2. Background

For the foundation on non-archimedean dynamics on $\mathsf{P}^1=\mathsf{P}^1(K)$ , see [Reference Baker and Rumely1, Reference Benedetto2, Reference Favre and Jonsson5, Reference Favre and Rivera-Letelier7, Reference Thuillier21], and the survey [Reference Jonsson10, Sections 1-4]. In what follows, we adopt notations from [Reference Okuyama14, Sections 2 and 3].

2·1. $\mathsf{H}^1$ as a hyperbolic space

We begin with continuing the exposition on $\mathsf{P}^1$ in Section 1. For every $a\in\mathbb{P}^1$ , $\#T_a\mathsf{P}^1=1$ . On the other hand,

\begin{equation*}\mathsf{H}^1_{\textrm{IV}}=\{\xi\in\mathsf{H}^1\;:\;\#T_\xi\mathsf{P}^1=1\}\quad\text{and}\quad\mathsf{H}^1_{\textrm{III}}=\{\xi\in\mathsf{P}^1\;:\;\#T_\xi\mathsf{P}^1=2\}.\end{equation*}

For each $a\in\mathbb{P}^1(K)$ , writing $a=[a_0\;:\;a_1]$ , where $\max\{|a_0|,|a_1|\}=1$ , the point $\tilde{a}\;:\!=\;[\tilde{a_0}\;:\;\tilde{a_1}]\in\mathbb{P}^1(k)$ is called the reduction of a modulo $\mathcal{M}_K$ , where $\mathcal{O}_K=B(0,1)$ is the ring of K-integers,

\begin{equation*} \mathcal{M}_K=\{z\in K\;:\;|z|<1\}(=U_{\overrightarrow{\xi_g0}}\cap K)\end{equation*}

is the unique maximal ideal of $\mathcal{O}_K$ , and $k=k_K=\mathcal{O}_K/\mathcal{M}_K$ is the residue field of K. For every $\xi\in\mathsf{H}^1_{\textrm{II}}$ , there is a bijection between $T_{\xi}\mathsf{P}^1$ and $\mathbb{P}^1(k)$ , so that $\#T_{\xi}\mathsf{P}^1=\#\mathbb{P}^1(k)>2$ ; for the Gauss (or canonical) point

\begin{equation*} \xi_g\;:\!=\;\xi_{\mathcal{O}_K}\end{equation*}

( $\textrm{diam}\;(\xi_g)=1\in|K^*|$ so $\xi_g\in\mathsf{H}^1_{\textrm{II}}$ ) in $\mathsf{P}^1$ , there is a canonical bijection $T_{\xi_g}\mathsf{P}^1\ni\overrightarrow{v}\leftrightarrow\tilde{a}\in\mathbb{P}^1(k)$ , where a is any point in $U_{\overrightarrow{v}}\cap\mathbb{P}^1(K)$ .

The hyperbolic (or big model) metric $\rho$ on $\mathsf{H}^1$ is defined as

\begin{align*}\rho(\xi,\xi')\;:\!=\;&2\log \textrm{diam}\;(\xi\wedge\xi')-\log \textrm{diam}\;\xi-\log \textrm{diam}\;(\xi')\\=&\begin{cases}\log\bigl(\textrm{diam}\;(\xi^\prime)/\textrm{diam}\;\xi\bigr) & (\xi,\xi^\prime\in\mathsf{H}^1,\xi\preceq\xi^\prime),\\\rho(\xi,\xi\wedge\xi^\prime)+\rho(\xi\wedge\xi^\prime,\xi^\prime) & (\xi,\xi^\prime\in\mathsf{H}^1,\text{ in general}). \end{cases}\end{align*}

The Berkovich hyperbolic space $(\mathsf{H}^1,\rho)$ is an $\mathbb{R}$ -tree having $\mathbb{P}^1$ as the ideal (or Gromov) boundary, and the topology of $(\mathsf{H}^1,\rho)$ is finer than the relative topology of $\mathsf{H}^1$ in $\mathsf{P}^1$ . For every $\xi\in\mathsf{P}^1$ , the action on $\mathsf{P}^1$ of a rational function $h\in K(z)$ on $\mathbb{P}^1$ of degree $>0$ induces the tangent map $h_*=(h_*)_{\xi}\;:\;T_{\xi}\mathsf{P}^1\to T_{h(\xi)}\mathsf{P}^1$ so that for every $\overrightarrow{v}=\overrightarrow{\xi\xi^\prime}\in T_{\xi}\mathsf{P}^1$ , if $\xi^\prime\in\mathsf{P}^1\setminus\{\xi\}$ is close enough to $\xi$ , then for any distinct $\xi^{\prime\prime},\xi^{\prime\prime\prime}\in[\xi,\xi^\prime]$ , we have $\rho(h(\xi^{\prime\prime}),h(\xi^{\prime\prime\prime}))/\rho(\xi^{\prime\prime},\xi^{\prime\prime\prime})\equiv m_{\overrightarrow{v}}(h)$ , where the constant $m_{\overrightarrow{v}}(h)\in\{1,\ldots,\deg h\}$ is called the directional local degree of h with respect to $\overrightarrow{v}$ , and $h_*(\overrightarrow{v})=\overrightarrow{h(\xi)h(\xi^\prime)}$ ; moreover,

(2·1) \begin{equation}h(U_{\overrightarrow{v}})=\text{either }\mathsf{P}^1\text{ or }U_{h_*(\overrightarrow{v})}.\end{equation}

The hyperbolic metric $\rho$ is independent of the choice of a projective coordinate on $\mathbb{P}^1$ ; indeed, $\rho$ is invariant under the restriction to $\mathsf{H}^1$ of the (extended analytic) action on $\mathsf{P}^1$ of the linear fractional transformations group $\textrm{PGL}(2,K)$ . Let us call an open subset A in $\mathsf{P}^1$ a (non-degenerating and finite open Berkovich) annulus in $\mathsf{P}^1$ if

\begin{equation*} A=A(\xi,\xi')\;:\!=\;U_{\overrightarrow{\xi\xi^\prime}}\cap U_{\overrightarrow{\xi^\prime\xi}}\quad\text{for some distinct }\xi,\xi'\in\mathsf{H}^1,\end{equation*}

and then the modulus of A is defined as

\begin{equation*} \operatorname{mod}A\;:\!=\;\rho(\xi,\xi')\in\mathbb{R}_{>0};\end{equation*}

the moduli of annuli are $\textrm{PGL}(2,K)$ -invariant and, for any annulus $A=A(\xi,\xi^\prime)$ in $\mathsf{P}^1$ and any $\xi^{\prime\prime}\in[\xi,\xi^\prime]\setminus\{\xi,\xi^\prime\}$ ,

(2·2) \begin{equation} \operatorname{mod}A=\operatorname{mod}\bigl(A(\xi,\xi'')\bigr)+\operatorname{mod}\bigl(A(\xi',\xi'')\bigr).\end{equation}

Similarly to concentric annuli, for a subset E in $\mathsf{P}^1$ , we say an annulus $A=A(\xi,\xi^\prime)$ separates E if $A\subset\mathsf{P}^1\setminus E$ , $E\cap(\mathsf{P}^1\setminus U_{\overrightarrow{\xi\xi^\prime}})\neq\emptyset$ , and $E\cap(\mathsf{P}^1\setminus U_{\overrightarrow{\xi^\prime\xi}})\neq\emptyset$ , and the condition (1·1) for this E is reformulated as

(2·3) \begin{equation} \sup\bigl\{\operatorname{mod}A\;:\;A\text{ is an annulus in }\mathsf{P}^1\text{ separating }E\bigr\}<\infty.\end{equation}

2·2. Chordal metric on $\mathbb{P}^1$ and the Hsia kernels on $\mathsf{P}^1$

Let $\pi\;:\;K^2\setminus\{(0,0)\}\to\mathbb{P}^1$ be the canonical projection. On $K^2$ , let $\|(p_0,p_1)\|$ 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 (normalised) chordal metric on $\mathbb{P}^1$ is defined as $[z,w]\;:\!=\;|p\wedge q|/(\|p\|\cdot\|q\|)$ on $\mathbb{P}^1\times\mathbb{P}^1$ , where $p\in\pi^{-1}(z),q\in\pi^{-1}(w)$ , so that

\begin{equation*} [z,w]=\frac{|z-w|}{\max\{1,|z|\}\max\{1,|w|\}}\le\max\{1,|z-w|\}\quad\text{on }K\times K\end{equation*}

and that the strong triangle inequality $[z,w]\le\max\{[z,w'],[w',w]\}$ for any $z,w,w'\in\mathbb{P}^1$ still holds. The topology of $(\mathbb{P}^1,[z,w])$ coincides with the relative topology of $\mathbb{P}^1$ in $\mathsf{P}^1$ . Let us denote by $\textrm{diam}_\#S$ the diameter of a (non-empty) subset S in $(\mathbb{P}^1,[z,w])$ .

The action on $(\mathbb{P}^1,[z,w])$ of a rational function $h\in K(z)$ of degree $>0$ is $1/|\textrm{Res}(h)|$ -Lipschitz continuous, where $\textrm{Res}(h)\in[1,+\infty)$ is the homogeneous resultant $\textrm{Res}(H_0,H_1)$ of a (non-degenerate homogeneous) lift $H=(H_0,H_1)\in(K[X_0,X_1]_{\deg h})^2$ of $h(=[H_0\;:\;H_1])$ normalised so that the maximum of the absolute values of all coefficients of $H_0,H_1$ equals 1 ([Reference Rumely and Winburn18, Theorem 0.1]). In particular, the action on $(\mathbb{P}^1,[z,w])$ of each element $m(z)=(az+b)/(cz+d)$ in $\textrm{PGL}(2,K)$ , where $\max\{|a|,|b|,|c|,|d|\}=1$ , is quasi-isometric in that for any $z,w\in\mathbb{P}^1$ , $|ad-bc|[z,w]\le[M(z),M(w)]\le |ad-bc|^{-1}[z,w]$ , so in particular the subgroup $\textrm{PGL}(2,\mathcal{O}_K)$ in $\textrm{PGL}(2,K)$ acts isometrically on $(\mathbb{P}^1,[z,w])$ .

The generalised Hsia kernel $[\xi,\xi^\prime]_g$ on $\mathsf{P}^1$ with respect to $\xi_g$ is the unique upper semicontinuous and separately continuous extension to $\mathsf{P}^1\times\mathsf{P}^1$ of the chordal metric function [z, w] on $\mathbb{P}^1\times\mathbb{P}^1$ . More generally, for each $\xi_0\in\mathsf{P}^1$ , the generalised Hsia kernel on $\mathsf{P}^1$ with respect to $\xi_0$ is defined as

\begin{equation*}[\xi,\xi']_{\xi_0}\;:\!=\;[\xi,\xi']_g/([\xi,\xi_0]_g[\xi',\xi_0]_g)\quad\text{on }\mathsf{P}^1\times\mathsf{P}^1,\end{equation*}

adopting the convention $1/0=0/(0^2)=+\infty$ ; indeed, denoting by $\xi\wedge_{\xi_0}\xi^\prime$ the unique point in $[\xi,\xi^\prime]\cap[\xi,\xi_0]\cap[\xi^\prime,\xi_0]$ for any $\xi,\xi^\prime,\xi_0\in\mathsf{P}^1$ (so in particular that $\xi\wedge_\infty\xi_0=\xi\wedge\xi^\prime$ ), the Gromov product on $(\mathsf{H}^1,\rho)$ with respect to a point $\xi_0\in\mathsf{H}^1$ is written as

(2·4) \begin{equation}\rho(\xi_0,\xi\wedge_{\xi_0}\xi')=-\log[\xi,\xi']_{\xi_0}+\rho(\xi_g,\xi_0)\quad\text{on }\mathsf{H}^1\times\mathsf{H}^1.\end{equation}

The generalised Hsia kernel function $[\xi,\xi']_{\xi_0}$ not only satisfies the strong triangle inequality

\begin{equation*} [\xi,\xi']_{\xi_0}\le\max\bigl\{[\xi,\xi'']_{\xi_0},[\xi'',\xi']_{\xi_0}\bigr\}\quad\text{for any }\xi,\xi'\in\mathsf{P}^1\end{equation*}

(the equality in which holds if $[\xi,\xi^{\prime\prime}]_{\xi_0}\neq[\xi^{\prime\prime},\xi^\prime]_{\xi_0}$ ), but also is $\textrm{PGL}(2,\mathcal{O}_K)$ -invariant in that for every $M\in\textrm{PGL}(2,\mathcal{O}_K)$ ,

(2·5) \begin{equation} [M(\xi),M(\xi')]_{M(\xi_0)}=[\xi,\xi']_{\xi_0}\quad\text{on }\mathsf{P}^1\times\mathsf{P}^1.\end{equation}

Let us denote $[\xi,\xi^\prime]_\infty$ also by $|\xi-\xi^\prime|_\infty$ , (the restriction to $\mathsf{A}^{\!1}\times\mathsf{A}^{\!1}$ of) which is the original Hsia kernel on $\mathsf{A}^{\!1}$ and restricts to $|z-w|$ on $K\times K$ , although $\xi-\xi^\prime$ itself is not necessarily defined. For any $\xi,\xi^\prime\in\mathsf{A}^{\!1}$ ,

(2·6) \begin{align} |\xi-\xi'|_\infty\bigl(=\sup\bigl\{|z-z^{\prime}|\;:\;z\in B_{\xi},z^{\prime}\in &B_{\xi^\prime}\bigr\}\quad\text{if }\xi,\xi'\not\in\mathsf{H}^1_{\textrm{IV}}\bigr)\\ &\;\;\;\;=\textrm{diam}\;(\xi\wedge\xi')\ge\max\{\textrm{diam}\;\xi,\textrm{diam}\;(\xi')\}.\end{align}

For every (non-empty) subset S in $\mathsf{A}^{\!1}$ , we also set

\begin{equation*} \textrm{diam}_\infty S\;:\!=\;\sup\bigl\{|\xi-\xi'|_\infty\;:\;\xi,\xi'\in S\bigr\}\in[0,+\infty],\end{equation*}

so in particular that for every $\xi\in\mathsf{A}^{\!1}$ , $\textrm{diam}\;\xi=\textrm{diam}_\infty\{\xi\}$ .

For every $\xi_0\in\mathsf{H}^1$ and every compact subset E in $\mathsf{A}^{\!1}$ , noting that $\log[\,\cdot\,,\xi_0]_g=-\rho(\xi_g,\,\cdot\wedge_{\xi_g}\xi_0)\ge-\rho(\xi_g,\xi_0)=\log[\xi_0,\xi_0]_g$ on $\mathsf{H}^1$ (by (2·4)), we have the uniform comparison

(2·7) \begin{equation} \bigl([\xi_0,\xi_0]_g\bigr)^2[\xi,\xi']_{\xi_0}\le |\xi-\xi'|_\infty\le\frac{[\xi,\xi^\prime]_{\xi_0}}{(\inf_E[\,\cdot\,,\infty]_g)^2}\quad\text{on }E\times E\end{equation}

between the kernel functions $|\xi-\xi^\prime|_\infty$ and $[\xi,\xi^\prime]_{\xi_0}$ restricted to $E\times E$ .

For every $\xi_0\in\mathsf{A}^{\!1}$ and every $r\ge \textrm{diam}\;(\xi_0)$ , set

\begin{align*} \mathsf{B}(\xi_0,r) & := \{\xi\in\mathsf{A}^{\!1}\;:\;|\xi-\xi_0|_\infty\le r\}\Bigl(=\mathsf{P}^1\setminus U_{\overrightarrow{\xi_{B(z_0,r)}\infty}}=\overline{B(z_0,r)}\quad\text{in }\mathsf{P}^1\\ &\;\;\;\quad\quad\quad\quad\quad\quad\quad\quad \text{writing }B_{\xi_0}=B(z_0,\textrm{diam}\;(\xi_0))\text{ for some }z_0\in K\text{ if }\xi_0\not\in\mathsf{H}^1_{\textrm{IV}}\Bigr).\end{align*}

Similarly, for every $a\in\mathbb{P}^1$ and every $r\in[0,1]$ , set

\begin{equation*} \mathsf{B}_\#(a,r)\;:\!=\;\{\xi\in\mathsf{P}^1\;:\;[\xi,a]_g\le r\}\bigl(=\overline{\{z\in\mathbb{P}^1\;:\;[z,a]\le r\}}\quad\text{in }\mathsf{P}^1\bigr),\end{equation*}

which is called a Berkovich chordal closed ball in $\mathsf{P}^1$ .

3. Proof of Theorem 1

Let $f\in K(z)$ be a rational function on $\mathbb{P}^1$ of degree $d>1$ having no potentially good reductions (otherwise $\mathsf{J}(f)$ is a singleton, so is not (uniformly) perfect). The following argument is an adaption of that in [Reference Mañé and da Rocha13], which seems most adaptable to non-archimedean setting among the (independent) originals [Reference Eremenko4, Reference Hinkkanen8, Reference Mañé and da Rocha13].

Suppose to the contrary that $\mathsf{J}(f)$ is not uniformly perfect, i.e., that there is a sequence $(A_j)$ of (non-degenerating and finite open Berkovich) annuli in $\mathsf{P}^1$ separating $\mathsf{J}(f)$ such that $\lim_{j\to\infty}\operatorname{mod}(A_j)=+\infty$ . Then without loss of generality, we assume that for every $j\in\mathbb{N}$ , $A_j=A(\xi_j,\xi^\prime_{j})$ , where

\begin{equation*} \xi_j=\xi_{B(a_j,|z_j|)}\quad\text{and}\quad\xi^\prime_{j}=\xi_{B(a_j,|z^{\prime}_{j}|)}\end{equation*}

for some $a_j\in\mathcal{O}_K=B(0,1)$ and some $z_j,z^{\prime}_{j}\in K^*$ , $|z_j|<|z^{\prime}_{j}|\le 1$ (so in particular $\lim_{j\to\infty}|z_j|=0$ ); indeed, (i) by (2·2), we assume that $\xi_g\not\in A_j$ for every $j\in\mathbb{N}$ , and in turn, (ii) by (2·2), the $\textrm{PGL}(2,K)$ -invariance of the moduli of annuli in $\mathsf{P}^1$ , and the $\textrm{PGL}(2,K)$ -equivariance of $\mathsf{J}(f)$ , taking a subsequence of $(A_j)$ and replacing f(z) with $1/f(1/z)$ , if necessary, we also assume that for every $j\in\mathbb{N}$ , $A_j=A(\xi_j,\xi^\prime_{j})$ for some $\xi_j,\xi^\prime_{j}\in\mathsf{H}^1$ satisfying $\xi_j\prec\xi^\prime_{j}\preceq\xi_g$ . Finally, (iii) by the density of $\mathsf{H}^1_{\textrm{II}}$ in $\mathsf{P}^1$ and the continuity of diam on any interval in $\mathsf{P}^1$ , we assume that for every $j\in\mathbb{N}$ , $\xi_j,\xi^\prime_{j}\in\mathsf{H}^1_{\textrm{II}}$ , without loss of generality.

Pick sequences $(w_j),(w^{\prime}_{j})$ in $K^*$ such that for every $j\in\mathbb{N}$ ,

\begin{equation*} \exp\bigl(-\operatorname{mod}(A_j)\bigr)=|z_j/z^{\prime}_{j}|<|w_j|<|w^{\prime}_{j}|< 1\end{equation*}

and that $\lim_{j\to\infty}|w^{\prime}_{j}|=\lim_{j\to\infty}|w_j|/|w^{\prime}_{j}|=0$ (so $\lim_{j\to\infty}|w_j|=0$ ). For every $j\in\mathbb{N}$ , since $|z_j|<|z^{\prime}_{j}w_j|<|z^{\prime}_{j}w^{\prime}_{j}|<|z^{\prime}_{j}|$ , we have

(3·1) \begin{equation} \emptyset\neq\bigl(\mathsf{J}(f)\cap(\mathsf{P}^1\setminus U_{\overrightarrow{\xi_j\xi^\prime_{j}}})\subset\bigr)\mathsf{J}(f)\cap U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}},\end{equation}

and then (see Benedetto [Reference Benedetto2, Proposition 12·9])

(3·2) \begin{equation}\mathsf{J}(f)\subset f^n(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})\quad\text{for }n\gg 1.\end{equation}

Let us also pick a subset $S\subset\mathbb{P}^1$ such that $\#S\ge 3$ as follows; since $\sup_{j\in\mathbb{N}}|z^{\prime}_{j}|\le 1$ and $\lim_{j\to\infty}|w_j|=0$ , we also have

(†) \begin{equation}\textrm{diam}_\infty(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})=\textrm{diam}\;(B(a_j,|z^{\prime}_{j}w_j|))=|z^{\prime}_{j}||w_j|\to 0\end{equation}

as $j\to\infty$ . Hence if (a) the sequence $(a_j)$ (in $\mathcal{O}_K$ ) has a limit point in $\mathcal{O}_K$ , then by (3·1), (†), and the strong triangle inequality, this limit point is in $\mathsf{J}(f)\cap\mathbb{P}^1$ , and then since $\mathsf{J}(f)\cap\mathbb{P}^1$ has no isolated points, we can pick $S\subset\mathsf{J}(f)\cap\mathbb{P}^1$ such that $\#S=3$ . Alternatively if (b) the sequence $(a_j)$ (in $\mathcal{O}_K$ ) has no limit points in $\mathcal{O}_K$ , then by (†) and the strong triangle inequality (and the completeness of K), we can fix $1\ll j_1<j_2<j_3$ such that $U_{\overrightarrow{\xi_{B(a_{j_1},|z^{\prime}_{j_1}w_{j_1}|)}a_{j_1}}}\cap U_{\overrightarrow{\xi_{B(a_{j_2},|z^{\prime}_{j_2}w_{j_2}|)}a_{j_2}}}=\emptyset$ and in turn that $(U_{\overrightarrow{\xi_{B(a_{j_1},|z^{\prime}_{j_1}w_{j_1}|)}a_{j_1}}}\cup U_{\overrightarrow{\xi_{B(a_{j_2},|z^{\prime}_{j_2}w_{j_2}|)}a_{j_2}}})\cap U_{\overrightarrow{\xi_{B(a_{j_3},|z^{\prime}_{j_3}w_{j_3}|)}a_{j_3}}}=\emptyset$ , and set $S\;:\!=\;\{a_{j_\ell}\;:\;\ell\in\{1,2,3\}\}$ .

Fix $L\gg 1$ so that $(0<)L^{-1}\cdot\min_{z,w\in S\;:\;\text{distinct}}[z,w]\le \textrm{diam}_\#(\mathsf{J}(f)\cap\mathbb{P}^1)$ and that the action on $(\mathbb{P}^1,[z,w])$ of f is L-Lipschitz continuous, and fix $c\in(0,L^{-1}\cdot\min_{z,w\in S\;:\;\text{distinct}}[z,w])$ . Then by (3·2), there exists a (unique) sequence $(m_j)$ in $\mathbb{N}$ tending to $\infty$ as $j\to\infty$ such that for every $j\gg 1$ ,

\begin{align*} \max_{n\in\{0,1,\ldots,m_j-1\}} & \textrm{diam}_{\#} \bigl(f^n(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})\cap\mathbb{P}^1\bigr) \lt c\\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \textrm{diam}_\#\bigl(f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})\cap\mathbb{P}^1\bigr)\bigl(\lt Lc\lt\min_{z,w\in S\;:\;\text{distinct}}[z,w]\le 1\bigr), \end{align*}

so in particular

(*) \begin{equation} \#\bigl((f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}}))\cap S\bigr)\le 1.\end{equation}

Moreover, by the choice of $(w_j)$ and $(w^{\prime}_{j})$ , for every $j\in\mathbb{N}$ , we have $U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j}}\setminus U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}}\subset A_j$ , which with $f(\mathsf{F}(f))\subset\mathsf{F}(f)$ also yields

(**) \begin{equation} f^{m_j}\bigl(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j}}\setminus U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}}\bigr)\subset f^{m_j}(A_j)\subset\mathsf{F}(f).\end{equation}

We claim that for $j\gg 1$ , we still have

\begin{equation*} \#\bigl((f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j}}))\cap S\bigr)\le 1;\end{equation*}

for, in the case (a), the claim holds by (^*) and (^**) since $S\subset\mathsf{J}(f)$ in this case. In the case (b), for $j\gg 1$ , by (2·1) and (^*), we have

(***) \begin{equation} f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})=U_{(f^{m_j})_*(\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j})}.\end{equation}

Then $f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})$ intersects $U_{\overrightarrow{\xi_{B(a_{j_\ell},|z^{\prime}_{j_\ell}w_{j_\ell}|)}\xi_{j_\ell}}}$ for at most one of $\ell\in\{1,2,3\}$ (by the strong triangle inequality), and then by (2·1), (3·1) and (^**) (and the strong triangle inequality), we even have

(****) \begin{equation} f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j}})=U_{(f^{m_j})_*(\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j})}.\end{equation}

Hence, if $\#\bigl\{\ell\in\{1,2,3\}\;:\;U_{(f^{m_j})_*(\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j})}\cap U_{\overrightarrow{\xi_{B(a_{j_\ell},|z^{\prime}_{j_\ell}w_{j_\ell}|)}a_{j_\ell}}}\neq\emptyset\bigr\}\le 1$ , then we are done. Otherwise, $U_{(f^{m_j})_*(\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j})}\supset U_{\overrightarrow{\xi_{B(a_{j_\ell},|z^{\prime}_{j_\ell}w_{j_\ell}|)}a_{j_\ell}}}$ for more than one of $\ell\in\{1,2,3\}$ (by the strong triangle inequality), and then

\begin{multline*}\emptyset=\mathsf{F}(f)\cap\mathsf{J}(f)\underset{\text{by }({}^{\ast\ast})}{\supset}\bigl(f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j}}\setminus U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})\bigr)\cap\mathsf{J}(f)\\\underset{\text{by }({}^{\ast\ast\ast})\&({}^{\ast\ast\ast\ast})}{\supset}\bigl(U_{(f^{m_j})_*(\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w^{\prime}_{j}|)}a_j})}\setminus U_{(f^{m_j})_*(\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j})}\bigr)\cap\mathsf{J}(f)\\\underset{\text{by }(^{\ast})}{\supset}(U_{\overrightarrow{\xi_{B(a_{j_\ell},|z^{\prime}_{j_\ell}w_{j_\ell}|)}a_{j_\ell}}})\cap\mathsf{J}(f)\underset{(3\cdot1)}{\neq}\emptyset\end{multline*}

for at least one of $\ell\in\{1,2,3\}$ (the final inclusion in which is also by the strong triangle inequality). This is impossible, and we are also done.

Once the claim is at our disposal, taking a subsequence if necessary, there is $S'\subset S$ such that $\#S'=2$ and that the family

\begin{equation*} g_j(z)\;:\!=\;f^{m_j}\bigl(a_j+z^{\prime}_{j}w^{\prime}_{j}z\bigr)\in K(z), \quad j\in\mathbb{N},\end{equation*}

restricted to $\mathcal{M}_K=U_{\overrightarrow{\xi_g0}}\cap K$ omits $S'$ . Then by Hsia’s Montel-type two points theorem (see Section 2·3), this family $\{g_j\;:\;\mathcal{M}_K\to(\mathbb{P}^1,[z,w])\}_{j\in\mathbb{N}}$ is equicontinuous at $z=0$ . Consequently, recalling that $\lim_{j\to\infty}|w_j/w^{\prime}_{j}|=0$ (from the choice of $(w_j),(w^{\prime}_{j})$ ), we must have

\begin{multline*} 0=\lim_{j\to\infty}\textrm{diam}_\#\bigl(g_j(B(0,|w_j/w^{\prime}_{j}|))\bigr) =\liminf_{j\to\infty}\textrm{diam}_\#\bigl(f^{m_j}(U_{\overrightarrow{\xi_{B(a_j,|z^{\prime}_{j}w_j|)}a_j}})\cap\mathbb{P}^1\bigr)\ge c>0,\end{multline*}

which is a contradiction. Now the proof of Theorem 1 is complete.

4. A (non-infinitesimal) lower capacity density property

Let K be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value $|\cdot|$ .

The logarithmic capacity $ \mathrm{Cap}_\infty E$ with pole $\infty$ of a compact subset E in $\mathsf{A}^{\!1}$ (see (5·1) for the definition) equals the transfinite diameter

\begin{equation*}d^{(\infty)}_\infty E\;:\!=\;\lim_{n\to\infty}\Bigg(\sup_{\xi_1,\ldots,\xi_n\in E}\Bigl(\prod_{i=1}^n\prod_{j\in\{1,\ldots,n\}\setminus\{i\}}\bigl|\xi_i-\xi_j\bigr|_\infty\Bigr)^{\frac{1}{n(n-1)}}\Bigg)\in\mathbb{R}_{\ge 0}\end{equation*}

of E with pole $\infty$ (the above limit is a decreasing one); in particular, $ \mathrm{Cap}_\infty \emptyset=0$ and $ \mathrm{Cap}_\infty E\le \mathrm{Cap}_\infty(E')$ if $E\subset E'$ . For every $\xi\in\mathsf{A}^{\!1}$ and every $r\ge \textrm{diam}\;\xi$ , we have

\begin{multline*} \mathrm{Cap}_\infty(\mathsf{B}(\xi,r))=r\Bigl(= \mathrm{Cap}_\infty(\{\xi_{B(z_0,r)}\})\\\text{writing }B_{\xi}=B(z_0,\textrm{diam}\;\xi)\text{ for some }z_0\in K\text{ if }\xi\not\in\mathsf{H}^1_{\textrm{IV}}\Bigr).\end{multline*}

More details on potential theory on $\mathsf{P}^1$ will be given in Section 5.

Definition 4·1. We say a compact subset E in $\mathsf{P}^1$ has the (non-infinitesimal) lower capacity density property (for some $c\in(0,1)$ ) if there is $c\in(0,1)$ such that for every $\xi\in E\setminus\{\infty\}$ and every $r\in(\textrm{diam}\;\xi,\textrm{diam}_\infty E)$ ,

(4·1) \begin{equation} \mathrm{Cap}_\infty\bigl(E\cap\mathsf{B}(\xi,r)\bigr)\ge cr.\end{equation}

The following potential theoretic precision/characterisation of compact subsets in $\mathsf{P}^1$ having the bounded moduli property for separating annuli (see (1·1)) is a non-archimedean counterpart to the fundamental Pommerenke [Reference Pommerenke16, Theorem 1] in complex analysis.

Among several proofs of [Reference Pommerenke16, Theorem 1] (see the survey [Reference Sugawa20]), the beautiful argument in Pommerenke’s original one seems most adaptable to non-archimedean setting.

Proof of Theorem 3. Let E be a compact subset in $\mathsf{P}^1$ , and set

\begin{equation*}c_E\;:\!=\;\sup\bigl\{\operatorname{mod}A\;:\;A\text{ is a concentric annulus in }\mathsf{A}^{\!1}\text{ separating }E\bigr\}\in\mathbb{R}_{\ge 0}\cup\{+\infty\}.\end{equation*}

$[\!\Leftarrow\!]$ For any (non-degenerating and finite Berkovich open) concentric annulus $A(\xi,\xi^\prime)$ (for some $\xi,\xi^\prime\in\mathsf{H}^1$ satisfying $\xi\prec\xi^\prime$ ) in $\mathsf{A}^{\!1}$ separating E and any $\textrm{diam}\;\xi<r<r'<\textrm{diam}\;(\xi^\prime)$ (so $r'\le \textrm{diam}_\infty E$ ), we have

\begin{equation*} \mathrm{Cap}_\infty\bigl(E\cap\mathsf{B}(\xi,r')\bigr)= \mathrm{Cap}_\infty\bigl(E\cap\mathsf{B}(\xi,r)\bigr)\le \mathrm{Cap}_\infty\bigl(\mathsf{B}(\xi,r)\bigr)=r.\end{equation*}

Hence if E has the (non-infinitesimal) lower capacity density property (for some $c\in(0,1)$ ), then $cr'\le r$ , i.e., $\log(r'/r)\le\log(1/c)$ . Making $r\searrow \textrm{diam}\;\xi$ and $r'\nearrow \textrm{diam}\;(\xi^\prime)$ , we have $\operatorname{mod}(A(\xi,\xi^\prime))\le\log(1/c)$ , so that $c_E\le\log(1/c)<+\infty$ .

$[\!\Rightarrow\!]$ Suppose $c_E\in\mathbb{R}_{\ge 0}$ . Fix $s\in(0,e^{-c_E})(\subset(0,1))$ , and pick

\begin{gather*} \xi_0\in E\quad\text{and}\quad r\in\bigl(\textrm{diam}\;(\xi_0),\textrm{diam}_\infty E\bigr).\end{gather*}

To any $j\in\mathbb{N}\cup\{0\}$ and any $\xi\in E\setminus\{\infty\}$ , we can associate a point $[\xi]_j\in E\setminus\{\infty\}$ satisfying

\begin{align*} \Bigg\{\!\!\begin{array}{ll} & [\xi]_j=\xi \;\;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \text{if }\textrm{diam}\;\xi\ge e^{-c_E}s^jr, \\[5pt]& [\xi]_j\in E\cap\bigl\{\xi^\prime\in\mathsf{A}^{\!1}\;:\;e^{-c_E}s^jr\le|\xi^\prime-\xi|_\infty\le s^jr\bigr\} \quad \text{if }\textrm{diam}\;\xi < e^{-c_E}s^jr; \end{array} \end{align*}

in the latter case, if $\xi\not\in\mathsf{H}^1_{\textrm{IV}}$ , then writing $\xi=B(a,\textrm{diam}\;\xi)$ for some $a\in K$ , we have $\{\xi^\prime\in\mathsf{A}^{\!1}\;:\;e^{-c_E}s^jr\le |\xi^\prime-\xi|_\infty\le s^jr\}=A(\xi_{B(a,e^{-c_E}s^jr)},\xi_{B(a,s^jr)})$ . The case that $\xi\in\mathsf{H}^1_{\textrm{IV}}$ , i.e., that $\xi$ is the cofinal equivalence class of a non-increasing and nesting sequence $(B(a_i,r_i))$ of K-closed disks having the empty intersection, is treated in a similar way.

Then inductively on $j\in\mathbb{N}$ , for every $j\in\mathbb{N}$ and every $(i_1,\ldots,i_j)\in\{0,1\}^j$ , we set $a_{i_1,\ldots,i_j,i_{j+1}}\in E\setminus\{\infty\}$ so that

\begin{align*} a_{i_1} = \Bigg\{\!\! \begin{array}{ll} &\xi_0 & \text{if }i_1=0, \\[5pt] &[\xi_0]_0\, & \text{if }i_1 = 1 \end{array} \quad \text{and}\quad a_{i_1, \ldots,i_j,i_{j+1}} = \Bigg\{\!\! \begin{array}{ll} & a_{i_1, \ldots,i_j} & \text{if }i_{j+1}=0,\\[5pt] & [a_{i_1, \ldots,i_j}]_j & \text{if }i_{j+1}=1, \end{array} \end{align*}

and for every $j\in\mathbb{N}$ , set

\begin{equation*} E_j\;:\!=\;\bigl\{a_{i_1,\ldots,i_j}\;:\;(i_1,\ldots,i_j)\in\{0,1\}^j\bigr\},\end{equation*}

so that $\xi_0\in E_j\subset E_{j+1}$ . For every $j\in\mathbb{N}$ , using the strong triangle inequality for the Hsia kernel $|\xi-\xi^\prime|_\infty$ repeatedly, we have

\begin{equation*} \sup_{(i_1,\ldots,i_j)\in\{0,1\}^j}|a_{i_1,\ldots,i_j}-\xi_0|_\infty\le\max\{\textrm{diam}\;(\xi_0),s^0r\}=r,\end{equation*}

so that $E_j\subset E\cap\mathsf{B}(\xi_0,r)$ .

For every $j\in\mathbb{N}$ and any distinct $(i_1,\ldots,i_j),(i^\prime_{1},\ldots,i^\prime_{j})\in\{0,1\}^j$ , writing

\begin{equation*} \xi=a_{i_1,\ldots,i_j}\quad\text{and}\quad\xi'=a_{i^\prime_{1},\ldots,i^\prime_{j}}\end{equation*}

(in $E_j$ ) for simplicity, set not only

\begin{multline*}m_*=m_*(\xi,\xi')\\\;:\!=\;\max\bigl\{m\in\{1,\ldots,j\}\;:\;i_k=i^\prime_{k}\text{ for any }k\in\{1,\ldots,m\}\bigr\}\in\{0,1,\ldots,j-1\}\end{multline*}

(under the convention $\max\emptyset=0$ ) but also

\begin{equation*} \xi_*=\xi_*(\xi,\xi')\;:\!=\;\begin{cases} a_{i_1,\ldots,i_{m_*}}=a_{i^\prime_{1},\ldots,i^\prime_{\phantom{1}\!\!\!m_*}}&\text{if }m_*>0,\\ \xi_0 &\text{if }m_*=0\end{cases}\in E_j.\end{equation*}

We claim that

\begin{equation*} |\xi-\xi'|_\infty>s^{m_*+1}r;\end{equation*}

indeed, replacing $(i_1,\ldots,i_j),(i^\prime_{1},\ldots,i^\prime_{j})$ if necessary, we have $i_{m_*+1}=0$ and $i^\prime_{m_*+1}=1$ , so that

\begin{equation*} a_{i_1,\ldots,i_{m_*},i_{m_*+1}}=\xi_*\quad\text{and}\quad a_{i^{\prime}_{1},\ldots,i^\prime_{\phantom{1}\!\!\!m_*},i^{\prime}_{m_*+1}}=[\xi_*]_{m_*}.\end{equation*}

If $\max\{\textrm{diam}\;(\xi_*),\textrm{diam}\;([\xi_*]_{m_*})\}>s^{m_*+1}r$ , then either $\xi=\xi_*$ or $\xi^\prime=[\xi_*]_{m_*}$ , and in turn, also using the lower estimate (2·6) of the kernel function $|\xi-\xi^\prime|_\infty$ , we have the desired estimate

\begin{equation*} |\xi-\xi'|_\infty\ge\max\bigl\{\textrm{diam}\;(\xi_*),\textrm{diam}\;([\xi_*]_{m_*})\bigr\}>s^{m_*+1}r\end{equation*}

in this case. Alternatively if $\max\{\textrm{diam}\;(\xi_*),\textrm{diam}\;([\xi_*]_{m_*})\}\le s^{m_*+1}r$ , then

\begin{equation*} \max_{m\in\{m_*+1,\cdots,j-1\}}\max\bigl\{\textrm{diam}\;(a_{i_1,\ldots,i_m}),\textrm{diam}\;(a_{i^{\prime}_{1},\ldots,i^\prime_{\phantom{1}\!\!\!m}})\bigr\}\le s^{m_*+1}r,\end{equation*}

so that also using the strong triangle inequality for the kernel function $|\xi-\xi^\prime|_\infty$ repeatedly, we compute

\begin{multline*} |\xi-\xi_*|_\infty\le\max_{m\in\{m_*+1,\ldots,j-1\}}\bigl|a_{i_1,\ldots,i_m,i_{m+1}}-a_{i_1,\ldots,i_m}\bigr|_\infty\\\le\max_{m\in\{m_*+1,\ldots,j-1\}}\max\bigl\{\textrm{diam}\;(a_{i_1,\ldots,i_m}),s^mr\bigr\}\le s^{m_*+1}r\end{multline*}

and, similarly,

\begin{equation*}\bigl|\xi'-[\xi_*]_{m_*}\bigr|_\infty\le\max_{m\in\{m_*+1,\ldots,j-1\}}\bigl|a_{i^\prime_{1},\ldots,i^\prime_{\phantom{1}\!\!\!m},i^\prime_{m+1}}-a_{i^\prime_{1},\ldots,i^\prime_{\phantom{1}\!\!\!m}}\bigr|_\infty\le s^{m_*+1}r.\end{equation*}

By the above two estimates and the equality in the strong triangle inequality for the kernel function $|\xi-\xi^\prime|_\infty$ , we have

\begin{multline*}|\xi-\xi'|_\infty=\max\Bigl\{|\xi-\xi_*|_\infty,\bigl|\xi_*-[\xi_*]_{m_*}\bigr|_\infty,\bigl|\xi'-[\xi_*]_{m_*}\bigr|_\infty\Bigr\}\\=\bigl|\xi_*-[\xi_*]_{m_*}\bigr|_\infty\ge e^{-c_E}s^{m_*}r>s^{m_*+1}r,\end{multline*}

which also yields the desired estimate in this case. Hence the claim holds.

Once the claim is at our disposal, for every $j\in\mathbb{N}$ and every $(i_1,\ldots,i_j)\in\{0,1\}^j$ , we compute

\begin{align*} &\prod_{(i^\prime_{1},\ldots,i^\prime_{j})\in\{0,1\}^j\setminus\{(i_1,\ldots,i_j)\}}|a_{i_1,\ldots,i_j}-a_{i^\prime_{1},\ldots,i^\prime_{j}}|_\infty\\=&\prod_{m=0}^{j-1}\Bigl(\prod_{(i^\prime_{1},\ldots,i^\prime_{j})\in\{0,1\}^j\setminus\{(i_1,\ldots,i_j)\}\;:\;m_*(a_{i_1,\ldots,i_j},a_{i^\prime_{1},\ldots,i^\prime_{\phantom{1}} \!\!\!\!\phantom{l}_{j} })=m}|a_{i_1,\ldots,i_j}-a_{i^\prime_{1},\ldots,i^\prime_{j}}|_\infty\Bigr)\\>&\prod_{m=0}^{j-1}(s^{m+1}r)^{2^{j-(m+1)}}=s^{\sum_{m=0}^{j-1}(m+1)2^{j-(m+1)}}\cdot r^{2^j-1},\end{align*}

so that, also recalling that $E_j\subset E\cap\mathsf{B}(\xi_0,r)$ and $\#\{0,1\}^j=2^j$ , we have

\begin{multline*} \mathrm{Cap}_\infty\bigl(E\cap\mathsf{B}(\xi_0,r)\bigr)=d^{(\infty)}_\infty\bigl(E\cap\mathsf{B}(\xi_0,r)\bigr)\\\ge\limsup_{j\to\infty}\Bigl(\bigl(s^{2^j\sum_{m=0}^{j-1}(m+1)2^{-(m+1)}}\cdot\, r^{2^j-1}\bigr)^{2^j}\Bigr)^{\frac{1}{2^j(2^j-1)}}=\bigl(s^{\sum_{m=0}^\infty\frac{m+1}{2^{m+1}}}\bigr)r=s^2r.\end{multline*}

Hence E has the (non-infinitesimal) lower capacity density property (for $c=e^{-2c_E}$ ).

5. A uniform Hölder continuity property

Let K be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value $|\cdot|$ .

Let us recall some details on logarithmic potential theory on $\mathsf{P}^1$ (see [Reference Baker and Rumely1, Section 6]). Recall that for every $\xi_0\in\mathsf{P}^1$ and every positive Radon measure $\nu$ on $\mathsf{P}^1$ , the logarithmic potential of $\nu$ on $\mathsf{P}^1$ with respect to $\xi_0$ (or with pole $\xi_0$ when $\xi_0\in\mathbb{P}^1$ ) is the function

\begin{equation*} p_{\xi_0,\nu}(\!\cdot\!)\;:\!=\;\int_{\mathsf{P}^1}\log[\cdot,\xi']_{\xi_0}\nu(\xi')\;:\;\mathsf{P}^1\to[{-}\infty,+\infty]\end{equation*}

(Definition 1·2). If either $\xi_0\in\mathsf{H}^1$ or $\textrm{supp}\;\nu\subset\mathsf{P}^1\setminus\{\xi_0\}$ , then $p_{\xi_0,\nu}$ is strongly upper semicontinuous on $\mathsf{P}^1$ in that for every $\xi\in\mathsf{P}^1$ , $\limsup_{\xi^\prime\to\xi}p_{\xi_0,\nu}(\xi^\prime)=p_{\xi_0,\nu}(\xi)$ , and satisfies

\begin{equation*}p_{\xi_0,\nu}(\xi_0)=-\log[\xi_0,\xi_0]_g=\max_{\mathsf{P}^1}p_{\xi_0,\nu}.\end{equation*}

If $\textrm{supp}\;\nu\subset\mathsf{P}^1\setminus\{\xi_0\}$ , then $p_{\xi_0,\nu}$ is not only subharmonic on $\mathsf{P}^1\setminus\{\xi_0\}$ but also harmonic on $\mathsf{P}^1\setminus(\{\xi_0\}\cup\textrm{supp}\;\nu)$ , and the logarithmic energy of $\nu$ with respect to $\xi_0$ (or with pole $\xi_0$ when $\xi_0\in\mathbb{P}^1$ ) is defined as

\begin{equation*} I_{\xi_0,\nu}\;:\!=\;\int_{\mathsf{P}^1}p_{\xi_0,\nu}\nu=\int_{\mathsf{P}^1\times\mathsf{P}^1}\log[\xi,\xi']_{\xi_0}(\nu\times\nu)(\xi,\xi')\in[{-}\infty,+\infty).\end{equation*}

For every subset C in $\mathsf{P}^1$ and every $\xi_0\in\mathbb{P}^1\setminus C$ , setting

\begin{equation*} V_{\xi_0}(C)\;:\!=\;\sup_{\nu}I_{\xi_0,\nu}\in[{-}\infty,+\infty),\end{equation*}

where $\nu$ ranges over all probability Radon measures on $\mathsf{P}^1$ supported by C (under the convention $\sup_{\emptyset}={-}\infty$ here), the logarithmic capacity of C with respect to $\xi_0$ (or with pole $\xi_0$ when $\xi_0\in\mathbb{P}^1$ ) is

(5·1) \begin{equation} \mathrm{Cap}_{\xi_0}(C)\;:\!=\;\exp\bigl(V_{\xi_0}(C)\bigr)\in\mathbb{R}_{\ge 0}.\end{equation}

For every compact subset E in $\mathsf{P}^1$ and every $\xi_0\in\mathbb{P}^1\setminus E$ , if $ \mathrm{Cap}_{\xi_0}(E)>0$ , then the following Frostman-type properties hold; there is a unique probability Radon measure $\nu$ on $\mathsf{P}^1$ , which is called the equilibrium mass distribution on E with respect to $\xi_0$ (or with pole $\xi_0$ when $\xi_0\in\mathbb{P}^1\setminus E$ ) and is denoted by $\nu_{\xi_0,E}$ , such that $\textrm{supp}\;\nu\subset E$ and that $I_{\xi_0,\nu}=V_{\xi_0}(E)$ ; then denoting by $D_{\xi_0}=D_{\xi_0,E}$ the component of $\mathsf{P}^1\setminus E$ containing $\xi_0$ , we have $\textrm{supp}\;(\nu_{\xi_0,E})\subset\partial(D_{\xi_0})$ ,

\begin{gather*}p_{\xi_0,\nu_{\xi_0,E}}\ge V_{\xi_0}(E)\text{ on }\mathsf{P}^1,\quad p_{\xi_0,\nu_{\xi_0,E}}>V_{\xi_0}(E)\text{ on }D_{\xi_0},\quad\text{and}\\p_{\xi_0,\nu_{\xi_0,E}}\equiv V_{\xi_0}(E)\text{ on }\mathsf{P}^1\setminus(D_{\xi_0}\cup F),\end{gather*}

and $p_{\xi_0,\nu_{\xi_0,E}}$ is continuous at every point in $\mathsf{P}^1\setminus F$ , where F is an $F_{\sigma}$ -subset in $(\partial(D_{\xi_0}))\cap\mathbb{P}^1$ satisfying $ \mathrm{Cap}_{\xi_0}(F)=0$ ; F could be chosen as $\emptyset$ if and only if $p_{\xi_0,\nu_{\xi_0,E}}$ is continuous at any point in $\partial(D_{\xi_0})$ , and then $\textrm{supp}\;(\nu_{\xi_0,E})=\partial(D_{\xi_0})$ . For the details on subharmonic functions on open subsets in $\mathsf{P}^1$ , see [Reference Baker and Rumely1, Sections 7 and 8] and [Reference Thuillier21, Section 3].

By the $\textrm{PGL}(2,\mathcal{O}_K)$ -invariance (2·5) of the generalised Hsia kernel functions $[\xi,\xi^\prime]_{\xi_0}$ on $\mathsf{P}^1$ , both the logarithmic capacity $ \mathrm{Cap}_{\xi_0}(E)$ of a compact subset E in $\mathsf{P}^1$ and the equilibrium mass distribution on E in $\mathsf{P}^1$ with respect to $\xi_0\in\mathsf{P}^1\setminus E$ are $\textrm{PGL}(2,\mathcal{O}_K)$ -invariant/equivariant in that for every $\xi_0\in\mathsf{P}^1\setminus E$ and every $M\in\textrm{PGL}(2,\mathcal{O}_K)$ ,

\begin{equation*} \mathrm{Cap}_{M(\xi_0)}(M(E))= \mathrm{Cap}_{\xi_0}E\end{equation*}

and, if in addition $ \mathrm{Cap}_{\xi_0}E>0$ , then $\nu_{M(\xi_0),M(E)}=M_*(\nu_{\xi_0,E})$ on $\mathsf{P}^1$ .

In computation, the so called Perron method is useful (see [Reference Thuillier21, Section 3·1·3]). For every $\xi_0\in\mathsf{H}^1$ , every $\overrightarrow{v}\in T_{\xi_0}\mathsf{P}^1$ , and every compact subset E in $U_{\overrightarrow{v}}$ , the zero-one extremal function

\begin{multline*}\xi\longmapsto h(\xi;\ E,U_{\overrightarrow{v}})\;:\!=\;\sup\bigl\{u(\xi)\;:\;u\text{ is subharmonic on }U_{\overrightarrow{v}},\, u|E\le 0,\,\text{and }u< 1\text{ on }U_{\overrightarrow{v}}\bigr\}\\:\;U_{\overrightarrow{v}}\to[{-}\infty,1)\end{multline*}

for E with respect to $U_{\overrightarrow{v}}$ is not only harmonic on $U_{\overrightarrow{v}}\setminus E$ but also $\equiv 0$ on the union of E and all components of $(U_{\overrightarrow{v}}\cup\{\xi_0\})\setminus E$ not containing $\xi_0$ , and the upper semicontinuous regularisation $h(\cdot;\,E,U_{\overrightarrow{v}})^*$ of $h(\cdot;\,E,U_{\overrightarrow{v}})$ on $U_{\overrightarrow{v}}$ is subharmonic on $U_{\overrightarrow{v}}$ and coincides with $h(\cdot;\,E,U_{\overrightarrow{v}})$ on $U_{\overrightarrow{v}}\setminus E$ . On the other hand, similarly, for every compact subset E in $\mathsf{P}^1$ and every $\xi_0\in\mathsf{H}^1\setminus E$ , if $ \mathrm{Cap}_{\xi_0}(E)>0$ , then the function

\begin{multline*}\xi\longmapsto P_{\xi_0,E}(\xi)\;:\!=\;\sup\bigl\{u(\xi)\;:\;u\text{ is subharmonic on }\mathsf{P}^1\setminus\{\xi_0\},\, u|E\le 0,\\\text{ and }u<-\log[\xi_0,\xi_0]_g-\log \mathrm{Cap}_{\xi_0}(E)\bigr\}\\:\;\mathsf{P}^1\to\bigl[{-}\infty,-\log[\xi_0,\xi_0]_g-\log \mathrm{Cap}_{\xi_0}(E)\bigr)\end{multline*}

is superharmonic on the $D_{\xi_0}=D_{\xi_0,E}$ , is harmonic on $D_{\xi_0}\setminus\{\xi_0\}$ , and is $\equiv 0$ on $\mathsf{P}^1\setminus D_{\xi_0}$ , and the upper semicontinuous regularisation $P_{\xi_0,E}^*$ of $P_{\xi_0,E}$ on $\mathsf{P}^1\setminus\{\xi_0\}$ is subharmonic on $\mathsf{P}^1\setminus\{\xi_0\}$ ; indeed,

(5·2) \begin{equation}P_{\xi_0,E}^*\equiv p_{\xi_0,\nu_{\xi_0,E}}-\log \mathrm{Cap}_{\xi_0}(E)\quad\text{on }\mathsf{P}^1,\end{equation}

which is nothing but the generalised Green function $G_{\xi_0,E}$ on $\mathsf{P}^1$ associated to E with respect to $\xi_0$ (or with pole $\xi_0$ when $\xi_0\in\mathbb{P}^1\setminus E$ ).

For every compact subset E in $\mathsf{A}^{\!1}$ , every $\xi_0\in\mathsf{H}^1\setminus E$ , every $a\in E\cap K$ , every $t>0$ , and every $R>1$ satisfying the following admissibility

(5·3) \begin{equation} \mathrm{Cap}_{\xi_0}\bigl(E\cap\mathsf{B}(a,t)\bigr)>0\quad\text{and}\quad\xi_{B(a,\,Rt)}\in[\xi_0,a],\end{equation}

we have not only

(5·4) \begin{multline}\frac{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}}{\sup_{U_{\overrightarrow{\xi_{B(a,\,Rt)}a}}}P_{\xi_0,E\cap\mathsf{B}(a,\;t)}}\le h\bigl(\,\cdot\,;\,E\cap\mathsf{B}(a,t),U_{\overrightarrow{\xi_{B(a,\,Rt)}a}}\bigr)\le\frac{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*}{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})}\\\text{on }U_{\overrightarrow{\xi_{B(a,\,Rt)}a}}\end{multline}

(cf. [Reference Siciak19, p.209]) using the maximum principle ([Reference Baker and Rumely1, Proposition 8·14]) for subharmonic functions (on $U_{\overrightarrow{\xi_{B(a,\,Rt)}a}}$ ), but also

(5·5) \begin{multline}P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})+\log \mathrm{Cap}_{\xi_0}\bigl(E\cap\mathsf{B}(a,t)\bigr)\\\Bigl(=p_{\xi_0,\nu_{\xi_0,E\cap\mathsf{B}(a,\;t)}}(\xi_{B(a,\,Rt)})\Bigr)=-\rho(\xi_0,\xi_{B(a,\,Rt)})+\rho(\xi_g,\xi_0)\end{multline}

using the identities (5·2) (applied to $E\cap\mathsf{B}(a,t)$ ) and $\log[\xi_{B(a,\,Rt)},\cdot\,]_{\xi_0}\equiv-\rho(\xi_0,\xi_{B(a,\,Rt)})+\rho(\xi_g,\xi_0)$ on $\mathsf{B}(a,t)$ (from (2·4)). The latter (5·5) in particular yields

(5·6) \begin{multline}P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})\Bigl(=-2\log \textrm{diam}\;(\xi_0\wedge\xi_{B(a,\,Rt)})+\log \textrm{diam}\;(\xi_0)+\log(Rt)\\+2\log \textrm{diam}\;(\xi_g\wedge\xi_0)-\log 1-\log \textrm{diam}\;(\xi_0)-\log \mathrm{Cap}_{\xi_0}\bigl(E\cap\mathsf{B}(a,t)\bigr)\Bigr)\\\le 2\log\bigl(\textrm{diam}\;(\xi_g\wedge\xi_0)/\textrm{diam}\;(\xi_0)\bigr)+\log\bigl(R/\bigl( \mathrm{Cap}_{\xi_0}(E\cap\mathsf{B}(a,t))/t\bigr)\bigr).\end{multline}

The following uniform Hölder continuity property of the generalised Green function on $\mathsf{P}^1$ associated to a non-empty compact subset in $\mathsf{P}^1$ satisfying the (non-infinitesimal) lower capacity density condition (4·1) is a non-archimedean counterpart to Lithner [Reference Lithner12] in complex analysis.

Theorem 4. Let K be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value. Then for every compact subset E in $\mathsf{P}^1$ having the (non-infinitesimal) lower capacity density property and every $\xi_0\in\mathsf{P}^1\setminus E$ , denoting by $D_{\xi_0}=D_{\xi_0,E}$ the component of $\mathsf{P}^1\setminus E$ containing $\xi_0$ , there are $\alpha>0$ and $\delta_0>0$ such that

\begin{equation*}\sup_{a\in(\partial(D_{\xi_0}))\cap\mathbb{P}^1}\sup_{\delta\in(0,\delta_0]}\frac{\sup_{\mathsf{B}_\#(a,\delta)}\bigl(p_{\xi_0,\nu_{\xi_0,E}}(\!\cdot\!)-\log \mathrm{Cap}_{\xi_0}(E)\bigr)}{\delta^\alpha}<+\infty.\end{equation*}

Proof. Pick a compact subset E in $\mathsf{P}^1$ , which is for a while not assumed to have the (non-infinitesimal) lower capacity density property, and pick a point $\xi_0\in\mathsf{P}^1\setminus E$ . Then by the above $\textrm{PGL}(2,\mathcal{O}_K)$ -invariance/equivariance of the kernel function $[\xi,\xi^\prime]_g$ , the (positive) logarithmic capacity, and the equilibrium mass distribution on a compact subset in $\mathsf{P}^1$ , we assume both

\begin{equation*} E\subset\mathsf{A}^{\!1}\end{equation*}

and $\xi_0\in D_\infty$ , and moreover $\xi_0=\infty$ when $\xi_0\in\mathbb{P}^1$ , without loss of generality. We even assume

\begin{equation*} \xi_0\in D_\infty\cap\mathsf{H}^1\end{equation*}

without loss of generality; for, by the strong triangle inequality for the kernel function $[\xi,\xi^\prime]_g$ , if $\xi_0\in\mathsf{H}^1$ is close enough to $\infty$ , then $[\xi,\xi^\prime]_{\xi_0}=[\xi,\xi^\prime]_\infty$ on $E\times E$ . In the following, we adapt Siciak’s Wiener-type argument in [Reference Siciak19] to non-archimedean setting.

(i) For this $\xi_0$ , any $t>0$ , any $R,r>1$ ( $R>r$ ), and any $a\in E\cap K$ satisfying the admissibility (5·3), by the latter inequality in (5·4), the maximum principle for subharmonic functions (on $U_{\overrightarrow{\xi_{B(a,\,rt)}a}}$ ), (5·5) for R,r, and (5·6), we compute

(5·7) \begin{multline}c(t)\;:\!=\;1-\sup_{U_{\overrightarrow{\xi_{B(a,\,rt)}a}}}h\bigl(\,\cdot\,;\,E\cap\mathsf{B}(a,t),U_{\overrightarrow{\xi_{B(a,\,Rt)}a}}\bigr)\\\biggl(\ge 1-\frac{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,rt)})}{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})}=\frac{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})-P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,rt)})}{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})}\\=\frac{-\rho(\xi_0,\xi_{B(a,\,Rt)})+\rho(\xi_0,\xi_{B(a,\,rt)})}{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})}=\frac{\rho(\xi_{B(a,\,Rt)},\xi_{B(a,\,rt)})}{P_{\xi_0,E\cap\mathsf{B}(a,\;t)}^*(\xi_{B(a,\,Rt)})}\biggr)\\\ge\frac{\log(R/r)}{2\log\bigl(\textrm{diam}\;(\xi_g\wedge\xi_0)/\textrm{diam}\;(\xi_0)\bigr)+\log\bigl(R/\bigl( \mathrm{Cap}_{\xi_0}(E\cap\mathsf{B}(a,t))/t\bigr)\bigr)}.\end{multline}

(ii) Fix $R>r>1$ (so $r/R\in(0,1)$ ), and for every $n\in\mathbb{N}$ , set

\begin{equation*} t_n\;:\!=\;\Bigl(\frac{r}{R}\Bigr)^n.\end{equation*}

Then for every $a\in E\cap K$ , every $n\in\mathbb{N}$ , and every $m\in\mathbb{N}\cup\{0\}$ , by an induction on m, we have

(5·8) \begin{equation}h\bigl(\,\cdot\,;\,\mathsf{B}(a,t_n)\cap E,U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}\bigr)\le e^{-\sum_{j=0}^m c(t_{n+j})}\quad\text{on }U_{\overrightarrow{\xi_{B(a,\,rt_{n+m})}a}};\end{equation}

indeed, for any subharmonic function u on $U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}$ satisfying $u|(\mathsf{B}(a,t_n)\cap E)\le 0$ and $u<1$ on $U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}$ , we have $u=1-(1-u)\le 1-c(t_n)\le e^{-c(t_n)}$ on $U_{\overrightarrow{\xi_{B(a,\,rt_n)}a}}=U_{\overrightarrow{\xi_{B(a,\,Rt_{n+1})}a}}$ , which yields not only

\begin{equation*} h(\cdot;\,\mathsf{B}(a,t_n)\cap E,U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}})\le e^{-c(t_n)}\quad\text{on }U_{\overrightarrow{\xi_{B(a,\,rt_n)}a}}=U_{\overrightarrow{\xi_{B(a,\,Rt_{n+1})}a}}\end{equation*}

but, together with $t_{n+1}\le t_n$ , also

\begin{multline*}\frac{h\bigl(\,\cdot\,;\,\mathsf{B}(a,t_n)\cap E,U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}\bigr)}{e^{-c(t_n)}}\Bigl(\le h\bigl(\,\cdot\,;\,\mathsf{B}(a,t_n)\cap E,U_{\overrightarrow{\xi_{B(a,\,Rt_{n+1})}a}}\bigr)\Bigr)\\\le h\bigl(\,\cdot\,;\,\mathsf{B}(a,t_{n+1})\cap E,U_{\overrightarrow{\xi_{B(a,\,Rt_{n+1})}a}}\bigr)\quad\text{on }U_{\overrightarrow{\xi_{B(a,\,Rt_{n+1})}a}}.\end{multline*}

Now an induction concludes the desired estimate (see [Reference Siciak19, p.212]).

(iii) Also fix $n\in\mathbb{N}$ so large that $Rt_n<\textrm{diam}\;(\xi_0)(\le \textrm{diam}\;(\xi_0\wedge a)$ for every $a\in E\cap K$ ); then for every $a\in E\cap K$ and every $m\in\mathbb{N}\cup\{0\}$ , the latter halves

\begin{equation*} \xi_{B(a,\,Rt_{n+m})},\xi_{B(a,\,rt_{n+m})}\in[\xi_0,a]\end{equation*}

in the admissibility (5·3) for $t=t_{n+m}$ and R,r hold.

To any $\delta\in(0,\min\{1,rt_{n+1}\}]$ , let us associate $m_\delta\in\mathbb{N}$ such that

\begin{equation*} \delta\in[rt_{n+m_\delta+1},rt_{n+m_\delta}],\end{equation*}

the lower bound of $\delta$ in which is equivalent to the inequality

\begin{equation*}-(m_\delta+1)\log(R/r)\le-\log\bigl((r/R)^nr\bigr)+\log\delta.\end{equation*}

(iv) Suppose now that the compact subset E (in $\mathsf{A}^{\!1}$ under the normalisation in (i)) has the (non-infinitesimal) lower capacity density property, for some $c\in(0,1)$ . Then using the latter half of the uniform comparison (2·7) between the kernel functions $|\xi-\xi^\prime|_\infty$ and $[\xi,\xi^\prime]_{\xi_0}$ on E, we have

\begin{multline*} c_0\;:\!=\;\inf_{\xi\in E}\inf_{t\in(\textrm{diam}\;\xi,\textrm{diam}_\infty E)}\frac{ \mathrm{Cap}_{\xi_0}(E\cap\mathsf{B}(\xi,t))}{t}\\\Bigl(\ge\inf_{\xi\in E}\inf_{t\in(\textrm{diam}\;\xi,\textrm{diam}_\infty E)}\frac{\bigl(\inf_E[\,\cdot\,,\infty]_g\bigr)^2 \mathrm{Cap}_{\infty}(E\cap\mathsf{B}(\xi,t))}{t}\Bigr)\ge c\cdot\!\bigl(\inf_E[\,\cdot\,,\!\infty]_g\bigr)^2\in(0,\!1)\end{multline*}

and set

\begin{equation*} \ell=\ell_E\;:\!=\;\frac{1}{2\log\bigl(\textrm{diam}\;(\xi_g\wedge\xi_0)/\textrm{diam}\;(\xi_0)\bigr)+\log(R/c_0)}\in\Bigl(0,\frac{1}{\log(R/c_0)}\Bigr).\end{equation*}

Then for every $a\in E\cap K$ , using (5·8) and (5·7), we have

(5·9) \begin{multline} h\bigl(\,\cdot\,;\,E\cap\mathsf{B}(a,t_n),U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}\bigr)\le\exp\Bigl(-\sum_{j=0}^{m_\delta}\ell\log(R/r)\Bigr)\\=e^{-\ell(m_\delta+1)\log(R/r)} \le \biggl(\frac{1}{(R/r)^nr}\biggr)^{\ell}\cdot\delta^{\ell}\quad\text{on }\mathsf{B}(a,\delta)(\subset U_{\overrightarrow{\xi_{B(a,\,rt_{n+m_{\delta}})}a}}).\end{multline}

(v) Let us complete the proof of Theorem 4. By (5·9) and the former half in (5·4), for any $a\in E\cap K$ , $P_{\xi_0,E\cap\mathsf{B}(a,\;t_n)}$ is continuous at a, and then so is $P_{\xi_0,E}$ since $P_{\xi_0,E}\le P_{\xi_0,E\cap\mathsf{B}(a,\;t_n)}$ on $\mathsf{P}^1$ . Then recalling the definition of $\ell=\ell_E$ , by (5·6), the maximum principle for subharmonic functions (applied to $P_{\xi_0,E\cap\mathsf{B}(a,\;t_n)}^*$ , which is $\ge P_{\xi_0,E\cap\mathsf{B}(a,\;t_n)}(\ge P_{\xi_0,E})$ , on $U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}$ ), and the former half in (5·4) again, we have

\begin{multline*} \ell\cdot\bigl(p_{\xi_0,\nu_{\xi_0,E}}-\log \mathrm{Cap}_{\xi_0}(E)\bigr)\Bigl(\le\frac{P_{\xi_0,E}}{P_{\xi_0,E\cap\mathsf{B}(a,\;t_n)}^*(\xi_{B(a,\,Rt_n)})}\Bigr)\\[5pt] \le h\bigl(\,\cdot\,;\,E\cap\mathsf{B}(a,t_n),U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}\bigr)\quad\text{on }\mathsf{B}(a,\delta)(\subset U_{\overrightarrow{\xi_{B(a,\,Rt_n)}a}}).\end{multline*}

This with (5·9) completes the proof, recalling the latter half of the uniform comparison (2·7) between the kernel functions $|\xi-\xi^\prime|_\infty$ and $[\xi,\xi^\prime]_g$ on E again.