2.1 Some reductions

We start with the following important reduction.

Proof. First we prove that it suffices to assume in each of the Theorems 1.1, 1.2 and 1.3 that $V\subset (\mathbb{P}^{1})^{n}$ is a hypersurface. Indeed, we assume Theorems 1.1, 1.2 and 1.3 hold for all hypersurfaces and we derive the same conclusion for all subvarieties of $(\mathbb{P}^{1})^{n}$ . So, let $V\subset (\mathbb{P}^{1})^{n}$ be an irreducible subvariety of dimension $D<n-1$ satisfying the hypotheses of either Theorem 1.1, or of Theorem 1.3, or hypothesis (1) (or (2)) of Theorem 1.2. Then there exist $D$ coordinate axes (without loss of generality, we assume they are $x_{1},\ldots ,x_{D}$ ) so that the projection $\unicode[STIX]{x1D70B}$ of $(\mathbb{P}^{1})^{n}$ onto its first $D$ coordinate axes remains dominant when restricted to $V$ . For each $j=D+1,\ldots ,n$ , we let $\unicode[STIX]{x1D70B}_{j}$ be the natural projection map of $(\mathbb{P}^{1})^{n}$ on coordinates $x_{1},\ldots ,x_{D},x_{j}$ , and we let $H_{j}:=\unicode[STIX]{x1D70B}_{j}(V)$ . Then $H_{j}\subset (\mathbb{P}^{1})^{D+1}$ is a hypersurface satisfying the hypotheses of Theorem 1.1, or of Theorem 1.3, or hypothesis (1) (or (2)) of Theorem 1.2 with respect to the coordinatewise action of the rational functions $f_{1},\ldots ,f_{D},f_{j}$ . Furthermore, for each $j=D+1,\ldots ,n$ , we let $\tilde{H}_{j}\subset (\mathbb{P}^{1})^{n}$ be the hypersurface $H_{j}\times (\mathbb{P}^{1})^{n-D-1}\subset (\mathbb{P}^{1})^{n}$ (i.e., we insert a copy of $\mathbb{P}^{1}$ on each coordinate axis not included in the set $\{1,\ldots ,D,j\}$ ). Then also $\tilde{H}_{j}\subset (\mathbb{P}^{1})^{n}$ is a hypersurface satisfying the hypotheses of either one of the three Theorems 1.1, 1.2 or 1.3. Let

(2.1.1) $$\begin{eqnarray}\tilde{H}:=\mathop{\bigcap }_{j=D+1}^{n}\tilde{H}_{j};\end{eqnarray}$$

clearly, $V\subset \tilde{H}$ and so, $D=\dim (V)\leqslant \dim (\tilde{H})$ .

Since $\dim (V)=D$ and $\unicode[STIX]{x1D70B}|_{V}:V\longrightarrow (\mathbb{P}^{1})^{D}$ is a dominant morphism, then we conclude that there exists a Zariski open subset $U\subset (\mathbb{P}^{1})^{D}$ such that for each $\unicode[STIX]{x1D6FC}\in U$ , the fibre $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D6FC})$ is finite. Therefore for each $\unicode[STIX]{x1D6FC}\in U$ and for each $j=D+1,\ldots ,n$ , we have that there exists a finite set $S_{\unicode[STIX]{x1D6FC},j}$ with the property that if $(a_{1},\ldots ,a_{n})\in \tilde{H}_{j}$ and $(a_{1},\ldots ,a_{D})=\unicode[STIX]{x1D6FC}$ , then $a_{j}\in S_{\unicode[STIX]{x1D6FC},j}$ . Thus for each $\unicode[STIX]{x1D6FC}\in U$ , we have that there exist finitely many points $(a_{1},\ldots ,a_{n})\in \tilde{H}$ such that $(a_{1},\ldots ,a_{D})=\unicode[STIX]{x1D6FC}$ . Hence $V$ is an irreducible component of $\tilde{H}$ ; moreover, any irreducible component $W$ of $\tilde{H}$ for which $\unicode[STIX]{x1D70B}|_{W}:W\longrightarrow (\mathbb{P}^{1})^{D}$ is a dominant morphism has dimension $D$ .

If Theorem 1.2 holds for hypersurfaces, then each hypersurface $\tilde{H}_{j}\subset (\mathbb{P}^{1})^{n}$ must have the form (1.3.1) since each one of these hypersurfaces satisfies the hypotheses of Theorem 1.2. Actually, since each $\tilde{H}_{j}$ is a hypersurface, then we must have

$$\begin{eqnarray}\tilde{H}_{j}=\unicode[STIX]{x1D70B}_{i,j}^{-1}(C_{i,j})\end{eqnarray}$$

for some curve $C_{i,j}\subset \mathbb{P}^{1}\times \mathbb{P}^{1}$ , which is preperiodic under the action of $(x_{i},x_{j})\mapsto (f_{i}^{\ell _{i}}(x_{i}),f_{j}^{\ell _{j}}(x_{j}))$ for some $\ell _{i},\ell _{j}\in \mathbb{N}$ with the property that $\deg (f_{i}^{\ell _{i}})=\deg (f_{j}^{\ell _{j}})$ . Because $V$ is an irreducible component of $\tilde{H}$ (see (2.1.1)), we obtain the desired conclusion in Theorem 1.2.

Now, if Theorems 1.1 or 1.3 hold for hypersurfaces, then each hypersurface $\tilde{H}_{j}\subset (\mathbb{P}^{1})^{n}$ is preperiodic under the action of $\unicode[STIX]{x1D6F7}:=(f_{1},\ldots ,f_{n})$ . Thus, also $\tilde{H}$ is preperiodic under the action of $\unicode[STIX]{x1D6F7}$ (see (2.1.1)). Combining the following facts:

• ∙ $\tilde{H}$ is preperiodic;

• ∙ $V$ is an irreducible component of $\tilde{H}$ ;

• ∙ each irreducible component $W$ of $\tilde{H}$ for which $\unicode[STIX]{x1D70B}|_{W}:W\longrightarrow (\mathbb{P}^{1})^{D}$ is a dominant morphism has dimension $D$ ; and

• ∙ each variety $\unicode[STIX]{x1D6F7}^{m}(V)$ (for $m\in \mathbb{N}$ ) projects dominantly onto $(\mathbb{P}^{1})^{D}$ ,

we obtain that $V$ itself must be preperiodic under the action of $\unicode[STIX]{x1D6F7}$ , as desired.

Now, once we reduced proving Theorems 1.1, 1.2 and 1.3 to the case $V\subset (\mathbb{P}^{1})^{n}$ is a hypersurface, we can reduce further to the special case when $V$ projects dominantly onto each subset of $(n-1)$ coordinate axes. Indeed, assuming otherwise, then (without loss of generality) we may assume $V=\mathbb{P}^{1}\times V_{0}$ for some hypersurface $V_{0}\subset (\mathbb{P}^{1})^{n-1}$ . Therefore, it suffices to prove Theorems 1.2 and 1.3 for the subvariety $V_{0}\subset (\mathbb{P}^{1})^{n-1}$ under the coordinatewise action of the rational functions $f_{2},\ldots ,f_{n}$ . A simple induction on $n$ finishes our proof. (Finally, as an aside, we observe that in light of [Reference Ghioca, Nguyen and YeGNY17, Theorem 1.1], then due to the reduction proved in Proposition 2.1, we have that Theorem 1.2 is equivalent with proving that if $n>2$ and also if each $f_{i}$ is non-exceptional, then there is no hypersurface $H\subset (\mathbb{P}^{1})^{n}$ projecting dominantly onto each subset of $(n-1)$ coordinate axes of $(\mathbb{P}^{1})^{n}$ such that $H$ contains a Zariski dense set of preperiodic points; this is exactly what we will be proving in Theorems 2.2 and 6.1.)◻

2.2 A technical result

The next result (proven in § 7) in conjunction with [Reference Ghioca, Nguyen and YeGNY17, Theorems 1.1 and 1.3] yields the conclusions of each of the three Theorems 1.1, 1.2 and 1.3.

2.3 Our main results as consequences of the technical result

We show next how to derive Theorems 1.1, 1.2 and 1.3 from Theorem 2.2.

Since both Theorems 1.1 and 1.3 have similar statements and proofs, we will show next in parallel how to derive these two results from Theorem 2.2.

Proof of both Theorems 1.1 and 1.3.

Again using Proposition 2.1, it suffices to prove Theorems 1.1 and 1.3 for irreducible hypersurfaces $V$ , which project dominantly onto each subset of $(n-1)$ coordinate axes of $(\mathbb{P}^{1})^{n}$ . The case $n=2$ in Theorem 1.3 was already proven in [Reference Ghioca, Nguyen and YeGNY17, Theorem 1.3]. On the other hand, note that [Reference Ghioca, Nguyen and YeGNY17, Equation (11)] yields that for a plane curve (neither horizontal, nor vertical) which contains infinitely many preperiodic points under the coordinatewise action of two rational functions $f_{1}$ and $f_{2}$ , if $f_{1}$ is conjugate to a monomial or $\pm$ Chebyshev polynomial, then $f_{2}$ must also be conjugated to a monomial or $\pm$ Chebyshev polynomial. Furthermore, if we assume $n>2$ , then Theorem 2.2 yields that:

1. (i) either for each $i=1,\ldots ,n$ we have that $f_{i}(x)=\unicode[STIX]{x1D708}_{i}^{-1}(x)\circ x^{\pm d}\circ \unicode[STIX]{x1D708}_{i}(x)$ or $f_{i}(x)=\unicode[STIX]{x1D708}_{i}^{-1}(x)\circ (\pm T_{d}(x))\circ \unicode[STIX]{x1D708}_{i}(x)$ for some automorphisms $\unicode[STIX]{x1D708}_{i}:\mathbb{P}^{1}\longrightarrow \mathbb{P}^{1}$ ;

2. (ii) or each $f_{i}$ is a Lattès map corresponding to some elliptic curve $E_{i}$ (for $i=1,\ldots ,n$ ).

Therefore, from now on, we work in both Theorems 1.1 and 1.3 under the assumption that either hypotheses (i) or (ii) above are met.

If condition (i) is satisfied, then at the expense of replacing $V$ by $\tilde{\unicode[STIX]{x1D708}}(V)$ , where $\tilde{\unicode[STIX]{x1D708}}$ is the automorphism of $(\mathbb{P}^{1})^{n}$ given by

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D708}}(x_{1},\ldots ,x_{n}):=(\unicode[STIX]{x1D708}_{1}(x_{1}),\ldots ,\unicode[STIX]{x1D708}_{n}(x_{n})),\end{eqnarray}$$

we may assume that each $f_{i}(x)$ is either $x^{\pm d}$ or $\pm T_{d}(x)$ . Next, let $\unicode[STIX]{x1D707}:\mathbb{G}_{m}^{n}\longrightarrow (\mathbb{P}^{1})^{n}$ be the morphism given by

$$\begin{eqnarray}\unicode[STIX]{x1D707}(x_{1},\ldots ,x_{n})=:(\unicode[STIX]{x1D707}_{1}(x_{1}),\ldots ,\unicode[STIX]{x1D707}_{n}(x_{n})),\end{eqnarray}$$

for rational functions $\unicode[STIX]{x1D707}_{i}$ which are:

• ∙ $\unicode[STIX]{x1D707}_{i}(x)=x$ if $f_{i}(x)=x^{\pm d}$ ; and

• ∙ $\unicode[STIX]{x1D707}_{i}(x)=x+(1/x)$ if $f_{i}(x)=\pm T_{d}(x).$

Then there exists an irreducible subvariety $W$ of $\unicode[STIX]{x1D707}^{-1}(V)\subset \mathbb{G}_{m}^{n}$ (projecting dominantly onto $V$ through the map $\unicode[STIX]{x1D707}$ ), which contains a Zariski dense set of preperiodic points under the action of $\unicode[STIX]{x1D6F7}:\mathbb{G}_{m}^{n}\longrightarrow \mathbb{G}_{m}^{n}$ given by

$$\begin{eqnarray}(x_{1},\ldots ,x_{n})\mapsto (\pm x_{1}^{\pm d},\ldots ,\pm x_{n}^{\pm d}).\end{eqnarray}$$

Hence, $W$ contains a Zariski dense set of torsion points of $\mathbb{G}_{m}^{n}$ . Laurent’s theorem [Reference LaurentLau84] (the original Manin–Mumford conjecture for tori) yields that $W$ is a subtorus, thus preperiodic under the action of $\unicode[STIX]{x1D6F7}$ . This proves that $V=\unicode[STIX]{x1D707}(W)$ is preperiodic under the action of

$$\begin{eqnarray}(x_{1},\ldots ,x_{n})\mapsto (f_{1}(x_{1}),\ldots ,f_{n}(x_{n})),\end{eqnarray}$$

as desired in the conclusion of Theorem 1.1. Furthermore, we note that in this case, the conclusion of Theorem 1.3 holds without the extra hypothesis regarding the preperiodicity of the tangent subspaces under the corresponding induced action; we will only need this extra assumption when dealing with hypothesis (ii) above, i.e., when the maps are Lattés.

Now, we assume condition (ii) is verified and so, each $f_{i}$ is a Lattès map which satisfies $p_{i}\,\circ \,\unicode[STIX]{x1D713}_{i}=f_{i}\circ p_{i}$ where $p_{i}:E_{i}\longrightarrow \mathbb{P}^{1}$ and $\unicode[STIX]{x1D713}_{i}:E_{i}\longrightarrow E_{i}$ are morphisms satisfying $\deg (\unicode[STIX]{x1D713}_{1})=\deg (\unicode[STIX]{x1D713}_{2})=\cdots =\deg (\unicode[STIX]{x1D713}_{n})$ because the Lattès maps $f_{i}$ have the same degree. Then there exists an irreducible component $W$ of $p^{-1}(V)\subset \tilde{E}:=\prod _{i=1}^{n}E_{i}$ (where $p:\tilde{E}\longrightarrow (\mathbb{P}^{1})^{n}$ is the morphism given by $p_{1}\times \cdots \times p_{n}$ ) with the property that it contains a Zariski dense set of (smooth) points $P$ which are preperiodic under the action of the endomorphism $\tilde{\unicode[STIX]{x1D713}}$ of $\tilde{E}$ given by $\unicode[STIX]{x1D713}_{1}\times \cdots \times \unicode[STIX]{x1D713}_{n}$ , and moreover, the tangent space of $W$ at $P$ is preperiodic under the induced action of $\tilde{\unicode[STIX]{x1D713}}$ on $\text{Gr}_{\dim (W)}(T_{\tilde{E},P})$ , where $T_{\tilde{E},P}$ is the tangent space of $\tilde{E}$ at $P$ and $\text{Gr}_{\dim (W)}(T_{\tilde{E},P})$ is the corresponding Grassmannian. Since each $\unicode[STIX]{x1D713}_{i}$ is an isogeny of $E_{i}$ of same degree, we get that $\tilde{\unicode[STIX]{x1D713}}$ is a polarizable endomorphism of $\tilde{E}$ and so, [Reference Ghioca, Tucker and ZhangGTZ11, Theorem 2.1] yields that $W$ is preperiodic under the action of $\tilde{\unicode[STIX]{x1D713}}$ . Therefore $V=p(W)$ is preperiodic under the action of

$$\begin{eqnarray}(x_{1},\ldots ,x_{n})\mapsto (f_{1}(x_{1}),\ldots ,f_{n}(x_{n})),\end{eqnarray}$$

as desired in the conclusion of Theorem 1.3.◻

2.4 Strategy for our proof

The remaining sections of our paper are dedicated to proving Theorem 2.2. The setup is as follows:

• ∙ $n>2$ and $H\subset (\mathbb{P}^{1})^{n}$ is a hypersurface projecting dominantly onto each subset of $(n-1)$ coordinate axes;

• ∙ $f_{1},\ldots ,f_{n}$ are rational functions of degrees larger than $1$ acting coordinatewise on $(\mathbb{P}^{1})^{n}$ ; and

• ∙ $H$ contains a Zariski dense set either of preperiodic points or of points of small height (see (2) in Theorem 1.2) under the action of $(x_{1},\ldots ,x_{n})\mapsto (f_{1}(x_{1}),\ldots ,f_{n}(x_{n}))$ .

If at least one of the functions $f_{i}$ is not exceptional, then we will derive a contradiction. Now, if some $f_{i}$ is conjugated to a monomial or $\pm$ Chebyshev polynomial, then we prove that each of the $n$ rational functions must be conjugated to a monomial or $\pm$ Chebyshev polynomial. Similarly, if one of the functions $f_{i}$ is a Lattés map, then we prove that each $f_{i}$ must be a Lattés map. We obtain this goal (see Theorem 6.1) by showing a similitude between the Julia sets of each one of the rational functions $f_{i}$ . In turn, the relation between the Julia sets is a consequence of a powerful equidistribution theorem for points of small height.

More precisely, using the equidistribution theorem of [Reference YuanYua08] for points of small height on a variety (see [Reference Chambert-LoirCha06] for the case of curves and also [Reference Baker and RumelyBR06] and [Reference Favre and Rivera-LetelierFR06] for the case of $\mathbb{P}^{1}$ ), we prove that under the above hypotheses for $H$ and $f_{i}$ , then the measures $\hat{\unicode[STIX]{x1D707}}_{i}$ induced on $H$ by the invariant measures corresponding to the dynamical systems

$$\begin{eqnarray}((\mathbb{P}^{1})^{n-1},f_{1}\times \cdots \times f_{i-1}\times f_{i+1}\times \cdots \times f_{n})\end{eqnarray}$$

are equal (for each $i=1,\ldots ,n$ ). Using a careful study of the local analytic maps which preserve (locally) the Julia set of a rational map (which is not exceptional), we obtain the conclusion of Theorem 1.2. Even though our arguments resemble the ones we employed in [Reference Ghioca, Nguyen and YeGNY17] to treat the case of plane curves (i.e., $n=2$ ), there are significant new complications in our analysis.

Indeed, using Yuan’s arithmetic equidistribution theorem [Reference YuanYua08] for points with small height on a space of dimension $n\geqslant 2$ , we first get connections for the $(n-1,n-1)$ -currents (coming from dynamics) on a hypersurface $H\subset (\mathbb{P}^{1})^{n}$ . From these connections, we are able to construct many symmetries for the aforementioned $(n-1,n-1)$ -current. A further analysis of the symmetries for such an $(n-1,n-1)$ -current yields additional symmetries of the Julia set on the one-dimensional slices of $(\mathbb{P}^{1})^{n}$ . Applying the rigidity of the symmetries of the Julia set on the one-dimensional slices, we are able to derive the rigidity of the symmetries of the entire $(n-1,n-1)$ -current, from which we derive the desired conclusion regarding $H$ and the dynamical system $(f_{1},\ldots ,f_{n})$ (see the proof of Theorem 6.1). It is precisely the study of the rigidity of this $(n-1,n-1)$ -current (for $n>2$ ) which provides the new proof of Medvedev’s result [Reference MedvedevMed07], which otherwise could not have been obtained from the arguments from our previous paper [Reference Ghioca, Nguyen and YeGNY17].

Also, in order to finish the proof of Theorem 2.2 by showing that the hypotheses of Theorem 6.1 are met, we need to know that for a hypersurface $H\subset (\mathbb{P}^{1})^{n}$ as in Theorem 2.2, for each point $(a_{1},\ldots ,a_{n})\in H$ ,

(2.4.1) $$\begin{eqnarray}\text{if }a_{1},\ldots ,a_{n-1}\text{ are preperiodic, then also }a_{n}\text{ is preperiodic}.\end{eqnarray}$$

If $n=2$ , this fact was known for quite some time (see [Reference MimarMim13] which publishes the findings of Mimar’s PhD thesis [Reference MimarMim97] from 20 years ago). However, if $n>2$ , in order to prove (2.4.1) (see our Proposition 5.2 in the case each $f_{i}$ and also $H$ are defined over $\bar{\mathbb{Q}}$ ), we need to use arithmetic versions of the Hodge index theorem proved by Faltings [Reference FaltingsFal84] and by Hriljac [Reference HriljacHri85] for arithmetic surfaces and proved by Moriwaki [Reference MoriwakiMor96] for higher dimensional arithmetic varieties, and also, we use crucially the new arithmetic Hodge index theorem proved by Yuan and Zhang [Reference Yuan and ZhangYZ17]. Furthermore, in order to derive (2.4.1) in the general case (over $\mathbb{C}$ ) we need a specialization argument based on a result of Yuan and Zhang [Reference Yuan and ZhangYZa, Reference Yuan and ZhangYZb] regarding the specialization of a Zariski dense set of preperiodic points for a polarizable endomorphism defined over a base curve.

3.1 The Julia set

Let $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ be a rational function defined over $\mathbb{C}$ of degree $d_{f}\geqslant 2$ . The Julia set $J_{f}$ is the set of points $x\in \mathbb{P}_{\mathbb{C}}^{1}$ for which the dynamics is chaotic under the iteration of $f$ . The Julia set $J_{f}$ is closed, non-empty and invariant under $f$ . Let $x$ be a periodic point in a cycle of exact period $n$ ; then the multiplier $\unicode[STIX]{x1D706}$ of this cycle (or of the periodic point $x$ ) is the derivative of $f^{n}$ at $x$ . A cycle is repelling if its multiplier has an absolute value greater than $1$ . All but finitely many cycles of $f$ are repelling, and repelling cycles are in the Julia set $J_{f}$ . Locally, at a repelling fixed point $x$ with multiplier $\unicode[STIX]{x1D706}$ , we can conjugate $f$ to the linear map $z\rightarrow \unicode[STIX]{x1D706}\cdot z$ near $z=0$ (note that $\unicode[STIX]{x1D706}\neq 0$ since the point is assumed to be repelling). For more details about the dynamics of a rational function, we refer the reader to Milnor’s book [Reference MilnorMil00].

There is a probability measure $\unicode[STIX]{x1D707}_{f}$ on $\mathbb{P}_{\mathbb{C}}^{1}$ associated with $f$ , which is the unique $f$ -invariant measure achieving maximal entropy $\log d_{f}$ ; see [Reference BrolinBro65, Reference LyubichLyu83, Reference Freire, Lopes and MañéFLM83, Reference MañéMañ83]. Also $\unicode[STIX]{x1D707}_{f}$ is the unique measure satisfying

(3.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{f}(f(A))=d_{f}\cdot \unicode[STIX]{x1D707}_{f}(A)\end{eqnarray}$$

for any Borel set $A\subset \mathbb{P}_{\mathbb{C}}^{1}$ with $f$ injective when restricted on $A$ . The support of $\unicode[STIX]{x1D707}_{f}$ is $J_{f}$ , and $\unicode[STIX]{x1D707}_{f}(x)=0$ for any $x\in \mathbb{P}_{\mathbb{C}}^{1}$ . Moreover, $\unicode[STIX]{x1D707}_{f}$ has continuous potential, in the sense that locally there is a continuous subharmonic function $u(x)$ such that the $(1,1)$ -current satisfies

$$\begin{eqnarray}dd^{c}u(x)=d\unicode[STIX]{x1D707}_{f}(x),\end{eqnarray}$$

and then (3.1.1) is equivalent to

$$\begin{eqnarray}dd^{c}u\circ f(x)=d_{f}\cdot d\unicode[STIX]{x1D707}_{f}(x).\end{eqnarray}$$

3.2 Measures on a hypersurface associated with a dynamical system

Let

$$\begin{eqnarray}\hat{f}(x_{1},\ldots ,x_{n}):=(f_{1}(x_{1}),\ldots ,f_{n}(x_{n}))\end{eqnarray}$$

be an endormorphism of $(\mathbb{P}_{\mathbb{C}}^{1})^{n}$ with $f_{i}$ being a rational function of degree $d_{i}\geqslant 2$ for $1\leqslant i\leqslant n$ . For $i=1,\ldots ,n$ , denote

(3.2.1) $$\begin{eqnarray}\tilde{f}_{i}:=(f_{1},\ldots ,f_{i-1},f_{i+1},\ldots ,f_{n})\end{eqnarray}$$

as an endormorphism of $(\mathbb{P}_{\mathbb{C}}^{1})^{n-1}$ with invariant measure

(3.2.2) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D707}}_{i}:=\unicode[STIX]{x1D707}_{f_{1}}\times \cdots \unicode[STIX]{x1D707}_{f_{i-1}}\times \unicode[STIX]{x1D707}_{f_{i+1}}\times \cdots \unicode[STIX]{x1D707}_{f_{n}}.\end{eqnarray}$$

Let $H\subset (\mathbb{P}_{\mathbb{C}}^{1})^{n}$ be an irreducible hypersurface projecting dominantly onto any subset of $(n-1)$ coordinates, i.e., the canonical projections $\hat{\unicode[STIX]{x1D70B}}_{i}:(\mathbb{P}^{1})^{n}\rightarrow (\mathbb{P}^{1})^{n-1}$ (where for each $i=1,\ldots ,n$ , $\hat{\unicode[STIX]{x1D70B}}_{i}$ is the projection of $(\mathbb{P}^{1})^{n}$ onto the $(n-1)$ coordinates forgetting the $i$ th coordinate axis) restrict to dominant morphisms $(\hat{\unicode[STIX]{x1D70B}}_{i})|_{H}:H\longrightarrow (\mathbb{P}^{1})^{n-1}$ . By abuse of notation, we denote the restriction $(\hat{\unicode[STIX]{x1D70B}}_{i})|_{H}$ also by $\hat{\unicode[STIX]{x1D70B}}_{i}$ . We define probability measures $\hat{\unicode[STIX]{x1D707}}_{i}$ (for $i=1,\ldots ,n$ ) on $H$ corresponding to the dynamical system $((\mathbb{P}_{\mathbb{C}}^{1})^{n-1},\tilde{f}_{i})$ . More precisely, for each $i=1,\ldots ,n$ , we pullback $\tilde{\unicode[STIX]{x1D707}}_{i}$ by $\hat{\unicode[STIX]{x1D70B}}_{i}$ to get a measure $\hat{\unicode[STIX]{x1D70B}}_{i}^{\ast }\unicode[STIX]{x1D707}_{i}$ on $H$ so that

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D70B}}_{i}^{\ast }\tilde{\unicode[STIX]{x1D707}}_{i}(A):=\tilde{\unicode[STIX]{x1D707}}_{i}(\hat{\unicode[STIX]{x1D70B}}_{i}(A))\end{eqnarray}$$

for any Borel set $A\subset H$ such that $\hat{\unicode[STIX]{x1D70B}}_{i}$ is injective on $A$ . Another way to interpret this is that for $t=(x_{1},\ldots ,x_{n})\in H$ , we have that $d(\hat{\unicode[STIX]{x1D70B}}_{i}^{\ast }\tilde{\unicode[STIX]{x1D707}}_{i})(t)$ is an $(n-1,n-1)$ -current on $H$ given by

$$\begin{eqnarray}d\hat{\unicode[STIX]{x1D70B}}_{i}^{\ast }\tilde{\unicode[STIX]{x1D707}}_{i}(t)=dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{i-1}(x_{i-1})\wedge dd^{c}u_{i+1}(x_{i+1})\wedge \cdots \wedge dd^{c}u_{n}(x_{n}),\end{eqnarray}$$

where $u_{j}$ is a locally defined continuous subharmonic function with $dd^{c}u_{j}=d\unicode[STIX]{x1D707}_{f_{j}}$ for each $j=1,\ldots ,n$ . Hence we get the probability measures on $H$ :

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D707}}_{i}:=\hat{\unicode[STIX]{x1D70B}}_{i}^{\ast }\tilde{\unicode[STIX]{x1D707}}_{i}/\text{deg}(\hat{\unicode[STIX]{x1D70B}}_{i})\quad \text{for }i=1,\ldots ,n.\end{eqnarray}$$

Similarly, one has that

$$\begin{eqnarray}\tilde{f}_{i}^{\ast }\tilde{\unicode[STIX]{x1D707}}_{i}=d_{1}\cdots d_{i-1}\cdot d_{i+1}\cdots d_{n}\cdot \tilde{\unicode[STIX]{x1D707}}_{i}\quad \text{for }i=1,\ldots ,n.\end{eqnarray}$$

3.3 Symmetries of the Julia set

Let $\unicode[STIX]{x1D701}$ be a meromorphic function on some disc $B(a,r)$ of radius $r$ centred at a point $a\in J_{f}$ . We say that $\unicode[STIX]{x1D701}$ is a symmetry on $J_{f}$ if it satisfies the following properties:

• ∙ $x\in B(a,r)\cap J_{f}$ if and only if $\unicode[STIX]{x1D701}(x)\in \unicode[STIX]{x1D701}(B(a,r))\cap J_{f}$ ; and

• ∙ if $J_{f}$ is either a circle, a line segment, or the entire sphere, there is a constant $\unicode[STIX]{x1D6FC}>0$ such that for any Borel set $A$ where $\unicode[STIX]{x1D701}|_{A}$ is injective, one has $\unicode[STIX]{x1D707}_{f}(\unicode[STIX]{x1D701}(A))=\unicode[STIX]{x1D6FC}\cdot \unicode[STIX]{x1D707}_{f}(A)$ .

A family ${\mathcal{S}}$ of symmetries of $J_{f}$ on $B(a,r)$ is said to be non-trivial if ${\mathcal{S}}$ is normal on $B(a,r)$ and no infinite sequence $\{\unicode[STIX]{x1D701}_{n}\}\subset {\mathcal{S}}$ converges to a constant function. A rational function is post-critically finite (sometimes called critically finite), if each of its critical points has finite forward orbit, i.e. all critical points are preperiodic. According to Thurston [Reference ThurstonThu85, Reference Douady and HubbardDH93], there is an orbifold structure on $\mathbb{P}^{1}$ corresponding to each post-critically finite map. A rational function is post-critically finite with parabolic orbifold if and only if it is exceptional; or equivalently its Julia set is smooth (a circle, a line segment or the entire sphere) with smooth maximal entropy measure on it; see [Reference Douady and HubbardDH93].

3.4 The height functions

Let $K$ be a number field and $\overline{K}$ be the algebraic closure of $K$ . The number field $K$ is naturally equipped with a set $\unicode[STIX]{x1D6FA}_{K}$ of pairwise inequivalent non-trivial absolute values, together with positive integers $N_{v}$ for each $v\in \unicode[STIX]{x1D6FA}_{K}$ such that:

• ∙ for each $\unicode[STIX]{x1D6FC}\in K^{\ast }$ , we have $|\unicode[STIX]{x1D6FC}|_{v}=1$ for all but finitely many places $v\in \unicode[STIX]{x1D6FA}_{K}$ ;

• ∙ every $\unicode[STIX]{x1D6FC}\in K^{\ast }$ satisfies the product formula

  (3.4.1) $$\begin{eqnarray}\prod _{v\in \unicode[STIX]{x1D6FA}_{K}}|\unicode[STIX]{x1D6FC}|_{v}^{N_{v}}=1.\end{eqnarray}$$

For each $v\in \unicode[STIX]{x1D6FA}_{K}$ , let $K_{v}$ be the completion of $K$ at $v$ , let $\overline{K}_{v}$ be the algebraic closure of $K_{v}$ and let $\mathbb{C}_{v}$ denote the completion of $\overline{K}_{v}$ . We fix an embedding of $\overline{K}$ into $\mathbb{C}_{v}$ for each $v\in \unicode[STIX]{x1D6FA}_{K}$ ; hence we have a fixed extension of $|\cdot |_{v}$ on $\overline{K}$ . If $v$ is Archimedean, then $\mathbb{C}_{v}\cong \mathbb{C}$ . Let $f\in K(z)$ be a rational function with degree $d\geqslant 2$ . There is a canonical height $\widehat{h}_{f}$ on $\mathbb{P}^{1}(\overline{K})$ given by

(3.4.2) $$\begin{eqnarray}\widehat{h}_{f}(x):=\frac{1}{[K(x):K]}\lim _{n\rightarrow \infty }\mathop{\sum }_{T\in \operatorname{Gal}(\overline{K}/K)\cdot X}~\mathop{\sum }_{v\in \unicode[STIX]{x1D6FA}_{K}}N_{v}\cdot \frac{\log \Vert F^{n}(T)\Vert _{v}}{d^{n}},\end{eqnarray}$$

where $F:K^{2}\rightarrow K^{2}$ and $X$ are homogenous lifts of $f$ and respectively $x\in \mathbb{P}^{1}(\overline{K})$ , while $\Vert (z_{1},z_{2})\Vert _{v}:=\max \{|z_{1}|_{v},|z_{2}|_{v}\}$ . By product formula (3.4.1), the height $\widehat{h}_{f}$ does not depend on the choice of the homogenous lift $F$ and therefore it is well defined. As proven in [Reference Call and SilvermanCS93], $\widehat{h}_{f}(x)\geqslant 0$ with equality if and only if $x$ is preperiodic under the iteration of $f$ .

4.1 Adelic metrized line bundle

Let ${\mathcal{L}}$ be an ample line bundle of an irreducible projective variety $V$ over a number field $K$ . As in § 3.4, $K$ is naturally equipped with absolute values $|\cdot |_{v}$ for $v\in \unicode[STIX]{x1D6FA}_{K}$ . A metric $\Vert \cdot \Vert _{v}$ on ${\mathcal{L}}$ is a collection of norms, one for each $t\in V(K_{v})$ , on the fibres ${\mathcal{L}}(t)$ of the line bundle, with

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}s(t)\Vert _{v}=|\unicode[STIX]{x1D6FC}|_{v}\Vert s(t)\Vert _{v}\end{eqnarray}$$

for any section $s$ of ${\mathcal{L}}$ . An adelic (semipositive) metrized line bundle $\overline{{\mathcal{L}}}=\{{\mathcal{L}},\{\Vert \cdot \Vert _{v}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}$ over ${\mathcal{L}}$ is a collection of metrics on ${\mathcal{L}}$ , one for each place $v\in \unicode[STIX]{x1D6FA}_{K}$ , satisfying certain continuity and coherence conditions; see [Reference ZhangZha95a, Reference ZhangZha95b, Reference Yuan and ZhangYZ17].

There are various adelic metrized line bundles; the simplest adelic (semipositive) metrized line bundle is the line bundle ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ equipped with metrics $\Vert \cdot \Vert _{v}$ (for each $v\in \unicode[STIX]{x1D6FA}_{K}$ ), which evaluated at a section $s:=u_{0}Z_{0}+u_{1}Z_{1}$ of ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ (where $u_{0},u_{1}$ are scalars and $Z_{0},Z_{1}$ are the canonical sections of ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ ) is given by

$$\begin{eqnarray}\Vert s([z_{0}:z_{1}])\Vert _{v}:=\frac{|u_{0}z_{0}+u_{1}z_{1}|_{v}}{\max \{|z_{0}|_{v},|z_{1}|_{v}\}}.\end{eqnarray}$$

Furthermore, we can define other metrics on ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ corresponding to a rational function $f$ of degree $d\geqslant 2$ defined over $K$ . We fix a homogenous lift $F:K^{2}\rightarrow K^{2}$ of $f$ with homogenous degree $d$ . For $j\geqslant 1$ , write $F^{j}=(F_{0,j},F_{1,j})$ . For each place $v\in \unicode[STIX]{x1D6FA}_{K}$ , we can define a metric on ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ as

(4.1.1) $$\begin{eqnarray}\Vert s([z_{0}:z_{1}])\Vert _{v,F,j}:=\frac{|u_{0}z_{0}+u_{1}z_{1}|_{v}}{\max \{|F_{0,j}(z_{0},z_{1})|_{v},|F_{1,j}(z_{0},z_{1})|_{v}\}^{1/d^{j}}},\end{eqnarray}$$

where $s=u_{0}Z_{0}+u_{1}Z_{1}$ with $u_{0},u_{1}$ scalars and $Z_{0},Z_{1}$ canonical sections of ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ . Hence $\{{\mathcal{O}}_{\mathbb{P}^{1}}(1),\{\Vert \cdot \Vert _{v,F,j}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}$ is an adelic metrized line bundle over ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ .

A sequence $\{{\mathcal{L}},\{\Vert \cdot \Vert _{v,j}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}_{j\geqslant 1}$ of adelic metrized line bundles over an ample line bundle ${\mathcal{L}}$ on a variety $V$ is convergent to $\{{\mathcal{L}},\{\Vert \cdot \Vert _{v}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}$ , if for all $j$ and all but finitely many $v\in \unicode[STIX]{x1D6FA}_{K}$ , we have that $\Vert \cdot \Vert _{v,j}=\Vert \cdot \Vert _{v}$ , and moreover, $\{\log (\Vert \cdot \Vert _{v,j}/\Vert \cdot \Vert _{v})\}_{j\geqslant 1}$ converges to $0$ uniformly on $V(\mathbb{C}_{v})$ for all $v\in \unicode[STIX]{x1D6FA}_{K}$ . The limit $\{{\mathcal{L}},\{\Vert \cdot \Vert _{v}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}$ is an adelic metrized line bundle. Also, the tensor product of two (adelic) metrized line bundles is again a (adelic) metrized line bundle.

A typical example of a convergent sequence of adelic metrized line bundles is $\{\{{\mathcal{O}}_{\mathbb{P}^{1}}(1),\{\Vert \cdot \Vert _{v,F,j}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}\}_{j\geqslant 1}$ which converges to the metrized line bundle denoted by

(4.1.2) $$\begin{eqnarray}\overline{{\mathcal{L}}}_{F}:=\{{\mathcal{O}}_{\mathbb{P}^{1}}(1),\{\Vert \cdot \Vert _{v,F}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}\end{eqnarray}$$

(see [Reference Baker and RumelyBR06] and also see [Reference ZhangZha95b, Theorem 2.2] for the more general case of a polarizable endomorphism $f$ of a projective variety).

As usual, we let $\tilde{f}=(f_{1},\ldots ,f_{n})$ with $f_{i}$ being a rational function of degree $d_{i}\geqslant 2$ defined over the number field $K$ for $1\leqslant i\leqslant n$ . Fix a homogenous lift $F_{i}$ for each $f_{i}$ and denote

$$\begin{eqnarray}\tilde{F}:=(F_{1},\ldots ,F_{n}).\end{eqnarray}$$

We let $\unicode[STIX]{x1D70B}_{i}$ be the $i$ th coordination projection map from $(\mathbb{P}^{1})^{n}$ to $\mathbb{P}^{1}$ . We construct an adelic metrized line bundle on $(\mathbb{P}^{1})^{n}$ as follows

(4.1.3) $$\begin{eqnarray}\overline{{\mathcal{L}}}_{\tilde{F}}:=\{{\mathcal{L}}_{\tilde{F}},\Vert \cdot \Vert _{v,\tilde{F}}\}:=(\unicode[STIX]{x1D70B}_{1}^{\ast }\overline{{\mathcal{L}}}_{F_{1}})\otimes (\unicode[STIX]{x1D70B}_{2}^{\ast }\overline{{\mathcal{L}}}_{F_{2}})\otimes \cdots \otimes (\unicode[STIX]{x1D70B}_{n}^{\ast }\overline{{\mathcal{L}}}_{F_{n}}),\end{eqnarray}$$

where the metric $\Vert \cdot \Vert _{v,\tilde{F}}$ on ${\mathcal{L}}_{\tilde{F}}$ is the one inherited from the metrics $\Vert \cdot \Vert _{v,F_{i}}$ on ${\mathcal{O}}_{\mathbb{P}^{1}}(1)$ for $1\leqslant i\leqslant n$ .

4.2 Equidistribution of small points

For a semipositive metrized line bundle $\overline{{\mathcal{L}}}$ on a (irreducible and projective) variety $V$ defined over a number field $K$ , the height for $t\in V(\overline{K})$ is given by

(4.2.1) $$\begin{eqnarray}\widehat{h}_{\overline{{\mathcal{L}}}}(t)=\frac{1}{|\text{Gal}(\overline{K}/K)\cdot t|}\mathop{\sum }_{y\in \operatorname{Gal}(\overline{K}/K)\cdot t}~\mathop{\sum }_{v\in \unicode[STIX]{x1D6FA}_{K}}-N_{v}\cdot \log \Vert s(y)\Vert _{v},\end{eqnarray}$$

where $|\text{Gal}(\overline{K}/K)\cdot t|$ is the number of points in the Galois orbits of $t$ , and $s$ is any meromorphic section of ${\mathcal{L}}$ with support disjoint from $\operatorname{Gal}(\overline{K}/K)\cdot t$ . A sequence of points $t_{j}\in V(\overline{K})$ is small if $\lim _{j\rightarrow \infty }\widehat{h}_{\overline{{\mathcal{L}}}}(t_{j})=\widehat{h}_{\overline{{\mathcal{L}}}}(V)$ , and is generic if no subsequence of $t_{j}$ is contained in a proper Zariski closed subset of $V$ ; see [Reference ZhangZha95b] for more details on constructing the height for any irreducible subvariety $Y$ of $V$ (which is denoted by $\widehat{h}_{\overline{{\mathcal{L}}}}(Y)$ ). We use the following equidistribution result due to Yuan [Reference YuanYua08] in the case of an arbitrary projective variety.

When $v$ is Archimedean, $V_{\mathbb{C}_{v}}^{an}$ corresponds to $V(\mathbb{C})$ and the curvature $c_{1}(\overline{{\mathcal{L}}})_{v}$ of the metric $\Vert \cdot \Vert _{v}$ is given by $c_{1}(\overline{{\mathcal{L}}})_{v}=(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}/\unicode[STIX]{x1D70B}i)\log \Vert \cdot \Vert _{v}$ . If $v$ is a non-Archimedean place, then $V_{\mathbb{C}_{v}}^{an}$ is the Berkovich space associated with $V(\mathbb{C}_{v})$ , and Chambert-Loir [Reference Chambert-LoirCha06] constructed an analog for the curvature on $V_{\mathbb{C}_{v}}^{an}$ . The precise meaning of the equidistribution statement in Theorem 4.1 is that

(4.2.2) $$\begin{eqnarray}\lim _{j\rightarrow \infty }\frac{1}{|\operatorname{Gal}(\overline{K}/K)\cdot t_{j}|}\mathop{\sum }_{y\in \operatorname{Gal}(\overline{K}/K)\cdot t_{j}}\unicode[STIX]{x1D6FF}_{y}=\unicode[STIX]{x1D707}_{v},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{y}$ is the point mass probability measure supported on $y\in V_{\mathbb{C}_{v}}^{an}$ , while the limit from (4.2.2) is the weak limit for the corresponding probability measures on the compact space $V_{\mathbb{C}_{v}}^{an}$ .

4.3 Some examples

For the dynamical system $(\mathbb{P}^{1},f)$ corresponding to a rational function $f$ defined over a number field $K$ and of degree $d_{f}\geqslant 2$ , at an Archimedean place $v$ , it is well known that the curvature of the limit of the metrized line bundles

$$\begin{eqnarray}\{{\mathcal{O}}_{\mathbb{P}^{1}}(1),\{\Vert \cdot \Vert _{v,F,j}\}_{v\in \unicode[STIX]{x1D6FA}_{K}}\}_{j\geqslant 1}\end{eqnarray}$$

is a $(1,1)$ -current given by $d\unicode[STIX]{x1D707}_{f}$ , which is independent on the choice of $F$ . Combining the definition (3.4.2) of the canonical height $\widehat{h}_{f}$ of $f$ , with the height (4.2.1) of points for an adelic metrized line bundle and the definition (4.1.1) and (4.1.2) of $\overline{{\mathcal{L}}}_{F}$ , we get

$$\begin{eqnarray}\widehat{h}_{\overline{{\mathcal{L}}}_{F}}(x)=\widehat{h}_{f}(x)\end{eqnarray}$$

which is independent of the choice for the lift $F$ of $f$ .

We conclude this section by noting that in the case of the metrized line bundle $\overline{{\mathcal{L}}}_{\tilde{F}}$ on $(\mathbb{P}^{1})^{n}$ associated with an endomorphism $\tilde{f}$ of $(\mathbb{P}^{1})^{n}$ (see § 4.1), at an Archimedean place $v$ , the $(n,n)$ -current satisfies the formula:

(4.3.1) $$\begin{eqnarray}c_{1}(\overline{{\mathcal{L}}}_{\tilde{F}})_{v}^{n}=n!\cdot d\tilde{\unicode[STIX]{x1D707}},\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D707}_{f_{1}}\times \cdots \times \unicode[STIX]{x1D707}_{f_{n}}$ is the invariant measure on $(\mathbb{P}_{\mathbb{C}_{v}}^{1})^{n}$ associated with the endomorphism $\tilde{f}=(f_{1},\ldots ,f_{n})$ . To see this, we first notice that since $v$ is Archimedean, then $\mathbb{C}_{v}=\mathbb{C}$ and so, by taking $(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}/\unicode[STIX]{x1D70B}i)\log \Vert \cdot \Vert _{v,\tilde{F}}$ we get

(4.3.2) $$\begin{eqnarray}c_{1}(\overline{{\mathcal{L}}}_{\tilde{F}})_{v}=dd^{c}(u_{1}(x_{1})+\cdots +u_{n}(x_{n})),\end{eqnarray}$$

where $u_{i}(x_{i})$ is a locally defined continuous subharmonic function on $\mathbb{P}_{\mathbb{C}_{v}}^{1}$ with $dd^{c}u_{i}=d\unicode[STIX]{x1D707}_{f_{i}}$ for $1\leqslant i\leqslant n$ . Hence

$$\begin{eqnarray}c_{1}(\overline{{\mathcal{L}}}_{\tilde{F}})_{v}^{n}=n!\cdot dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{n}(x_{n})=n!\cdot d\tilde{\unicode[STIX]{x1D707}},\end{eqnarray}$$

and so, the equality from (4.3.1) follows. Moreover, for a point $t=(a_{1},\ldots ,a_{n})\in (\mathbb{P}^{1})^{n}(\overline{K})$ , from (4.1.3) we see that

(4.3.3) $$\begin{eqnarray}\widehat{h}_{\overline{{\mathcal{L}}}_{\tilde{F}}}(t)=\widehat{h}_{f_{1}}(a_{1})+\cdots +\widehat{h}_{f_{n}}(a_{n}).\end{eqnarray}$$

5.1 Adelic metrized line bundles on the hypersurface

For each $i=1,\ldots ,n$ , as in (3.2.1), we let $\tilde{f}_{i}$ be the endomorphism of $(\mathbb{P}^{1})^{n-1}$ given by forgetting the $i$ th coordinate axis (along with the action of $f_{i}$ ) in the dynamical system $((\mathbb{P}^{1})^{n},\hat{f})$ . Let $\tilde{F}_{i}$ be a homogenous lift of $\tilde{f}_{i}$ as in § 3.2 and then similar to (4.1.3), we construct an adelic metrized line bundle $\overline{{\mathcal{L}}}_{\tilde{F}_{i}}$ on $(\mathbb{P}^{1})^{n-1}$ such that when $v$ is Archimedean, we have

$$\begin{eqnarray}c_{1}(\overline{{\mathcal{L}}}_{\tilde{F}_{i}})_{v}^{n-1}=(n-1)!\cdot d\tilde{\unicode[STIX]{x1D707}}_{i}\end{eqnarray}$$

(for each $1\leqslant i\leqslant n$ ), where the probability measure $\tilde{\unicode[STIX]{x1D707}}_{i}$ on $(\mathbb{P}_{\mathbb{C}_{v}}^{1})^{n-1}$ is the one appearing in (3.2.2).

For each $i=1,\ldots ,n$ , we recall from § 3.2 that the projection

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D70B}}_{i}:H\longrightarrow (\mathbb{P}^{1})^{n-1}\end{eqnarray}$$

is the one given by forgetting the $i$ th coordinates; $\hat{\unicode[STIX]{x1D70B}}_{i}$ is a finite, dominant morphism (due to our assumption on $H$ ). We let

(5.1.1) $$\begin{eqnarray}\overline{{\mathcal{L}}}_{\hat{F}_{i}}:=\hat{\unicode[STIX]{x1D70B}}_{i}^{\ast }\overline{{\mathcal{L}}}_{\tilde{F}_{i}}\end{eqnarray}$$

be an adelic metrized line bundle on $H$ , which is the pullback of the adelic metrized line bundle $\overline{{\mathcal{L}}}_{\tilde{F}_{i}}$ (on $(\mathbb{P}^{1})^{n-1}$ ) by the morphism $\hat{\unicode[STIX]{x1D70B}}_{i}$ .

5.2 Height functions on the hypersurface

For each $i=1,\ldots ,n$ and each $t=(a_{1},\ldots ,a_{n})\in H(\overline{K})$ , using (4.3.3) we conclude that

(5.2.1) $$\begin{eqnarray}\widehat{h}_{\overline{{\mathcal{L}}}_{\hat{F}_{i}}}(t)=\widehat{h}_{f_{1}}(a_{1})+\cdots +\widehat{h}_{f_{i-1}}(a_{i-1})+\widehat{h}_{f_{i+1}}(a_{i+1})+\cdots +\widehat{h}_{f_{n}}(a_{n}).\end{eqnarray}$$

Hence $\widehat{h}_{\overline{{\mathcal{L}}}_{\hat{F}_{i}}}(t)\geqslant 0$ with equality if and only if $a_{j}$ is preperiodic under $f_{j}$ for each $j\neq i$ with $1\leqslant j\leqslant n$ . So, if the set of all $t\in H(\overline{K})$ for which $\widehat{h}_{\overline{{\mathcal{L}}}_{\hat{F}_{i}}}(t)=0$ is Zariski-dense on $H$ , then each essential minima $e_{j}(\overline{{\mathcal{L}}}_{\hat{F}_{i}})$ (for $j=1,\ldots ,n$ , defined as in [Reference ZhangZha95b]) are equal to $0$ . Therefore, using the inequality from [Reference ZhangZha95b, Theorem 1.10], we conclude that

(5.2.2) $$\begin{eqnarray}\widehat{h}_{\overline{{\mathcal{L}}}_{\hat{F}_{i}}}(H)=0.\end{eqnarray}$$

5.3 Equal measures on the hypersurface

Now we are ready to prove the following result.

Theorem 5.1. Suppose that there is a generic sequence of points $t_{j}=(x_{1,j},\ldots ,x_{n,j})\in H(\overline{K})$ such that

$$\begin{eqnarray}\lim _{j\rightarrow \infty }\widehat{h}_{f_{1}}(x_{1,j})+\cdots +\widehat{h}_{f_{n}}(x_{n,j})=0.\end{eqnarray}$$

Then $\hat{\unicode[STIX]{x1D707}}_{1}=\hat{\unicode[STIX]{x1D707}}_{2}=\cdots =\hat{\unicode[STIX]{x1D707}}_{n}$ .

5.4 Preperiodic points on hypersurfaces

In this section we prove the following important result; we thank Xinyi Yuan and Shouwu Zhang for several very helpful conversations regarding its proof.

We prove first that hypothesis (2) in Proposition 5.2 yields the desired conclusion, and then we prove that part (1) may be reduced to part (2) in Proposition 5.2 through a specialization result of Yuan and Zhang [Reference Yuan and ZhangYZa, Reference Yuan and ZhangYZb].

Proof of Proposition 5.2 assuming hypothesis (2) holds.

Since the case $n=2$ was proven in [Reference MimarMim97] (see also [Reference MimarMim13]), from now on, we assume $n>2$ . We assume each $f_{i}\in \bar{\mathbb{Q}}(x)$ and also that $H$ is defined over $\bar{\mathbb{Q}}$ .

We use the notation as in § 5.1; so, we consider the adelic metrics $\overline{{\mathcal{L}}}_{\hat{F}_{i}}$ (for $i=1,\ldots ,n$ ) on $H$ , defined as in (5.1.1). For the sake of simplifying our notation, we will denote from now on the tensor product of two line bundles ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ as ${\mathcal{M}}_{1}+{\mathcal{M}}_{2}$ . We denote by $\overline{\text{Pic}}(H)$ the group of (adelic) metrized line bundles on $H$ .

Lemma 5.3. There exist real numbers $c_{i}$ (for $i=1,\ldots ,n$ ) not all equal to $0$ such that the metrized line bundle

(5.4.1) $$\begin{eqnarray}\overline{{\mathcal{L}}}_{0}:=c_{1}\cdot \overline{{\mathcal{L}}}_{\hat{F}_{1}}+\cdots +c_{n}\overline{{\mathcal{L}}}_{\hat{F}_{n}}\in \overline{\text{Pic}}(H)\otimes \mathbb{R}\end{eqnarray}$$

has the property that $\overline{{\mathcal{L}}}_{0}\cdot x=\widehat{h}_{\overline{{\mathcal{L}}}_{0}}(x)=0$ for each $x\in H(\bar{\mathbb{Q}})$ .

Proof of Lemma 5.3.

We thank Shouwu Zhang for suggesting to us the proof of this lemma, which follows the idea used in the proof of [Reference Yuan and ZhangYZ17, Theorem 4.13].

We let $\hat{{\mathcal{L}}}_{i}\in \text{Pic}(H)$ be the line bundle supporting $\overline{{\mathcal{L}}}_{\hat{F}_{i}}$ , i.e.,

$$\begin{eqnarray}\hat{{\mathcal{L}}}_{i}:=\unicode[STIX]{x1D70B}_{1}^{\ast }{\mathcal{O}}_{\mathbb{P}^{1}}(1)\otimes \cdots \otimes \unicode[STIX]{x1D70B}_{i-1}^{\ast }{\mathcal{O}}_{\mathbb{P}^{1}}(1)\otimes \unicode[STIX]{x1D70B}_{i+1}^{\ast }{\mathcal{O}}_{\mathbb{P}^{1}}(1)\otimes \cdots \otimes \unicode[STIX]{x1D70B}_{n}^{\ast }{\mathcal{O}}_{\mathbb{P}^{1}}(1),\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}_{j}$ is the induced projection map of $H$ onto the $j$ th coordinate axis of $(\mathbb{P}^{1})^{n}$ (for each $j=1,\ldots ,n$ ).

Claim 5.4. There exist real constants $c_{1},\ldots ,c_{n}$ (not all equal to $0$ ) such that the line bundle ${\mathcal{L}}_{0}:=\sum _{i=1}^{n}c_{i}\hat{{\mathcal{L}}}_{i}\in \text{Pic}(H)\otimes \mathbb{R}$ is numerically trivial.

Proof of Claim 5.4.

The main ingredient in our proof is a result on arithmetic intersections, which generalizes the classical Hodge index theorem (see [Reference Yuan and ZhangYZ17, Theorem 5.20]). We let

(5.4.2) $$\begin{eqnarray}{\mathcal{L}}_{1}:=\mathop{\sum }_{i=1}^{n}\hat{{\mathcal{L}}}_{i}\in \text{Pic}(H);\end{eqnarray}$$

then ${\mathcal{L}}_{1}$ is ample (since it is the pullback of an ample line bundle on $(\mathbb{P}^{1})^{n}$ under the natural inclusion map $H{\hookrightarrow}(\mathbb{P}^{1})^{n}$ ). We find the real numbers $c_{i}$ so that ${\mathcal{L}}_{0}:=\sum _{i=1}^{n}c_{i}\hat{{\mathcal{L}}}_{i}$ satisfies the following two conditions:

1. (A) ${\mathcal{L}}_{0}\cdot {\mathcal{L}}_{1}^{n-2}=0$ ;

2. (B) ${\mathcal{L}}_{0}^{2}\cdot {\mathcal{L}}_{1}^{n-3}=0$ .

Condition (A) above yields a linear relation between the unknowns $c_{i}$ . On the other hand, condition (B) yields a quadratic form in the variables $c_{i}$ . This quadratic form is not positive-definite since (from the generalization of the Hodge index theorem proven in [Reference FaltingsFal84, Reference HriljacHri85, Reference MoriwakiMor96]) we know that generically, for any line bundle ${\mathcal{M}}$ satisfying ${\mathcal{M}}\cdot {\mathcal{L}}_{1}^{n-2}=0$ , we have that ${\mathcal{M}}^{2}\cdot {\mathcal{L}}_{1}^{n-3}\leqslant 0$ . Also, this quadratic form is not negative definite since ${\mathcal{L}}_{1}^{2}\cdot {\mathcal{L}}_{1}^{n-3}={\mathcal{L}}_{1}^{n-1}>0$ (because ${\mathcal{L}}_{1}$ is ample). Therefore, there exist real numbers $c_{i}$ , not all equal to $0$ such that ${\mathcal{L}}_{0}$ satisfies both conditions (A) and (B) above. Then [Reference Yuan and ZhangYZ17, Theorem 5.20] yields that ${\mathcal{L}}_{0}$ is numerically trivial, as claimed.◻

Now let $c_{1},\ldots ,c_{n}\in \mathbb{R}$ satisfy the conclusion of Claim 5.4 and define

$$\begin{eqnarray}\overline{{\mathcal{L}}}_{0}:=\mathop{\sum }_{i=1}^{n}c_{i}\overline{{\mathcal{L}}}_{\hat{F}_{i}}\in \overline{\text{Pic}}(H)\otimes \mathbb{R}.\end{eqnarray}$$

We consider next the adelic metrized line bundle $\overline{{\mathcal{L}}}_{1}:=\sum _{i=1}^{n}\overline{{\mathcal{L}}}_{\hat{F}_{i}}$ ; note that the generic fibre of $\overline{{\mathcal{L}}}_{1}$ is the ample line bundle ${\mathcal{L}}_{1}$ from (5.4.2). Using our hypothesis (2) from Proposition 5.2, i.e., the existence of a Zariski dense set of points on $H$ of height tending to $0$ , we obtain that each of the successive minima $e_{j}(\overline{{\mathcal{L}}}_{1})=0$ for $j=0,\ldots ,n-1$ . Note that for each $j=0,\ldots ,n$ , we have

$$\begin{eqnarray}e_{j}(\overline{{\mathcal{L}}}_{1}):=\sup _{\substack{ Y\subset H \\ \text{codim}_{H}(Y)=j}}\inf _{x\in (H\setminus Y)(\bar{\mathbb{Q}})}\widehat{h}_{\overline{{\mathcal{L}}}_{1}}(x)\end{eqnarray}$$

and so, indeed hypothesis (2) of Proposition 5.2 yields that $e_{j}(\overline{{\mathcal{L}}}_{1})=0$ . In particular, $e_{n}(\overline{{\mathcal{L}}}_{1})=0$ and thus $\overline{{\mathcal{L}}}_{1}^{n}=0$ . The exact same argument applied for each $i_{1},i_{2}=1,\ldots ,n$ and for each $m_{1},m_{2}\in \mathbb{N}$ to the metrized line bundle $\overline{{\mathcal{L}}}_{i_{1},i_{2},m_{1},m_{2}}:=\overline{{\mathcal{L}}}_{1}+m_{1}\overline{{\mathcal{L}}}_{\hat{F}_{i_{1}}}+m_{2}\overline{{\mathcal{L}}}_{\hat{F}_{i_{2}}}$ yields again

(5.4.3) $$\begin{eqnarray}(\overline{{\mathcal{L}}}_{1}+m_{1}\overline{{\mathcal{L}}}_{\hat{F}_{i_{1}}}+m_{2}\overline{{\mathcal{L}}}_{\hat{F}_{i_{2}}})^{n}=0.\end{eqnarray}$$

Keeping $i_{1}$ and $i_{2}$ fixed and letting $m_{1}$ and $m_{2}$ vary in $\mathbb{N}$ , we see that (5.4.3) yields that $\overline{{\mathcal{L}}}_{1}^{j_{0}}\cdot \overline{{\mathcal{L}}}_{\hat{F}_{i_{1}}}^{j_{1}}\cdot \overline{{\mathcal{L}}}_{\hat{F}_{i_{2}}}^{j_{2}}=0$ for non-negative integers $j_{0},j_{1},j_{2}$ such that $j_{0}+j_{1}+j_{2}=n$ . Hence

(5.4.4) $$\begin{eqnarray}\overline{{\mathcal{L}}}_{0}^{2}\cdot \overline{{\mathcal{L}}}_{1}^{n-2}=0;\end{eqnarray}$$

moreover, because the numbers $c_{i}$ satisfy the construction from Claim 5.4 (see condition (A) in the proof of the aforementioned Claim), we also have that

(5.4.5) $$\begin{eqnarray}{\mathcal{L}}_{0}\cdot {\mathcal{L}}_{1}^{n-2}=0.\end{eqnarray}$$

Furthermore, since each $\overline{{\mathcal{L}}}_{\hat{F}_{i}}$ is semipositive, we obtain that (with the terminology from [Reference Yuan and ZhangYZ17]) $\overline{{\mathcal{L}}}_{0}$ is $\overline{{\mathcal{L}}}_{1}$ -bounded, i.e., there exists $m\in \mathbb{N}$ (any integer larger than $\max _{i}|c_{i}|$ would work) such that both $m\cdot \overline{{\mathcal{L}}}_{1}-\overline{{\mathcal{L}}}_{0}$ and $m\cdot \overline{{\mathcal{L}}}_{1}+\overline{{\mathcal{L}}}_{0}$ are semipositive.

Since $\overline{{\mathcal{L}}}_{1}$ may not necessarily be arithmetically positive, we alter $\overline{{\mathcal{L}}}_{1}$ by adding to it an arbitrarily positive metrized line bundle $\unicode[STIX]{x1D704}^{\ast }({\mathcal{C}})$ where ${\mathcal{C}}$ is a positive metrized line bundle on $\text{Spec}(\bar{\mathbb{Q}})$ and $\unicode[STIX]{x1D704}:H\longrightarrow \text{Spec}(\bar{\mathbb{Q}})$ is the structure morphism (for a similar application, see the proof of [Reference Yuan and ZhangYZ17, Theorem 4.13]). Then $\overline{{\mathcal{L}}}_{0}$ would still be $\overline{{\mathcal{L}}}_{1}^{\prime }$ -bounded with respect to this new metrized line bundle $\overline{{\mathcal{L}}}_{1}^{\prime }:=\overline{{\mathcal{L}}}_{1}+\unicode[STIX]{x1D704}^{\ast }({\mathcal{C}})$ . Because the generic fibre of $\overline{{\mathcal{L}}}_{0}$ is numerically trivial (according to our choice of the numbers $c_{i}$ satisfying the conclusion of Claim 5.4), then (5.4.4) and (5.4.5) yield

(5.4.6) $$\begin{eqnarray}{\mathcal{L}}_{0}\cdot ({\mathcal{L}}_{1}^{\prime })^{n-1}=0\quad \text{and}\quad \overline{{\mathcal{L}}}_{0}^{2}\cdot (\overline{{\mathcal{L}}}_{1}^{\prime })^{n-2}=0.\end{eqnarray}$$

Thus the hypotheses of [Reference Yuan and ZhangYZ17, Theorem 3.2] are verified and so, we obtain that the metrized line bundle $\overline{{\mathcal{L}}}_{0}$ is itself numerically trivial, i.e., $\widehat{h}_{\overline{{\mathcal{L}}}_{0}}(x)=0$ for each $x\in H(\bar{\mathbb{Q}})$ . This concludes the proof of Lemma 5.3.◻

So, by Lemma 5.3, there exist suitable constants $c_{i}\in \mathbb{R}$ (for $i=1,\ldots ,n$ ), not all equal to $0$ such that the metrized line bundle $\overline{{\mathcal{L}}}_{0}:=c_{1}\cdot \overline{{\mathcal{L}}}_{\hat{F}_{1}}+\cdots +c_{n}\overline{{\mathcal{L}}}_{\hat{F}_{n}}\in \overline{\text{Pic}}(H)\otimes \mathbb{R}$ is numerically trivial on $H$ and therefore, for each $\unicode[STIX]{x1D6FC}\in H(\bar{\mathbb{Q}})$ , we have that $\overline{{\mathcal{L}}}_{0}\cdot \unicode[STIX]{x1D6FC}=0$ , i.e.,

(5.4.7) $$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}c_{i}\cdot \widehat{h}_{\overline{{\mathcal{L}}}_{\hat{F}_{i}}}(\unicode[STIX]{x1D6FC})=0.\end{eqnarray}$$

Since not all $c_{i}$ are equal to $0$ , then there exists some $i_{0}\in \{1,\ldots ,n\}$ with the property that

(5.4.8) $$\begin{eqnarray}c_{1}+\cdots +c_{i_{0}-1}+c_{i_{0}+1}+\cdots +c_{n}\neq 0.\end{eqnarray}$$

Now, for any $\unicode[STIX]{x1D6FC}:=(a_{1},\ldots ,a_{n})\in H(\bar{\mathbb{Q}})$ and for any $i=1,\ldots ,n$ , we have that

(5.4.9) $$\begin{eqnarray}\widehat{h}_{\overline{{\mathcal{L}}}_{\hat{F}_{i}}}(\unicode[STIX]{x1D6FC})=\widehat{h}_{f_{1}}(a_{1})+\cdots +\widehat{h}_{f_{i-1}}(a_{i-1})+\widehat{h}_{f_{i+1}}(a_{i+1})+\cdots +\widehat{h}_{f_{n}}(a_{n}),\end{eqnarray}$$

as shown in (5.2.1). Now, if

$$\begin{eqnarray}\widehat{h}_{f_{1}}(a_{1})=\cdots =\widehat{h}_{f_{i_{0}-1}}(a_{i_{0}-1})=\widehat{h}_{f_{i_{0}+1}}(a_{i_{0}+1})=\cdots =\widehat{h}_{f_{n}}(a_{n})=0,\end{eqnarray}$$

then (5.4.7), (5.4.8) and (5.4.9) yield that also $\widehat{h}_{f_{i_{0}}}(a_{i_{0}})=0$ , as claimed in the conclusion of Proposition 5.2. This concludes our proof of Proposition 5.2 assuming each rational function $f_{i}$ along with the hypersurface $H$ are defined over $\bar{\mathbb{Q}}$ .◻

Proof of Proposition 5.2 assuming hypothesis (1) holds.

We let $K\subset \mathbb{C}$ be a finitely generated extension of $\bar{\mathbb{Q}}$ such that each $f_{i}\in K(x)$ and also $H$ is defined over $K$ . We argue by induction on $r:=\text{trdeg}_{\bar{\mathbb{Q}}}K$ ; the case $r=0$ is already proved using Proposition 5.2 with hypothesis (2). Hence, we assume the conclusion of Proposition 5.2 holds whenever $r<s$ (for some $s\in \mathbb{N}$ ) and we prove that it also holds when $r=s$ . We know there exists an infinite sequence $S$ of points $\unicode[STIX]{x1D6FC}_{j}\in H(\mathbb{C})$ such that $\unicode[STIX]{x1D6FC}_{j}$ has its $i$ th coordinate preperiodic under the action of $f_{i}$ (for each $i=1,\ldots ,n$ ). Also, we let

$$\begin{eqnarray}d:=\deg (f_{1})=\deg (f_{2})=\cdots =\deg (f_{n}).\end{eqnarray}$$

Then we let $K_{0}$ be a subfield $\bar{\mathbb{Q}}\subset K_{0}\subset K$ such that $\text{trdeg}_{K_{0}}K=1$ and we let ${\mathcal{C}}$ be a curve defined over $K_{0}$ whose function field is $K$ (at the expense of replacing both $K_{0}$ and $K$ by finite extensions, we may assume ${\mathcal{C}}$ is a projective, smooth, geometrically irreducible curve). We fix some algebraic closures $\overline{K_{0}}\subset \overline{K}$ of our fields.

There exists a Zariski dense, open subset $C\subseteq {\mathcal{C}}$ such that we may view each $f_{i}$ as a base change of an endomorphism $f_{i,C}$ of $\mathbb{P}_{C}^{1}$ ; similarly, $H$ is the base change of a hypersurface $H_{C}\subset (\mathbb{P}_{C}^{1})^{n}$ , while $S$ is the base change of a subset $S_{C}\subset H_{C}$ . For each geometric point $t\in C(\overline{K_{0}})$ , the objects $H_{C}$ , $f_{i,C}$ and $S_{C}$ have reductions $H_{t}$ , $f_{i,t}$ and respectively $S_{t}$ , such that $S_{t}\subset H_{C}$ consists of points with their $i$ th coordinate preperiodic under the action of $f_{i,C}$ , for each $i=1,\ldots ,n$ .

Let $C_{0}$ be the Zariski dense, open subset of $C$ satisfying the conclusion of Claim 5.5. At the expense of perhaps shrinking $C_{0}$ to a smaller, dense, open subset, we may assume that

(5.4.10) $$\begin{eqnarray}\deg (f_{i,t})=d>1\quad \text{for all }i=1,\ldots ,n\text{ and each }t\in C_{0}(\overline{K_{0}}).\end{eqnarray}$$

For each $t\in C_{0}(\bar{\mathbb{Q}})$ , our inductive hypothesis (which can be applied since each $f_{i}$ and also $H$ are defined over $\overline{K_{0}}$ and $\text{trdeg}_{\bar{\mathbb{Q}}}\overline{K_{0}}<s$ ) yields the existence of some index $i_{t}\in \{1,\ldots ,n\}$ which has the property that for each $\unicode[STIX]{x1D6FC}\in H_{t}(\bar{\mathbb{Q}})$ , if we know that the $j$ th coordinate of $\unicode[STIX]{x1D6FC}$ is preperiodic under the action of $f_{j,t}$ for each $j\in \{1,\ldots ,n\}\setminus \{i_{t}\}$ , then also the $i_{t}$ th coordinate of $\unicode[STIX]{x1D6FC}$ is preperiodic under the action of $f_{i_{t},t}$ .

Let $h_{{\mathcal{C}}}(\cdot )$ be a height function for the points on ${\mathcal{C}}(\overline{K_{0}})$ corresponding to a divisor of degree $1$ on ${\mathcal{C}}$ , constructed with respect to the Weil height on $\overline{K_{0}}$ . Note that if $\text{trdeg}_{\mathbb{Q}}K_{0}\geqslant 1$ , then we construct the Weil height on the function field $K_{0}/\bar{\mathbb{Q}}$ as in [Reference Bombieri and GublerBG06]. At the expense of replacing $C_{0}$ by an infinite subset $U_{0}$ for which

(5.4.11) $$\begin{eqnarray}\sup _{t\in U_{0}}h_{{\mathcal{C}}}(t)=+\infty ,\end{eqnarray}$$

we may even assume that for each $t\in U_{0}$ , there is the same index $i_{0}:=i_{t}\in \{1,\ldots ,n\}$ satisfying the above property. We show next that this index $i_{0}$ would satisfy the conclusion of Proposition 5.2 for $H$ .

Indeed, let $\unicode[STIX]{x1D6FC}=(a_{1},\ldots ,a_{n})\in H(\overline{K})$ with the property that for each $j\in \{1,\ldots ,n\}\setminus \{i_{0}\}$ , we have that $a_{j}$ is preperiodic under the action of $f_{j}$ . Then for each $t\in U_{0}$ we have that each $a_{j,t}$ (for $j\in \{1,\ldots ,n\}\setminus \{i_{0}\}$ ) is preperiodic for $f_{j,t}$ and so, also $a_{i_{0},t}$ is preperiodic under the action of $f_{i_{0},t}$ . Therefore, the canonical height

(5.4.12) $$\begin{eqnarray}\widehat{h}_{f_{i_{0},t}}(a_{i_{0},t})=0,\end{eqnarray}$$

where $\widehat{h}_{f_{i_{0},t}}(\cdot )$ is the canonical height corresponding to the rational function $f_{i_{0},t}$ (which has degree larger than $1$ ; see (5.4.10)), constructed using the Weil height on $\overline{K_{0}}$ . Using [Reference Call and SilvermanCS93, Theorem 4.1], we have that

(5.4.13) $$\begin{eqnarray}\lim _{h_{{\mathcal{C}}}(t)\rightarrow \infty }\frac{\widehat{h}_{f_{i_{0},t}}(a_{i_{0},t})}{h_{{\mathcal{C}}}(t)}=\widehat{h}_{f_{i_{0}}}(a_{i_{0}}),\end{eqnarray}$$

where $\widehat{h}_{f_{i_{0}}}(\cdot )$ is the canonical height of $f_{i_{0}}$ constructed with respect to the function field $K/K_{0}$ . Equations (5.4.11), (5.4.12) and (5.4.13) yield that

(5.4.14) $$\begin{eqnarray}\widehat{h}_{f_{i_{0}}}(a_{i_{0}})=0.\end{eqnarray}$$

If $f_{i_{0}}\in K(x)$ is not isotrivial over $K_{0}$ , then [Reference BakerBak09] (see also [Reference BenedettoBen05] for the case of polynomials) yields that (5.4.14) is equivalent with saying that $a_{i_{0}}$ is preperiodic under the action of $f_{i_{0}}$ , as desired. Now, if $f_{i_{0}}$ is isotrivial over $K_{0}$ , then there exists a linear transformation

$$\begin{eqnarray}\unicode[STIX]{x1D708}:\mathbb{P}^{1}\longrightarrow \mathbb{P}^{1}\quad (\text{defined over }\overline{K})\end{eqnarray}$$

such that $\unicode[STIX]{x1D708}^{-1}\circ f_{i_{0}}\circ \unicode[STIX]{x1D708}\in \overline{K_{0}}(x)$ . If $\unicode[STIX]{x1D708}^{-1}(a_{i_{0}})\in \overline{K_{0}}$ , then since we know there exists even a single specialization $t$ such that $a_{i_{0},t}$ is preperiodic for $f_{t}$ , we get that also $a_{i_{0}}$ is preperiodic for $f_{i_{0}}$ . On the other hand, if $\unicode[STIX]{x1D708}^{-1}(a_{i_{0}})\notin \overline{K_{0}}$ , then $\unicode[STIX]{x1D708}^{-1}(a_{i_{0}})$ cannot be preperiodic for $\unicode[STIX]{x1D708}^{-1}\circ f_{i_{0}}\circ \unicode[STIX]{x1D708}\in \overline{K_{0}}(x)$ and so, $a_{i_{0}}$ is not preperiodic for $f_{i_{0}}$ , contradiction. This concludes the proof of Proposition 5.2 under hypothesis (1).◻

6.1 Statement of our theorem

Theorem 6.1. Let $n\geqslant 2$ , let $f_{i}$ be rational functions defined over $\mathbb{C}$ of degrees $d_{i}>1$ (for $1\leqslant i\leqslant n+1$ ), and let $H\subset (\mathbb{P}^{1})^{n+1}$ be an irreducible hypersurface defined over $\mathbb{C}$ which projects dominantly onto each subset of $n$ coordinate axes. For each $i=1,\ldots ,n+1$ , let $\tilde{f}_{i}$ be the coordinatewise action on $(\mathbb{P}^{1})^{n}$ given by

$$\begin{eqnarray}(x_{1},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{n+1})\mapsto (f_{1}(x_{1}),\ldots ,f_{i-1}(x_{i-1}),f_{i+1}(x_{i+1}),\ldots ,f_{n+1}(x_{n+1})).\end{eqnarray}$$

Let $\hat{\unicode[STIX]{x1D707}}_{i}$ be the measures on $H$ induced from the dynamical systems $((\mathbb{P}^{1})^{n},\tilde{f}_{i})$ and assume that $\hat{\unicode[STIX]{x1D707}}_{i}=\hat{\unicode[STIX]{x1D707}}_{n+1}$ for $1\leqslant i\leqslant n$ . Also assume that there is a point $(\tilde{a},b_{0})\in H\,\cap \,\mathbb{C}^{n+1}$ with $\tilde{a}=(a_{1},\ldots ,a_{n})$ , such that:

• ∙ $a_{i}$ is a repelling fixed point of $f_{i}$ for $1\leqslant i\leqslant n$ ; and

• ∙ $b_{1}:=f_{n+1}(b_{0})$ is a fixed point of $f_{n+1}$ ; and

• ∙ there is a holomorphic germ $h(\tilde{x})$ at $\tilde{a}$ with $h(\tilde{a})=b_{0}$ , and $(\tilde{x},h(\tilde{x}))\in H(\mathbb{C})$ for all $\tilde{x}\in \mathbb{C}^{n}$ in a small (complex analytic) neighbourhood of $\tilde{a}$ . Moreover, for each $i=1,\ldots ,n$ we have that

  $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{i}:=\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x_{i}}(\tilde{a})\neq 0.\end{eqnarray}$$

Then the functions $f_{i}$ must be exceptional, and moreover, they are

• ∙ either all of them conjugate to monomials and $\pm$ Chebyshev polynomials,

• ∙ or all of them Lattès maps.

6.2 Julia sets and invariant measures

From the assumptions of Theorem 6.1, the multiplier

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{i}:=f_{i}^{\prime }(a_{i})\end{eqnarray}$$

has absolute value $|\unicode[STIX]{x1D706}_{i}|>1$ for $1\leqslant i\leqslant n$ . So, each $a_{i}$ is in the support of the Julia set $J_{f_{i}}$ of $f_{i}$ , for $i=1,\ldots ,n$ . Thus $(\tilde{a},b_{0})$ is in the support of $\hat{\unicode[STIX]{x1D707}}_{n+1}$ and because $\hat{\unicode[STIX]{x1D707}}_{n}=\hat{\unicode[STIX]{x1D707}}_{n+1}$ , we get that $(\tilde{a},b_{0})$ must be in the support of $\hat{\unicode[STIX]{x1D707}}_{n}$ . Therefore, $b_{0}$ must be in the support $J_{f_{n+1}}$ of $\unicode[STIX]{x1D707}_{f_{n+1}}$ . Hence $b_{1}=f_{n+1}(b_{0})\in J_{f_{n+1}}$ and so, it has multiplier

$$\begin{eqnarray}\unicode[STIX]{x1D70C}:=f_{n+1}^{\prime }(b_{1})\end{eqnarray}$$

of absolute value $|\unicode[STIX]{x1D70C}|\geqslant 1$ . Let $j_{0}$ be the local degree of the map $f_{n+1}(x)$ at $x=b_{0}$ , and let $g(x)$ be a holomorphic germ on $\mathbb{P}^{1}$ at $b_{0}$ which is one of the following branches

(6.2.1) $$\begin{eqnarray}g(x):=f_{n+1}^{-1}\circ f_{n+1}\circ f_{n+1}(x)\end{eqnarray}$$

satisfying $g(b_{0})=b_{0}$ . Although there are $j_{0}$ different choices for $g(x)$ , in the rest of this section we fix our choice $g(x)$ for such a branch. An easy computation shows that

(6.2.2)

is a $j_{0}$ th root of the multiplier $\unicode[STIX]{x1D70C}$ of $f_{n+1}$ at $b_{1}$ . Since $\unicode[STIX]{x1D707}_{f_{n+1}}$ admits no atoms on $\mathbb{P}^{1}$ and $\unicode[STIX]{x1D707}_{f_{n+1}}(f_{n+1}(A))=d_{n+1}\cdot \unicode[STIX]{x1D707}_{f_{n+1}}(A)$ for any Borel set $A$ with $f_{n+1}$ being injective on $A$ , the definition of $g(x)$ yields that

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{f_{n+1}}(g(A))=d_{n+1}\cdot \unicode[STIX]{x1D707}_{f_{n+1}}(A)\end{eqnarray}$$

for any Borel set $A$ in a small neighbourhood of $b_{0}$ .

Proof of Lemma 6.2.

We first assume that $|\unicode[STIX]{x1D706}|\leqslant 1$ and then prove the lemma by deriving a contradiction. Using (6.2.2) and the fact that $|\unicode[STIX]{x1D70C}|\geqslant 1$ , we get that $|\unicode[STIX]{x1D706}|=1$ .

Pick a positive integer $m$ with $d_{n}<d_{n+1}^{m}$ . Let

(6.2.3) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{00}(\tilde{x}):=(\text{}\underline{x},h(\text{}\underline{x},f_{n}(x_{n})))\quad \text{and}\quad \unicode[STIX]{x1D6F7}_{11}(\tilde{x}):=(\text{}\underline{x},g^{m}\circ h(\tilde{x}))\end{eqnarray}$$

be functions locally defined in a neighbourhood of $\tilde{a}\in \mathbb{C}^{n}$ , mapping that small neighbourhood of $\tilde{a}$ into a neighbourhood of $(\text{}\underline{a},b_{0})\in \mathbb{C}^{n}$ . Since $\hat{\unicode[STIX]{x1D707}}_{n}=\hat{\unicode[STIX]{x1D707}}_{n+1}$ , there exists some $c>0$ with

(6.2.4) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{00}^{\ast }(\tilde{\unicode[STIX]{x1D707}}_{n})=c\cdot d_{n}\cdot \tilde{\unicode[STIX]{x1D707}}_{n+1}\quad \text{and}\quad \unicode[STIX]{x1D6F7}_{11}^{\ast }(\tilde{\unicode[STIX]{x1D707}}_{n})=c\cdot d_{n+1}^{m}\cdot \tilde{\unicode[STIX]{x1D707}}_{n+1}.\end{eqnarray}$$

The measures $\tilde{\unicode[STIX]{x1D707}}_{n}$ and $\tilde{\unicode[STIX]{x1D707}}_{n+1}$ (defined in (3.2.2)) appearing in (6.2.4) are restricted on some small neighbourhood of $\tilde{a}$ (respectively of $(\text{}\underline{a},b_{0})$ ). Let $A$ be the polydisc given by $A:=D_{n-1}(\text{}\underline{a},r_{1})\times D(a_{n},r_{2})$ for very small $r_{2}$ and much smaller $r_{1}$ . We claim that $\unicode[STIX]{x1D6F7}_{11}(A)\subset \unicode[STIX]{x1D6F7}_{00}(A)$ . To see this, let $r_{2}$ be very small and we see that $f_{n}(D(a_{n},r_{2}))\sim D(a_{n},|\unicode[STIX]{x1D706}_{n}|r_{2})$ . As $|\unicode[STIX]{x1D706}|=|g^{\prime }(b_{0})|=1<|\unicode[STIX]{x1D706}_{n}|$ and $\unicode[STIX]{x1D6FD}_{n}=(\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x_{n})(\tilde{a})\neq 0$ , using (6.2.3) we can pick some very small $r_{2}$ and a much smaller $r_{1}$ such that

(6.2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{11}(A)\subset D_{n-1}(\text{}\underline{a},r_{1})\times D(b_{0},r_{2}\cdot |\unicode[STIX]{x1D6FD}_{n}|\cdot |\unicode[STIX]{x1D706}_{n}|^{1/2})\subset \unicode[STIX]{x1D6F7}_{00}(A).\end{eqnarray}$$

However, combining (6.2.4) with $d_{n}<d_{n+1}^{m}$ gives

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D707}}_{n}(\unicode[STIX]{x1D6F7}_{11}(A))>\tilde{\unicode[STIX]{x1D707}}_{n}(\unicode[STIX]{x1D6F7}_{00}(A)),\end{eqnarray}$$

which is a contradiction. This concludes the proof of Lemma 6.2.◻

6.3 A special sequence of tuples of positive integers

Now since $|\unicode[STIX]{x1D706}|>1$ and $|\unicode[STIX]{x1D706}_{i}|>1$ for $1\leqslant i\leqslant n$ , we can pick a sequence of tuples of positive integers $(j_{\ell },j_{1,\ell },\ldots ,j_{n,\ell })$ such that $j_{\ell }\rightarrow \infty$ as $\ell \rightarrow \infty$ and moreover,

(6.3.1) $$\begin{eqnarray}\lim _{\ell \rightarrow \infty }\inf \frac{|\unicode[STIX]{x1D706}_{1}^{j_{1,\ell }}|}{|\unicode[STIX]{x1D706}^{j_{\ell }}|},\ldots ,\lim _{\ell \rightarrow \infty }\inf \frac{|\unicode[STIX]{x1D706}_{n-1}^{j_{1,\ell }}|}{|\unicode[STIX]{x1D706}^{j_{\ell }}|}\geqslant \lim _{\ell \rightarrow \infty }\frac{\unicode[STIX]{x1D706}_{n}^{j_{n,\ell }}}{\unicode[STIX]{x1D706}^{j_{\ell }}}=1.\end{eqnarray}$$

It will be useful later in our argument (see Lemma 6.8) that our sequence of tuples $(j_{\ell },j_{1,\ell },\ldots ,j_{n,\ell })$ satisfies the following arithmetic property in addition to (6.3.1). We want that for every $N\in \mathbb{N}$ , there exist $\ell _{2}>\ell _{1}>N$ such that

(6.3.2) $$\begin{eqnarray}j_{\ell _{2}}=j_{\ell _{1}}\quad \text{and}\quad j_{i,\ell _{2}}=j_{i,\ell _{1}}\quad \text{for }2\leqslant i\leqslant n\text{ while }j_{1,\ell _{2}}=j_{1,\ell _{1}}+1.\end{eqnarray}$$

In order to achieve (6.3.2), we may replace the original sequence of tuples $\{(j_{\ell },j_{1,\ell },\ldots ,j_{n,\ell })\}_{\ell =1}^{\infty }$ by the larger sequence $\{(j_{\ell }^{\prime },j_{1,\ell }^{\prime },\ldots ,j_{n,\ell }^{\prime })\}_{\ell =1}^{\infty }$ for which

$$\begin{eqnarray}j_{2\ell -1}^{\prime }=j_{2\ell }^{\prime }=j_{\ell }\quad \text{and}\quad j_{i,2\ell -1}^{\prime }=j_{i,2\ell }^{\prime }=j_{i,\ell }\end{eqnarray}$$

for $i=2,\ldots ,n$ , while

$$\begin{eqnarray}j_{1,2\ell -1}^{\prime }=j_{1,\ell }\quad \text{and}\quad j_{1,2\ell }^{\prime }=j_{1,\ell }+1\end{eqnarray}$$

and still the new sequence $\{(j_{\ell }^{\prime },j_{1,\ell }^{\prime },\ldots ,j_{n,\ell }^{\prime })\}_{\ell =1}^{\infty }$ satisfies (6.3.1) and the fact that $j_{\ell }\rightarrow \infty$ as $\ell \rightarrow \infty$ . For the sake of simplifying our notation, we will denote our new sequence of tuples also as $\{(j_{\ell },j_{1,\ell },\ldots ,j_{n,\ell })\}_{\ell =1}^{\infty }$ , but we note that this sequence of tuples satisfies (6.3.2).

6.4 Local symmetries for the Julia sets

From [Reference MilnorMil00], we know we can conjugate $f_{i}$ (for $1\leqslant i\leqslant n$ ) and $g$ to linear maps in small neighbourhoods of the repelling fixed points. More precisely, there exist holomorphic germs $\unicode[STIX]{x1D719}_{i}$ at $x=0$ satisfying

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{i}(0)=a_{i},\unicode[STIX]{x1D719}_{n+1}(0)=b_{0},\unicode[STIX]{x1D719}_{i}^{\prime }(0)=\unicode[STIX]{x1D719}_{n+1}^{\prime }(0)=1\quad \text{for }1\leqslant i\leqslant n\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{i}^{-1}\circ f_{i}\circ \unicode[STIX]{x1D719}_{i}(x)=\unicode[STIX]{x1D706}_{i}\cdot x\quad \text{for }1\leqslant i\leqslant n,\end{eqnarray}$$

while

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{n+1}^{-1}\circ g\circ \unicode[STIX]{x1D719}_{n+1}(x)=\unicode[STIX]{x1D706}\cdot x.\end{eqnarray}$$

We notice that for $(x_{1},\ldots ,x_{n})$ in a neighbourhood of $\tilde{a}\in \mathbb{C}^{n}$ , we have an equality of germs:

$$\begin{eqnarray}\displaystyle g^{j_{\ell }}\circ h\circ (f_{1}^{-j_{1,\ell }}(x_{1}),\ldots ,f_{n}^{-j_{n,\ell }}(x_{n}))=\unicode[STIX]{x1D719}_{n+1}\circ \biggl(\unicode[STIX]{x1D706}^{j_{\ell }}\cdot h_{\unicode[STIX]{x1D719}}\biggl(\frac{\unicode[STIX]{x1D719}_{1}^{-1}(x_{1})}{\unicode[STIX]{x1D706}_{1}^{j_{1,\ell }}},\ldots ,\frac{\unicode[STIX]{x1D719}_{n}^{-1}(x_{n})}{\unicode[STIX]{x1D706}_{n}^{j_{n,\ell }}}\biggr)\biggr), & & \displaystyle \nonumber\end{eqnarray}$$

where $h_{\unicode[STIX]{x1D719}}:=\unicode[STIX]{x1D719}_{n+1}^{-1}\circ h\circ (\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{n})$ and $f_{i}^{-1}$ is the germ of a branch of the inverse of $f_{i}$ at $x_{i}=a_{i}$ with $f_{i}^{-1}(a_{i})=a_{i}$ . So, using also (6.3.1), then for very small $r_{0}>0$ and all $\tilde{x}$ in the ball $B(\tilde{a},r_{0})\subset \mathbb{C}^{n}$ of radius $r_{0}$ , the map

$$\begin{eqnarray}\tilde{x}\mapsto g^{j_{\ell }}\circ h\circ (f_{1}^{-j_{1,\ell }},\ldots ,f_{n}^{-j_{n,\ell }})(\tilde{x})\end{eqnarray}$$

is well defined and uniformly bounded on $B(\tilde{a},r_{0})$ for all $\ell \geqslant 1$ . Next, we construct the function

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\tilde{x}):=(x_{1},\ldots ,x_{n-1},h(\tilde{x}))\end{eqnarray}$$

for $\tilde{x}=(x_{1},\ldots ,x_{n})$ , which is locally one-to-one at $\tilde{x}=\tilde{a}$ since $\unicode[STIX]{x1D6FD}_{n}=(\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x_{n})(\tilde{a})\neq 0$ . Shrinking $r_{0}$ if necessary, we let

(6.4.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{\ell }(\tilde{x}) & := & \displaystyle \unicode[STIX]{x1D6F9}^{-1}\circ (x_{1},\ldots ,x_{n-1},g^{j_{\ell }}\circ h\circ (f_{1}^{-j_{1,\ell }},\ldots ,f_{n}^{-j_{n,\ell }})(\tilde{x}))\nonumber\\ \displaystyle & =: & \displaystyle (x_{1},\ldots ,x_{n-1},h_{\ell }(\tilde{x}))\end{eqnarray}$$

for all $\tilde{x}\in B(\tilde{a},r_{0})$ and all $\ell \geqslant 1$ , where $h_{\ell }$ is some local analytic function on $B(\tilde{a},r_{0})$ satisfying (6.4.1).

Proof of Lemma 6.3.

Since $\tilde{x}\mapsto g^{j_{\ell }}\circ h\circ (f_{1}^{-j_{1,\ell }},\ldots ,f_{n}^{-j_{n,\ell }})(\tilde{x})$ is uniformly bounded on $B(\tilde{a},r_{0})$ for all $\ell \geqslant 1$ , then that $h_{\ell }$ (defined as in (6.4.1)) is uniformly bounded on $B(\tilde{a},r_{0})$ , i.e., there exist $R>0$ such that

$$\begin{eqnarray}h_{\ell }(B(\tilde{a},r_{0}))\subset B(b_{0},R)\subset \mathbb{C}\end{eqnarray}$$

for all $\ell \geqslant 1$ . Hence $h_{\ell }$ is a distance non-increasing map from $B(\tilde{a},r)$ (with respect to the Bergman metric) to $B(b_{0},R)$ (with respect to the hyperbolic metric). Thus $\{h_{\ell }(\tilde{x})\}_{\ell \geqslant 1}$ is equicontinuous on $B(\tilde{a},r_{0})$ , or equivalently, $\{h_{\ell }(\tilde{x})\}_{\ell \geqslant 1}$ is a normal family.◻

From Lemma 6.3, we can pick a subsequence of $\{\unicode[STIX]{x1D6F9}_{\ell }\}_{\ell \geqslant 1}$ which converges uniformly on $B(\tilde{a},r_{0})$ . By passing to a subsequence, without loss of generality, we can assume that the sequence $\{\unicode[STIX]{x1D6F9}_{\ell }\}_{\ell \geqslant 1}$ itself converges uniformly to

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{0}(\tilde{x}):=\lim _{\ell \rightarrow \infty }\unicode[STIX]{x1D6F9}_{\ell }(\tilde{x})\end{eqnarray}$$

and satisfies (6.3.2) with $\unicode[STIX]{x1D6F9}_{0}(\tilde{a})=\tilde{a}$ and $\unicode[STIX]{x1D6F9}_{0}(\tilde{x})=:(x_{1},\ldots ,x_{n-1},h_{0}(\tilde{x}))$ . Since

$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}h_{0}}{\unicode[STIX]{x2202}x_{n}}(\tilde{a})=\lim _{\ell \rightarrow \infty }\frac{\unicode[STIX]{x2202}h_{\ell }}{\unicode[STIX]{x2202}x_{n}}(\tilde{a})=\lim _{\ell \rightarrow \infty }\frac{\unicode[STIX]{x1D706}^{j_{\ell }}}{\unicode[STIX]{x1D706}_{n}^{j_{n,\ell }}}=1\neq 0,\end{eqnarray}$$

the map $\unicode[STIX]{x1D6F9}_{0}$ is locally one-to-one at $\tilde{x}=\tilde{a}$ . Shrinking $r_{0}$ if necessary, we can further assume that the sequence of maps

(6.4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{0}^{-1}\circ \unicode[STIX]{x1D6F9}_{\ell }(\tilde{x})=:(x_{1},\ldots ,x_{n-1},\hbar _{\ell }(\tilde{x}))=:\unicode[STIX]{x1D6F7}_{\ell }(\tilde{x})\end{eqnarray}$$

converges uniformly to the identity map on $B(\tilde{a},r_{0})$ as $\ell \rightarrow \infty$ . The next goal is to show that $\unicode[STIX]{x1D6F7}_{\ell }$ is the identity map for all large $\ell$ ; see Lemma 6.7.

6.5 Equal currents

Proposition 6.4. Let $r_{1}$ and $r_{2}$ be positive real numbers and let $u_{1},\ldots ,u_{n}$ and $u$ be continuous subharmonic functions on $D(0,r_{1})$ , respectively on $D(0,r_{2})$ . Let $\unicode[STIX]{x1D703}$ be a holomorphic map from $D_{n}(\tilde{0},r_{1})$ to $D(0,r_{2})$ and moreover, assume the following two $(n,n)$ -currents satisfy the relation:

$$\begin{eqnarray}dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{n}(x_{n})=c_{0}\cdot dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{n-1}(x_{n-1})\wedge dd^{c}u\circ \unicode[STIX]{x1D703}(\tilde{x})\end{eqnarray}$$

on $D_{n}(\tilde{0},r_{1})$ for some constant $c_{0}>0$ . Then for any given point $\text{}\underline{\unicode[STIX]{x1D6FC}}$ in the support of $dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{n-1}(x_{n-1})$ , we have the following equality of $(1,1)$ -currents on $D(0,r_{1})$ :

$$\begin{eqnarray}dd^{c}u_{n}(x_{n})=c_{0}\cdot dd^{c}u\circ \unicode[STIX]{x1D703}(\text{}\underline{\unicode[STIX]{x1D6FC}},x_{n}).\end{eqnarray}$$

Proof of Proposition 6.4.

Let $\text{}\underline{\unicode[STIX]{x1D6FC}}$ be a point in the support of $dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{n-1}(x_{n-1})$ . It suffices to show that for any $C^{\infty }$ real function $\unicode[STIX]{x1D711}$ with compact support on $D(0,r_{1})$ , one has

$$\begin{eqnarray}\int _{D(0,r_{1})}\unicode[STIX]{x1D711}(x_{n})\,dd^{c}u_{n}(x_{n})=c_{0}\int _{D(0,r_{1})}\unicode[STIX]{x1D711}(x_{n})\,dd^{c}u\circ \unicode[STIX]{x1D703}(\text{}\underline{\unicode[STIX]{x1D6FC}},x_{n}).\end{eqnarray}$$

To see this, we let $\text{}\underline{\unicode[STIX]{x1D707}}$ be the measure on $D_{n-1}(\text{}\underline{0},r_{1})$ with

$$\begin{eqnarray}d\text{}\underline{\unicode[STIX]{x1D707}}(\text{}\underline{x}):=c_{0}\cdot dd^{c}u_{1}(x_{1})\wedge \cdots \wedge dd^{c}u_{n-1}(x_{n-1})\end{eqnarray}$$

and let $\tilde{\unicode[STIX]{x1D707}}$ be the measure on $D_{n}(\tilde{0},r_{1})$ with

$$\begin{eqnarray}d\tilde{\unicode[STIX]{x1D707}}(\tilde{x}):=d\text{}\underline{\unicode[STIX]{x1D707}}(\text{}\underline{x})\wedge \frac{dx_{n}\wedge d\bar{x}_{n}}{-4\unicode[STIX]{x1D70B}i}\end{eqnarray}$$

.

For each small positive real number $r$ , we let $\unicode[STIX]{x1D702}_{r}(\text{}\underline{x})$ be a $C^{\infty }$ -function on $D_{n-1}(\text{}\underline{0},r_{1})$ satisfying the properties:

• ∙ $0\leqslant \unicode[STIX]{x1D702}_{r}\leqslant 1$ ;

• ∙ $\unicode[STIX]{x1D702}_{r}$ is supported on $D_{n-1}(\text{}\underline{\unicode[STIX]{x1D6FC}},r)$ ; and

• ∙ $\unicode[STIX]{x1D702}_{r}=1$ on $D_{n-1}(\text{}\underline{\unicode[STIX]{x1D6FC}},r/2)$ .

From the proportionality assumption of the two $(n,n)$ -currents, we get

(6.5.1) $$\begin{eqnarray}\displaystyle \frac{1}{c_{0}}\biggl(\int \unicode[STIX]{x1D702}_{r}\,d\text{}\underline{\unicode[STIX]{x1D707}}\biggr)\int \unicode[STIX]{x1D711}\,dd^{c}u_{n} & = & \displaystyle \frac{1}{c_{0}}\int \unicode[STIX]{x1D702}_{r}(\text{}\underline{x})\unicode[STIX]{x1D711}(x_{n})\,d\text{}\underline{\unicode[STIX]{x1D707}}(\text{}\underline{x})\wedge dd^{c}u_{n}(x_{n})\nonumber\\ \displaystyle & = & \displaystyle \int \unicode[STIX]{x1D702}_{r}(\text{}\underline{x})\unicode[STIX]{x1D711}(x_{n})\,d\text{}\underline{\unicode[STIX]{x1D707}}(\text{}\underline{x})\wedge dd^{c}u\circ \unicode[STIX]{x1D703}(\tilde{x})\nonumber\\ \displaystyle & = & \displaystyle \int u\circ \unicode[STIX]{x1D703}(\tilde{x})\,d\text{}\underline{\unicode[STIX]{x1D707}}\wedge dd^{c}(\unicode[STIX]{x1D702}_{r}\unicode[STIX]{x1D711})\nonumber\\ \displaystyle & = & \displaystyle \int \unicode[STIX]{x1D702}_{r}(\text{}\underline{x})u\circ \unicode[STIX]{x1D703}(\tilde{x})\triangle \unicode[STIX]{x1D711}(x_{n})\,d\tilde{\unicode[STIX]{x1D707}}(\tilde{x})\end{eqnarray}$$

where $\triangle$ is the Laplacian and the right-hand side is integrated over the domain $D_{n}(\tilde{0},r_{1})$ . Similarly we derive that

$$\begin{eqnarray}\biggl(\int \unicode[STIX]{x1D702}_{r}\,d\text{}\underline{\unicode[STIX]{x1D707}}\biggr)\int \unicode[STIX]{x1D711}\,dd^{c}u\circ \unicode[STIX]{x1D703}(\text{}\underline{\unicode[STIX]{x1D6FC}},x_{n})=\int \unicode[STIX]{x1D702}_{r}(\text{}\underline{x})u\circ \unicode[STIX]{x1D703}(\text{}\underline{\unicode[STIX]{x1D6FC}},x_{n})\triangle \unicode[STIX]{x1D711}(x_{n})\,d\tilde{\unicode[STIX]{x1D707}}(\tilde{x}).\end{eqnarray}$$

Now let

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{r}(\tilde{x}):=\unicode[STIX]{x1D702}_{r}(\text{}\underline{x})\cdot (u\circ \unicode[STIX]{x1D703}(\text{}\underline{\unicode[STIX]{x1D6FC}},x_{n})-u\circ \unicode[STIX]{x1D703}(\tilde{x}))\cdot \triangle \unicode[STIX]{x1D711}(x_{n})\end{eqnarray}$$

which is supported on $D_{n-1}(\text{}\underline{\unicode[STIX]{x1D6FC}},r)\times D(0,r_{1})$ . Hence as $u\circ \unicode[STIX]{x1D703}$ is continuous and $\unicode[STIX]{x1D711}$ has compact support on $D(0,r_{1})$ , there exist constants $\unicode[STIX]{x1D716}_{r}\rightarrow 0$ as $r\rightarrow 0$ such that for any $\tilde{x}\in D_{n}(\tilde{0},r_{1})$ , we have

$$\begin{eqnarray}|\unicode[STIX]{x1D6E9}_{r}(\tilde{x})|\leqslant \unicode[STIX]{x1D702}_{r}(\text{}\underline{x})\cdot \unicode[STIX]{x1D716}_{r}.\end{eqnarray}$$

Consequently

$$\begin{eqnarray}\biggl|\frac{1}{c_{0}}\int \unicode[STIX]{x1D711}\,dd^{c}u_{n}-\int \unicode[STIX]{x1D711}\,dd^{c}u\circ \unicode[STIX]{x1D703}(\text{}\underline{\unicode[STIX]{x1D6FC}},x_{n})\biggr|\leqslant \frac{\int _{D_{n}(\tilde{0},r_{1})}\unicode[STIX]{x1D702}_{r}(\text{}\underline{x})\cdot \unicode[STIX]{x1D716}_{r}\,d\tilde{\unicode[STIX]{x1D707}}(\tilde{x})}{\int _{D_{n-1}(\text{}\underline{0},r_{1})}\unicode[STIX]{x1D702}_{r}(\text{}\underline{x})\,d\text{}\underline{\unicode[STIX]{x1D707}}(\text{}\underline{x})}=\unicode[STIX]{x1D716}_{r}\cdot c_{1}\end{eqnarray}$$

with $c_{1}=\int _{D(0,r_{1})}1\cdot (dx_{n}\wedge d\bar{x}_{n})/(-4\unicode[STIX]{x1D70B}i)$ . Now letting $r\rightarrow 0$ , the conclusion in Proposition 6.4 follows.◻