Dynamical Belyi maps and arboreal Galois groups

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. This allows us to study the behavior of the monodromy groups under specialization of the maps, and to derive applications to dynamical sequences.


Introduction
Let f : P 1 F → P 1 F be a rational map defined over a number field F . The Galois theory of the iterates f n = f • · · · • f : P 1 F → P 1 F was first studied in the work of Odoni ( [13] ), and has applications both in number theory and in arithmetic dynamics. Specializing the iterates f n at an F -rational place a ∈ P 1 F , we obtain an infinite tower of number fields (K n,a ) n≥1 . Many recent papers study the question of the distribution of the primes of F that ramify in this tower.
We denote the Galois group of f n by G n,F = G n,F (f ) and the Galois group of its specialization at a by G n,a = G n,a,F (f ). For places a avoiding a finite subset of P 1 F , the limit G ∞,a = lim ← −n G n,a acts on the infinite d-ary regular tree T ∞ , where d = deg(f ). We obtain an arboreal representation, and hence an embedding of G ∞,a into the automorphism group Aut(T ∞ ) of the tree; this is why we call the group G ∞,a an arboreal Galois group. A paper by Jones ( [9] ) provides a survey of the theory of arboreal representations. A central question is to characterize when the index [Aut(T ∞ ) : G ∞,a ] is finite ( [9, Question 1.1] ). As Jones discusses, this question may be considered as an analog of Serre's open image theorem. We take a different perspective and study a class of rational maps for which the index [Aut(T ∞ ) : G ∞,a ] is infinite.
The maps we study in this paper are Belyi maps f : P 1 F → P 1 F with exactly 3 ramification points, which we assume to be 0, 1, ∞. We normalize the coordinate on the target projective line such that the ramification points are fixed by f . A map f satisfying these properties is called a normalized (dynamical) Belyi map and is completely determined by its (combinatorial) type (Definition 1.1.2), which tabulates the ramification indices. In particular, all maps may be defined over F = Q (Proposition 1.1.3). All normalized Belyi maps are so-called post-critically finite (PCF) maps, since the forward orbit of each ramification point is preperiodic. A result of Jones and Pink ( [9, Theorem 3.1]) states that the index [Aut(T ∞ ) : G ∞,a ] is infinite for PCF maps, and hence for ours, as well.
The arboreal Galois group G ∞,a of a specialization of a PCF map is a mysterious group, which is hard to describe in general. The case that f is a polynomial of degree deg(f ) = 2 has been extensively studied, see e.g. [12], [15]. In [3], the authors consider the case of the cubic polynomial f (x) = −2x 3 +3x 2 , providing the first completely explicit result on its Galois theory. This cubic polynomial is the easiest example of a normalized Belyi map; the class of normalized Belyi maps contains maps of arbitrarily large degree, which are not necessarily polynomial.
A related problem, which is very interesting but difficult in general, is to determine the primes that ramify in the tower of number fields (K n,a ) n≥1 . For PCF maps f , it is known that only finitely many primes ramify in the whole tower (K n,a ) n≥1 . This was proven in [1,Theorem 1.1] in the case that f is a polynomial and in [4,Theorem 1] in the general case; see also [10,Theorem 3.2].
The set of primes that ramify in (K n,a ) n≥1 contains as subset the primes for which f has bad reduction. In our previous paper [2] we proved a rather general result on the reduction of normalized Belyi maps (Proposition 1.2.4). Every map in this class can be computed explicitly; Proposition 1.1.4 provides other participants Jacqueline Anderson, Neslihan Girgin, and Michelle Manes for many useful discussions. Our previous paper [2], which evolved from work done at the conference Women in Numbers Europe 2, is a collaboration with them. We are grateful to Alexander Hulpke for his help with some group-theoretic computations. We gratefully acknowledge the support of the Mathematisches Forschungsinstitut Oberwolfach, where part of the work was carried out. The second and third authors were supported by an AMS -Simons Travel Grant.
1. Dynamical Belyi maps 1.1. Normalized Belyi maps of type (d; e 1 , e 2 , e 3 ). In this section we introduce the class of Belyi maps that we will study in this paper. Recall that a Belyi map is a finite cover f : X → P 1 of smooth projective curves over C that is branched exactly at x 1 = 0, x 2 = 1, and x 3 = ∞. A Belyi map has genus zero if X has genus g(X) = 0. This notion was introduced in [18]. In this case the nth iterate of f , which we denote by f n , is also a Belyi map. All dynamical Belyi maps are post-critically finite; a map f : P 1 → P 1 is post-critically finite (PCF) if each of its ramification points has finite forward orbit. Note that normalized (single-cycle genus-0) Belyi maps f are dynamical, and hence PCF. They even satisfy the stronger condition of being conservative, which means that they are PCF maps such that each of their ramification points is a fixed point.
All Belyi maps considered in this paper are assumed to have genus zero. We therefore omit this from the notation in the rest of the paper. A Beyi map of type (d; e 1 , e 2 , e 3 ) is automatically single cycle. We therefore will not mention this explicitly in the rest of the paper.
To exclude trivial cases we always assume that all normalized Belyi maps of type (d; e 1 , e 2 , e 3 ) have exactly three ramification points, i.e. e i > 1 for all 1 ≤ i ≤ 3. For simplicity we will moreover always assume that e 1 ≤ e 2 ≤ e 3 . Permuting the e i corresponds to changing coordinates on both projective lines simultaneously, therefore these inequalities pose no restriction. An abstract type is a tuple (d; e 1 , e 2 , e 3 ) such that 2 ≤ e 1 ≤ e 2 ≤ e 3 ≤ d and such that (1.1.1) holds.
The following result is classical; a proof can be found in [2,Proposition 1]. Note that the normalization condition completely fixes the coordinates on both projective lines. where Proof. This is [2, Propositions 2 and 3].
Let f be a normalized Belyi map of type (d; e 1 , e 2 , e 3 ). Associated with f is a generating system (g 1 , g 2 , g 3 ), which describes the quotient of the topological fundamental group π 1 (P 1 \ {0, 1, ∞}, * ) corresponding to f . In our situation, a generating system consists of three permutations g i ∈ S d for 1 ≤ i ≤ 3, where g i is a single cycle of length e i , that satisfy the relation g 1 g 2 g 3 = 1. This motivates the terminology single cycle. The generating system for f is unique up to simultaneous conjugacy by elements of S d . The group G(f ) := g 1 , g 2 , g 3 is the Galois group of the Galois closure of f . Note that G(f ) denotes the Galois group of the Belyi map f : show that the triple g is weakly rigid in the sense that it is unique up to uniform conjugacy by S d . However, in the case that G(f ) S d the generating system is not unique up to uniform conjugacy by G(f ), i.e. the triple is not rigid. In this case G(f ) is not necessarily the Galois group of f over Q, even though the map f is defined over Q. We discuss this phenomenon in more detail in Section 2.4. We refer to [16,Chapter 2] for a general introduction to this topic. (1) Assume C = (6; 4, 4, 5). Then the Galois group G(f ) of the Galois closure of f is isomorphic to either S d or A d , depending on whether at least one e i is even or not.

Reduction of normalized Belyi maps.
In this section we recall from [2, Section 4] the definition of and some results on the reduction of normalized Belyi maps.
We identify the cover f with the rational function that defines it. Without loss of generality we may assume the following hold: (2) The polynomials f 1 , f 2 satisfy gcd(f 1 , f 2 ) = 1; (3) The greatest common divisor of all coefficients of f is 1. The following proposition lists some properties of the reduction of a normalized Belyi map. The proof uses the assumption that f is a normalized Belyi map of type C in an essential way, see [2,Example 4]. Proposition 1.2.2. Let f be a normalized Belyi map of type C = (d; e 1 , e 2 , e 3 ) and let p be a prime. ( Proof. The statement is not stated explicitly in [2] in this form, but it follows immediately from the proof of [2,Proposition 4] and [2,Remark 6]. (1) A normalized Belyi map f has bad reduction at a prime p if deg(f ) has strictly smaller degree than f . Otherwise, f has good reduction at p. (2) A normalized Belyi map f has good monomial reduction at p if it has good reduction at p and its reduction is A normalized Belyi map f has good separable reduction at p if it has good reduction at p and its reduction is a separable rational map.
If a normalized Belyi map f has good separable reduction at p then its reduction f is also a normalized Belyi map and f has the same type as f . This follows from the proof of [2,Proposition 5]. Good monomial reduction is a special case of good inseparable reduction. In Proposition 3.1.2 we use it to characterize the irreducibility of specializations of f and its iterates.
The following result allows us to prescribe the reduction of a normalized Belyi map by purely combinatorial conditions on its ramification indices. (1) Assume that p > d. Then f has good separable reduction at p.  We label the vertices of the tree as follows: the root of the tree corresponds to the level 0, and has the empty label (). The vertices at level i are labeled as (ℓ 1 , . . . , ℓ i ) with ℓ j ∈ {1, . . . , d}. Here (ℓ 1 , . . . , ℓ i−1 ) is the unique vertex at level i − 1 which is connected to (ℓ 1 , . . . , ℓ i ) by an edge. For the vertices of T n at level n, also called the leaves, we additionally use the numbering . . , d n } instead of (ℓ 1 , . . . , ℓ n ).
Since Aut(T n ) acts faithfully on the leaves of the tree T n , the choice of the numbering induces an injective group homomorphism (2.1.2) ι n : Aut(T n ) ֒→ S d n .
In this paper we use the convention that permutations act from the right. Rather than considering Aut(T n ) as a subgroup of S d n it is more convenient for our purposes to view Aut(T n ) as a subgroup of the n-fold iterated wreath product of S d by itself. The structure of Aut(T n ) as n-fold iterated wreath product is This isomorphism is induced by the decomposition of T n as the subtree T 1 (consisting of the levels 0 and 1), and d copies of T n−1 each consisting of the complete subtree of T n with root (j) for j ∈ {1, . . . , d}.
We remark that Aut(T 1 ) ≃ S d , but that the iterated wreath product Aut(T n ) is a strict subgroup of S d n for n ≥ 2. Equation (2.1.3) allows us to write the elements of Aut(T n ) as tuples Using this identification, an element of Aut(T n ) acts on vertices of T n as In other words, τ permutes the d complete subtrees isomorphic to T n−1 , and σ j acts on the complete subtree with root (j). Let (ℓ 1 , . . . , ℓ m ) be a vertex of T n at level m ≤ n − 1 and let i = 1 + m k=1 (ℓ k − 1)d k−1 . We denote the complete subtree of T n with root (ℓ 1 , . . . , ℓ m ) by S (T i n ) = {(σ, τ ) ∈ Aut(T n ) | (σ, τ ) acts trivially outside T i n ). We may view S (T i n ) as a subgroup of S d n via ι n . For future reference, we note that (2.1.8) Here − denotes the trivial permutation. For every m ≤ n we write π m for the natural projection which corresponds to restricting the action of an element of Aut(T n ) to the subtree T m consisting of the levels 0, 1, . . . , m.
We call sgn 2 the wreath-product sign.
Using (2.1.8), one may check that sgn 2 is a group homomorphism. We define a subgroup E n of Aut(T n ) using the wreath-product sign. We show in Corollary 2.2.4 below that the Galois group of f n is a subgroup of E n . (2) Assume that d is odd. Then the wreath-product sign on Aut(T n ) agrees with the restriction of the usual sign on S d 2 via the embedding ι n from (2.1.2): Proof. The definition of E n (Definition 2.1.2) implies that Since E 1 = Aut(T 1 ), the definition of the wreath-product sign (2.1.10) implies that
This finishes the proof of Statement (2).
Note that it follows from the proof of Lemma 2.1.3.(2) that for d even the wreath-product sign (2.1.9) is not compatible with the natural sign on S d 2 .
2.2. A generating system for f n . In this section we only assume that f is a dynamical Belyi map that is normalized so that We start by defining the Galois groups that are the central object of study in this paper. We then determine a generating system for f n in terms of a generating system for f . Let F be a field of definition of f . Since f is a Belyi map, we may assume that F is a number field. (In the case that f is a normalized Belyi map of type (d; e 1 , e 2 , e 3 ), i.e. in the single-cycle case, it follows from Proposition 1.1.3 that we may take F = Q. When discussing the single-cycle case we always choose F = Q.) Write K 0 := F (t) for the function field of P 1 F . Since F is a field of definition for f n for all n, the field F is integrally closed in the extension of function fields corresponding to the map f n : P 1 F → P 1 F , which we denote by K n /K 0 . We choose a normal closure M n /K 0 of K n /K 0 such that M n contains M n−1 for any n ≥ 1.
The extension of function fields K n ⊗ F Q/Q(t) corresponds to f n : P 1 Q → P 1 Q considered as map over the algebraic closure Q of the number field F . Note that M n ⊗ F Q/Q(t) is a normal closure of We say that a rational function f is irreducible if [K 1 : K 0 ] = deg(f ). It is not true in general that the composition of two irreducible rational functions is again irreducible, but it is true if we compose an irreducible rational function with itself. This follows for example from [4,Prop. 2].
Note that G 1,Q = G(f ) as defined in Section 1.1. It follows from the definitions that In general it is not true that G n,Q = G n,Q , see Remark 2.4.1. In the case that we have equality in (2.2.1) we say that the Galois extension M n ⊗ F Q/Q(t) descends to Q(t) or in short that the group G n Q descends to Q.
Our convention that the normal closure M n contains M m for all m ≤ n implies that G n,Q , and hence G n,Q , naturally has the structure of a wreath product. Identifying the sheets of f n above a chosen base point in P 1 (Q) \ {0, 1, ∞} with the leaves of the tree T n yields an inclusion In the rest of the paper we fix this inclusion. Our next goal is to determine a generating system (g 1,n , g 2,n , g 3,n ) for f n for all n in terms of a fixed generating system (g 1,1 , g 2,1 , g 3,1 ) for f . Since G n,Q = g 1,n , g 2,n , we can use this to determine the group G n,Q . We refer to Theorem 2.3.1 for the precise result.
.) The analogous statements with 0 replaced by 1 or ∞ and e 1 by e 2 or e 3 , respectively, also hold. This description determines the cycle type of the group elements of a generating system of f n considered as elements of S d n . For our purposes we need more precise information, which the following proposition supplies.
Proof. This follows by considering the image of the sheets of the Belyi map under f n = f • f n−1 , using the notation introduced in (2.1.4).
In the single-cycle case a generating system for f was given in Lemma 1.1.5. (2).
Proof. We have already seen that G n,Q ⊆ Aut(T n ). Since G n,Q is generated by the g j,n , it suffices to check that g j,n ∈ E n for j = 1, 2, 3.
The inductive definition of g j,n given in Proposition 2.2.3 implies that π 2 (g j,n ) = g j,2 and the explicit expression for g j,2 implies that sgn 2 (g j,2 ) = 1 for j ∈ {1, 2, 3}. The statement for n = 2 follows. The statement for arbitrary n ≥ 2 follows by induction from the definition of E n (Definition 2.1.2).
The group N i n,Q is naturally a permutation group on the d letters (ℓ 1 , . . . , ℓ n−1 , s) for s = 1, . . . , d; We therefore obtain an identification With this identification, we may write In the rest of this section, we restrict to the single-cycle case and fix a normalized Belyi map f of type C = (d; e 1 , e 2 , e 3 ). We use Proposition 2.2.3 to construct suitable elements in N n,Q . This is a first step towards determining G n,Q in Theorem 2.3.1.
(2) Write ρ i j for the component of α j,n in S (T i n ) using the notation from (2.2.3). Then (3) Conjugation by the elements of G n,Q acts transitively on N 1 n,Q , . . . , N d n−1 n,Q for all n ≥ 2.
Proof. Let i ∈ {1, 2, 3} and n ≥ 2 arbitrary. Using (2.1.8) one computes that Expression (2.2.4) for α j,n implies that α j,n fixes all vertices on level one. If α j,n−1 ∈ N n−1,Q then α j,n fixes all vertices of the tree on levels 2, . . . , n−1, as well. It follows that α j,n ∈ N n,Q . Statement (1) of the proposition is vacuous for n = 1. The statement therefore follows by induction on n.
Moreover, α j,n acts as α j,n−1 on the subtree T n (ℓ 1 ) of T n with root (ℓ 1 ) if ℓ 1 ∈ supp(g j,1 ) and acts trivially otherwise. Statement (2) therefore also follows by induction on n.
Statement (3) follows by induction, as well, since In the following proposition we exclude two types in small degree. For C = (6; 4, 4, 5) we have that G 1,Q is S 5 (embedded as a transitive group in S 6 ), hence is isomorphic to neither S 6 nor A 6 . For C = (4; 3, 3, 3) we have that G 1,Q ≃ A 4 . However in this case Proposition 2. Proof. Lemma 2.2.7(3) states that G n,Q acts transitively on the set of subgroups N i n,Q for i = 1, . . . d n−1 . Therefore it suffices to prove the proposition for a specific value of i. We prove that N i 0 n,Q is a non-trivial normal subgroup of G 1,Q for some value i 0 , by showing it is normal and contains a non-trivial element β n . Since A d is simple for d ≥ 5 and N i n,Q ≤ A d for all i by construction, it follows that N i 0 n,Q = A d . In the remaining cases d ∈ {3, 4} the statement can be shown by treating each type separately.
where σ i = α j,n−1 ∈ G n−1 for i in the support of g j,1 and trivial otherwise. Since the generating system is weakly rigid, to prove the claim it suffices to prove that the element β n is a non-trivial element in N i 0 n for the generating system of Lemma 1.1.5 (2). We use this generating system for the rest of this proof.
With this choice we have By induction we find that the component ρ i of β n in N i n,Q satisfies Hence β n ∈ N i 0 n,Q as claimed. To show that β n is non-trivial, it suffices to show that [g 1,1 , [g 2,1 , g 3,1 ]] is non-trivial. This is an explicit calculation using the generating system in Lemma 1.1.5 (2). For instance, one checks that e 3 is not sent to itself. We conclude that β n is non-trivial, and Claim 1 follows.
Claim 2: N i 0 n,Q is a normal subgroup of A d . We first note that Definitions 2.1.1 and 2.2.6 imply that N i 0 n,Q is a subgroup of A d . It follows from (2.1.8) that the conjugates of β n by the elements α j,n ∈ N n are also in N i 0 n,Q for j ∈ {1, 2, 3}. The group G 1,Q is generated by the g j,1 . Since g j,1 is the component of α j,n in N i 0 n,Q for j ∈ {1, 2, 3}, the group As explained in the beginning of the proof, the statement for d ≥ 5 follows from Claims 1 and 2. The remaining cases can be checked separately.
2.3. Determination of G n,Q for normalized Belyi maps. Let f be a normalized Belyi map of type (d; e 1 , e 2 , e 3 ). In this section, we completely determine the group structure of the groups G n,Q (Definition 2.2.1) as a subgroup of Aut(T n ). We refer to Lemma 1.1.5 for the description of G 1,Q . (2) Assume that G 1,Q ≃ A d , i.e., that all e j are odd. Then G n,Q is the n-fold iterated wreath product of A d with itself.
The key step in the proof of Theorem 2.3.1 is determining the size of the group G n,Q . Let χ denote the following homomorphism, induced by the sgn function on each of the components N i n,Q : where the discrete logarithm dlog −1 sends 1 → 0 and −1 → 1.
Note that the assumption that E n−1 ≃ G n−1,Q for n = 1 states that Therefore, χ induces an injection For the remainder of the proof we identify N n,Q with its image in F d n−1

2
. Since e 1 + e 2 + e 3 = 2d + 1 is odd and we assume that G 1,Q ≃ S d , at least one of the e j is even. It follows that there is a unique j such that e j is odd. Let s ∈ {1, 2, 3} be one of the indices such that e s is even. Since at most one of the e j equals d, we may assume that e s = d.
Claim: The elements ξ I for I satisfying properties (1) and (2) above generate H . Fix k, ℓ and i such that 1 ≤ k < ℓ ≤ d and 1 ≤ j ≤ d n−2 . Let ξ k,ℓ;j denote the vector whose entries are 1 in the positions k + (j − 1)d and ℓ + (j − 1)d and 0 otherwise. Note that ξ k,ℓ;j ∈ H .
To prove the claim it suffices to show that for all 1 ≤ j ≤ d n−2 and for all 1 ≤ k < ℓ ≤ d, the vectors ξ k,l;j are in the linear hull of the ξ I . Using that G n,Q acts transitively on the blocks, it suffices to prove this for j = 1. Since N 1 n,Q ⊆ G n,Q acts transitively on B(1) by Proposition 2.2.8, we may moreover assume that (k, ℓ) = (1, 2). In other words, it suffices to show that ξ 1,2;1 ∈ H . i=2 I (i)) ∪ I ′ (1), we see that ξ 1,2;1 = ξ I + ξ I ′ ∈ H . This proves the claim.
The claim inductively implies that This proves the lemma.
Now we are ready to prove Theorem 2.3.1.
Proof of Theorem 2.3.1. In Corollary 2.2.4 we have shown that G n,Q is a subgroup of E n for all n ≥ 1.
To prove Statement (1), it suffices to show that the groups G n,Q and E n have the same cardinality. Since G n,Q is a subgroup of Aut(T n ) for all n ≥ 1, the definition of N n,Q implies that The last equality is the statement of Lemma 2.3.2. The expression for [Aut(T n ) : as well. It follows that |G n,Q | = |E n |. This proves the theorem in the first case.
In this case G n,Q is a subgroup of the n-fold iterated wreath product of A d with itself. The statement of the theorem in this case therefore follows by induction on n using Proposition 2.2.8.
This finishes the proof of the theorem.

Descent.
In this section we determine the groups G n,Q (Definition 2.2.1) under certain conditions. Recall from (2.2.1) that G n,Q ⊆ G n,Q ⊆ Aut(T n ), for all n ≥ 1.
Using the notation from Section 1.2, we have that G 1,Q ≃ A d if and only if the discriminant ∆(f ) of x e 1 f 1 − tf 2 , considered as polynomial in x, is a square in Q(t), and G 1,Q ≃ S d otherwise. For simplicity, we restrict to the case that f is a polynomial, i.e. that e 3 = d. In this case, a formula for the discriminant of f is given in [1,Proposition 3.1]: where ℓ(f ) denotes the leading coefficient of f . Proposition 1.1.4 (1) gives an explicit expression for ℓ(f ). The formula for the discriminant, together with the expression for ℓ(f ), implies that if f is a polynomial and d is odd, then ∆(f − t) is never a square in Q(t). We note that a formula for ∆(f − t) in the non-polynomial case may be deduced from [4,Proposition 1]. This is similar to the proof of Proposition 2.4.2 below. For simplicity we only determine G n,Q in the case that G 1,Q ≃ S d here, but one may treat the case that G 1,Q ≃ A d using similar methods, though the statements are not quite as nice in this case.
The structure of Aut(T n ) as an iterated wreath product implies that the case n = 2 plays a key role in determining G ∞,Q . We therefore treat the case n = 2 first. The case of arbitrary n is treated in Proposition 2.4.4. In the rest of this section we only treat the case that d is odd. Lemma 2.1.3 (2) states that in this case the wreath-product sign agrees with the usual sign on Aut(T 2 ). Since the difference between G 2,Q and Aut(T 2 ) can be expressed in terms of the wreath-product sign (Lemma 2.3.2), we obtain a concrete criterion for checking whether G 2,Q equals G 2,Q in terms of discriminants in the case that d is odd.
Recall that f n denotes the nth iterate of f . We write f n (x) = g n (x)/h n (x), where g n , h n ∈ Z[x] are relatively prime as polynomials in Z[x]. In the terminology of Section 1.2 we have g 1 (x) = x e 1 f 1 (x) and h 1 (x) = f 2 (x). For convenience, we define a third polynomial f 1 by The assumption that f is a normalized Belyi map of type (d; e 1 , e 2 , e 3 ) implies that f 1 is indeed a polynomial and that it has degree d − e 2 . For our purposes we want to determine the discriminant of g 2 (x) − th 2 (x) as a polynomial in Q[t] up to squares. We call ∆(g 2 (x) − th 2 (x)) the discriminant of f 2 for convenience. In the following lemma, the notation ≡ 2 signifies that the equality holds modulo squares in Q(t). For polynomials ϕ, ψ we denote the leading coefficient of ϕ by ℓ(ϕ), and the resultant of ϕ and ψ by Res(ϕ, ψ) .
Proof. The formula above is a more explicit version of [4, Proposition 1] in the special case we are considering here. Loc. cit. implies that where the symbols occurring in this formula are defined below. We note that this is essentially [4, Equation (5)], but that formula contains a few very minor typographical errors. The correct formula is the last formula on page 276. The authors of [4] do not explicitly compute the sign (−1) Σ 2 , but it can be easily computed by keeping track of the signs in the steps of their proof of [4, Proposition 1]. One finds that Let R(f 2 ) denote the affine ramification locus of f 2 . In our situation we have  The integer D 2 is defined as the leading coefficient of the polynomial g 2 (x) ′ h 2 (x) − g 2 (x)h 2 (x) ′ , where ′ denotes derivation with respect to x. To compute it, it is useful to write Here Lemma 3], where a similar expression for D 2 is given.) The result [4, Proposition 2] yields an expression for Res(g 2 , h 2 ). In our situation we find It remains to compute ρ∈R(f 2 ) (g 2 (ρ) − th 2 (ρ)) m 2 (ρ) . We divide the product into five terms, according to the subdivision of R(f 2 ) in (2.4.2).
Terms I and II: ρ = 0, 1. Equation (2.4.3) implies that g 2 (0) = 0 and h 2 (0) = f 2 (0) d+1 . The contribution of ρ = 0 to the product is therefore . Similarly, we find that the contribution of ρ = 1 is Term III: roots of f 1 . We first note that the roots ρ of f 1 satisfy h 2 (ρ) = f 2 (ρ) d f 2 (0) and g 2 (ρ) = 0. We find that the corresponding contribution to the product is Term IV: roots of f 1 . Similarly as for Term III, we find that the contribution of the roots of f 1 to the product is Here we have used that f 2 (ρ) = g 2 (ρ)/h 2 (ρ) = 1 and h 2 (ρ) = f 2 (ρ) d f 2 (1) for all roots of f 1 . By definition, we have (x − 1) e 2 f 1 = x e 1 f 1 − f 2 . Using this relation and [4, Lemma 1], together with some standard properties of the resultant recalled in [4] immediately above the lemma, we find that Substituting this into the previous expression for the contribution and using that ℓ( f 1 ) = ℓ(f 1 ), we find The roots ρ of f 2 satisfy f 2 (ρ) = ∞, and hence h 2 (ρ) = f 2 (ρ) = 0 and g 2 (ρ) = ρ e 1 d ℓ(f 1 )f 1 (ρ) d .
For the contribution to the product we therefore find Here we used that ρ: The proposition follows by putting all terms together and reducing modulo squares in Q(t).
The following corollary interprets the result of Proposition 2.4.2 for the two families of normalized Belyi maps of Proposition 1.1.4. We remark that its proof only uses very minimal information on the explicit expression of f given in Proposition 2.4.2. This illustrates that it should be possible to obtain similar statements for other families of normalized Belyi maps in the single-cycle case. As explained above, we restrict to the case that d is odd, since only in this case we may interpret the usual sign as wreath-product sign.

Corollary 2.4.3. Assume that d is odd.
(1) Let f be a normalized Belyi map of type (d; e 1 , e 2 , e 3 = d), i.e. f is a polynomial. Then the discriminant of f 2 is a square.
Proof. Part (1) follows from Proposition 2.4.2 by noting that f 2 = 1 in the case that f is a polynomial. Let f be as in (2), i.e. e 1 = e 3 . Proposition 1.1.4 (2) implies that ℓ(f 1 ) = f 2 (0). Part (2) follows, since d and e 2 are both odd. Proof. We prove the statement by induction on n.
The assumption that at least one e j is even implies that G 1,Q ≃ S d (Lemma 1.1.5). It follows that G 1,Q = G 1,Q = Aut(T 1 ) ≃ S d , and the statement for n = 1 holds.
Assume that the statement holds for n − 1, i.e. we have G n−1,Q = G n−1,Q . Theorem 2.3.1 implies that

The induction hypothesis implies that
The second inclusion follows from the structure of G n,Q as a wreath product induced by the decomposition f n = f • f n−1 . Lemma 2.1.3 (1) implies that [E n−1 ≀ E 1 : E n ] = 2. We conclude that G n,Q equals either G n,Q = E n or E n−1 ≀ E 1 . The definition of the wreath-product sign (Definition 2.1.1) implies that we can distinguish the two possible groups by considering their images under π 2 . Since π 2 (G n,Q ) = G 2,Q and π 2 (G n,Q ) = G 2,Q , it suffices to consider the case n = 2.
Lemma 2.1.3. (2) implies that E 2 is the kernel of sgn 2 = sgn •ι 2 , where sgn is the usual sign on

Specialization
In this section we prove some explicit results on the specialization of normalized Belyi maps f . For any n ≥ 1, Hilbert's Irreducibility Theorem implies that there exists a non-empty Zariski-open set H n = H n (f ) ⊆ P 1 (Q), called a Hilbert set, such that specializing the parameter t to a ∈ H n does not change the Galois group. In this section we determine explicit elements a ∈ H n (f ) for all n. This means that we get an explicit tower of number fields (K n,a ) n≥1 (Definition 3.1.1 by specializing to these values of a. 3.1. Irreducibility and ramification conditions. Throughout this section, we fix a normalized Belyi map f of type C := (d; e 1 , e 2 , e 3 ). Recall that we write f both for the rational function f (x) ∈ Q(x) and the corresponding map f : P 1 → P 1 : x → t = f (x). As in the beginning of Section 1.2 we write f (x) = x e 1 f 1 (x)/f 2 (x) and assume that the polynomials f i satisfy (1)-(3) introduced there.
Recall that for any n ≥ 1 we write K n /Q(t) for the extension of function fields corresponding to the map f n : P 1 Q → P 1 Q and that G n,Q denotes the Galois group of the Galois closure of this extension (Definition 2.2.1).
Definition 3.1.1. Let a ∈ P 1 (Q)\{0, 1, ∞} such that the numerator x e 1 f 1 − af 2 of f n − a is irreducible for all n ≥ 1 and define K n,a to be the splitting field of x e 1 f 1 − af 2 over K 0,a := Q. We denote by G n,a the Galois group of the normal closure of K n,a /K 0,a . In particular, G n,a is a transitive subgroup of S d n for all n ≥ 1.
Recall that an explicit criterion for good monomial reduction was given in Proposition 1.2.4. (2).
. Since f satisfies (1)-(3) from Section 1.2 and we assume it has good monomial reduction at p, we have that is an Eisenstein polynomial for p, hence irreducible. Here we have also used that f (0) = 0 and ν p (a) = 1. Moreover, the numerator and the denominator of f a (x) are relatively prime. The statement follows for n = 1.
Similarly, for n > 1 arbitrary we find The same argument as for n = 1 therefore also applies to the case of arbitrary n.
For a ∈ P 1 (Q) \ {0, 1, ∞} such that the numerator of f n − a is irreducible, i.e. such that [K n,a : Q] = [K n : Q(t)] = d n , there exists an isomorphism between the field extensions K n,a ⊗ Q Q(t)/Q(t) and K n /Q(t). This isomorphism induces an inclusion of Galois groups (3.1.1) G n,a ֒→ G n,Q .
In the rest of this section we fix these inclusions for all n. For more details we refer to [ Proposition 3.1.3 below provides partial information on the ramification of K n,a /Q for suitable choices of a. Here we use the reduction of f at a prime q for which f has good separable reduction. In Proposition 1.2.4(1) we showed that this holds if q > d. The idea of the proof of Proposition 3.1.3 is similar to that of Proposition 3.1.2. What we show is that if a ≡ 0 (mod q), then the ramification of q in K n,a /Q is the same as the ramification above t = 0 in the iteration f n : P 1 → P 1 , which was described in Remark 2.2.2. We give similar statements for the ramification above the other branch points t = 1, ∞. From this, we deduce the existence of concrete elements in G n,a ; see Lemma 3.1.7 below for the precise statement.
Proposition 3.1.3. Let f be a normalized Belyi map of type (d; e 1 , e 2 , e 3 ) and let q be a prime of good separable reduction for f . Let a ∈ P 1 (Q) \ {0, 1, ∞} such that [K n,a : Q] = d n for all n ≥ 1. Assume additionally that ν q (a) > 0.
For every n ≥ 1, there is a unique ramified prime q n in K n,a /K n−1,a lying above q n−1 . The ramification index of q n in K n,a /K n−1,a is e 1 . All other primes of K n,a over q are unramified in K n,a /K n−1,a .
Proof. We only prove the statement in the case that ν q (a) > 0; the other two cases are similar.
The assumption that f has good separable reduction at q implies that this reduction satisfies where f 1 and f 2 are separable polynomials of degree d−e 1 and d−e 3 , respectively, which are relatively prime. In addition, the fact that the reduction f of f at q is still of type (d; e 1 , e 2 , e 3 ) implies that f 1 (0) = 0 and f 2 (0) = 0. It follows that there is a unique prime q 1 of K n,1 above q 0 := q that is ramified. Moreover, the ramification index of this prime is e 1 . The statement of the proposition for n = 1 and ν q (a) > 0 follows. More precisely, the assumption that f has good separable reduction at q implies that the Newton polygon of x e 1 f 1 − af 2 has two q-adic slopes: a slope 0 with multiplicity d − e 1 and a slope −1/ν q (a) with multiplicity e 1 . It follows that the root a 1 ∈ K 1,a of f (x) − a satisfies ν q 1 (a 1 ) > 0. (In fact, we have that ν q (a) = ν q 1 (a 1 ).) We conclude that the prime q 1 and the rational function f (x) − a 1 ∈ K 1,a (x) satisfy the hypothesis of the proposition, as well. By induction, we conclude that there exists a unique ramified prime q n in K n,a /K n−1,a . Moreover, the prime q n lies above the prime q n−1 , and its ramification index is e 1 .
Let q ′ 1 of K 1,a be an unramified prime above q 0 . In this case it follows that ν q ′ 1 (a 1 ) = 0. We conclude that all primes above q ′ 1 in K 2,a are unramified in K 2,a /K 1,a . By induction, we conclude that if a prime q ′ n above q is unramified in K n,a /K n−1,a then all primes above it in the tower of number fields are unramified. This concludes the proof.  . Choose a ∈ P 1 (Q)\{0, 1, ∞} and distinct primes p, q 1 , q 2 , q 3 such that the following hold: f has good monomial reduction at p and good separable reduction at q 1 , q 2 , q 3 , and we have Remark 3.1.5. The results in this paper can be used to construct towers of number fields that are branched at an explicit finite set of primes. Fix a normalized Belyi map f of type (d; e 1 , e 2 , e 3 ) and a value a ∈ P 1 (Q) \ {0, 1, ∞} such that [K n,a : Q] = d n for all n ≥ 1. Construct the tower Q = K 0,a ⊆ K 1,a ⊆ K 2,a ⊆ . . . of number fields. We denote the set of rational primes in Q by P.
Recall from the introduction that there is a finite set P ⊆ P such that K n,a /K 0,a is unramified outside primes lying above P. (This is [4, Theorem 1], using that normalized Belyi maps are post-critically finite. ) We sketch what we can say about the finite set P in our situation. (This is a more precise version of [4, Section 5], using the results on the reduction of normalized Belyi maps from [2].) A subtle point is that there is a difference between the reduction of a rational function f ∈ Z[x] (defined by reducing the coefficients modulo p as in Definition 1.2.1) and reduction of the cover f : P 1 Q → P 1 Q . A model over Spec(Z p ) of the cover f ⊗ Q Q p is required to be finite and flat. Reducing the rational function f ∈ Z[x] modulo p by reducing its coefficients may yield a rational function of strictly smaller degree. This happens if and only if f has bad reduction to characteristic p > 0 in the sense of Definition 1.2.3.
The results of [2,Section 4] can be interpreted as saying that these notions are closely related for normalized Belyi maps in the single-cycle case. Namely, the rational function f has good separable reduction at p in the sense of Definition 1.2.3 if and only if the Galois closure of the map f : P 1 Qp → P 1 Qp has potentially good reduction, meaning that there exist an extension L/Q p and a model of f ⊗ Qp L over Spec(O L ) whose special fiber is a separable Galois cover of P 1 branched at three points. This is very special to the case of normalized single-cycle Belyi maps. Recall that we have chosen f ∈ Z(x) so that the conditions (1)-(3) in Section 1.2 are satisfied. As in [4, Section 5] we let P bad be the set of rational primes for which the fiber of f at p has degree strictly less than p or is inseparable. (The third case of [4,Section 5], in which the ramification points coalesce modulo p, does not occur in our case, by Proposition 1.2.2(2).) Proposition 1.2.4(1) implies that We can determine this set more precisely for a given type: deg(f ) < deg(f ) at p if and only if p divides the leading coefficient of f . If f has good inseparable reduction at p, then p| deg(f ). In fact one can show that f has either bad or good inseparable reduction at the primes p dividing deg(f ). (This may be deduced from [2,Prop. 5].) It follows from [4, Theorem 2] that we may take P = P bad ∪ P a , where Using the explicit expressions in Proposition 1.1.4 it is easy to find many more results along these lines.
Lemma 3.1.7 below translates Conditions (3.1.4) into a statement on the existence of certain elements h j,n,a ∈ G n,a . We start by setting up some notation. Since G n,a ⊆ G n,Q ⊆ Aut(T n ), we may define subgroups of G n,a analogous to the subgroups N n,Q and N i n,Q defined in Definitions 2.2.5 and 2.2.6. Definition 3.1.6. Define N n,a := ker(G n,a → G n−1,a ) and N i n,a := N i n ∩ S (T i n ). Analogous to (2.2.3), we may write elements of N n,a as tuples (ρ 1 , . . . , ρ d n−1 ), where ρ i ∈ N i n,a ⊆ S (T i n ). Lemma 3.1.7. Let f be a normalized Belyi map satisfying Conditions (3.1.4) for a choice of a, p, q 1 , q 2 , q 3 . For n ≥ 2 and j ∈ {1, 2, 3} there exist elements h j,n ∈ G n,a such that the following conditions hold.
(1) For n ≥ 2 and j ∈ {1, 2, 3}, there exist elements h j,n ∈ G n,a that are conjugate under G n,Q to g j,n . (2) For n = 1, the element h j,1 ∈ G 1,a is a pure cycle of length e j . Proof. For n ≥ 1 and j ∈ {1, 2, 3} let h j,n ∈ G n,a be a generator of the inertia group of q j in G n,a . Then Proposition 3.1.3 implies that h j,n is conjugate in G n,a to g j,n . Hence Statement (1) holds. For n = 1, the h j,1 are pure e j -cycles for j = 1, 2, 3, proving Statement (2). Moreover, it is clear that we may choose the h j,n for varying n consistingly, guaranteeing that Statement (3) holds.
Arguing as in the proof of Lemma 2.2.7, we conclude that the elements α j,n,a are contained in N n,a for j ∈ {1, 2, 3}. Statement (1) implies that α j,n,a has the same cycle type as α j,n (defined in Lemma 2.2.7) when considered as an element of S d n . Statement (4) follows therefore immediately from Lemma 2.2.7.

3.2.
Comparing G n,a and G n,Q . In this section we compare the Galois groups G n,a ⊆ G n,Q and give sufficient conditions on a for these groups to be equal for all n ≥ 1. The key step is to show that the geometric Galois group G n,Q is a subgroup of G n,a for all n if Conditions (3.1.4) are satisfied. We show this using the explicit elements of G n,a we produced in Lemma 3.1.7 and by arguing as in Sections 2.2 and 2.3. (1) We have that G 1,Q ⊆ G 1,a ⊆ S d . In particular, G 1,a ≃ S d in the case that G 1,Q ≃ S d .
(2) For n ≥ 2 and 1 ≤ i ≤ d n−1 , the image of the projection map Proof. The existence of the prime p of good monomial reduction implies that the Galois group G 1,a ⊆ S d of K 1,a /Q is a transitive group on d letters (Proposition 3.1.2). The conditions on a with respect to the primes q i imply that G 1,a contains elements h 1,1 , h 2,1 , h 3,1 , which are pure cycles of length e 1 , e 2 , e 3 respectively, where e 1 + e 2 + e 3 = 2d + 1 (Lemma 3.1.7(2)). We argue as in the proof of [11,Theorem 5.3] to show that G 1,a contains a subgroup isomorphic to G 1,Q . We start by proving that G 1,a is primitive. To reach a contradiction, suppose that G 1,a is not primitive. Since G 1,a is transitive on d letters, there exists a number m|d (with 1 < m < d) and a division of {1, 2, . . . , d} into m blocks of length d ′ := d/m such that every element of G 1,a either has order strictly less than m and acts trivially on the blocks, or has order mk for some k ≥ 1. Now we distinguish the following cases.
• If all e i are strictly less than m, then 2d + 1 = e 1 + e 2 + e 3 < 3m ≤ 3d/2, since m ≤ d/2, by assumption. We obtain a contradiction. • If all e i are a multiple of m, then 2d + 1 = e 1 + e 2 + e 3 is divisible by m. Since m|d we obtain a contradiction. • Assume that exactly one of the e i is strictly less than m; this is necessarily e 1 , since 1 < e 1 ≤ e 2 ≤ e 3 . Write e 2 = mk 2 and e 3 = mk 3 for some k 2 , k 3 ≥ 1. We obtain 2d + 1 = e 1 + m(k 2 + k 3 ). Since m|d, we have e 1 ≡ 1 (mod m). This implies that e 1 > m, and we obtain a contradiction. • If exactly two of the e i (that is, e 1 and e 2 ) are strictly less than m, then a similar argument shows that e 1 + e 2 ≡ 1 (mod m) so e 1 + e 2 = 1 + m ≤ 1 + d/2. But then e 3 = (2d + 1) − (e 1 + e 2 ) ≥ 3d/2 > d, which again yields a contradiction. Hence G 1,a is primitive, as claimed. For d > 10, the group G 1,a contains a cycle of length e ≤ (d−e)! (namely, h 1,1 ). Hence, by Williamsons's Theorem [17], we have that G 1,a is isomorphic to either A d or S d . If at least one of the e i is even, then G 1,a is isomorphic to S d . In both cases we therefore have that G 1,a contains G 1,Q (Lemma 1.1.5(1)). For d ≤ 10, the statement follows from the case-by-case analysis in the proof of [11,Theorem 5.3]. Statement (1) follows.
The group N n,a contains the elements α 1,n,a , α 2,n,a , α 3,n,a from Lemma 3.1.7 (4). Recall that G n,a is a transitive groups on d n letters. In particular, it follows that conjugation by G n,a acts transitively on the d n−1 blocks of d indices, where the ith block corresponds to the vertices of T i n . It therefore suffices to prove Statement (2) for i = 1.
Replacing α j,n,a by a conjugate under G n,a , we may assume that the component α 1 j,n,a in S (T 1 n ) is non-trivial. Note that the group elements we conjugate by may depend on j ∈ {1, 2, 3}. We conclude that the image of N n,a under projection to S (T 1 n ) contains an e 1 -cycle, an e 2 -cycle, and an e 3 -cycle. We denote this image by G 1 n,a . The group G 1 n,a may be identified with the Galois group of the specialization of f at a point b with f n−1 (b) = a. The corresponding field extension K 1,b of K 0,b ≃ Q defined by f may therefore be identified with a subextension of K n,a /K n−1,a . Since [K n,a : K 0,a ] = d n (Proposition 3.1.2), it follows that [K 1,b : K 0,b ] = d. We conclude that G 1 n,a is a transitive group on d letters. The argument from the proof of Statement (1) shows that G 1,Q ⊆ G 1 n,a ⊆ S d . This proves Statement (2). Theorem 3.2.2 below extends the conclusion of Proposition 3.2.1(1) that G 1,Q ⊆ G 1,a under Conditions (3.1.4) to arbitrary n. Phrased differently, we show that we do not need additional conditions on a in passing from n = 1 to arbitrary n. We exclude exactly the same types as in Theorem 2.3.1.
Here we have also used that 2d + 1 = e 1 + e 2 + e 3 . (For the generating system from Lemma 1.1.5(2), the unique integer in the intersection is e 3 .) In the case that G 1,Q ≃ S d we have that G 1,Q = G 1,a ≃ S d , and in this case we may even assume that h j,1 = g j,1 for all j ∈ {1, 2, 3}. More precisely, from the fact that the generating system from Lemma 1.1.5(2) is weakly rigid it follows that we may assume that h j,1 = σ −1 g j,1 σ for all j ∈ {1, 2, 3} and some σ ∈ S d which is independent of j. The proof of Lemma 2.2.7(2) may be applied to the elements α j,n,a defined in Lemma 3.1.7(4). In the current situation we conclude that the component α i j,n,a of α j,n,a in S (T i n ) satisfies α i j,n,a = an e 1 -cycle if i ∈ supp(h h,1 ), trivial otherwise.
The following claim is analogous to Claim 1 in the proof of Proposition 2.2.8.
Claim 2 follows from Claim 1 as in the proof of Proposition 2.2.8 (Claim 2). The proof of Lemma 2.3.2 applies in this case as well: in the proof of that lemma, we only use the fact that exactly (e j ) n−1 components of the element α j,n ∈ N n,Q are non-trivial. In the current situation, this property follows from Lemma 3.1.7(4). We conclude that N n,a contains a subgroup isomorphic to N n,Q . The statement of the theorem follows as in the proof of Theorem 2.3.1.
The following straightforward corollary of Theorem 3.2.2 summarizes the relation between the groups G n,Q , G n,a , G n,Q in the case that all assumptions we have imposed at various places in this paper hold. Recall that we gave explicit conditions guaranteeing that G n,Q = G n,Q for all n ≥ 1 in Proposition 2.4.4. Proof. We have G n,Q ⊆ G n,a ⊆ G n,Q ⊆ Aut(T n ) for all n ≥ 1: the first inclusion is Theorem 3.2.2, the second one is Equation (3.1.1, and the third holds by definition. The assumption that G n,Q = G n,Q for all n ≥ 1 therefore implies that all three groups are equal. The second statement on the index follows from Lemma 2.1.3(1).

Applications to dynamical sequences
Let f be a normalized Belyi map of degree d such that G 1,Q ≃ S d . By Theorem 2.3.1, the iterates f n then have geometric monodromy groups G n,Q ≃ E n for all n ≥ 1, where the groups E n are defined in Definition 2.1.2. We prove in Theorem 4.1.2 that when E 1 ≃ S d and for any d ≥ 3, the proportion of elements of E n ≃ G n that fix a leaf on level n tends to zero as n tends to infinity. This generalizes [3, Theorem 5.1], where the authors prove the same for the Belyi map f (x) = −2x 3 + 3x 2 of degree d = 3; our proof is inspired by theirs.
Choose a ∈ P 1 (Q) \ {0, 1, ∞} and distinct primes p, q 1 , q 2 , q 3 such that Conditions (3.1.4) hold. Then G n,Q ⊆ G n,a ⊆ G n,Q for all n ≥ 1, by Theorem 3.2.2. Further assume that G n,Q ≃ G n,Q for all n ≥ 1, so that these inclusions all become equalities. For f of degree d ≥ 3, Theorem 4.1.2 then shows that the proportion of elements of G n,a that fix a leaf tends to zero as n tends to infinity. In Corollary 4.2.1 we derive some arithmetic dynamical consequences from this theorem. A n,id = |{ elements of E n which act as id on T 1 }|, A ′ n,id = |{ elements of E n,fix which act as id on T 1 }|, and for 2 ≤ k ≤ d, let A n,k = |{ elements of E n that act as as τ with |supp(τ )| = k on T 1 }|, A ′ n,k = |{ elements of E n,fix that act as as τ with |supp(τ )| = k on T 1 }|. We see that |E n | = A n,id + d i=2 A n,i and |E n,fix | = A ′ n,id + d−1 i=2 A ′ n,i . Recall from Section 2 that Finally, we note that for any n ≥ 1, exactly half of the elements of E n−1 ≀ E 1 are contained in E n by Definition 2.1.2.  3. Let f be a normalized Belyi map of degree d ≥ 3, such that G n,Q ≃ G n,Q ≃ E n for all n ≥ 1. If we choose a ∈ P 1 (Q) \ {0, 1, ∞} and distinct primes p, q 1 , q 2 , q 3 such that Conditions (3.1.4) hold, then G n,Q ≃ G n,a ≃ G n,Q for all n ≥ 1, and Theorem 4.1.2 implies that |G n,a,fix |/|G n | → 0 as n → ∞, where G n,a,fix = { elements of G n,a which fix a leaf }.

Dynamical sequences.
For a set S of prime numbers, let δ(S) denote its natural density if it exists. A dynamical sequence in a field K is a sequence {c i } i≥0 with c i ∈ K such that c i = f (c i−1 ) for some map f : K → K. Prime divisors of entries of such sequences were first studied using Galois theory by Odoni in [14]. For four particular quadratic maps f , Jones [8] shows the density of prime divisors in the dynamical sequence for f is zero; Gottesman and Tang [6] show non-zero densities can also occur for quadratic maps. More general treatments of higher degree maps can be found in e.g. [5], [7]. (1) Choose a ∈ P 1 (Q) \ {0, 1, ∞} and distinct primes p, q 1 , q 2 , q 3 , such that Conditions (3.1.4) hold. Consider the sequence {a i } i≥0 where a 0 ∈ P 1 (Q) \ {0, 1, ∞, a} and a i = f (a i−1 ) for all i ≥ 1. Then δ(S) = 0, where S is the set of primes p ∈ Q such that a i ≡ a (mod p) for some i ≥ 0.
(2) Suppose that for any of d − e 1 non-zero preimages of zero under f , there exist distinct primes p, q 1 , q 2 , q 3 such that Conditions (3.1.4) hold. Consider the sequence {b i } i≥0 where b 0 ∈ P 1 (Q)\{0, 1, ∞, and their preimages under f } and b i = f (b i−1 ) for all i ≥ 1. Then δ(T ) = 0, where T is the set of primes p ∈ Q such that p|b i for some i ≥ 0.

Proof.
(1) For any n ≥ 1, the set S n of primes not dividing a such that a i ≡ a (mod p) for some 0 ≤ i ≤ n − 1 is finite. Further, if a i ≡ a (mod p) for some i ≥ n, then the rational map f n (x) − a has a rational root over Z/pZ (namely, a i−n ). It follows from the Chebotarev density theorem that δ(S) ≤ δ({p ∈ Q : p ∈ S n and f n (x) − a has a root modulo p}) = |E n,fix | |E n | , so the result follows by letting n → ∞ and using Theorem 4.1.2. (2) We may ignore the finitely many primes of bad or good inseparable reduction for f , as well as the finitely many primes dividing b 0 . By assumption, b j = 0 for any j ≥ 0. Let p be a prime such that b i ≡ 0 (mod p) for some, without loss of generality minimal, value of i ≥ 1. Writing f (x) = x e 1 f 1 (x)/f 2 (x) as before, we see that f 1 (b i−1 ) ≡ 0 (mod p) and hence that b i−1 is congruent (but not equal) to one of the d − e 1 non-zero preimages c 1 , . . . , c d−e 1 of zero under f . Since by assumption G n,c j ≃ G n ≃ E n for all n ≥ 1 and all 1 ≤ j ≤ d − e 1 , the result now follows by observing that δ(T ) = δ({p ∈ Q : b i ≡ c j (mod p) for some i ≥ 1 and some 1 ≤ j ≤ d − e 1 }) and applying (1).