text{an}}(K)$; and

  4. (iv)

preserves the types of each Berkovich point.

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###### Proof.

Since $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ therefore this follows from the same
results for $\phi_{1*}$ and $\phi_{2*}$, namely, Theorem 3.4 Theorem 3.6,
[Ben19, Theorem 7.4, Corollary 7.9, Corollary 7.16]. ∎

The following theorem generalises [Ben19, Theorem 7.8] for rational maps to
skew products.

######  Theorem 3.8.

Let $\phi_{*}$ be a non-constant skew product over $K$, let
$W\subseteq\mathbb{P}^{1}_{\text{an}}(K)$ be a connected Berkovich affinoid,
and let $W_{I}=W\cap\mathbb{P}^{1}(K)$ be the corresponding connected affinoid
in $\mathbb{P}^{1}(K)$. Then $\phi_{*}(W)$ is the Berkovich connected affinoid
of the same type (if any) corresponding to $\phi_{*}(W_{I})$, and
$\phi_{*}^{-1}(W)$ is the Berkovich affinoid of the same type (if any)
corresponding to $\phi_{*}^{-1}(W_{I})$. Moreover, the following hold.

  1. (a)

$\partial(\phi_{*}(W))\subseteq\phi_{*}(\partial W)$.

  2. (b)

Each of the connected components $V_{1},\dots,V_{m}$ of $\phi_{*}^{-1}(W)$ is
a connected Berkovich affinoid mapping onto $W$.

  3. (c)

For each $i=1,\dots,m$,

$\phi_{*}(\partial V_{i})=\partial W\text{ and
}\phi_{*}(\operatorname{int}V_{i})=\operatorname{int}W,$

where $\operatorname{int}X$ denotes the interior of the set $X$.

  4. (d)

If $W$ is open, then $\phi_{*}(\partial V_{i})\cap W=\emptyset$.

A key omission from this theorem states that each map $\phi_{*}:V_{i}\to W$ is
$d_{j}$-to-$1$ counting multiplicity, with
$d_{1}+\cdots+d_{m}=\operatorname{rdeg}(\phi)$. This will follow later in
subsection 3.6 when we have a good notion of multiplicity, called local
degree. Overall, one can compare with the classical case Theorem 2.13.

###### Proof.

Since $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ this follows from Theorem 3.6 and
[Ben19, Theorem 7.8]. ∎

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The following generalises [Ben19, Corollary 7.9].

######  Corollary 3.9 (Properness Criterion).

Let $\phi_{*}$ be a non-constant skew product, and let
$U\subseteq\mathbb{P}^{1}_{\text{an}}$ be an open connected Berkovich
affinoid. Then the following are equivalent.

  1. (a)

$\phi_{*}(U)\cap\phi_{*}(\partial U)=\emptyset$.

  2. (b)

$U$ is a connected component of $\phi_{*}^{-1}(\phi_{*}(U))$.

The next theorem generalises [Ben19, Theorem 7.12] and is important for
understanding local ramification or _degrees_ , as explained in the next
subsection.

######  Theorem 3.10.

Let $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ be a non-constant skew product with
scale factor $q$, and let $\zeta=\zeta(a,r)\in\mathbb{P}^{1}_{\text{an}}$ be a
point of Type II or III. Let $\lambda<1$ be large enough so that

$\phi_{2}(y)=\sum_{n\in\mathbb{Z}}b_{n}(y-a)^{n}$

converges on the annulus

$U_{\lambda}=\left\\{\lambda r<|y-a|<r\right\\}$

and such that $\phi_{2}(y)-b_{0}$ has both the inner and outer Weierstrass
degrees equal to $d$. Setting $s={b_{d}}^{q}r^{dq}$, we have

$\phi_{*}(U_{\lambda})=\begin{cases}\left\\{\lambda^{dq}s<|y-\phi_{1*}(b_{0})|<s\right\\}&d>0\\\
\left\\{s<|y-\phi_{1*}(b_{0})|<\lambda^{dq}s\right\\}&d<0\end{cases}$

Similarly if instead $\lambda>1$ and $\phi_{2}(y)$ converges on the annulus

$V_{\lambda}=\left\\{r<|y-a|<r\lambda\right\\}$

and such that $\phi_{2}(y)-b_{0}$ has both the inner and outer Weierstrass
degrees equal to $d$, then we have

$\phi_{*}(V_{\lambda})=\begin{cases}\left\\{s<|y-\phi_{1*}(b_{0})|<\lambda^{dq}s\right\\}&d>0\\\
\left\\{\lambda^{dq}s<|y-\phi_{1*}(b_{0})|<s\right\\}&d<0\end{cases}$

Moreover $r\in\left|K^{\times}\right|\iff s\in\left|K^{\times}\right|$, and

$\phi_{*}(\zeta)=\zeta(\phi_{1*}(b_{0}),s).$

###### Proof.

From [Ben19, Theorem 7.12] we know this result for rational maps, meaning that

$\phi_{2*}(U_{\lambda})=\begin{cases}\left\\{\left|b_{d}\right|(\lambda
r)^{d}<\left|y-b_{0}\right|<\left|b_{d}\right|r^{d}\right\\}&d>0\\\
\left\\{\left|b_{d}\right|r^{d}<\left|y-b_{0}\right|<\left|b_{d}\right|(\lambda
r)^{d}\right\\}&d<0\end{cases}$

The action of $\phi_{1*}$ is to scale by $q$, in particular Theorem 3.4 shows
that

$\phi_{1*}\left(\left\\{R_{1}<\left|z-b\right|<R_{2}\right\\}\right)=\left\\{R_{1}^{q}<\left|z-\phi_{1*}(b)\right|<R_{2}^{q}\right\\}$

therefore

$\phi_{*}(U_{\lambda})=\phi_{1*}\circ\phi_{2*}(U_{\lambda})=\begin{cases}\left\\{\left|b_{d}\right|^{q}(\lambda
r)^{dq}<\left|y-\phi_{1*}(b_{0})\right|<\left|b_{d}\right|^{q}r^{dq}\right\\}&d>0\\\
\left\\{\left|b_{d}\right|^{q}r^{dq}<\left|y-\phi_{1*}(b_{0})\right|<\left|b_{d}\right|^{q}(\lambda
r)^{dq}\right\\}&d<0\end{cases}$

Similar proves the $\lambda>1$ case. ∎

###  3.5. Local Degrees in Directions

The aim of the next few subsections is to generalise the theory of local
degrees from rational maps to skew products; compare with Benedetto [Ben19,
§7.3, 7.4, 7.5].

######  Definition 3.4.

Let $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ be a skew product. The _relative
degree_ of $\phi_{*}$ (or $\phi$),
$\operatorname{rdeg}(\phi_{*})=\operatorname{rdeg}(\phi)$ is the degree of the
rational function $\phi_{2}(y)=\phi^{*}(y)\in K(y)$.

Suppose that $\phi_{2}$ has the Taylor series at $y=a$ given by

$\phi_{2}(y)=b_{0}+b_{d}(y-a)^{d}+\mathcal{O}\left((y-a)^{d+1}\right),$

where $b_{d}\neq 0$. Then _algebraic multiplicity_ of $\phi$ (and $\phi_{2}$)
at $a$, $\deg_{a}(\phi)=\deg_{a}(\phi_{2})$ is $d$, the degree of the first
non-zero term of the Taylor series $\phi_{2}(y)-b_{0}$ about $y=a$,
equivalently, it is the multiplicity of $a$ as a root of the equation
$\phi_{2}(y)=\phi_{2}(a)$.

In the case of a $k$-rational skew product $\phi$, this
$\operatorname{rdeg}(\phi)$ is the degree of $\phi_{2}(x,y)$ with respect to
$y$ only; this justifies the use of ‘relative’. We expand the definition of
map on directions and local degrees to skew products, again using annuli.

######  Definition 3.5.

Let $\phi_{*}=\phi_{1*}\circ\phi_{2*}$ be a non-constant skew product,
$\zeta\in\mathbb{P}^{1}_{\text{an}}$ and ${\bf v}\in
T_{\zeta}\mathbb{P}^{1}_{\text{an}}$ be a direction at $\zeta$. We define the
_local degree of $\phi_{*}$ at $\zeta$ in the direction ${\bf v}$_,
$\deg_{\zeta,{\bf v}}(\phi)$, and a direction $\phi_{\\#}({\bf v})\in
T_{\phi_{*}(\zeta)}\mathbb{P}^{1}_{\text{an}}$ at $\phi_{*}(\zeta)$ as
follows.

  * •

For a Type I point $\zeta=a$ and the unique direction ${\bf v}$ at $\zeta$,
$\phi_{\\#}({\bf v})$ is the unique direction at $\phi_{*}(\zeta)=b_{0}$, and
the local degree at $a$ in this direction is the algebraic multiplicity of
$\phi_{2}$ at $a$

$\deg_{\zeta,{\bf v}}(\phi)=\deg_{a}(\phi_{2}).$

  * •

For a Type II/III point $\zeta=\zeta(a,r)$ and a direction ${\bf
v}=\vec{v}(a)$, then $\phi_{\\#}({\bf v})$ is the direction at
$\phi_{*}(\zeta)$ containing $\phi_{*}(U)$ where $U=\left\\{\lambda
r<|\zeta-a|<r\right\\}$ is the annulus in Theorem 3.10 and the local degree in
the direction of ${\bf v}$ will be the common Weierstrass degree

$\deg_{\zeta,{\bf v}}(\phi)=\operatorname{wdeg}_{a,r}(\phi_{2}).$

In the case where the direction ${\bf v}=\vec{v}(a)$ contains $\infty$, we use