Value distribution of derivatives in polynomial dynamics

TL;DR

This paper proves the equidistribution of derivatives of polynomial iterates towards harmonic measures, extending to non-archimedean, arithmetic, and complex Hénon-type polynomial automorphisms, revealing deep distributional properties.

Contribution

It establishes new equidistribution results for derivatives of polynomial iterates in complex, non-archimedean, and arithmetic settings, including Hénon maps.

Findings

Derivatives' pull-backs equidistribute towards harmonic measure.

Results extend to Berkovich spaces and number fields.

Distribution of eigenvalues in Hénon maps analyzed.

Abstract

For every $m\in\mathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $\mathbb{C}\setminus\{0\}$ under the $m$-th order derivatives of the iterates of a polynomials $f\in \mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field $k$ for a sequence of effective divisors on $\mathbb{P}^1(\overline{k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a H\'enon-type polynomial automorphism of $\mathbb{C}^2$ has a given eigenvalue.

Equations

$$\displaystyle K(f):=\Bigl{\{}z\in\mathbb{C}:\limsup_{n\to\infty}|f^{n}(z)|<\infty\Bigr{\}}$$

$$g_f-\frac{\log \max \{1,|f^n|\}}{d^n}=O(d^{-n})\text{as }n\to \infty$$

$${\mu }_f:=\Delta g_f+{\delta }_{\infty }\text{on }{\mathbb{P}}^1,$$

$$E(f):=\{a\in {\mathbb{P}}^1:\#\bigcup _{n\in \mathbb{N}\cup \{0\}}{f}^{-n}(a)<\infty \},$$

$$\underset{n\to \infty }{\lim }\frac{(f^n{)}^{\ast }{\delta }_a}{d^n}={\mu }_f\text{weakly on }{\mathbb{P}}^1.$$

$$\displaystyle\lim_{n\to\infty}\frac{\bigl{(}(f^{n})^{(m)}\bigr{)}^{*}\delta_{a}}{d^{n}-m}=\mu_{f}\quad\text{weakly on }\mathbb{P}^{1}.$$

$$\displaystyle\lim_{n\to\infty}\frac{\bigl{(}(f^{n})^{(m)}\bigr{)}^{*}\delta_{a}}{d^{n}-m}=\mu_{f}\quad\text{weakly on }\mathsf{P}^{1}.$$

$$\underset{n\to \infty }{\lim }{\hat{h}}_f([(f^n{)}^{(m)}=a])=0$$

$$\lim_{n\to\infty}\sup_{v\in M_{k}}N_{v}\int_{\mathsf{P}^{1}(\mathbb{C}_{v})\times\mathsf{P}^{1}(\mathbb{C}_{v})\setminus\operatorname{diag}_{\mathbb{P}^{1}(\mathbb{C}_{v})}}(\log|\mathcal{S}-\mathcal{S}^{\prime}|_{v}-g_{f,v}(\mathcal{S})-g_{f,v}(\mathcal{S}^{\prime}))\\ \biggl{(}\Bigl{(}\frac{\bigl{(}(f^{n})^{(m)}\bigr{)}^{*}\delta_{a}}{d^{n}-m}-\mu_{f,v}\Bigr{)}\times\Bigl{(}\frac{\bigl{(}(f^{n})^{(m)}\bigr{)}^{*}\delta_{a}}{d^{n}-m}-\mu_{f,v}\Bigr{)}\biggr{)}(\mathcal{S},\mathcal{S}^{\prime})=0$$

$$\displaystyle\lim_{n\to\infty}\frac{\bigl{(}(f^{n})^{(m)}\bigr{)}^{*}\delta_{a}}{d^{n}-m}=\mu_{f,v}\quad\text{weakly on }\mathsf{P}^{1}(\mathbb{C}_{v}).$$

$$L_{\infty }:=\{t=0\}={\mathbb{P}}^2\setminus {\mathbb{C}}^2$$

$$\displaystyle K^{+}:=\Bigl{\{}(z,w)\in\mathbb{C}^{2}:\limsup_{n\to\infty}\|f^{n}(z,w)\|<\infty\Bigr{\}}.$$

$$g^{+}:=\underset{n\to \infty }{\lim }\frac{\log \max \{1,\Vert f^n\Vert \}}{d^n}\text{on }{\mathbb{C}}^2.$$

$$\underset{n\to \infty }{\lim }\frac{[\det (D(f^n)-\lambda I_2)]}{d^n-1}=T^{+}\text{on }{\mathbb{P}}^2$$

$$\displaystyle I_{\infty}(f):=\Bigl{\{}z\in\mathbb{P}^{1}:\lim_{n\to\infty}f^{n}(z)=\infty\Bigr{\}}$$

$$J(f)=\partial K(f)=\partial I_{\infty }(f).$$

$$\displaystyle(f^{n})^{(m)}=\bigl{(}(e^{O(1)}\cdot d^{n})^{m}+O(d^{(m-1)n})\bigr{)}\cdot f^{n}\quad\text{as }n\to\infty$$

$$\underset{n\to \infty }{\lim }\frac{\log |(f^n{)}^{(m)}-a|}{d^n-m}=g_f$$

$$\displaystyle\frac{(f^{n})^{\prime}}{f^{n}}(z)=d^{n}\cdot\bigl{(}1+O(\psi(z)^{-d^{n}})\bigr{)}\cdot\frac{\psi^{\prime}}{\psi}(z)\quad\text{as }n\to\infty$$

$$\displaystyle\frac{(f^{n})^{\prime}(z)}{f^{n}(z)}=\frac{d^{n}\cdot\psi(z)^{d^{n}-1}\cdot\psi^{\prime}(z)}{\psi^{-1}\bigl{(}\psi(z)^{d^{n}}\bigr{)}\cdot\psi^{\prime}(f^{n}(z))}=d^{n}\cdot\frac{\frac{\psi(z)^{d^{n}}}{\psi^{-1}(\psi(z)^{d^{n}})}}{\psi^{\prime}(f^{n}(z))}\cdot\frac{\psi^{\prime}(z)}{\psi(z)}$$

$$\frac{\psi (z{)}^{d^n}}{{\psi }^{-1}(\psi (z{)}^{d^n})}=$$

$$=$$

$${\psi }^{\prime }(f^n(z))=$$

$$=$$

$$\displaystyle\frac{(f^{n})^{\prime}}{f^{n}}=\frac{((f^{n-M})^{\prime}\circ f^{M})\cdot(f^{M})^{\prime}}{f^{n-M}\circ f^{M}}=d^{n-M}\cdot\Bigl{(}\frac{\psi^{\prime}}{\psi}\circ f^{M}\cdot(f^{M})^{\prime}\Bigr{)}+o(1)\quad\text{as }n\to\infty$$

$$\displaystyle\frac{(f^{n})^{(m)}}{f^{n}}=\biggl{(}d^{n-M}\cdot\frac{\psi^{\prime}}{\psi}\circ f^{M}\cdot(f^{M})^{\prime}\biggr{)}^{m}+O(d^{(m-1)n})\quad\text{as }n\to\infty$$

$$\displaystyle\frac{(f^{n})^{(m)}}{f^{n}}-\frac{(f^{n})^{(m-1)}\cdot(f^{n})^{\prime}}{f^{n}\cdot f^{n}}=\biggl{(}\frac{(f^{n})^{(m-1)}}{f^{n}}\biggr{)}^{\prime}=O(d^{n(m-1)})\quad\text{as }n\to\infty$$

$$\frac{(f^{n})^{(m)}}{f^{n}}=\frac{(f^{n})^{(m-1)}\cdot(f^{n})^{\prime}}{f^{n}\cdot f^{n}}+O(d^{(m-1)n})\\ =\Bigl{(}\Bigl{(}d^{n-M}\cdot\frac{\psi^{\prime}}{\psi}\circ f^{M}\cdot(f^{M})^{\prime}\Bigr{)}^{m-1}+O(d^{(m-2)n})\Bigr{)}\cdot\Bigl{(}d^{n-M}\cdot\frac{\psi^{\prime}}{\psi}\circ f^{M}\cdot(f^{M})^{\prime}+O(1)\Bigr{)}\\ +O(d^{(m-1)n})\quad\text{as }n\to\infty$$

$$\frac{\log |(f^n{)}^{(m)}-a|}{d^n-m}\le \frac{\log (2\max \{|(f^n{)}^{(m)}|,|a|\})}{d^n-m}\le g_f+O(1)\text{as }n\to \infty$$

$$\displaystyle S_{f^{n}}:=\frac{(f^{n})^{\prime\prime\prime}}{(f^{n})^{\prime}}-\frac{3}{2}\Bigl{(}\frac{(f^{n})^{\prime\prime}}{(f^{n})^{\prime}}\Bigr{)}^{2}=-2d^{2n}\cdot(\partial_{z}g_{f})^{2}+O(d^{n})\quad\text{and}$$

$$T_{f^n}:=\frac{(f^n{)}^{\prime \prime }}{(f^n{)}^{\prime }}=2d^n\cdot {\partial }_z{g}_f+O(1)\text{as }n\to \infty$$

Full text

We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb{C}^{2}$ has a given eigenvalue.

1. Introduction

Let $f\in\mathbb{C}[z]$ be a polynomial of degree $d>1$. The filled-in Julia set

[TABLE]

of $f$ is a non-polar compact subset in $\mathbb{C}$. Let $g_{f}$ be the Green function of $K(f)$ with pole at $\infty$, regarding $\mathbb{P}^{1}$ as $\mathbb{C}\cup\{\infty\}$ (see e.g. [22, §4.4]). We extend $g_{f}$ as $=0$ on $K(f)$. For every $n\in\mathbb{N}$, the difference $g_{f}-(\log\max\{1,|f^{n}|\})/d^{n}$ on $\mathbb{C}$ is harmonic and bounded near $\infty$ so it admits a harmonic extension across $\infty$, and we have the estimate

[TABLE]

on $\mathbb{P}^{1}$ uniformly.

Let us denote by $\delta_{a}$ the Dirac measure on $\mathbb{P}^{1}$ at each $a\in\mathbb{P}^{1}$. The harmonic measure of $K(f)$ with pole at $\infty$ is the probability measure

[TABLE]

which has no atoms on $\mathbb{P}^{1}$ and is supported by $\partial K(f)$. The exceptional set of $f$ is defined as

[TABLE]

which consists of $\infty$ ($f^{-1}(\infty)=\{\infty\}$) and at most one point $b\in\mathbb{C}$ ($f^{-1}(b)=\{b\}$). For every $h\in\mathbb{C}(z)$ of $\deg h>0$ and every $a\in\mathbb{P}^{1}$, by the definition of the pullback operator $h^{*}$, we have $h^{*}\delta_{a}=\sum_{w\in h^{-1}(a)}(\deg_{w}h)\delta_{a}$ on $\mathbb{P}^{1}$, where $\deg_{w}h$ is the local degree of $h$ at $w$.

Brolin [7] studied the value distribution of the iteration sequence $(f^{n}:\mathbb{P}^{1}\to\mathbb{P}^{1})$ of $f$ and established that for every $a\in\mathbb{C}\setminus E(f)$,

[TABLE]

This equidistribution of pullbacks of points under iterations initiated the study of value distribution of complex dynamics (see e.g. [22, §6.5], [6, §VIII], [9, 24]). In [14, §2] and [20, Theorem 1], a similar equidistribution statement replacing $f^{n}$ with the first order derivative $(f^{n})^{\prime}$ of $f^{n}$ has been proved first for $a\in\mathbb{C}$ outside a polar set and then for any $a\in\mathbb{C}^{*}$, respectively.

Our aim is to contribute to the study of the parallelism between the value distribution of the sequence of higher derivatives (or jets) of the iterations of $f$ and the value distribution of higher derivatives (or jets) of meromorphic mappings (cf. [25]), extending the results mentioned above to several different settings; higher derivatives of polynomials over various valued fields and Hénon-type polynomial automorphisms of $\mathbb{C}^{2}$.

1.1. Over the field $\mathbb{C}$ of complex numbers

Let $f\in\mathbb{C}[z]$ be a polynomial of degree $d>1$. For every $h\in\mathbb{C}[z]$ and every $m\in\mathbb{N}$, we write the $m$-th order derivative $\frac{\mathrm{d}^{m}}{\mathrm{d}z^{m}}h(z)$ of $h$ as $h^{(m)}$.

Our first principal result is the following.

Theorem 1**.**

Let $f\in\mathbb{C}[z]$ be a polynomial of degree $d>1$, and $m\in\mathbb{N}$. Then for every $a\in\mathbb{C}\setminus\{0\}$,

[TABLE]

In Theorem 1, the values $a=0,\infty$ need to be excluded as for every $n\in\mathbb{N}$, $((f^{n})^{(m)})^{*}\delta_{\infty}/(d^{n}-m)=\delta_{\infty}\neq\mu_{f}$ and, if $b\in E(f)\cap\mathbb{C}$, then for every $n\in\mathbb{N}$, $((f^{n})^{(m)})^{*}\delta_{0}/(d^{n}-m)=\delta_{b}\neq\mu_{f}$. An affine coordinate on $\mathbb{C}$ is fixed in Theorem 1, but note that $A^{*}(((f^{n})^{(m)})^{*}\delta_{a}-(d^{n}-m)\cdot\mu_{f})=(((A\circ f\circ A^{-1})^{n})^{(m)})^{*}\delta_{(A^{\prime})^{m-1}(a)}-(d^{n}-m)\cdot\mu_{A\circ f\circ A^{-1}}$ on $\mathbb{P}^{1}$ for any affine transformation $A$ on $\mathbb{C}$.

The equidistribution (1.2) for $m>1$ was expected in [14, §2.4], at least when $f$ has no Siegel disks. As seen in the proof below, (1.2) follows only by an analysis of $(f^{n})^{(m)}$ on $\mathbb{P}^{1}\setminus K(f)$ in this case. This analysis is not difficult for $m=1$ by the chain rule, but for $m>1$ it requires to deal carefully with the higher order derivatives of the Böttcher coordinates of $f$ near $\infty$. An extra and more involved effort is required to treat the situation on $K(f)$ under the presence of Siegel disks of $f$ in general.

1.2. Over a non-archimedean complete valued field $K$

Let $K$ be an algebraically closed field. We say that an absolute value $|\cdot|$ on $K$ is non-trivial if $|K|\not\subset\{0,1\}$ and that it is non-archimedean if the strong triangle inequality $|z+w|\leq\max\{|z|,|w|\}$ holds for any $z,w\in K$. For the details on the Berkovich projective line $\mathsf{P}^{1}=\mathsf{P}^{1}(K)$, the canonical action of $f$ on $\mathsf{P}^{1}$, and the equilibrium (or canonical) measure $\mu_{f}$ of $f$ on $\mathsf{P}^{1}$, see Subsection 3.1 below. By convention, we say $f$ has no potentially good reductions if $\mu_{f}(\{\mathcal{S}\})=0$ for any $\mathcal{S}\in\mathsf{P}^{1}\setminus\mathbb{P}^{1};$ this definition coincides with the usual algebraic one $($cf. [3, Corollary 10.33]$)$.

Our second principal result is a non-archimedean counterpart of Theorem 1.

Theorem 2**.**

Let $K$ be an algebraically closed field of characteristic [math] that is complete with respect to a non-trivial and non-archimedean absolute value. Let $m\in\mathbb{N}$ and $f\in K[z]$ be a polynomial of degree $d>1$ having no potentially good reductions. Then for every $a\in K$,

[TABLE]

The no potentially good reductions assumption allows us to deal with the Berkovich filled-in Julia set $\mathsf{K}(f)$ of $f$. The analysis on $\mathsf{P}^{1}\setminus\mathsf{K}(f)$ in the proof is similar to that in the archimedean case, using the (non-archimedean) Böttcher coordinate near $\infty$ and a non-archimedean potential theory instead (see [21]).

1.3. Over a product formula field $k$

Let $k$ be a field. We denote by $\overline{k}$ an algebraic closure of $k$. An effective $k$-divisor $\mathcal{Z}$ on $\mathbb{P}^{1}(\overline{k})$ is the scheme theoretic vanishing of some $P\in\bigcup_{d\in\mathbb{N}}k[z_{0},z_{1}]_{d}$. Then, $\mathcal{Z}$ is supported by $\overline{k}$ (regarding $\mathbb{P}^{1}(\overline{k})$ as $\overline{k}\cup\{\infty\}$) if and only if $P(z_{0},z_{1})=z_{0}^{\deg p}p(z_{1}/z_{0})$ for some $p(z)\in k[z]$ of degree $>0$ (identifying $[z_{0}:z_{1}]$ with $z_{1}/z_{0}$ when $z_{0}\neq 0$, that is, $\infty=[0:1]$ as the convention in [13]), which is unique up to multiplication in $k^{*}=k\setminus\{0\}$ and is called a representative of $\mathcal{Z}$.

A field $k$ is a product formula field if $k$ is equipped with a (possibly uncountable) family $M_{k}$ of (not necessarily all) places of $k$, a family $(|\cdot|_{v})_{v\in M_{k}}$ of non-trivial absolute values $|\cdot|_{v}$ representing $v$, and a family $(N_{v})_{v\in M_{k}}$ in $\mathbb{N}$ satisfying the product formula property in that, for every $z\in k^{*}$,

$|z|_{v}=1$ for all but finitely many $v\in M_{k}$, and $\prod_{v\in M_{k}}|z|^{N_{\nu}}_{v}=1$.

A place $v\in M_{k}$ is said to be finite (resp. infinite) if $|\cdot|_{v}$ is non-archimedean (resp. archimedean). If $M_{k}$ contains an infinite place of $v$, then $k$ is (isomorphic to) a number field (so there are at most finitely many infinite places of a product formula field). For each $v\in M_{k}$, let $k_{v}$ be the completion of $k$ with respect to $|\cdot|_{v}$. Then $|\cdot|_{v}$ extends to $\overline{k_{v}}$. Let $\mathbb{C}_{v}$ be the completion of $\overline{k_{v}}$ with respect to $|\cdot|_{v}$ (so $|\cdot|_{v}$ extends to $\mathbb{C}_{v}$) and fix an embedding of $\overline{k}$ to $\mathbb{C}_{v}$ extending that of $k$ to $k_{v}$. By convention, the dependence of a local quantity induced by $|\cdot|_{v}$ on each $v\in M_{k}$ is emphasized by adding the suffix to it, like $k_{v}$ and $\mathbb{C}_{v}$.

Let $\hat{h}_{f}(\mathcal{Z})$ be the Call-Silverman canonical height of an effective $k$-divisor $\mathcal{Z}$ on $\mathbb{P}^{1}(\overline{k})$ (see Subsection 3.2 below for the definition). The following is our third principal result.

Theorem 3**.**

Let $k$ be a product formula field of characteristic [math], and let $f\in k(z)$ be a polynomial of degree $d>1$ and $m\in\mathbb{N}$. Then for every $a\in k$, denoting by $[(f^{n})^{(m)}=a]$ the effective $k$-divisor on $\mathbb{P}^{1}(\overline{k})$ whose representative is $(f^{n})^{(m)}-a\in k[z]$, we have the $(g_{f,v})_{v\in M_{k}}$-small heights property

[TABLE]

of the sequence $([(f^{n})^{(m)}=a])_{n}$ of effective $k$-divisors on $\mathbb{P}^{1}(\overline{k})$.

Assume, in addition, that $k$ is a number field and $a\in k^{*}$, then the uniform asymptotically $(g_{f,v})_{v\in M_{k}}$-Fekete configuration property

[TABLE]

of $([(f^{n})^{(m)}=a])$ holds, so in particular, for every $v\in M_{k}$,

[TABLE]

The proof is based on an adelic equidistribution result for effective divisors on $\mathbb{P}^{1}(\overline{k})$ having small diagonals and small heights ([18]).

1.4. The derivatives of the iterates of a Hénon-type polynomial

automorphism of $\mathbb{C}^{2}$

Let $[t:z:w]$ be the homogeneous coordinate on $\mathbb{P}^{2}$, endowed with the Fubini-Study form. Identifying $\mathbb{C}^{2}$ with $\{t=1\}$, we let

[TABLE]

be the line at infinity in $\mathbb{P}^{2}$. We fix the orthonormal frame $(\partial_{z},\partial_{w})$ of the tangent space $T\mathbb{C}^{2}$ of $\mathbb{C}^{2}$, so that for a polynomial endomorphism $f$ of $\mathbb{C}^{2}$, the derivative $\mathrm{d}f$ of $f$ is identified with the $\mathrm{M}(2,\mathbb{C})$-valued function $(z,w)\mapsto(Df)_{(z,w)}$. Here, a polynomial automorphism of $\mathbb{C}^{2}$ is a polynomial endomorphism of $\mathbb{C}^{2}$ whose inverse exists and is a polynomial endomorphism of $\mathbb{C}^{2}$.

Recall some basic facts on a Hénon-type polynomial automorphism $f$ of $\mathbb{C}^{2}$ of degree $d>1$ ([4, 10]). The Jacobian determinant $J_{f}:=\det(Df)\in\mathbb{C}[z,w]$ of $f$ is a non-zero constant on $\mathbb{C}^{2}$, so for every $n\in\mathbb{N}$, the Jacobian determinant $J_{f^{n}}=\det(D(f^{n}))\in\mathbb{C}[z,w]$ of $f^{n}$ on $\mathbb{C}^{2}$ is equal to the non-zero constant $J_{f}^{n}$. This $f$ extends to a birational self-map on $\mathbb{P}^{2}$, which is still denoted by $f$ for simplicity, so that both the indeterminacy loci $I^{+},I^{-}$ of $f,f^{-1}$ are singletons in $L_{\infty}$, that $I^{-}\neq I^{+}$ (so often normalized as $I^{+}=\{[0:0:1]\},I^{-}=\{[0:1:0]\}$), and that $I^{-}=f(L_{\infty}\setminus I^{+})$. Moreover, the unique point in $I^{-}$ is a superattracting fixed point of $f|(\mathbb{P}^{1}\setminus I^{+})$, and the attractive basin $B^{+}$ of $f|(\mathbb{P}^{1}\setminus I^{+})$ associate to $I^{-}$ satisfies $B^{+}\setminus\mathbb{C}^{2}=L_{\infty}\setminus I^{+}$. Let $\|\cdot\|$ be the Euclidean norm on $\mathbb{C}^{2}$. The filled-in Julia set of $f$ is defined by

[TABLE]

Then $\overline{K^{+}}=K^{+}\cup I^{+}$ in $\mathbb{P}^{2}$ and $\mathbb{P}^{2}=\overline{K^{+}}\cup B^{+}$ (see e.g. [10, Proposition 5.5]). The Green function $g^{+}$ of $f$ is the locally uniform limit

[TABLE]

It is continuous and plurisubharmonic on $\mathbb{C}^{2}$, it is $>0$ and pluriharmonic on $B^{+}$, and it is $\equiv 0$ on $K^{+}$. The Green current $T^{+}$ of $f$ is defined as the trivial extension of $\mathrm{d}\mathrm{d}^{c}g^{+}$ on $\mathbb{C}^{2}$ to $\mathbb{P}^{2}$. It is a positive closed $(1,1)$-current on $\mathbb{P}^{2}$ and moreover of mass $1$ ([10, Lemma 6.3]).

For a non-constant polynomial $P\in\mathbb{C}[z,w]$, let $[P]$ be the current of integration along the hypersurface in $\mathbb{P}^{2}$ defined by the zeros of (the homogenized) $P$ in $\mathbb{P}^{2}$, taking into account their multiplicities. The mass of $[P]$ equals $\deg P$ by Bézout’s theorem. Let $I_{2}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ be the identity matrix in $\mathrm{M}(2,\mathbb{C})$.

Our final principal result is the following.

Theorem 4**.**

Let $f$ be a Hénon-type polynomial automorphism of $\mathbb{C}^{2}$ of degree $d>1$ and $\lambda\in\mathbb{C}^{*}=\mathbb{C}\setminus\{0\}$. Then for every $n\in\mathbb{N}$, $\det(D(f^{n})-\lambda I_{2})\in\mathbb{C}[z,w]$ is of degree $d^{n}-1$, and

[TABLE]

as currents.

In the proof, we show the $L^{1}_{\mathrm{loc}}$-convergence of a sequence of potentials of $[\det(D(f^{n})-\lambda I_{2})]/(d^{n}-1)$ towards $g^{+}$ on $B^{+}$ as $n\to\infty$ using the first order partial derivatives of $g^{+}$. The pleasant uniqueness of $T^{+}$ among all positive closed $(1,1)$-currents on $\mathbb{P}^{2}$ of mass $1$ which are supported by $\overline{K^{+}}$ ([12]; see also [10, Theorem 6.5]) allows us to deal with $K^{+}$.

Organization of the article

In Section 2, we treat the field $\mathbb{C}$ of complex numbers. In Subsection 2.1, we recall some notion and facts from complex dynamics. In Subsection 2.2, we give a proof of Theorem 1 and, in Subsection 2.3, we give a simpler treatment for the cases $m=1,2$. In Section 3, we treat a non-archimedean field $K$ and a product formula field $k$. In Subsections 3.1 and 3.2, we recall a background from non-archimedean and arithmetic dynamics, respectively, and in Subsection 3.3, we show Theorems 2 and 3. In Section 4, we show Theorem 4 in a slightly more general form.

2. Proof of Theorem 1

2.1. Background from complex dynamics

Let $f\in\mathbb{C}[z]$ be a polynomial of degree $d>1$. The superattractive basin

[TABLE]

of $f$ associated to the superattracting fixed point $\infty$ of $f$ (regarding $\mathbb{P}^{1}$ as $\mathbb{C}\cup\{\infty\}$) is a domain in $\mathbb{P}^{1}$ containing $\infty$, and coincides with $\mathbb{P}^{1}\setminus K(f)$. Let $C(f)$ be the critical set of $f$ (as a branched self-covering of $\mathbb{P}^{1}$) which consists of $\infty$ and all the zeros of $f^{\prime}$ on $\mathbb{C}$. The set $\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f)\setminus\{\infty\})$ is bounded in $\mathbb{C}$.

The topology of $\mathbb{P}^{1}$ coincides with the induced one from the chordal metric on $\mathbb{P}^{1}$. The Julia set $J(f)$ of $f$ is defined as the set of all $z\in\mathbb{P}^{1}$ at which the family $(f^{n}:\mathbb{P}^{1}\to\mathbb{P}^{1})_{n\in\mathbb{N}}$ is not normal. The Fatou set $F(f)$ of $f$ is defined by $\mathbb{P}^{1}\setminus J(f)$ and a component of $F(f)$ is called a Fatou component of $f$. Both $J(f)$ and $F(f)$ are totally invariant under $f$ and

[TABLE]

Any Fatou component of $f$ is either $I_{\infty}(f)$ or a component of the interior of $K(f)$ and is mapped properly to a Fatou component of $f$. Any Fatou component of $f$ other than $I_{\infty}(f)$ is simply connected. A Fatou component $W$ of $f$ is said to be cyclic under $f$ if there is $p\in\mathbb{N}$ such that $f^{p}(W)=W$. If in addition the restriction $f^{p}:W\to W$ is injective, $W$ is called a Siegel disk of $f$ and then there exists a holomorphic injection $h:W\to\mathbb{C}$ such that for some $\alpha\in\mathbb{R}\setminus\mathbb{Q}$, $h\circ f^{p}=e^{2i\pi\alpha}\cdot h$ on $W$. In particular, $h(V)=\{|w|<r\}$ for some $r>0$, $v_{0}:=h^{-1}(0)$ is fixed by $f^{p}$, and $(f^{p})^{\prime}(v_{0})=e^{2i\pi\alpha}$. For more details on complex dynamics, see e.g. [17].

2.2. Proof of Theorem 1

Let $f\in\mathbb{C}[z]$ be a polynomial of degree $d>1$. Fix $m\in\mathbb{N}$.

Lemma 2.1**.**

We have

[TABLE]

on $I_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$ locally uniformly. Moreover, for every $a\in\mathbb{C}$, the family \bigl{(}(\log|(f^{n})^{(m)}-a|)/(d^{n}-m)\bigr{)}_{n} of subharmonic functions on $\mathbb{C}$ is locally uniformly bounded from above on $\mathbb{C}$ and

[TABLE]

locally uniformly on $I_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$.

Proof.

Fixing $r\gg 1$, there exists a biholomorphism $w=\psi(z)$ from $\mathbb{P}^{1}\setminus\{g_{f}\leq r\}$ to $\mathbb{P}^{1}\setminus\{|w|\leq e^{r}\}$, which is called a Böttcher coordinate near $\infty$ associated to $f$, such that $\psi(f(z))=\psi(z)^{d}$ on $\mathbb{P}^{1}\setminus\{g_{f}\leq r\}$. Then $\psi(\infty)=\infty$, $\psi^{\prime}\neq 0$ on $\mathbb{C}\setminus\{g_{f}\leq r\}$, and letting $\iota:\mathbb{P}^{1}\to\mathbb{P}^{1}$ be the involution $z\mapsto 1/z$ (regarding $1/0$ as $\infty$), $(\iota\circ\psi\circ\iota)^{\prime}(0)=1/(\iota\circ\psi^{-1}\circ\iota)^{\prime}(0)\neq 0$.

We first claim that

[TABLE]

on $\mathbb{C}\setminus\{g_{f}\leq r\}$ uniformly; indeed, for every $n\in\mathbb{N}$, since $\psi(f^{n}(z))=\psi(z)^{d^{n}}$ on $\mathbb{C}\setminus\{g_{f}\leq r\}$, we have $f^{n}(z)=\psi^{-1}(\psi(z)^{d^{n}})$ and $\psi^{\prime}(f^{n}(z))\cdot(f^{n})^{\prime}(z)=d^{n}\cdot\psi(z)^{d^{n}-1}\cdot\psi^{\prime}(z)$ on $\mathbb{C}\setminus\{g_{f}\leq r\}$, so that

[TABLE]

on $\mathbb{C}\setminus\{g_{f}\leq r\}$. Moreover, we have

[TABLE]

on $\mathbb{C}\setminus\{g_{f}\leq r\}$ uniformly and, since $(\iota\circ\psi\circ\iota)^{\prime}(1/f^{n}(z))=-\frac{\psi^{\prime}(f^{n}(z))\cdot\{-(f^{n}(z)^{2})\}}{\psi(f^{n}(z))^{2}}$ on $\mathbb{C}\setminus\{g_{f}\leq r\}$ by the chain rule, we also have

[TABLE]

on $\mathbb{C}\setminus\{g_{f}\leq r\}$ uniformly. Hence the claim holds.

For any domain $D\Subset I_{\infty}(f)\cap K$ and any $M\in\mathbb{N}\cup\{0\}$ so large that $f^{M}(D)\subset\mathbb{P}^{1}\setminus\{g_{f}\leq r\}$, by (2.3), we have

[TABLE]

on some open neighborhood of $\overline{D}$ uniformly. Let us show by induction that for any $m\in\mathbb{N}$,

[TABLE]

on some open neighborhood of $\overline{D}$ uniformly; we have just seen (2.4) for $m=1$ on some open neighborhood of $\overline{D}$ uniformly, so assume that $m>1$ and that (2.4) for $m-1$ holds on some open neighborhood of $\overline{D}$ uniformly. Then using Cauchy’s estimate, we have

[TABLE]

on some open neighborhood of $\overline{D}$ uniformly, which with (2.4) for both $1$ and $m-1$ on some open neighborhood of $\overline{D}$ uniformly yields

[TABLE]

on some open neighborhood of $\overline{D}$ uniformly. This yields (2.4) for $m$ on some open neighborhood of $\overline{D}$ uniformly and concludes the induction. Now, if in addition $D\Subset I_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$, so $\inf_{D}|\frac{\psi^{\prime}}{\psi}\circ f^{M}\cdot(f^{M})^{\prime}|>0$, then the estimate (2.4) yields the asymptotic estimate (2.1).

Fix $a\in\mathbb{C}$. The final locally uniform convergence (2.2) follows from (2.1) and (1.1). Then, for every $R>0$ so large that $\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f)\setminus\{\infty\})\subset\{|z|<R\}$, we also have

[TABLE]

on $\{|z|=R\}$ uniformly. Hence by the maximum principle for subharmonic functions, we deduce that the family $((\log|(f^{n})^{(m)}-a|)/(d^{n}-m))_{n}$ is locally uniformly bounded from above on $\mathbb{C}$. ∎

Remark 2.2* *(the Schwarzian and pre-Schwarzian derivatives $S_{f^{n}},T_{f^{n}}$

of $f^{n}$).

The expression of $(f^{n})^{(m)}$ given by (2.4) in the proof of Lemma 2.1 also quantifies Ye [26, Theorems 1.1 and 3.3] as

[TABLE]

on $I_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$ locally uniformly. Indeed, recall that $g_{f}=\log|\psi|$ so $\partial_{z}g_{f}=\psi^{\prime}/(2\psi)$ on $\mathbb{C}\setminus\{g_{f}\leq r\}$, and $g_{f}\circ f=d\cdot g_{f}$ so $(\partial_{z}g_{f})\circ f^{M}\cdot(f^{M})^{\prime}=d^{M}\cdot\partial_{z}g_{f}$ on $I_{\infty}(f)$. Hence (2.4) is rewritten as

[TABLE]

on $\overline{D}$ uniformly. This for $m\in\{1,2,3\}$ yields the above asymptotics of $S_{f^{n}}$ and $T_{f^{n}}$.

Fix $a\in\mathbb{C}$, and let us continue the proof of Theorem 1. By the final two assertions in Lemma 2.1, applying to $((\log|(f^{n})^{(m)}-a|)/(d^{n}-m))_{n}$ a compactness principle (see [15, Theorem 4.1.9(a)]) for a family of subharmonic functions on a domain in $\mathbb{R}^{N}$, there are a sequence $(n_{j})$ in $\mathbb{N}$ tending to $+\infty$ as $j\to\infty$ and a subharmonic function $\phi$ on $\mathbb{C}$ such that

[TABLE]

($m_{2}$ denotes the (real $2$-dimensional) Lebesgue measure on $\mathbb{C}$). By (2.2), we have $\phi\equiv g_{f}$ $m_{2}$-a.e. on $I_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$, and in turn on $I_{\infty}(f)$ by the subharmonicity of $\phi-g_{f}$ on $I_{\infty}(f)\cap\mathbb{C}$. Then also by $I_{\infty}(f)=\{g_{f}>0\}$, the subharmonicity of $\phi$ on $\mathbb{C}$, and the maximum principle for subharmonic functions, we have $\phi\leq\max_{\{g_{f}=\epsilon\}}\phi=\max_{\{g_{f}=\epsilon\}}g_{f}=\epsilon$ on $K(f)=\{g_{f}=0\}\subset\{g_{f}<\epsilon\}$ for every $\epsilon>0$, and in turn $\phi\leq 0$ on $K(f)$. By the upper semicontinuity of $\phi-g_{f}$ on $\mathbb{C}$, the subset $\{\phi<g_{f}\}$ is open in $\mathbb{C}$.

Lemma 2.3**.**

If $a\neq 0$, then $\phi=g_{f}$ on $\mathbb{C}$.

Proof.

Suppose that $\{\phi<g_{f}\}\neq\emptyset$ and let us show $a=0$. By $\phi\equiv g_{f}$ on $I_{\infty}(f)$, there is a Fatou component $U\subset K(f)$ of $f$ containing a component $W$ of $\{\phi<g_{f}\}$. Since $\phi\leq g_{f}=0$ on $U$, we in fact have $U=W$ by the maximum principle for subharmonic functions.

(I)

Taking a subsequence of $(n_{j})$ if necessary, there is a locally uniform limit

[TABLE]

We claim that

[TABLE]

on $U$, so in particular we can say $g\in\mathbb{C}[z]$ (of degree $\leq m$); indeed, for any domain $D\Subset U=W$, by Hartogs’s lemma for a sequence of subharmonic functions on a domain in $\mathbb{R}^{N}$ (see [15, Theorem 4.1.9(b)]), we have

[TABLE]

Then g^{(m)}=\bigl{(}\lim_{j\to\infty}(f^{n_{j}})\bigr{)}^{(m)}=\lim_{j\to\infty}\bigl{(}(f^{n_{j}})^{(m)}\bigr{)}\equiv a on $D$, so the claim holds. In the case that $g$ is constant, we have $g^{(m)}\equiv 0=a$, so we are done.

(II)

Let us assume that $g$ is non-constant. Then by Hurwitz’s theorem and Fatou’s classification of cyclic Fatou components of $f$ (see, e.g., [17, §16]), there is $N\in\mathbb{N}$ such that $V:=f^{n_{N}}(U)=g(U)(\supset g(\overline{D}))$ is a Siegel disk of $f$. Setting $p:=\min\{n\in\mathbb{N}:f^{n}(V)=V\}$, for any $j\geq N$, we have $p|(n_{j}-n_{N})$ and there is a holomorphic injection $h:V\to\mathbb{C}$ such that for some $\alpha\in\mathbb{R}\setminus\mathbb{Q}$, setting $\lambda:=e^{2i\pi\alpha}\in\partial\mathbb{D}$, we have $h\circ f^{p}=\lambda\cdot h$ on $V$. Hence for every $j\geq N$,

[TABLE]

Taking a subsequence of $(n_{j})$ if necessary, the limit

[TABLE]

also exists and then

[TABLE]

Set $v_{0}:=h^{-1}(0)$ and fix $z_{0}\in U\cap f^{-n_{N}}(v_{0})$, so that $f^{p}(v_{0})=v_{0}=g(z_{0})$ and $(f^{p})^{\prime}(v_{0})=\lambda$. For every $0<r\ll 1$, $\{|w|<2r\}\Subset h(V)$, and letting $D_{r}$ be a component of $(h\circ f^{n_{N}})^{-1}(\{|w|<r\})$ containing $z_{0}$, the restriction $h\circ f^{n_{N}}:D_{r}\setminus\{z_{0}\}\to\{0<|w|<r\}$ is an unramified covering of degree $\deg_{z_{0}}(f^{n_{N}})=\deg_{z_{0}}g$. Hence, the restriction $h\circ g:D_{r}\setminus\{z_{0}\}\to\{0<|w|<r\}$ is also an unramified covering of the same degree as that of $h\circ f^{n_{N}}|D_{r}$ by Hurwitz’s theorem. Let us denote by $h^{-1}$ the holomorphic inverse of the biholomorphism $h:V\to h(V)\subset\mathbb{C}$.

Let us see by induction that for any $\ell\in\mathbb{N}$,

[TABLE]

indeed, for every $j\geq N$, applying Cauchy’s integration formula to $f^{n_{j}}-g$ on $D_{r}$, by $g^{(m)}\equiv a$, (2.7), and (2.7*′*), we have

[TABLE]

on $D_{r}$, where recalling $h\circ f^{n_{N}}(\partial D_{r})=\{|w|=r\}$ and $\{|w|<2r\}\Subset h(V)$ and applying Cauchy’s estimate to the holomorphic function $h^{-1}|\{w^{\prime}\in\mathbb{C}:|w^{\prime}-w|\leq r\}$ for each $|w|=r$, the $O(\lambda^{(n_{j}-n_{N})/p}-\lambda_{0})$ term is estimated as

[TABLE]

so the implicit constant of it is independent of $z\in D_{r}$ and $\zeta\in\partial D_{r}$. On the other hand, for every $z\in D_{r}$, by (2.6) and [20, (3.8)], we also have

[TABLE]

Hence also by Cauchy’s integration formula, we have

[TABLE]

on $D_{r}$, that is, (2.8) holds for $\ell=1$.

Next, suppose that (2.8) holds for $1,\ldots,\ell-1$. Then applying Cauchy’s integration formula to $((h^{-1})^{(k)}(\lambda_{0}\cdot h\circ f^{n_{N}}(z))\cdot(h\circ f^{n_{N}}(z))^{k})^{(m)}\equiv 0$ on $D_{r}$ for $k\in\{1,\ldots,\ell-1\}$, also by (2.9), we have

[TABLE]

on $D_{r}$, where recalling $h\circ f^{n_{N}}(\partial D_{r})=\{|w|=r\}$ and $\{|w|<2r\}\Subset h(V)$ and applying Cauchy’s estimate to the holomorphic function $h^{-1}|\{w^{\prime}\in\mathbb{C}:|w^{\prime}-w|\leq r\}$ for each $|w|=r$, the $O(\lambda^{(n_{j}-n_{N})/p}-\lambda_{0})$ term is estimated as

[TABLE]

so the implicit constant of it is independent of $z\in D_{r}$ and $\zeta\in\partial D_{r}$. Hence by (2.10) again, also using Cauchy’s integration formula, we have

[TABLE]

that is, (2.8) holds for $\ell$ and concludes the induction.

Once this claim (2.8) is at our disposal, for every $\ell\in\mathbb{N}$, there is $P_{\ell}\in\mathbb{C}[z]$ of degree $<m$ such that

[TABLE]

Then recalling $(h\circ f^{n_{N}})(z_{0})=0$, for every $\ell\geq m$, we have $P_{\ell}\equiv P_{\ell}(z_{0})=0$; for, otherwise, we must have $m>\deg P_{\ell}\geq\deg_{z_{0}}P_{\ell}\geq\ell\geq m$, which is a contradiction. Consequently, also by (2.7*′*) and $(h\circ f^{n_{N}})(D_{r}\setminus\{z_{0}\})=\{0<|w|<r\}$, for every $\ell\geq m$,

[TABLE]

which implies that there is $Q\in\mathbb{C}[z]$ (of degree $<m$) such that $h^{-1}\equiv Q$ on $\{0<|w|<r\}$ since $h\circ g:D_{r}\setminus\{z_{0}\}\to\{0<|w|<r\}$ is an unramified covering. Then $\deg Q>0$.

On the other hand, we also have

[TABLE]

and in turn $f^{p}(Q(w))=Q(\lambda w)$ in $\mathbb{C}[w]$ by the identity theorem for holomorphic functions. Then $Q\in\mathbb{C}[w]$ must be constant since $\deg(f^{p})=d^{p}>1$. This contradicts $\deg Q>0$.

Hence $g$ is constant, and the proof of Lemma 2.3 is complete. ∎

Using Lemma 2.3, the $L^{1}_{\operatorname{loc}}(\mathbb{C},m_{2})$-convergence (2.5), a continuity of the Laplacian $\Delta$, and the equalities

[TABLE]

for each $j\in\mathbb{N}$ and $\Delta g_{f}=\mu_{f}$ on $\mathbb{C}$, whenever $a\in\mathbb{C}\setminus\{0\}$, we conclude the desired weak convergence (1.2) on $\mathbb{C}$, and in turn on $\mathbb{P}^{1}$ since $\operatorname{supp}\mu_{f}\subset\mathbb{C}$. Now the proof of Theorem 1 is complete. ∎

2.3. On the proof of Theorem 1 for the first and second orders derivatives

In step (II) of the proof of Lemma 2.3 in Section 2.1, it might be interesting to show that $a=0$ by direct computations in the case where $g$ is non-constant, instead of showing that $g$ is constant by contradiction. We include herewith such proofs in (II)’ and (II)” below for the first and second orders derivatives cases $m=1,2$, respectively.

(II)’

Here, assume that $m=1$ and that $g$ is non-constant. For any $j\geq N$, differentiating both sides in (2.7), by the chain rule, we have

[TABLE]

so that evaluating them at $z=z_{0}$, also by $h^{\prime}(v_{0})\neq 0$, we have

[TABLE]

(here $m=1$). Hence for any $j\geq N$, we have

[TABLE]

On the other hand, by (2.6) (here $m=1$) and [20, (3.8)], we have

[TABLE]

Hence we have

[TABLE]

which with $a=\lambda_{0}\cdot(f^{n_{N}})^{\prime}(z_{0})$ yields $a=0$. ∎

(II)”

Now, assume that $m=2$ and that $g$ is non-constant. For any $j\geq N$, differentiating both sides in (2.7) twice, by the chain rule, we have

[TABLE]

on $U$, so that evaluating them at $z=z_{0}$, also by $h^{\prime}(v_{0})\neq 0$, we have

[TABLE]

and in turn making $j\to\infty$,

[TABLE]

(here $m=2$ so $a=g^{\prime\prime}(z_{0})$). Hence for any $j\geq N$, subtracting (2.15) from (2.13) and then eliminating $(f^{n_{j}})^{\prime}(z_{0})$ and $g^{\prime}(z_{0})$ by (2.12) and (2.14), the above four equalities yield

[TABLE]

which is rewritten as

[TABLE]

On the other hand, by (2.6) (here $m=2$) and [20, (3.8)], we have

[TABLE]

Hence making $j\to\infty$ in (2.16), we must have

[TABLE]

which with (2.16) in turn yields

[TABLE]

for any $j\geq N$. Then by (2.17) again, from (2.16*′*), we have

[TABLE]

which with (2.18) and (2.14) yields

[TABLE]

Consequently, by (2.15), (2.19), (2.20), and $h^{\prime}(v_{0})\neq 0$, we have $a=0$. ∎

3. Proofs of Theorems 2 and 3

3.1. Non-archimedean dynamics of polynomials of degree $>1$

Let $K$ be an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value $|\cdot|$. The Berkovich projective line $\mathsf{P}^{1}=\mathsf{P}^{1}(K)$ is a compact augmentation of the classical projective line $\mathbb{P}^{1}=\mathbb{P}^{1}(K)$ and is also locally compact, Hausdorff, and uniquely arcwise connected. Let us see more details. As a set, the Berkovich affine line $\mathsf{A}^{1}=\mathsf{A}^{1}(K)$ is the set of all multiplicative seminorms $K[z]$ which restricts to $|\cdot|$ on $K$. We write an element of $\mathsf{A}^{1}$ like $\mathcal{S}$ and denote it by $[\cdot]_{\mathcal{S}}$ as a multiplicative seminorm on $K[z]$. A $K$-closed disk is a subset in $K$ written as $B(a,r):=\{z\in K:|z-a|\leq r\}$ for some $a\in K$ and $r\geq 0$; by the strong triangle inequality, for any $b\in B(a,r)$, we have $B(b,r)=B(a,r)$, and for any two $K$-closed disks $B,B^{\prime}$ having non-empty intersection, we have either $B\subset B^{\prime}$ or $B\supset B^{\prime}$. By Berkovich’s representation [5], any element $\mathcal{S}\in\mathsf{A}^{1}$ is induced by a non-increasing and nesting sequence $(B_{n})$ of $K$-closed disks in that

[TABLE]

In particular, each point $a\in K$ is regarded as an element of $\mathsf{A}^{1}$ induced by the (constant sequence of the) $K$-closed disk $B(a,0)=\{a\}$, and more generally, each $K$-closed disk $B$ is regarded as an element of $\mathsf{A}^{1}$ induced by (the constant sequence of) $B$. In particular, $K$ is regarded as a subset of $\mathsf{A}^{1}$. The relative topology of $\mathsf{A}^{1}$ is the weakest topology such that for any $\phi\in K[z]$, $\mathsf{A}^{1}\ni\mathcal{S}\mapsto[\phi]_{\mathcal{S}}\in\mathbb{R}_{\geq 0}$ is continuous, and then $\mathsf{A}^{1}$ is a locally compact, uniquely arcwise connected, Hausdorff topological space. The action on $K$ of a polynomial $h\in K[z]$ continuously extends to $\mathsf{A}^{1}$ as

[TABLE]

preserving $K$ and $\mathsf{A}^{1}\setminus K$ if in addition $\deg h>0$.

As a set, $\mathsf{P}^{1}$ is nothing but $\mathsf{A}^{1}\cup\{\infty\}$, regarding $\mathbb{P}^{1}$ as $K\cup\{\infty\}$, and as a topological space, $\mathsf{P}^{1}$ is identified with the one-point compactification of $\mathsf{A}^{1}$. An ordering $\leq_{\infty}$ on $\mathsf{A}^{1}$ is defined so that for any $\mathcal{S},\mathcal{S}^{\prime}\in\mathsf{A}^{1}$, $\mathcal{S}\leq_{\infty}\mathcal{S}^{\prime}$ if and only if $[\cdot]_{\mathcal{S}}\leq_{\infty}[\cdot]_{\mathcal{S}^{\prime}}$ on $K[z]$, and this $\leq_{\infty}$ extends to the ordering on $\mathsf{P}^{1}$ so that $\mathcal{S}\leq_{\infty}\infty$ for every $\mathcal{S}\in\mathsf{P}^{1}$. For any $\mathcal{S},\mathcal{S}^{\prime}\in\mathsf{P}^{1}$, if $\mathcal{S}\leq_{\infty}\mathcal{S}^{\prime}$, then set $[\mathcal{S},\mathcal{S}^{\prime}]=[\mathcal{S}^{\prime},\mathcal{S}]:=\{\mathcal{S}^{\prime\prime}\in\mathsf{P}^{1}:\mathcal{S}\leq_{\infty}\mathcal{S}^{\prime\prime}\leq_{\infty}\mathcal{S}^{\prime}\}$, and in general, we have $[\mathcal{S},\infty]\cap[\mathcal{S}^{\prime},\infty]=[\mathcal{S}\wedge_{\infty}\mathcal{S}^{\prime},\infty]$, for some (unique) $\mathcal{S}\wedge_{\infty}\mathcal{S}^{\prime}\in\mathsf{P}^{1}$, and then set $[\mathcal{S},\mathcal{S}^{\prime}]:=[\mathcal{S},\mathcal{S}\wedge_{\infty}\mathcal{S}^{\prime}]\cup[\mathcal{S}\wedge_{\infty}\mathcal{S}^{\prime},\mathcal{S}^{\prime}]$. These closed intervals $[\mathcal{S},\mathcal{S}^{\prime}]\subset\mathsf{P}^{1}$ make $\mathsf{P}^{1}$ an “$\mathbb{R}$-”tree in the sense of Jonsson [16, Definition 2.2]. For any $\mathcal{S}\in\mathsf{P}^{1}$, the equivalence class $T_{\mathcal{S}}\mathsf{P}^{1}:=(\mathsf{P}^{1}\setminus\{\mathcal{S}\})/\sim$ is defined so that for any $\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\in\mathsf{P}^{1}\setminus\{\mathcal{S}\}$, $\mathcal{S}^{\prime}\sim\mathcal{S}^{\prime\prime}$ if $[\mathcal{S},\mathcal{S}^{\prime}]\cap[\mathcal{S},\mathcal{S}^{\prime\prime}]=[\mathcal{S},\mathcal{S}^{\prime}\wedge_{\mathcal{S}}\mathcal{S}^{\prime\prime}]$ for some (unique) $\mathcal{S}^{\prime}\wedge_{\mathcal{S}}\mathcal{S}^{\prime\prime}\in\mathsf{P}^{1}\setminus\{\mathcal{S}\}$. An element $\boldsymbol{v}$ of $T_{\mathcal{S}}\mathsf{P}^{1}$ is called a direction of $\mathsf{P}^{1}$ at $\mathcal{S}$, which is denoted by $U(\boldsymbol{v})$ as a subset in $\mathsf{P}^{1}\setminus\{\mathcal{S}\}$ and, if $\mathcal{S}^{\prime}\in U(\boldsymbol{v})$, also by $\overrightarrow{\mathcal{S}\mathcal{S}^{\prime}}$. A non-empty subset in $\mathsf{P}^{1}$ is called a simple domain if it is the intersection of some finitely many elements of $\{U(\boldsymbol{v}):\mathcal{S}\in\mathsf{P}^{1},\boldsymbol{v}\in T_{\mathcal{S}}\mathsf{P}^{1},\#T_{\mathcal{S}}\mathsf{P}^{1}>1\}$. The topology of $\mathsf{P}^{1}$ has an open basis consisting of all simple domains in $\mathsf{P}^{1}$.

The point $[\cdot]_{\mathcal{O}_{K}}$ in $\mathsf{P}^{1}$, where $\mathcal{O}_{K}:=\{z\in K:|z|\leq 1\}$ is the ring of $K$-integers, is called the Gauss or canonical point in $\mathsf{P}^{1}$ and is denoted by $\mathcal{S}_{\operatorname{can}}$. Let us denote the continuous extension of $|\cdot|$ to $\mathsf{A}^{1}$ by the same $|\cdot|$ for simplicity. More generally, let $|\mathcal{S}-\mathcal{S}^{\prime}|$ be the Hsia kernel on $\mathsf{A}^{1}$, which is the upper semicontinuous and separately continuous extension to $\mathsf{A}^{1}\times\mathsf{A}^{1}$ of the function $|z-w|$ on $K\times K$ (although $\mathcal{S}-\mathcal{S}^{\prime}$ itself is undefined unless $\mathcal{S},\mathcal{S}^{\prime}\in K$), and then the function $\log|\mathcal{S}-\mathcal{S}^{\prime}|-\log\max\{1,|\mathcal{S}|\}-\log\max\{1,|\mathcal{S}^{\prime}|\}$ on $\mathsf{P}^{1}\times\mathsf{P}^{1}$ is the generalized Hsia kernel on $\mathsf{P}^{1}$ with respect to $\mathcal{S}_{\operatorname{can}}$, which is the upper semicontinuous and separately continuous extension to $\mathsf{P}^{1}\times\mathsf{P}^{1}$ of the (normalized) chordal metric on $\mathbb{P}^{1}$ ([3, §4.4]).

The function $\log\max\{1,|\cdot|\}$ on $\mathsf{A}^{1}=\mathsf{P}^{1}\setminus\{\infty\}$ extends superharmonically near $\infty$ so that

[TABLE]

Here, the Laplacian on $\mathsf{P}^{1}$ is denoted by $\Delta:=\Delta_{\mathsf{P}^{1}}$ (in [3] the opposite sign convention on $\Delta$ is adopted). A function $g:\mathsf{P}^{1}\to\mathbb{R}\cup\{\pm\infty\}$ is said to be $\delta_{\mathcal{S}_{\operatorname{can}}}$-subharmonic if

[TABLE]

is a probability Radon measure on $\mathsf{P}^{1}$; for example, $-\log\max\{1,|\cdot|\}$ is a $\delta_{\mathcal{S}_{\operatorname{can}}}$-subharmonic function on $\mathsf{P}^{1}$. If in addition $g$ is an $\mathbb{R}$-valued continuous function on $\mathsf{P}^{1}$, then the function

[TABLE]

is constant on $\mathsf{P}^{1}$ (see [3, Proposition 8.70]).

The continuous action on $\mathbb{P}^{1}$ of a rational function $h\in K(z)$ canonically extends to $\mathsf{P}^{1}$. If in addition $h$ is non-constant, then the action of $h$ on $\mathsf{P}^{1}$ preserves both $\mathbb{P}^{1}$ and $\mathsf{P}^{1}\setminus\mathbb{P}^{1}$ and is open and surjective. The local degree function $w\mapsto\deg_{w}h$ on $\mathbb{P}^{1}$ also canonically extends to an upper semi-continuous function on $\mathsf{P}^{1}$, satisfying $\sum_{\mathcal{S}^{\prime}\in h^{-1}(\mathcal{S})}\deg_{\mathcal{S}^{\prime}}h=\deg h$ for each $\mathcal{S}\in\mathsf{P}^{1}$. In particular, the action of $h$ on $\mathsf{P}^{1}$ induces the pull-back action on the space of Radon measures on $\mathsf{P}^{1}$ so that, letting $\delta_{\mathcal{S}}$ be the Dirac measure on $\mathsf{P}^{1}$ at each $\mathcal{S}\in\mathsf{P}^{1}$, $h^{*}\delta_{\mathcal{S}}=\sum_{\mathcal{S}^{\prime}\in h^{-1}(\mathcal{S})}(\deg_{\mathcal{S}^{\prime}}h)\delta_{\mathcal{S}^{\prime}}$ on $\mathsf{P}^{1}$.

Let $f\in K[z]$ be a polynomial of degree $d>1$. The Berkovich filled-in Julia set of $f$ is

[TABLE]

which is a compact subset in $\mathsf{A}^{1}$, and the escape rate function of $f$ on $\mathsf{A}^{1}$ is the limit $g_{f}:=\lim_{n\to\infty}(\log\max\{1,|f^{n}|\})/d^{n}$ on $\mathsf{A}^{1}$. The difference $g_{f}-(\log\max\{1,|f^{n}|\})/d^{n}$ on $\mathsf{A}^{1}$ is harmonic and bounded on a neighborhood of $\infty$, so it extends harmonically across $\infty$ (see e.g. [3, §7]), and we have the estimate

[TABLE]

on $\mathsf{P}^{1}$ uniformly. The function $g_{f}$ is continuous, subharmonic, and $\geq 0$ on $\mathsf{A}^{1}$, it is harmonic and $>0$ on $\mathsf{A}^{1}\setminus\mathsf{K}(f)$, and it is $=0$ on $\mathsf{K}(f)$. The equilibrium (or canonical) measure of $f$ is the probability Radon measure

[TABLE]

which is supported exactly by $\partial\mathsf{K}(f)$. The Berkovich superattractive basin

[TABLE]

of $f$ associated to the superattracting fixed point $\infty$ of $f$ is a domain in $\mathsf{P}^{1}$ containing $\infty$, and coincides with $\mathsf{P}^{1}\setminus\mathsf{K}(f)$. Let $C(f)$ be the (classical) critical set of $f$, which consists of $\infty$ and all the (at most $d-1$) zeros of $f^{\prime}$ on $K$. Then $\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f)\setminus\{\infty\})$ is bounded in $K$.

The Berkovich Julia set of $f$ is defined as

[TABLE]

The Berkovich Fatou set $\mathsf{F}(f)$ of $f$ is defined by $\mathsf{P}^{1}\setminus\mathsf{J}(f)$, and a component of $\mathsf{F}(f)$ is called a Berkovich Fatou component of $f$. Both $\mathsf{J}(f)$ and $\mathsf{F}(f)$ are totally invariant under $f$ and any Berkovich Fatou component of $f$ is either $\mathsf{I}_{\infty}(f)$ or a component of the interior of $\mathsf{K}(f)$.

Set $c_{d}:=\lim_{K\ni z\to\infty}f(z)/z^{d}\in K^{*}=K\setminus\{0\}$. Since $g_{f}-\log\max\{1,|\cdot|\}$ is an $\mathbb{R}$-valued continuous and $\delta_{\mathcal{S}_{\operatorname{can}}}$-subharmonic function on $\mathsf{P}^{1}$ and satisfies $\Delta(g_{f}-\log\max\{1,|\cdot|\})+\delta_{\mathcal{S}_{\operatorname{can}}}=\mu_{f}$ on $\mathsf{P}^{1}$, by (3.4), the function $\mathcal{S}\mapsto\int_{\mathsf{P}^{1}}\log|\mathcal{S}-\mathcal{S}^{\prime}|\mu_{f}(\mathcal{S}^{\prime})-g_{f}(\mathcal{S})$ is constant on $\mathsf{P}^{1}$. This with (3.5) yields the identity

[TABLE]

For more details on the harmonic analysis and dynamics on $\mathsf{P}^{1}$, see [3, 11].

3.2. Arithmetic dynamics of polynomials of degree $>1$

Let $k$ be a product formula field as in Subsection 1.3. Let $f\in k[z]$ be a polynomial of degree $d>1$. For each $v\in M_{k}$, we obtain $g_{f,v}$ and $\mu_{f,v}$ on $\mathsf{P}^{1}(\mathbb{C}_{v})$ from the action of $f$ on $\mathbb{P}^{1}(\mathbb{C}_{v})$. Writing $f(z)$ as $\sum_{j=0}^{d}c_{j}z^{j}\in k[z]$, so $c_{d}\in k^{*}$, there is a finite set $E_{f}$ containing all the infinite places of $k$ such that for every $v\in M_{k}\setminus E_{f}$, $|c_{d}|_{v}=1,|c_{0}|_{v},\ldots,|c_{d-1}|_{v}\leq 1$, and moreover, $g_{f,v}=\log\max\{1,|\cdot|_{v}\}$ and $\mu_{f,v}=\delta_{\mathcal{S}_{\operatorname{can},v}}$ on $\mathsf{P}^{1}(\mathbb{C}_{v})$.

Recall that an embedding of $\overline{k}$ to $\mathbb{C}_{v}$ is fixed for each $v\in M_{k}$. The Call-Silverman $f$-canonical height of an effective $k$-divisor $\mathcal{Z}$ on $\mathbb{P}^{1}(\overline{k})$ supported by $\overline{k}$ is

[TABLE]

where $p\in k[z]$ is a representative of $\mathcal{Z}$ and the naive height

[TABLE]

of $\mathcal{Z}$ is in fact a finite sum by a standard argument involving the ramification theory of valuations (or [18, Lemma 2.3]). For every $v\in M_{k}$, setting $a_{p}:=p^{(\deg p)}/(\deg p)!\in k^{*}$, we have $\log|p(\cdot)|_{v}=\sum_{z\in\overline{k}:p(z)=0}(\deg p)\log|\cdot-z|_{v}+\log|a_{p}|_{v}$ on $\mathsf{A}^{1}(\mathbb{C}_{v})$, integrating both sides in which against $\mu_{f,v}$ over $\mathsf{P}^{1}(\mathbb{C}_{v})$, also by (3.6), we have

[TABLE]

Consequently, also by the product formula property of $k$, the defining equality (3.7) of $\hat{h}_{f}(\mathcal{Z})$ is rewritten as the Mahler-type formula

[TABLE]

(cf. [18, (1.1)]). For more details on canonical heights on $\mathsf{P}^{1}$, see [1, 2, 11, 8]. For the treatment of effective divisors rather than Galois conjugacy classes, which are effective divisors represented by irreducible polynomials, see [18].

3.3. Proofs of Theorems 2 and 3

Let $K$ be an algebraically closed field of characteristic [math] that is complete with respect to a non-trivial and non-archimedean absolute value $|\cdot|$. Let $f\in K[z]$ be a polynomial of degree $d>1$, and fix $m\in\mathbb{N}$.

The following is a non-archimedean counterpart to Lemma 2.1.

Lemma 3.1**.**

We have

[TABLE]

on $\mathsf{I}_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$ locally uniformly. Moreover, for every $a\in K$, the family \bigl{(}(\log|(f^{n})^{(m)}-a|)/(d^{n}-m)-\log\max\{1,|\cdot|\}\bigr{)}_{n} of $\delta_{\mathcal{S}_{\operatorname{can}}}$-subharmonic functions on $\mathsf{P}^{1}$ is locally uniformly bounded from above on $\mathsf{P}^{1}$ and

[TABLE]

on $\mathsf{I}_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$ locally uniformly.

Proof.

Fixing $r\gg 1$, there is a $($rigid$)$ biholomorphism $w=\psi(z)$ from $\mathbb{P}^{1}\setminus\{g_{f}\leq r\}$ to $\mathbb{P}^{1}\setminus\{|w|\leq e^{r}\}$, which is called a (non-archimedean) Böttcher coordinate near $\infty$ associated to $f$, such that $\psi(f(z))=\psi(z)^{d}$ on $\mathbb{P}^{1}\setminus\{g_{f}\leq r\}$ (see Rivera-Letelier [23, the proof of Proposition 3.3(ii)]). Then $\psi(\infty)=\infty$ and $\psi^{\prime}\neq 0$ on $\mathbb{P}^{1}\setminus\{g_{f}\leq r\}$. By a computation similar to that in the proof of Lemma 2.1, we have

[TABLE]

on $K\setminus\{g_{f}\leq r\}$ uniformly.

For any simple domain $D\Subset I_{\infty}(f)\cap\mathsf{A}^{1}$ and any $M\in\mathbb{N}\cup\{0\}$ so large that $f^{M}(D)\subset\mathsf{P}^{1}\setminus\{g_{f}\leq r\}$, from (2.3*′*), we also have

[TABLE]

on $D\cap\mathbb{P}^{1}$ uniformly. Fix now $m\in\mathbb{N}$. Then noting that, by the definition of a simple domain, there is $0<\epsilon\ll 1$ such that $B(z,\epsilon)\subset D\cap\mathbb{P}^{1}$ for any $z\in D\cap\mathbb{P}^{1}$, an induction which is similar to that in the proof of Lemma 2.1 and involves the almost straightforward (non-archimedean) Cauchy’s estimate for (rigid) analytic functions on those disks $B(z,\epsilon)$ yields

[TABLE]

on $D\cap\mathbb{P}^{1}$ uniformly. If in addition $D\Subset I_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$, so $\inf_{D}|\frac{\psi^{\prime}}{\psi}\circ f^{M}\cdot(f^{M})^{\prime}|>0$, then this (2.4*′*) yields the asymptotic estimate (2.1*′*) on $D\cap\mathbb{P}^{1}$ uniformly, and in turn on $D$ uniformly by the continuity of $|(f^{n})^{(m)}/f^{n}|$ on $D$ and the density of $\mathbb{P}^{1}$ in $\mathsf{P}^{1}$.

Also fix $a\in K$. The locally uniform convergence (2.2*′*) on $\mathsf{I}_{\infty}(f)\setminus\bigcup_{n\in\mathbb{N}\cup\{0\}}f^{-n}(C(f))$ follows from the estimate (2.1*′*). In particular, for $R\gg 1$, letting $\mathcal{S}_{R}\in[0,\infty]\setminus\mathbb{P}^{1}$ be the point in $\mathsf{P}^{1}\setminus\mathbb{P}^{1}$ induced by the (constant sequence of the) $K$-closed disk $B(0,R)$ (so $T_{\mathcal{S}_{R}}\mathsf{P}^{1}\supset\{\overrightarrow{\mathcal{S}_{R}0},\overrightarrow{\mathcal{S}_{R}\infty}\}$), we have the convergence (2.2*′*) at $\mathcal{S}=\mathcal{S}_{R}$, and in turn, by the maximum principle for subharmonic functions (cf. [3, Proposition 8.14]), the family \bigl{(}\log|(f^{n})^{(m)}-a|/(d^{n}-m)\bigr{)}_{n} is uniformly bounded from above on $U(\overrightarrow{\mathcal{S}_{R}0})$ (whose boundary is $\{\mathcal{S}_{R}\}$). Similarly, for $R\gg 1$, noting that \log\bigl{|}(f^{n})^{(m)}/f^{n}\bigr{|} is a subharmonic function on $U(\overrightarrow{\mathcal{S}_{R}\infty})$ (whose boundary is $\{\mathcal{S}_{R}\}$), by the maximum principle for subharmonic functions (and (3.5)), we have

[TABLE]

on $U(\overrightarrow{\mathcal{S}_{R}\infty})$ uniformly. Hence the family \bigl{(}(\log|(f^{n})^{(m)}-a|)/(d^{n}-m)-\log\max\{1,|\cdot|\}\bigr{)}_{n} is locally uniformly bounded from above on $\mathsf{P}^{1}$. ∎

Fix also $a\in K$. By the second and the last assertions in Lemma 3.1, a compactness principle for a family of $\delta_{\mathcal{S}_{\operatorname{can}}}$-subharmonic functions on $\mathsf{P}^{1}$ (cf. [11, Proposition 2.18], [3, Proposition 8.57]) yields a sequence $(n_{j})$ in $\mathbb{N}$ tending to $\infty$ as $j\to\infty$ and a function $\phi:\mathsf{P}^{1}\to\mathbb{R}\cup\{-\infty\}$ such that

[TABLE]

and that

[TABLE]

is a probability Radon measure on $\mathsf{P}^{1}$. By (2.2*′*), we have $\phi\equiv 0$ on $\mathsf{I}_{\infty}(f)\setminus\mathbb{P}^{1}$, and in turn $\phi\equiv 0$ on $\mathsf{I}_{\infty}(f)$ by the subharmonicity of $\phi=(\phi+g_{f})-g_{f}$ on $\mathsf{I}_{\infty}(f)\cap\mathsf{A}^{1}$ and the maximum principle for subharmonic functions (cf. [3, Proposition 8.14]). Then also by $\mathsf{I}_{\infty}(f)=\{g_{f}>0\}$, the subharmonicity of $\phi+g_{f}$ on $\mathsf{A}^{1}$, and the maximum principle for subharmonic functions again, we have $\phi+g_{f}\leq\max_{\{g_{f}=\epsilon\}}(\phi+g_{f})=0+\epsilon=\epsilon$ on $\mathsf{K}(f)=\{g_{f}=0\}\subset\{g_{f}<\epsilon\}$ for every $\epsilon>0$, so that $\phi(=\phi+g_{f})\leq 0$ on $\mathsf{K}(f)$. Then we also have $\phi\equiv 0$ on $\mathsf{J}(f)=\partial\mathsf{K}(f)$.

Let us see that

[TABLE]

which will be used in the proof of Theorem 3; indeed,

[TABLE]

where the first inequality is by $\operatorname{supp}\mu_{f}=:\mathsf{J}(f)$, and the second one is by a version of Hartogs’s lemma for a sequence of $\delta_{\mathcal{S}_{\operatorname{can}}}$-subharmonic functions on $\mathsf{P}^{1}$ (cf. [11, Proposition 2.18], [3, Proposition 8.57]).

Proof of Theorem 2.

Let us continue the above argument. Suppose now that the open subset $\{\phi<0\}$ is non-empty. Then since $\phi\equiv 0$ on $\mathsf{I}_{\infty}(f)$, there is a Berkovich Fatou component $U$ of $f$ other than $\mathsf{I}_{\infty}(f)$ (so $U\Subset\mathsf{A}^{1}$) such that $U\cap\{\phi<0\}\neq\emptyset$, and $\partial U$ is a singleton, say $\{\mathcal{S}_{0}\}$, in $\mathsf{P}^{1}\setminus\mathbb{P}^{1}$ (see [21, Lemma 2.1]). Moreover, $\phi\equiv 0$ on $\partial U\subset\mathsf{J}(f)$.

Assume in addition that $f$ has no potentially good reductions. Then in particular, $\mu_{f}(\partial U)(=\mu_{f}(\{\mathcal{S}_{0}\}))=0$. Now setting

[TABLE]

and checking that the function $\psi+g_{f}$ is subharmonic on $\mathsf{A}^{1}$, we conclude $\psi\equiv 0$ on $\mathsf{P}^{1}$ by an argument similar to that in [21, Proof of Theorem 1] involving a Bedford-Taylor-type domination principle (see [21, §4]). This contradicts $U\cap\{\phi<0\}\neq\emptyset$.

Hence $\phi\equiv 0$ on $\mathsf{P}^{1}$ under the no potentially good reductions condition on $f$. Then (1.3) follows from the equality

[TABLE]

and a continuity of the Laplacian $\Delta$. ∎

Proof of Theorem 3.

Let $k$ be a product formula field of characteristic [math] and let $f\in k[z]$ be a polynomial of degree $d>1$. Recall that, writing $f(z)$ as $\sum_{j=0}^{d}c_{j}z^{j}\in k[z]$, so $c_{d}\in k^{*}$, there is a finite subset $E_{f}$ in $M_{k}$ containing all the infinite places of $k$ such that for every $v\in M_{k}\setminus E_{f}$,

[TABLE]

and moreover, $g_{f,v}=\log\max\{1,|\cdot|_{v}\}$ and $\mu_{f,v}=\delta_{\mathcal{S}_{\operatorname{can},v}}$ on $\mathsf{P}^{1}(\mathbb{C}_{v})$, regarding $f\in\mathbb{C}_{v}[z]$.

Fix $m\in\mathbb{N}$ and $a\in k$. For every $n\in\mathbb{N}$, $(f^{n})^{(m)}\in(\mathbb{Z}[c_{0},\ldots,c_{d}])[z]$ by induction. By the product formula property of $k$, there is an at most finite (and possibly empty) subset $E_{a}$ in $M_{k}$ such that for every $v\in M_{k}\setminus E_{a}$, $|a|_{v}\in\{0,1\}$. Then for every $n\in\mathbb{N}$ and every $v\in M_{k}\setminus(E_{f}\cup E_{a})$, we have

[TABLE]

(see (3.1) and (3.2) for the first equality), which with the second assertions in Lemmas 3.1 and 2.1 (for finite and infinite $v\in M_{k}$, respectively) implies that

[TABLE]

Now by the Mahler-type formula (3.7*′*), Fatou’s lemma, and (3.8), we have

[TABLE]

which with the non-negativity (3.7) of $\hat{h}_{f}$ yields the small $(g_{f,v})_{v\in M_{k}}$-heights property (1.4) of the sequence $([(f^{n})^{(m)}=a])_{n}$ of effective $k$-divisors on $\mathbb{P}^{1}(\overline{k})$.

We note that $\deg[(f^{n})^{(m)}=a]=d^{n}-m\to\infty$ as $n\to\infty$ and that, whenever $v\in M_{k}$ is infinite, we have $\mathbb{C}_{v}\cong\mathbb{C}$. Suppose now that $k$ is a number field and that $a\in k^{*}$, and choose an infinite place $v\in M_{k}$ of $k$. Then from the equidistribution (1.2) of $(((f^{n})^{(m)})^{*}\delta_{a}/(d^{n}-m))_{n}$ towards $\mu_{f,v}$, which has no atoms, on $\mathsf{P}^{1}(\mathbb{C}_{v})\cong\mathbb{P}^{1}(\mathbb{C})$, we have $\sup_{w\in\mathbb{P}^{1}(\overline{k}):(f^{n})^{(m)}(w)=a}\deg_{w}((f^{n})^{(m)})=o((\deg[(f^{n})^{(m)}=a]))$ as $n\to\infty$, so in particular the small diagonal property

[TABLE]

of $([(f^{n})^{(m)}=a])_{n}$. Now the uniform asymptotically $(g_{f,v})_{v\in M_{k}}$-Fekete configuration property (1.5) of $([(f^{n})^{(m)}=a])_{n}$ holds (see [19, Theorem 1]), so in particular the adelic equidistribution (1.6) holds. ∎

4. Proof of Theorem 4

Let us first show a slightly more general equidistribution statement (1.7*′*) under the following normalization (4.1) below. Let $f$ be a Hénon-type polynomial automorphism of $\mathbb{C}^{2}$ of degree $d>1$ normalized as

[TABLE]

Then the function

[TABLE]

extends pluriharmonically to an open neighborhood of $L_{\infty}\setminus I^{+}$ in $\mathbb{P}^{2}$ ([10, Theorem 6.1]). Moreover, for every $n\in\mathbb{N}$, writing $f^{n}$ as

[TABLE]

we have $\deg P_{n}=\deg_{z}P_{n}=d^{n}>\deg Q_{n}$ ([10, Proposition 5.11]), and then

[TABLE]

on $B^{+}\cap\mathbb{C}^{2}$ locally uniformly, recalling also that $\lim_{n\to\infty}f^{n}=[0:1:0]$ on $B^{+}$ locally uniformly.

Fix a $2\times 2$ matrix $A=\begin{pmatrix}a_{1}&a_{2}\\ a_{3}&a_{4}\end{pmatrix}\in\mathrm{M}(2,\mathbb{C})$ satisfying the condition

[TABLE]

so that for every $n\in\mathbb{N}$,

[TABLE]

is indeed of degree $d^{n}-1$.

Lemma 4.1**.**

For each $j\in\{z,w\}$,

[TABLE]

on $B^{+}\cap\mathbb{C}^{2}$ locally uniformly.

Proof.

Pick any open concentric bidisks $D\Subset D^{\prime}\Subset B^{+}\cap\mathbb{C}^{2}$, and fix $j\in\{z,w\}$. Let us write $D,D^{\prime}$ as $D_{1}\times D_{2},D_{1}^{\prime}\times D_{2}^{\prime}$, respectively.

By the former half in (4.2), we have $\inf_{D^{\prime}}|P_{n}|>0$ if $n\gg 1$. We claim that

[TABLE]

on $\overline{D}$ uniformly; indeed, for every $z\in\overline{D_{1}}$, using Poisson’s integral of the function $w\mapsto g^{+}(z,w)-d^{-n}\log|P_{n}(z,w)|$ on $\partial D_{2}^{\prime}$, the former half in (4.2) yields the asymptotic estimate (4.6) on $\{z\}\times\overline{D_{2}}$ uniformly, and moreover, the implicit constant in $O$ depends only on $D$. Hence the claim holds. In particular, the former half in (4.5) holds.

Similarly, using the latter half in (4.2) twice and Cauchy’s integral of the function $Q_{n}/P_{n}$ on $\partial{D_{1}^{\prime}}\times\partial{D_{2}^{\prime}}$, we also have

[TABLE]

on $\overline{D}$ uniformly, which together with (4.6) and $\sup_{D}|\partial_{j}g^{+}|<\infty$ yields

[TABLE]

on $\overline{D}$ uniformly. Hence the latter half in (4.5) also holds. ∎

By the pluriharmonicity of $g^{+}$ on $B^{+}$, the function $a_{4}\partial_{z}g^{+}-a_{3}\partial_{w}g^{+}$ is holomorphic on $B^{+}\cap\mathbb{C}^{2}$. Set

[TABLE]

Recall the assumption that $a_{4}\neq 0$.

Lemma 4.2**.**

$Y$* is an analytic hypersurface in $B^{+}\cap\mathbb{C}^{2}$, no irreducible component of which is horizontal, i.e., $\{w=w_{0}\}$ for some $w_{0}\in\mathbb{C}$.*

Proof.

Let us first show that $Y$ is not equal to $B^{+}\cap\mathbb{C}^{2}$. Suppose to the contrary that $a_{4}\partial_{z}g^{+}-a_{3}\partial_{w}g^{+}\equiv 0$ on $B^{+}\cap\mathbb{C}^{2}$. Then letting $L$ be the complex affine line $w=-(a_{3}/a_{4})z$ in $\mathbb{C}^{2}$, there is $c\in\mathbb{R}$ such that $g^{+}\equiv c$ on $L\cap B^{+}$. On the other hand, since the projective line $\overline{L}$ in $\mathbb{P}^{2}$ intersects $L_{\infty}$ at $[0:1:-a_{3}/a_{4}]\in L_{\infty}\setminus I^{+}$, near which $g^{+}(z,w)-\log\max\{1,|z|\}$ extends pluriharmonically, we must have $c=g^{+}(z,w)=\log\max\{1,|z|\}+O(1)\to\infty$ as $L\cap B^{+}\ni(z,w)\to[0:1:-a_{3}/a_{4}]$. This is a contradiction. Hence the former assertion holds.

The latter assertion is shown similarly noting that the closure of any horizontal line intersects $L_{\infty}$ at $[0:1:0]\in L_{\infty}\setminus I^{+}$. ∎

Recall the computation (4.4) of the polynomial $\det(D(f^{n})-A)\in\mathbb{C}[z,w]$ of degree $d^{n}-1$. For every $n\in\mathbb{N}$, set

[TABLE]

which is a plurisubharmonic function on $\mathbb{C}^{2}$ and satisfies $\mathrm{d}\mathrm{d}^{c}\phi_{n}=[\det(D(f^{n})-A)]/(d^{n}-1)$ as currents on $\mathbb{C}^{2}$ by the Poincaré-Lelong formula.

Lemma 4.3**.**

We have $\phi_{n}=g^{+}+O(nd^{-n})$ as $n\to\infty$ on $B^{+}\cap(\mathbb{C}^{2}\setminus Y)$ locally uniformly. Moreover, the family $(\phi_{n})_{n}$ is locally uniformly bounded from above on $\mathbb{C}^{2}$.

Proof.

First, pick any open bidisk $D\Subset B^{+}\cap(\mathbb{C}^{2}\setminus Y)$. Then by (4.5) and the former half in (4.2), we have

[TABLE]

on $\overline{D}$ uniformly, and then using the former half in (4.2) again and $D\Subset B^{+}\cap(\mathbb{C}^{2}\setminus Y)$, we have

[TABLE]

on $\overline{D}$ uniformly. Hence the former assertion holds.

Fix $(z_{0},w_{0})\in\mathbb{C}^{2}$. By $L_{\infty}\setminus I^{+}\subset B_{+}$ and the latter half in Lemma 4.2, we have $\{|z-z_{0}|=r\}\times\{|w-w_{0}|=\epsilon\}\subset B^{+}\cap(\mathbb{C}^{2}\setminus Y)$ for $r\gg 1$ and $0<\epsilon\ll 1$, so that by the former assertion and the maximum principle for the plurisubharmonic function $\phi_{n}$ on $\mathbb{C}^{2}$, we have

[TABLE]

Hence the latter assertion also holds. ∎

Let us see

[TABLE]

as currents. First, let $\tilde{S}=\lim_{j\to\infty}[\det(D(f^{n_{j}})-A)]/(d^{n_{j}}-1)$ be any limit point, which is also a positive closed $(1,1)$-current on $\mathbb{P}^{2}$ of mass $1$, of the sequence $([\det(D(f^{n})-A)]/(d^{n}-1))_{n}$ of positive closed $(1,1)$-currents on $\mathbb{P}^{2}$ of masses $1$. On the other hand, by Lemma 4.3 and the compactness principle for plurisubharmonic functions on a domain in $\mathbb{C}^{N}$, taking a subsequence of $(n_{j})$ if necessary, there is a plurisubharmonic function $\phi$ on $\mathbb{C}^{2}$ such that $\phi=\lim_{j\to\infty}\phi_{n_{j}}$ in $L^{1}_{\mathrm{loc}}(\mathbb{C}^{2},m_{4})$, where $m_{4}$ is the Lebesgue measure on $\mathbb{C}^{2}$. Then we have $\tilde{S}|\mathbb{C}^{2}=\mathrm{d}\mathrm{d}^{c}\phi$ on $\mathbb{C}^{2}$ and, by the former half in Lemma 4.3, the plurisubharmonicity of $\phi$ on $\mathbb{C}^{2}$, and the pluriharmonicity of $g^{+}$ on $B^{+}$, we also have $\phi\equiv g^{+}$ on $B^{+}\cap\mathbb{C}^{2}$. Hence $\operatorname{supp}(\tilde{S}|\mathbb{C}^{2})\subset K^{+}$. Next, let $S$ be the trivial extension of $\mathrm{d}\mathrm{d}^{c}\phi$ to $\mathbb{P}^{2}$ across $L_{\infty}$. It is a positive closed $(1,1)$-current on $\mathbb{P}^{2}$ (cf. [10, Theorem 2.7]) and supported by $\overline{K^{+}}=K^{+}\cup I^{+}$. Then by the uniqueness of $T^{+}$ mentioned above among such currents, there is $c\geq 0$ such that $S=c\cdot T^{+}$ on $\mathbb{P}^{2}$. Moreover, for the current of integration $[L]$ along any projective line $L\subset\mathbb{P}^{2}\setminus I^{+}$ other than $L_{\infty}$ and passing through $I^{-}$, if $R\gg 1$, then we have $\phi\equiv g^{+}$ on $\{(z,w)\in\mathbb{C}^{2}:\|(z,w)\|>R-1\}\cap L\subset B^{+}$, and in turn, recalling the definition of $S,T^{+}$ and using Stokes’s formula, we have

[TABLE]

(cf. [10, Proof of Lemma 6.3]). Hence $S=T^{+}$ on $\mathbb{P}^{2}$. Consequently, $S|\mathbb{C}^{2}=T^{+}|\mathbb{C}^{2}=\mathrm{d}\mathrm{d}^{c}\phi=\tilde{S}|\mathbb{C}^{2}$ on $\mathbb{C}^{2}$, and then $\tilde{S}\geq S$ on $\mathbb{P}^{2}$ by their construction. Since both $\tilde{S},S$ are of masses $1$, we conclude that $\tilde{S}=S=T^{+}$ on $\mathbb{P}^{2}$. Hence (1.7*′*) holds.

Proof of Theorem 4.

Let $f$ be a Hénon-type polynomial automorphism of $\mathbb{C}^{2}$ of degree $d>1$. Fix $\lambda\in\mathbb{C}^{*}$, and set $A=\lambda I_{2}\in\mathrm{M}(2,\mathbb{C})$. Then using the chain rule and the equivariance of $T^{+}$ under affine coordinate changes on $\mathbb{C}^{2}$, we can assume that $f$ satisfies the normalization (4.1), without loss of generality. Noting also that $A=\lambda I_{2}$ satisfies the condition (4.3), the desired (1.7) as currents on $\mathbb{P}^{2}$ is nothing but (1.7*′*) as currents on $\mathbb{P}^{2}$ for this $A=\lambda I_{2}$. ∎

Acknowledgement**.**

The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (C), 15K04924. The authors were partially supported by Invitational Fellowships for Research in Japan (Short-term S18024) JSPS BRIDGE Fellowship 2018, and thank for the hospitality of Université de Picardie Jules Verne and Kyoto Institute of Technology, where they visited each other in 2018 and this work grew up.