1. Introduction

Let $K$ be a number field and $\alpha \in K$. We study the dynamical system attached to a polynomial $f \in K[X]$ of degree $d \geq 2$ and denote by ${f^{\circ n}}$ the $n$th iterate of $f$ defined by $f^{\circ 0}(X) = X$ and $f^{\circ n}(X) = f^{\circ n-1}(f(X))$ for $n \in \mathbb {Z}_{\geq 1}$. We consider the small orbit $\mathcal {S}_\alpha$ of a point $\alpha$ that was introduced by McMullen and Sullivan [Reference McMullen and SullivanMS98, 6.1] as part of their investigations of the dynamics of holomorphic maps. It is given by

\begin{align*} \mathcal{S}_\alpha= \{\beta \in \bar{\mathbb{Q}}; f^{\circ n}(\beta) = {f^{\circ n}}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*}

This $\mathcal {S}_\alpha$ is an infinite set unless $\alpha$ is a super-attractive fixed point of degree $d$ (see [Reference MilnorMil06] for a definition). In order to have a more traditional diophantine analogue we may consider $\mathcal {S}_\alpha$ to be the set of ‘torsion translates’ of $\alpha$ for the dynamical system attached to $f$. For example, in the more classical setting of the multiplicative group $\mathbb {G}_m$ an element $x$ of $\alpha \mu _{\infty }$ (where $\mu _{\infty }$ is the group of roots of unity) satisfies $x^{n} = \alpha ^{n}$ for some positive integer $n$. The study of the equation $x^{n} = y^{n}$ where $y$ varies in a finitely generated group is integral to go from the Mordellic part to Lang's full conjecture in $\mathbb {G}_m^{n}$ (see [Reference LaurentLau84]). In fact, by employing Kummer theory one reduces the conjecture to the study of this equation (and, thus, Manin–Mumford with a certain uniformity). The same approach is used in all known proofs of Mordell–Lang to go from the Mordellic part to the divisible hull (for example, in the ultimate resolution of the conjecture for semi-abelian varieties by McQuillan [Reference McQuillanMcQ95]).

In this article, we obtain a structure theorem for the set $\mathcal {S}_\alpha$ that is analogous to what Manin–Mumford delivers in the context of algebraic groups (and also achieve a certain uniformity). We also use a similar strategy in this context to show, quite analogously as in the group context, that we can go from the small orbit of $\alpha$ to its grand orbit

\begin{align*} \mathcal{G}_\alpha = \{\beta \in \bar{\mathbb{Q}}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\}, \end{align*}

albeit under the condition that the orbit of $\alpha$ tends to infinity at a finite place of good reduction co-prime to $d$. These results may be viewed as new cases of the Zilber–Pink (or unlikely intersection) conjecture on dynamical systems as formulated in [Reference Ghioca and NguyenGN16, p. 1438] as we explain in the following. In order to ease notation and deal with trivial cases and those covered by previous results, we introduce two definitions (one of which appeared previously in the literature).

Our definition of exceptional agrees with that in [Reference Ghioca, Nguyen and YeGNY19] for polynomials. For the notion of pre-periodic varieties we refer the reader to the established literature [Reference Medvedev and ScanlonMS14, Reference Ghioca, Nguyen and YeGNY19]. Our main theorem is the following.

Theorem 1 Let $K$ be a number field and $\mathcal {C} \subset \mathbb {A}^{2}$ be an irreducible plane curve that is not fibral. Let $f \in K[X]$ of degree $d \geq 2$ be a non-exceptional polynomial and $\alpha \in K$ be a point that is not pre-periodic under $f$. If

\begin{align*} \mathcal{C}(\bar{\mathbb{Q}})\cap \mathcal{S}_\alpha^{2} \end{align*}

contains infinitely many points, then there exists $n \in \mathbb {Z}_{\geq 0}$ such that

\begin{align*} (f^{\circ n},f^{\circ n})(\mathcal{C}) = \Delta, \end{align*}

where $\Delta \subset \mathbb {A}^{2}$ is the diagonal.

If $\mathcal {C}$ is fibral, then the statement is trivial and $\mathcal {C}$ is, up to permuting coordinates, of the form $\{\gamma \}\times \mathbb {A}^{1}$ for $\gamma \in \mathcal {S}_\alpha$. If $f$ is an exceptional polynomial, then Manin–Mumford for the multiplicative group (see § 2) answers most questions one might have about Zariski-closures of subsets of $\mathcal {S}_\alpha ^{2}$. If $\alpha$ is pre-periodic for $f$, then $\mathcal {S}_\alpha$ is contained in the set of pre-periodic points of $f$ and in this case the dynamical Bogomolov conjecture for split rational maps proven by Ghioca et al. [Reference Ghioca, Nguyen and YeGNY19, Theorem 1.1] applies. We combine these theorems in a unified statement in Theorem 5. We remark that $\Delta$ is fixed by $(f,f)$ but not all pre-periodic curves in $\mathbb {A}^{2}$ are pull-backs of the diagonal (see, for example, [Reference Medvedev and ScanlonMS14]).

Moreover an extension of Theorem 1 to either $f$ being exceptional or $\alpha$ being pre-periodic fails. A whimsical explanation is that we have more structure lurking in the background if either of these conditions hold. More concretely, if $f = X^{d}$ and thus exceptional, we can take any (co-prime) integers $l,k$ and consider the co-set in $\mathbb {G}_m^{2}$ defined by

(1)\begin{equation} X^{\ell}Y^{k} =\alpha^{\ell+k}. \end{equation}

Then any of these co-sets contain infinitely many points in $\mathcal {S}_\alpha ^{2}$, but if we take $\ell,k$ to be co-prime to $d$ say, it does not define a curve as in the conclusion of the theorem. If $f$ is non-exceptional, then the only super-attracting fixed point of degree $d$ is at infinity. Thus, if $f$ is non-exceptional and $\alpha$ is pre-periodic, say $f^{\circ \ell }(\alpha ) = f^{\circ k}(\alpha )$ with $\ell > k$, then the rational curve given by $(f^{\circ \ell }(X), f^{\circ k}(X))$ provides a counterexample to a possible extension of Theorem 1. Thus, Theorem 1 is the best possible.

Another way of putting Theorem 1 is to say that the only non-trivial algebraic relations that can hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ are those that define $\mathcal {S}_\alpha$ in the first place. It is straightforward to check that the reverse also holds true. Each curve that appears in the conclusion of Theorem 1 also contains infinitely many points on $\mathcal {S}_\alpha ^{2}$.

It is not foreseeable that what is obtained in this article can be achieved with previously established methods such as applying known equidistribution results or Siegel's theorem on integral points. This is because the usual basic assumptions such as small canonical height or rationality of the points under consideration are not fulfilled in the cases we consider.

Our methods are close to transcendence techniques which introduce entirely new techniques into the study of dynamical systems. However, they also lead to a new and elementary proof of Manin–Mumford for $\mathbb {G}_m^{n}$ as was demonstrated in [Reference SchmidtSch03]. We expect there to be more applications of the methods used in this article.

In the assumption of Theorem 1, $\alpha$ is not pre-periodic and, thus, has positive canonical height [Reference Ghioca, Nguyen and YeGNY19]. This implies that there exists a place $v$ of $K$ for which $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ (where we denote by $|\cdot |_v$ the corresponding $v$-adic absolute value) and, consequently, for which the local canonical height of $\alpha$ is positive. All points in $\mathcal {S}_\alpha$ have the same local canonical height at all places of $K$ and if there is an archimedean place at which the local height is positive, then Theorem 1 follows from Theorem 3; if there is such a non-archimedean place, then it follows from Proposition 4.1.

In fact, in each case, we are able to show a stronger version that allows for some uniformity (see Proposition 4.1 and Theorem 10). If $v$ can be chosen to be finite and of good reduction co-prime to $d$, we can go from small orbits to grand orbits. This is quite similar to the case of finitely generated to finite rank groups in Lang's conjecture on semi-abelian varieties that we alluded to previously.

We say that $v$ is a finite place of good reduction (for $f$) if $f = a_0X^{d} + \sum _{i = 1}^{d}a_iX^{d-i}$ and $|a_i/a_0|_v \leq 1$ for $i = 1, \ldots, n$. We say that $v$ is co-prime to $d$ if $|d|_v = 1$.

Theorem 2 Let $K,\mathcal {C}, \Delta$ and $f$ be as in Theorem 1. Further suppose that $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of $K$ of good reduction that is co-prime to $d$. If

\begin{align*} \mathcal{C}(\bar{\mathbb{Q}})\cap \mathcal{G}_\alpha^{2} \end{align*}

contains infinitely many points, then there exist $n,m \in \mathbb {Z}_{\geq 0}$ such that

\begin{align*} (f^{\circ n},f^{\circ m})(\mathcal{C}) = \Delta. \end{align*}

We may view this result as resolving the Zilber–Pink conjecture as formulated in [Reference Ghioca and NguyenGN16] for the curve $\mathcal {C} \times \{\alpha \} \subset (\mathbb {P}_1)^{3}$ and the dynamical system $(f,f,f)$.

As an example of an application of Theorem 2 we can take the non-exceptional polynomial $f = X^{2} -1$ and $\alpha = 1/3$. Then $\mathcal {G}_\alpha$ is dense in the filled Julia set of $f$ (in $\mathbb {C}$) and Theorem 2 shows that the only algebraic relations that can hold between infinitely many distinct pairs of elements in the grand orbit of $1/3$ are the same that define the grand orbit. We could also take the same $f$ and $\alpha = 5/3$ and then $\mathcal {G}_\alpha$ is dense in the complement of the Julia set and our theorem still holds. This gives a hint at the algebraic rigidity of the extended tree formed by the grand orbit under $f$ at a non-pre-periodic point; at least if $f$ is non-exceptional. In contrast if $f = X^{d}$, then we can take the same co-set as previously (1) with $k,\ell$ co-prime to $d$ to obtain a counterexample to an extension of Theorem 2 to exceptional polynomials. Again one could phrase the theorem as stating that the only non-trivial algebraic relations that hold between infinitely many pairs of points in $\mathcal {G}_\alpha$ are those that define the set itself. In addition, in § 9 we show that Theorem 2 cannot be extended to $\alpha$ being a pre-periodic point. For the proof of Theorem 2 we also rely on work of Ghioca, Nguyen and Ye on the dynamical Bogomolov conjecture [Reference Ghioca, Nguyen and YeGNY19, Theorem 1.1]. We want to point out that we really use the Bogomolov property, not just the Manin–Mumford property in their work.

If there is an infinite place $v$ of $K$ at which $f^{\circ n}(\alpha )$ tends to infinity, we deduce Theorem 1 from a new rigidity theorem (Theorem 3) that holds for non-exceptional polynomials. However, we need some notation first.

For any complex polynomial $f \in \mathbb {C}[X]$ of degree $d \geq 2$, we let

\begin{align*} g_f(z) = \lim_{n \rightarrow \infty} \frac{\log^{+}(|f^{\circ n}(z)|)}{d^{n}}, \end{align*}

where $\log ^{+}(x) = \log \max \{1, x\}$ for $x \in \mathbb {R}_{\geq 0}$. This $g_f$ is the Green function attached to the Julia set of $f$ and a continuous function on $\mathbb {C}$. For $r > 0$, we define the equipotential curve of $r$ to be the level set

(2)\begin{equation} L_r = \{z \in \mathbb{C}; g_f(z) = r \}. \end{equation}

We note that for $f = X^{d}$ it holds that $L_r$ is the circle of radius $\exp (r)$ and that for all $f$ the level set at 0 is the filled Julia set of $f$. Thus, $\alpha \in \mathbb {C}$ satisfies $|f^{\circ n}(\alpha )| \rightarrow \infty$ with the usual absolute value if and only if $g_f(\alpha ) >0$.

The following is a characterisation of curves that intersect $L_r^{2}$ in infinitely many points.

Theorem 3 Let $f$ be a non-exceptional polynomial with complex coefficients. If a complex connected plane curve $\mathcal {C} \subset \mathbb {A}^{2}$, that is not fibral, has infinite intersection with $L_r^{2}$ (for some $r > 0$), then $\mathcal {C}$ is a component of a curve defined by

\begin{align*} f^{\circ n}(X) - L\circ f^{\circ n}(Y) = 0 \end{align*}

for some $n \in \mathbb {Z}_{\geq 0}$ and some linear polynomial $L(X)$ such that $L\circ f^{\circ n} = f^{\circ n} \circ L$. Here $X,Y$ are the affine coordinates on $\mathbb {A}^{2}$.

Owing to the fact that $L_r$ is a piecewise real analytic set one can interpret Theorem 3 as a characterisation of all algebraic correspondences that preserve an equipotential curve $L_r$. These sorts of rigidity theorems often play a key role in the proofs in arithmetic dynamics. For example, in the proof of a case of the dynamical André–Oort conjecture in [Reference Ghioca, Krieger, Nguyen and YeGKNY17, Reference Favre and GauthierFG04], a key ingredient is a classification of algebraic correspondences preserving (generalised) Mandelbrot set.

Theorem 3 does not hold without the requirement on $f$ to be non-exceptional. For example, taking $f = X^{d}$ graphs of Möbius transformations sending $L_r$ to $L_r$ provide an ample supply of counterexamples (see, for example, the introduction of [Reference Bays and HabeggerBH15]). These counterexamples are made possible by the fact that $L_r$ is semi-algebraic. This suggests that $L_r$ should be a transcendental curve if $f$ is not exceptional and we can support this intuition with the following theorem whose proof is straightforward given the tools used in the proof of Theorem 3.

We pause to mention that we thought it is worth recording this theorem because it is analogous to Fatou's theorem [Reference FatouFat20, p. 250]: Fatou's theorem characterises exceptional polynomials as those whose Julia set is a (piecewise) analytic arc, whereas our theorem characterises exceptional polynomials through properties of $L_r$. As the curves $L_r$ are always piecewise real analytic, we need a finer criterion and Theorem 3 shows that this finer criterion is (real) algebraicity. It is also worth pointing out that Fatou's theorem itself is crucial for the proof of Theorems 3 and 4.

Another important characterisation of exceptional polynomials that we make use of is due to the combined work of Levin [Reference LevinLev91] and Douady, Hubbard and Thurston [Reference Douady and HubbardDH93]. This characterises polynomials whose Julia set has a normal family of symmetries as being exceptional. These theorems allow us to show that ‘irrational rotations’ of the basin of infinity of $f$ with connected Julia set cannot be algebraic (see Lemma 5.9).

Finally, for $f$ with disconnected Julia set, we introduce considerations that seem to be entirely absent from the existing literature. Namely we study the monodromy action on Mahler functions that appear as the inverse of the famous Böttcher coordinates. This requires careful analysis of Puiseux expansions among other things. Indeed, somewhat surprisingly in the disconnected case we do not need to use any deep rigidity theorem from complex dynamics. Instead, monodromy alone seems to contain sufficient information in this case.

As a final remark on the rigidity theorems we note that the techniques used to prove Theorems 3 and 4 potentially extend to higher dimensions to prove that for a non-exceptional polynomials the components of the intersection of an algebraic variety with powers of $L_r$ must have low dimension. This might be of interest in connection with dynamical questions and equidistribution on powers of $L_r$.

At this point it seems appropriate to put Theorems 1 and 2 into the context of the dynamical literature as well. Our paper (and, in particular, Theorem 2) may be viewed as a first attempt to unify the dynamical Mordell–Lang conjecture and the dynamical Manin–Mumford conjecture.

A dynamical Mordell–Lang conjecture was first formulated by Ghioca and Tucker in [Reference Ghioca and TuckerGT09] and then further expanded on in the book [Reference Bell, Ghioca and TuckerBGT16] where the authors also discuss intersections of orbits. In [Reference Ghioca and TuckerGT09] the authors also use properties of attracting fixed points in their proof. A dynamical Manin–Mumford conjecture was first formulated by Zhang and later corrected and refined by Ghioca et al. [Reference Ghioca, Tucker and ZhangGTZ11] as well as by Ghioca and Tucker [Reference Ghioca and TuckerGT21].

On the dynamical Manin–Mumford side we mention some results that mostly concern the projective line $\mathbb {P}_1$ and its powers $\mathbb {P}_1^{n}$. As alluded to previously, Ghioca, Nguyen and Ye have proven the dynamical Manin–Mumford conjecture for split endomorphisms of $\mathbb {P}_1^{n}$. This work relied on the equidistribution theorem by Yuan on adelic heights [Reference YuanYua08]. Similarly Baker and Demarco used equidistribution techniques to prove a Manin–Mumford theorem for families of dynamical systems [Reference Baker and DeMarcoBD11]. This is by no means an exhaustive list of the results obtained by using the powerful equidistribution theorem combined with deep insights into the intricate geometry and arithmetic of dynamical systems. Worth mentioning is also the resolution of some cases of a dynamical André–Oort conjecture with these techniques [Reference Ghioca, Krieger, Nguyen and YeGKNY17].

On the dynamical Mordell–Lang side, the works that are perhaps closest in spirit to ours are [Reference Ghioca, Tucker and ZieveGTZ08, Reference Ghioca, Tucker and ZieveGTZ12] as they treat intersections of varieties with orbits of polynomials.

From the point of view of height theory, Theorem 1 may also be viewed as an intermediate between [Reference Ghioca, Nguyen and YeGNY19] and [Reference Ghioca, Tucker and ZieveGTZ08] as we are treating points of constant canonical height whereas the cited works treat points of small and unbounded canonical height, respectively.

In order to incorporate exceptional polynomials and periodic points in our statement we need some care to encode essentially group theoretic information in dynamical terms (see [Reference Ghioca and TuckerGT21] for related problems). What is new in our context compared with [Reference Ghioca and TuckerGT21] is that we need to define an analogue of co-sets in the dynamical setting. For this we introduce some notation. We formulate a conjecture for powers of the projective line here and in § 9 a conjecture for all projective varieties.

For an endomorphism $g$ of $\mathbb {P}_1$ we let $\Xi _g$ be $\mathbb {P}_1\setminus \text {S}_g$ where $\text {S}_g$ is the set of super-attracting fixed points of $g$ of degree $\deg (g)$ (see [Reference MilnorMil06]). The variety $\Xi _g$ is always quasi-affine. It is a natural space on which $g$ acts. We say that two morphisms commute if they commute by composition.

Definition 2 Let $\mathcal {K}$ be an algebraically closed field of characteristic zero and $n$ a positive integer. We say that a variety $\mathcal {V} \subset \mathbb {P}_1^{n}$ defined over $\mathcal {K}$ is stratified by an endomorphism $g$ of $\mathbb {P}_1$ defined over $\mathcal {K}$ if there exists an endomorphism $h$ of $\mathbb {P}_1$ and a finite morphism $\varphi :\Xi _h \rightarrow \Xi _g$ with the property $g\circ \varphi = \varphi \circ h$ such that the following holds: there exists a variety $\mathcal {V}_\varphi \subset \Xi _h^{n}$ and a morphism $H: \Xi _h^{n} \rightarrow \Xi _h^{n}$ commuting with $(h,\ldots, h):\Xi _h^{n} \rightarrow \Xi _h^{n}$ such that

\begin{align*} (\varphi, \ldots, \varphi)(\mathcal{V}_\varphi) = \mathcal{V}\cap \Xi_g^{n}, \quad (\varphi, \ldots, \varphi) \circ H(\mathcal{V}_\varphi) = \{\beta \} \times \Xi_g^{\dim(\mathcal{V})} \end{align*}

for some $\beta \in \Xi _g^{n-\dim (\mathcal {V})}(\mathcal {K})$.

Definition 2 is quite technical. However, it tries to capture the fact that if a rational map comes from an algebraic group, the group structure gives rise to ‘many’ endomorphisms that commute with the given rational map in purely dynamical terms. A simple example is that $\mathcal {V}$ is stratified by $g$ if the projection of $\mathcal {V}$ to a factor of $\mathbb {P}_1$ is constant. In the exceptional cases we can think of $\Xi _h$ as coming from an algebraic group and $H$ as coming from a group morphism. Note that for $g= X^{2}$ any co-set of $\mathbb {G}_m^{n}$ defines a variety that is stratified by $g$. We obtain the following from Theorem 1, [Reference Ghioca, Nguyen and YeGNY19] and Manin–Mumford.

Following a suggestion of Ghioca, we formulate a conjecture for intersections of varieties with grand orbits. It can be viewed as a natural dynamical analogue of Lang's conjecture on finite rank groups.

For an endomorphism $g$ of $\mathbb {P}_1$ and $\alpha \in \mathbb {P}_1(\mathcal {K})$ we set $\mathcal {G}_\alpha (g)$ to be the grand orbit of $\alpha$ under $g$. Let $X = \mathbb {P}_1^{n}$ and $\underline {g} = (g_1, \ldots, g_n)$ where $g_1,\ldots, g_n$ are endomorphisms of $\mathbb {P}_1$ of degree at least 2 defined over $\mathcal {K}$ as previously. Let $\underline {\alpha } = (\alpha _1,\ldots, \alpha _n)\in \mathbb {P}_1^{n}(\mathcal {K})$ and let $V$ be a hypersurface in $X$.

Conjecture 1 Suppose that

\begin{align*} V \cap \mathcal{G}_{\alpha_1}(g_1)\times \cdots \times \mathcal{G}_{\alpha_n}(g_n) \end{align*}

contains infinitely many points. Then $V$ is either pre-periodic under $\underline {h} = (h_1,\ldots, h_n)$ where $h_1, \ldots, h_n$ are endomorphisms commuting with an iterate of $g_1, \ldots, g_n$, respectively, and such that at least one of $h_1, \ldots, h_n$ has degree at least 2 or $V$ is stratified by $g_i$ for some $i \in \{1,\ldots, n\}$.

From Theorem 2 it follows that we can treat part of Conjecture 1.

We call a curve $\mathcal {C} \subset \mathbb {A}^{2}$ special if it is a component of curve defined by $f^{\circ n}(X)- f^{\circ n}(Y) = 0$ for some $n \geq 0$ or if $\mathcal {C}$ is of the form $\{\beta \} \times \mathbb {A}^{1}$ or $\mathbb {A}^{1}\times \{\beta \}$ for some $\beta \in \mathcal {S}_\alpha$.

It is also possible to put Theorem 1 in the more classical context of unlikely intersections. More precisely, we define the set of special subvarieties $\text {Sp}_f$ of $\mathbb {A}^{3}$ with coordinates $(X_1,X_2,X_3)$ to be Boolean combinations of varieties defined by

\begin{align*} f^{\circ n}(X_i) = f^{\circ n}(X_j), \quad n \in \mathbb{Z}_{\geq 0},\quad i,j \in \{1,2,3 \},\ i \neq j. \end{align*}

Theorem 1 implies that for $\alpha$ that is not pre-periodic under $f$ and $f$ non-exceptional, the intersection

\begin{align*} \mathcal{C} \times \{\alpha\} \cap\bigg(\bigcup_{V \in \text{Sp}_f, \text{codim}(V) > 1} V\bigg) \end{align*}

is finite unless $\mathcal {C}\times \{\alpha \}$ is contained in a special subvariety of positive codimension.

It is useful to keep this terminology and we often refer to a special curve in $\mathbb {A}^{2}$ to be either of the form $\{\beta \}\times \mathbb {A}^{1}$ or $\mathbb {A}^{1}\times \{\beta \}$ for some $\beta \in \mathcal {S}_\alpha$ or if $\mathcal {C}$ is a component of the variety defined by $f^{\circ n}(X) = f^{\circ n}(Y)$, respectively.

We now summarise the discussion of our methods and compare them with the Pila–Zannier method which is often used in special circumstances. As mentioned previously, we localise at a place $v$ for which $\alpha$ lies in its associated basin of infinity. If the place turns out to be archimedean we reduce Theorem 1 to a rigidity theorem (Theorem 3) that is really an analytic property of the dynamical system. If $v$ is non-archimedean, however, we make use of the ‘uniformisation’ of the dynamical system by the inverse of the Böttcher map. Quite similarly as in the Pila–Zannier method, we study the pullback of the curve $\mathcal {C}$ by this uniformisation and show a functional transcendence result for this pull-back. However, we only show that the pull-back is not contained in a co-set of $\mathbb {G}_m^{2}$ unless $\mathcal {C}$ is special (Theorems 7 and 8). We essentially study pairs of roots of unity on our pull-back and these are of bounded height but not of bounded degree. This rules out the possibility to directly apply counting of rational points in the Pila–Wilkie style.

Our approach is different and seems to be particularly suited for non-archimedean completions. To a fixed solution $(\beta _1,\beta _2) \in \mathcal {S}_\alpha ^{2}\cap \mathcal {C}$ we construct an auxiliary function that vanishes at the pull-back of the whole Galois orbit of $(\beta _1,\beta _2)$. We then apply Poisson–Jensen to bound its zero set, which does not care about the degree of the point. Our method is suited for generalisations and we explore it further in the function field case in future work. As we showed in [Reference SchmidtSch03], and have already mentioned, it leads to a new and quite elementary proof of (the classical) Manin–Mumford in the multiplicative group.

As we often work with roots of unity, we set $\mu _{\infty }$ to be the set of all roots of 1 and $\mu _{d^{\infty }}$ the set of roots of unity whose order divides a power of $d$. Finally, for a positive integer $N$ we set $\mu _N$ to be the set of the $N$th root of 1. For a field $\mathcal {K}$ (often $K$) we set $\mathcal {K}_v$ to be its completion by the valuation $v$ and set $\bar {\mathcal {K}}$ to be its algebraic closure. We denote by $h,H:\bar {\mathbb {Q}} \rightarrow \mathbb {R}$ the logarithmic, multiplicative, respectively, (naive) Weil height [Reference Bombieri and GublerBG06].

Here is how we proceed. In the next section, we reduce the last two theorems in this section to Theorems 1 and 2. Then we introduce the main players in our proof which are the Mahler functions defined by $f$. In § 4 we give the proof of a uniform version of Theorem 1 when $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place. In § 5 we give the proof of Theorem 3 when the Julia set of $f$ is connected and in § 6 when it is disconnected. We summarise the proof of Theorem 1 in § 7 and in § 8 we give the proof of Theorem 2. We conclude in the final section with some remarks and give a uniform version of Theorem 3.