A Survey of Non-Archimedean Dynamics

1. Introduction

In complex dynamics, one considers a rational function $f\in \mathbb{C}(z)$ as a map from the Riemann sphere $\mathbb{P}^1(\mathbb{C})\coloneq \mathbb{C}\cup \{\infty \}$ to itself. One then studies the iterates of $f$, given by

$$\begin{equation*} f^n \coloneq \underbrace{f\circ \cdots \circ f}_{n\text{ times}} \end{equation*}$$

as they act on the sphere. A rich theory follows — see, for example, the expositions in CG93Mil06 — with famous fractal pictures that exemplify beautiful theorems.

In the past few decades, a younger, parallel theory has developed for the case that we replace $\mathbb{C}$ with $p$-adic and, more generally, non-archimedean fields. Three principal motivations have driven the non-archimedean theory: seeking dynamical phenomena for comparison and contrast with the complex theory, as in HY83RL03; applying local field results to number-theoretic questions, as in BR10Sil07; and analyzing families of complex dynamical systems, especially at degeneration points, as in DMF14Kiw15.

Some of these motivations involve the Laurent series, Puiseux series, and Levi-Civita fields we will discuss in Examples 1.2 and 1.5, while others focus more on the $p$-adic fields we will discuss in Examples 1.3 and 1.6. The unifying theme is to work over non-archimedean fields, which we now describe.

1.1. Absolute values

A field is a set equipped with two binary operations $+$ and $\cdot$ satisfying all the usual algebraic axioms; the prototypical example is the field $\mathbb{Q}$ of rational numbers. In analysis, one works with the real line $\mathbb{R}$, a completion of $\mathbb{Q}$, formed by adjoining limits of all Cauchy sequences. To do so, however, one first needs the absolute value function $|\cdot |$, since the definitions of limits and Cauchy-ness of sequences both use the absolute value. But the familiar absolute value is only one example of a real-valued function that satisfies many properties important in analysis proofs, as follows.

Definition 1.1.

Let $K$ be a field. An absolute value on $K$ is a function $|\cdot |:K\to \mathbb{R}$ such that for all $x,y\in K$,

•

$|x| \geq 0$, with equality if and only if $x=0$,

•

$|xy| = |x|\cdot |y|$,

•

$|x+y|\leq |x| + |y|$.

We say that $|\cdot |$ is non-archimedean if in fact

$$\begin{equation} |x+y|\leq \max \{|x|,|y|\} \quad \text{for all} \quad x,y\in K. \tag{1.1}{} \end{equation}$$

The trivial absolute value on $K$ is given by $|x|=1$ for all $x\in K^{\times } \coloneq K\smallsetminus \{0\}$ (and $|0| \coloneq 0$ by the first axiom above). We will restrict our attention to nontrivial absolute values, i.e., those for which there is some $x\in K$ with $|x|\neq 0,1$.

The familiar absolute value on $\mathbb{R}$ is an archimedean absolute value, because it does not satisfy the non-archimedean triangle inequality 1.1. Our focus in this article will concern fields equipped with a non-archimedean (and nontrivial) absolute value.

Example 1.2.

Let $\mathbb{C}(T)$ be the field of rational functions with complex coefficients. Let $\operatorname {ord}_0(f)$ denote the order of vanishing of $f\in \mathbb{C}(T)^{\times }$ at $T=0$. (If $f(0)\neq 0,\infty$, we set $\operatorname {ord}_0(f)=0$; if $f$ has a pole at $T=0$, we set $\operatorname {ord}_0(f)= - \operatorname {ord}_0(1/f)$.) Define $|\cdot |_0: \mathbb{C}(T) \to \mathbb{R}$ by

$$\begin{equation*} |f|_0 = \begin{cases} \displaystyle e^{-\operatorname {ord}_0(f)} & \text{ if } f\neq 0, \\ 0 & \text{ if } f=0. \end{cases} \end{equation*}$$

It is an exercise to check that $|\cdot |_0$ is a non-archimedean absolute value on $\mathbb{C}(T)$ according to Definition 1.1. If $f$ has a zero at $T=0$, it is “small” — and the greater the order of the zero, the smaller $|f|_0$ is — and if $f$ has a pole there, it is “big.”

Example 1.3.

Fix a prime number $p\geq 2$. The $p$-adic absolute value on $\mathbb{Q}$ is defined by

$$\begin{equation*} \bigg | \frac{r}{s} p^n \bigg |_p \coloneq p^{-n} \quad \text{for } r,s\in \mathbb{Z}\text{ not divisible by } p, \end{equation*}$$

with $|0|_p \coloneq 0$. Thus, a rational number whose numerator is divisible by $p$ is $p$-adically small — the higher the power, the smaller the $p$-adic absolute value — and one whose denominator is divisible by $p$ is $p$-adically big. Thus, $13$ and $389/5$ are very close together $3$-adically, because:

$$\begin{equation*} \bigg | \frac{389}{5} - 13 \bigg |_3 = \bigg | \frac{324}{5} \bigg |_3 = \bigg | \frac{4}{5} \cdot 3^4 \bigg |_3 = \frac{1}{3^4} = \frac{1}{81} . \end{equation*}$$

As foreign as $|\cdot |_p$ may seem at first, it is again an exercise to show that it is a non-archimedean absolute value on $\mathbb{Q}$.

1.2. Non-archimedean fields

Given a field $K$ equipped with an absolute value $|\cdot |_v$, we can mimic basic definitions from real analysis. We say that a sequence $\{a_n\}_{n\geq 0} \subseteq K$:

•

is Cauchy if for every real $\varepsilon >0$, there exists $N\geq 0$ such that for all $m,n\geq N$, we have $|a_m - a_n|_v < \varepsilon$,

•

converges to $a\in K$ if for every real $\varepsilon >0$, there exists $N\geq 0$ such that for all $n\geq N$, we have $|a_n - a|_v < \varepsilon$.

We say that $K$ is complete if every Cauchy sequence converges. If $K$ is complete with respect to a non-archimedean absolute value, we say $K$ is a non-archimedean field. It is another exercise to prove the following series convergence test that many calculus students wish were true for $\mathbb{R}$.

Proposition 1.4.

Let $K$ be a (complete) non-archimedean field with absolute value $|\cdot |_v$. Then for any sequence $\{a_n\}_{n\geq 0}\subseteq K$, we have

$$\begin{equation*} \sum _{n=0}^{\infty } a_n \text{ converges} \quad \text{if and only if} \quad \lim _{n\to \infty } a_n = 0 , \end{equation*}$$

where in both cases, convergence is with respect to $|\cdot |_v$.

Even if $K$ is not complete, we can construct a completion $K_v\supseteq K$ by adjoining limits of all Cauchy sequences.

In archimedean analysis, one proceeds from $\mathbb{Q}$ to $\mathbb{R}$ by completion, and then to $\mathbb{C}$ by algebraic closure. Curiously, the algebraic closure $\overline{K}_v$ of $K_v$ often fails to be complete, but fortunately we may complete once more, and the resulting field $\mathbb{C}_v$ is, like $\mathbb{C}$, both complete and algebraically closed.

Example 1.5.

The completion of $\mathbb{C}(T)$ with respect to $|\cdot |_0$ is the field

$$\begin{equation*} \mathbb{C}((T)) = \bigg \{ \sum _{n\geq n_0} a_n T^n \; \text{ and } \; a_n\in \mathbb{C}\bigg \} \end{equation*}$$

of formal Laurent series with complex coefficients. (That is, we do not worry about convergence issues, and we allow at worst a pole at $T=0$, not an essential singularity.) The completion $\mathbb{L}$ of the algebraic closure $\overline{\mathbb{C}((T))}$ is the Levi-Civita field. (Elements of $\overline{\mathbb{C}((T))}$ are called Puiseux series, and sometimes elements of $\mathbb{L}$ are informally also called Puiseux series.)

The field $\mathbb{L}$ consists of all formal sums

$$\begin{equation*} \sum _{i\geq 0} a_i T^{q_i} \end{equation*}$$

with $a_i\in \mathbb{C}$ and $q_i\in \mathbb{Q}$ for which the sequence $q_0 < q_1 < \cdots$ is an unbounded, strictly increasing sequence of rational numbers. We say that $\mathbb{L}$ is infinitely ramified over $\mathbb{C}((T))$ because of the introduction of arbitrarily large denominators in the rational powers $T^{m/n}$ that appear.

Example 1.6.

The completion of $\mathbb{Q}$ with respect to $|\cdot |_p$ is the field $\mathbb{Q}_p$ of $p$-adic rationals. Since $\lim _{n\to \infty } p^n= 0$ with respect to $|\cdot |_p$, any power series in $p$ with integer coefficients will converge in $\mathbb{Q}_p$. Combining this idea with base-$p$ expansion of positive integers, it is not difficult to characterize $\mathbb{Q}_p$ as the set of all formal Laurent series in $p$ with coefficients in the set $\{0,1,\ldots ,p-1\}$, i.e.,

$$\begin{equation*} \mathbb{Q}_p= \bigg \{ \sum _{n\geq n_0} a_n p^n \; \text{ and } \; a_n\in \{0,1,\ldots ,p-1\} \bigg \} . \end{equation*}$$

In contrast to the Laurent series of Example 1.5, addition and multiplication in $\mathbb{Q}_p$ involve carrying digits as in base-$p$ arithmetic.

When we pass to the completion $\mathbb{C}_p$ of the algebraic closure $\overline{\mathbb{Q}}_p$, then besides increasing ramification by allowing rational powers $p^{m/n}$ as in Example 1.5, we also see an infinite extension of the residue field. Roughly speaking, this means that we go from the original set of coefficients $\{0,1,\ldots ,p-1\}$, which may be identified with the $p$-element field $\mathbb{F}_p$, to an infinite set corresponding to the algebraic closure $\overline{\mathbb{F}}_p$. Unfortunately, there is no explicit description of the elements of $\mathbb{C}_p$ as there was for $\mathbb{L}$, because the existence of the algebraic closure $\overline{\mathbb{Q}}_p$ relies on the use of non-constructive tools such as the Axiom of Choice.

Throughout this paper, we declare that

$\mathbb{C}_v$ denotes an algebraically closed and

complete non-archimedean field

with absolute value $|\cdot |_v$

We remark that the set $|\mathbb{C}_v^{\times }|_v$ of absolute values attained by nonzero elements of $\mathbb{C}_v$ is a dense subgroup of the positive real line $(0,\infty )$ under multiplication, but it is usually not the whole of $(0,\infty )$. Indeed, in Example 1.5, we have $|\mathbb{L}^{\times }|_0 = e^{\mathbb{Q}}$, and in Example 1.6, we have $|\mathbb{C}_p^{\times }|_p = p^{\mathbb{Q}}$.

1.3. Non-archimedean disks

For any point $a\in \mathbb{C}_v$ and any positive real number $r>0$, we define the open and closed disks of radius $r$ centered at $a$ by

$$\begin{align*} D(a,r) & \coloneq \{x\in \mathbb{C}_v\, : \, |x-a|_v < r\}, \\ \overline{D}(a,r) & \coloneq \{x\in \mathbb{C}_v\, : \, |x-a|_v \leq r\}, \end{align*}$$

respectively. That is, we define disks according to the metric given by $d(x,y)=|x-y|_v$, which induces a topology on $\mathbb{C}_v$. Observe that if $r\in |\mathbb{C}_v^{\times }|_v$, then $D(a,r)\subsetneq \overline{D}(a,r)$, in which case we call these two disks rational open and rational closed, respectively. On the other hand, if $r\not \in |\mathbb{C}_v^{\times }|_v$, then $D(a,r) = \overline{D}(a,r)$, and we call this disk irrational. Non-archimedean disks, and the topology on $\mathbb{C}_v$, have several properties not shared by their archimedean cousins, but which are easy to prove:

(a)

Any point of a disk is its center. That is, if $b\in D(a,r)$, then $D(b,r)=D(a,r)$; and if $b\in \overline{D}(a,r)$, then $\overline{D}(b,r)=\overline{D}(a,r)$.

However, the radius of a disk $D$ is well-defined, as $r=\sup \{|x-y|_v : x,y\in D\}$.

(b)

If two disks have nonempty intersection, then one contains the other.

(c)

All disks are both open and closed topologically. But any disk is exactly one of the three types: rational open, rational closed, or irrational.

(d)

$\mathbb{C}_v$ is not locally compact.

(e)

$\mathbb{C}_v$ is totally disconnected. That is, the only nonempty connected subsets of $\mathbb{C}_v$ are singletons.

Properties (c), (d), and (e) above follow from the fact that any rational closed disk may be written as a disjoint union of open disks. For example, in the Levi-Civita field $\mathbb{L}$, it is easy to check that

$$\begin{equation} \overline{D}(0,1) = \bigcup _{a\in \mathbb{C}} D(a, 1) \tag{1.2}{} \end{equation}$$

and that the union in 1.2 is a disjoint union, partitioning the set of Levi-Civita series in $\overline{D}(0,1)$ according to constant term. Thus, speaking in very loose terms, $\mathbb{C}_v$ topologically resembles an infinite disjoint union of Cantor sets.

Lest you think that $\mathbb{C}_v$ is all pathology, here is one lovely property of non-archimedean disks. Let $D\subseteq \mathbb{C}_v$ be a disk, and let $f\in \mathbb{C}_v[z]$ be a nonconstant polynomial. Then the image $f(D)$ is also a disk — not just homeomorphic to a disk, but actually equal to a disk.

2. Basic Dynamics on $\mathbb{C}_v$

As in complex dynamics, a rational function $f(z)\in \mathbb{C}_v(z)$ maps the projective line $\mathbb{P}^1(\mathbb{C}_v)\coloneq \mathbb{C}_v\cup \{\infty \}$ to itself, and we wish to study the iterates $f^n$. For any point $x\in \mathbb{P}^1(\mathbb{C}_v)$, the sets

$$\begin{align*} \operatorname {Orb}_f^+(x) & \coloneq \{ f^n(x) : n\geq 0\}, \\ \operatorname {Orb}_f^-(x) & \coloneq \{ y \in \mathbb{P}^1(\mathbb{C}_v): \\ & \quad \quad \quad f^n(y) = x \text{ for some } n \geq 0 \} \end{align*}$$

are called the forward orbit and the backward orbit of $x$, respectively.

Writing $f\in \mathbb{C}_v(z)$ as $f=g/h$ for relatively prime polynomials $g,h\in \mathbb{C}_v[z]$, we define the degree of $f$ to be

$$\begin{equation*} \deg f \coloneq \max \{\deg g, \deg h \} . \end{equation*}$$

It is often convenient to change coordinates on $\mathbb{P}^1(\mathbb{C}_v)$ via a degree-one rational function, i.e., via a Möbius transformation $\theta \in \operatorname {PGL}(2,\mathbb{C}_v)$, given by

$$\begin{equation*} \theta (z) = \frac{az+b}{cz+d} \quad \quad a,b,c,d\in \mathbb{C}_v, \;\; ad-bc\neq 0 . \end{equation*}$$

If we change coordinates by $\theta$ on both the domain and range of the map $f:\mathbb{P}^1(\mathbb{C}_v)\to \mathbb{P}^1(\mathbb{C}_v)$, then $f$ becomes the conjugated map $\theta \circ f \circ \theta ^{-1}$.

Naturally, we are most interested in dynamical phenomena that are not specific to a particular choice of coordinate. For instance, a point $x\in \mathbb{P}^1(\mathbb{C}_v)$ is

•

fixed by $f$ if $f(x)=x$.

•

periodic under $f$ if there is some $n\geq 1$ such that $f^n(x)=x$. In that case, we say $x$ has period $n$; and if $n$ is the smallest such integer, we say $x$ has exact period $n$.

•

preperiodic under $f$ if there are integers $n>m\geq 0$ such that $f^n(x)=f^m(x)$, i.e., if some $f^m(x)$ is periodic, or equivalently, if the forward orbit $\operatorname {Orb}_f^+(x)$ is finite.

•

exceptional if the backward orbit $\operatorname {Orb}_f^-(x)$ is finite.

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a critical point, or ramification point, if $f'(x)=0$. (At least for $x, f(x) \neq \infty$.)

All of these properties are coordinate-independent, in the sense that $x$ has the given property for $f$ if and only if $\theta (x)$ has the same property for $\theta \circ f \circ \theta ^{-1}$.

If $x$ is periodic under $f$ of exact period $n$, then the multiplier of $x$, defined to be $\lambda \coloneq (f^n)'(x)$, is also coordinate-independent. Locally near $x$, we have

$$\begin{equation} f^n(z)\approx x + \lambda (z-x). \tag{2.1}{} \end{equation}$$

If $|\lambda |_v< 1$, we say $x$ is attracting, because 2.1 implies that nearby points in $\mathbb{P}^1(\mathbb{C}_v)$ approach $x$ under iteration of $f^n$. Similarly if $|\lambda |_v>1$, we say $x$ is repelling. If $|\lambda |_v=1$, we say $x$ is indifferent, in which case one can show that $f^n$ is an isometry on a neighborhood of $x$. This last fact may surprise readers familiar with complex dynamics, where indifferent periodic points exhibit more intricate behavior; but the isometry here is a simple consequence of 2.1 and the non-archimedean triangle inequality 1.1.

3. The Berkovich Line

As observed by Rivera-Letelier in RL03, the appropriate setting for non-archimedean dynamics is not the classical projective line $\mathbb{P}^1(\mathbb{C}_v)$, but rather the Berkovich projective line $\mathbb{P}^1_{\mathrm{an}}$, which is a topological space that contains $\mathbb{P}^1(\mathbb{C}_v)$ as a dense subset, first described in Ber90. The Berkovich line $\mathbb{P}^1_{\mathrm{an}}$ is both compact and path-connected, in stark contrast with $\mathbb{P}^1(\mathbb{C}_v)$.

The precise definition of $\mathbb{P}^1_{\mathrm{an}}$ involves multiplicative seminorms on the polynomial ring $\mathbb{C}_v[z]$ — for details, see Chapter 6 of Ben19 or Chapters 1–2 of BR10 — but here we describe the space more intuitively. In particular, there are four types of points in $\mathbb{P}^1_{\mathrm{an}}$:

•

The points in the classical projective line $\mathbb{P}^1(\mathbb{C}_v)=\mathbb{C}_v\cup \{\infty \}$ are of Type I.

•

Each closed disk $D=\overline{D}(a,r) \subseteq \mathbb{C}_v$ gives a point of $\mathbb{P}^1_{\mathrm{an}}$, which we denote $\zeta (a,r)$. If $D$ is a rational closed disk, then $\zeta (a,r)$ is of Type II; if $D$ is irrational, then $\zeta (a,r)$ is of Type III.

•

Type IV points correspond to decreasing chains of disks $D_1\supsetneq D_2\supsetneq \cdots$ with empty intersection, a perhaps surprising phenomenon that occurs in some complete non-archimedean fields, including $\mathbb{L}$ and $\mathbb{C}_p$.

Figure 1 is a rough sketch of $\mathbb{P}^1_{\mathrm{an}}$. The line segments you see are truly line segments, i.e., homeomorphic copies of the real interval $[0,1]$. For example, the points on the line segment running between the two type I points $0$ and $\infty$ are the points $\zeta (0,r)$ of types II and III, for $0<r<\infty$. These include the type II point $\zeta (0,1)$, corresponding to the closed unit disk $\overline{D}(0,1)$, which is called the Gauss point. Emanating from the Gauss point, or indeed from any type II point, are infinitely many branches, corresponding to the infinitely many elements of the residue field (plus one extending towards $\infty$), or equivalently corresponding to the infinitely many open disks in the disjoint union 1.2.

Figure 1.

The Berkovich projective line.

Thus, for example, we may proceed along the line segment from $0$ to $\zeta (0,1)$, which consists of points of the form $\zeta (0,r)$, by increasing $r$ from $0$ to $1$. Then, because $\overline{D}(0,1)=\overline{D}(1,1)$, we may proceed from $\zeta (0,1)=\zeta (1,1)$ to the type I point $1$ via the line segment consisting of points of the form $\zeta (1,r)$, with $r$ decreasing from $1$ back down to $0$. Similarly, the type I points $b,c\in \mathbb{C}_p$ in Figure 1 satisfy $0<|b|<|c|=1$, and we can travel from one to the other as follows. First, proceed along the line segment from $b$ to $\xi \coloneq \zeta (b,|b|)=\zeta (0,|b|)$ via points of the form $\zeta (b,r)$ with $0<r\leq |b|$; then from there to $\zeta (0,1)=\zeta (c,1)$ via points of the form $\zeta (0,r)=\zeta (b,r)$ with $|b|\leq r\leq 1$; and finally to $c$ via points of the form $\zeta (c,r)$, with $r$ decreasing back to $0$. The tracing of such paths illustrates, at least conceptually, that $\mathbb{P}^1_{\mathrm{an}}$ is path-connected.

Along any segment, there is a dense collection of type II points, because as noted earlier, $|\mathbb{C}_v^{\times }|$ is dense in $(0,\infty )$, and each radius $r\in |\mathbb{C}_v^{\times }|$ gives a type II point. Moreover, as we have just seen, there is infinite branching at each type II point. Thus, $\mathbb{P}^1_{\mathrm{an}}$ has the structure of an infinitely-branched tree: along each segment, there are infinitely many points of branching, each with an infinite number of branches. The type I points, which may be viewed as lying around the outer rim of the figure, sit at the ends of many of the branches. (The type IV points, which we will mostly brush under the rug here, lie at the ends of other branches.)

Figure 1 may appear to conflict with our earlier claim that the set $\mathbb{P}^1(\mathbb{C}_v)$ of Type I points is dense in $\mathbb{P}^1_{\mathrm{an}}$. However, the topology on $\mathbb{P}^1_{\mathrm{an}}$ is weaker than the picture might suggest. In particular, for any Type II point $\zeta$ and any open set $U$ containing $\zeta$, all but finitely many of the branches emanating from $\zeta$ must be completely contained in $U$.

Any rational function $f\in \mathbb{C}_v(z)$ has a unique extension from $\mathbb{P}^1(\mathbb{C}_v)$ to a continuous map $f:\mathbb{P}^1_{\mathrm{an}}\to \mathbb{P}^1_{\mathrm{an}}$. If $f$ is nonconstant, this map is surjective, preserving types of points. For example, if $f\in \mathbb{C}_v[z]$ is a nonconstant polynomial and $\zeta$ is a point of Type II or III corresponding to a disk $D$, then $f(\zeta )$ is the point of the same type corresponding to the disk $f(D)$. For a fully general description of $f(\zeta )$ for $f\in \mathbb{C}_v(z)$ and $\zeta \in \mathbb{P}^1_{\mathrm{an}}$, see Chapter 7 of Ben19.

Each open disk $D(a,r)\subseteq \mathbb{C}_v$ has a Berkovich version $D_{\mathrm{an}}(a,r)$ whose set of type I points is precisely $D(a,r)$. Similarly, a closed disk $\overline{D}(a,r)$ has a Berkovich version $\overline{D}_{\mathrm{an}}(a,r)$. Moreover, we define an open disk in $\mathbb{P}^1_{\mathrm{an}}$ containing $\infty$ to be the complement of a (Berkovich) closed disk, and a closed disk in $\mathbb{P}^1_{\mathrm{an}}$ containing $\infty$ to be the complement of a (Berkovich) open disk. Through the rest of this paper, a “disk” will refer to a Berkovich disk in this sense, i.e., a disk in $\mathbb{P}^1_{\mathrm{an}}$, possibly containing the point $\infty$.

4. Dynamics on $\mathbb{P}^1_{\mathrm{an}}$

Throughout this section, let $f\in \mathbb{C}_v(z)$ be a rational function of degree $d\geq 2$, and consider its iterates $f^n$ as functions from $\mathbb{P}^1_{\mathrm{an}}$ to itself.

4.1. Periodic points

As in Section 2, points of $\mathbb{P}^1_{\mathrm{an}}$ may or may not be fixed, periodic, or preperiodic under $f$. Moreover, Berkovich points of types II, III, or IV that are periodic may be classified as indifferent or repelling, as follows. Suppose $\zeta \in \mathbb{P}^1_{\mathrm{an}}\smallsetminus \mathbb{P}^1(\mathbb{C}_v)$ is periodic of period $n$. If $\zeta$ has a small neighborhood $U$ that maps into itself, then $\zeta$ is indifferent; otherwise $\zeta$ is repelling, although the reader should be cautioned that even in that case, $\zeta$ does not generally “repel” individual nearby points. (On the other hand, periodic points of types II–IV cannot be considered attracting, because it turns out that if $\bigcap _{i\geq 1} f^{ni}(U)=\{\zeta \}$, then $\zeta$ must be of type I.) It would take us too far afield to define this classification formally — although in practice it is straightforward — so we will settle for an example.

Example 4.1.

Let $f(z)=a_0 + a_1 z + \cdots + a_d z^d$ be a polynomial with $|a_i|_v \leq 1$ for all $i$. Suppose that $|a_m|_v=1$ for some $m\geq 1$, with $|a_i|_v < 1$ for $i>m$. Then $f$ fixes the Gauss point $\zeta (0,1)$, which is indifferent if $m=1$, but repelling if $m\geq 2$.

Indeed, if $m=1$, then there is some small enough $r>1$ such that $f$ maps the open disk $D_{\mathrm{an}}(0,r)$ into itself. On the other hand, if $m\geq 2$, then for any $r>1$, we have

$$\begin{equation*} f\big ( D_{\mathrm{an}}(0,r) \big ) \supseteq D_{\mathrm{an}}(0,r^m) \supsetneq D_{\mathrm{an}}(0,r), \end{equation*}$$

and hence $\bigcup _{n\geq 0} f^n(D_{\mathrm{an}}(0,r))=\mathbb{P}^1_{\mathrm{an}}\smallsetminus \{\infty \}$.

4.2. Fatou and Julia sets

The two qualitatively different behaviors in Example 4.1 inspire the following definition.

Definition 4.2.

Let $f\in \mathbb{C}_v(z)$. The Julia set of $f$ is the set $\mathcal{J}_{\mathrm{an},f}$ of points $\zeta \in \mathbb{P}^1_{\mathrm{an}}$ with the following property: for every neighborhood $U$ of $\zeta$, the union of iterates $\bigcup _{n\geq 0} f^n(U)$ omits only finitely many points of $\mathbb{P}^1_{\mathrm{an}}$. Its complement