2.1 The measure of maximal entropy

For each rational map $f: {\mathbb {P}}^1\to {\mathbb {P}}^1$ of degree $d \geq 2$, there is a unique probability measure $\mu _f$ of maximal entropy. Its support is equal to the Julia set of $f$, and it is characterized by the properties that it has no atoms, so $\mu _f(\{z\}) = 0$ for all $z \in {\mathbb {P}}^1({\mathbb {C}})$, and $\frac {1}{d}f^* \mu _f = \mu _f$, meaning that

\[ \frac{1}{d} \int_{{\mathbb{P}}^1} \bigg(\sum_{f(x)=y} \varphi(x) \bigg) \mu_f(y) = \int_{{\mathbb{P}}^1} \varphi(x) \mu_f(x) \]

for all continuous functions $\varphi$ on ${\mathbb {P}}^1$ (see [Reference MañéMañ83, Reference Freire, Lopes and MañéFLM83, Reference LyubichLyu83a]).

2.2 Exceptional maps

We say that a map $f : {\mathbb {P}}^1\to {\mathbb {P}}^1$ of degree $d\geq 2$ is exceptional if it is the quotient of an affine transformation of ${\mathbb {C}}$; see [Reference MilnorMil06] for details. Every exceptional $f$ is conjugate by an element of $\operatorname {Aut} {\mathbb {P}}^1 \simeq \mathrm {PSL}_2{\mathbb {C}}$ to a power map $z^{\pm d}$, a Tchebyshev polynomial $\pm T_d$, or it is a Lattès map, meaning that it is the quotient of a map on an elliptic curve. The exceptional maps are distinguished by properties of their measures $\mu _f$. In each case, the Julia set $J(f)$ is a real submanifold of ${\mathbb {P}}^1({\mathbb {C}})$ (with boundary, in the case of the Tchebyshev polynomials), and the measure $\mu _f$ is absolutely continuous with respect to the Hausdorff measure on $J(f)$. Zdunik proved the converse: the exceptional maps are the only maps for which this absolute continuity can hold [Reference ZdunikZdu90].

2.3 Preperiodic points and the maximal measure

It is well known that the preperiodic points of $f$ are uniformly distributed with respect to the measure $\mu _f$. That is, defining discrete measures

\[ \mu_{n,m} = \frac{1}{d^n} \sum_{f^n(z)=f^m(z)} \delta_z \]

in ${\mathbb {P}}^1$ for every pair of integers $n > m\geq 0$, then for any sequence $(n_k,m_k)$ of integers $n_k > m_k\geq 0$ with $\max \{n_k,m_k\} \to \infty$ as $k\to \infty$, the measures $\mu _{n_k, m_k}$ converge weakly to the measure $\mu _f$ (see [Reference MañéMañ83, Reference Freire, Lopes and MañéFLM83, Reference LyubichLyu83a]).

But the preperiodic points determine the measure $\mu _f$ in a stronger sense, without ordering them by period or orbit length.

Remark 2.2 The main theorem of [Reference Levin and PrzytyckiLP97] states that if $\mu _f = \mu _g$, and if $f$ and $g$ are non-exceptional, then there exist iterates $f^n$ and $g^m$ and positive integers $\ell$ and $k$ so that

\[ (g^{-m}\circ g^{m}) \circ g^{\ell m} = (f^{-n}\circ f^{n}) \circ f^{k n} \]

for some (possibly multi-valued) branches of the inverse $g^{-m}$ and $f^{-n}$. It follows that $\mathrm {Preper}(f) = \mathrm {Preper}(g)$; see [Reference Levin and PrzytyckiLP97, Theorem A and Remark 2].

As the equivalences of Theorem 2.1 are not stated this way in the literature, we outline the proof ingredients.

Remark 2.3 In [Reference PakovichPak21], Pakovich proved that each $f \in \mathrm {Rat}_d$ of degree $d \geq 4$ with $2d-2$ distinct critical values satisfies

\[ \{g \in \mathrm{Rat}_d: \mu_g = \mu_f\} = \{f\}. \]

As these form a Zariski-dense and -open subset of $\mathrm {Rat}_d$, this implies, when combined with Theorem 2.1 for degrees $d \geq 4$, the set of pairs $(f,g)$ with $|\mathrm {Preper}(f) \cap \mathrm {Preper}(g)| = \infty$ is not Zariski dense in the space $\mathrm {Rat}_d\times \mathrm {Rat}_d$ of all pairs with $d\geq 4$.

2.4 Why not periodic points?

It is reasonable to ask why we work with all preperiodic points and not the subset $\mathrm {Per}(f)$ of periodic points of $f$, which are also uniformly distributed with respect to $\mu _f$. Of course, the uniform bound on $|\mathrm {Preper}(f)\cap \mathrm {Preper}(g)|$ in Theorem 1.1 is stronger than a bound on $|\mathrm {Per}(f)\cap \mathrm {Per}(g)|$, but there are two underlying reasons for our focus on preperiodic points. First, if we were to replace $\mathrm {Preper}(f)$ with $\mathrm {Per}(f)$, then the equivalences of Theorem 2.1 would break down. For example, if $f$ and $g$ are Lattès maps induced from $P\mapsto 2P$ and $P \mapsto 3P$ on the same elliptic curve via the same projection to ${\mathbb {P}}^1$, then $\mu _f = \mu _g$ with $|\mathrm {Per}(f) \cap \mathrm {Per}(g)| = \infty$ but $\mathrm {Per}(f) \not = \mathrm {Per}(f)$. In addition, there are many non-exceptional examples with $\mu _f = \mu _g$ but $|\mathrm {Per}(f) \cap \mathrm {Per}(g)| < \infty$, such as $f(z) = z^2 + c$ and $g(z) = -(z^2+c)$ for $c \in {\mathbb {C}}$. On the other hand, it follows from the proof of [Reference Levin and PrzytyckiLP97, Theorem A] that if $\mu _f = \mu _g$ for non-exceptional $f$ and $g$, and if there exists just one common repelling periodic point $x \in \mathrm {Per}(f) \cap \mathrm {Per}(g)$, then $\mathrm {Per}(f) = \mathrm {Per}(g)$; see [Reference YeYe15, Theorem 1.5]. A second reason is our method of proof and original motivation for this project. For maps $f: {\mathbb {P}}^1\to {\mathbb {P}}^1$ defined over $\overline {\mathbb {Q}}$, the preperiodic points for $f$ are precisely the points in ${\mathbb {P}}^1(\overline {\mathbb {Q}})$ for which the canonical height $\hat {h}_f$ vanishes [Reference Call and SilvermanCS93]. Though somewhat hidden in this article, much of our analysis is, fundamentally, about properties of these height functions.

2.5 Stability and the bifurcation current

Let $\Lambda$ be a connected complex manifold and $f: \Lambda \times {\mathbb {P}}^1 \to \Lambda \times {\mathbb {P}}^1$ a holomorphic map defined by $(\lambda, z)\mapsto (\lambda, f_\lambda (z))$ where each $f_\lambda$ has degree $d \geq 2$. The measures $\mu _{f_\lambda }$ can be packaged together into a positive $(1,1)$-current on the total space $\Lambda \times {\mathbb {P}}^1$ as follows. Let $\omega$ be a smooth and positive $(1,1)$-form on ${\mathbb {P}}^1$ with $\int _{{\mathbb {P}}^1} \omega = 1$, and consider $p^*\omega$ on $\Lambda \times {\mathbb {P}}^1$, where $p: \Lambda \times {\mathbb {P}}^1 \to {\mathbb {P}}^1$ is the projection. The dynamical Green current for $f$ on $\Lambda \times {\mathbb {P}}^1$ is

(2.1)\begin{equation} \hat{T}_f = \lim_{n\to\infty} \frac{1}{d^n} (f^n)^* (p^*\omega). \end{equation}

Then $\hat {T}_f$ is a closed, positive $(1,1)$-current on $\Lambda \times {\mathbb {P}}^1$ with continuous potentials, and the slice current $\hat {T}_f|_{\{\lambda \}\times {\mathbb {P}}^1}$ coincides with the measure $\mu _{f_\lambda }$. See, for example, [Reference Dujardin and FavreDF08, § 3]. If $\Lambda = \mathrm {Rat}_d$ is the space of all maps of degree $d$, then we simply denote this current by $\hat {T}$, as in the introduction.

We say the family $f_\lambda$, for $\lambda \in \Lambda$, is stable if the Julia sets of $f_\lambda$ are moving holomorphically with $\lambda$; see [Reference Mañé, Sad and SullivanMSS83, Reference LyubichLyu83b, Reference McMullenMcM94] for background. Following [Reference DeMarcoDeM01, Reference Dujardin and FavreDF08], the bifurcation current for $f$ is defined by

(2.2)\begin{equation} T_{f, \mathrm{bif}} = (\pi_\Lambda)_* ( \hat{T}_f \wedge [C] ), \end{equation}

where $\pi _\Lambda : \Lambda \times {\mathbb {P}}^1 \to \Lambda$ is the projection and $[C]$ is the current of integration along the critical locus of $f$; it is a closed and positive $(1,1)$-current on $\Lambda$ with continuous potentials. In [Reference DeMarcoDeM01], it is proved that $T_{f, \mathrm {bif}} = 0$ if and only if the family is stable.

For algebraic families, namely where $\Lambda$ is a smooth quasiprojective complex algebraic variety, and $f$ is a morphism, McMullen proved in [Reference McMullenMcM87] that the family $\{f_\lambda : \lambda \in \Lambda \}$ is stable if and only if either $f$ is isotrivial or $f$ is a family of flexible Lattès maps. (The family $f$ is isotrivial if all $f_\lambda$ are conjugate by elements of $\operatorname {Aut}{\mathbb {P}}^1$.) Specifically, McMullen proved that if $f$ is stable but not isotrivial, then each critical point of $f_\lambda$ is preperiodic for all $\lambda \in \Lambda$. In [Reference Dujardin and FavreDF08], Dujardin and Favre extended this result by studying the iterates of each critical point independently. Namely, if $c: \Lambda \to {\mathbb {P}}^1$ parameterizes a critical point for $f_\lambda$, they introduced the current

(2.3)\begin{equation} \hat{T}_{f,c} := (\pi_\Lambda)_* (\hat{T}_f \wedge [\Gamma_c]) \end{equation}

on $\Lambda$, where $\Gamma _c \subset \Lambda \times {\mathbb {P}}^1$ is the graph of $c$, and they proved that $\hat {T}_{f,c} = 0$ if and only if $f$ is isotrivial or $c$ is persistently preperiodic for $f_\lambda$ (see [Reference Dujardin and FavreDF08, Theorems 2.5 and 3.2]).

2.6 Stability of a marked point

Assume that $\Lambda$ is a smooth quasiprojective complex algebraic variety. Suppose that $a \in {\mathbb {P}}^1(k)$ is any point defined over the function field $k = {\mathbb {C}}(\Lambda )$ defining a holomorphic map $a: \Lambda \to {\mathbb {P}}^1$. The pair $(f,a)$ is isotrivial if both $f$ and $a$, after changing coordinates by Möbius transformation defined over a finite extension of $k = {\mathbb {C}}(\Lambda )$, become independent of the parameter $\lambda$. In other words, working over ${\mathbb {C}}$, the group $\operatorname {Aut} {\mathbb {P}}^1$ acts on pairs $(f,a) \in \mathrm {Rat}_d \times {\mathbb {P}}^1$ by $A\cdot (f,a) = (A\circ f \circ A^{-1}, A\circ a)$, and a pair $(f,a)$ defined over $k$ is isotrivial if the associated map $\Lambda \to (\mathrm {Rat}_d\times {\mathbb {P}}^1)/\operatorname {Aut}{\mathbb {P}}^1$ is constant.

Similarly to (2.3), we can define $\hat {T}_{f,a} := (\pi _\Lambda )_* (\hat {T}_f \wedge [\Gamma _a])$ by intersecting the graph $\Gamma _a$ with $\hat {T}_f$ in $\Lambda \times {\mathbb {P}}^1$. Then $\hat {T}_{f,a} = 0$ if and only if the pair $(f,a)$ is either isotrivial or persistently preperiodic [Reference DeMarcoDeM16, Theorem 1.4]. (Strictly speaking, the theorem there is only proved for $\Lambda$ of dimension 1, but it holds more generally, and it is not formulated in terms of the current $\hat {T}_{f,a}$. The equivalence between the stability condition there and the vanishing of $\hat {T}_{f,a}$ is proved in [Reference DeMarcoDeM03, Theorem 9.1].) This characterization of stability was reproved and extended to a more general setting by Gauthier and Vigny in [Reference Gauthier and VignyGV19]. We need the following consequence.

3.1 Proof of Theorem 3.1 when dimension of $S$ is 1

For simplicity, we first present the proof when $S$ is a curve. Let $\Omega$ be any Zariski-open subset of $S \times {\mathbb {P}}^1$. It suffices to show there exists a single point $(s_0, z_0) \in \Omega$ for which $z_0 \in \mathrm {Preper}(f_{s_0}) \cap \mathrm {Preper}(g_{s_0})$. Choose any irreducible, algebraic curve $P \subset S\times {\mathbb {P}}^1$ parameterizing a periodic point of $f$ that intersects $\Omega$. Note that there are infinitely many choices of such curves, because $f_s$ has infinitely many periodic points for every $s \in S$. In fact, there exists a choice of $s$ so that all periodic points of sufficiently large period for $f_s$ will lie in $\Omega$ (a fact that will be relevant to our argument). We may view the curve $P$ as the graph of a point in ${\mathbb {P}}^1(k')$ for some finite extension $k'$ of $k = {\mathbb {C}}(S)$. Let $S'\to S$ denote a finite branched covering map so that $k' = {\mathbb {C}}(S')$. By construction, the pair $(f,P)$ is persistently preperiodic over $S'$.

Now assume that the pair $(g, P)$ is not isotrivial. If the pair $(g,P)$ is also persistently preperiodic for $g$, then we are done. Otherwise, by Theorem 2.4, there exists $s_0\in S'$ at which $P_{s_0}$ is preperiodic for $g_{s_0}$. This completes the proof under this assumption of non-isotriviality of $(g,P)$.

If $(g,P)$ is isotrivial, it is convenient to pass to a further finite branched cover $S' \to S$, if necessary, and change coordinates so that the family $g_s$ is independent of $s \in S'$ and the point $P_s$ is constant. In these new coordinates, if the pair $(g,P)$ is persistently preperiodic, the proof is complete. If the pair $(g,P)$ is isotrivial but with infinite orbit, we then repeat the argument with a different choice of curve $P$. If the pair $(g,P)$ is isotrivial for all curves $P$ parameterizing points of a given large period for $f$, then each of these periodic points for $f$ is constant in these coordinates over $S'$. In particular, by an interpolation argument, $f$ itself must be a constant family. More precisely, choosing any period $N > 2d+1$, we would be able to find a set of distinct points $z_1, z_2, \ldots, z_N \in {\mathbb {P}}^1({\mathbb {C}})$ so that $f_s(z_i) = z_{i+1}$ for all $s \in S'$ and all $i = 1, \ldots, N-1$, and this would imply that the maps $f_s$ are constant in $s$ (see [Reference DeMarcoDeM16, Lemma 2.5]). In other words, the pair $(f,g)$ is isotrivial, violating the hypothesis. This completes the proof.

3.2 Proof of Theorem 3.1 for any base $S$

Let $m = \dim _{\mathbb {C}} S$. Let $\Omega$ be any Zariski-open subset of $S \times ({\mathbb {P}}^1)^m$. It suffices to show that there exists a single point $(s_0, z_1, z_2, \ldots, z_m) \in \Omega$ for which

\[ z_i \in \mathrm{Preper}(f_{s_0})\cap \mathrm{Preper}(g_{s_0}) \]

for all $i = 1, \ldots, m$. Indeed, this shows Zariski density of the preperiodic points of $\Phi ^{(m)}$ in $S \times \Delta ^m$. For $S \times \Delta ^\ell$ with $\ell < m$, we observe that the projection, forgetting some factors of $\Delta$, will still be Zariski dense, which will complete the proof.

The proof proceeds by induction on the dimension of $S$.

First let $P_1$ denote an irreducible subvariety in $S \times ({\mathbb {P}}^1)^m$ of codimension 1 having nontrivial intersection with $\Omega$, and for which $z_1$ is periodic for $f_s$ for all $(s, z_1, \ldots, z_m) \in P_1$. As the periodic points of $f$ are Zariski dense in $S \times {\mathbb {P}}^1$, we can always find such a $P_1$. Projecting $P_1$ to the $z_1$ coordinate, we may view this $P_1$ as single marked point defined on a finite (branched) cover $S'$ of $S$. Now consider the pair $(g,P_1)$ over $S'$, abusing the notation slightly to identify $P_1$ with its projection. If this pair is isotrivial over $S'$, then there is a change of coordinates (passing to a further finite branched cover of $S'$ if necessary) so that the pair is constant. In this case, we select a different periodic point $P_1$ for $f$. If all periodic points of large period for $f$ lead to isotrivial pairs for $g$, then we carry out an interpolation argument as in § 3.1 to deduce that $f$ must also be constant in the new coordinates over $S'$. In other words, the pair $(f,g)$ is isotrivial, a contradiction.

Thus, we may assume that there exists a periodic point $P_1$ for $f$ so that $P_1 \cap \Omega \not = \emptyset$ and the pair $(g, P_1)$ is not isotrivial over $S$. It follows from Theorem 2.4 that the pair $(g,P_1)$ is either persistently preperiodic, in which case we let $x_1$ be any element of $P_1\cap \Omega$, or there exists a point $x_1 = (s_1, z_1, \ldots, z_m) \in P_1 \cap \Omega$ so that $z_1$ is preperiodic to a repelling cycle of $g_{s_1}$. We then consider an irreducible subvariety $P_1' \subset P_1$ containing $x_1$ along which $z_1$ is persistently preperiodic for $g$. Note that the codimension of $P_1'$ is at most 1 and its projection to $S$ will have codimension at most 1. If the codimension is 1, we let $S_1$ be its projection to the base. If this codimension is 0, we replace $P_1'$ with its intersection with $\pi ^{-1}(S_1)$ for an arbitrarily chosen irreducible subvariety $S_1$ of codimension 1 in $S$ passing through $s_1$. Therefore, $P_1'$ has nonempty intersection with $\Omega$, the projection of $P_1'$ to the base $S$ has dimension $m-1$, and the $z_1$-coordinate of $P_1'$ is persistently preperiodic for both $f$ and $g$. If $S_1$ is singular, we replace it with the regular part, so that it will be a smooth and irreducible quasiprojective variety of dimension $m-1$.

We now repeat the process, beginning with a subvariety $P_2$ of codimension 1 in $P_1'$, having nonempty intersection with $\Omega$, and for which the second coordinate $z_2$ is periodic for $f$ over all of $S_1$, with $z_2$ not identically equal to $z_1$ throughout $P_2$. Projecting to the $z_2$-coordinate, we consider the associated pair $(g,P_2)$, passing to a further finite branched cover $S' \to S$ if needed. If this pair is isotrivial, we replace $P_2$ with another choice of periodic point for $f$; as above, if all periodic points of sufficiently large period for $f$ lead to isotrivial pairs $(g,P_2)$, then the pair $(f,g)$ would be isotrivial along $S_1$. Here we use the assumption that $(f,g)$ is maximally non-isotrivial, not just non-isotrivial. So we can find a $P_2$ that has nonempty intersection with $\Omega$ and so that, when projecting to the $z_2$-coordinate, the pair $(f,P_2)$ is persistently periodic and the pair $(g,P_2)$ is not isotrivial.

In this way, we inductively reduce the problem to the argument of § 3.1. This completes the proof of Theorem 3.1.

4.1 The bifurcation current and measure for families of endomorphisms

Suppose that $S$ is a smooth and irreducible quasiprojective variety defined over ${\mathbb {C}}$. A family of ( $k$-dimensional) polarized dynamical systems $(X\to S, \Phi, \mathcal {L})$ is given by a family of complex projective varieties $X\to S$, flat over $S$ with smooth fibers $X_s$ of dimension $k$ over each $s\in S$, a regular map $\Phi : X\to X$ that preserves the fibers $X_s$, and a relatively ample line bundle $\mathcal {L}$ on $X$ such that for each $s\in S$, we have $(\Phi |_{X_s})^{*}(\mathcal {L}|_{X_s})\simeq (\mathcal {L}|_{X_s})^{\otimes d}$ for some $d>0$.

As explained, for instance, in [Reference Gauthier and VignyGV19, § 2.3], to such a family we can associate a dynamical Green current, denoted by $\hat {T}_{\Phi }$, as follows. We let $\hat {\omega }$ be a smooth positive $(1,1)$-form on $X$ cohomologous to a multiple of $\mathcal {L}$ such that $\omega _s :=\hat {\omega }|_{X_{s}}$ is Kähler for all $s\in S$ and

(4.1)\begin{equation} \int_{X_{s}}\omega^k_{s}=1 \end{equation}

for each $s\in S$, where $k = \dim X_s$. The sequence $d^{-n}(\Phi ^n)^{*}(\hat {\omega })$ converges weakly to a closed positive $(1,1)$-current $\hat {T}_{\Phi }$ with continuous potentials.

Suppose $(X \to S, \Phi, \mathcal {L})$ is a family of $k$-dimensional polarized dynamical systems. Suppose that $Y$ is closed subvariety of $X$ of codimension equal to $r$, defining a flat family over $S$. As defined in [Reference Gauthier and VignyGV19], the bifurcation current for the triple $(X, Y, \Phi )$ is defined by

(4.2)\begin{equation} \hat{T}_{\Phi, Y} := \pi_* (\hat{T}_{\Phi}^{\wedge (k-r+1)} \wedge [Y]), \end{equation}

where $\pi :X \to S$ is the projection. The bifurcation measure is given by

(4.3)\begin{equation} \mu_{\Phi, Y} := (\hat{T}_{\Phi, Y})^{\wedge (\dim S)} \end{equation}

on $S$. The wedge powers are well defined because the current has continuous potentials. Also as a consequence of having a continuous potential, the bifurcation measure $\mu _{\Phi, Y}$ does not charge pluripolar sets in $S$ (see [Reference KlimekKli91, Proposition 4.6.4]).

Example 4.4 For any algebraic family of pairs $\Phi = (f,g)$ over a smooth and irreducible complex quasiprojective variety $S$, with projection $\pi : S\times ({\mathbb {P}}^1)^2 \to S$, we let $X = S \times ({\mathbb {P}}^1\times {\mathbb {P}}^1)$ and set $Y = S \times \Delta$, where $\Delta \subset {\mathbb {P}}^1 \times {\mathbb {P}}^1$ is the diagonal. Then we refer to $\hat {T}_{\Phi, Y}$ as the pairwise-bifurcation current associated to $\Phi$ and denote it by $\hat {T}_{\Phi,\Delta }$. In the notation of Example 4.2, we have

\begin{align*} \hat{T}_{\Phi, \Delta} &= \pi_* (\hat{T}_{\Phi}^{\wedge 2} \wedge [S \times \Delta]) \\ &= \pi_* ( p_1^*\hat{T}_f \wedge p_2^* \hat{T}_g \wedge [S \times \Delta] ) \\ &= \pi_*' ( \hat{T}_f \wedge \hat{T}_g ), \end{align*}

where $\pi '$ denotes the projection from $S \times {\mathbb {P}}^1$ to $S$. In particular, when $S$ is the total space $\mathrm {Rat}_d\times \mathrm {Rat}_d$, this agrees with the pairwise-bifurcation current $\hat {T}_{\Delta }$ defined in (1.1). The pairwise-bifurcation measure is defined as

\[ \mu_{\Phi, \Delta} = (\hat{T}_{\Phi, \Delta})^{\wedge (\dim S)}. \]

4.2 Non-degeneracy and equidistribution

Suppose $(X \to S, \Phi, {\mathcal {L}})$ is a family of $k$-dimensional polarized dynamical systems. Suppose that $Y$ is a closed subvariety of $X$ of codimension equal to $r$, defining a flat family of subvarieties in $X$. We say that the triple $(X, Y, \Phi )$ is non-degenerate if the current

\[ \hat{T}_{\Phi}^{\wedge (k-r + \dim S)} \wedge [Y] \]

is nonzero on $X$. This is an exact analog of the notion of non-degeneracy introduced by Habegger in [Reference HabeggerHab13] and studied in general by Gao [Reference GaoGao20a] for subvarieties in families of abelian varieties, where the Betti form is replaced by $\hat {T}_{\Phi }$ on the total space $X$. This notion of non-degeneracy agrees with the one introduced by Yuan and Zhang [Reference Yuan and ZhangYZ21, § 6.2.2] as they demonstrate in [Reference Yuan and ZhangYZ21, Lemma 5.4.4].

For a family of hypersurfaces $Y$, Gauthier, Taflin, and Vigny recently observed a relation between the bifurcation measure and non-degeneracy on a fiber power of $X$ (see [Reference Gauthier, Taflin and VignyGTV23]; compare [Reference YuanYua21, Lemma 4.1]). For any integer $m\geq 1$, let

\[ \Phi^{(m)}: X^{(m)} \to X^{(m)} \]

be the fiber power of $\Phi$ acting on the $m$th fiber power of $X$ over $S$. It is polarized by the line bundle $p_1^* {\mathcal {L}}\otimes \cdots \otimes p_m^* {\mathcal {L}}$, where $p_i: X^{(m)} \to X$ is the projection to the $i$th factor. Let $Y^{(m)}$ be the corresponding fiber power of $Y$ over $S$. We continue to denote the projection to $S$ by $\pi$.

Proposition 4.5 [Reference Gauthier, Taflin and VignyGTV23, Proposition 1.4]

Suppose $(X \to S, \Phi, {\mathcal {L}})$ is a family of $k$-dimensional polarized dynamical systems, and let $m = \dim S$. Assume that $Y$ is a closed hypersurface in $X$, defining a flat family of hypersurfaces over $S$. Then the bifurcation measure $\mu _{\Phi, Y}$ on $S$ satisfies

\[ \mu_{\Phi, Y} =(\hat{T}_{\Phi, Y})^{\wedge m} = \pi_* ( \hat{T}_{\Phi^{(m)}}^{\wedge (mk)} \wedge [Y^{(m)}] ). \]

In particular, the triple $(X^{(m)}, Y^{(m)}, \Phi ^{(m)})$ is non-degenerate if and only if the bifurcation measure $\mu _{\Phi, Y}$ is nonzero.

Proof. The first equality is simply the definition of $\mu _{\Phi,Y}$, and we need to prove the second. Since the dimension of each fiber of $X \to S$ is $k$, it follows that $\hat {T}_\Phi ^{\wedge (k+1)} = 0$ on $X$. Let $m = \dim S$, and note that the fibers of $X^{(m)}\to S$ have dimension $mk$ and $Y^{(m)}$ has codimension $m$. Let $p_j: X^{(m)} \to X$ be the projection to the $j$th factor. Then there is a constant $C>0$ so that

\begin{align*} \hat{T}_{\Phi^{(m)}}^{\wedge (mk)} &= C (p_1^* \hat{T}_\Phi + \cdots + p_m^* \hat{T}_\Phi)^{\wedge (mk)} \\ &= C \frac{(mk)!}{(k!)^m} \bigwedge_{j=1}^m p_j^* \hat{T}_\Phi^{\wedge k}. \end{align*}

Because of the normalization that $\hat {T}_{\Phi ^{(m)}}^{\wedge (mk)}$ be a probability measure on each slice over $t \in S$, we must take $C = {(k!)^m}/{(mk)!}$. On the other hand, we have

\[ [Y^{(m)}] = \bigwedge_{j=1}^m p_j^*[Y], \]

so that

\[ \hat{T}_{\Phi^{(m)}}^{\wedge (mk)} \wedge [Y^{(m)}] = \bigwedge_{j=1}^m \, p_j^* (\hat{T}_\Phi^{\wedge k} \wedge [Y] ) \]

and the conclusion follows.

Now assume that the triple $(X, Y, \Phi )$ is defined over a number field and is non-degenerate. Yuan and Zhang recently proved an equidistribution theorem in [Reference Yuan and ZhangYZ21] for points of small fiber-wise canonical height in $Y$, extending a result of Kühne [Reference KühneKüh21]. A closely related result has also recently been obtained by Gauthier [Reference GauthierGau21]. A sequence of points $y_n \in Y(\overline {\mathbb {Q}})$ is said to be generic if no subsequence lies in a proper, Zariski-closed subset of $Y$.

Theorem 4.6 [Reference Yuan and ZhangYZ21, Theorem 6.2.3]

Suppose $(X \to S, \Phi, {\mathcal {L}})$ is a family of $k$-dimensional polarized dynamical systems over smooth, irreducible, quasiprojective $S$, defined over a number field $K$. Suppose that the triple $(X, Y, \Phi )$ is non-degenerate, where $Y\!$ is a closed subvariety of $X$ of codimension $r$, defining a flat family over $S$, and also defined over $K$. Then for any generic sequence of preperiodic points of $\Phi$ in $Y(\overline {K})$, their $\operatorname {Gal}(\overline {K}/K)$-orbits are uniformly distributed with respect to the measure $\hat {T}_{\Phi }^{\wedge (k-r+\dim S)} \wedge [Y]$ on $X({\mathbb {C}})$. More precisely, given any continuous function $\varphi$ with compact support in $X$, and given any generic sequence $\{y_n\}$ of points in $Y(\overline {K})$ that are preperiodic for $\Phi$, we have

\[ \frac{1}{\# \operatorname{Gal}(\overline{K}/K)\cdot y_n} \sum_{y \in \operatorname{Gal}(\overline{K}/K)\cdot y_n} \varphi(y) \longrightarrow \frac{1}{\operatorname{vol}(Y)} \int_{X({\mathbb{C}})} \varphi (\hat{T}_{\Phi}^{\wedge (k-r+\dim S)} \wedge [Y]), \]

as $n\to \infty$ where $\operatorname {vol}(Y) = \int _{X({\mathbb {C}})} \hat {T}_{\Phi }^{\wedge (k-r+\dim S)} \wedge [Y]$.

4.3 A repelling-cycle criterion to show non-degeneracy

Suppose $(\pi : X \to S, \Phi, {\mathcal {L}})$ is a family of $k$-dimensional polarized dynamical systems. Now suppose that $Y$ is a closed subvariety of $X$ of codimension $m = \dim S$ in $X$, defining a flat family of subvarieties with codimension $m$. We say that a point $y_0 \in Y({\mathbb {C}})$ is a rigid repeller for $(X, Y, \Phi )$ if:

1. (1) some iterate $x_0 = \Phi ^{n_0}(y_0)$ is a repelling periodic point for $\Phi _{s_0}$, where $s_0 = \pi (y_0)$;

2. (2) the point $y_0$ lies in the support of the equilibrium measure $\mu _{s_0} := \big (\hat {T}_\Phi |_{X_{s_0}}\big )^k$ in the fiber $X_{s_0}$; and

3. (3) there is a holomorphic section $\eta$ over a neighborhood of $s_0$ in $S$ parameterizing a repelling periodic point of $\Phi _s$, with $\eta (s_0) = x_0$, so that $x_0$ is an isolated point of the intersection of $\Phi ^{n_0}(Y)$ with the image of $\eta$.

The next proposition is a minor modification of [Reference Gauthier and VignyGV19, Lemma 4.8], and the proof is very similar to that of [Reference Berteloot, Bianchi and DupontBBD18, Proposition 3.7].

Proposition 4.8 Suppose $(X \to S, \Phi, {\mathcal {L}})$ is a family of $k$-dimensional polarized dynamical systems over quasiprojective $S$. Suppose that $Y$ is a closed subvariety of $X$ of codimension equal to $\dim S$, defining a flat family over $S$. Suppose there exists a rigid repeller for $(X, Y, \Phi )$ at $y_0 \in Y({\mathbb {C}})$. Then $y_0$ lies in the support of the (nonzero) measure

\[ (\hat{T}_{\Phi})^{\wedge k} \wedge [Y] \]

on $X({\mathbb {C}})$. In particular, the triple $(X, Y, \Phi )$ is non-degenerate.

Proof. Let $d \geq 2$ be the polarization degree of $\Phi$. Let $x_0 = \Phi ^{n_0}(y_0) \in \Phi ^{n_0}(Y)$ be a repelling periodic point in the orbit of $y_0$. Let $s_0 = \pi (x_0) \in S$. Since $y_0$ is in the support of the equilibrium measure $\mu _{s_0} = (\hat {T}_\Phi |_{X_{s_0}})^k$, it follows that $x_0$ is also in this support. Let $p$ be the period of $x_0$. Let $\eta$ denote the parameterization of the nearby repelling periodic points over a neighborhood $U$ of $s_0$, and let $\Gamma _\eta$ denote its image in $X$. By hypothesis, $x_0$ is an isolated point of the intersection of $\Gamma _\eta$ with $\Phi ^{n}(Y_0)$.

Shrinking $U$ if necessary, there exists a tubular neighborhood $N$ of $\Gamma _\eta$ in $\pi ^{-1}(U)$ and a constant $K>1$ so that

\[ d_{X_s}(\Phi_s^p(x), \Phi_s^p(\eta(s))) \geq K \, d_{X_s}(x, \eta(s)) \]

for all $x \in N \cap X_s$ and all $s \in U$ and for any reasonable choice of distance function $d_{X_s}$ on the fibers. In particular, there exists a nested sequence of tubular neighborhoods $N_n \subset N$ around $\Gamma _\eta \cap \pi ^{-1}(U)$, for $n\geq 1$, so that $\Phi ^{np}: N_n \to N$ is proper and one-to-one. Then for all integers $n\geq 0$,

\begin{align*} d^{npk} \int_{N_n} \hat{T}_\Phi^{\wedge k} \wedge [\Phi^{n_0}(Y)] &= \int_{N_n} (\Phi^{np})^* \hat{T}_\Phi^{\wedge k} \wedge [\Phi^{n_0}(Y)] \\ &= \int_N \hat{T}_\Phi^{\wedge k} \wedge (\Phi^{np})_* [\Phi^{n_0}(Y)] \\ &= \int_N \hat{T}_\Phi^{\wedge k} \wedge [\Phi^{n_0+np}(Y)]. \end{align*}

On the other hand, we have

\[ \lim_{n\to\infty} \chi_N [\Phi^{n_0+np}(Y)] = \alpha [X_{s_0}\cap N] \]

for some $\alpha >0$, in the weak sense of currents, where $\chi _N$ is the indicator function. Indeed, since $N\cap \Phi ^{n_0}(Y) \cap \Gamma _\eta = \{x_0\}$, the vertical expansion of $\Phi ^p$ shows that the limit must be supported in the fiber $X_{s_0}$. As the limit current is closed and positive and nonzero, it must be (a scalar multiple of) the current of integration along $X_{s_0}\cap N$. Finally, since $x_0$ is in the support of the measure $\mu _{s_0} = \hat {T}_\Phi ^{\wedge k}|_{X_{s_0}}$, we have

\[ \int_N \hat{T}_\Phi^{\wedge k} \wedge [X_{s_0}\cap N] > 0 \]

so that

\[ \int_N \hat{T}_\Phi^{\wedge k} \wedge [\Phi^{n_0+np}(Y)] > 0 \]

for all sufficiently large $n$.

Now choose an open set $V$ around $y_0$ so that $\Phi ^{n_0}: V \to N$ is proper. Then

\begin{align*} d^{n_0k} \int_V \hat{T}_\Phi^{\wedge k} \wedge [Y] &= \int_V (\Phi^n)^* \hat{T}_\Phi^{\wedge k} \wedge [Y] \\ &= \int_{N} \hat{T}_\Phi^{\wedge k} \wedge (\Phi^{n_0})_* [Y] \\ &\geq \int_{N} \hat{T}_\Phi^{\wedge k} \wedge [\Phi^{n_0}(Y)] \\ &\geq \int_{N_n} \hat{T}_\Phi^{\wedge k} \wedge [\Phi^{n_0}(Y)] \end{align*}

for all $n\geq 1$. Therefore,

\[ \int_V \hat{T}_\Phi^{\wedge k} \wedge [Y] > 0. \]

Now let $S$ be a smooth and irreducible complex quasiprojective variety. Recall that an algebraic family of pairs $\Phi = (f,g)$ was defined in § 3, and the pairwise-bifurcation current was defined in Example 4.4.

5.1 Product structure

Let $\Phi = (f,g)$ act on ${\mathbb {P}}^1\times {\mathbb {P}}^1$, defined over the field $k = \overline {\mathbb {Q}}(S)$. Let $\Delta \subset {\mathbb {P}}^1\times {\mathbb {P}}^1$ be the diagonal. For $m = \dim S$, we let $\Phi ^{(m)}$ denote the product map acting on $({\mathbb {P}}^1 \times {\mathbb {P}}^1)^m$ over $k$. Following [Reference Mavraki and SchmidtMS22], we consider the product of $({\mathbb {P}}^1\times {\mathbb {P}}^1)^{m}$ with another copy of ${\mathbb {P}}^1$, acted on by $f$ or by $g$. This defines maps over ${\mathbb {C}}$ as

\begin{gather*} (\Phi^{(m)},f): S \times ({\mathbb{P}}^1)^{2m + 1} \to S \times ({\mathbb{P}}^1)^{2m + 1}\\ (s, z_1, \ldots, z_{2m + 1}) \mapsto (s, f_s(z_1), g_s(z_2), f_s(z_3), \ldots, g_s(z_{2 m}), f_s(z_{2m + 1})) \end{gather*}

and

\begin{gather*} (\Phi^{(m)},g): S \times ({\mathbb{P}}^1)^{2m + 1} \to S \times ({\mathbb{P}}^1)^{2m +1}\\ (s, z_1, \ldots, z_{2m + 1}) \mapsto (s, f_s(z_1), g_s(z_2), f_s(z_3), \ldots, g_s(z_{2 \dim S}), g_s(z_{2m + 1})). \end{gather*}

Let

\[ p_1: S \times ({\mathbb{P}}^1\times{\mathbb{P}}^1)^{m}\times {\mathbb{P}}^1 \to S \times ({\mathbb{P}}^1\times{\mathbb{P}}^1)^{m} \]

be the projection forgetting the final factor of ${\mathbb {P}}^1$. Let

\[ p_2: S \times ({\mathbb{P}}^1\times{\mathbb{P}}^1)^{m}\times {\mathbb{P}}^1 \to S \times {\mathbb{P}}^1 \]

denote the projection forgetting the intermediate factor.

Proposition 5.3 Let $S$ be an irreducible quasiprojective complex algebraic variety of dimension $m$, and suppose $\Phi = (f,g)$ is an algebraic family of pairs of degree $d>1$ over $S$. We have

\[ M_f := \hat{T}_{(\Phi^{(m)},f)}^{\wedge(2m + 1)} \wedge [S \times \Delta^{m}\times {\mathbb{P}}^1] = p_1^*R \wedge p_2^*\hat{T}_f, \]

where

\[ R := \hat{T}_{\Phi^{(m)}}^{\wedge 2m} \wedge [S \times \Delta^{m}]. \]

Similarly for $g$ and

\[ M_g := \hat{T}_{(\Phi^{(m)},g)}^{\wedge(2m + 1)} \wedge [S \times \Delta^{m}\times {\mathbb{P}}^1]. \]

Consequently, if the bifurcation measure $\mu _{\Phi, \Delta }$ is nonzero for a family of pairs $\Phi = (f,g)$ parameterized by $S$, then $M_f$ and $M_g$ are nonzero.

Proof. Suppose that $f$ is an algebraic family of maps on ${\mathbb {P}}^1$ over $S$. Then $\hat {T}_f$ on $S \times {\mathbb {P}}^1$ satisfies $\hat {T}_f^{\wedge 2} = 0$. It follows that, for any fiber product of such maps, $(f_1, \ldots, f_\ell )$ on $({\mathbb {P}}^1)^\ell$ over $S$, we have

\[ \hat{T}_{(f_1, \ldots, f_\ell)}^{\wedge \ell}= q_1^* \hat{T}_{f_1} \wedge \cdots \wedge q_m^* \hat{T}_{f_\ell} \]

for the projections $q_i: S\times ({\mathbb {P}}^1)^\ell \to S\times {\mathbb {P}}^1$. In the setting of the proposition, it follows that

\[ \hat{T}_{\Phi^{(m)}}^{\wedge(2m)} = q_1^* \hat{T}_f \wedge q_2^* \hat{T}_g \wedge \cdots \wedge q_{2m - 1}^* \hat{T}_f \wedge q_{2m}^* \hat{T}_g \]

and

\[ \hat{T}_{(\Phi^{(m)},f)}^{\wedge(2m + 1)} = p_1^* \hat{T}_{\Phi^{(m)}}^{\wedge(2m)} \wedge p_2^* \hat{T}_f. \]

The first statements of the proposition follow. Finally, since $\Delta \subset {\mathbb {P}}^1\times {\mathbb {P}}^1$ is a hypersurface, we know from Proposition 4.5 that $\mu _{\Phi, \Delta }$ is nonzero if and only if

\[ R = \hat{T}_{\Phi^{(m)}}^{\wedge(2m)} \wedge [S \times \Delta^m] > 0. \]

In this case, we see immediately that $M_f$ and $M_g$ are also nonzero.

5.2 Proof of Theorem 5.1

Recall that a sequence of points $z_n$ in a variety $Z$ is said to be generic if no subsequence lies in a proper, Zariski-closed subset of $Z$.

Let $m=\dim _{\mathbb {C}} S$. Note that the set

\[ \{(s, x_1, \ldots, x_{m+1}) \in (S \times ({\mathbb{P}}^1)^{m+1})(\overline{\mathbb{Q}}): x_i \in \mathrm{Preper}(f_s) \cap \mathrm{Preper}(g_s) \mbox{ for each } i\} \]

in the statement of Theorem 5.1 is naturally identified with the set

\[ \mathrm{Preper}(\Phi^{(m+1)}) \cap (S(\overline{\mathbb{Q}}) \times \Delta^{m+1}) \subset S\times ({\mathbb{P}}^1\times{\mathbb{P}}^1)^{m+1}, \]

for the $(m+1)$th fiber power of $\Phi$ over $S$.

Let $K$ be a number field over which $\Phi$ and $S$ are defined. Suppose, towards a contradiction, that $\mathrm {Preper}(\Phi ^{(m+1)})$ is Zariski dense in $S(\overline {K})\times \Delta ^{m+1}$. Via the projection of $\Delta \subset {\mathbb {P}}^1\times {\mathbb {P}}^1$ to the component ${\mathbb {P}}^1$'s, these preperiodic points of $\Phi ^{(m+1)}$ in $S\times \Delta ^{m+1}$ project to define a generic sequence of points in $(S\times \Delta ^{m}\times {\mathbb {P}}^1)(\overline {K})$ that are preperiodic for both the maps $(\Phi ^{(\dim S)}, f)$ and $(\Phi ^{(\dim S)}, g)$. Since the pairwise-bifurcation measure $\mu _{\Phi, \Delta }$ is nonzero on $S({\mathbb {C}})$, we know from Proposition 5.3 that the measures $M_g$ and $M_f$ are nonzero on $( S\times ({\mathbb {P}}^1\times {\mathbb {P}}^1)^m\times {\mathbb {P}}^1)({\mathbb {C}})$. In other words, setting

\[ X = S\times({\mathbb{P}}^1\times{\mathbb{P}}^1)^m\times {\mathbb{P}}^1 \quad\mbox{and}\quad Y = S\times\Delta^{m}\times {\mathbb{P}}^1, \]

the triples $(X, Y, (\Phi ^{(m)},f))$ and $(X, Y, (\Phi ^{(m)},g))$ are non-degenerate. By Theorem 4.6, it follows that the $\operatorname {Gal}(\overline {K}/K)$-orbits of these preperiodic points must be uniformly distributed with respect to the measures $M_f$ and $M_g$. Consequently, we have $M_f = M_g$ in $X({\mathbb {C}})$.

Now let $p_1: S \times ({\mathbb {P}}^1\times {\mathbb {P}}^1)^{m}\times {\mathbb {P}}^1 \to S \times ({\mathbb {P}}^1\times {\mathbb {P}}^1)^{m}$ be the projection forgetting the final factor of ${\mathbb {P}}^1$, as in Proposition 5.3. By slicing $M_f$ and $M_g$, we conclude that

(5.1)\begin{equation} \int_{S\times({\mathbb{P}}^1)^{2m}} \int_{{\mathbb{P}}^1}\varphi(t,x) \, d\mu_{f_{\pi(t)}}(x) \, dR(t) = \int_{S\times({\mathbb{P}}^1)^{2m}} \int_{{\mathbb{P}}^1}\varphi(t,x) \, d\mu_{g_{\pi(t)}}(x) \, dR(t) \end{equation}

for every continuous and compactly supported function $\varphi$ on $S\times ({\mathbb {P}}^1)^{2m+1}$ and the $R$ of Proposition 5.3. Here, $\pi : S\times ({\mathbb {P}}^1)^m \to S$ denotes the projection to the base, and $\mu _{f_{\pi (t)}}$ and $\mu _{g_{\pi (t)}}$ are the measures of maximal entropy introduced in § 2.1. But since $\pi _*R = \mu _{\Phi, \Delta }$, we infer that

\[ \mu_{f_s} = \mu_{g_s} \]

on ${\mathbb {P}}^1$ for $\mu _{\Phi, \Delta }$-almost every parameter $s \in S({\mathbb {C}})$. Indeed, suppose there exists $b \in S({\mathbb {C}})$ with $\mu _{f_b} \not = \mu _{g_b}$ and so that $\mu _{\Phi,\Delta }(U) >0$ for every open neighborhood $U$ of $b$. Then we can find a continuous function $\psi$ on ${\mathbb {P}}^1({\mathbb {C}})$ such that $\int \psi \mu _{f_b} \neq \int \psi \mu _{g_b}$. By continuity of the measures we find that $\int \psi \mu _{f_t} \neq \int \psi \mu _{g_t}$ for all $t$ in a neighborhood $U$ of $b$. Therefore, setting $\varphi (t, x_1, \ldots, x_{2m}, x_{2m+1}) = h_b(t) \psi (x_{2m+1})$ on $S \times ({\mathbb {P}}^1)^{2m+1}$ for a bump function $h_b$ supported in $U$, the equality (5.1) will fail.

Now we use the hypothesis that $f$ and $g$ are not both conjugate to a power map $z^{\pm d}$ over all of $S$. Let $V \subset S$ be the (possibly empty) proper subvariety over which the maps $f$ and $g$ are both conjugate to $\pm z^d$. It follows from Theorem 2.1 that $\mathrm {Preper}(f_s) = \mathrm {Preper}(g_s)$ for all $s\in (\operatorname {supp} \mu _{\Phi, \Delta } \setminus V)({\mathbb {C}})$. As $\mu _{\Phi,\Delta }$ does not charge pluripolar sets, we conclude that $\mathrm {Preper}(f_s) = \mathrm {Preper}(g_s)$ for all $s \in (S \setminus V)({\mathbb {C}})$. Indeed, the preperiodic points of $f$ or $g$ each form a countable union of hypersurfaces in $(S\setminus V) \times {\mathbb {P}}^1$. For each irreducible hypersurface $P \subset S\times {\mathbb {P}}^1$ which is preperiodic for $f$, its intersection with $\mathrm {Preper}(g)$ contains all of $P \cap \pi ^{-1}(\operatorname {supp} \mu _{\Phi, \Delta } \setminus V)$ and so cannot lie in a countable union of hypersurfaces of $P$. Therefore $P$ must be persistently preperiodic for $g$. This shows that $\mathrm {Preper}(f_s) \subset \mathrm {Preper}(g_s)$ for all $s \in (S\setminus V)({\mathbb {C}})$; equality follows from the same argument in reverse.

Again using Theorem 2.1, it follows that $\mu _{f_s} = \mu _{g_s}$ for all $s \in (S\setminus V)({\mathbb {C}})$. By continuity of the equilibrium measures, we conclude that $\mu _{f_s} = \mu _{g_s}$ for all $s\in S({\mathbb {C}})$. Now fix a small open $D$ in the base $S$, and let $U$ be a continuous potential for $\hat {T}_f - \hat {T}_g$ for $(s,z) \in D \times {\mathbb {P}}^1$. As $\hat {T}_f$ and $\hat {T}_g$ have the same slice measures for every $s$, we see that $U$ depends only on the variable $s \in D$. As $\hat {T}_f^{\wedge 2} = \hat {T}_g^{\wedge 2} = 0$, we compute

\[ \hat{T}_{\Phi, \Delta} = \pi'_*(\hat{T}_f \wedge \hat{T}_g) = -\frac12 dd^c \pi'_* (U dd^c U) = -\frac12 (dd^c)_s \bigg( \int_{{\mathbb{P}}^1} U(s,z) (dd^c)_z U(s,z) \bigg) = 0, \]

for the projection $\pi ': S \times {\mathbb {P}}^1 \to S$ (as in Example 4.4). The current $\hat {T}_{\Phi, \Delta }$ vanishes on all open $D \subset S$, so we may conclude that $\mu _{\Phi, \Delta } = 0$ on all of $S({\mathbb {C}})$, contradicting our hypothesis.

5.3 Proof of Theorem 1.8

We assume there exists a rigid $m$-repeller for the family $(f,g)$ over $S$, where $m = \dim S$. Corollary 4.9 implies that the bifurcation measure $\mu _{\Phi, \Delta }$ is nonzero on the space $S$. Finally, Theorem 5.1 gives us the result we desire.

5.4 Proof of Theorem 5.2

Recall that a sequence of points $s_n \in S$ is said to be generic if no subsequence lies in a proper, Zariski-closed subset of $S$. We start with the following lemma.

Proof. Fix $m \geq 1$. For each $n$, let

\[ M(n) \leq \# \; \mathrm{Preper}(f_{s_n}) \cap \mathrm{Preper}(g_{s_n}) \in \mathbb{N} \cup \{\infty\} \]

so be chosen so that $M(n) \to \infty$ as $n\to \infty$. The points of $\mathrm {Preper}(f_{s_n}) \cap \mathrm {Preper}(g_{s_n})$ determine a configuration of $M(n)^m$ points in $\Delta ^m$ that are preperiodic for $\Phi _{s_n}^{(m)}$. Note that this collection of points is symmetric under permutation of the coordinates on $\Delta ^m$.

Let $Z$ be the Zariski closure of these points in $S \times \Delta ^m$. The symmetry of the preperiodic points in $\Delta ^m$ implies that $Z$ is also symmetric under permuting the $m$ coordinates of $\Delta ^m$. Moreover, because the sequence $\{s_n\}$ is Zariski dense in $S$, we see that $\pi (Z) = S$ for the projection $\pi : S\times \Delta ^m \to S$.

Now suppose that $Z$ is not all of $S\times \Delta ^m$. Then, over a Zariski-open subset $U\subset S$, $Z$ is contained in a family of hypersurfaces in $\Delta ^m \simeq ({\mathbb {P}}^1)^m$ over $U$ that are symmetric under permuting the $m$ coordinates. Shrinking $U$ if necessary, these hypersurfaces will have a well-defined degree $(r, \ldots, r)$ for some $r \geq 1$; that is, each projection that forgets one component $\Delta$ will be of degree $r$. But since the sequence $\{s_n\}$ is generic, this implies that $M(n) \leq r$ for all sufficiently large $n$. This is a contradiction.

Now to prove Theorem 5.2, let $S$ be a smooth, irreducible quasiprojective variety parameterizing an algebraic family of pairs $(f,g)$, all defined over $\overline {\mathbb {Q}}$ as in its statement. By Theorem 5.1 and in view of Lemma 5.4 we infer that there exists a strict Zariski closed $V\subset S$ defined over $\overline {\mathbb {Q}}$ and $M\in {\mathbb {R}}$ such that

(5.2)\begin{align} \#\mathrm{Preper}(f_s)\cap\mathrm{Preper}(g_s)\le M, \end{align}

for all $s\in (S\setminus V)(\overline {\mathbb {Q}})$. Write $U:=S\setminus V$. We want to show that $M$ can be chosen so that (5.2) holds for all $s\in U({\mathbb {C}})$. Fix $s_0\in U({\mathbb {C}})\setminus U(\overline {\mathbb {Q}})$ and let $P_{s_0}:= \mathrm {Preper}(f_{s_0})\cap \mathrm {Preper}(g_{s_0})$. Let $L$ be a finitely generated subfield of ${\mathbb {C}}$ (with transcendence degree at least $1$) such that $\Phi _{s_0}$ is defined over $L$. Thus, there is a quasiprojective variety $X$ over $\overline {\mathbb {Q}}$ of finite type with function field $L$ over which we can extend $\Phi _{s_0}$ (viewing $s_0$ as an element of $L$) to an endomorphism $\Phi _X: X\times {\mathbb {P}}^1\times {\mathbb {P}}^1\to X\times {\mathbb {P}}^1\times {\mathbb {P}}^1$ defined over $\overline {\mathbb {Q}}$. We also extend $P_{s_0}$ to $P_X\subset X\times {\mathbb {P}}^1\times {\mathbb {P}}^1$. Note that each specialization $(\Phi _X)_t = (f_t, g_t)$ for $t\in X(\overline {\mathbb {Q}})$ is naturally identified with some $\Phi _s$ for $s \in U(\overline {\mathbb {Q}})$. Thus, for each $t\in X(\overline {\mathbb {Q}})$ we have a uniform bound on $\# \mathrm {Preper}(f_t)\cap \mathrm {Preper}(g_t)$. But clearly the specializations of the distinct points in $P_X$ remain distinct at some $t\in X(\overline {\mathbb {Q}})$. The proof is complete.