On a Characterization of Polynomials Among Rational Functions in Non-Archimedean Dynamics

Abstract

We study a question on characterizing polynomials among rational functions of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.

1 Introduction

Let K be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value $|\cdot |$. The Berkovich projective line ${\textsf {P}}^1={\textsf {P}}^1(K)$ is, as a topological augmentation of the (classical) projective line ${\mathbb {P}}^1={\mathbb {P}}^1(K)=K\cup \{\infty \}$, a compact, locally compact, uniquely arcwise connected, and Hausdorff topological space. The set ${\textsf {H}}^1:={\textsf {P}}^1\setminus {\mathbb {P}}^1$ is called the Berkovich upper half space in ${\textsf {P}}^1$.

Let $f\in K(z)$ be a rational function of degree $d>1$. For every $n\in {\mathbb {N}}$, set $f^n:=f\circ f^{n-1}$, where $f^0:={\mathrm {Id}}_{{\mathbb {P}}^1}$. The action of f on ${\mathbb {P}}^1$ uniquely extends to a continuous endomorphism on ${\textsf {P}}^1$, which is still open, surjective, and fiber-discrete, and preserves both ${\mathbb {P}}^1$ and ${\textsf {H}}^1$. Let us define the Berkovich Julia set ${{\mathsf {J}}}(f)$ of f by the set of all points ${\mathcal {S}}\in {\textsf {P}}^1$ such that for any open neighborhood U of ${\mathcal {S}}$ in ${\textsf {P}}^1$,

$$\begin{aligned} {\textsf {P}}^1\setminus E(f)\subset \bigcup _{n\in {\mathbb {N}}}f^n(U), \end{aligned}$$

where the set $E(f):=\{a\in {\mathbb {P}}^1:\#\bigcup _{n\in {\mathbb {N}}}f^{-n}(a)<\infty \}$ is called the (classical) exceptional set of f and is at most countable subset in ${\mathbb {P}}^1$. The local degree function $\deg _{\, \cdot }f$ on ${\mathbb {P}}^1$ also canonically extends to ${\textsf {P}}^1$, and this extended local degree function $\deg _{\,\cdot }(f)$ induces a canonical pullback operator $f^*$ from the space of all Radon measures on ${\textsf {P}}^1$ to itself (see Sect. 2.2 below). Corresponding to the construction of the unique maximal entropy measure in complex dynamics (studied since Lyubich [20], Freire–Lopes–Mañé [15], Mañé [23]), the f-canonical measure $\mu _f$ on ${\textsf {P}}^1$ has been constructed as the unique probability Radon measure $\nu$ on ${\textsf {P}}^1$ such that

$$\begin{aligned} f^*\nu =d\cdot \nu \text { on }{\textsf {P}}^1\quad \text {and that}\quad \nu (E(f))=0, \end{aligned}$$

so in particular $\mu _f$ is invariant under f in that $f_*\mu _f=\mu _f$ on ${\textsf {P}}^1$. The support of $\mu _f$ coincides with ${{\mathsf {J}}}(f)$ and is the minimal non-empty and closed subset in ${\textsf {P}}^1$ backward invariant under f [14]. The Berkovich Fatou set of f is defined by

$$\begin{aligned} {\mathsf {F}}(f):={\textsf {P}}^1\setminus {{\mathsf {J}}}(f), \end{aligned}$$

and each component of ${\mathsf {F}}(f)$ is called a Berkovich Fatou component of f. We note that $E(f)\subset {\mathsf {F}}(f)$. A Berkovich Fatou component of f is mapped properly to a Berkovich Fatou component of f under f, and the preimage of a Berkovich Fatou component of f under f is the union of at most d Berkovich Fatou components of f.

Notation 1.1

For every $z\in {\mathsf {F}}(f)\cap {\mathbb {P}}^1$, let $D_z=D_z(f)$ be the Berkovich Fatou component of f containing z.

For any $z\in {\mathsf {F}}(f)\cap {\mathbb {P}}^1$, the compact subset ${\textsf {P}}^1\setminus D_z$ in ${\textsf {P}}^1$ is of logarithmic capacity $>0$ with pole z, or equivalently, there is the unique equilibrium mass distribution $\nu _{z,{\textsf {P}}^1\setminus D_z}$ on ${\textsf {P}}^1\setminus D_z$ with pole z, which is in fact supported by $\partial D_z\subset {{\mathsf {J}}}(f)$ (we will recall some details on the logarithmic potential theory on ${\textsf {P}}^1$ in Sect. 2.4 below). If $f(\infty )=\infty \in {\mathsf {F}}(f)$, then $\nu _{\infty ,{\textsf {P}}^1\setminus D_\infty }$ is invariant under f in that

$$\begin{aligned} f_*(\nu _{\infty ,{\textsf {P}}^1\setminus D_\infty })=\nu _{\infty ,{\textsf {P}}^1\setminus D_\infty } \quad \text {on }{\textsf {P}}^1 \end{aligned}$$

(see Lemma 4.7 below). If moreover $f\in K[z]$ or equivalently $f^{-1}(\infty )=\{\infty \}$, then $\infty \in E(f)$, $f^{-1}(D_\infty )=D_\infty$, and we can see

$$\begin{aligned} \mu _f=\nu _{\infty ,{\textsf {P}}^1\setminus D_{\infty }}\quad \text {on }{\textsf {P}}^1 \end{aligned}$$

(since Brolin [9] in complex dynamics). Let $\delta _{{\mathcal {S}}}$ be the Dirac measure on ${\textsf {P}}^1$ at ${\mathcal {S}}\in {\textsf {P}}^1$.

Our aim is to study whether polynomials can be characterized among rational functions of degree $>1$ using potential theory in non-archimedean setting, corresponding to the studies [19, 21, 22, 25, 29, 30] in complex dynamics. Concretely, we study the following question on a characterization of polynomials among rational functions in non-archimedean dynamics.

Question

Let $f\in K(z)$ be a rational function of degree $>1$, and suppose that $f(\infty )=\infty \in {\mathsf {F}}(f)$ (so in particular $f(D_\infty )=D_\infty$) and that ${{\mathsf {J}}}(f)\not \subset {\textsf {H}}^1$. Then, are the statements

$$\begin{aligned} \text {(i) } f\in K[z]\quad \text {and}\quad \text {(ii) } \mu _f=\nu _{\infty ,{\textsf {P}}^1\setminus D_\infty } \ on\ {\textsf {P}}^1 \end{aligned}$$

equivalent?

The corresponding question in complex dynamics has been answered affirmatively (Lopes[21]).

Here are a few comments on this Question. We already mentioned that (i) implies (ii) (without assuming ${{\mathsf {J}}}(f)\not \subset {\textsf {H}}^1$). It is not difficult to construct such $f\in K(z)\setminus K[z]$ of degree $>1$ that $f(D_\infty )=D_\infty$, that $f(\infty )\ne \infty \in {\mathsf {F}}(f)$, that ${{\mathsf {J}}}(f)\not \subset {\textsf {H}}^1$, and that $\mu _f=\nu _{\infty ,{\textsf {P}}^1\setminus D_\infty }$ on ${\textsf {P}}^1$ (e.g., Remark 6.5 below). On the other hand, if ${{\mathsf {J}}}(f)\subset {\textsf {H}}^1$, then for any $g\in K(z)$ of the same degree as that of f which is close enough to f (in the coefficients topology), both the Berkovich Julia set ${{\mathsf {J}}}(g)$ of g and the action of g on ${{\mathsf {J}}}(g)$ are same as those of f (cf. [14, Sect. 5.3]). Since there is $f\in K[z]$ of degree $>1$ satisfying ${{\mathsf {J}}}(f)\subset {\textsf {H}}^1$ (e.g., such f that has a potentially good reduction, see below a characterization of this condition), for any such f and any $b\in K$, if $0<|b|\ll 1$, then the small perturbation $f_b(z):=f(z)/(bz+1)\in K(z)\setminus K[z]$ of $f=f/1$ in K(z) is of the same degree as that of f and satisfies that $f_b(\infty )=\infty \in {\mathsf {F}}(f_b)$, that ${{\mathsf {J}}}(f_b)={{\mathsf {J}}}(f)\subset {\textsf {H}}^1$, and that $\mu _{f_b}=\nu _{\infty ,{\textsf {P}}^1\setminus D_\infty (f_b)}$ on ${\textsf {P}}^1$.

Recall that f has a potentially good reduction if and only if there exists a point ${\mathcal {S}}\in {\textsf {H}}^1$ such that

$$\begin{aligned} f^{-1}({\mathcal {S}})=\{{\mathcal {S}}\}; \end{aligned}$$

then ${{\mathsf {J}}}(f)=\{{\mathcal {S}}\}(\subset {\textsf {H}}^1$ so $\infty \in {\mathsf {F}}(f)$) and $\mu _f=\nu _{\infty ,{{\textsf {P}}}^1\setminus D_\infty }=\delta _{{\mathcal {S}}}$ on ${{\textsf {P}}}^1$ (see also Remark 3.2 below). We say f has no potentially good reductions if f does not have a potentially good reduction.

We already mentioned that the total invariance $f^{-1}(D_\infty )=D_\infty$ of $D_\infty$ under f is a necessary condition for $f\in K[z]$. Our first result is the following more general statement, under no potentially good reductions:

Theorem 1

Let K be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value. Let $f\in K(z)$ be a rational function of degree $>1$. If $\infty \in {{\mathsf {F}}}(f)$, $f(D_\infty )=D_\infty$, $\mu _f=\nu _{\infty ,{{\textsf {P}}}^{1}\setminus D_\infty }$ on ${{\textsf {P}}}^1$, and f has no potentially good reductions, then

$$\begin{aligned} f^{-1}(D_\infty )=D_\infty . \end{aligned}$$

Our second result is that even if we assume in addition ${{\mathsf {J}}}(f)\subset {\mathbb {P}}^1$, the latter statement (ii) does not necessarily imply the former (i) in Question.

Pick a prime number p. The p-adic norm $|\cdot |_p$ on ${\mathbb {Q}}$ is normalized so that for any $m,\ell \in {\mathbb {Z}}\setminus \{0\}$ not divisible by p and any $r\in {\mathbb {Z}}$, $\bigl |\frac{m}{\ell }p^r\bigr |_p=p^{-r}$. The completion ${\mathbb {Q}}_p$ of $({\mathbb {Q}},|\cdot |_p)$ is still a field, and the extended norm $|\cdot |_p$ on ${\mathbb {Q}}_p$ extends to an algebraic closure $\overline{{\mathbb {Q}}_p}$ of ${\mathbb {Q}}_p$ as a norm. The completion ${\mathbb {C}}_p$ of $(\overline{{\mathbb {Q}}_p},|\cdot |_p)$ is still an algebraically closed field, and the extended norm $|\cdot |_p$ on ${\mathbb {C}}_p$ is a non-trivial and non-archimedean absolute value on ${\mathbb {C}}_p$. The completion ${\mathbb {Z}}_p$ of $({\mathbb {Z}},|\cdot |_p)$ is a complete discrete valued local ring and has the unique maximal ideal $p{\mathbb {Z}}_p$, and coincides with the ring of ${\mathbb {Q}}_p$-integers $\{z\in {\mathbb {Q}}_p:|z|_p\le 1\}$. In particular, the residual field of ${\mathbb {Q}}_p$ is ${\mathbb {F}}_p$.

The following counterexample of the implication (ii)$\Rightarrow$(i) in Question is suggested to the authors by Juan Rivera-Letelier:

Theorem 2

Pick a prime number p, and set

$$\begin{aligned} f(z):=\frac{z^p-1}{p}\in {\mathbb {Q}}[z]\quad \text {and}\quad A(z):=\frac{az+b}{cz+d}\in \mathrm {PGL}(2,{\mathbb {Z}}_p). \end{aligned}$$

If $c\ne 0$ and (a, b, c, d) is close enough to (1, 0, 0, 1) in $({\mathbb {Z}}_p)^4$, then there is an attracting fixed point $z_A$ of $f\circ A$ in ${\mathbb {C}}_p\setminus {\mathbb {Z}}_p$ (so $z_A\in {{\mathsf {F}}}(f\circ A))$ such that

$$\begin{aligned} {{\mathsf {J}}}(f\circ A)= & {} {\mathbb {Z}}_p={{\textsf {P}}}^{1}({\mathbb {C}}_p)\setminus D_{z_A}(f\circ A)\quad \text {and}\\ \nu _{z_A,{\mathbb {Z}}_p}= & {} \nu _{\infty ,{\mathbb {Z}}_p}\quad \text {on }{{{\textsf {P}}}}^{1}({\mathbb {C}}_p). \end{aligned}$$

Then setting $m_A(z):=1/(z-z_A)\in \mathrm {PGL}(2,{\mathbb {C}}_p)$, the rational function $g_A(z):=m_A\circ (f\circ A)\circ m_A^{-1}\in {\mathbb {C}}_p(z)$ is of degree p and satisfies $g_A\not \in {\mathbb {C}}_p[z]$, $g_A(\infty )=\infty \in {{\mathsf {F}}}(g_A)$, ${{{\mathsf {J}}}}(g_A)\subset {\mathbb {P}}^{1}({\mathbb {C}}_p)$, and

$$\begin{aligned} \mu _{g_A}=\nu _{\infty ,{{\textsf {P}}}^{1}({\mathbb {C}}_p)\setminus D_\infty (g_A)}\quad \text {on }{{\textsf {P}}}^{1}({\mathbb {C}}_p). \end{aligned}$$

1.1 Organization of this Article

In Sects. 2 and 3, we prepare background material from potential theory and dynamics, respectively. In Sect. 4, we make preparatory computations from potential theory and give a proof of the invariance of $\nu _{\infty ,{{\textsf {P}}}^{1}\setminus D_\infty }$ under f when $f(\infty )=\infty \in {{\mathsf {F}}}(f)$. In Sects. 5 and 6, we show Theorems 1 and 2, respectively.

2 Background from Potential Theory on ${\textsf {P}}^1$

Let K be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value $|\cdot |$; in general, a norm $|\cdot |$ on a field k is non-trivial if $|k|\not \subset \{0,1\}$, and is non-archimedean if $|\cdot |$ satisfies the strong triangle inequality

$$\begin{aligned} |x+y|\le \max \{|x|,|y|\}\quad \text {for any }x,y\in k. \end{aligned}$$

For the foundation of potential theory on ${{\textsf {P}}}^1={{\textsf {P}}}^1(K)$, see [5, Sects. 5, 8] , [12, Sect. 7], [13, Sect. 3], [33], and the survey [18, Sects. 1–4], and the book [6, Sect. 13]. In what follows, we adopt a presentation from [28, Sects. 2, 3].

Notation 2.1

Let

$$\begin{aligned} \pi :K^2\setminus \{(0,0)\}\rightarrow {\mathbb {P}}^1={\mathbb {P}}^1(K)=K\cup \{\infty \} \end{aligned}$$

be the canonical projection such that

$$\begin{aligned} \pi (p_0,p_1)= {\left\{ \begin{array}{ll} p_1/p_0 &{}\text {if }p_0\ne 0,\\ \infty &{}\text {if }p_0=0, \end{array}\right. } \end{aligned}$$

following the convention on coordinate of ${\mathbb {P}}^1$ from the book [16].

On $K^2$, let $\Vert (p_0,p_1)\Vert$ be the maximum norm $\max \{|p_0|,|p_1|\}$. With the wedge product $(p_0,p_1)\wedge (q_0,q_1):=p_0q_1-p_1q_0$ on $K^2$, the normalized chordal metric [z, w] on ${\mathbb {P}}^1$ is the function

$$\begin{aligned} {[}z,w]:=\frac{|p\wedge q|}{\Vert p\Vert \cdot \Vert q\Vert }(\le 1) \end{aligned}$$

on ${\mathbb {P}}^1\times {\mathbb {P}}^1$, where $p\in \pi ^{-1}(z),q\in \pi ^{-1}(w)$.

2.1 Berkovich Projective Line ${\textsf {P}}^1$

A (K-closed) disk in K is a subset in K written as $\{z\in K:|z-a|\le r\}$ for some $a\in K$ and some $r\ge 0$. By the strong triangle inequality, two decreasing infinite sequences of disks in K either infinitely nest or are eventually disjoint. This alternative induces the cofinal equivalence relation among decreasing (or more precisely, nesting and non-increasing) infinite sequences of disks in K, and the set of all cofinal equivalence classes ${\mathcal {S}}$ of decreasing infinite sequences $(B_n)$ of disks in K together with $\infty \in {\mathbb {P}}^1$ is, as a set, nothing but ${{\textsf {P}}}^1$ ([7, p. 17]); if $B_{{\mathcal {S}}}:=\bigcap _n B_n\ne \emptyset$, then $B_{{\mathcal {S}}}$ is itself a disk in K, and we also say ${\mathcal {S}}$ is represented by $B_{{\mathcal {S}}}$. For example, the canonical (or Gauss) point ${\mathcal {S}}_{{\text {can}}}$ in ${{\textsf {P}}}^1$ is represented by the the ring of K-integers

$$\begin{aligned} \mathcal {O}_K:=\{z\in K:|z|\le 1\}, \end{aligned}$$

and each $z\in K$ is represented by the disk $\{z\}$ in K. The above alternative between two (decreasing infinite sequences of) disks in K also induces a canonical ordering $\succeq$ on ${{\textsf {P}}}^1$ so that $\infty$ is the unique maximal element in $({{\textsf {P}}}^1,\succeq )$ and that for every ${\mathcal {S}},{\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty \}$ satisfying $B_{{\mathcal {S}}},B_{{\mathcal {S}}'}\ne \emptyset$, ${\mathcal {S}}\succeq {\mathcal {S}}'$ iff $B_{{\mathcal {S}}}\supset B_{{\mathcal {S}}'}$ (the description of $\succeq$ is a little complicated unless $B_{{\mathcal {S}}},B_{{\mathcal {S}}'}\ne \emptyset$), and equips ${{\textsf {P}}}^1$ with a (profinite) tree structure. The topology of ${{\textsf {P}}}^1$ coincides with the weak (or observer) topology on ${{\textsf {P}}}^1$ as a (profinite) tree, so that ${{\textsf {P}}}^1$ is compact and uniquely arcwise-connected, and contains both ${\mathbb {P}}^1$ and ${\textsf {H}}^1$ as dense subsets. For the details on the tree structure on ${{\textsf {P}}}^1$, see e.g. [18, Sect. 2].

2.2 Action of Rational Functions on ${\textsf {P}}^1$

Let $h\in K(z)$ be a rational function. The action of h on ${\mathbb {P}}^1$ uniquely extends to a continuous endomorphism on ${{\textsf {P}}}^1$. Suppose in addition that $\deg h>0$. Then the extended action of h on ${{\textsf {P}}}^1$ is surjective and open, has discrete (so finite) fibers, and preserves both ${\mathbb {P}}^1$ and ${\textsf {H}}^1$, and the local degree function $z\mapsto \deg _zh$ on ${\mathbb {P}}^1$ also canonically extends to ${{\textsf {P}}}^1$ so that for every ${\mathcal {S}}\in {{\textsf {P}}}^1$,

$$\begin{aligned} \sum _{{\mathcal {S}}'\in h^{-1}({\mathcal {S}})}\deg _{{\mathcal {S}}'}h=\deg h. \end{aligned}$$

The action of h on ${{\textsf {P}}}^1$ induces the push-forward operator $h_*$ on the space of all continuous functions on ${{\textsf {P}}}^1$ to itself and, by duality, also the pullback operator $h^*$ on the space of all Radon measures on ${{\textsf {P}}}^1$ to itself; for every continuous test function $\phi$ on ${{\textsf {P}}}^1$, $(h_*\phi )(\cdot )=\sum _{{\mathcal {S}}'\in h^{-1}(\cdot )}(\deg _{{\mathcal {S}}'}h)\cdot \phi ({\mathcal {S}}')$ on ${{\textsf {P}}}^1$, and for every ${\mathcal {S}}\in {{\textsf {P}}}^1$, $h^*\delta _{{\mathcal {S}}} =\sum _{{\mathcal {S}}'\in h^{-1}({\mathcal {S}})}(\deg _{{\mathcal {S}}'}h)\cdot \delta _{{\mathcal {S}}'}$ on ${{\textsf {P}}}^1$. For more details, see [5, Sect. 9], [14, Sect. 2.2].

2.3 Kernel Functions and the Laplacian on ${\textsf {P}}^1$

The generalized Hsia kernel $[{\mathcal {S}},{\mathcal {S}}']_{{\text {can}}}$ on ${{\textsf {P}}}^1$ with respect to ${\mathcal {S}}_{{\text {can}}}$ is a unique upper semicontinuous and separately continuous extension of the chordal distance function ${\mathbb {P}}^1\times {\mathbb {P}}^1\ni (z,z')\mapsto [z,z']$ to ${{\textsf {P}}}^1\times {{\textsf {P}}}^1$.

More generally, for every $z_0\in {\mathbb {P}}^1$, the generalized Hsia kernel

$$\begin{aligned} {[}{\mathcal {S}},{\mathcal {S}}']_{z_0} := {\left\{ \begin{array}{ll} \displaystyle \frac{[{\mathcal {S}},{\mathcal {S}}']_{{\text {can}}}}{[{\mathcal {S}},z_0]_{{\text {can}}}\cdot [{\mathcal {S}}',z_0]_{{\text {can}}}} &{} \text {on }({{\textsf {P}}}^1\setminus \{z_0\})\times ({{\textsf {P}}}^1\setminus \{z_0\})\\ +\infty &{} \text {on }(\{z_0\}\times {{\textsf {P}}}^1)\cup ({{\textsf {P}}}^1\times \{z_0\}) \end{array}\right. } \end{aligned}$$

on ${{\textsf {P}}}^1$ with respect to $z_0$ is a unique upper semicontinuous and separately continuous extension of the function $({\mathbb {P}}^1\setminus \{z_0\})\times ({\mathbb {P}}^1\setminus \{z_0\})\ni (z,z')\mapsto [z,z']/([z,z_0]\cdot [z',z_0])$ as a function ${{\textsf {P}}}^1\times {{\textsf {P}}}^1\rightarrow [0,+\infty ]$. In particular, the function

$$\begin{aligned} |{\mathcal {S}}-{\mathcal {S}}'|_{\infty }:=[{\mathcal {S}},{\mathcal {S}}']_\infty \end{aligned}$$

on ${{\textsf {P}}}^1\times {{\textsf {P}}}^1$ extends the distance function $K\times K\ni (z,z')\mapsto |z-z'|$ to $({{\textsf {P}}}^1\setminus \{\infty \})\times ({{\textsf {P}}}^1\setminus \{\infty \})$, jointly upper semicontinuously and separately continuously, and the function

$$\begin{aligned} |{\mathcal {S}}|_\infty :=|{\mathcal {S}}-0|_\infty (=[{\mathcal {S}},0]_\infty )\quad \text {on }{{\textsf {P}}}^1 \end{aligned}$$

extends the norm function $K\ni z\mapsto |z|$ to ${{\textsf {P}}}^1\setminus \{\infty \}$ continuously (see [13, Sect. 3.4], [5, Sect. 4.4]).

Let $\Omega _{{\text {can}}}$ be the Dirac measure $\delta _{{\mathcal {S}}_{{\text {can}}}}$ on ${{\textsf {P}}}^1$ at ${\mathcal {S}}_{{\text {can}}}$. The Laplacian $\Delta$ on ${{\textsf {P}}}^1$ is normalized so that for each ${\mathcal {S}}'\in {{\textsf {P}}}^1$,

$$\begin{aligned} \Delta \log [\cdot ,{\mathcal {S}}']_{{\text {can}}}=\delta _{{\mathcal {S}}'}-\Omega _{{\text {can}}} \end{aligned}$$

on ${{\textsf {P}}}^1$, and then, for every $z_0\in {\mathbb {P}}^1$ and every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{z_0\}$, $\Delta \log [\cdot ,{\mathcal {S}}']_{z_0}=\delta _{{\mathcal {S}}'}-\delta _{z_0}$ on ${{\textsf {P}}}^1$. For the details on the construction and properties of $\Delta$, see [5, Sect. 5], [12, Sect. 7.7], [14, Sect. 2.4], [33, Sect. 3]; in [5, 33], the opposite sign convention for $\Delta$ is adopted.

2.4 Logarithmic Potential Theory on ${\textsf {P}}^1$

For every $z\in {\mathbb {P}}^1$ and every positive Radon measure $\nu$ on ${{\textsf {P}}}^1$ supported by ${{\textsf {P}}}^1\setminus \{z\}$, the logarithmic potential of $\nu$ on ${{\textsf {P}}}^1$ with pole z is the function

$$\begin{aligned} p_{z,\nu }(\cdot ):=\int _{{{\textsf {P}}}^1}\log [\cdot ,{\mathcal {S}}']_{z}\nu ({\mathcal {S}}')\quad \text {on }{{\textsf {P}}}^1, \end{aligned}$$

and the logarithmic energy of $\nu$ with pole z is defined by

$$\begin{aligned} I_{z,\nu }:=\int _{{{\textsf {P}}}^1}p_{z,\nu }\nu \in [-\infty ,+\infty ). \end{aligned}$$

Then $p_{z,\nu }:{{\textsf {P}}}^1\rightarrow [-\infty ,+\infty ]$ is upper semicontinuous, and in fact is strongly upper semicontinuous in that for every ${\mathcal {S}}\in {{\textsf {P}}}^1$,

$$\begin{aligned} \limsup _{{\mathcal {S}}'\rightarrow {\mathcal {S}}}p_{z,\nu }({\mathcal {S}}')=p_{z,\nu }({\mathcal {S}}) \end{aligned}$$

(2.1)

([5, Proposition 6.12]).

For every non-empty subset C in ${{\textsf {P}}}^1$ and every $z\in {\mathbb {P}}^1\setminus C$, we say C is of logarithmic capacity $>0$ with pole z if

$$\begin{aligned} V_z(C):=\sup _{\nu }I_{z,\nu }>-\infty , \end{aligned}$$

where $\nu$ ranges over all probability Radon measures on ${{\textsf {P}}}^1$ supported by C; otherwise, we say C is of logarithmic capacity 0 with pole z. For every non-empty compact subset C in ${{\textsf {P}}}^1$ of logarithmic capacity $>0$ with pole $z\in {\mathbb {P}}^1\setminus C$, there is a unique probability Radon measure $\nu$ on ${{\textsf {P}}}^1$, which is called the equilibrium mass distribution on C with pole z and is denoted by $\nu _{z,C}$, such that ${\text {supp}}\nu \subset C$ and that $I_{z,\nu }=V_z(C)$, and then (i) $\nu _{z,C}(E)=0$ for any subset E in C of logarithmic capacity 0 with pole z, (ii) letting $D_z$ be the component of ${{\textsf {P}}}^1\setminus C$ containing z, we have

$$\begin{aligned} {\text {supp}}\nu _{z,C}\subset \partial D_z,\quad p_{z,\nu _{z,C}}\ge & {} I_{z,\nu _{z,C}}\text { on }{{\textsf {P}}}^1,\quad p_{z,\nu _{z,C}}>I_{z,\nu _{z,C}}\text { on }D_z,\quad \text {and}\\ p_{z,\nu _{z,C}}\equiv & {} I_{z,\nu _{z,C}}\text { on }{{\textsf {P}}}^1\setminus (D_z\cup E), \end{aligned}$$

where E is a possibly empty $F_{\sigma }$-subset in $\partial D_z$ of logarithmic capacity 0 with pole z, (iii) if in addition $p_{z,\nu _{z,C}}$ is continuous on ${{\textsf {P}}}^1\setminus {\{z\}}$, then

$$\begin{aligned} {\text {supp}}\nu _{z,C}=\partial D_z\quad \text {and}\quad p_{z,\nu _{z,C}}\equiv I_{z,\nu _{z,C}}\text { on }{{\textsf {P}}}^1\setminus D_z, \end{aligned}$$

and (iv) for any probability Radon measure $\nu '$ supported by C, we have

$$\begin{aligned} \inf _{{\mathcal {S}}\in C}p_{z,\nu '}\le I_{z,\nu _{z,C}}\le \sup _{{\mathcal {S}}\in C}p_{z,\nu '} \end{aligned}$$

(2.2)

(see [5, Sects. 6.2, 6.3]).

We list a few observations:

Observation 2.2

For every $a\in K\setminus \{0\}$ and every $b\in K$, setting $\ell (z):=az+b\in \mathrm {PGL}(2,K)$, we have $\log |\ell ({\mathcal {S}})-\ell ({\mathcal {S}}')|_\infty =\log |{\mathcal {S}}-{\mathcal {S}}'|_\infty +\log |a|$ on $K\times K$, and in turn on ${{\textsf {P}}}^1\times {{\textsf {P}}}^1$. In particular, for every non-empty compact subset C in ${{\textsf {P}}}^1\setminus \{\infty \}$ of logarithmic capacity $>0$ with pole $\infty$, we have $I_{\infty ,\nu _{\infty ,\ell (C)}}=I_{\infty ,\nu _{\infty ,C}}+\log |a|$ and $\ell _*(\nu _{\infty ,C})=\nu _{\infty ,\ell (C)}$ on ${{\textsf {P}}}^1$.

Observation 2.3

Since the involution $\iota (z)=1/z\in \mathrm {PGL}(2,{\mathcal {O}}_K)$ acts on $({\mathbb {P}}^1,[z,w])$ isometrically, for any $z_0\in {\mathbb {P}}^1$, we have $[\iota ({\mathcal {S}}),\iota ({\mathcal {S}}')]_{\iota (z_0)}=[{\mathcal {S}},{\mathcal {S}}']_{z_0}$ on ${\mathbb {P}}^1\times {\mathbb {P}}^1$, and in turn on ${{\textsf {P}}}^1\times {{\textsf {P}}}^1$. Hence for any non-empty compact subset C in ${{\textsf {P}}}^1$ and any $z\in {{\textsf {P}}}^1\setminus C$, if C is of logarithmic capacity $>0$ with pole z, then $V_z(C)=V_{\iota (z)}(\iota (C))$ and $\iota _*(\nu _{z,C})=\nu _{\iota (z),\iota (C)}$ on ${{\textsf {P}}}^1$.

Observation 2.4

For every $z\in {\mathbb {P}}^1$, the strong triangle inequality $[{\mathcal {S}},{\mathcal {S}}'']_{z}\le \max \{[{\mathcal {S}},{\mathcal {S}}']_{z},[{\mathcal {S}}',{\mathcal {S}}'']_{z}\}$ for ${\mathcal {S}},{\mathcal {S}}',{\mathcal {S}}''\in {{\textsf {P}}}^1$ still holds (see [5, Proposition 4.10]). Hence for every non-empty compact subset C in ${{\textsf {P}}}^1\setminus \{\infty \}$ and every $z\in {\mathbb {P}}^1\setminus C$ so close to $\infty$ that $[z,\infty ]<\inf _{{\mathcal {S}}\in C}[{\mathcal {S}},z]_{{\text {can}}}$, we have $[\cdot ,\infty ]_{{\text {can}}}=[\cdot ,z]_{{\text {can}}}$ on C, which yields $[{\mathcal {S}},{\mathcal {S}}']_\infty =[{\mathcal {S}},{\mathcal {S}}']_{z}$ on $C\times C$, so if in addition C is of logarithmic capacity $>0$ with pole $\infty$, then $V_{\infty }(C)=V_z(C)$ and $\nu _{\infty ,C}=\nu _{z,C}$ on ${{\textsf {P}}}^1$.

2.5 Potential Theory with a Continuous Weight on ${\textsf {P}}^1$

A continuous weight g on ${{\textsf {P}}}^1$ is a continuous function on ${{\textsf {P}}}^1$ such that

$$\begin{aligned} \mu ^g:=\Delta g+\Omega _{{\text {can}}} \end{aligned}$$

is a probability Radon measure on ${{\textsf {P}}}^1$. Then $\mu ^g$ has no atoms on ${\mathbb {P}}^1$, or more strongly, $\mu ^g(E)=0$ for any subset E in ${{\textsf {P}}}^1$ of logarithmic capacity 0 with some (indeed any) point in ${{\textsf {P}}}^1\setminus E$.

For a continuous weight g on ${{\textsf {P}}}^1$, the g-potential kernel on ${{\textsf {P}}}^1$ (the negative of an Arakelov Green kernel function on ${{\textsf {P}}}^1$ relative to $\mu ^g$ [5, Sect. 8.10] ) is an upper semicontinuous function

$$\begin{aligned} \Phi _g({\mathcal {S}},{\mathcal {S}}'):=\log [{\mathcal {S}},{\mathcal {S}}']_{{\text {can}}}-g({\mathcal {S}})-g({\mathcal {S}}')\quad \text {on }{{\textsf {P}}}^1\times {{\textsf {P}}}^1. \end{aligned}$$

(2.3)

For every Radon measure $\nu$ on ${{\textsf {P}}}^1$, the g-potential of $\nu$ on ${{\textsf {P}}}^1$ is the function

$$\begin{aligned} U_{g,\nu }(\cdot ):=\int _{{{\textsf {P}}}^1}\Phi _g(\cdot ,{\mathcal {S}}')\nu ({\mathcal {S}}')\quad \text {on }{{\textsf {P}}}^1, \end{aligned}$$

and the g-energy of $\nu$ is defined by

$$\begin{aligned} I_{g,\nu }:=\int _{{{\textsf {P}}}^1}U_{g,\nu }\nu \in [-\infty ,+\infty ). \end{aligned}$$

The g-equilibrium energy $V_g$ of (the whole) ${{\textsf {P}}}^1$ is the supremum of the g-energy functional $\nu \mapsto I_{g,\nu }$, where $\nu$ ranges over all probability Radon measures on ${{\textsf {P}}}^1$. Then $V_g\in \mathbb {R}$ since $I_{g,\Omega _{{\text {can}}}}>-\infty$. As in the logarithmic potential theory presented in the previous subsection, there is a unique probability Radon measure $\nu ^g$ on ${{\textsf {P}}}^1$, which is called the g-equilibrium mass distribution on ${{\textsf {P}}}^1$, such that $I_{g,\nu ^g}=V_g$. In fact

$$\begin{aligned} U_{g,\nu ^g}\equiv V_g\quad \text {on }{{\textsf {P}}}^1 \quad \text {and}\quad \nu ^g=\mu ^g\quad \text {on }{{\textsf {P}}}^1 \end{aligned}$$

(see [5, Theorem 8.67, Proposition 8.70]).

A continuous weight g on ${{\textsf {P}}}^1$ is a normalized weight on ${{\textsf {P}}}^1$ if $V_g=0$. For a continuous weight g on ${{\textsf {P}}}^1$, $\overline{g}:=g+V_g/2$ is the unique normalized weight on ${{\textsf {P}}}^1$ satisfying $\mu ^{\overline{g}}=\mu ^g$.

3 Background from Dynamics on ${\textsf {P}}^1$

For a potential-theoretic study of dynamics of a rational function of degree $>1$ on ${{\textsf {P}}}^1={{\textsf {P}}}^1(K)$, see [5, Sect. 10], [14, Sect. 3], [18, Sect. 5], and [6, Sect. 13]. In the following, we adopt a presentation from [28, Sect. 8.1].

3.1 Canonical Measure and the Dynamical Green Function of f on ${\textsf {P}}^1$

Let $f\in K(z)$ be a rational function of degree $d>1$. We call $F\in (K[p_0,p_1]_d)^2$ a lift of f if

$$\begin{aligned} \pi \circ F=f\circ \pi \end{aligned}$$

on $K^2\setminus \{(0,0)\}$, where for each $j\in {\mathbb {N}}\cup \{0\}$, $K[p_0,p_1]_j$ is the set of all homogeneous polynomials in $K[p_0,p_1]$ of degree j, as usual. A lift $F=(F_0,F_1)$ of f is unique up to multiplication in $K\setminus \{0\}$. Setting $d_0:=\deg F_0(1,z)$ and $d_1:=\deg F_1(1,z)$ and letting $c^F_0,c^F_1\in K\setminus \{0\}$ be the coefficients of the maximal degree terms of $F_0(1,z),F_1(1,z)\in K[z]$, respectively, the homogeneous resultant

$$\begin{aligned} {\text {Res}}F= (c^F_0)^{d-d_1}\cdot (c^F_1)^{d-d_0}\cdot R\bigl (F_0(1,\cdot ),F_1(1,\cdot )\bigr )\in K \end{aligned}$$

of F does not vanish, where $R(P,Q)\in K$ is the usual resultant of $(P,Q)\in (K[z])^2$ (for the details on ${\text {Res}}F$, see e.g. [32, Sect. 2.4]).

Let F be a lift of f, and for every $n\in {\mathbb {N}}\cup \{0\}$, set $F^n=F\circ F^{n-1}$ where $F^0:={\mathrm {Id}}_{K^2}$. Then for every $n\in {\mathbb {N}}$, $F^n$ is a lift of $f^n$, and the function

$$\begin{aligned} T_{F^n}:=\log \Vert F^n\Vert -d^n\cdot \log \Vert \cdot \Vert \end{aligned}$$

on $K^2\setminus \{(0,0)\}$ descends to ${\mathbb {P}}^1$ and in turn extends continuously to ${{\textsf {P}}}^1$, satisfying the equality $\Delta T_{F^n}=(f^n)^*\Omega _{{\text {can}}}-d^n\cdot \Omega _{{\text {can}}}$ on ${{\textsf {P}}}^1$ (see, e.g., [26, Definition 2.8]). The dynamical Green function of F on ${{\textsf {P}}}^1$ is the uniform limit $g_F:=\lim _{n\rightarrow \infty }T_{F^n}/d^n$ on ${{\textsf {P}}}^1$, which is a continuous weight on ${{\textsf {P}}}^1$. The energy formula

$$\begin{aligned} V_{g_F}=-\frac{\log |{\text {Res}}F|}{d(d-1)} \end{aligned}$$

is due to DeMarco [11] for archimedean K by a dynamical argument, and due to Baker–Rumely [4] when f is defined over a number field; see Baker [2, Appendix A] or the present authors [29, Appendix] for a simple and potential-theoretic proof of this remarkable formula, for general K. The f-canonical measure is the probability Radon measure

$$\begin{aligned} \mu _f:=\Delta g_F+\Omega _{{\text {can}}}\quad \text {on }{{\textsf {P}}}^1. \end{aligned}$$

The measure $\mu _f$ is independent of the choice of the lift F of f, has no atoms in ${\mathbb {P}}^1$, and satisfies the f-balanced property $f^*\mu _f=d\cdot \mu _f$ (so in particular $f_*\mu _f=\mu _f$) on ${{\textsf {P}}}^1$. For more details, see [5, Sect. 10], [10, Sect. 2], [14, Sect. 3.1].

The dynamical Green function $g_f$ of f on ${{\textsf {P}}}^1$ is the unique normalized weight on ${{\textsf {P}}}^1$ such that $\mu ^{g_f}=\mu _f$. By the above energy formula on $V_{g_F}$ and

$$\begin{aligned} {\text {Res}}(cF)=c^{2d}\cdot {\text {Res}}F\quad \text {for every }c\in K\setminus \{0\}, \end{aligned}$$

there is a lift F of f normalized so that $V_{g_F}=0$ or equivalently that $g_F=g_f$ on ${{\textsf {P}}}^1$, and such a normalized lift F of f is unique up to multiplication in $\{z\in K:|z|=1\}$. By $g_f=g_F=\lim _{n\rightarrow \infty }T_{F^n}/d^n$ on ${{\textsf {P}}}^1$ for a normalized lift F of f, for every $n\in {\mathbb {N}}$, we have $g_{F^n}=g_{f^n}=g_f$ on ${{\textsf {P}}}^1$ and $\mu _{f^n}=\mu _f$ on ${{\textsf {P}}}^1$. We note that $g_f\circ f=d\cdot \lim _{n\rightarrow \infty }T_{F^{n+1}}/d^{n+1}-T_F=d\cdot g_f-T_F$ on ${\mathbb {P}}^1$, that is,

$$\begin{aligned} d\cdot g_f-g_f\circ f=T_F \end{aligned}$$

(3.1)

on ${\mathbb {P}}^1$, and in turn on ${{\textsf {P}}}^1$ by the density of ${\mathbb {P}}^1$ in ${{\textsf {P}}}^1$ and the continuity of both sides on ${{\textsf {P}}}^1$ (cf. [27, Proof of Lemma 2.4]).

3.2 Fundamental Properties of $\mu _f$

Recall the definition of ${{\mathsf {J}}}(f)$ in Sect. 1. The characterization of $\mu _f$ as the unique probability Radon measure $\nu$ on ${{\textsf {P}}}^1$ such that $\nu (E(f))=0$ and that $f^*\nu =d\cdot \nu$ on ${{\textsf {P}}}^1$ is a consequence of the following equidistribution theorem: for every probability Radon measure $\mu$ on ${{\textsf {P}}}^1$, if $\mu (E(f))=0$, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{(f^n)^*\mu }{d^n}=\mu _f\quad weakly~on~{{\textsf {P}}}^1. \end{aligned}$$

(3.2)

This foundational result is due to Favre and Rivera-Letelier [14] (for a purely potential-theoretic proof, see also Jonsson [18]) and is a non-archimedean counterpart to Brolin [9], Lyubich [20], Freire et al. [15].

Remark 3.1

The classical Julia set ${{\mathsf {J}}}(f)\cap {\mathbb {P}}^1$ of f coincides with the set of all points in ${\mathbb {P}}^1$ at each of which the family $\bigl (f^n:({\mathbb {P}}^1,[z,w])\rightarrow ({\mathbb {P}}^1,[z,w])\bigr )_{n\in {\mathbb {N}}}$ is not locally equicontinuous (see, e.g., [5, Theorem 10.67] ).

The equality ${\text {supp}}\mu _f={{\mathsf {J}}}(f)$ holds; the inclusion ${{\mathsf {J}}}(f)\subset {\text {supp}}\mu _f$ follows from the definition of ${{\mathsf {J}}}(f)$, the balanced property $f^*\mu _f=d\cdot \mu _f$ on ${{\textsf {P}}}^1$, and ${\text {supp}}\mu _f\not \subset E(f)$ (or more precisely, recalling that E(f) is an at most countable subset in ${\mathbb {P}}^1$ and that $\mu _f$ has no atoms in ${\mathbb {P}}^1$). The opposite inclusion ${\text {supp}}\mu _f\subset {{\mathsf {J}}}(f)$ follows from the definition of ${{\mathsf {J}}}(f)$ and the above equidistribution theorem.

Remark 3.2

(see, e.g., [5, Corollary 10.33]) If $\mu _f$ has an atom in ${{\textsf {P}}}^1$, then f has a potentially good reduction, so in particular ${{\mathsf {J}}}(f)$ is a singleton in ${\textsf {H}}^1$.

For every $n\in {\mathbb {N}}$, by ${\text {supp}}\mu _f={{\mathsf {J}}}(f)$ and $\mu _{f^n}=\mu _f$ on ${{\textsf {P}}}^1$, we also have ${{\mathsf {J}}}(f^n)={{\mathsf {J}}}(f)$. For every $m\in \mathrm {PGL}(2,K)$, we have $m_*\mu _f=\mu _{m\circ f\circ m^{-1}}$ on ${{\textsf {P}}}^1$, $m({{\mathsf {J}}}(f))={{\mathsf {J}}}(m\circ f\circ m^{-1})$, and $m({{\mathsf {F}}}(f))={{\mathsf {F}}}(m\circ f\circ m^{-1})$.

3.3 Root Divisors on ${\mathbb {P}}^1$ and the Proximity Functions on ${\textsf {P}}^1$

For any distinct $h_1,h_2\in K(z)$, let $[h_1=h_2]$ be the effective (K-)divisor on ${\mathbb {P}}^1$ defined by all solutions to the equation $h_1=h_2$ in ${\mathbb {P}}^1$ taking into account their multiplicities, which is also regarded as the Radon measure

$$\begin{aligned} \sum _{w\in {\mathbb {P}}^1}({\text {ord}}_w[h_1=h_2])\cdot \delta _w \end{aligned}$$

on ${{\textsf {P}}}^1$. The function ${\mathbb {P}}^1\ni z\mapsto [h_1(z),h_2(z)]$ between $h_1$ and $h_2$ uniquely extends to a continuous function ${\mathcal {S}}\mapsto [h_1,h_2]_{{\text {can}}}({\mathcal {S}})$ on ${{\textsf {P}}}^1$ (see, e.g., [26, Proposition 2.9]), so that for every continuous weight g on ${{\textsf {P}}}^1$, (the $\exp$ of) the function

$$\begin{aligned} \Phi (h_1,h_2)_g({\mathcal {S}}):=\log [h_1,h_2]_{{\text {can}}}({\mathcal {S}})-g(h_1({\mathcal {S}}))-g(h_2({\mathcal {S}})) \quad \text {on }{{\textsf {P}}}^1 \end{aligned}$$

(3.3)

is a unique continuous extension of (the $\exp$ of) the function ${\mathbb {P}}^1\ni z\mapsto \Phi _g(h_1(z),h_2(z))$.

4 Potential-Theoretic Computations

Let $f\in K(z)$ be a rational function of degree $d>1$.

Lemma 4.1

(Riesz’s decomposition for the pullback of an atom) For every ${\mathcal {S}}\in {{\textsf {P}}}^1$,

$$\begin{aligned} \Phi _{g_f}(f(\cdot ),{\mathcal {S}})=U_{g_f,f^*\delta _{\mathcal {S}}}(\cdot ) \quad \text {on }{{\textsf {P}}}^1. \end{aligned}$$

(4.1)

Proof

Fix a lift F of f normalized so that $g_F=g_f$ on ${{\textsf {P}}}^1$. Fix $w\in {\mathbb {P}}^1$ and $W\in \pi ^{-1}(w)$. Choose a sequence $(q_j)_{j=1}^{d}$ in $K^2\setminus \{(0,0)\}$ such that $F(p_0,p_1)\wedge W\in K[p_0,p_1]_d$ factors as $F(p_0,p_1)\wedge W=\prod _{j=1}^{d}((p_0,p_1)\wedge q_j)$ in $K[p_0,p_1]$. This together with (3.1) and the definition of $T_F$ implies

$$\begin{aligned}&\Phi _{g_f}(f\circ \pi ,w)-U_{g_f,f^*\delta _w}\circ \pi \\&\quad =\bigl (\log |F(\cdot )\wedge W|-\log \Vert F\Vert -\log \Vert W\Vert -(g_f\circ f)(\pi (\cdot ))-g_f(w)\bigr )\\&\qquad -\sum _{j=1}^d\bigl (\log |\cdot \wedge q_j|-\log \Vert \cdot \Vert -\log \Vert q_j\Vert -g_f\circ \pi -g_f(\pi (q_j))\bigr )\\&\quad =\bigl (\log |F(\cdot )\wedge W|-\sum _{j=1}^d\log |\cdot \wedge q_j|\bigr ) -\bigl ((g_f\circ f)(\pi (\cdot ))+d\cdot g_f\circ \pi \bigr )\\&\qquad -(\log \Vert F\Vert -d\cdot \log \Vert \cdot \Vert )\\&\qquad -(g_f(w)+\log \Vert W\Vert ) +\sum _{j=1}^{d}(g_f(\pi (q_j))+\log \Vert q_j\Vert )\\&\quad \equiv -(g_f(w)+\log \Vert W\Vert ) +\sum _{j=1}^{d}(g_f(\pi (q_j))+\log \Vert q_j\Vert )=:C\quad \text {on }K^2\setminus \{0\}, \end{aligned}$$

so $\Phi _{g_f}(f(\cdot ),w)-U_{g_f,f^*\delta _w}(\cdot )\equiv C$ on ${\mathbb {P}}^1$, and in turn on ${{\textsf {P}}}^1$ by the density of ${\mathbb {P}}^1$ in ${{\textsf {P}}}^1$ and the continuity of (the $\exp$ of) both sides on ${{\textsf {P}}}^1$. Integrating both sides against $\mu _f$ over ${{\textsf {P}}}^1$, since $\int _{{{\textsf {P}}}^1}U_{g_f,f^*\delta _w}\mu _f =\int _{{{\textsf {P}}}^1}U_{g_f,\mu _f}(f^*\delta _w)=0$ (by $U_{g_f,\mu _f}\equiv 0$) and $f_*\mu _f=\mu _f$, we have

$$\begin{aligned} C=\int _{{{\textsf {P}}}^1}\Phi _{g_f}(f(\cdot ),w)\mu _f =U_{g_f,f_*\mu _f}(w)=U_{g_f,\mu _f}(w)=0. \end{aligned}$$

This completes the proof of (4.1) in the case ${\mathcal {S}}=w\in {\mathbb {P}}^1$.

Fix ${\mathcal {S}}_0\in {\textsf {H}}^1$. By the density of ${\mathbb {P}}^1$ in ${{\textsf {P}}}^1$, we can choose a sequence $(w_n)$ in ${\mathbb {P}}^1$ tending to ${\mathcal {S}}_0$ as $n\rightarrow \infty$. Then $\lim _{n\rightarrow \infty }f^*\delta _{w_n}=f^*\delta _{{\mathcal {S}}_0}$ weakly on ${{\textsf {P}}}^1$ and, for every $n\in {\mathbb {N}}$, applying (4.1) to ${\mathcal {S}}=w_n\in {\mathbb {P}}^1$, we have $\Phi _{g_f}(f(\cdot ),w_n)=U_{g_f,f^*\delta _{w_n}}(\cdot )$ on ${{\textsf {P}}}^1$. Hence, for each ${\mathcal {S}}'\in {\textsf {H}}^1$, by the continuity of both $\Phi _{g_f}(f({\mathcal {S}}'),\cdot )$ and $\Phi _{g_f}({\mathcal {S}}',\cdot )$ on ${{\textsf {P}}}^1$, we have

$$\begin{aligned} \Phi _{g_f}(f({\mathcal {S}}'),{\mathcal {S}}_0) =\lim _{n\rightarrow \infty }\Phi _{g_f}(f({\mathcal {S}}'),w_n) =\lim _{n\rightarrow \infty }U_{g_f,f^*\delta _{w_n}}({\mathcal {S}}') =U_{g_f,f^*\delta _{{\mathcal {S}}_0}}({\mathcal {S}}'). \end{aligned}$$

This completes the proof of (4.1) by the density of ${\textsf {H}}^1$ in ${{\textsf {P}}}^1$ and the continuity of (the $\exp$ of) both $\Phi _{g_f}(f(\cdot ),{\mathcal {S}}_0)$ and $U_{g_f,f^*\delta _{{\mathcal {S}}_0}}(\cdot )$ on ${{\textsf {P}}}^1$. $\square$

The following computation is an application of Lemma 4.1. We include a proof of it although it will not be used in this article.

Lemma 4.2

(Riesz’s decomposition for the fixed points divisor on ${\mathbb {P}}^1)$

$$\begin{aligned} \Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}=U_{g_f,[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}\quad \text {on }{{\textsf {P}}}^{1}. \end{aligned}$$

(4.2)

Proof

Fix a lift F of f normalized so that $g_F=g_f$ on ${{\textsf {P}}}^1$. Choose a sequence $(q_j)_{j=1}^{d+1}$ in $K^2\setminus \{(0,0)\}$ so that $(F\wedge {\mathrm {Id}}_{{\mathbb {P}}^1})(p_0,p_1)\in K[p_0,p_1]_{d+1}$ factors as $(F\wedge {\mathrm {Id}}_{{\mathbb {P}}^1})(p_0,p_1)=\prod _{j=1}^{d+1}((p_0,p_1)\wedge q_j)$ in $K[p_0,p_1]$, which with (3.1) implies

$$\begin{aligned} \Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}-U_{g_f,[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}\equiv \sum _{j=1}^{d+1}(g_f(\pi (q_j))+\log \Vert q_j\Vert )=:C \end{aligned}$$

on ${\mathbb {P}}^1$, and in turn on ${{\textsf {P}}}^1$ by the density of ${\mathbb {P}}^1$ in ${{\textsf {P}}}^1$ and the continuity of (the $\exp$ of) both sides on ${{\textsf {P}}}^1$. Integrating both sides against $\mu _f$ over ${{\textsf {P}}}^1$, since $\int _{{{\textsf {P}}}^1}U_{g_f,[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}\mu _f=\int _{{{\textsf {P}}}^1}U_{g_f,\mu _f}[f={\mathrm {Id}}_{{\mathbb {P}}^1}]=0$ (by $U_{g_f,\mu _f}\equiv 0$), we have $C=\int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f$, so that we first have

$$\begin{aligned} \Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f} =U_{g_f,[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}+\int _{{{\textsf {P}}}^1} \Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f\quad \text {on }{{\textsf {P}}}^1. \end{aligned}$$

Fix $z_0\in {\mathbb {P}}^1\setminus ({\text {supp}}[f={\mathrm {Id}}_{{\mathbb {P}}^1}])$. Using the above equality twice, by $f_*[f={\mathrm {Id}}_{{\mathbb {P}}^1}]=[f={\mathrm {Id}}_{{\mathbb {P}}^1}]$ on ${{\textsf {P}}}^1$ and (4.1), we have

$$\begin{aligned}&\Phi _{g_f}(f(z_0),z_0)-\int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f\\ =&U_{g_f,[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}(z_0) =U_{g_f,f_*[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}(z_0) =\int _{{{\textsf {P}}}^1}\Phi _{g_f}(z_0,\cdot )(f_*[f={\mathrm {Id}}_{{\mathbb {P}}^1}])(\cdot )\\ =&\int _{{{\textsf {P}}}^1}\Phi _{g_f}(z_0,f(\cdot ))[f={\mathrm {Id}}_{{\mathbb {P}}^1}](\cdot ) =\int _{{{\textsf {P}}}^1}U_{g_f,f^*\delta _{z_0}}[f={\mathrm {Id}}_{{\mathbb {P}}^1}]\\ =&\int _{{{\textsf {P}}}^1}U_{g_f,[f={\mathrm {Id}}_{{\mathbb {P}}^1}]}(f^*\delta _{z_0}) =\int _{{{\textsf {P}}}^1}\Bigl (\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f} -\int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f\Bigr )(f^*\delta _{z_0})\\ =&\int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}(f^*\delta _{z_0}) -d\cdot \int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f, \end{aligned}$$

and moreover, $\int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}(f^*\delta _{z_0}) =U_{g_f,f^*\delta _{z_0}}(z_0)=\Phi _{g_f}(f(z_0),z_0)$ by (4.1). Hence $(d-1)\int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f=0$, and in turn since $d>1$,

$$\begin{aligned} \int _{{{\textsf {P}}}^1}\Phi (f,{\mathrm {Id}}_{{\mathbb {P}}^1})_{g_f}\mu _f=0. \end{aligned}$$

(4.3)

This completes the proof. $\square$

From now on, we focus on the case where $\infty \in {{\mathsf {F}}}(f)$. We adopt the following convention when no confusion would be caused:

Convention

For every probability Radon measure $\nu$ supported by ${{\textsf {P}}}^1\setminus \{\infty \}$, we denote $p_{\infty ,\nu }$ and $I_{\infty ,\nu }$ by $p_\nu$ and $I_\nu$, respectively, for simplicity.

Since ${\text {supp}}\mu _f={{\mathsf {J}}}(f)\subset {{\textsf {P}}}^1\setminus D_\infty$, the equality (4.5) below implies that ${{\textsf {P}}}^1\setminus D_\infty$ is of logarithmic capacity $>0$ with pole $\infty$.

Lemma 4.3

Suppose that $\infty \in {{\mathsf {F}}}(f)$. Then

$$\begin{aligned} p_{\mu _f}= & {} g_f-\log [\cdot ,\infty ]_{{\text {can}}}+\frac{I_{\mu _f}}{2}\quad \text {on }{{\textsf {P}}}^1, \end{aligned}$$

(4.4)

$$\begin{aligned} I_{\mu _f}= & {} -2\cdot g_f(\infty )>-\infty ,\text { and} \end{aligned}$$

(4.5)

$$\begin{aligned} \Phi _{g_f}(\cdot ,\infty )= & {} -p_{\mu _f}+I_{\mu _f}\quad \text {on }{{\textsf {P}}}^1. \end{aligned}$$

(4.6)

Proof

Suppose $\infty \in {{\mathsf {F}}}(f)$. Then we have ${\text {supp}}\mu _f={{\mathsf {J}}}(f)\subset {{\textsf {P}}}^1\setminus D_\infty$ and

$$\begin{aligned} 0=V_{g_f}=\int _{{{\textsf {P}}}^1\times {{\textsf {P}}}^1}\Phi _{g_f}(\mu _f\times \mu _f) =I_{\mu _f}-2\cdot \int _{{{\textsf {P}}}^1}(g_f-\log [\cdot ,\infty ]_{{\text {can}}})\mu _f, \end{aligned}$$

so that $I_{\mu _f}=2\cdot \int _{{{\textsf {P}}}^1}(g_f-\log [\cdot ,\infty ]_{{\text {can}}})\mu _f$, which with

$$\begin{aligned} 0\equiv U_{g_f,\mu _f}=p_{\mu _f}-(g_f-\log [\cdot ,\infty ]_{{\text {can}}}) -\int _{{{\textsf {P}}}^1}(g_f-\log [\cdot ,\infty ]_{{\text {can}}})\mu _f\quad \text {on }{{\textsf {P}}}^1 \end{aligned}$$

yields (4.4). By (4.4) and $\log [z,\infty ]=\log [z,0]-\log |z|$ on ${\mathbb {P}}^1\setminus \{\infty \}$, we have

$$\begin{aligned} g_f(\infty )=\lim _{z\rightarrow \infty }\bigl ((p_{\mu _f}(z)-\log |z|)+\log [z,0]\bigr ) -\frac{I_{\mu _f}}{2}=-\frac{I_{\mu _f}}{2}, \end{aligned}$$

so that (4.5) holds. By (4.4) and (4.5), we have $\Phi _{g_f}(\cdot ,\infty )=\log [\cdot ,\infty ]_{{\text {can}}}-g_f-g_f(\infty ) =(-p_{\mu _f}+I_{\mu _f}/2)+I_{\mu _f}/2=-p_{\mu _f}+I_{\mu _f}$ on ${{\textsf {P}}}^1$, so (4.6) also holds. $\square$

Let $F=(F_0,F_1)\in (K[p_0,p_1]_d)^2$ be a normalized lift of f, and $c_0^F,c_1^F\in K\setminus \{0\}$ be the coefficients of the maximal degree terms of $F_0(1,z),F_1(1,z)\in K[z]$, respectively. No matter whether $\infty \in {{\mathsf {F}}}(f)$, by the equality $[z,\infty ]=1/\Vert (1,z)\Vert$ on ${\mathbb {P}}^1$ and the definition of $T_F$, we have

$$\begin{aligned} T_F=-\log [f(\cdot ),\infty ]_{{\text {can}}} +\log |F_0(1,\cdot )|_\infty +d\cdot \log [\cdot ,\infty ]_{{\text {can}}} \end{aligned}$$

on ${\mathbb {P}}^1\setminus (\{\infty \}\cup f^{-1}(\infty ))$, and in turn on ${{\textsf {P}}}^1\setminus (\{\infty \}\cup f^{-1}(\infty ))$ by the density of ${\mathbb {P}}^1$ in ${{\textsf {P}}}^1$ and the continuity of both sides on ${{\textsf {P}}}^1\setminus (\{\infty \}\cup f^{-1}(\infty ))$. By (3.1), this equality is rewritten as

$$\begin{aligned} d\cdot (g_f-\log [\cdot ,\infty ]_{{\text {can}}})-(g_f\circ f-\log [f(\cdot ),\infty ]_{{\text {can}}}) =\log |F_0(1,\cdot )|_{\infty } \end{aligned}$$

(4.7)

on ${{\textsf {P}}}^1\setminus (\{\infty \}\cup f^{-1}(\infty ))$.

Lemma 4.4

(Pullback formula for $p_{\mu _f}$ under f) If $\infty \in {{\mathsf {F}}}(f)$, then

$$\begin{aligned} \log |{F}_0(1,\cdot )|_\infty =d\cdot p_{\mu _f}-p_{\mu _f}\circ f-(d-1)\frac{I_{\mu _f}}{2} \end{aligned}$$

(4.8)

on ${{\textsf {P}}}^1\setminus (\{\infty \}\cup f^{-1}(\infty ));$ moreover, for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty ,f(\infty )\}$,

$$\begin{aligned}&p_{\mu _f}({\mathcal {S}}')-\int _{{{\textsf {P}}}^1\setminus \{\infty \}}p_{\mu _f}(f^*\delta _{{\mathcal {S}}'}) +(d-1)I_{\mu _f} \nonumber \\&\quad = -\int _{{{\textsf {P}}}^1}\log |{F}_0(1,\cdot )|_\infty \frac{f^*\delta _{{\mathcal {S}}'}}{d} +(d-1)\frac{I_{\mu _f}}{2}, \end{aligned}$$

(4.9)

and similarly

$$\begin{aligned} \int _{{{\textsf {P}}}^1\setminus \{\infty \}}p_{\mu _f}(f^*\delta _\infty )-(d-1)I_{\mu _f} =-\log |c_0^{{F}}|-(d-1)\frac{I_{\mu _f}}{2}. \end{aligned}$$

(4.10)

Proof

Suppose $\infty \in {{\mathsf {F}}}(f)$. Then for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty ,f(\infty )\}$, by (4.7) and (4.4), we have (4.8). Integrating both sides in (4.8) against $f^*\delta _{{\mathcal {S}}'}/d$ over ${{\textsf {P}}}^1$, we have (4.9). Similarly, integrating both sides in (4.8) against $\mu _f$ over ${{\textsf {P}}}^1$, also by $f_*\mu _f=\mu _f$ and $I_{\mu _f}:=\int _{{{\textsf {P}}}^1}p_{\mu _f}\mu _f$, we have

$$\begin{aligned}&\log |c_0^F|+\int _{{{\textsf {P}}}^1\setminus \{\infty \}}p_{\mu _f}(f^*\delta _\infty ) =\int _{{{\textsf {P}}}^1}\log |F_0(1,\cdot )|_\infty \mu _f\\&\quad =d\cdot I_{\mu _f}-\int _{{{\textsf {P}}}^1}(p_{\mu _f}\circ f)\mu _f-(d-1)\frac{I_{\mu _f}}{2} =(d-1)\frac{I_{\mu _f}}{2}, \end{aligned}$$

so (4.10) also holds. $\square$

If $f(\infty )=\infty$, then $F(0,1)=(0,c^F_1)$, so that by the homogeneity of F, for every $n\in {\mathbb {N}}$, $F^n(0,1)=(0,(c^F_1)^{(d^n-1)/(d-1)})$ and that

$$\begin{aligned} g_f(\infty )=\lim _{n\rightarrow \infty }\frac{T_{F^n}(\infty )}{d^n} =\lim _{n\rightarrow \infty }\frac{\log \Vert F^n(0,1)\Vert }{d^n}-\log \Vert (0,1)\Vert =\frac{\log |c^F_1|}{d-1}. \end{aligned}$$

Lemma 4.5

If $f(\infty )=\infty \in {{\mathsf {F}}}(f)$, then

$$\begin{aligned} I_{\mu _f}=-\frac{2}{d-1}\log |c^F_1| \end{aligned}$$

(4.11)

and, for every ${\mathcal {S}}'\in {{\textsf {P}}}^1$,

$$\begin{aligned} \int _{{{\textsf {P}}}^1\setminus \{\infty \}}p_{\mu _f}(f^*\delta _{{\mathcal {S}}'})-(d-1)I_{\mu _f} ={\left\{ \begin{array}{ll} p_{\mu _f}({\mathcal {S}}') &{} \text {if }{\mathcal {S}}'\ne \infty ,\\ \displaystyle \log \biggl |\frac{c_1^F}{c_0^F}\biggr | &{} \text {if }{\mathcal {S}}'=\infty . \end{array}\right. } \end{aligned}$$

(4.12)

Proof

Suppose that $f(\infty )=\infty \in {{\mathsf {F}}}(f)$. Then by the above computation of $g_f(\infty )$ and (4.5), we have (4.11). Moreover, for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty \}$, using (4.6) twice and (4.1) (and the assumption $f(\infty )=\infty$), we compute

$$\begin{aligned}&-p_{\mu _f}({\mathcal {S}}')+I_{\mu _f}=\Phi _{g_f}(\infty ,{\mathcal {S}}')= \Phi _{g_f}(f(\infty ),{\mathcal {S}}')\\&\quad =\int _{{{\textsf {P}}}^1}\Phi _{g_f}(\infty ,\cdot )(f^*\delta _{{\mathcal {S}}'}) =-\int _{{{\textsf {P}}}^1}p_{\mu _f}(f^*\delta _{{\mathcal {S}}'})+d\cdot I_{\mu _f}, \end{aligned}$$

so (4.12) holds for ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty \}$. Finally, (4.12) for ${\mathcal {S}}'=\infty$ holds by (4.10) and (4.11). $\square$

Let us now focus on $\nu _\infty =\nu _{\infty ,{{\textsf {P}}}^1\setminus D_{\infty }}$ when $\infty \in {{\mathsf {F}}}(f)$. Then $f(\infty )\in {{\mathsf {F}}}(f)$ and, since ${\text {supp}}\nu _\infty \subset \partial D_\infty \subset {{\mathsf {J}}}(f)={\text {supp}}\mu _f$, we have

$$\begin{aligned} {\text {supp}}(f_*\nu _\infty )\subset f({{\mathsf {J}}}(f))={{\mathsf {J}}}(f)={\text {supp}}\mu _f\subset {{\textsf {P}}}^1\setminus D_\infty . \end{aligned}$$

Lemma 4.6

Suppose that $\infty \in {{\mathsf {F}}}(f)$. Then for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty ,f(\infty )\}$,

$$\begin{aligned}&p_{f_*\nu _\infty }({\mathcal {S}}') -\int _{{{\textsf {P}}}^1}p_{\nu _\infty }(f^*\delta _{{\mathcal {S}}'}) +d\cdot I_{\nu _\infty }-\int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f \nonumber \\&\quad =p_{\mu _f}({\mathcal {S}}')-\int _{{{\textsf {P}}}^1}p_{\mu _f}(f^*\delta _{{\mathcal {S}}'})+(d-1)I_{\mu _f} \end{aligned}$$

(4.13)

and, if in addition $\nu _\infty$ is invariant under f in that $f_*\nu _\infty =\nu _\infty$ on ${{\textsf {P}}}^1$, then

$$\begin{aligned}&p_{\nu _\infty }({\mathcal {S}}')-\int _{{{\textsf {P}}}^1}p_{\nu _\infty }(f^*\delta _{{\mathcal {S}}'}) +(d-1)\cdot I_{\nu _\infty } \nonumber \\&\quad =p_{\mu _f}({\mathcal {S}}')-\int _{{{\textsf {P}}}^1}p_{\mu _f}(f^*\delta _{{\mathcal {S}}'})+(d-1)I_{\mu _f}. \end{aligned}$$

(4.14)

Proof

Suppose that $\infty \in {{\mathsf {F}}}(f)$. Then for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty ,f(\infty )\}$, using (4.4) repeatedly and (4.1), we have

$$\begin{aligned}&p_{f_*\nu _\infty }({\mathcal {S}}') =\int _{{{\textsf {P}}}^1}\log |{\mathcal {S}}'-\cdot |_\infty (f_*\nu _\infty ) =\int _{{{\textsf {P}}}^1}\log |{\mathcal {S}}'-f(\cdot )|_\infty \nu _\infty \\&=\int _{{{\textsf {P}}}^1}\Bigl (\Phi _{g_f}(f(\cdot ),{\mathcal {S}}')+\bigl (p_{\mu _f}(f(\cdot )) -\frac{I_{\mu _f}}{2}\bigr )+\bigl (p_{\mu _f}({\mathcal {S}}')-\frac{I_{\mu _f}}{2}\bigr )\Bigr )\nu _\infty \\&=\int _{{{\textsf {P}}}^1}\Bigl (\int _{{{\textsf {P}}}^1}\Phi _{g_f}(\cdot ,{\mathcal {S}})(f^*\delta _{{\mathcal {S}}'}) ({\mathcal {S}})\Bigr )\nu _\infty +\int _{{{\textsf {P}}}^1}(p_{\mu _f}\circ f)\nu _\infty +p_{\mu _f}({\mathcal {S}}')-I_{\mu _f}\\&=\int _{{{\textsf {P}}}^1}\biggl (\int _{{{\textsf {P}}}^1}\Bigl (\log |{\mathcal {S}}-\cdot |_\infty -\bigl (p_{\mu _f}({\mathcal {S}})-\frac{I_{\mu _f}}{2}\bigr )-\bigl (p_{\mu _f}(\cdot ) -\frac{I_{\mu _f}}{2}\bigr )\Bigr )(f^*\delta _{{\mathcal {S}}'})({\mathcal {S}})\biggr )\nu _\infty \\&\quad +\int _{{{\textsf {P}}}^1}(p_{\mu _f}\circ f)\nu _\infty +p_{\mu _f}({\mathcal {S}}')-I_{\mu _f}\\&=\int _{{{\textsf {P}}}^1}p_{\nu _\infty }(f^*\delta _{{\mathcal {S}}'}) +\int _{{{\textsf {P}}}^1}(p_{\mu _f}\circ f-d\cdot p_{\mu _f})\nu _\infty \\&\quad +p_{\mu _f}({\mathcal {S}}')-\int _{{{\textsf {P}}}^1}p_{\mu _f}(f^*\delta _{{\mathcal {S}}'}) +(d-1)I_{\mu _f}. \end{aligned}$$

Moreover, by Fubini’s theorem and $p_{\nu _\infty }\equiv I_{\nu _\infty }$ on ${{\textsf {P}}}^1\setminus D_\infty$, we also have

$$\begin{aligned}&\int _{{{\textsf {P}}}^1}(p_{\mu _f}\circ f-d\cdot p_{\mu _f})\nu _\infty \\&\quad =\int _{{{\textsf {P}}}^1}p_{\mu _f}(f_*\nu _\infty )-d\cdot \int _{{{\textsf {P}}}^1}p_{\mu _f}\nu _\infty =\int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f-d\cdot I_{\nu _\infty }, \end{aligned}$$

which completes the proof of (4.13).

If in addition $f_*\nu _\infty =\nu _\infty$ on ${{\textsf {P}}}^1$, then by the identity $p_{\nu _\infty }\equiv I_{\nu _\infty }$ on ${{\textsf {P}}}^1\setminus (D_\infty \cup E)$, where E is an $F_\sigma$-subset in $\partial D_\infty$ of logarithmic capacity 0 with pole $\infty$, and by the vanishing $\mu _f(E)=0$ (from (4.5)), we also have

$$\begin{aligned} \int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f= \int _{{{\textsf {P}}}^1}(p_{\nu _\infty })\mu _f=I_{\nu _\infty }, \end{aligned}$$

(4.15)

which completes the proof of (4.14). $\square$

Lemma 4.7

(Invariance of $\nu _\infty$ under f) If $f(\infty )=\infty \in {{\mathsf {F}}}(f)$, then $f_*\nu _\infty =\nu _\infty$ on ${{\textsf {P}}}^1$ and, for every ${\mathcal {S}}'\in {{\textsf {P}}}^1$,

$$\begin{aligned} \int _{{{\textsf {P}}}^1\setminus \{\infty \}}p_{\nu _\infty } (f^*\delta _{{\mathcal {S}}'})-(d-1)I_{\nu _\infty } ={\left\{ \begin{array}{ll} p_{\nu _\infty }({\mathcal {S}}') &{} \text {if }{\mathcal {S}}'\ne \infty ,\\ \displaystyle \log \biggl |\frac{c^F_1}{c^F_0}\biggr | &{} \text {if }{\mathcal {S}}'=\infty . \end{array}\right. } \end{aligned}$$

(4.16)

Proof

Suppose that $f(\infty )=\infty \in {{\mathsf {F}}}(f)$. Then for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty \}$, by (4.13) and (4.12), we have

We claim that

$$\begin{aligned} p_{f_*\nu _\infty }\equiv \int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f\quad \text {on }{{\mathsf {J}}}(f); \end{aligned}$$

(4.17)

for, by the equality (4.13’) and $p_{\nu _\infty }\ge I_{\nu _\infty }$ on ${{\textsf {P}}}^1$ (and Fubini’s theorem and (4.4)), we have

$$\begin{aligned} p_{f_*\nu _\infty }\ge \int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f >-\infty \quad \text {on }{{\textsf {P}}}^1\setminus \{\infty \}, \end{aligned}$$

so that $p_{f_*\nu _\infty }\equiv \int _{{{\textsf {P}}}^1}p_{\mu _f}(f_*\nu _\infty )$ $\mu _f$-a.e. on ${{\textsf {P}}}^1$. Hence the claim follows by the strong upper semicontinuity (2.1) of $p_{f_*\nu _\infty }$ on ${{\textsf {P}}}^1$ and ${{\mathsf {J}}}(f)={\text {supp}}\mu _f$, also recalling Remark 3.2.

Once the identity (4.17) is at our disposal, using also the maximum principle for the subharmonic function $p_{f_*\nu _\infty }$ and the latter inequality in (2.2), we have

$$\begin{aligned} p_{f_*\nu _\infty } \equiv \int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f =\sup _{{{\mathsf {J}}}(f)}p_{f_*\nu _\infty }\ge \sup _{{{\textsf {P}}}^1\setminus D_\infty }p_{f_*\nu _\infty }\ge I_{\nu _\infty }\quad \text {on }{{\mathsf {J}}}(f), \end{aligned}$$

and integrating both sides of this inequality against $f_*\nu _\infty$, we have $I_{f_*\nu _\infty }\ge I_{\nu _\infty }$ or equivalently

$$\begin{aligned} f_*\nu _\infty =\nu _\infty \quad \text {on }{{\textsf {P}}}^1. \end{aligned}$$

Then (4.16) holds for every ${\mathcal {S}}'\in {{\textsf {P}}}^1\setminus \{\infty \}$ by (4.14) and (4.12). Finally, integrating both sides in (4.8) against $\nu _\infty$ over ${{\textsf {P}}}^1$, by (4.15) and Fubini’s theorem, we compute

$$\begin{aligned}&\log |c_0^F|+\int _{{{\textsf {P}}}^1\setminus \{\infty \}}p_{\nu _\infty }(f^*\delta _\infty ) =\int _{{{\textsf {P}}}^1}\log |F_0(1,\cdot )|_\infty \nu _\infty \\&\quad =d\cdot I_{\nu _\infty } -\int _{{{\textsf {P}}}^1}(p_{\mu _f}\circ f)\nu _\infty -(d-1)\frac{I_{\mu _f}}{2}\\&\quad =d\cdot I_{\nu _\infty }-\int _{{{\textsf {P}}}^1}(p_{f_*\nu _\infty })\mu _f -(d-1)\frac{I_{\mu _f}}{2} =(d-1)I_{\nu _\infty }-(d-1)\frac{I_{\mu _f}}{2}, \end{aligned}$$

which with (4.11) yields (4.16) for ${\mathcal {S}}'=\infty$. $\square$

Remark 4.8

All the computations in this Section are also valid for $K={\mathbb {C}}$.

Remark 4.9

The f-invariance of $\nu _{\infty }$ in Lemma 4.7 is a non-archimedean counterpart to Mañé and da Rocha [22, p. 253, before Corollary 1]. Their argument was based on solving Dirichlet problem using the Poisson kernel on $D_\infty \cup \partial D_\infty$. A similar machinery has been only partly developed in the potential theory on ${{\textsf {P}}}^1$ (see [5, Sects. 7.3, 7.6]).

5 Proof of Theorem 1

Let $f\in K(z)$ be a rational function of degree $d>1$, and $F=(F_0,F_1)\in (K[p_0,p_1]_d)^2$ be a normalized lift of f. When $\infty \in {{\mathsf {F}}}(f)$, let us still denote $\nu _{{{\textsf {P}}}^1\setminus D_\infty }=\nu _{\infty ,{{\textsf {P}}}^1\setminus D_\infty }$ by $\nu _\infty$ for simplicity. If $\mu _f=\nu _\infty$ on ${{\textsf {P}}}^1$, then not only $p_{\mu _f}=p_{\nu _\infty }>I_{\nu _\infty }=I_{\mu _f}$ on $D_\infty$ but, by the continuity of $p_{\mu _f}$ on ${{\textsf {P}}}^1\setminus \{\infty \}$ (by (4.4)), also $p_{\mu _f}=p_{\nu _\infty }\equiv I_{\nu _\infty }=I_{\mu _f}$ on ${{\textsf {P}}}^1\setminus D_\infty$.

Suppose that $\infty \in {{\mathsf {F}}}(f)$, $f(D_\infty )=D_\infty$ (so $D_\infty \subset f^{-1}(D_\infty )$), and $\mu _f=\nu _\infty$ on ${{\textsf {P}}}^1$. Then by (4.8) and $p_{\mu _f}\equiv I_{\mu _f}$ on ${{\textsf {P}}}^1\setminus D_\infty$, we have

$$\begin{aligned} \log |F_0(1,\cdot )|_\infty \equiv (d-1)\frac{I_{\mu _f}}{2}=:I_0 \quad \text {on }{{\textsf {P}}}^1\setminus f^{-1}(D_\infty ). \end{aligned}$$

(5.1)

Let ${\mathcal {S}}_0$ be the point in ${\textsf {H}}^1$ represented by the disk $\{z\in K:|z|\le e^{I_0}\}$ in K.

Suppose also that $f^{-1}(D_\infty )\setminus D_\infty \ne \emptyset$. Then $\deg F_0(1,z)>0$. The subset

$$\begin{aligned} U_\infty :=\{{\mathcal {S}}\in {{\textsf {P}}}^1:|F_0(1,{\mathcal {S}})|_\infty >e^{I_0}\} \end{aligned}$$

in ${{\textsf {P}}}^1$ is the component of ${{\textsf {P}}}^1\setminus (F_0(1,\cdot ))^{-1}({\mathcal {S}}_0)$ containing $\infty$, and $\partial U_\infty =(F_0(1,\cdot ))^{-1}({\mathcal {S}}_0)$. By (5.1), we have $U_\infty \subset f^{-1}(D_\infty )$, and in turn

$$\begin{aligned} U_\infty \subset D_\infty . \end{aligned}$$

For every $w\in f^{-1}(\infty )\setminus \{\infty \} =(F_0(1,\cdot ))^{-1}(0)\subset \{{\mathcal {S}}\in {{\textsf {P}}}^1:|F_0(1,{\mathcal {S}})|_\infty <e^{I_0}\}$, let $D_w$ (resp. $U_w$) be the component of $f^{-1}(D_\infty )$ (resp. the component of $\{{\mathcal {S}}\in {{\textsf {P}}}^1:|F_0(1,{\mathcal {S}})|_\infty <e^{I_0}\}$) containing w. Then $U_w$ is the component of ${{\textsf {P}}}^1\setminus (F_0(1,\cdot ))^{-1}({\mathcal {S}}_0)$ containing w, and $\partial U_w$ is a singleton in $(F_0(1,\cdot ))^{-1}({\mathcal {S}}_0)=\partial U_\infty$. For every $w\in f^{-1}(\infty )\cap D_\infty$, $D_w=D_\infty$.

We claim that $\partial D_\infty$ is a singleton say $\{{\mathcal {S}}_\infty \}$ in ${\textsf {H}}^1$ and, moreover, that for every $w\in f^{-1}(\infty )\setminus D_\infty (\ne \emptyset$ under the assumption that $f^{-1}(D_\infty )\setminus D_\infty \ne \emptyset$),

$$\begin{aligned} \partial D_w=\partial D_\infty (=\{{\mathcal {S}}_\infty \}); \end{aligned}$$

indeed, for every $w\in f^{-1}(\infty )\setminus D_\infty$, we not only have $D_w\subset U_w$ (since otherwise, we must have $\emptyset \ne D_w\cap U_\infty \subset D_w\cap D_\infty$ so $D_w=D_\infty$, which contradicts $w\not \in D_\infty$) but also $U_w\subset D_w$ (by (5.1)), so that $U_w=D_w$. This together with $\partial U_w\subset \partial U_\infty$ and $U_\infty \subset D_\infty$ yields

$$\begin{aligned} \partial D_w=\partial U_w\subset \partial D_\infty \end{aligned}$$

(since otherwise, we must have $\emptyset \ne U_w\cap D_\infty =D_w\cap D_\infty$ so $D_w=D_\infty$, which contradicts $w\not \in D_\infty$). Hence the claim holds since $f(\partial U_w)=f(\partial D_w)=\partial D_\infty$ is a singleton in ${\textsf {H}}^1$.

Once the claim is at our disposal, we compute

$$\begin{aligned}&f^{-1}(\{{\mathcal {S}}_\infty \})=f^{-1}(\partial D_\infty ) \subset \bigcup _{w\in f^{-1}(\infty )}\partial D_w\\&\quad =\Bigl (\bigcup _{w\in f^{-1}(\infty )\cap D_\infty }\partial D_w\Bigr ) \cup \Bigl (\bigcup _{w\in f^{-1}(\infty )\setminus D_\infty }\partial D_w\Bigr ) =\{{\mathcal {S}}_\infty \}\cup \{{\mathcal {S}}_\infty \}=\{{\mathcal {S}}_\infty \}, \end{aligned}$$

so f has a potential good reduction. $\square$

6 Proof of Theorem 2

Pick a prime number p, and let us denote $|\cdot |_p$ by $|\cdot |$ for simplicity. Set

$$\begin{aligned} f(z):=\frac{z^p-z}{p}\in {\mathbb {Q}}[z]\quad \text {and}\quad A(z):=\frac{az+b}{cz+d}\in \mathrm {PGL}(2,{\mathbb {Z}}_p). \end{aligned}$$

If $|c|<1$, then $|ad-bc|=|ad|=1$, so that $|a|=|d|=1$.

Let ${{\mathsf {J}}}(f\circ A)$ and ${{\mathsf {F}}}(f\circ A)$ denote the Berkovich Julia and Fatou sets in ${{\textsf {P}}}^1({\mathbb {C}}_p)$ of $f\circ A$ as an element of ${\mathbb {C}}_p(z)$ of degree p, respectively.

6.1 Computing ${\mathsf {J}}(f\circ A)$

The fact that $\mathsf {J}(f)$ coincides with the classical Julia set of f (see Remark 3.1), which is $\mathbb {Z}_p$, is well known (see e.g., [17, Example 4.11], [6, Example 5.30]). In this subsection, more general facts will be established.

Lemma 6.1

If $|c|<1$, then $(f\circ A)^{-1}({\mathbb {Z}}_p)={\mathbb {Z}}_p$.

Proof

We first claim that for every $z \in \mathbb {Z}$, $p\cdot f(z)=z^p-z\equiv 0$ modulo $p{\mathbb {Z}}$; indeed, when is obvious if $z=0$ modulo $p{\mathbb {Z}}$, and is the case by Fermat’s Little Theorem when $z\ne 0$ modulo $p{\mathbb {Z}}$. By this claim, we have $f({\mathbb {Z}})\subset {\mathbb {Z}}$ (cf. [34]), and in turn $f({\mathbb {Z}}_p)\subset {\mathbb {Z}}_p$ by the continuity of the action of f on ${\mathbb {Q}}_p$ and the density of ${\mathbb {Z}}$ in ${\mathbb {Z}}_p$. Next, we claim that $f^{-1}({\mathbb {Z}}_p)\subset {\mathbb {Z}}_p$ or equivalently that for every $w\in {\mathbb {Z}}_p$, $f^{-1}(w)\subset {\mathbb {Z}}_p$; indeed, setting $W(X):=X^p-X-pw\in {\mathbb {Z}}_p[X]$ of degree p, we have already seen that the reduction $\overline{W}(X)=X^p-X\in {\mathbb {F}}_p[X]$ of W modulo $p{\mathbb {Z}}_p$ has p distinct roots $\overline{0},\ldots ,\overline{p-1}$ in ${\mathbb {F}}_p$. Hence by Hensel’s lemma (see, e.g., [24, Corollary 1 in Sect. 5.1], [8, Sect. 3.3.4, Proposition 3]), W(X) also has p distinct roots in ${\mathbb {Z}}_p$, and has no other roots in $\overline{{\mathbb {Q}}_p}$, so the claim holds. We have seen that $f^{-1}({\mathbb {Z}}_p)={\mathbb {Z}}_p$.

Suppose now that $|c|<1$. Then for every $z\in {\mathbb {Z}}_p$, we have $|cz|<1=|d|$, so that $|A(z)|=|az+b|/|cz+d|=|az+b|\le 1$. Hence $A({\mathbb {Z}}_p)\subset {\mathbb {Z}}_p$, and similarly $A^{-1}({\mathbb {Z}}_p)\subset {\mathbb {Z}}_p$ since $A^{-1}(z)=(dz-b)/(-cz+a)\in \mathrm {PGL}(2,{\mathbb {Z}}_p)$ and $|-c|=|c|<1$. Now we conclude that $(f\circ A)^{-1}({\mathbb {Z}}_p)=A^{-1}({\mathbb {Z}}_p)={\mathbb {Z}}_p$. $\square$

Lemma 6.2

If $|b|\ll 1$ and $|c|\ll 1$, then $f\circ A$ has an attracting fixed point $z_A$ in ${\mathbb {P}}^1({\mathbb {C}}_p)\setminus {\mathbb {Z}}_p$, which tends to $\infty$ as $(a,b,c,d)\rightarrow (1,0,0,1)$ in $({\mathbb {Z}}_p)^4$. Moreover, if in addition $c\ne 0$, then $z_A\in {\mathbb {C}}_p\setminus {\mathbb {Z}}_p$ and $(f\circ A)^{-1}(z_A)\ne \{z_A\}$.

Proof

Since $f^{-1}(\infty )=\{\infty \}$ and $\deg f=p>1$, the former assertion holds also noting that $({\mathrm {Id}}_{{\mathbb {P}}^1({\mathbb {C}}_p)})'\equiv 1\ne 0$ and applying an implicit function theorem to the equation $(f\circ A)(z)=z$ near $(z,a,b,c,d)=(\infty ,1,0,0,1)$ in ${\mathbb {P}}^1({\mathbb {C}}_p)\times ({\mathbb {Z}}_p)^4$ (see, e.g., [1, (10.8)]). Moreover, since $f'(z)=z^{p-1}-p^{-1}$ and $f''(z)=(p-1)z^{p-2}$, the point $A^{-1}(\infty )=-d/c$ is the unique point $z\in {\mathbb {P}}^1({\mathbb {C}}_p)$ such that $\deg _z(f\circ A)=p(=\deg (f\circ A))$, and on the other hand, if in addition $c\ne 0$, then the point $A^{-1}(\infty )$ is $\ne \infty$ and is not fixed by $f\circ A$. Hence the latter assertion holds also noting that $(f\circ A)(\infty )\ne \infty$ if in addition $c\ne 0$. $\square$

Consequently, if $|b|\ll 1$ and $|c|\ll 1$, then

$$\begin{aligned} {{\mathsf {J}}}(f\circ A)={\mathbb {Z}}_p={{\textsf {P}}}^1({\mathbb {C}}_p)\setminus D_{z_A}(f\circ A); \end{aligned}$$

(6.1)

indeed, by Lemma 6.1 (and (3.2)), if $|c|<1$, then ${{\mathsf {J}}}(f\circ A)\subset {\mathbb {Z}}_p$. If in addition $|b|\ll 1$ and $|c|\ll 1$, then by Lemma 6.2 (and ${\mathbb {Z}}_p\subset {\mathbb {C}}_p$), we have ${{\mathsf {F}}}(f\circ A)=D_{z_A}(f\circ A)$, which is an (immediate) attractive basin of f (see [31, Théorème de Classification]) associated with $z_A\in {\mathbb {P}}^1({\mathbb {C}}_p)\setminus {\mathbb {Z}}_p$, and in turn have ${{\mathsf {J}}}(f\circ A)={\mathbb {Z}}_p$ since $(f\circ A)({\mathbb {Z}}_p)\subset {\mathbb {Z}}_p$ by Lemma 6.1.

6.2 Computing Energies and Measures

Since

$$\begin{aligned} {\text {Res}}\bigl (p^{1/2}\cdot \bigl (z_0^p,z_0^pf(z_1/z_0)\bigr )\bigr ) =(p^{1/2})^{2p}\cdot (1^{p-p}\cdot (p^{-1})^{p-0}\cdot 1)=1, \end{aligned}$$

the pair

$$\begin{aligned} F(z_0,z_1):=p^{1/2}\cdot \bigl (z_0^p, z_0^pf(z_1/z_0)\bigr )\in ({\mathbb {Q}}[z_0,z_1]_p)^2 \end{aligned}$$

is a normalized lift of f. Noting that $|{\text {Res}}(az_0+bz_1,cz_0+dz_1)|=|ad-bc|=1$ and using a formula for the homogeneous resultant of the composition of homogeneous polynomial maps (see, e.g., [32, Exercise 2.12]), we also have $\bigl |{\text {Res}}\bigl (F(az_0+bz_1,cz_0+dz_1)\bigr )\bigr | =\bigl |({\text {Res}}F)^1\cdot ({\text {Res}}(az_0+bz_1,cz_0+dz_1))^{p^2}\bigr | =1$, so that

$$\begin{aligned} F_A(z_0,z_1):= & {} F(az_0+bz_1,cz_0+dz_1)\\= & {} p^{1/2}\cdot \biggl ((az_0+bz_1)^p, \frac{(cz_0+dz_1)^p-(az_0+bz_1)^{p-1}(cz_0+dz_1)}{p}\biggr ) \\&\quad \in ({\mathbb {Q}}_p[z_0,z_1]_p)^2 \end{aligned}$$

is a normalized lift of $f\circ A$. For every $n\in {\mathbb {N}}$, write

$$\begin{aligned} (F_A)^n=\bigl (F_{A,0}^{(n)},F_{A,1}^{(n)}\bigr )\in ({\mathbb {Q}}_p[z_0,z_1]_{p^n})^2. \end{aligned}$$

Lemma 6.3

If $|b|<1$ and $|c|<1$, then

$$\begin{aligned} g_{f\circ A}(\infty ) \biggl (=\sum _{j=1}^\infty \Bigl (\frac{\log \Vert (F_A)^j(0,1)\Vert }{p^j} -\frac{\log \Vert (F_A)^{j-1}(0,1)\Vert }{p^{j-1}}\Bigr )\biggr ) =\frac{\log p}{2(p-1)}. \end{aligned}$$

Proof

Suppose that $|b|<1$ and $|c|<1$(, and recall $|p|=p^{-1}<1$). Then for every $(z_0,z_1)\in {\mathbb {C}}_p^2$, if $|z_0|<|z_1|$, then

$$\begin{aligned} |cz_0+dz_1|=|dz_1|=|z_1|>\max \{|az_0|,|bz_1|\}\ge |az_0+bz_1| \end{aligned}$$

so

$$\begin{aligned} |F_{A,0}^{(1)}(z_0,z_1)|&<|F_{A,1}^{(1)}(z_0,z_1)|\quad \text {and}\\ \Vert F_A(z_0,z_1)\Vert&=|F_{A,1}^{(1)}(z_0,z_1)|=p^{1/2}|cz_0+dz_1|^p\\&= p^{1/2}|dz_1|^p=p^{1/2}|z_1|^p =p^{1/2}\Vert (z_0,z_1)\Vert ^p. \end{aligned}$$

Hence inductively, for every $n\in {\mathbb {N}}$, we have $|F_{A,0}^{(n)}(0,1)|<|F_{A,1}^{(n)}(0,1)|$, and moreover

$$\begin{aligned}&\sum _{j=1}^n\Bigl (\frac{\log \Vert (F_A)^j(0,1)\Vert }{p^j} -\frac{\log \Vert (F_A)^{j-1}(0,1)\Vert }{p^{j-1}}\Bigr ) =\sum _{j=1}^n\frac{\frac{1}{2}\log p}{p^j}\\&\quad =\Bigl (\frac{1}{2}\log p\Bigr )\frac{(1/p)(1-1/p^n)}{1-1/p}\rightarrow \Bigl (\frac{1}{2}\log p\Bigr )\frac{1}{p-1} \end{aligned}$$

as $n\rightarrow \infty$. $\square$

Lemma 6.4

If (a, b, c, d) is close enough to (1, 0, 0, 1) in $({\mathbb {Z}}_p)^4$, then

$$\begin{aligned} \mu _{f\circ A}=\nu _{\infty ,{\mathbb {Z}}_p}=\nu _{z_A,{\mathbb {Z}}_p}\quad \text {on }{{\textsf {P}}}^1({\mathbb {C}}_p). \end{aligned}$$

Proof

If $|b|\ll 1$ and $|c|\ll 1$, then by (6.1) and ${\mathbb {Z}}_p\subset {\mathbb {C}}_p$, we have

$$\begin{aligned} \infty \in {{\mathsf {F}}}(f\circ A)=D_{z_A}(f\circ A)={{\textsf {P}}}^1({\mathbb {C}}_p)\setminus {\mathbb {Z}}_p. \end{aligned}$$

Then by (4.5) and Lemma 6.3, we have

$$\begin{aligned} I_{\infty ,\mu _{f\circ A}}=-2\cdot \biggl (\frac{\log p}{2(p-1)}\biggr ) =\log p^{\frac{-1}{p-1}}, \end{aligned}$$

and in particular, recalling $\nu _{\infty ,{\mathbb {Z}}_p}=\mu _f$ on ${{\textsf {P}}}^1({\mathbb {C}}_p)$, also $I_{\infty ,\nu _{\infty ,{\mathbb {Z}}_p}}=I_{\infty ,\mu _f}=\log p^{\frac{-1}{p-1}}$ (for a non-dynamical and more direct computation of $I_{\infty ,\nu _{\infty ,{\mathbb {Z}}_p}}$, see [3]). Now the first equality holds by the uniqueness of the equilibrium mass distribution on the non-polar compact subset ${\mathbb {Z}}_p$ in ${{\textsf {P}}}^1({\mathbb {C}}_p)$. The second equality holds since $z_A$ tends to $\infty$ as $(a,b,c,d)\rightarrow (1,0,0,1)$ in $({\mathbb {Z}}_p)^4$ (by Lemma 6.2), also recalling Observation 2.4. $\square$

Remark 6.5

If $0<|c|\ll 1$ and $|b|\ll 1$, then $(f\circ A)(\infty )\ne \infty \in {{\mathsf {F}}}(f\circ A)$, $(f\circ A)(D_\infty (f\circ A))=D_\infty (f\circ A)$, ${{\mathsf {J}}}(f\circ A)\not \subset {\textsf {H}}^1$ (indeed ${{\mathsf {J}}}(f\circ A)\subset {\mathbb {C}}_p$), and $\mu _{f\circ A}=\nu _{\infty ,{{\textsf {P}}}^1\setminus D_\infty }$ on ${{\textsf {P}}}^1$.

6.3 Conclusion

If $|b|\ll 1$ and $0<|c|\ll 1$, then setting $m_A(z):=\frac{1}{z-z_A}\in \mathrm {PGL}(2,{\mathbb {C}}_p)$, the rational function

$$\begin{aligned} g_A:=m_A\circ (f\circ A)\circ m_A^{-1}\in {\mathbb {C}}_p(z) \end{aligned}$$

is of degree p and satisfies $g_A(\infty )=\infty ,|g_A'(\infty )|<1, g_A^{-1}(\infty )\ne \{\infty \}$, and $\infty \in m_A(D_{z_A}(f\circ A))=D_\infty (g_A)$. If moreover (a, b, c, d) is close enough to (1, 0, 0, 1) in $({\mathbb {Z}}_p)^4$, then also recalling Observations 2.2 and 2.3, we have

$$\begin{aligned} \mu _{g_A}= & {} (m_A)_*\mu _{f\circ A} =(m_A)_*\nu _{\infty ,{\mathbb {Z}}_p}=(m_A)_*\nu _{z_A,{\mathbb {Z}}_p}\\= & {} (m_A)_*\nu _{z_A,{{\textsf {P}}}^1\setminus D_{z_A}(f\circ A)} =\nu _{\infty ,{{\textsf {P}}}^1\setminus D_{\infty }(g_A)} \quad \text {on }{{\textsf {P}}}^1({\mathbb {C}}_p). \end{aligned}$$

Now the proof of Theorem 2 is complete. $\square$