2.1 Preliminaries and basic notation

A product formula field is a field k together with a set $\mathcal {M}_k$ consisting of pairwise inequivalent non-trivial absolute values, and a unique positive integer $N_v$ associated to each element of $\mathcal {M}_k$ such that the following holds.

• • For each $\alpha \in k^*$ , we have $|\alpha |_v=1$ for all but finitely many places $v\in \mathcal {M}_k$ , and the product formula holds,

  (2.1) $$ \begin{align} \prod_{v \in \mathcal{M}_k} |\alpha|_v^{N_v} \ = \ 1. \end{align} $$

In what follows, we often refer to the elements of $\mathcal {M}_k$ as places of k. Important examples of product formula fields include number fields and function fields of smooth projective curves. Let $\overline {k}$ and $k^{\mathrm {sep}}$ be the algebraic and respectively separable closure of k. If the characteristic of k is zero, then $\overline {k}=k^{\mathrm { sep}}$ . For each $v \in \mathcal {M}_k$ , let $k_v$ be the completion of k with respect to $|\cdot |_v$ , $\overline {k}_v$ be an algebraic closure of $k_v$ , and ${\mathbb C}_v$ denote the completion of $\overline {k}_v$ . We also let ${\mathbb P}^{1, an}_{v}$ be the Berkovich projective line over ${\mathbb C}_v$ . This is a canonically defined path-connected compact Hausdorff space containing ${\mathbb P}^1({\mathbb C}_v)$ as a dense subspace. For each $v\in \mathcal {M}_k$ , we fix an embedding of $\overline {k}$ into ${\mathbb C}_v$ . We remark here that if v is archimedean, we have ${\mathbb C}_v \simeq {\mathbb C}$ and ${\mathbb P}^{1, an}_{v} \simeq {\mathbb P}^1({\mathbb C})$ .

For each $v \in \mathcal {M}_k$ , there is a distribution-valued Laplacian operator $\Delta $ on ${\mathbb P}^{1, an}_{v}$ . For its definition and some examples, we refer the reader to [Reference Baker and Rumely7, Ch. 5]. An important example is the Laplacian of $\log ^{+}|z|_v:= \max \{\log |z|_v, 0\}$ . Note that the function $\log ^+|z|_v$ , which is originally defined on ${\mathbb P}^1({\mathbb C}_v)\setminus \{\infty \}$ , extends naturally to a continuous real valued function defined on ${\mathbb P}^{1, an}_{v} \backslash \{ \infty \}$ . The Laplacian of its extension, also denoted by $\log ^{+}|z|_v$ and taking images in $\mathbb {R}\cup \{+\infty \}$ , is

(2.2) $$ \begin{align} \Delta \log^+|z|_v = \delta_{\infty} - \unicode{x3bb}_v, \end{align} $$

where $\unicode{x3bb} _v$ is the uniform probability measure on the complex unit circle $\{ |z|_v = 1 \}$ when v is archimedean and a point mass at the Gauss point of ${\mathbb P}^{1, an}_{v}$ when v is non-archimedean.

A probability measure $\mu _v$ on ${\mathbb P}^{1, an}_{v}$ is said to have continuous potentials if $\mu _v - \unicode{x3bb} _v = \Delta g$ for some continuous function $g : {\mathbb P}^{1, an}_{v} \to {\mathbb R}$ . We call the function g a potential of $\mu _v$ and note that any two potentials of $\mu _v$ differ by a constant.

If $\mu _v$ has continuous potentials, then there is a function $g^0_{\mu _v}: {\mathbb P}^{1, an}_{v} \to {\mathbb R}\cup \{+\infty \}$ such that $g: {\mathbb P}^{1, an}_{v} \to {\mathbb R}$ , defined by

$$ \begin{align*} g(z)=\log^+|z|_v-g^0_{\mu_v}(z), \end{align*} $$

is a potential for $\mu _v$ and is continuous. We call the function $G^{\mathrm {hom}}_{\mu _v}(x,y)$ uniquely determined by $g^0_{\mu _v}$ and defined as

(2.3) $$ \begin{align} G^{\mathrm{hom}}_{\mu_v}(x,y): =\begin{cases} g^0_{\mu_v}(x/y)+\log|y|_v &\text{ for } x,y\in {\mathbb C}_v \text{ and } y\neq 0,\\ \log|x|_v-g(\infty)&\text{ for } x\in {\mathbb C}_v, x\neq 0, \text{ and } y=0,\\ -\infty &\text{ for } x=y=0, \end{cases} \end{align} $$

where $g(\infty )=\lim _{z\to \infty }(\log ^+|z|_v-g^0_{\mu _v}(z))$ , a homogeneous potential of $\mu _v$ . If we further require that the set

$$ \begin{align*} \{(x,y)\in {\mathbb C}_{v}^2 : G^{\mathrm{hom}}_{\mu_v}(x,y)\leq 0\} \end{align*} $$

has homogeneous capacity equal to $1$ (see [Reference Baker and Rumely6, §3.3] and [Reference DeMarco14] for the definition of homogeneous capacity), then the homogeneous potential is uniquely determined. We denote it by $G_{\mu _v}$ and call it the normalized homogeneous potential of $\mu _v$ . Further, we write

(2.4) $$ \begin{align} M_{\mu_v}:=\{(x,y)\in {\mathbb C}_{v}^2 : G_{\mu_v}(x,y)\leq 0\}. \end{align} $$

Note that any homogeneous potential of $\mu _v$ differs from $G_{\mu _v}$ by a constant. An important property of a homogeneous potential $G^{\mathrm {hom}}_{\mu _v}$ is that it scales logarithmically,

(2.5) $$ \begin{align} G^{\mathrm{hom}}_{\mu_v}(\alpha x,\alpha y)=G^{\mathrm{hom}}_{\mu_v}(x, y)+\log|\alpha|_v. \end{align} $$

For example, from (2.2), we get that the normalized homogeneous potential of $\unicode{x3bb} _v$ is

$$ \begin{align*} G_{\unicode{x3bb}_v}(x,y)=\log \|(x,y)\|_v\text{ and } M_{\unicode{x3bb}_v}=\bar{D}^2(0,1)\subset {\mathbb C}_v^2, \end{align*} $$

where $\|\cdot \|_v$ is the maximum norm defined as $\|(x,y)\|_v:=\max \{|x|_v, |y|_v\}.$

Finally, we point out that for each probability measure $\mu _v$ on ${\mathbb P}^{1, an}_{v}$ with continuous potentials, we have a unique normalized Arakelov–Green function $g_{\mu _v} : {\mathbb P}^{1, an}_{v} \times {\mathbb P}^{1, an}_{v} \to {\mathbb R} \cup \{ +\infty \}$ . This is characterized by the differential equation $\Delta _x g_{\mu _v}(x,y) = \delta _y - \mu _v$ and the normalization

(2.6) $$ \begin{align} \iint g_{\mu_v}(x,y) \,d\mu_v(x) \,d\mu_v(y) = 0. \end{align} $$

Note that for points $(x,y)\in {\mathbb C}_v^2$ , the normalized Arakelov–Green function $g_{\mu _v}$ is given by

(2.7) $$ \begin{align} g_{\mu_v}(x,y) = -\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}), \end{align} $$

where $\tilde {x}=(x_1,x_2)\in {\mathbb C}_v^2$ and $\tilde {y}=(y_1,y_2)\in {\mathbb C}_v^2$ are lifts of x and y respectively and $|\tilde {x}\wedge \tilde {y}|_v:=|x_1y_2-y_1x_2|_v$ ; see [Reference Baker and Rumely6, Reference Baker and Rumely7] for more details.

2.2 Quasi-adelic measure and canonical height function

Let $\mu _v$ be a probability measure on ${\mathbb P}^{1, an}_{v}$ with continuous potentials. We define the outer radius and inner radius for $\mu _v$ as

$$ \begin{align*} r_{\mathrm{out}}(\mu_v)&:=\inf \{r>0 : M_{\mu_v} \subset \bar{D}(0, r)\times \bar{D}(0,r)\},\\ r_{\mathrm{in}}(\mu_v)&:=\sup \{r>0 : \bar{D}(0, r)\times \bar{D}(0,r) \subset M_{\mu_v}\}. \end{align*} $$

A quasi-adelic measure on ${\mathbb P}^1$ with respect to a product formula field k is a collection ${\mathbb \mu } = \{ \mu _v \}_{v \in \mathcal {M}_k}$ of probability measures on ${\mathbb P}^{1, an}_{v}$ , one for each $v \in \mathcal {M}_k$ , such that:

• • $\mu _v$ has continuous potentials for each $v \in \mathcal {M}_k$ , and

• • $\prod _{v\in \mathcal {M}_k} r_{\mathrm {in}}(\mu _v)^{N_v}>0$ and $\prod _{v\in \mathcal {M}_k} r_{\mathrm {out}}(\mu _v)^{N_v} <\infty $ .

If $\rho ,\rho '$ are probability measures on ${\mathbb P}^{1, an}_{v}$ , we define the $\mu _v$ -energy of $\rho $ and $\rho '$ as

$$ \begin{align*} ( \rho, \rho' )_{\mu_v} := \frac{1}{2} \iint_{{\mathbb P}^{1, an}_{v} \times {\mathbb P}^{1, an}_{v} \backslash \mathrm{Diag}} g_{\mu_v}(x,y)\, d\rho(x)\, d\rho'(y). \end{align*} $$

The $\mu _v$ -energy of $\rho $ is defined as $I_{\mu _v}(\rho ) := (\rho ,\rho )_{\mu _v}$ .

We can now define the height associated with a quasi-adelic measure. Let $S\subset {\mathbb P}^1(k^{\mathrm {sep}})$ be a finite $\operatorname {Gal}(k^{\mathrm {sep}}/k)$ -invariant set with cardinality $|S|>1$ . Let $\tilde {S}\subset {\mathbb C}_v^2$ be a lift of S, that is, a $\operatorname {Gal}(k^{\mathrm {sep}}/k)$ -invariant set consisting of lifts $\tilde {x}\in k^{\mathrm {sep}}\times k^{\mathrm {sep}} \setminus \{(0,0)\}$ of elements $x\in S$ ; in particular, $|\tilde {S}|=|S|$ . We denote by $[S]_v$ the discrete probability measure on ${\mathbb P}^{1, an}_{v}$ supported equally on all elements of S. The canonical height of S associated to a quasi-adelic measure ${\mathbb \mu } = \{ \mu _v \}_{v \in M_k}$ is a number given by

(2.8) $$ \begin{align} \hat{h}_{\mu}(S) :&= \frac{|S|}{|S|-1}\sum_{v\in \mathcal{M}_k} N_v \cdot ([S]_v, [S]_v)_{\mu_v} \nonumber\\ &= \frac{|S|}{|S|-1} \sum_{v\in \mathcal{M}_k} \frac{N_v }{2|S|^2}\sum_{x,y\in S, x\not=y} g_{\mu_v}(x,y) \nonumber\\ &= \sum_{ x,y\in S, x\not=y } \sum_{v\in \mathcal{M}_k} \frac{N_v \cdot ( -\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}) )}{2|S|(|S|-1)}\nonumber\\ &= \frac{1}{2|S|(|S|-1)} \sum_{\tilde{x},\tilde{y}\in \tilde{S}, x\not=y} \; \sum_{v\in \mathcal{M}_k} N_v \cdot (G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}) ),\quad \text{by (2.1)}\nonumber\\ &= \frac{1}{|S|} \cdot \sum_{\tilde{x}\in \tilde{S}}\sum_{v\in \mathcal{M}_k} N_v\cdot G_{\mu_v}(\tilde{x}). \end{align} $$

Here, the constants $N_v$ are the same as those appearing in the product formula. Therefore, we have

(2.9) $$ \begin{align} \hat{h}_{\mu}(S)= \frac{1}{|S|} \cdot \sum_{\tilde{x}\in \tilde{S}}\sum_{v\in \mathcal{M}_k} N_v \cdot G_{\mu_v}(\tilde{x}). \end{align} $$

Equation (2.9) allows us to extend the definition of our height to the case $|S|=1$ . If $x\in k^{\mathrm {sep}}$ , we may take S to be equal to $\operatorname {Gal}(k^{\mathrm {sep}}/ k)\cdot x$ and use (2.9) to define the canonical height of x as

$$ \begin{align*} \hat{h}_{\mu}(x) :=\hat{h}_{\mu}(\operatorname{Gal}(k^{\mathrm{sep}}/ k)\cdot x). \end{align*} $$

In the next proposition, we show that our height is finite and well defined.

Proof. That $\hat {h}_{\mu }(S)$ is independent of the choice of lift $\tilde {S}$ of S follows from the product formula of k once we recall that the homogeneous potential $G_{\mu _v}$ scales logarithmically; see (2.5). To see that $\hat {h}_{\mu }(S)$ is a finite number, notice that from the definition of inner and outer radii, we have

$$ \begin{align*}\log r_{\mathrm{in}}(\mu_v) \leq\frac{1}{|S|}\sum_{\tilde{x}\in \tilde{S}} (\log\|\tilde{x}\|_v-G_{\mu_v}(\tilde{x}))\leq \log r_{\mathrm{out}}(\mu_v). \end{align*} $$

Summing this inequality over all places $v\in \mathcal {M}_k$ , we get

$$ \begin{align*} \sum_{v\in \mathcal{M}_k} N_v\log r_{\mathrm{in}}(\mu_v) \leq\sum_{v\in \mathcal{M}_k} \frac{N_v}{|S|}\sum_{\tilde{x}\in \tilde{S}} (\log\|\tilde{x}\|_v-G_{\mu_v}(\tilde{x}))\leq \sum_{v\in \mathcal{M}_k} N_v\log r_{\mathrm{out}}(\mu_v), \end{align*} $$

which, using the definition of our height, yields

(2.10) $$ \begin{align} \log \prod_{v\in \mathcal{M}_k} r_{\mathrm{in}}(\mu_v)^{N_v}\leq \frac{1}{|S|}\sum_{v\in \mathcal{M}_k}\sum_{\tilde{x}\in \tilde{S}} N_v\log\|\tilde{x}\|_v -\hat{h}_{\mathbb{\mu}}(S)\leq \log \prod_{v\in \mathcal{M}_k} r_{\mathrm{out}}(\mu_v)^{N_v}. \end{align} $$

Recall that $\mu $ is a quasi-adelic measure. Thus, $\log \prod _{v\in \mathcal {M}_k} r_{\mathrm {in}}(\mu _v)^{N_v}$ and $ \log \prod _{v\in \mathcal {M}_k} r_{\mathrm {out}}(\mu _v)^{N_v}$ are real numbers. The proposition follows.

2.3 Quasi-adelic set

In this section, we introduce the notion of a quasi-adelic set. This has a geometric interpretation; hence in many applications, it is easier to manipulate than a quasi-adelic measure. Analogous to the notion of the homogeneous filled Julia set in [Reference Baker and Rumely6, §3.2], we define a homogeneous set with continuous potential as

(2.11) $$ \begin{align} M_v:=\{(x,y)\in {\mathbb C}_v^2 : G_{v}(x,y)\leq 0\}, \end{align} $$

where $G_{v}$ is a homogeneous potential for a probability measure on ${\mathbb P}^{1, an}_{v}$ having continuous potentials. In what follows, we write $G_{M_v}$ for $G_v$ as in (2.11), thinking of $G_{M_v}$ as the potential associated to the homogeneous set $M_v$ with continuous potential. We also denote the measure corresponding to $G_{M_v}$ , by $\mu _{M_v}$ . When v is an archimedean place, $M_v$ having continuous potential is equivalent to saying that $M_v\subset {\mathbb C}_v^2\simeq {\mathbb C}^2$ is a compact, circled, and pseudoconvex set, or that $G_{M_v}$ is a continuous and plurisubharmonic function satisfying:

1. (1) $G_{M_v}(\alpha z) =G_{M_v}(z) + \log |\alpha |_v$ for all $\alpha \in {\mathbb C}_v$ ; and

2. (2) $G_{M_v}(z) = \log \|z\|_v + O(1)$ .

See [Reference DeMarco14]. We point out here that there are many homogeneous sets with continuous potential. If $F_n: {\mathbb C}_v^2\to {\mathbb C}_v^2$ is a sequence of homogeneous polynomials with $\deg (F_n)\geq 1$ such that the sequence of functions $\{{\log \| F_n\|_v}/{\deg (F_n)}\}_{n\geq 1}$ converges uniformly to $G_v$ on ${\mathbb C}_v^2\backslash \{(0, 0)\}$ , then $G_v$ is a homogeneous potential for some probability measure with continuous potentials on ${\mathbb P}^{1, an}_{v}$ ; see [Reference Baker and Rumely6, §3]. Hence, $M_v=\{(x,y) \in {\mathbb C}_v^2: G_v(x,y)\leq 0\}$ is a homogeneous set with continuous potential. As seen in [Reference DeMarco, Wang and Ye23, $\S$ 2], its capacity can be computed by the following limit:

$$ \begin{align*} \operatorname{Cap}(M_v)=\lim_{n\to \infty} |{\operatorname{Res}}(F_n)|_v^{-{1}/{\deg(F_n)^2}}. \end{align*} $$

Analogous to the definition of the radii of $\mu _v$ , we define the outer and inner radii of $M_v$ as

$$ \begin{align*} r_{\mathrm{out}}(M_v):=\inf \{r>0 : M_{v} \subset \bar{D}(0, r)\times \bar{D}(0,r)\}, \end{align*} $$

$$ \begin{align*} r_{\mathrm{in}}(M_v):=\sup \{r>0 : \bar{D}(0, r)\times \bar{D}(0,r) \subset M_{v} \}. \end{align*} $$

A product $\prod _{v\in \mathcal {M}_k}r_v^{N_v}$ , with $r_v> 0$ for each $v\in \mathcal {M}_k$ , converges strongly if

$$ \begin{align*} \sum_{v\in \mathcal{M}_k}N_v\cdot |{\log}\ r_v|<\infty. \end{align*} $$

We define a quasi-adelic set (with respect to a product formula field k) to be a collection ${\mathbb M} = \{ M_v \}_{v \in \mathcal {M}_k}$ of sets such that the following hold.

• • For each $v\in \mathcal {M}_k$ , the set $M_v$ is a homogeneous set with continuous potential.

• • The products $\prod _{v\in \mathcal {M}_k} r_{\mathrm {out}}(M_v)^{N_v}$ and $\prod _{v\in \mathcal {M}_k} r_{\mathrm {in}}(M_v)^{N_v}$ converge strongly.

Note that there is a unique probability measure $\mu _{M_v}$ with continuous potential associated to a homogeneous set $M_v$ with continuous potential. Hence, a quasi-adelic set ${\mathbb M} = \{ M_v \}_{v \in M_k}$ gives a measure ${\mathbb \mu _{\mathbb M}} = \{ \mu _{M_v} \}_{v \in M_k}$ on ${\mathbb P}^1$ . In the next theorem, proved in §3, we will see that this measure is also quasi-adelic.

2.4 Some properties

A number field k is naturally equipped with a set of inequivalent absolute values $\mathcal {M}_k$ and positive integers $\{N_v\}_{v\in \mathcal {M}_k}$ making it a product formula field. For any $x\in k$ , the logarithmic Weil height of x is defined as follows:

(2.13) $$ \begin{align} h(x):=\frac{1}{[k:{\mathbb Q}]}\sum_{v\in \mathcal{M}_k} N_v\cdot \log^+|x|_v. \end{align} $$

It is well defined and does not depend on the embedding of a number field $k\hookrightarrow \overline {{\mathbb Q}}$ . We show that the canonical height associated to a quasi-adelic measure differs from a multiple of the Weil height by a bounded amount.

Proposition 2.3. Let k be a number field. Suppose $\mu =\{\mu _{v}\}_{v\in \mathcal {M}_k}$ is a quasi-adelic measure. Then the canonical height $\hat {h}_{\mu }$ is bounded by the logarithmic Weil height h on ${\mathbb P}^1(\overline {k})$ as

$$ \begin{align*} \log \prod_{v\in \mathcal{M}_k} r_{\mathrm{in}}(\mu_v)^{N_v}\leq [k:{\mathbb Q}]h(x)-\hat{h}_{\mathbb{\mu}}(x)\leq \log \prod_{v\in \mathcal{M}_k} r_{\mathrm{out}}(\mu_v)^{N_v}, \end{align*} $$

for all $x\in \overline {k}$ .

Proof. Recall that $\hat {h}_{\mathbb {\mu }}(x)=\hat {h}_{\mathbb {\mu }}(\operatorname {Gal}(\overline {k}/ k)\cdot x)$ . The proposition follows by applying (2.10) for $S=\operatorname {Gal}(\overline {k}/ k)\cdot x$ , once one notices that then

$$ \begin{align*} \frac{1}{|S|}\sum_{v\in \mathcal{M}_k}\sum_{\tilde{x}\in \tilde{S}} N_v\log\|\tilde{x}\|_v = [k: {\mathbb Q}]h(x). \\[-48pt] \end{align*} $$

Finally, we note that similar to the set of adelic metrized line bundles which is closed under taking tensor products, we have that the set of quasi-adelic metrics is closed under taking certain linear combinations. For example, the average of two quasi-adelic measures is a quasi-adelic measure.

3.1 Proof of quasi-adelic arithmetic equidistribution Theorem 1.1

By assumption, $\{S_n\}_{n\geq 1}$ is a sequence of subsets of ${\mathbb P}^1(k^{\mathrm {sep}})$ which are $\operatorname {Gal}(k^{\mathrm {sep}}/k)$ -invariant and the cardinality $|S_n|$ tends to infinity. For each $v\in \mathcal {M}_k$ , the $\mu _v$ -energy of the probability measure $[S_n]_v$ on ${\mathbb P}^{1, an}_{v}$ is given by

$$ \begin{align*} ([S_n]_v, [S_n]_v)_{\mu_v}&=\frac{1}{2|S_n| ^2}\sum_{x\neq y\in S_n} g_{\mu_v}(x,y)\\ &=\frac{1}{2|S_n| ^2}\sum_{x\neq y\in S_n} (-\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y})), \end{align*} $$

where $\tilde {x}\in {\mathbb C}_v^2$ and $\tilde {y}\in {\mathbb C}_v^2$ are lifts of x and y respectively. We begin with a lemma, which follows the idea from [Reference Baker and Rumely6, Reference Favre and Rivera-Letelier29]. It asserts that the sequence $\{S_n\}_{n\ge 1}$ is pseudo-equidistributed with respect to $g_{\mu _v}$ ; compare with [Reference Baker and Rumely6, Definition 4.4], [Reference Baker and Rumely6, Theorem 4.6], and [Reference Favre and Rivera-Letelier29, p. 349].

Lemma 3.1. Let k be a product formula field and let ${\mathbb \mu } = \{ \mu _v \}_{v \in \mathcal {M}_k}$ be a quasi-adelic measure. Suppose that $S_n$ is a sequence of $\operatorname {Gal}(k^{\mathrm {sep}}/k)$ -invariant subsets of ${\mathbb P}^1(k^{\mathrm {sep}})$ such that $|S_n|\to \infty $ and $\lim \sup _{n\to \infty }\hat {h}_{{\mathbb \mu }}(S_n) \le 0$ . Then for each $v\in \mathcal {M}_k$ , we have

$$ \begin{align*} \lim_{n\to \infty} ([S_n]_v, [S_n]_v)_{\mu_v}=0. \end{align*} $$

Proof. First we will show that for each $v\in \mathcal {M}_k$ , we have

(3.1) $$ \begin{align} \liminf_{n\to \infty} ([S_n]_v, [S_n]_v)_{\mu_v}\geq 0. \end{align} $$

For this, we follow the proof of [Reference Baker and Rumely6, Lemma 3.17]. From the definition of the homogeneous capacity, we have

(3.2) $$ \begin{align} \liminf_{n\to\infty} \inf_{\tilde x_1, \tilde x_2, \ldots, \tilde x_n \in M_{\mu_v}}\frac{1}{n(n-1)}\sum_{i\neq j}-\log|\tilde{x_i}\wedge \tilde{x}_j|_v\geq -\log \operatorname{Cap}(M_{\mu_v})=0. \end{align} $$

Let $\epsilon>0$ be an arbitrary small number. Since $\{|\alpha |_v: \alpha \in {\mathbb C}_v\}$ is dense in ${\mathbb R}_{\ge 0}$ , we can choose lifts of $x, y\in S_n$ , denoted by $\tilde {x}, \tilde {y} \in M_{\mu _v}$ , such that

$$ \begin{align*} -\epsilon<G_{\mu_v}(\tilde{x})\leq 0\quad\text{and}\quad-\epsilon<G_{\mu_v}(\tilde{y})\leq 0. \end{align*} $$

Now, the definition of the energy function and (3.2) yield

$$ \begin{align*} \liminf_{n\to\infty} ([S_n]_v, [S_n]_v)_{\mu_v} &=\liminf_{n\to\infty} \frac{1}{2|S_n| ^2}\sum_{x\neq y\in S_n} (-\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}))\\ &\geq -\epsilon+\liminf_{n\to\infty} \bigg(-\frac{1}{2|S_n| ^2}\sum_{x\neq y\in S_n} \log|\tilde{x} \wedge \tilde{y}|_v\bigg) \geq -\epsilon. \end{align*} $$

Shrinking $\epsilon $ to zero, (3.1) follows. It remains to prove that for each $v\in \mathcal {M}_k$ , we have

$$ \begin{align*} \limsup_{n\to \infty}([S_n]_v, [S_n]_v)_{\mu_v}\leq 0. \end{align*} $$

Let $v_0\in \mathcal {M}_k$ be a fixed place. Let $\epsilon>0$ . We will show that

(3.3) $$ \begin{align} \limsup_{n\to \infty}([S_{n}]_{v_0}, [S_{n}]_{v_0})_{\mu_{v_0}}\le \epsilon. \end{align} $$

The conclusion then follows letting $\epsilon \to 0$ . For each $v\in \mathcal {M}_k$ and any $\delta>0$ , we lift $x,y\in S_n$ to $\tilde {x}, \tilde {y}\in M_{\mu _v}$ such that

$$ \begin{align*} G_{\mu_v}(\tilde{x})\geq -\delta, G_{\mu_v}(\tilde{y})\geq -\delta. \end{align*} $$

Since we know that $M_{\mu _v}$ is bounded by the polydisc with outer radius $r_{\mathrm {out}}(\mu _v)$ , we have $\|\tilde {x}\|_v, \|\tilde {y}\|_v\leq r_{\mathrm {out}}(\mu _v)$ . Hence, for a non-archimedean place $v\in \mathcal {M}_k$ , we have

$$ \begin{align*} \log|\tilde{x} \wedge \tilde{y}|_v \leq \log r_{\mathrm{out}}(\mu_v)^2, \end{align*} $$

and

(3.4) $$ \begin{align} ([S_n]_v, [S_n]_v)_{\mu_v}&=\frac{1}{2|S_n| ^2}\sum_{x\neq y\in S_n} (-\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}))\nonumber\\ &\geq \bigg(\frac{1-|S_n|}{|S_n|}\bigg)\delta- \sum_{x\neq y\in S_n}\frac{ \log r_{\mathrm{out}}(\mu_v)}{| S_n|^2}\nonumber\\ &=\bigg(\frac{1-|S_n|}{|S_n|}\bigg)(\delta+\log r_{\mathrm{out}}(\mu_v)). \end{align} $$

Shrinking $\delta $ to zero, we get that for all non-archimedean places $v\in \mathcal {M}_k$ :

(3.5) $$ \begin{align} ([S_n]_v, [S_n]_v)_{\mu_v}\geq \bigg(\frac{1-|S_n|}{|S_n|}\bigg)\log r_{\mathrm{out}}(\mu_v). \end{align} $$

It is well known that for a product formula field k, there are only finitely many archimedean places; see [Reference Artin1, Ch. 12, Theorem 3]. Since $\mu =\{\mu _v\}_{v\in \mathcal {M}_k}$ is quasi-adelic, the product $\prod _{v\in \mathcal {M}_k} r_{\mathrm {out}}(\mu _v)^{N_v}$ converges. Hence, we can choose a set $\mathcal {M}_k'\subset \mathcal {M}_k$ such that the following hold.

• • $\mathcal {M}_k":=\mathcal {M}_k\backslash ( \mathcal {M}_k'\cup \{v_0\})$ has only finitely many places.

• • All places in $\mathcal {M}_k'$ are non-archimedean and $v_0\not \in \mathcal {M}_k'$ .

• • $\sum _{v\in \mathcal {M}_k'}N_v\cdot \log r_{\mathrm {out}}(\mu _v)\leq \epsilon $ .

Now the definition of the canonical height $\hat h_{\mu }$ gives

$$ \begin{align*} N_{v_0}\cdot ([S_{n}]_{v_0}, [S_{n}]_{v_0})_{\mu_{v_0}}&=\bigg(\frac{|S_{n}|-1}{|S_{n}|}\bigg)\hat h_{\mu}(S_{n})-\sum_{v\in \mathcal{M}_k\backslash \{v_0\}} N_v \cdot([S_{n}]_v, [S_{n}]_v)_{\mu_v}\\ &=\bigg(\frac{|S_{n}|-1}{|S_{n}|}\bigg)\hat h_{\mu}(S_{n})-\sum_{v\in \mathcal{M}_k"} N_v\cdot ([S_{n}]_v, [S_{n}]_v)_{\mu_v}\\&\quad-\sum_{v\in \mathcal{M}_k'} N_v \cdot ([S_{n}]_v, [S_{n}]_v)_{\mu_v}. \end{align*} $$

This in turn, upon using (3.5), implies

$$ \begin{align*} N_{v_0}\cdot ([S_{n}]_{v_0}, [S_{n}]_{v_0})_{\mu_{v_0}}&\leq\bigg(\frac{|S_{n}|-1}{|S_{n}|}\bigg) \hat h_{\mu}(S_{n})\\&\quad-\sum_{v\in \mathcal{M}_k"} N_v \cdot ([S_{n}]_v, [S_{n}]_v)_{\mu_v}+\bigg(\frac{| S_{n}|-1}{| S_{n}|}\bigg)\cdot\epsilon. \end{align*} $$

Since the set $\mathcal {M}_k"$ contains only finitely many places and $\lim \sup _{n\to \infty }\hat {h}_{\mu }(S_n)\le 0$ , by the lower bound in (3.1), taking the superior limit in the above inequality yields

(3.6) $$ \begin{align} \limsup_{n\to\infty} N_{v_0}\cdot ([S_{n}]_{v_0}, [S_{n}]_{v_0})_{\mu_{v_0}}&\leq \limsup_{n\to\infty} \bigg(\bigg(\frac{|S_{n}|-1}{|S_{n}|}\bigg)\hat h_{\mu}(S_{n})\nonumber\\&\quad-\sum_{v\in \mathcal{M}_k"} N_v\cdot ([S_{n}]_v, [S_{n}]_v)_{\mu_v}+ \epsilon\bigg)\nonumber\\ &\leq \limsup_{n\to\infty} \hat h_{\mu}(S_{n})+\epsilon\leq \epsilon, \end{align} $$

as claimed. This finishes the proof of the lemma.

The proof of Theorem 1.1 now follows as in [Reference Baker and Rumely6, Theorem 4.9]; see also [Reference Favre and Rivera-Letelier29, Propositions 2.11 and 4.12]. The set of probability measures on ${\mathbb P}^{1, an}_{v}$ is compact in the weak topology. Hence, to show that $[S_n]_v$ converges weakly to $\mu _v$ as $n\to \infty $ , it suffices to show that any convergent subsequence of $[S_n]_v$ converges to $\mu _v$ . Passing to a subsequence if necessary, we may assume that $[S_n]_v$ converges to some $\nu _v$ ,

$$ \begin{align*} \lim_{n\to \infty}[S_n]_v=\nu_v. \end{align*} $$

We have to show that $\nu _v=\mu _v$ . By Lemma 3.1, the $\mu _v$ -energy $I_{\mu _v}(\nu _v)$ of $\nu _v$ satisfies

$$ \begin{align*} 0&=\lim_{n\to \infty}([S_{n}]_v, [S_{n}]_v)_{\mu_v}=\lim_{n\to\infty} \frac{1}{2} \iint_{{\mathbb P}^{1, an}_{v} \times {\mathbb P}^{1, an}_{v} \backslash \mathrm{Diag}} g_{\mu_v}(x,y) \,d[S_n]_v(x)\,d[S_n]_v(y)\\ &\geq \frac{1}{2} \iint_{{\mathbb P}^{1, an}_{v} \times {\mathbb P}^{1, an}_{v}} g_{\mu_v}(x,y) \,d\nu_v(x)\,d\nu_v(y) \quad \text{by [6, Lemma 3.26]}\\ &=I_{\mu_v}(\nu_v). \end{align*} $$

Note, our assumption that the measures $\mu _v$ have continuous potentials implies that they are $\log $ -continuous in the sense of [Reference Baker and Rumely6, Definition 3.20]. By the energy minimization principle [Reference Baker and Rumely6, Theorem 3.25], where $\mu _v$ is the unique probability measure on ${\mathbb P}^{1, an}_{v}$ minimizing the $\mu _v$ -energy function $I_{\mu _v}(\cdot )$ and $I_{\mu _v}(\mu _v)=0\geq I_{\mu _v}(\nu _v)$ , we get that $I_{\mu _v}(\mu _v)=I_{\mu _v}(\nu _v)$ and $\nu _v=\mu _v$ . This finishes the proof of Theorem 1.1.

3.2 Proof of Theorem 2.2

First, we show that $\mathbb M=\{M_v\}_{v\in \mathcal {M}_k}$ is quasi-adelic implies that ${\mathbb \mu } = \{ \mu _{M_v} \}_{v \in M_k}$ is quasi-adelic. Assume that $\mathbb M=\{M_v\}_{v\in \mathcal {M}_k}$ is a quasi-adelic set. For any $r>0$ , let

$$ \begin{align*} rM_v:=\{(\alpha x, \alpha y) : (x,y)\in M_v, \alpha \in {\mathbb C}_v \text{ with } |\alpha|_v\leq r\}. \end{align*} $$

From the definition of the capacity, we have $\operatorname {Cap}(rM_v)=r^2\operatorname {Cap}(M_v)$ . Since $G_{M_v}$ is a homogeneous potential for $\mu _{M_v}$ , the normalized homogeneous potential $G_{\mu _{M_v}}$ is given by

(3.7) $$ \begin{align} G_{\mu_{M_v}}(x,y)=G_{M_v}(x,y)+\tfrac{1}{2}\log \operatorname{Cap}(M_v), \end{align} $$

and $M_{\mu _{M_v}}\!=({1}/{\sqrt {\operatorname {Cap}(M_v)}})M_v$ . As a consequence,

(3.8) $$ \begin{align} r_{\mathrm{in}}(\mu_{M_v})=\frac{r_{\mathrm{in}}(M_v)}{\sqrt{\operatorname{Cap}(M_v)}},\quad r_{\mathrm{out}}(\mu_{M_v})=\frac{r_{\mathrm{out}}(M_v)}{\sqrt{\operatorname{Cap}(M_v)}}. \end{align} $$

Moreover, as $\operatorname {Cap}(\bar {D}^2(0, r))=r^2$ ,

(3.9) $$ \begin{align} r_{\mathrm{in}}(M_v)\leq \sqrt{\operatorname{Cap}(M_v)}\leq r_{\mathrm{out}}(M_v). \end{align} $$

Then by (3.8),

$$ \begin{align*} \frac{r_{\mathrm{in}}(M_v)}{r_{\mathrm{out}}(M_v)}\leq r_{\mathrm{in}}(\mu_{M_v})\leq 1\leq r_{\mathrm{out}}(\mu_{M_v})\leq \frac{r_{\mathrm{out}}(M_v)}{r_{\mathrm{in}}(M_v)}. \end{align*} $$

As $\mathbb M=\{M_v\}_{v\in \mathcal {M}_k}$ is quasi-adelic, the products of inner and outer radii converge strongly. Then the above inequalities imply that the products of the inner and outer radii of $\mu =\{\mu _{M_v}\}_{v\in \mathcal {M}_k}$ converge, that is, $\mu =\{\mu _{M_v}\}_{v\in \mathcal {M}_k}$ is quasi-adelic.

Now assume that $\bar {D}^2(0,r^{\prime }_v)\subset M_v\subset \bar {D}^2(0,r_v)$ and $\prod _{v\in \mathcal {M}_k}{r^{\prime }_v}^{N_v}$ , $\prod _{v\in \mathcal {M}_k}{r_v}^{N_v}$ converge strongly. Then the products $\prod _{v\in \mathcal {M}_k} r_{\mathrm {out}}(M_v)^{N_v}$ and $\prod _{v\in \mathcal {M}_k} r_{\mathrm {in}}(M_v)^{N_v}$ converge strongly, since

$$ \begin{align*} r^{\prime}_v\leq r_{\mathrm{in}}(M_v)\leq r_{\mathrm{out}}(M_v)\leq r_v. \end{align*} $$

Hence, $\mathbb M=\{M_v\}_{v\in \mathcal {M}_k}$ is quasi-adelic. Moreover, by (3.9), the product of the capacities converges strongly. Then the last formula for the canonical height is clear from (2.8) and (3.7). This finishes the proof of Theorem 2.2.

3.3 A finiteness property

The following proposition is the analog of the last assertion of [Reference Favre and Rivera-Letelier29, Theorem 1] for quasi-adelic measures. It will be useful in the last section in proving the equidistribution of parameters t with small height with respect to $\hat {h}_{f_{c(t)}}(c(t))$ .

Proposition 3.2. Let k be a product formula field. Suppose $\mathbb {\mu }$ is a quasi-adelic measure. Then for any $\delta>0$ , there are at most finitely many $x\in k^{\mathrm {sep}}$ with

$$ \begin{align*} \hat{h}_{\mu}(x)<-\delta. \end{align*} $$

Proof. Assume to the contrary that there are infinitely many $x_i\in k^{\mathrm {sep}}$ with $\hat {h}_{\mu }(x_i)<-\delta $ . Let $S_n=\bigcup _{i=1}^n \operatorname {Gal}(k^{\mathrm {sep}}/k)\cdot x_i$ . Then $S_n$ is $\operatorname {Gal}(k^{\mathrm {sep}}/k)$ -invariant and $|S_n|\to \infty $ . By (2.8), we get $\hat {h}_{\mu }(S_n)\leq -\delta $ . Moreover, from (3.1), we see that for each $v\in \mathcal {M}_k$ ,

$$ \begin{align*} \liminf_{n\to\infty} ([S_n]_v, [S_n]_v)_{\mu_v}\geq 0. \end{align*} $$

Replacing $\hat {h}_{\mu } (S_{n_j})$ in (3.6) by $\hat {h}_{\mu }(S_n)$ and letting $\epsilon $ tend to zero, we get

$$ \begin{align*} \limsup_{n\to \infty}([S_n]_v, [S_n]_v)_{\mu_v}\leq -\delta, \end{align*} $$

which is a contradiction.

4.1 A dynamical pair on ${\mathbb P}^1$

Let k be a product formula field and let $K=k(t)$ . Recall that a dynamical pair on ${\mathbb P}^1$ is a pair $(f, c)\in K(z)\times K$ with $d:=\deg _z f\geq 2$ . We say that the pair $(f, c)$ is isotrivial if there is a family of Möbius transformations $M_t(z)\in \overline K(z)$ such that both $M_t\circ f_t\circ M_t^{-1}(z)$ and $M_t(c(t))$ are independent of t. Moreover, we say that $(f, c)$ is preperiodic if the starting point $c\in K$ is preperiodic under $f\in K(z)$ , that is, if there are integers $m>n\geq 0$ with $f^n(c)=f^m(c)\in K$ .

Recall further that we have the Call–Silverman canonical height function associated to $f\in K(z)$ , denoted by $\hat {h}_f:{\mathbb P}^1(\overline {K})\to {\mathbb P}^1(\overline {K})$ , determined uniquely by the properties $\hat {h}_f(f(c))=d\cdot \hat {h}_f(c)$ and $\hat {h}_f(c)=h(c)+O(1)$ . Alternatively, for $c\in K$ , we can compute the canonical height as

$$ \begin{align*} \hat{h}_f(c):=\lim_{n\to \infty}\frac{\deg_t f^n(c)}{d^n}. \end{align*} $$

We note that $\hat {h}_f(c)\geq 0$ and equality holds if and only if $(f, c)$ is either isotrivial or preperiodic; see [Reference Baker2, Reference DeMarco17].

4.2 Homogenization

Let $(f, c)\in K(z)\times K$ be a dynamical pair with degree $d:=\deg _z f\geq 2$ . In what follows, we choose lifts of $f_t(z)={P_t(z)}/{Q_t(z)}$ and $c(t)={A(t)}/{B(t)}$ . For the lift of f, we write

$$ \begin{align*} F_{t_1,t_2}(z,w)=(P_{t_1,t_2}(z,w),Q_{t_1,t_2}(z,w)), \end{align*} $$

where $P_{t_1,t_2}, Q_{t_1,t_2}$ are homogeneous polynomials in $(z,w)$ of degree d, with the coefficients of homogeneous polynomials in $(t_1,t_2)$ of the same degree that are relatively prime. For a lift of c, we write

$$ \begin{align*}C(t_1,t_2)=(A(t_1,t_2),B(t_1,t_2)),\end{align*} $$

where A and B are homogeneous polynomials in $k[t_1,t_2]$ of the same degree and have no common linear factor in $\overline {k}[t_1,t_2]$ . Moreover, we define

$$ \begin{align*} (A_{C,n}(t_1,t_2),B_{C,n}(t_1,t_2)):=F^n_{t_1,t_2}(C(t_1,t_2)). \end{align*} $$

That is,

$$ \begin{align*} A_{C,0}(t_1,t_2)=A(t_1,t_2)\text{ and }B_{C,0}(t_1,t_2)=B(t_1,t_2), \end{align*} $$

while for all $n\ge 0$ ,

(4.1) $$ \begin{align} A_{C,n+1}(t_1,t_2)&=P_{t_1,t_2}(A_{C,n}(t_1,t_2),B_{C,n}(t_1,t_2)),\nonumber\\ B_{C,n+1}(t_1,t_2)&=Q_{t_1,t_2}(A_{C,n}(t_1,t_2),B_{C,n}(t_1,t_2)). \end{align} $$

Note that $f_t^n(c(t))={A_{C,n}(t,1)}/{B_{C,n}(t,1)}$ . We often work with dehomogenized coordinates and when doing so, we identify ${\mathbb P}^1(\overline {k})$ with ${\mathbb A}^1(\overline {k})\cup \{\infty \}$ as follows. A point $[t_1:t_2]\in {\mathbb P}^1(\overline {k})$ is seen as $t_1/t_2\in {\mathbb A}^1(\overline {k})$ when $t_2\neq 0$ and as $\infty $ when $t_2=0$ .

We say that $f\in K(z)$ degenerates at $t\in {\mathbb A}^1(\overline {k})\cup \{\infty \}$ if $\deg _z(f_t)<d$ and write $\mathrm {Sing}(f)$ for the set of degenerating parameters. Notice that the set $\mathrm {Sing}(f)$ is precisely the zero locus of the resultant of the homogeneous polynomials $P_{t_1, t_2}, Q_{t_1, t_2}$ in $(z,w)$ , which we denote by

$$ \begin{align*} \operatorname{Res}(F_{t_1,t_2}):=\operatorname{Res}_{(z,w)}(P_{t_1,t_2},Q_{t_1,t_2})\in k[t_1,t_2]\setminus\{0\}. \end{align*} $$

We remark here that $ \operatorname {Res}(F_{t_1,t_2})$ is a homogeneous polynomial in $t_1,t_2$ . Thus,

$$ \begin{align*} \mathrm{Sing}(f)=\{[\alpha_1:\alpha_2]\in {\mathbb P}^1(\overline{k}):\operatorname{Res}(F_{\alpha_1, \alpha_2})=0\}. \end{align*} $$

We also work with a lift of $f^n(c)$ defined by coprime homogeneous polynomials. To write the greatest common divisor of $F_{t_1,t_2}^n(C(t_1,t_2))$ , for each $\alpha \in {\mathbb P}^1(\overline {k})$ , we let

$$ \begin{align*} u_{\alpha}(t_1,t_2) = \begin{cases} t_1-\alpha t_2& \mbox{ if}\ \alpha\in{\mathbb A}^1(\overline{k}),\\ t_2 & \mbox{ if}\ \alpha=\infty. \\ \end{cases} \end{align*} $$

Moreover, we let $m_{C,n}(\alpha )$ be the maximal integer $m\in {\mathbb N}$ with $u_{\alpha }^m|A_{C,n}$ and $u_{\alpha }^m|B_{C,n}$ . Then,

$$ \begin{align*} \mathrm{gcd} (F^n_{t_1, t_2}(C(t_1,t_2)))=\prod_{\substack{\alpha\in\mathrm{Sing}(f)}}u_{\alpha}(t_1,t_2)^{m_{C,n}(\alpha)}. \end{align*} $$

We point out here that for each $\alpha \in \mathrm {Sing}(f)$ , the sequence $\{m_{C,n}(\alpha )/d^n\}_{n\in {\mathbb N}}$ converges as $n\to \infty $ since $\{m_{C,n+1}(\alpha )-d\cdot m_{C,n}(\alpha )\}_{n\in {\mathbb N}}$ is uniformly bounded. We associate a lift of $f^n(c)$ with lifts $F_{t_1,t_2}$ and C of f and c respectively, defined by

$$ \begin{align*} F_{C,n}(t_1,t_2):=F^n_{t_1,t_2}(C(t_1,t_2))/\mathrm{gcd} (F^n_{t_1, t_2}(C(t_1,t_2)). \end{align*} $$

This is given by coprime homogeneous polynomials in the variables $(t_1,t_2)$ and $\deg F_{C,n}(t_1, t_2)=\deg _t f_t^n(c(t))$ .

Next, we introduce measures associated with each dynamical pair.

4.3 Measure associated to a dynamical pair: a family of rescaled ‘bifurcation’ measures

Let k be a product formula field and $K=k(t)$ as before. In this section, we introduce the canonical measure $\mu _{f,c}=\{\mu _{f,c,v}\}_{v\in \mathcal {M}_k}$ associated to a non-isotrivial and non-preperiodic dynamical pair $(f,c)$ . Recall that for each $v\in \mathcal {M}_k$ and $z,w\in \mathbb {C}_v$ , we write $\|(z,w)\|_v=\max \{|x|_v,|y|_v\}$ . Since $(f,c)$ is not preperiodic and non-isotrivial, one has $\hat {h}_{f}(c)\neq 0$ and in particular $\deg F_{C, n}\neq 0$ for sufficiently large n. To see this, in the case of f being non-isotrivial, we refer to [Reference Baker2, Theorem 1.6]; however, when f is isotrivial but $(f,c)$ is non-isotrivial, an easy argument following [Reference Baker and Rumely7, Lemma 10.1] indicates that $\hat {h}_f(c)>0$ . For n sufficiently large, we let

$$ \begin{align*} G_{F, C, n, v}(t_1, t_2):=\frac{\log \| F_{C, n}(t_1, t_2)\|_v}{\deg F_{C, n}}. \end{align*} $$

By a telescopic argument as in [Reference Branner and Hubbard8] or [Reference Fornaess and Sibony30], this sequence converges locally uniformly on ${\mathbb C}_v^{2}\setminus (\{(0,0)\}\cup \pi ^{-1}(\mathrm {Sing}(f)))$ . We denote its limit by

$$ \begin{align*} G_{F,C,v}(t_1,t_2):=\lim_{n\to \infty} G_{F, C, n, v}(t_1, t_2). \end{align*} $$

Note that for non-archimedean places, $G_{F,C,v}$ extends to the product of Berkovich affine lines $\mathbb {A}^{1, an}_v\times \mathbb {A}^{1, an}_v\setminus \{(0,0)\}$ ; see [Reference Baker and Rumely7, Ch. 10]. By slight abuse of notation, we also denote its extension by $G_{F,C,v}$ . Let $\pi :{\mathbb C}_v^2\setminus \{(0,0)\}\to \mathbb {P}^1({\mathbb C}_v)$ be the standard projection map.

We define a measure $\mu _{f,c,v}$ on $\mathbb {P}^1({\mathbb C}_v)\setminus \mathrm {Sing}(f)$ by

(4.2) $$ \begin{align} \mu_{f,c,v}:=\pi_{*}dd^c_{(t_1,t_2)} G_{F,C,v}(t_1,t_2). \end{align} $$

When v is archimedean so that ${\mathbb C}_v\simeq {\mathbb C}$ and ${\mathbb P}^{1, an}_{v}\simeq {\mathbb P}^1({\mathbb C})$ , this measure is a rescaled bifurcation measure. Indeed, the bifurcation measure associated to $(f,c)$ is defined on ${\mathbb P}^1({\mathbb C}_v)\backslash \mathrm {Sing}(f)$ by

$$ \begin{align*}\pi_{*} dd^c_{(t_1,t_2)}\bigg(\lim_{n\to \infty}\frac{1}{d^n}\log\|F_{t_1,t_2}^n(C(t_1,t_2))\|_v\bigg).\end{align*} $$

It is an important measure in dynamics as it is supported exactly at the set of non-degenerating parameters $t=\unicode{x3bb} $ at which $(f, c)$ is unstable, that is, the sequence of holomorphic functions $\{t\mapsto f_t^n(c(t))\}$ is not normal near $\unicode{x3bb} $ . Recall here that the family $f_t$ is stable at $\unicode{x3bb} $ if the Julia set moves holomorphically for a small perturbation of t at $\unicode{x3bb} $ , or equivalently if the dynamical pairs $(f, c)$ (upon passing to a finite branched cover of ${\mathbb P}^1$ ) are stable at $\unicode{x3bb} $ for each critical point c; see [Reference Mañé, Sad and Sullivan41, Reference McMullen45]. We refer the reader to [Reference DeMarco13, Reference DeMarco14, Reference Dujardin and Favre24] for more details.

In the following proposition, we show that the definition of $\mu _{f,c,v}$ extends naturally to give a probability measure on ${\mathbb P}^{1, an}_{v}$ for each $v\in \mathcal {M}_k$ . When $\deg F_{C, n}$ is not zero, we write

$$ \begin{align*} \mu_{f,c,n,v}:=\pi_{*}dd^c_{(t_1,t_2)} G_{F,C,n,v}(t_1,t_2) \end{align*} $$

to denote the probability measure on ${\mathbb P}^{1, an}_{v}$ associated to $F_{C, n}$ for $v\in \mathcal {M}_k$ . This measure is independent of the choice of lifts for f and c.

Proof. First we assume that $v\in \mathcal {M}_k$ is an archimedean place, so that ${\mathbb C}_v\simeq {\mathbb C}$ and ${\mathbb P}^{1, an}_{v}\simeq {\mathbb P}^1({\mathbb C})$ . Thus, we may work on ${\mathbb P}^1({\mathbb C})$ . Notice that $G_{F, C, n, v}(t_1, t_2)$ is a plurisubharmonic function on ${\mathbb C}^2\backslash \{(0, 0)\}$ and recall that the sequence $G_{F, C, n, v}$ converges locally uniformly on ${\mathbb C}^{2}\setminus (\{(0,0)\}\cup \pi ^{-1}(\mathrm {Sing}(f)))$ . Hence, the sequence of probability measures $\mu _{f,c,n, v}$ converges weakly to the rescaled bifurcation measure $\mu _{f,c,v}$ on ${\mathbb P}^1({\mathbb C})\backslash \mathrm {Sing}(f)$ . Since the space of probability measures on ${\mathbb P}^1({\mathbb C})$ is compact in the weak topology, to show that $\mu _{f,c,n, v}$ has a unique limit, it suffices to prove that for any convergent subsequence of $\mu _{f,c,n, v}$ , the limit admits no point mass on $\mathrm {Sing}(f)$ . Without loss of generality, we may assume that $0\in \mathrm {Sing}(f)$ . We have to show that for any $\epsilon>0$ , there is a radius $r>0$ and an integer $N>0$ , such that for all $n\geq N$ , $\mu _{f,c,n, v}(D(0, r))<\epsilon .$ Suppose that this is not the case. Then we may find integers $n_j\to \infty $ and a sequence of radii $r_{n_j}\to 0$ such that

$$ \begin{align*} \mu_{f,c,n_j, v}(D(0, r_{n_j}))\to \epsilon_0>0, \end{align*} $$

as $j\to \infty $ . Let $P_{n_j}(t)$ be a potential function of $\mu _{f,c,n_j, v}|_{D(0, r_{n_j})}$ . We have

$$ \begin{align*} P_{n_j}(t):=\int \log |t-s|_v\,d(\mu_{f,c,n_j, v}|_{D(0, r_{n_j})})\to \epsilon_0\log|t|_v \end{align*} $$

locally uniformly on a punctured disk centered at $0$ . Hence, the sequence of subharmonic functions $G_{F, C, n_j, v}(t, 1)-P_{n_j}(t)$ converges locally uniformly to a subharmonic function $G_{F, C, v}(t, 1)-\epsilon _0 \log |t|_v$ on a punctured disk. So we can find some $L_0>0$ and $r_0>0$ , such that for all big $n_j$ , we have

$$ \begin{align*} \sup_{|t|=r_0}(G_{F, C, n_j, v}(t, 1)-P_{n_j}(t))<L_0. \end{align*} $$

From [Reference DeMarco17, Proposition 3.1], one has $G_{F, C, v}(t, 1)=o(\log |t|_v).$ Then for very small t, we can find $n_j$ big enough such that

$$ \begin{align*} G_{F, C, n_j, v}(t, 1)-P_{n_j}(t)>-\frac{\epsilon_0}{2} \log|t|_v>L_0, \end{align*} $$

which is a contradiction as the subharmonic function $G_{F, C, n_j, v}(t, 1)-P_{n_j}(t)$ achieves its maximal value on the boundary of $D(0, r_0)$ . Since all $G_{F, C, n, v}$ are bounded above uniformly near $0$ , $G_{F, C,v}(t, 1)$ is bounded above and subharmonic on the punctured disk centered at $0$ . By [Reference Ransford47, Theorem 3.6.1], $G_{F, C, v}(t,1)$ has a unique extension to a subharmonic function in a disk centered at $0$ , with $G_{F, C, v}(0,1):=\limsup _{t\to 0} G_{F, C, v}(t,1)$ . Because $G_{F, C, v}(t, 1)=o(\log |t|_v)$ and $\mu _{f, c, v}(\{0\})=0$ , the extended subharmonic function is a potential of $\mu _{f, c, v}$ near $0$ . Hence, $\mu _{f, c, v}$ has continuous potential if and only if $G_{F, C, v}(t_1, t_2)$ can be extended continuously. For properties of subharmonic functions, we refer the reader to [Reference Ransford47].

Assume now that $v\in \mathcal {M}_k$ is non-archimedean. Each $F_{C, n, v}$ determines a probability measure $\mu _{f,c,n, v}$ with continuous potential on ${\mathbb P}^{1, an}_{v}$ defined on ${\mathbb P}^1({\mathbb C}_v)$ by

$$ \begin{align*} g_{f,c,n, v}(x) =\log\|\tilde{x}\|_v- G_{F, C, n, v}(\tilde{x}). \end{align*} $$

The measure is given by

(4.3) $$ \begin{align} \mu_{f,c, n, v}:=\triangle g_{f,c,n, v}+\unicode{x3bb}_v. \end{align} $$

Here, $\unicode{x3bb} _v$ is the probability measure supported on the Gauss point. For any neighborhood $U\subset {\mathbb P}^1({\mathbb C}_v)$ of $\mathrm {Sing}(f)$ , the sequence $G_{F, C, n, v}(\tilde x)$ converges uniformly for $x\in {\mathbb P}^1({\mathbb C}_v)\backslash U$ . Since ${\mathbb P}^1({\mathbb C}_v)$ is dense in ${\mathbb P}^{1, an}_{v}$ , we also have that for any neighborhood of $U^{an}$ of $\mathrm {Sing}(f)$ in ${\mathbb P}^{1, an}_{v}$ , the function $g_{f,c,n,v}(x)$ converges uniformly on ${\mathbb P}^{1, an}_{v}\backslash U^{an}$ as $n\to \infty $ . Hence, from (4.3), we see that the limit

$$ \begin{align*} g_v(x):=\lim_{n\to \infty}g_{f,c,n, v}(x)=\log\|\tilde x\|_v-G_{F, C, v}(\tilde x) \end{align*} $$

is an element of BVD $({\mathbb P}^{1, an}_{v})$ (see [Reference Baker and Rumely7, Definition 5.11]), with

$$ \begin{align*} \mu_{f, c, v}-\unicode{x3bb}_v:=\triangle g_v. \end{align*} $$

It is clear that the probability measure $\mu _{f, c, v}$ is the unique limit of $\{\mu _{f,c,n, v}\}$ on ${\mathbb P}^{1, an}_{v}$ , with potential $g_v$ . Since the potential function of $\mu _{f, c, v}$ is unique up to a constant, we have that $\mu _{f, c, v}$ has a continuous potential if and only if $g_v$ can be extended continuously to $\mathrm {Sing}(f)$ , or equivalently if and only if $G_{F, C, v}$ can be extended continuously on ${\mathbb C}_v^2\backslash \{(0, 0)\}$ .

We are indebted to Laura DeMarco for sharing the idea of the following proposition.

Proposition 4.2. Let $(f,c)\in K(z)\times K$ be a non-isotrivial and non-preperiodic dynamical pair and $v\in \mathcal {M}_k$ . We write $M_{F,C,v}=\{(\unicode{x3bb} _1, \unicode{x3bb} _2)\in {\mathbb C}_v^2\setminus \{(0,0)\} : G_{F,C,v}(\unicode{x3bb} _1, \unicode{x3bb} _2) \leq 0\}$ . Suppose that $G_{F,C,v}$ extends continuously to ${\mathbb C}_v^2\backslash \{(0, 0)\}$ . Then,

$$ \begin{align*} \operatorname{Cap}(M_{F,C,v})\leq \liminf_{n\to \infty} |\operatorname{Res} F_{C, n}|_v^{-1/\deg(F_{C,n})^2}. \end{align*} $$

Proof. Let $v\in \mathcal {M}_k$ and write $G_{F, C, n, v}(t_1, t_2):={\log \|F_{C,n}(t_1, t_2)\|_v}/{\deg F_{C,n}}$ and

$$ \begin{align*} M_{n,v}:=\{(s_1, s_2)\in {\mathbb C}_v^2\setminus\{(0,0)\} : G_{F, C, n, v}(s_1, s_2) \leq 0\}. \end{align*} $$

From [Reference DeMarco, Wang and Ye23, Proposition 2.1], we have $\operatorname {Cap}(M_{n,v})=|\operatorname {Res}(F_{C,n})|_v^{-1/\deg (F_{C,n})^2}$ . We are going to prove that for any $\epsilon \in |{\mathbb C}_v^{*}|_v$ , there exists an $N\in {\mathbb N}$ such that for any $n\geq N$ , we have

(4.4) $$ \begin{align} G_{F, C, n, v}(t_1, t_2)-G_{F,C,v}(t_1,t_2) < \epsilon. \end{align} $$

This will imply that $M_{F,C,v}\subset e^{\epsilon } M_{n,v}$ and hence, by the monotonicity of the homogeneous capacity, $\operatorname {Cap}(M_{F,C,v})\leq e^{2\epsilon } \operatorname {Cap}(M_{n,v})$ for all $n\ge N$ . Since $|{\mathbb C}_v^{*}|_v$ is dense in ${\mathbb R}_{\ge 0}$ , the proposition follows. Note that $G_{F, C, n, v}(t_1, t_2)$ converges locally uniformly to $G_{F,C,v}(t_1,t_2)$ away from $\pi ^{-1}(\mathrm {Sing}(f))$ . Hence, it suffices to prove that (4.4) holds in a small neighborhood of $\pi ^{-1}(\mathrm {Sing}(f))$ . To this end, we may assume without loss of generality that $0\in \mathrm {Sing}(f)$ and show that there exists an $r>0$ , such that for all $t\in {\mathbb C}_v$ with $|t|_v\leq r$ , we have $G_{F,C,n,v}(t,1)< G_{F,C,v}(t,1)+\epsilon $ for large n. Since $G_{F,C,v}(t, 1)$ is continuous, we can choose r small enough such that for $|t|_v\leq r$ , we have $|G_{F,C,v}(t, 1)-G_{F,C,v}(0, 1)|<\epsilon /3$ . Moreover, enlarging N if necessary, we may further assume that $G_{F,C,n,v}(t,1)<G_{F,C,v}(t,1)+\epsilon /3<G_{F,C,v}(0,1)+2\epsilon /3$ , when $|t|_v=r$ . Then, since $G_{F, C, n, v}(t, 1)$ is subharmonic, by the maximum principle (see [Reference Baker and Rumely7, Proposition 8.14] when v is non-archimedean), we get

$$ \begin{align*} G_{F, C, n, v}(t, 1)< G_{F,C,v}(0,1)+2\epsilon/3<G_{F,C,v}(t,1)+\epsilon \end{align*} $$

for all $t\in {\mathbb C}_v$ with $|t|_v\leq r$ and $n\ge N$ , as claimed.

4.4 A generic dynamical pair is not adelic

Let k be a number field or the function field of a smooth projective curve defined over a field (of arbitrary characteristic) and let $\alpha \in \overline {k}$ (or $\alpha =\infty \in {\mathbb P}^1(\overline {k})$ ). To simplify our notation, for a homogeneous polynomial $H(x,y)$ defined over a ring R and an element $\chi \in R$ , we write $H(\chi ):=H(\chi ,1)$ and $H(\infty ):=H(1,0)$ . In particular,

$$ \begin{align*} P_{\alpha}(z, w)&:=P_{\alpha, 1}(z, w)\quad\text{and}\quad Q_{\alpha}(z,w):= Q_{\alpha,1}(z,w)\text{, when }\alpha\in\overline{k}\text{ and }\\ P_{\infty}(z,w)&:=P_{1,0}(z,w)\text{, }Q_{\infty}(z,w):=Q_{1,0}(z,w) \end{align*} $$

We let

$$ \begin{align*} R_{\alpha}(z,w):=\operatorname{gcd} (P_{\alpha}(z, w),Q_{\alpha}(z,w)), \end{align*} $$

with $R_{\alpha }(z,1)$ being a monic polynomial. Then

(4.5) $$ \begin{align} P_{\alpha}=R_{\alpha}\cdot P_{\alpha}^{*}\quad\text{and}\quad Q_{\alpha}=R_{\alpha}\cdot Q_{\alpha}^{*}, \end{align} $$

where $P_{\alpha }^*(z,w),Q_{\alpha }^*(z,w)$ are homogeneous polynomials with no common linear factor over $\overline {k}$ . Note that $R_{\alpha }\neq 1$ if and only if $\alpha \in \mathrm {Sing}(f)$ . We define the set of $\alpha $ -holes of f by

$$ \begin{align*} \mathcal{H}_{f,\alpha}:=\{[\unicode{x3bb}_1:\unicode{x3bb}_2]\in {\mathbb P}^1(\overline{k}) : R_{\alpha}(\unicode{x3bb}_1,\unicode{x3bb}_2)=0\}. \end{align*} $$

Abusing notation slightly, we often view $\mathcal {H}_{f,\alpha }$ as a subset of $\bar {k}\cup \{\infty \}$ identifying ${\mathbb P}^1(\overline {k})$ with $\bar {k}\cup \{\infty \}$ . Under this identification, $\mathcal {H}_{f,\alpha }$ consists of $t\in \bar {k}$ such that $R_{\alpha }(t,1)=0$ together with possibly $\infty $ in the event that $R_{\alpha }(1,0)=0$ .

Next, we define our notion of an $\alpha $ -generic dynamical pair $(f,c)$ and subsequently state our theorem.

With this terminology in place, Theorem 1.3 reduces to the following statement.

Before we proceed to the proof of Theorem 4.5, let us see how it implies Theorem 1.3. First we make a useful observation.

We now establish some preliminary results toward the proof of Theorem 4.5. Let $S\subset \mathcal {M}_k$ be a finite set containing all the archimedean places (if they exist). We denote the set of S-integers of k by

$$ \begin{align*} \mathcal{O}_{S,k}:=\{\alpha\in k : |\alpha|_v\le 1\text{ for all }v\notin S\}. \end{align*} $$

If $\varphi \in k(z)$ is a rational map and $\alpha \in \overline {k}$ , we denote the orbit of $\alpha $ under the action of $\varphi $ as

$$ \begin{align*}\mathcal{O}_{\varphi}(\alpha)=\{\varphi^n(\alpha) : n\in{\mathbb N}\}. \end{align*} $$

The following theorem will play a crucial role in our proofs. We thank Patrick Ingram for referring us to it.

We point out here that an analog of Theorem 4.6 also holds for function fields of curves; see [Reference Carney, Hindes and Tucker11, Reference Huang, Sun and Wang38] for the case of characteristic $0$ and p. Here we need to assume that $\varphi $ is not isotrivial.

Lemma 4.8. Let k be either a number field or a function field of a smooth projective curve. Let $\varphi \in k(z)$ be a rational map of degree at least $2$ such that $\varphi ^2(z)\notin k[z]$ . If k is a function field, assume that $\varphi $ is not isotrivial. Let $(\alpha ,\beta )\in k^2\setminus \{(0,0)\}$ be such that ${\alpha }/{\beta }$ is not preperiodic for $\varphi $ . Let $\{a_n\}$ and $\{b_n\}$ be the sequences defined as follows. For a choice of coprime homogeneous polynomials $P,Q\in k[z,w]$ such that $\varphi =[P:Q]$ , we let

$$ \begin{align*} a_0&=\alpha \quad\text{and}\quad b_0=\beta, \quad\text{and for all }n\ge 0, \end{align*} $$

$$ \begin{align*} a_{n+1}&=P(a_n,b_n), \quad b_{n+1}=Q(a_n,b_n). \end{align*} $$

Then there are infinitely many non-archimedean places $v\in \mathcal {M}_k$ such that $|b_n|_v<1$ for some $n\in \mathbb {N}$ .

Lemma 4.9. Let $(f,c)\in k(t)(z)\times k(t)$ be a dynamical pair and let $\alpha \in \mathrm {Sing}(f)$ be such that $\deg (f_{\alpha })\ge 1$ . Assume that for all $n\in {\mathbb N}$ , we have $f_{\alpha }^n(c(\alpha ))\notin \mathcal {H}_{f,\alpha }$ . Then,

$$ \begin{align*}m_{C,n}(\alpha)=0 \end{align*} $$

for all $n\in {\mathbb N}$ . In particular, for all $n\in {\mathbb N}$ , we have

$$ \begin{align*} \mathrm{gcd} (F^n_{t_1, t_2}(C(t_1,t_2)))= \prod_{\substack{\beta\in\mathrm{Sing}(f)\setminus\{\alpha\}}}u_{\beta}(t_1,t_2)^{m_{C,n}(\beta)}. \end{align*} $$

Proof. Let $\alpha \in {\mathbb P}^1(\overline {k})$ be as in the statement of the lemma. Note that

(4.6) $$ \begin{align} m_{C,0}(\alpha)=\min\{\operatorname{ord}_{\alpha}A, \operatorname{ord}_{\alpha}B\}=0. \end{align} $$

We will show that $m_{C,n}(\alpha )=0$ for all $n\in {\mathbb N}$ . Assume the contrary and let $N\in {\mathbb N}$ be the smallest integer such that $m_{C,N}(\alpha )>0$ . By (4.6), we have $N\ge 1$ . Since $m_{C,N}(\alpha )>0$ , (4.1) yields

$$ \begin{align*} P_{\alpha}(A_{C,N-1}(\alpha),B_{C,N-1}(\alpha))=Q_{\alpha}(A_{C,N-1}(\alpha),B_{C,N-1}(\alpha))=0. \end{align*} $$

Hence, by (4.5), we get that either

(4.7) $$ \begin{align} P^*_{\alpha}(A_{C,N-1}(\alpha),B_{C,N-1}(\alpha))=Q^{*}_{\alpha}(A_{C,N-1}(\alpha),B_{C,N-1}(\alpha))=0 \end{align} $$

or

(4.8) $$ \begin{align} R_{\alpha}(A_{C,N-1,\alpha}(\alpha),B_{C,N-1,\alpha}(\alpha))=0. \end{align} $$

If (4.7) is true, then $(A_{C,N-1}(\alpha ),B_{C,N-1}(\alpha ))\neq (0,0)$ is a common zero of $P^*_{\alpha }$ , $Q^*_{\alpha }$ , contradicting the fact that they do not share a common linear factor.

If however (4.8) holds, then

(4.9) $$ \begin{align} \frac{A_{C,N-1}(\alpha)}{B_{C,N-1}(\alpha)}\in \mathcal{H}_{f,\alpha}. \end{align} $$

Nonetheless, by our assumption, $f_{\alpha }^n(c(\alpha ))\notin \mathcal {H}_{f,\alpha }$ for all $n\le N-1$ . Moreover, the minimality of N implies that $m_{C,n}(\alpha )=0$ for all $n\le N-1$ . Hence,

$$ \begin{align*} f_{\alpha}^{n}(c(\alpha))=\frac{A_{C,n}(\alpha)}{B_{C,n}(\alpha)} \end{align*} $$

for all $n\le N-1$ . Thus, (4.9) yields

$$ \begin{align*} f_{\alpha}^{N-1}(c(\alpha))=\frac{A_{C,N-1}(\alpha)}{B_{C,N-1}(\alpha)}\in \mathcal{H}_{f,\alpha}, \end{align*} $$

contradicting our assumption that the orbit of $c(\alpha )$ under iteration by $f_{\alpha }$ never meets the set $\mathcal {H}_{f,\alpha }$ . In both cases, we got a contradiction. Thus, the lemma follows.

Before stating the next proposition, we recall that $\infty $ is a totally ramified fixed point of a rational map $\psi \in k(z)$ if and only if $\psi \in k[z]$ .

Proposition 4.10. Let $(f,c)\in k(t)(z)\times k(t)$ be a dynamical pair with $\deg _z f\ge 3$ . Assume that f is not isotrivial. Let $\alpha \in \mathrm {Sing}(f)$ be such that $(f,c)$ is $\alpha $ -generic. Then there are infinitely many $v\in \mathcal {M}_{k}$ such that for some $n_v\in {\mathbb N}$ , we have

$$ \begin{align*} \max\{|A_{C,n_v}(\alpha)|_v,|B_{C,n_v}(\alpha)|_v\}<1. \end{align*} $$

Proof. Let $\alpha \in \mathrm {Sing}(f)$ be such that $(f,c)$ is $\alpha $ -generic. To simplify the notation, throughout this proof, we write $a_n:=A_{C,n}(\alpha )$ and $b_n:=B_{C,n}(\alpha )$ . Since (P3) holds, by Lemma 4.9 we know that not both of $a_n$ and $b_n$ are zero. We have

(4.10) $$ \begin{align} a_{n+1}&=P_{\alpha}(a_n,b_n)=R_{\alpha}(a_n,b_n)P^{*}_{\alpha}(a_n,b_n),\nonumber\\ b_{n+1}&=Q_{\alpha}(a_n,b_n)=R_{\alpha}(a_n,b_n)Q^{*}_{\alpha}(a_n,b_n), \end{align} $$

for all $n\in {\mathbb N}$ . We define auxiliary sequences $\{a^*_n\}$ and $\{b^*_n\}$ as $a^{*}_0:=a_0, b^{*}_0:=b_0$ and

$$ \begin{align*} a^*_{n+1}:=P^{*}_{\alpha}(a^*_n,b^*_n),\quad b^*_{n+1}:=Q^{*}_{\alpha}(a^*_n,b^*_n)\quad\text{for } n\ge 1. \end{align*} $$

Then, for all $n\in {\mathbb N}$ , we have

(4.11) $$ \begin{align} a_{n}=\prod_{i=0}^{n-1}R_{\alpha}(a^*_i,b^*_i)^{d^{n-1-i}}\cdot a^*_n\nonumber,\\ b_{n}=\prod_{i=0}^{n-1}R_{\alpha}(a^*_i,b^*_i)^{d^{n-1-i}}\cdot b^*_n. \end{align} $$

Let $S\subset \mathcal {M}_{k}$ be a finite set of places, containing the archimedean ones, such that for all $v\notin S$ , the coefficients of $P_{\alpha }^{*}$ and $Q_{\alpha }^{*}$ are v-adic integers, $|\mathrm {Res}(P_{\alpha }^{*},Q_{\alpha }^{*})|_v=1$ and $\max \{|a^*_0|_v,|b^*_0|_v\}=1$ . Then invoking [Reference Baker and Rumely7, Lemma 10.1], for all $n\in \mathbb {N}$ and $v\notin S$ , we have

(4.12) $$ \begin{align} \max\{|a^*_n|_v,|b^*_n|_v\}=1. \end{align} $$

We may enlarge the set S, if necessary, to assume that the elements of $\mathcal {H}_{f,\alpha }\cap \overline {k}$ are v-adic integers for all $v\notin S$ . Thus, also the coefficients of $R_{\alpha }$ are v-adic integers for $v\notin S$ . Combining then (4.11) with (4.12), we get that

(4.13) $$ \begin{align} \max\{|a_{n+1}|_v,|b_{n+1}|_v\}\le |R_{\alpha}(a^*_n,b^*_n)|_v\le |u_{\rho}(a^*_n,b^*_n)|_v \end{align} $$

for all $v\notin S$ and $n\in {\mathbb N}$ and for any $\rho \in \mathcal {H}_{f,\alpha }$ . Now let $\rho \in \mathcal {H}_{f,\alpha }$ be as in (P2). We claim that there are infinitely many $v\in \mathcal {M}_{k}$ such that

(4.14) $$ \begin{align} |u_{\rho}(a^*_{n}, b^*_{n})|_v<1 \end{align} $$

for some $n\in \mathbb {N}$ . By (4.13), it is clear that this suffices to prove this proposition. To prove (4.14), we use Lemma 4.8. If $\rho =\infty $ , our claim follows. Otherwise, let $M_{\rho }(z,w)=(w+\rho z,z)$ and $(\hat {P_{\alpha }^{*}},\hat {Q_{\alpha }^{*}})=M_{\rho }^{-1}\circ (P_{\alpha }^{*},Q_{\alpha }^{*})\circ M_{\rho }$ . Consider the morphism $\hat g_{\alpha }:{\mathbb P}^1\to {\mathbb P}^1$ defined by

$$ \begin{align*} [z:w]\mapsto[\hat{P}_{\alpha}^{*}(z,w):\hat{Q}_{\alpha}^{*}(z,w)]. \end{align*} $$

Since $\rho \in \mathcal {H}_{f,\alpha }$ is not a totally ramified fixed point of $f^2_{\alpha }$ as in (P2), we know that $\infty $ is not a totally ramified fixed point of $\hat g^2_{\alpha }$ . Moreover, by (P4), we have $|\mathcal {O}_{\hat g_{\alpha }}({b_0}/{a_0-\rho b_0})|=\infty $ and by (P1), we have $\deg (\hat g_{\alpha })\ge 2$ . Thus, by Lemma 4.8 applied to the rational map $\hat g_{\alpha }$ and $(b_0,a_0-\rho b_0)$ , our claim follows. This finishes our proof.

We are now ready to prove the main result of this section: In most cases, $(f,c)$ is not adelic.

4.5 Proof of Theorem 4.5

Let f and c be as in the statement of the theorem. Let $\alpha \in \mathrm {Sing}(f)$ be such that $(f,c)$ is $\alpha $ -generic and assume to the contrary that $(f,c)$ is adelic. Then there is a finite set $S\subset \mathcal {M}_k$ such that for all $v\notin S$ , we have

(4.15) $$ \begin{align} G_{F, C, v}(t_1,t_2)=\log \|(t_1,t_2)\|_v+c_v \end{align} $$

for a constant $c_v$ . Denote by $\mathcal {P}\subset \mathcal {M}_k$ the infinite set of places satisfying the conclusion of Proposition 4.10. In other words, for $v\in \mathcal {P}$ , there exists $n_v\in {\mathbb N}$ such that

$$ \begin{align*} \max\{|A_{C,n_v}(\alpha)|_v,|B_{C,n_v}(\alpha)|_v\}<1. \end{align*} $$

Enlarging the set S if necessary, we may further assume that for all $v\notin S$ , the following hold.

• (S1) $|\alpha |_v=1$ if $\alpha \neq 0,\infty $ .

• (S2) For all $\beta \in \mathrm {Sing}(f)\setminus \{\alpha \}$ , we have $|u_{\beta }(\alpha )|_v=1$ and if $\beta \neq 0$ also $|u_{\beta }(0,1)|_v=1$ .

• (S3) The coefficients of $F_{t_1,t_2}(z,w)\in k[t_1,t_2,z,w]$ are v-adic integers.

• (S4) $|\operatorname {Res}_{(t_1,t_2)}(A,B)|_v=1$ and the coefficients of $C(t_1,t_2)=(A, B)$ are v-adic integers.

• (S5) All coefficients of $\operatorname {Res}(F_{t_1,t_2})$ are v-adic units.

We aim to prove that (4.15) does not hold for places in the infinite set $\mathcal {P}\setminus S$ , thus leading to a contradiction. To do so, we will evaluate (4.15) at two distinct points that yield distinct values for $c_v$ when $v\in \mathcal {P}\setminus S$ .

Let $v\in \mathcal {P}\setminus S$ . In the rest of this proof, for $t_0\in {\mathbb C}_v$ , we write

$$ \begin{align*} T_0 = \begin{cases} (t_0,1) &\mbox{ if } \ \alpha\in{\mathbb A}^1(\overline{k}),\\ (1,t_0) &\mbox{ if } \ \alpha=\infty.\\ \end{cases} \end{align*} $$

View both $T_0$ and $\alpha $ as elements of ${\mathbb P}^1$ . Since by Lemma 4.9 either $A_{C,n_v}(\alpha )$ or $B_{C,n_v}(\alpha )$ is non-zero and v is non-archimedean, we may choose $T_0\neq (0,1)$ sufficiently close to $\alpha $ in the v-adic topology to be such that the following hold.

• (T1) $0<|u_{\alpha }(T_0)|_v<1$ .

• (T2) $\max \{|A_{C,n_v}(T_0)|_v,|B_{C,n_v}(T_0)|_v\}\le \max \{|A_{C,n_v}(\alpha )|_v,|B_{C,n_v}(\alpha )|_v\}$ .

Next, we show that evaluating (4.15) at $T_0$ gives $c_v<0$ . To this end, recall that

(4.16) $$ \begin{align} F_{C,n}(t_1,t_2)=\bigg(\frac{A_{C,n}(t_1,t_2)}{\mathrm{gcd} (F^n_{t_1, t_2}(C(t_1,t_2))},\frac{B_{C,n}(t_1,t_2)}{\mathrm{gcd} (F^n_{t_1, t_2}(C(t_1,t_2))}\bigg), \end{align} $$

where by Lemma 4.9, we have

$$ \begin{align*} g_{C,n}(t_1,t_2):=\mathrm{gcd} (F^n_{t_1, t_2}(C(t_1,t_2)) = \prod_{\substack{\beta\in\mathrm{Sing}(f)\setminus\{\alpha\}}}u_{\beta}(t_1,t_2)^{m_{C,n}(\beta)}. \end{align*} $$

Combining (S2) and (T1), the ultrametric inequality gives $|u_{\beta }(T_0)|_v=1$ for all $\beta \in \mathrm {Sing}(f)\setminus \{\alpha \}$ . This in turn yields $|g_{C,n}(T_0)|_v=1$ for all $n\in {\mathbb N}$ . Thus, evaluating (4.16) at $T_0$ , we have

(4.17) $$ \begin{align} \|F_{C,n}(T_0)\|_v=\|F_{T_0}^n(C(T_0))\|_v=\|(A_{C,n}(T_0),B_{C,n}(T_0))\|_v. \end{align} $$

Note that by (S1) and (T1), we have $\|T_0\|_v\le 1$ . Combining this with (S3), (4.17) yields

$$ \begin{align*} \|F_{C,n+1}(T_0)\|_v\le \|F_{C,n}(T_0)\|_v^{d}. \end{align*} $$

An easy argument by induction and (T2) yield that for all $n\ge n_v$ , we have

$$ \begin{align*} \|F_{C,n}(T_0)\|_v\le\max\{|A_{C,n_v}(\alpha)|_v,|B_{C,n_v}(\alpha)|_v\}^{d^{n-n_v}}<1, \end{align*} $$

where the last inequality follows from our assumption that $v\in \mathcal {P}$ . We now have

(4.18) $$ \begin{align} c_v=\lim_{n\to\infty}\frac{\log\|F_{C,n}(T_0)\|_v}{\deg(F_{C,n})}\le\frac{\log\max\{|A_{C,n_v}(\alpha)|_v,|B_{C,n_v}(\alpha)|_v\}}{d^{n_v}\cdot \hat{h}_f(c)}<0, \end{align} $$

as claimed. We point out here that by our assumption f is not isotrivial. Hence, our property (P4) guarantees that $\hat {h}_f(c)\neq 0$ ; see [Reference Baker2, Reference DeMarco17].

However, we can choose $S_0=(s_0,1)\in {\mathbb C}^2_v$ to be such that for all $\beta \in \mathrm {Sing}(f)$ , we have $|s_0|_v=|u_{\beta }(S_0)|_v=1$ . Then, upon using (S3), we get that the coefficients of $F_{S_0}$ are v-adic integers. Moreover, by (S5) and since $|u_{\beta }(S_0)|_v=1$ for all $\beta \in \mathrm {Sing}(f)$ , we have $|\operatorname {Res}_{(z,w)}(F_{S_0})|_v=1$ . Therefore, [Reference Baker and Rumely7, Lemma 10.1] yields that $\|F_{S_0}(z,w)\|_v=\|(z,w)\|_v^d$ . Since by (S4) we have that C has good reduction and moreover $\|S_0\|_v=1$ , another application of [Reference Baker and Rumely7, Lemma 10.1] yields $\|C(S_0)\|_v=1$ . Thus, an easy argument by induction yields $\|F^n_{S_0}(C(S_0))\|_v=1$ . By our choice of $S_0$ , we have $|g_{C,n}(S_0)|_v=1$ for all $n\in {\mathbb N}$ . Hence, $\|F^n_{S_0}(C(S_0))\|_v=\|F_{C,n}(S_0)\|_v=1$ for all $n\in {\mathbb N}$ . Therefore,

$$ \begin{align*} c_v=\lim_{n\to\infty}\frac{\log\|F_{C,n}(S_0)\|_v}{\deg(F_{C,n})}=0. \end{align*} $$

This contradicts (4.18) and finishes the proof of our theorem.

5.1 Homogeneous lifts

Throughout the rest of this section, we fix a primitive $\ell $ th root of unity $\unicode{x3bb} $ with order at least $2$ . It is more convenient to work with the $\ell $ th iterate of $g_{\unicode{x3bb} , t}(z)$ , which we denote by

$$ \begin{align*} f_t(z):=g_{\unicode{x3bb}, t}^{\ell}(z). \end{align*} $$

It is easy to see that $g_{\unicode{x3bb} }$ degenerates at $t=\infty $ . Thus, $\mathrm {Sing}(f)=\{\infty \}$ . Throughout this section, we write

$$ \begin{align*} d:=2^{\ell}, \quad d_1:=2^{\ell-1}-1, \quad d_2:=2^{\ell-1}. \end{align*} $$

At times, we also use notation introduced in §4. We fix a homogeneous lift of $g_{\unicode{x3bb} , t}(z)$ as

$$ \begin{align*} G_{t_1,t_2}(z, w):=(t_2\unicode{x3bb} zw, t_1zw+t_2(z^2+w^2)). \end{align*} $$

The following proposition enables us to show that $g^{\ell }_{\unicode{x3bb} }$ degenerates to $f_{\infty }(z):=z/(z^2+1)$ at $t=\infty $ .

Proposition 5.1. Let $\unicode{x3bb} $ be an $\ell $ th primitive root of unity. The $\ell $ th iterate of $G_{t_1,t_2}$ is given by

$$ \begin{align*} G^{\ell}_{t_1,t_2}(z, w)=t_2^{d_2}\cdot (\tau \cdot (t_1zw)^{d_1}zw+t_2(\ldots), \tau \cdot (t_1zw)^{d_1} (z^2+w^2)+t_2(\ldots)), \end{align*} $$

for some non-zero $\tau \in {\mathbb Z}[\unicode{x3bb} ]$ . In particular, $f=g^{\ell }_{\unicode{x3bb} }$ degenerates to $f_{\infty }(z)=z/(z^2+1)$ at $t=\infty $ .

Proof. We prove this proposition by induction. Notice that, inductively for $2\leq n\leq \ell $ , one has $G^n_{t_1, t_2}=(P_n, Q_n)$ , where

(5.1) $$ \begin{align} P_n(z,w)&=t_2^{2^{n-1}}((zw)^{2^{n-1}}\alpha_n t_1^{2^{n-1}-1} +(zw)^{2^{n-1}-1}(z^2+w^2)\eta_{n}t_2t_1^{2^{n-1}-2}+t_2^{2}(\ldots))\nonumber,\\ Q_n(z,w)&=t_2^{2^{n-1}-1}( (zw)^{2^{n-1}}\beta_{n} t_1^{2^{n-1}} +(zw)^{2^{n-1}-1}(z^2+w^2)\tau_{n} t_2t_1^{2^{n-1}-1}+t_2^{2}(\ldots)), \end{align} $$

for constants $\alpha _{n}, \beta _n, \eta _n$ , and $\tau _n$ depending on $\unicode{x3bb} $ . From the iteration formula, we get $\alpha _2=\unicode{x3bb} ^2, \eta _2=\unicode{x3bb} ^2, \beta _{2}=1+\unicode{x3bb} ,\tau _{2}=\unicode{x3bb} +2$ and for all $n\ge 2$ , we have

$$ \begin{align*} \begin{cases} \alpha_{n+1}=\unicode{x3bb} \cdot \alpha_{n} \cdot \beta_{n},\\ \beta_{n+1}= \beta_{n}\cdot(\alpha_{n}+ \beta_{n}),\\ \end{cases} \text{and}\quad \begin{cases} \eta_{n+1}=\unicode{x3bb} \cdot ( \alpha_{n} \cdot \tau_{n}+\beta_n\cdot \eta_n),\\ \tau_{n+1}= \alpha_n\cdot \tau_n+\beta_n\cdot \eta_n+2\beta_n\cdot \tau_n.\\ \end{cases} \end{align*} $$

Consequently, for $n\geq 3$ , we have

$$ \begin{align*} \alpha_{n}&=\unicode{x3bb}^n\cdot \prod_{i=1}^{n-2}(1+\unicode{x3bb}+\cdots +\unicode{x3bb}^i)^{2^{n-2-i}},\\ \beta_{n}&=(1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{n-1})\cdot \prod_{i=1}^{n-2}(1+\unicode{x3bb}+\cdots +\unicode{x3bb}^i)^{2^{n-2-i}}. \end{align*} $$

Since $\unicode{x3bb} $ is an $\ell $ th primitive root of unity, we have $\alpha _{\ell }\neq 0$ and $\beta _{\ell }=0$ . It remains to show that

(5.2) $$ \begin{align} \tau:=\tau_{\ell}=\alpha_{\ell}. \end{align} $$

Let $z_0$ be a primitive $6$ th root of unity and notice that ${z_0}/({z_0^2+1})=1$ . Since $\beta _{\ell }=0$ , from the expression of $G^{\ell }_{t_1, t_2}(z_0, 1)$ in (5.1), we get

$$ \begin{align*} \lim_{t_1=1, t_2\to 0}\frac{P_{\ell}(z_0,1)}{Q_{\ell}(z_0, 1)}\to \frac{ \alpha_{\ell}}{\tau_{\ell}}, \end{align*} $$

or equivalently,

(5.3) $$ \begin{align} \lim_{t\to \infty} g_{\unicode{x3bb}, t}^{\ell}(z_0)= \frac{ \alpha_{\ell}}{\tau_{\ell}}. \end{align} $$

We are going to show that this limit is equal to one, and hence equation (5.2) follows. Notice that

$$ \begin{align*} g_{\unicode{x3bb}, t}(z_0)=\frac{\unicode{x3bb} z_0}{1+t z_0+z_0^2}=\frac{\unicode{x3bb}}{t}\cdot \bigg(1-\frac{1}{t}+o\bigg(\frac{1}{t}\bigg)\bigg) \end{align*} $$

for $t\to \infty $ . Using the expression of $g_{\unicode{x3bb} , t}(z)=\unicode{x3bb} \cdot z/(1+t\cdot z+z^2)$ , inductively we get

$$ \begin{align*} g^n_{\unicode{x3bb}, t}(z_0)=\frac{1}{t}\cdot \frac{\unicode{x3bb}^n}{1+\unicode{x3bb}+\cdots +\unicode{x3bb}^{n-1}}\cdot \bigg(1-\frac{1}{1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{n-1}}\cdot \frac{1}{t}+o\bigg(\frac{1}{t}\bigg)\bigg) \end{align*} $$

for all $1\leq n\leq \ell -1$ as $t\to \infty $ . Consequently, we have

$$ \begin{align*}\small{ \begin{aligned} g^{\ell}_{\unicode{x3bb}, t}(z_0)&=g_{\unicode{x3bb}, t}( g^{\ell-1}_{\unicode{x3bb}, t}(z_0))\\ &=g_{\unicode{x3bb}, t}\bigg(\frac{1}{t}\cdot \frac{\unicode{x3bb}^{\ell-1}}{1+\unicode{x3bb}+\cdots +\unicode{x3bb}^{\ell-2}}\cdot \bigg(1-\frac{1}{1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{\ell-2}}\cdot \frac{1}{t}+o\bigg(\frac{1}{t}\bigg)\bigg)\bigg)\\ &=\frac{{1}/{t}\cdot {\unicode{x3bb}\cdot \unicode{x3bb}^{\ell-1}}/({1+\unicode{x3bb}+\cdots +\unicode{x3bb}^{\ell-2}})\cdot (1-{1}/({1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{\ell-2}})\cdot {1}/{t}+o({1}/{t}))}{1+t\cdot {1}/{t}\cdot {\unicode{x3bb}^{\ell-1}}/({1+\unicode{x3bb}+\cdots +\unicode{x3bb}^{\ell-2}})\cdot (1-{1}/({1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{\ell-2}})\cdot {1}/{t}+o({1}/{t}))+o({1}/{t})}\\ &=\frac{{1}/{t}\cdot {1}/({1+\unicode{x3bb}+\cdots +\unicode{x3bb}^{\ell-2}})\cdot (1-{1}/({1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{\ell-2}})\cdot {1}/{t}+o({1}/{t}))}{1+(-1)\cdot (1-{1}/({1+\unicode{x3bb}+\cdots+\unicode{x3bb}^{\ell-2}})\cdot {1}/{t}+o({1}/{t}))+o({1}/{t^2})}, \end{aligned}} \end{align*} $$

where in the last equality, we used the fact that $\unicode{x3bb} ^{\ell }=1$ and $1+\unicode{x3bb} +\cdots +\unicode{x3bb} ^{\ell -1}=0$ . Letting now $t\to \infty $ , we get $g^{\ell }_{\unicode{x3bb} , t}(z_0)\to 1$ . Combining this with (5.3), we get (5.2). The proposition follows.

Let us now fix a lift of $f_t$ in homogeneous coordinates $(t_1,t_2)$ as

(5.4) $$ \begin{align} F_{t_1,t_2}(z,w):&=(P_{t_1,t_2}(z,w),Q_{t_1,t_2}(z,w)):=G^{\ell}_{t_1,t_2}(z, w)/(\tau\cdot t_2^{d_2})\nonumber\\ &= ( (t_1zw)^{d_1}zw+t_2(\ldots), (t_1zw)^{d_1} (z^2+w^2)+t_2(\ldots)). \end{align} $$

Notice that at the point at infinity, our lift specializes to the map

$$ \begin{align*}F_{1,0}(z,w)=( (zw)^{d_1}zw, (zw)^{d_1} (z^2+w^2)), \end{align*} $$

which is a homogeneous lift of

$$ \begin{align*} f_{\infty}(z)=\frac{z}{z^2+1}. \end{align*} $$

Keeping the notation as in §4, we have $R_{f,\infty }(z,w)=(zw)^{d_1}$ , and hence $\mathcal {H}_{f,\infty }=\{0,\infty \}$ . We let k be a number field containing $\unicode{x3bb} $ , so that $f\in k(t)(z)$ . We fix a starting point $c\in k(t)$ satisfying $0,\infty \notin \mathcal {O}_{f_{\infty }}(c(\infty ))$ so that condition (P3) in Definition 4.4 holds for the pair $(f,c)$ . We also fix a homogeneous lift of the starting point c, with coefficients in $\mathcal {O}_k$ , as

$$ \begin{align*} C(t_1,t_2):=(A(t_1,t_2),B(t_1,t_2)), \end{align*} $$

and write

$$ \begin{align*} F_{C,n}(t_1,t_2)=F^n_{t_1,t_2}(C(t_1,t_2))/\mathrm{gcd}(F^n_{t_1, t_2}(C(t_1,t_2)))=(A_{C,n}(t_1,t_2),B_{C,n}(t_1,t_2)). \end{align*} $$

Lemma 5.2. For all $n\in {\mathbb N}$ , we have $\mathrm {gcd}(F^n_{t_1, t_2}(C))=1$ . Hence,

$$ \begin{align*} F_{C,n}(t_1,t_2)=F^n_{t_1,t_2}(C(t_1,t_2)). \end{align*} $$

Proof. As $\mathrm {Sing}(f)=\{+\infty \}$ , $\deg (f_{\infty })=2$ , and we have assumed that $f^n_{\infty }(c(\infty ))\notin \mathcal {H}_{f,\infty }=\{0,\infty \}$ , this follows from Lemma 4.9. Notice that here,

$$ \begin{align*} \operatorname{gcd}(F^n_{t_1,t_2}(C(t_1,t_2)))=\prod_{\beta\in\mathrm{Sing}(f)\setminus\{\infty\}}u_{\beta}(t_1,t_2)^{m_{C,n}(\beta)}=1, \end{align*} $$

as the product is empty.

When there is no scope for confusion, we use $a_n$ and $b_n$ to denote $A_{C,n}(1,0)$ and $B_{C,n}(1,0)$ respectively. From Lemma 5.2, we see that

(5.5) $$ \begin{align} a_{n+1}&=(a_nb_n)^{d_1}\cdot a_nb_n\nonumber,\\ b_{n+1}&=(a_nb_n)^{d_1}\cdot (a^2_n+b^2_n) \end{align} $$

for all $n\in {\mathbb N}$ . We also make use of auxiliary sequences $\{a^*_{n}\}, \{b^*_{n}\}\subset k$ defined by

(5.6) $$ \begin{align} a_0^{*}:=a_0&,\quad b_0^{*}:=b_0\nonumber,\\ a_{n+1}^*:=a_n^*b_n^*&, \quad b^*_{n+1}:={a_n^*}^2+{b_n^*}^2\quad\text{for }n\ge 1. \end{align} $$

Notice that if we let

(5.7) $$ \begin{align} \alpha_n:=\prod_{i=0}^{n-1}(a_i^{*}b_i^{*})^{ d_1\cdot d^{n-i-1}}, \end{align} $$

then for $n\geq 1$ , we have $a_n=\alpha _na_n^{*}$ and $b_n=\alpha _nb_n^{*}$ .

5.2 Continuity of the escape rate

To prove that $(f,c)$ is quasi-adelic, we need to first show that the escape rate $G_{F, C, v}$ is a continuous function.

Before we proceed to the proof of this theorem, we establish some lemmata. First we let

$$ \begin{align*}F_{C,n}(1,s)=(A_{C,n}(1,s),B_{C,n}(1,s))=(A_{C,n}(1,0)+sp_n(s),B_{C,n}(1,0)+s q_n(s)).\end{align*} $$

Proof. By [Reference Baker and Rumely7, Lemma 10.1], the recursive definition of $\{a_n^{*}\},\{b_n^{*}\}$ in (5.6) implies that there is a set of constants $\{ L_v : v\in \mathcal {M}_k\}$ and a finite set $S\subset \mathcal {M}_k$ such that $L_v=1\text { for all }v\notin S$ and for all $v\in \mathcal {M}_k$ , we have $L_v\ge 1$ and

(5.8) $$ \begin{align} \max\{|a_n^{*}|_v,|b_n^{*}|_v\}&\le L_v^{2^n}\quad\text{for all }n\in{\mathbb N}. \end{align} $$

Now let $L=\prod _{\substack {v\in \mathcal {M}_k}}L_v^{N_v}$ . Invoking the product formula, (5.8) yields

(5.9) $$ \begin{align} \min\{|a_n^{*}|_v,|b^{*}_n|_v\}\ge \frac{1}{L^{2^n}}. \end{align} $$

Moreover, (5.8) implies

(5.10) $$ \begin{align} \max\{|a_n^{*}|_v,|b^{*}_n|_v\}\le L^{2^n}. \end{align} $$

In particular, (5.9) and (5.10) yield that $\lim _{n\to \infty }({\log |a_n^{*}|_v}/{d^n})=\lim _{n\to \infty }({\log |b_n^{*}|_v}/ {d^n})=0$ and for $\{\alpha _n\}$ as in (5.7), the sequence

$$ \begin{align*}\frac{\log|\alpha_n|_v}{d^n}=\sum_{i=0}^{n-1}d_1\cdot\frac{ \log|a_i^{*}b_i^{*}|_v}{d^{i+1}}\end{align*} $$

converges. Denoting its limit by $\gamma _v$ , we have established the following:

$$ \begin{align*}\gamma_v=\lim_{n\to\infty}\frac{\log|a_n|_v}{d^n}=\lim_{n\to\infty}\frac{\log|b_n|_v}{d^n}=\lim_{n\to\infty}\frac{\log|\alpha_n|_v}{d^n}.\end{align*} $$

The first part of the lemma follows. Now let $c_n$ and $e_n$ be the constant terms of the polynomials $p_n(s)$ and $q_n(s)$ respectively. From the recursive definition of $F_{C, n}$ as in Lemma 5.2, we see that there are homogeneous $\Phi , \Psi \in k[z,w]$ of degree d and $\Phi _i, \Psi _i\in k[z,w]$ for $i=1, 2$ of degree $d-1$ , such that

(5.11) $$ \begin{align} c_{n+1}=\Phi(a_n, b_n)+c_n\cdot \Phi_1(a_n, b_n)+e_n\cdot \Phi_2(a_n, b_n)\nonumber,\\ e_{n+1}=\Psi(a_n, b_n)+c_n\cdot \Psi_1(a_n, b_n)+e_n\cdot \Psi_2(a_n, b_n) \end{align} $$

for all $n\geq 0$ . Define auxiliary sequences $\{c^*_{n}\}, \{e^*_{n}\}\subset k$ as $c_n=\alpha _n\cdot c^{*}_n$ and $e_n=\alpha _n \cdot d^{*}_n$ . We will show that

(5.12) $$ \begin{align} \limsup_{n\to\infty}\frac{\log|c^{*}_n|_v}{d^n}\text{, }\limsup_{n\to\infty}\frac{\log|e_n^{*}|_v}{d^n}\le 0. \end{align} $$

Having proved this, the second part of our lemma will follow, since

$$ \begin{align*} \limsup_{n\to\infty}\frac{\log|c_n|_v}{d^n}\text{, }\limsup_{n\to\infty}\frac{\log|e_n|_v}{d^n}\le \lim_{n\to\infty}\frac{|\alpha_n|_v}{d^n}=\gamma_v. \end{align*} $$

To prove (5.12), first notice that by (5.7), we have ${\alpha ^d_n}/{\alpha _{n+1}}={1}/{(a_n^{*}b_n^{*})^{d_1}}$ . The recursive formulas in (5.11) can now be written as

(5.13) $$ \begin{align} c^*_{n+1}=\frac{\Phi(a^*_n, b^*_n)+c^*_n\cdot \Phi_1(a^*_n, b^*_n)+e^*_n\cdot \Phi_2(a^*_n, b^*_n)}{(a^*_n b^*_n)^{d_1}}\nonumber,\\[4pt] e^*_{n+1}=\frac{\Psi(a^*_n, b^*_n)+c^*_n\cdot \Psi_1(a^*_n, b^*_n)+e^*_n\cdot \Psi_2(a^*_n, b^*_n)}{(a^*_n b^*_n)^{d_1}}. \end{align} $$

Let $L_{n,v}:=\max \{|c^*_n|_v,|e^*_n|_v\}$ . By (5.8) and (5.9), we get that there is some constant $L_0\ge 1$ such that

$$ \begin{align*} \max\bigg\{|\Phi(a^*_n, b^*_n)|_v,|\Phi_i(a^*_n, b^*_n)|_v, |\Psi(a^*_n, b^*_n)|_v, |\Psi_i(a^*_n, b^*_n)|_v, \frac{1}{ |a^*_n b^*_n|_v^{d_1}}\bigg\}\le L_0^{2^n} \end{align*} $$

for $i=1, 2$ and all $v\in \mathcal {M}_k$ . Combining this with (5.13), we see that there is some $r\ge 1$ such that

$$ \begin{align*} L_{n+1,v}\le r^{2^n}\max\{1,L_{n, v}\} \end{align*} $$

for all $n\in {\mathbb N}$ . An easy argument by induction now yields that $L_{n,v}\le r^{n2^n}\max \{1,L_{0, v}\}$ for all $n\in {\mathbb N}$ . Therefore,

$$ \begin{align*} \limsup_{n\to\infty}\frac{\log L_{n,v}}{d^n}\le 0, \end{align*} $$

and (5.12) follows. This finishes our proof.