Quasi-adelic measures and equidistribution on $\P^1$
Hexi Ye
We generalize adelic measure to quasi-adelic measure on $\PP^1$, and show that arithmetic equidistribution of small points holds for quasi-adelic measure. In the end, we briefly describe an application of the arithmetic equidistribution theorem for quasi-adelic measures.
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\begin{document}


\title{Quasi-adelic measures and equidistribution on $\P^1$}

\author{Hexi Ye}
\address{Department of Mathematics, University of British Columbia, Canada}
\email{yehexi@math.ubc.ca}

\date{\today}
\begin{abstract}
We generalize adelic measure  to quasi-adelic measure on $\PP^1$, and show that arithmetic equidistribution of small points holds for quasi-adelic measure. In the end, we briefly describe an application of the arithmetic equidistribution theorem for quasi-adelic measures.\end{abstract}

\maketitle

\thispagestyle{empty}

\section{Introduction}

Let $k$ be a product formula field, such as a number field or function field of a curve. Adelic measures on $\P^1$ were introduced in \cite{BRbook, FRL:equidistribution}, where the authors proved independently the arithmetic equidistribution of points with small heights, on the Berkovich compactification of $\P^1(k)$. Similar notions (the adelic metrized line bundle \cite{Zhang:line, Zhang:metrics}) appear as a hypothesis in the equidistribution results on more general varieties \cite{CLoir, Yuan}. In this article, we examine the equidistribution of small points on $\P^1$ in a more general setting, that of quasi-adelic measures. 

Roughly, a quasi-adelic measure is a collection of probability measures $\mu_v$ on the Berkovich projective line $\P^1_{\Berk,v}$, indexed by the places v of k, so that their potential functions satisfy a summability condition.  By contrast, an adelic measure is required to have ``trivial" contributions at all but finitely many places.  The measures determine a height function $\hat{h}_\mu$ on $\P^1(\ksep)$, given by the sum of the potential functions of the measures ${\mu_v}$.  From the height point of view, the height of a quasi-adelic measure allow nontrivial contributions from local heights for infinitely many places, which is prohibited for the heights of adelic measures. Precise definitions are given in Section \ref{definitions}. 

%Adelic measures on $\PP^1$ were introduced in \cite{BRbook, FRL:equidistribution},  where the authors proved independently the arithmetic equidistribution of points with small heights. The equidistribution results in \cite{BRbook, FRL:equidistribution} by Baker-Rumely and Favre-Rivera-Letelier apply on $\P^1$, while the treatments of more general varieties by Chambert-Loir and Yuan are given in \cite{CLoir, Yuan}. The arithmetic equidistribution theorem plays an important role in the recent study of dynamics and number theory, see \cite{BD:preperiodic, BD:polyPCF, DWY:Lattes, Ghioca:Hsia:Tucker, GHT:preprint, GKN}. In this article, we concentrate on the study of equidistribution on $\P^1$.  

The arithmetic equidistribution theorems play an important role in the recent study of complex dynamics and number theory; see, e.g. \cite{BD:preperiodic, DWY:Lattes, GHT:preprint, Favre:Gauthier, Ullmo:Bogomolov, Zhang:Bogomolov}. However, in attempts to apply the theory to more general families of rational functions on $\P^1$, we discovered that the ``adelic" hypothesis of the equidistribution theorems will not always be satisfied \cite{DWY:Per1}. The main result of this article, which is used to prove one of the main results of \cite{DWY:Per1}, extends the equidistribution theorems (and proofs) of  \cite{BRbook, FRL:equidistribution} to the quasi-adelic setting:
\begin{theorem} 
\label{arithmetic equidistribution}
Let $k$ be a product formula field and ${\mathbb \mu} = \{ \mu_v \}_{v \in M_k}$ be a quasi-adelic measure.  Suppose $S_n$ is a sequence of $\Gal(\ksep/k)$-invariant subsets of $\PP^1(\ksep)$ with $|S_n|\to \infty$ and $\hhat_{{\mathbb \mu}}(S_n) \to 0$,
as $n \to \infty$.  Then for each $v \in \cM_k$,
 the sequence of probability measures $[S_n]_v$, weighted equally on the points in $S_n$,  converges weakly to $\mu_v$ on $\PP^1_{\Berk,v}$ as $n \to \infty$.  \end{theorem}

%\noindent {\bf Outline of the article.}

%Elliptic curves are the most important abelian varieties. Let $E_t:=\{(x,y): y^2=x(x-1)(x-t)\}$ be the Legendre family of elliptic curves for $t\in \C\backslash\{0, 1\}$. Motivated by the recent work on unlikely intersections, Masser and Zannier \cite{Masser:Zannier:CR,Masser:Zannier, Masser:Zannier:2} proved the following deep result: there are only finitely many parameters $t\in \C$, for which the points $P_t=(a,\sqrt{a(a-1)(a-t)})$ and $Q_t=(b, \sqrt{b(b-1)(b-t)})$ are both torsion on $E_t$, with $a\neq b$ being constants in $\Qbar\backslash\{0,1\}$. The motivations of this result are provided at length in \cite{Masser:Zannier:CR,Masser:Zannier, Masser:Zannier:2,Zannier:book}. The articles \cite{BD:preperiodic, BD:polyPCF, Ghioca:Hsia:Tucker,  GKN, GNY:preprint} are somehow inspired by Masser and Zannier's result, and a key ingredient in these articles is the equidistribution of small points for adelic measures. More recently, we found a new proof in \cite{DWY:Lattes} for Masser and Zannier's result above, using dynamical and potential-theoretical methods. And a key ingredient in the new proof is the equidistribution theorem for adelic measures. Points on $E_t(\Qbar)$ have zero N\'eron-Tate height if and only if they are torsion.  The advantage of using the equidistribution theorem in the new proof is that we obtained stronger ({\em Bogomolov type}, compare \cite{Ullmo:Bogomolov,Zhang:Bogomolov}) result: there are only finitely many $t\in \Qbar\backslash\{0, 1\}$ such that both $P_t$ and $Q_t$ have small N\'eron-Tate height on the elliptic curve $E_t$. 

%The equidistribution theorem for adelic measures on $\P^1$ is very powerful in applications. However, when we studied the distribution of postcritically finite maps in the moduli space of quadratic rational maps, many measures arisen there are not adelic , see \cite{DWY:Per1}. In order to have applications in more general contents, we generalize adelic measure to quasi-adelic measure in this article, and show that the arithmetic equidistribution theorem holds for quasi-adelic measures too. We follow the language and ideas of \cite{Baker:Rumely:equidistribution,BRbook}. 


\noindent {\bf Acknowledgements.}
 We would like to thank Matthew Baker, Laura DeMarco, Dragos Ghioca and Xiaoguang Wang for many helpful comments and suggestions. 

 %For each adelic measure, there is a canonical height associated to it. The heights associated to adelic measures have good connection with varies well known height functions (depending on which context the measure arised), such as Neron-Tate height for elliptic curves, Call-Silverman canonical heights and Weil heights etc, see \cite{??}. 
 %\section{Setup and main results}
\section{Preliminaries and some properties}\label{definitions}
In this section, we introduce quasi-adelic measures, quasi-adelic sets, define canonical heights and show some properties. 
 \subsection{Preliminaries and basic notations}
A {\em product formula field} is a field $k$ together with 
\begin{itemize}
\item a set $\cM_k$ of pairwise inequivalent non-trivial absolute values on $k$ (we call them places). And a unique positive integer $N_v$ associated to each place $v\in \cM_k$. 
\item for each $\alpha \in k^*$, $|\alpha|_v=1$ for all but finitely many places $v\in \cM_k$, and $\alpha$ satisfies the following {\em product formula}: 
\begin{equation} \label{product formula}
\prod_{v \in \cM_k} |\alpha|_v^{N_v} \ = \ 1 \ .
\end{equation}
\end{itemize}
The most important product formula fields are number fields and function fields of curves. Let $\kbar$ be the algebraic closure of $k$, and $\ksep$ be the separable closure of $k$. If the characteristic of $k$ is zero, then it is well known that $\kbar=\ksep$. For each $v \in \cM_k$, let $k_v$ be the completion of $k$ at $v$, let $\kvbar$ be an algebraic closure of $k_v$, and let $\CC_v$ denote the completion of $\kvbar$.  We let $\PP^1_{\Berk,v}$ denote the Berkovich projective line over $\CC_v$, which is a canonically defined path-connected compact Hausdorff space containing $\PP^1(\CC_v)$ as a dense subspace. For each $v$, we fixed an embedding of $\kbar$ to $\C_v$.  If $v$ is archimedean, then $\CC_v \cong \CC$ and $\PP^1_{\Berk,v} = \PP^1(\CC)$.

For each $v \in \cM_k$ there is a distribution-valued Laplacian operator $\Delta$ on $\PP^1_{\Berk,v}$.  For example, the function
$\log^+|z|_v:= \max\{\log  |z|_v, 0\}$ on $\PP^1(\CC_v)$ extends naturally to a continuous real valued function $\PP^1_{\Berk,v} \backslash \{ \infty \} \to \RR$ and
\begin{equation}\label{lambdav potential}
\Delta \log^+|z|_v = \delta_{\infty} - \lambda_v,
\end{equation}
where $\lambda_v$ is the uniform probability measure on the complex unit circle $\{ |z| = 1 \}$ when $v$ is archimedean and
$\lambda_v$ is a point mass at the Gauss point of $\PP^1_{\Berk,v}$ when $v$ is non-archimedean.

A probability measure $\mu_v$ on $\PP^1_{\Berk,v}$ is said to have {\em continuous potentials} if $\mu_v - \lambda_v = \Delta g$
for some continuous function $g : \PP^1_{\Berk,v} \to \RR$.  If $\mu_v$ has continuous potentials,  then there is a unique function $g^o_{\mu_v}: \PP^1_{\Berk,v} \to \RR\cup\{+\infty\}$ such that 
    $$g(z)=\log^+|z|_v-g^o_{\mu_v}(z)$$
is a continuous potential for $\mu_v$, and the set 
\begin{equation}\label{Kv set}
M_{\mu_v}:=\{(x,y)\in \C_{v}^2 | G_{\mu_v}(x,y)\leq 0\}
\end{equation}
has capaticy $\capacity(M_{\mu_v})=1$; see \cite{D:lyap} and \cite[\S 3.3]{Baker:Rumely:equidistribution} for the definition of homogeneous capacity. Here $G_{\mu_v}(x,y)$ is a function uniquely determined by $g^o_{\mu_v}$ as follows
 \begin{equation}
G_{\mu_v}(x,y): =\begin{cases}
g^o_{\mu_v}(x/y)+\log|y|_v, \textup{ for $x,y\in \C_v$ and $y\neq 0
$}\\
\log|x|_v-g(\infty),\textup{ for $x\in \C_v$, $x\neq 0$ and $y=0$}\\
-\infty , \textup{ for $x=y=0$},
\end{cases}
\end{equation}
where $g(\infty)=\lim_{z\to \infty}(\log^+|z|_v-g^o_{\mu_v}(z))$. The function $G_{\mu_v}$ {\em scales logarithmically}, in the sense that 
   $$G_{\mu_v}(\alpha x,\alpha y)=G(x, y)+\log|\alpha|_v.$$
We call the continuous function $G_{\mu_v}$ on $\C_v^2\backslash \{(0, 0)\}$ {\em normalized homogeneous potential} for $\mu_v$. And homogeneous potentials of $\mu_v$ are all functions differ from $G_{\mu_v}$ by a constant.   For example, from (\ref{lambdav potential}), we know the normalized homogeneous potential for $\lambda_v$ is 
  $$G_{\lambda_v}(x,y)=\log \|(x,y)\|_v\textup{, and } M_{\lambda_v}=\bar{D}^2(0,1)\subset \C_v^2,$$
where $\|\cdot\|_v$ is the maximal norm, i.e. $\|(x,y)\|_v:=\max\{|x|_v, |y|_v\}.$
 
For each probability measure $\mu_v$ on $\PP^1_{\Berk, v}$ with continuous potentials, there is a unique {\em normalized Arakelov-Green function} $g_{\mu_v} : \PP^1_{\Berk,v} \times \PP^1_{\Berk,v} \to \RR \cup \{ +\infty \}$ which is characterized by the differential equation $\Delta_x g_{\mu_v}(x,y) = \delta_y - \mu_v$ and the normalization
\begin{equation} \label{normalization}
	\iint g_{\mu_v}(x,y) d\mu_v(x) d\mu_v(y) = 0.
\end{equation}
For points $(x,y)\in \C_v^2$, the normalized Arakelov-Green function $g_{\mu_v}$ is exactly given by: 
\begin{equation}  \label{explicit g}
g_{\mu_v}(x,y) = -\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}),
\end{equation}
where $\tilde{x}, \tilde{y}$ are any lifts of $x$ and $y$, and  $|\tilde{x}\wedge \tilde{y}|:=|x_1y_2-y_1x_2|$ for $\tilde{x}=(x_1,x_2)$ and $\tilde{y}=(y_1,y_2)$. Since $\PP^1(\C_v)$ is dense in $\PP^1_{\Berk, v}$, by continuity, it is easy to see that the function $g_{\mu_v}$ on $\PP^1_{\Berk,v} \times \PP^1_{\Berk,v}$ is uniquely determined by $G_{\mu_v}$.  The fact that the capacity of the set $M_{\mu_v}$ defined in (\ref{Kv set}) is one guarantees the function $g_{\mu_v}$ defined in (\ref{explicit g}) satisfies the integral formula (\ref{normalization}); see \cite{Baker:Rumely:equidistribution,BRbook} for more details. 
%&&&&&&&&&&&&&&
%&&&&&&&&&&&

\subsection{Quasi-adelic measure and canonical height function}
Let $\mu_v$ be a probability measure on $\PP^1_{\Berk, v}$ with continuous potentials. We define the {\em outer radius} and  {\em inner radius} for $\mu_v$ as
   $$r_{out}(\mu_v):=\inf \{r>0| M_{\mu_v} \subset \bar{D}(0, r)\times \bar{D}(0,r)\}$$
    $$r_{in}(\mu_v):=\sup \{r>0| \bar{D}(0, r)\times \bar{D}(0,r) \subset M_{\mu_v}\}.$$

A {\em quasi-adelic measure} on $\PP^1$ (with respect to a product formula field $k$) is a collection ${\mathbb \mu} = \{ \mu_v \}_{v \in M_k}$ of probability measures
on $\PP^1_{\Berk,v}$, one for each $v \in \cM_k$, such that
\begin{itemize}
\item  $\mu_v$ has continuous potentials for all $v \in \cM_k$; and 
\item $\prod_{v\in \cM_k} r_{in}(\mu_v)^{N_v}>0$ and  $\prod_{v\in \cM_k} r_{out}(\mu_v)^{N_v} <\infty$. 
\end{itemize}
\begin{remark} Since  $\capacity(M_{\mu_v})=1$ and $\capacity(\bar{D}^2(0,r))=r^2$, the radii $0< r_{in}(\mu_v)\leq 1\leq  r_{out}(\mu_v)$. The measure ${\mathbb \mu} = \{ \mu_v \}_{v \in M_k}$ is {\em adelic} if we replace the  second condition by $\mu_v=\lambda_v$ for all but finitely many $v\in \cM_k$, or equivalently $r_{in}(\mu_v)= r_{out}(\mu_v)=1$ for all but finitely many places $v$; see \cite{BRbook, FRL:equidistribution}. 
\end{remark}

If $\rho,\rho'$ are probability measures on $\PP^1_{\Berk,v}$, we define
$$( \rho, \rho' )_{\mu_v} := \frac{1}{2} \iint_{\PP^1_{\Berk,v} \times \PP^1_{\Berk,v} \backslash {\rm Diag}} g_{\mu_v}(x,y) d\rho(x) d\rho'(y).$$
and the {\em $\mu_v$-energy} of $\rho$ is defined as
$$I_{\mu_v}(\rho) := \frac{1}{2} \iint_{\PP^1_{\Berk,v} \times \PP^1_{\Berk,v}} g_{\mu_v}(x,y) d\rho(x) d\rho(y).$$

 Let $S\subset \PP^1(\ksep)$ be a finite set which is Gal$(\ksep/k)$-invariant and $|S|>1$. Here $|S|$ is the cardinality of the set $S$. We denote $[S]_v$ as the discrete probability measure on $\PP^1_{\Berk,v}$, supported
equally on all elements of $S$. The {\em canonical height} of $S$ associated to an adelic measure ${\mathbb \mu} = \{ \mu_v \}_{v \in M_k}$ is an number given by 
\begin{eqnarray}
\hhat_{\mu}(S) :&=& \frac{|S|}{|S|-1}\sum_{v\in \cM_k} N_v \cdot ([S]_v, [S]_v)_{\mu_v} \nonumber \\
	&=&  \frac{|S|}{|S|-1}  \sum_{v\in \cM_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 \cM_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)} \textup{, by (\ref{product formula})} \nonumber \\
	&=&  \frac{1}{2|S|(|S|-1)} \sum_{x,y\in S, x\not=y}  \; \sum_{v\in \cM_k} N_v \cdot (G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y}) ) \nonumber %\mbox{ by the product formula} 
			 \\
     &=&  \frac{1}{|S|}  \cdot \sum_{x\in S}\sum_{v\in \cM_k} N_v G_{\mu_v}(\tilde{x}),   \label{height definition}
\end{eqnarray}
where $\tilde{x}, \tilde{y}$ are fixed lifts of $x,y$ in ${\ksep}\times \ksep$. Hence, from (\ref{height definition}), for any $x\in \ksep$, let $S=\Gal(\ksep\backslash k)\cdot x$, we can defined the canonical height for $x$: 
   $$\hhat_{\mu}(x) :=\hhat_{\mu}(S).$$
The constants $N_v$ are the same as those appearing in the product formula (\ref{product formula}).
   
\begin{remark} The canonical height defined here is slightly different from the one appeared in \cite{BRbook, FRL:equidistribution}, but agrees with the one defined in \cite{DWY:Lattes, DWY:Per1}. The factor $\frac{|S|}{|S|-1}$ included here makes  the canonical height a natural extension to all points in $\PP^1(\ksep)$. 
\end{remark}


 %&&&&&&&&&&
 %&&&&&&&&&


\subsection{ Quasi-adelic set} Next, we introduce quasi-adelic sets. In many applications, it is much easier to deal with  quasi-adelic set other than quasi-adelic measure.

Analog to the definition of the {\em homogeneous filled Julia set} in \cite[\S 3.2]{Baker:Rumely:equidistribution}, we define  {\em homogeneous set with continuous potential} as: 
   $$M_v:=\{(x,y)\in \C_v^2| ~ G_{M_v}(x,y)\leq 0\},$$
with $G_{M_v}$ being a homogeneous potential for a probability measure (denote as $\mu_{M_v}$) with continuous potentials on $\P^1_{\Berk, v}$. 


When $v$ is an archimedean place (see \cite{D:lyap}), it is equivalent to say that  $M_v\subset \C_v^2\cong \C^2$ is a compact, circled and pseudoconvex set, or $G_{M_v}$ is a continuous and plurisubharmonic function satisfying
\begin{enumerate}
\item	 $G_{M_v}(\alpha z) =G_{M_v}(z) + \log|\alpha|_v$ for all $\alpha \in \C_v$, and
\item $G_{M_v}(z) = \log\|z\|_v + O(1)$.
\end{enumerate}

In reality, there are many homogeneous sets with continuous potential. For example, suppose we have a sequence of non-degenerate homogeneous polynomials $F_n: \C_v^2\to \C_v^2$, such that the sequence $\frac{\log \| F_n\|_v}{\deg (F_n)}$ converges uniformly to $G_v$ on $\C_v^2\backslash \{(0, 0)\}$. Then $G_v$ is a homogeneous potential for some probability measure with continuous potentials on $\P^1_{\Berk, v}$; see \cite[\S 3]{Baker:Rumely:equidistribution}. Hence $M_v=\{\tilde{x} \in \C_v^2: G_v(\tilde{x})\leq 0\}$ is a homogeneous set with continuous potential,  and the capacity is computed by the limit of the resultants (see \cite{DWY:Lattes}): 
   $$\capacity(M_v)=\lim_{n\to \infty} |\Res(F_n)|_v^{-\frac{1}{\deg(F_n)^2}}.$$


%$$M_v = \{z\in\C_v^2: G_{M_v}(z) \leq 0\}.$$



Analog to the definition of the radii of $\mu_v$, we define the {\em outer} and {\em inner} radii of $M_v$ as
   $$r_{out}(M_v):=\inf \{r>0| M_{v} \subset \bar{D}(0, r)\times \bar{D}(0,r)\}.$$
    $$r_{in}(M_v):=\inf \{r>0| \bar{D}(0, r)\times \bar{D}(0,r) \subset M_{v} \}.$$
A product $\prod_{v\in \cM_k}r_v^{N_v}$ {\em converges strongly} with $r_v> 0$ for each $v\in \cM_k$ if 
   $$\sum_{v\in \cM_k}N_v\cdot |\log r_v|<\infty.$$

A {\em quasi-adelic set}  (with respect to a product field $k$)  ${\mathbb M} = \{ M_v \}_{v \in M_k}$ is a collection of sets, one for each $v \in \cM_k$, such that 
\begin{itemize}
\item $M_v$ is a homogeneous set with continuous potential, and
\item the products $\prod_{v\in \cM_k} r_{out}(M_v))^{N_v}$ and $\prod_{v\in \cM_k} r_{in}(M_v)^{N_v}$ converge strongly.
\end{itemize} 


For each homogeneous set $M_v$ with continuous potential, there is a unique probability measure $\mu_{M_v}$ with continuous potentials associated to it. Hence, from a quasi-adelic set ${\mathbb M} = \{ M_v \}_{v \in M_k}$, we get a measure ${\mathbb \mu} = \{ \mu_{M_v} \}_{v \in M_k}$ on $\PP^1$. 

\begin{theorem}\label{quasi-adelic set}
Let $k$ be a product field and $\mathbb M=\{M_v\}_{v\in  \cM_k}$ be a collection of homogeneous sets with continuous potential, one for each $v\in \cM_k$.  One has
\begin{itemize}
\item If the set $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is quasi-adelic, then the corresponding measure ${\mathbb \mu} = \{ \mu_{M_v} \}_{v \in M_k}$ is quasi-adelic.  
\item Suppose that for each $v\in \cM_k$, there are radii $r'_v, r_v$ with $\bar{D}^2(0,r'_v)\subset M_v\subset \bar{D}^2(0,r_v)$
and $\prod_{v\in \cM_k}{r'_v}^{N_v}$, $\prod_{v\in \cM_k}{r_v}^{N_v}$ converging strongly. Then the set $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is quasi-adelic. Moreover, the product $\prod_{v\in \cM_k}\capacity(M_v)^{N_v}$ converges strongly, and for any $\Gal$($\ksep/k$)-invariant $S\subset \P^1(\ksep)$
  $$\hhat_{\mu}(S)= \frac{1}{|S|}  \cdot \sum_{x\in S}\sum_{v\in \cM_k} N_v G_{M_v}(\tilde{x})+\frac{1}{2}\log \prod_{v\in \cM_k}\capacity(M_v)^{N_v}.$$ 

\end{itemize}

\end{theorem}
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\subsection{Some properties}
Linear combination of quasi-adelic measures is a quasi-adelic measure. 
\begin{theorem} \label{linear height} Let $k$ be a product formula field. Suppose $\mathbb{\mu}_i=\{\mu_{v, i}\}_{v\in \cM_k}$ are distinct  quasi-adelic meausres, for $1\leq i\leq m$ with $m\geq 2$. Let $0< \lambda_i<1$ be non-negative numbers with  $\sum_{i=1}^m \lambda_i=1$. Then $\mu=\{\mu_{v}\}_{v\in \cM_k}:=\sum_{i=1}^m \lambda_i \mu_i$ is a quasi-adelic measure, and there is an $L>0$ depending on $\mu_i$ and $\lambda_i$ with 
    $$\sum_{i=1}^m\lambda_i\hhat_{\mu_i}= \hhat_\mu+L.$$
\end{theorem}
\proof Let $G_{\mu_{v,i}}$ be the normalized homogeneous potential for $\mu_{v, i}$. Then
 \begin{equation}\label{linear homogeneous}
 G_v:=\sum_{i=1}^m\lambda_iG_{\mu_{v,i}}
 \end{equation}
is a homogeneous potential function (may not be normalized) for the probability measure $\mu_v=\sum_{i=1}^m\lambda_i \mu_{v, i}$ on $\PP^1_{\Berk, v}$. Since $M_{\mu_{v,i}}$ has inner and outer radii $r_{in}(\mu_{v,i}), r_{out}(\mu_{v,i})$. We know that 
   $$M_v:=\{(x,y)\in\C_v^2| G_v(x,y)\leq 0\}$$
 has inner and outer radii satisfying 
   $$ r_{in}(M_v)\geq  \min \{r_{in}(\mu_{v, 1}), r_{in}(\mu_{v, 2}), \cdots, r_{in}(\mu_{v, m})\},$$
   $$ r_{out}(M_v)\leq  \max \{r_{out}(\mu_{v, 1}), r_{out}(\mu_{v, 2}), \cdots, r_{out}(\mu_{v, m})\}.$$
Because $\mu_i$ is quasi-adelic, the above inequalities indicates that $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is a quasi-adelic set. Hence by Theorem \ref{quasi-adelic set}, $\mu=\{\mu_{v}\}_{v\in \cM_k}$ is a quasi-adelic measure. 


Let $g_{\mu_{i,v}}$ be the normalized Arakelov-Green function for $\mu_{i,v}$, the component of $\mu_i$ for a place $v\in \cM_k$. 
By \cite[Theorem 5.8]{Baker:Rumely:equidistribution}, 
$$ \int \int g_{\mu_{i,v}}d\mu_v(x) d\mu_v(y)\geq 0$$
with equality if and only if $\mu_{i, v}=\mu_v$. Hence
   $$L_v:=\frac{1}{2}\int \int \sum_{i=1}^m \lambda_i g_{\mu_{i,v}}d\mu_v(x) d\mu_v(y)\geq 0$$
with equality if and only if  $\mu_{i, v}=\mu_v$ for $i=1, \cdots, m$. 
Then by the definition of Arakelov-Green function, 
   $$ g_{\mu_{v}}=\sum_{i=1}^m \lambda_i g_{\mu_{i,v}}-L_v  $$
is the normalized Arakelov-Green function for $\mu_v$. 
Consequently, one has 
   $$\sum_i\lambda_i\hhat_{\mu_i}= \hhat_\mu+L,$$
with $L=\sum_{v\in \cM_k} N_v\cdot L_v>0$ since $\mu_i$ are distinct.\qed

The canonical height function for a quasi-adelic measure is ``almost" non-negative, in the following sense
\begin{prop}\label{finiteness}
Let $k$ be a product formula field. Suppose $\mathbb{\mu}$ is a quasi-adelic meausre. Then for any $\delta>0$, there are only finitely many $x\in \ksep$ with 
   $$\hhat_{\mu}(x)<-\delta.$$
\end{prop}
\proof We prove it by contradiction. Assume that there are infinitely many $x\in \ksep$ such that $\hhat_{\mu}(x)<-\delta$. Let $S_n=\cup_{i=1}^n \Gal(\ksep/k)\cdot x_n$. Then $S_n$ is $\Gal(\ksep/k)$-invariant and $|S_n|\to \infty$. Moreover, by  (\ref{height definition}), $\hhat_{\mu}(S_n)\leq -\delta$. From the proof of Lemma \ref{energy limit}, it is not hard to see that for all $v\in \cM_k$,
   $$\lim_{n\to \infty} \inf ([S_n]_v, [S_n]_v)_{\mu_v}\geq 0$$
and 
   $$\lim_{n\to \infty} \sup ([S_n]_v, [S_n]_v)_{\mu_v}\leq -\delta.$$
So we get a contradiction. \qed  
\begin{remark} 
For quasi-adelic measures, the canonical heights don't  need to be always non-negative on $\PP^1(\ksep)$; see Example \ref{negative height example}. 
\end{remark}
 \medskip 
 
For any number field $k$, it is naturally equipped with a set of inequivalent norms $\cM_k$ and positive integers $\{N_v\}_{v\in \cM_k}$ making it a product formula field.  For any $x\in k$, the {\em logarithmic Weil height} of $x$ is
  $$h(x):=\frac{1}{[k:\Q]}\sum_{v\in \cM_k} N_v\log^+|x|_v.$$
The logarithmic Weil height is well defined over $\Qbar$, and doesn't depend on the embedding of a number field $k\to \Qbar$. A good property of a adelic measure is that its canonical height is bounded by the Weil height. For quasi-adelic measures over a number field $k$, we have a similar result: 
\begin{prop} \label{equivalent heights} Let $k$ be a number field. Suppose  $\mu=\{\mu_{v}\}_{v\in \cM_k}$ is a quasi-adelic meausre. Then the canonical height $\hhat_{\mu}$ is bounded by the logarithmic Weil height $h$ on $\PP^1(\Qbar)$, i.e. 
     $$\log \prod_{v\in \cM_k} r_{in}(\mu_v)^{N_v}\leq [k:\Q]h(x)-\hhat_\mathbb{\mu}(x)\leq \log \prod_{v\in \cM_k} r_{out}(\mu_v)^{N_v}$$ 
for all $x\in \kbar$. 
\end{prop}
\proof  Let $x_o$ be an element in $\kbar$ and its Galois orbits $S=\Gal(\kbar\backslash k)\cdot x_o$. For any lift $\tilde{x}\in \kbar^2$ of $x\in \kbar$, by the definition of outer and inner radii, 
   $$\log r_{in}(\mu_v) \leq\frac{1}{|S|}\sum_{x\in S} (\log\|\tilde{x}\|_v-G_{\mu_v}(\tilde{x}))\leq \log r_{out}(\mu_v)$$
Consequently, 
   $$\sum_{v\in \cM_k} N_v\log r_{in}(\mu_v) \leq\sum_{v\in \cM_k} \frac{N_v}{|S|}\sum_{x\in S} (\log\|\tilde{x}\|_v-G_{\mu_v}(\tilde{x}))\leq \sum_{v\in \cM_k} N_v\log r_{out}(\mu_v)$$
By (\ref{height definition}),
$$\log \prod_{v\in \cM_k} r_{in}(\mu_v)^{N_v}\leq [k: \Q]h(x_o)-\hhat_\mathbb{\mu}(x_o)\leq \log \prod_{v\in \cM_k} r_{out}(\mu_v)^{N_v}.$$ 
\qed


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\section{Proof of  main theorems}
In this section, we give proofs for Theorem \ref{arithmetic equidistribution} and  \ref{quasi-adelic set}. 
\subsection{Proof of Theorem \ref{arithmetic equidistribution}} By assumption,  $\{S_n\}_{n\geq 1}$ is a sequence of subsets of $\PP^1(\ksep)$, which are Gal$(\ksep/k)$-invariant and $|S_n|$ tends to infinity. For any $v\in \cM_k$, the $\mu_v$-energy of the probability measure $[S_n]_v$ on $\PP^1_{\Berk, v}$ 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} \left(-\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y})\right)
\end{align*}
with $\tilde{x}$ and $\tilde{y}$ being any lifts of $x$ and $y$. We have the following lemma: 

\begin{lemma}\label{energy limit}
Take $\{S_n\}_{n\geq 0}$ and $\mu=\{\mu_v\}_{v\in \cM_k}$ satisfying the same conditions as in Theorem \ref{arithmetic equidistribution}. For any place $v\in \cM_k$, one has
   $$\lim_{n\to \infty} ([S_n]_v, [S_n]_v)_{\mu_v}=0.$$
\end{lemma}
\proof First, it can be shown that for any $v\in \cM_k$, 
\begin{equation}\label{inf limit}
\lim_{n\to \infty}\inf  ([S_n]_v, [S_n]_v)_{\mu_v}\geq 0.
\end{equation} 
For this, we follow the proof of \cite[Lemma 3.24]{Baker:Rumely:equidistribution}. By the definition of capacity, 
\begin{equation}\label{22222}\lim_{n\to \infty} \inf \inf_{x_1, x_2, \cdots, x_n \in M_{\mu_v}}\frac{1}{n(n-1)}\sum_{i\neq j}-\log|\tilde{x_i}\wedge \tilde{x}_j|\geq -\log \capacity(M_{\mu_v})=0.
\end{equation}
   Fix an arbitrary small $\epsilon>0$. For any $x, y\in S_n$, since $\{|x|_v: x\in \C_v\}$ is dense in $\R^+$, we can take $\tilde{x}, \tilde{y} \in M_{\mu_v}$ as lifts  of $x$ and $y$, such that 
    $$-\epsilon<G_{\mu_v}(\tilde{x})\leq 0,  ~-\epsilon<G_{\mu_v}(\tilde{y})\leq 0.$$
Hence by the definition of energy function and (\ref{22222})
     \begin{align*}\lim_{n\to \infty}\inf ([S_n]_v, [S_n]_v)_{\mu_v} &=\lim_{n\to \infty}\inf\frac{1}{2|S_n|   ^2}\sum_{x\neq y\in S_n} \left(-\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y})\right)\\
     &\geq -\epsilon+\lim_{n\to \infty}\inf(-\frac{1}{2|S_n|   ^2}\sum_{x\neq y\in S_n} \log|\tilde{x} \wedge \tilde{y}|_v) \geq -\epsilon. 
\end{align*}

Second, we still need to show that for all $v\in \cM_k$, one has
   $$\lim_{n\to \infty}\sup ([S_n]_v, [S_n]_v)_{\mu_v}\leq 0.$$ 
 
Assume that this lemma fails, then there exist an $\epsilon>0$, $v_o\in \cM_k$ and a sequence of strictly increasing integers $\{n_j\}_{j\geq 1}$, such that 
\begin{equation}\label{big limit} N_{v_o}\cdot ([S_{n_j}]_{v_o}, [S_{n_j}]_{v_o})_{\mu_{v_o}}>3\epsilon, \textup{ for all $j\geq 1$}.
\end{equation}

For any $v\in \cM_k$ and any $\delta>0$, we lift $x,y\in S_n$ to $\tilde{x}, \tilde{y}\in M_{\mu_v}$ such that 
      $$G_{\mu_v}(\tilde{x})\geq -\delta, ~ G_{\mu_v}(\tilde{y})\geq -\delta.$$ 
Since we know that $M_{\mu_v}$ is bounded by the polydisc with outer radius $r_{out}(\mu_v)$, then $\|\tilde{x}\|_v, \|\tilde{y}\|_v\leq r_{out}(\mu_v)$ and hence
  $$\log|\tilde{x} \wedge \tilde{y}|_v \leq \log r_{out}(\mu_v)^2$$
  when $v$ is non-archimedean. 
For non-archimedean place $v\in \cM_k$, one has 
 \begin{align}([S_n]_v, [S_n]_v)_{\mu_v}&=\frac{1}{2|S_n|   ^2}\sum_{x\neq y\in S_n} \left(-\log|\tilde{x} \wedge \tilde{y}|_v + G_{\mu_v}(\tilde{x}) + G_{\mu_v}(\tilde{y})\right)\\
      &\geq \frac{1-|S_n|}{|S_n|}\delta- \frac{1}{| S_n|   ^2}\sum_{x\neq y\in S_n} \log r_{out}(\mu_v)=\frac{1-|S_n|}{|S_n|}\left(\delta+\log r_{out}(\mu_v)\right)\end{align}
We shrink $\delta$ to zero, then the above inequality becomes 
\begin{equation}\label{1111}([S_n]_v, [S_n]_v)_{\mu_v}\geq \frac{1-|S_n|}{|S_n|}\log r_{out}(\mu_v)
\end{equation}
as long as $v$ is non-archimedean.

It is well known that for product formula field $k$, there are only finite many archimedean places; see \cite[Chapter 12, Theorem 3]{Ar}. Since $\mu=\{\mu_v\}_{v\in \cM_k}$ is quasi-adelic, the product $\prod_{v\in \cM_k} r_{out}(\mu_v)^{N_v}$ converges. Hence we can choose a set $\cM_k'\subset \cM_k$ such that: 
\begin{itemize}
\item $\cM_k'':=\cM_k\backslash( \cM_k'\cup \{v_o\})$ only has finitely many places.
\item all places in $\cM_k'$ are non-archimedean and $v_o\not \in \cM_k'$.  
\item $\sum_{v\in \cM_k'}N_v\cdot \log r_{out}(\mu_v)\leq \epsilon$. 
\end{itemize}
Hence, from the definition of the canonical height function $\hat h_\mu$, we have
 \begin{align*} N_{v_o}([S_{n_j}]_{v_o}, [S_{n_j}]_{v_o})_{\mu_{v_o}}&=\hat h_{\mu}(S_{n_j})-\sum_{v\in \cM_k\backslash \{v_o\}} N_v \cdot([S_{n_j}]_v, [S_{n_j}]_v)_{\mu_v}\\
  &=\hat h_{\mu}(S_{n_j})-\sum_{v\in \cM_k''} N_v ([S_{n_j}]_v, [S_{n_j}]_v)_{\mu_v}-\sum_{v\in \cM_k'} N_v  ([S_{n_j}]_v, [S_{n_j}]_v)_{\mu_v}\\
  &\leq \hat h_{\mu}(S_{n_j})-\sum_{v\in \cM_k''} N_v \cdot ([S_{n_j}]_v, [S_{n_j}]_v)_{\mu_v}+\frac{(| S_{n_j}|-1) \epsilon}{| S_{n_j}|}\textup{, by (\ref{1111})}\\
  &\leq \hat h_{\mu}(S_{n_j})-\sum_{v\in \cM_k''} N_v \cdot ([S_{n_j}]_v, [S_{n_j}]_v)_{\mu_v}+ \epsilon.
\end{align*}
Since the set $\cM_k''$ has only finitely many places and the height of $S_{n_j}$ tends to zero,  by (\ref{inf limit}), the inferior limit of the above inequality
  $$\lim_{j\to\infty} \sup N_{v_o}\cdot ([S_{n_j}]_{v_o}, [S_{n_j}]_{v_o})_{\mu_{v_o}}\leq \lim_{j\to \infty}\sup \left(-\sum_{v\in \cM_k''} N_v \cdot ([S_{n_j}]_v, [S_{n_j}]_v)_{\mu_v}+ \epsilon\right)\leq \epsilon, $$
which contradicts to our assumption (\ref{big limit}). We finish the proof of this lemma. \qed

The set of probability measures on $\PP^1_{\Berk,v}$ is compact in the weak topology. Hence, to show $[S_n]_v$ converges weakly to $\mu_v$ as $n\to \infty$, it suffices to show that any convergent subsequence of  $[S_n]_v$ converges to $\mu_v$. Without loss of generality, we assume that $[S_n]_v$ converges to some $\nu_v$, i.e. 
   $$\lim_{n\to \infty}[S_n]_v=\nu_v.$$
By Lemma \ref{energy limit}, for the $\mu_v$-energy $I_{\mu_v}(\nu_v)$ of $\nu_v$,
\begin{align*} 0&=\lim_{n\to \infty}([S_{n}]_v, [S_{n}]_v)_{\mu_v}=\lim_{n\to\infty} \frac{1}{2} \iint_{\PP^1_{\Berk,v} \times \PP^1_{\Berk,v} \backslash {\rm Diag}} g_{\mu_v}(x,y) d[S_n]_v(x)d[S_n]_v(y)\\
    &\geq \frac{1}{2} \iint_{\PP^1_{\Berk,v} \times \PP^1_{\Berk,v}} g_{\mu_v}(x,y)  d\nu_v(x)d\nu_v(y) \textup{, by \cite[Lemma 3.46]{Baker:Rumely:equidistribution}}\\
    &=I_{\mu_v}(\nu_v).
\end{align*}

By \cite[Theorem 3.45]{Baker:Rumely:equidistribution}, $\mu_v$ is the unique probability measure on $\PP^1_{\Berk, v}$ minimizing the  $\mu_v$-energy function $I_{\mu_v}(\cdot)$. Then from above discussion, $I_{\mu_v}(\mu_v)=0\geq I_{\mu_v}(\nu_v)$ guarantees $I_{\mu_v}(\mu_v)=I_{\mu_v}(\nu_v)$. Hence $\nu_v=\mu_v$. We finish the proof of Theorem \ref{arithmetic equidistribution}. 
 
\subsection{Proof of Theorem \ref{quasi-adelic set}} We show first that $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is quasi-adelic implies that ${\mathbb \mu} = \{ \mu_{M_v} \}_{v \in M_k}$ is quasi-adelic.   We assume that $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is a quasi-adelic set.  For any $r>0$, let
   $$rM_v:=\{(\alpha x, \alpha y)| (x,y)\in M_v, \alpha \in \C_v \textup{ with  } |\alpha|_v\leq r\}.$$
For the definition of capacity, it is clear $\capacity(rM_v)=r^2\capacity(M_v)$. 
Since $G_{M_v}$ is a homogeneous potential for $\mu_{M_v}$, the normalized homogeneous potential is 
\begin{equation}\label{potential relation}
G_{\mu_{M_v}}(x,y)=G_{M_v}(x,y)+\frac{1}{2}\log \capacity(M_v),
\end{equation}
and $M_{\mu_{M_v}}=\frac{1}{\sqrt{\capacity(M_v)}}M_v$. As a consequence, 
\begin{equation}\label{radii capacity}r_{in}(\mu_{M_v})=\frac{r_{in}(M_v)}{\sqrt{\capacity(M_v)}}, ~ r_{out}(\mu_{M_v})=\frac{r_{out}(M_v)}{\sqrt{\capacity(M_v)}}
\end{equation}
Moreover, as $\capacity(\bar{D}^2(0, r))=r^2$ for polydiscs, one has 
\begin{equation}\label{capacity bound}
 r_{in}(M_v)\leq \sqrt{\capacity(M_v)}\leq r_{out}(M_v).
\end{equation}
Then by (\ref{radii capacity}), 
     $$\frac{r_{in}(M_v)}{r_{out}(M_v)}\leq r_{in}(\mu_{M_v})\leq 1\leq  r_{out}(\mu_{M_v})\leq \frac{r_{out}(M_v)}{r_{in}(M_v)}$$
As $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is quasi-adelic, the products of inner and outer radii converge strongly. Then the above inequalities implies the products of inner and outer radii of $\mu=\{\mu_{M_v}\}_{v\in \cM_k}$ converge, i.e.  $\mu=\{\mu_{M_v}\}_{v\in \cM_k}$ is quasi-adelic. 

Second, assume that $\bar{D}^2(0,r'_v)\subset M_v\subset \bar{D}^2(0,r_v)$
and $\prod_{v\in \cM_k}{r'_v}^{N_v}$, $\prod_{v\in \cM_k}{r_v}^{N_v}$ converge strongly. Then the products $\prod_{v\in \cM_k} r_{out}(M_v)^{N_v}$ and $\prod_{v\in \cM_k} r_{in}(M_v)^{N_v}$ converge strongly, since
  $$r'_v\leq r_{in}(M_v)\leq r_{out}(M_v)\leq r_v.$$
Hence $\mathbb M=\{M_v\}_{v\in  \cM_k}$ is quasi-adelic by definition. Moreover, by (\ref{capacity bound}), the product of the capacities converges strongly, and then  the last formula for the canonical height is clear by (\ref{height definition}) and (\ref{potential relation}). We finish the proof of Theorem \ref{quasi-adelic set}. 


\section{Applications}
In this section, we briefly describe an application in \cite{DWY:Per1}, and give some examples of quasi-adelic measures. 

An interesting question in dynamics is to classify algebraic subvarities of ${\rm M}_d$ (the moduli space of degree $d$ rational maps on $\PP^1(\C)$) with Zariski-dense postcritially finite maps (PCF for short); see \cite[Conjecture 1.4]{BD:polyPCF} and \cite[\S 6.5]{Silverman:moduli}. A rational map is PCF if all its critical points are preperiodic under iteration.  Milnor \cite{Milnor:quad} introduced lines $\Per_1(\lambda)\subset {\rm M}_2\simeq \C^2$, with $\Per_1(\lambda)$ being the set of conjugacy classes $[f]$ such that $f'(p)=\lambda$ for some fixed point $p$ of $f$. In  \cite[Theorem 1.1]{DWY:Per1}, we addressed a special case of Conjecture 1.4 in \cite{BD:polyPCF}, by showing that PCF maps are Zariski-dense in $\Per_1(\lambda)$ if and only if $\lambda=0$. A key ingredient is the arithmetic equidistribution theorem for quasi-adelic measures. 

Fix $0\neq \lambda\in \Qbar$, the family $\Per_1^{cm}(\lambda)$ parametrized by $t$ 
    $$f_t(z):=\frac{\lambda z}{z^2+tz+1}$$
is a double cover of $\Per_1(\lambda)\subset {\rm M}_2$, with marked critical points $\pm 1$. We lift $f_t$ to homogeneous function on $\C^2$ with homogeneous parameters $(t_1, t_2)$
   $$F_{t_1, t_2}(z_1, z_2):=(\lambda t_2 z_1^2, t_2 z_1^2+t_1 z_1 z_2+ t_2z_2^2).$$
Iterate the homogeneous lifts $(\pm 1, 1)$ of the critical points $\pm 1$, and get rid of common factors, we get
  $$F^{\pm}_n(t_1, t_2)=F^n_{t_1, t_2}(\pm 1, 1)/{\rm gcd} F^n_{t_1, t_2}(\pm 1, 1).$$
In the rest of this article, we fix $\lambda\in \Qbar$ with $\lambda\neq 0$ and not being a root of unity. Let $k=\Q(\lambda)$.  Then $F^\pm_n(t_1, t_2)$ is a non-degenerated homogeneous function over $\C_v^2$ for each $v\in \cM_k$ with degree $\deg F_n^\pm$, i.e. 
   $$F^{\pm}_n(t_1, t_2): \C_v^2\to \C_v^2.$$
In \cite[Theorem 5.1]{DWY:Per1}, it was proved that $\frac{\log^+\|F_n^\pm(t_1, t_2)\|_v}{\deg F_n^\pm}$ converges uniformly to a function $G_v^\pm$ on $\C_v^2\backslash \{(0,0)\}$, for all $v\in \cM_k$, i.e.
   $$G_v^\pm(t_1, t_2):=\lim_{n\to \infty} \frac{\log^+\|F_n^\pm(t_1, t_2)\|_v}{\deg F_n^\pm}.$$
Hence $G_v^\pm$ are homogeneous continuous potentials for some probability measures (denoted as $\mu_v^\pm$) on $\PP^1_{\Berk, v}$. Moreover, it was shown (in Proposition 6.3 and Section 7.2 of \cite{DWY:Per1}) that there are $0<r_v'\leq r_v$ such that $\bar{D}^2(0,r'_v)\subset M_v^\pm \subset \bar{D}^2(0,r_v)$, and $\prod_{v\in \cM_k}{r'_v}^{N_v}$, $\prod_{v\in \cM_k}{r_v}^{N_v}$ converge strongly, where $M_v^\pm$ are sets defined as 
   $$M^\pm_v:=\{(t_1, t_2)\in \C_v^2| G_v^\pm(t_1,t_2)\leq 0\}.$$
By Theorem \ref{quasi-adelic set}, $\mathbb M^\pm:=\{M_v^\pm\}_{v\in\cM_k}$ are quasi-adelic sets, and then 
   $$\mu^\pm:=\{\mu_v^\pm\}_{v\in \cM_k}$$
are quasi-adelic measures. Hence the equidistribution theorem holds for measures $\mu^\pm=\{\mu_v^\pm\}_{v\in \cM_k}$. The measure $\mu^\pm:=\{\mu_v^\pm\}_{v\in \cM_k}$  is adelic if and only if $\mu_v^\pm=\lambda_v$ for all but finitely many $v\in \cM_k$, or equivalently $M_v^\pm=\bar{D}^2(0, \sqrt{\capacity(M_v^\pm)})$ for all but finitely many $v\in \cM_k$. Next, we show that $\mu^\pm$ are not adelic measures.

\begin{prop} For any $\lambda\in \Qbar\backslash\{0\}$ and $\lambda$ is not a root of unity, both $\mu^\pm$ are quasi-adelic but not adelic. Actually, for all but finitely many places $v\in\cM_k$, $\mu^\pm_v\neq \lambda_v$, where $k=\Q(\lambda)$. 
\end{prop}
\proof It  suffices to show that for any non-archimedean place $v\in \cM_k$ with $|\lambda|_v=1$ and $|1-\lambda|_v=1$, we have $\mu^\pm_v\neq \lambda_v$, i.e. $\mu^\pm_v$ is not the point mass measure supported on the Gauss point in $\PP^1_{\Berk,v}$. 

Let $\mathcal{O}:=\{ x\in k:~ |x|_v\leq 1\}$ be the valuation ring of $k$ and $\mathfrak{m}\subset \mathcal{O}$ be its unique maximal ideal which is given by $\mathfrak{m}:=\{x\in \mathcal{O}:~ |x|_v<1\}$. The residue field $\mathcal{O}/\mathfrak{m}$ is of finite order $m:=|\mathcal{O}/\mathfrak{m}|<\infty$. Since $|\lambda|_v=1$,  $\lambda\not \equiv 0 \textup{ mod } \mathfrak{m}$. Hence $\lambda^{m-1}\equiv 1\textup{ mod } \mathfrak{m}$, i.e. $1-\lambda^{m-1}\in \mathfrak{m}$ or equivalently $|1-\lambda^{m-1}|_v<1$. And because $|1-\lambda|_v=1$ by assumption, the integer $m\geq 3$.  Now we can show that  $M_v^\pm\neq \bar{D}^2(0, \sqrt{\capacity(M_v^\pm)})$, which is the same as $\mu^\pm_v\neq \lambda_v$. By \cite[Theorem 5.1]{DWY:Per1}, $G_v^\pm(1,0)=\gamma_v(\lambda)$, where 
   $$\gamma_v(\lambda)=\frac{1}{2}\sum_{i=1}^\infty\frac{1}{2^i}\log|1+\lambda+\cdots+\lambda^i|_v<0$$
since $|1+\lambda+\cdots+\lambda^i|_v\leq 1$ for all $i\geq 1$ and $|1+\lambda+\cdots+\lambda^{m-2}|_v=\frac{|1-\lambda^{m-1}|_v}{|1-\lambda|_v}<1$. Hence $G_v^\pm(1,0)<0$. Also, from the proof of \cite[Proposition 6.3]{DWY:Per1}, we know that $G_v^\pm(0,1)=0$. Then from the definition of $M_v^\pm$, $M_v^\pm\neq \bar{D}^2(0, \sqrt{\capacity(M_v^\pm)})$. Otherwise we must have $G_v(x,y)=\log \|(x,y)\|_v+\frac{1}{2}\log \capacity(M_v^\pm)$. Hence $G_v^\pm(0,1)=G_v^\pm(1,0)$, which is a contradiction.\qed

%\begin{remark} For $\lambda$ being a root of unity, it (a little more delicate) can be shown that the measures associate to the critical points $\pm1$ are quasi-adelic but not adelic. 
%\end{remark}
\medskip

We define the {\em Call-Silverman} canonical heights $\hhat^\pm$ on the parameter space $t\in\Per_1^{cm}(\lambda)(\Qbar)$ as 
   $$\hhat^{\pm}(t):=\hhat_{f_t}(\pm 1):=\lim_{n\to \infty}\frac{1}{2^n}h(f_t^n(\pm 1)),$$
where $h$ is the logarithmic Weil height and $\hhat_{f_t}$ is the canonical height of the morphism $f_t$. By \cite[Corollary 1.1.1]{Silverman:moduli}, $\hhat^{\pm}(t)\geq 0$ on $\PP^1(\Qbar)$ with equality if and only if the critical point $\pm 1$ is preperiodic for the morphism $f_t$. In  \cite[Proposition 7.6]{DWY:Per1}, it was shown that 
    $$\hhat_{\mu^\pm}(t) =2[k:\Q]\cdot \hhat^\pm(t) \textup{  with $k=\Q(\lambda).$}$$
Then by \cite[Lemma 2.1]{DWY:Per1}, there are infinitely many $t\in \kbar$ with $\hhat_{\mu^+}(t)=0$ (or $\hhat_{\mu^-}(t)=0$). 

Let $\mu$ be the average of $\mu^\pm$, i.e. 
  $$\mu=\{\mu_v\}_{v\in \cM_k}:=\frac{1}{2}\mu^++\frac{1}{2}\mu^-=\{\frac{1}{2}\mu^+_v+\frac{1}{2}\mu^-_v\}_{v\in \cM_k}$$
By Theorem \ref{linear height}, $\mu=\{\mu_v\}_{v\in \cM_k}$ is quasi-adelic, and 
   $$\hhat_\mu(t)=\frac{1}{2}\hhat_{\mu^+}(t)+\frac{1}{2}\hhat_{\mu^-}(t)-L$$
for a constant $L>0$ since $\mu_v^+\neq \mu_v^-$ when $v$ is archimedean; see \cite[Theorem 1.2]{DWY:Per1}. The function $f_t$ is PCF if and only if $\hhat_{\mu^+}(t)=\hhat_{\mu^-}(t)=0$, or equivalently if and only if 
   $$\hhat_\mu(t)=-L.$$
But there are only finitely many such $t$ by Proposition \ref{finiteness}. 
    


\begin{example}\label{negative height example} Let $\lambda=2$. When $t=0$, both the critical points $\pm 1$ are fixed by $f_0(z)=2z/(z^2+1)$. Hence $\hhat_{\mu^+}(0)=\hhat_{\mu^-}(0)=0$. As a consequence,
   $$\hhat_\mu(0)=\frac{1}{2}\hhat_{\mu^+}(0)+\frac{1}{2}\hhat_{\mu^-}(0)-L=-L<0$$
is negative. So canonical heights associated to quasi-adelic measures can be negative on some points in $\PP^1(\ksep)$.  \end{example}

\begin{remark} When $v\in \cM_k$ is archimedean, the measure $\mu_v$, as a component of $\mu=\{\mu_v\}_{v\in \cM_k}$, is the normalized (total mass $1$) bifurcation measure on $\Per_1^{cm}(\lambda)$, which is a well known object in complex dynamics. And the support of $\mu_v$ is the set of parameters $t$, roughly speaking, where the dynamics of $f_t$ is not stable for small perturbation of $t$. See \cite{D:current,D:lyap, Dujardin:Favre:critical} for details. 
\end{remark}



%\noindent {\bf Acknowledgements.}










%\bibliographystyle{../tex/bib/math}
%\bibliography{../tex/bib/math}
\def\cprime{$'$}
\begin{thebibliography}{GHT2}


\bibitem[Ar]{Ar}
E. ~Artin.
\newblock{\em Algebraic numbers and algebraic functions}. 
\newblock AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the 1967 original. 


\bibitem[BD1]{BD:preperiodic}
M.~Baker and L.~DeMarco.
\newblock {Preperiodic points and unlikely intersections}.
\newblock {\em Duke Math. J.} {\bf 159}(2011), 1--29.

\bibitem[BD2]{BD:polyPCF}
M.~Baker and L.~DeMarco.
\newblock {Special curves and postcritically-finite polynomials}.
\newblock {\em Forum of Math. Pi} {\bf 1}(2013), 35 pp.

\bibitem[BR1]{Baker:Rumely:equidistribution}
M.~Baker and R.~Rumely.
\newblock {Equidistribution of small points, rational dynamics, and potential
  theory}.
\newblock {\em Ann. Inst. Fourier (Grenoble)} {\bf 56}(2006), 625--688.


\bibitem[BR2]{BRbook}
M.~Baker and R.~Rumely.
\newblock {\em Potential theory and dynamics on the {B}erkovich projective
  line}, volume 159 of {\em Mathematical Surveys and Monographs}.
\newblock American Mathematical Society, Providence, RI, 2010.

\bibitem[CL]{CLoir}
A. ~Chambert-Loir.
\newblock{Mesures et\'equidistribution sur les espaces de Berkovich}.
\newblock{\em J. Reine Angew. Math.} {\bf 595} (2006), 215--235.

\bibitem[De1]{D:current}
L.~DeMarco.
\newblock {Dynamics of rational maps: a current on the bifurcation locus}.
\newblock {\em Math. Res. Lett.} {\bf 8}(2001), 57--66.


\bibitem[De2]{D:lyap}
L.~DeMarco.
\newblock {Dynamics of rational maps: {L}yapunov exponents, bifurcations, and
  capacity}.
\newblock {\em Math. Ann.} {\bf 326}(2003), 43--73.

\bibitem[DF]{Dujardin:Favre:critical}
R.~Dujardin and C.~Favre.
\newblock {Distribution of rational maps with a preperiodic critical point}.
\newblock {\em Amer. J. Math.} {\bf 130}(2008), 979--1032.

\bibitem[DWY1]{DWY:Lattes}
L.~DeMarco, X.~Wang, and H.~Ye.
\newblock {Torsion points and the Latt\`es family}.
\newblock To appear, {\em Amer. J. Math.}.

\bibitem[DWY2]{DWY:Per1}
L.~DeMarco, X.~Wang, and H.~Ye.
\newblock {Bifurcation measures and quadratic rational maps}.
\newblock {\em Preprint, {\em 2014}}.

\bibitem[FG]{Favre:Gauthier}
C.~Favre and T.~Gauthier.
\newblock {Distribution of postcritically finite polynomials}.
\newblock {\em Preprint, {\em 2013}}.

\bibitem[FRL]{FRL:equidistribution}
C.~Favre and J.~Rivera-Letelier.
\newblock {\'{E}quidistribution quantitative des points de petite hauteur sur
  la droite projective}.
\newblock {\em Math. Ann.} {\bf 335}(2006), 311--361.


%\bibitem[GHT1]{Ghioca:Hsia:Tucker}
%D.~Ghioca, L.-C.~Hsia, and T.~Tucker.
%\newblock {Preperiodic points for families of polynomials}.
%\newblock {\em {\em To appear}, Algebra and Number Theory}.

\bibitem[GHT]{GHT:preprint}
D.~Ghioca, L.-C. Hsia, and T.~Tucker.
\newblock {Preperiodic points for families of rational maps}.
\newblock {\em {\em To appear}, Proceedings of the London Mathematical Society}.

%\bibitem[GKN]{GKN}
%D.~Ghioca, H.~Krieger, and K.~Nguyen.
%\newblock{\em A case of the dynamical Andr\'e-Oort Conjecture}
%\newblock{\em Preprint}, {\em 2014}. 

%\bibitem[GNY]{GNY:preprint}
%D.~Ghioca, K. Nguyen, and H.~Ye.
%\newblock {The dynamical Andr\'e-Oort conjecture for rational curves in the moduli space of uncritical polynomials of degree $d$}.
%\newblock {\em Preprint, {\em 2014}}.

\bibitem[Mi1]{Milnor:quad}
J.~Milnor.
\newblock {Geometry and dynamics of quadratic rational maps}.
\newblock {\em Experiment. Math.} {\bf 2}(1993), 37--83.
\newblock With an appendix by the author and Lei Tan.

%\bibitem[MZ1]{Masser:Zannier:CR}
%D.~Masser and U.~Zannier.
%\newblock {Torsion anomalous points and families of elliptic curves}.
%\newblock {\em C. R. Math. Acad. Sci. Paris} {\bf 346}(2008), 491--494.

%\bibitem[MZ2]{Masser:Zannier}
%D.~Masser and U.~Zannier.
%\newblock {Torsion anomalous points and families of elliptic curves}.
%\newblock {\em Amer. J. Math.} {\bf 132}(2010), 1677--1691.

%\bibitem[MZ3]{Masser:Zannier:2}
%D.~Masser and U.~Zannier.
%\newblock {Torsion points on families of squares of elliptic curves}.
%\newblock {\em Math. Ann.} {\bf 352}(2012), 453--484.

\bibitem[Si]{Silverman:moduli}
J.~H.~Silverman.
\newblock {\em Moduli spaces and arithmetic dynamics}, volume~30 of {\em CRM
  Monograph Series}.
\newblock American Mathematical Society, Providence, RI, 2012.

\bibitem[Ull]{Ullmo:Bogomolov}
E.~Ullmo.
\newblock {Positivit\'e et discr\'etion des points alg\'ebriques des courbes}.
\newblock {\em Ann. of Math. (2)}, {\bf 147}(1998), 167---179.

\bibitem[Yu]{Yuan}
X. ~Yuan.
\newblock{Big line bundles over arithmetic varieties}.
\newblock{\em Invent. Math.} {\bf 173} (2008), no. 3, 603--649.

%\bibitem[Za]{Zannier:book}
%U.~Zannier.
%\newblock {\em Some problems of unlikely intersections in arithmetic and
%  geometry}, volume 181 of {\em Annals of Mathematics Studies}.
%\newblock Princeton University Press, Princeton, NJ, 2012.
%\newblock With appendixes by David Masser.

\bibitem[Zh1]{Zhang:line}
S. ~Zhang.
\newblock{Positive line bundles on arithmetic varieties}.
\newblock{\em J. Am. Math. Soc.} {\bf 8}(1995), 187--221. 

\bibitem[Zh2]{Zhang:metrics}
S. ~Zhang.
\newblock{Small points and adelic metrics}.
\newblock{\em J. Algebr. Geom.} {\bf 4}(1995), 281--300. 

\bibitem[Zh3]{Zhang:Bogomolov}
S. ~Zhang.
\newblock {Equidistribution of small points on abelian varieties}.
\newblock {\em Ann. of Math. (2)}, {\bf 147}(1998), 159--165.

\end{thebibliography}















\end{document}



1502.04660.extracted_bib
[Ar]{Ar E. ~Artin. \newblock{\em Algebraic numbers and algebraic functions. \newblock AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the 1967 original.
1502.04660.extracted_bib
[BD1]{BD:preperiodic M.~Baker and L.~DeMarco. \newblock {Preperiodic points and unlikely intersections. \newblock {\em Duke Math. J. {159(2011), 1--29.
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[BD2]{BD:polyPCF M.~Baker and L.~DeMarco. \newblock {Special curves and postcritically-finite polynomials. \newblock {\em Forum of Math. Pi {1(2013), 35 pp.
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[BR1]{Baker:Rumely:equidistribution M.~Baker and R.~Rumely. \newblock {Equidistribution of small points, rational dynamics, and potential theory. \newblock {\em Ann. Inst. Fourier (Grenoble) {56(2006), 625--688.
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[BR2]{BRbook M.~Baker and R.~Rumely. \newblock {\em Potential theory and dynamics on the {Berkovich projective line, volume 159 of {\em Mathematical Surveys and Monographs. \newblock American Mathematical Society, Providence, RI, 2010.
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[CL]{CLoir A. ~Chambert-Loir. \newblock{Mesures et\'equidistribution sur les espaces de Berkovich. \newblock{\em J. Reine Angew. Math. {595 (2006), 215--235.
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[De2]{D:lyap L.~DeMarco. \newblock {Dynamics of rational maps: {Lyapunov exponents, bifurcations, and capacity. \newblock {\em Math. Ann. {326(2003), 43--73.
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[DF]{Dujardin:Favre:critical R.~Dujardin and C.~Favre. \newblock {Distribution of rational maps with a preperiodic critical point. \newblock {\em Amer. J. Math. {130(2008), 979--1032.
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[DWY1]{DWY:Lattes L.~DeMarco, X.~Wang, and H.~Ye. \newblock {Torsion points and the Latt\`es family. \newblock To appear, {\em Amer. J. Math..
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[DWY2]{DWY:Per1 L.~DeMarco, X.~Wang, and H.~Ye. \newblock {Bifurcation measures and quadratic rational maps. \newblock {\em Preprint, {\em 2014.
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[FG]{Favre:Gauthier C.~Favre and T.~Gauthier. \newblock {Distribution of postcritically finite polynomials. \newblock {\em Preprint, {\em 2013.
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[FRL]{FRL:equidistribution C.~Favre and J.~Rivera-Letelier. \newblock {\'{Equidistribution quantitative des points de petite hauteur sur la droite projective. \newblock {\em Math. Ann. {335(2006), 311--361. %
1502.04660.extracted_bib
[GHT1]{Ghioca:Hsia:Tucker %D.~Ghioca, L.-C.~Hsia, and T.~Tucker. %\newblock {Preperiodic points for families of polynomials. %\newblock {\em {\em To appear, Algebra and Number Theory.
1502.04660.extracted_bib
[GHT]{GHT:preprint D.~Ghioca, L.-C. Hsia, and T.~Tucker. \newblock {Preperiodic points for families of rational maps. \newblock {\em {\em To appear, Proceedings of the London Mathematical Society. %
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[GKN]{GKN %D.~Ghioca, H.~Krieger, and K.~Nguyen. %\newblock{\em A case of the dynamical Andr\'e-Oort Conjecture %\newblock{\em Preprint, {\em 2014. %
1502.04660.extracted_bib
[GNY]{GNY:preprint %D.~Ghioca, K. Nguyen, and H.~Ye. %\newblock {The dynamical Andr\'e-Oort conjecture for rational curves in the moduli space of uncritical polynomials of degree $d$. %\newblock {\em Preprint, {\em 2014.
1502.04660.extracted_bib
[Mi1]{Milnor:quad J.~Milnor. \newblock {Geometry and dynamics of quadratic rational maps. \newblock {\em Experiment. Math. {2(1993), 37--83. \newblock With an appendix by the author and Lei Tan. %
1502.04660.extracted_bib
[MZ1]{Masser:Zannier:CR %D.~Masser and U.~Zannier. %\newblock {Torsion anomalous points and families of elliptic curves. %\newblock {\em C. R. Math. Acad. Sci. Paris {346(2008), 491--494. %
1502.04660.extracted_bib
[MZ2]{Masser:Zannier %D.~Masser and U.~Zannier. %\newblock {Torsion anomalous points and families of elliptic curves. %\newblock {\em Amer. J. Math. {132(2010), 1677--1691. %
1502.04660.extracted_bib
[MZ3]{Masser:Zannier:2 %D.~Masser and U.~Zannier. %\newblock {Torsion points on families of squares of elliptic curves. %\newblock {\em Math. Ann. {352(2012), 453--484.
1502.04660.extracted_bib
[Si]{Silverman:moduli J.~H.~Silverman. \newblock {\em Moduli spaces and arithmetic dynamics, volume~30 of {\em CRM Monograph Series. \newblock American Mathematical Society, Providence, RI, 2012.
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[Ull]{Ullmo:Bogomolov E.~Ullmo. \newblock {Positivit\'e et discr\'etion des points alg\'ebriques des courbes. \newblock {\em Ann. of Math. (2), {147(1998), 167---179.
1502.04660.extracted_bib
[Yu]{Yuan X. ~Yuan. \newblock{Big line bundles over arithmetic varieties. \newblock{\em Invent. Math. {173 (2008), no. 3, 603--649. %
1502.04660.extracted_bib
[Za]{Zannier:book %U.~Zannier. %\newblock {\em Some problems of unlikely intersections in arithmetic and % geometry, volume 181 of {\em Annals of Mathematics Studies. %\newblock Princeton University Press, Princeton, NJ, 2012. %\newblock With appendixes by David Masser.
1502.04660.extracted_bib
[Zh1]{Zhang:line S. ~Zhang. \newblock{Positive line bundles on arithmetic varieties. \newblock{\em J. Am. Math. Soc. {8(1995), 187--221.
1502.04660.extracted_bib
[Zh2]{Zhang:metrics S. ~Zhang. \newblock{Small points and adelic metrics. \newblock{\em J. Algebr. Geom. {4(1995), 281--300.
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[Zh3]{Zhang:Bogomolov S. ~Zhang. \newblock {Equidistribution of small points on abelian varieties. \newblock {\em Ann. of Math. (2), {147(1998), 159--165.