Configuration of the Crucial Set for a Quadratic Rational Map

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $\varphi(z) \in K(z)$ have degree two. We describe the crucial set of $\varphi$ in terms of the multipliers of $\varphi$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $\varphi$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set.


Introduction
Let K be an algebraically closed field, complete with respect to a non-Archimediean absolute value | · | v . Let O = O K denote its ring of integers and m = m K ⊆ O its maximal ideal. Let k = O/m denote the residue field. We assume that | · | v and the logarithm log v are normalized so that ord m (x) = − log v |x| v . We will typically drop the dependence on v and m in the notation and simply write | · | and ord. Let P 1 K denote the Berkovich projective line over K; it is a compact, uniquely path connected Hausdorff space which contains P 1 (K) as a dense subset.
Let ϕ(z) ∈ K(z) be a rational map of degree d ≥ 2. In [8], Rumely found a canonical way to assign non-negative integer weights w ϕ (P ) to points in P 1 K , for which the sum of the weights is d − 1. The set of points which receive weight is called the crucial set of ϕ, and the probability measure ν ϕ := 1 (W2) P is a 'Berkovich multiplicatively indifferent fixed point' of ϕ which is a branch point of Γ Fix ; (W3) P is a 'Berkovich additively indifferent fixed point' of ϕ belonging to Γ Fix ; (W4) P is a branch point of Γ Fix which is moved by ϕ.
Here we have added the word 'Berkovich' for emphasis; usually it will be omitted. For a quadratic function, d − 1 = 1, so there is a unique point ξ which receives weight. It turns out that this is the Minimal Resultant Locus (see [9]), the point where ϕ has "best possible" reduction. According as (W1) through (W4) holds at ξ, we will say that a quadratic map ϕ has potential good reduction, potential multiplicative reduction, potential additive reduction, or potential constant reduction.
In this paper, we first determine for which quadratic maps ϕ the point ξ satisfies (W1), (W2), (W3), or (W4); this is accomplished in Propositions 3.1 and 3.4, where we show that the reduction type of ϕ at ξ is determined by the multipliers at the classical fixed points.
We apply this to study the image of ϕ in M 2 , the moduli space of degree 2 rational maps. Using geometric invariant theory, Silverman [12] constructed M d as a scheme over Z for all d ≥ 2, and for d = 2 showed there is a canonical isomorphism s : M 2 → A 2 . (Milnor had shown this earlier over C; see [7,Lemma 3.1].) This leads to a natural compactification of M 2 as M 2 ∼ = P 2 .
For a quadratic map ϕ(z) ∈ K(z), let [ϕ] ∈ M 2 (K) denote the point corresponding to ϕ. We will base-change to O and regard M 2 and M 2 as schemes over O. The isomorphism s : M 2 → A 2 is given by s([ϕ]) = (σ 1 (ϕ), σ 2 (ϕ)) where σ 1 , σ 2 are the first and second symmetric functions in the multipliers at the fixed points of ϕ. We identify A 2 (K) with { [x : y : 1]} ⊂ P 2 (K). Given a point P ∈ P 2 (K), we write P ∈ P 2 (k) for the specialization of P modulo m.
For arbitrary d ≥ 2, the connection between the crucial set and M d was first noted in [8], where it was shown that points in the barycenter of ν ϕ correspond to conjugates of ϕ having semi-stable reduction in the sense of geometric invariant theory. The main result of this paper is the following theorem, which says that for quadratic functions, the crucial set determines a stratification of M 2 (K) compatible with specialization of [ϕ] to M 2 (k): Theorem 1.1. Let K be a complete, algebraically closed non-Archimedean field. Let ϕ be a degree two rational map over K, and let ξ denote the unique point in the crucial set of ϕ. Then   The fact that s([ϕ]) ∈ A 2 (k) if and only if ϕ has potential good reduction had previously been shown by D. Yap in her thesis [13,Thm. 3.0.3]. Our theorem may be considered a strengthening of Yap's result. One also notes the parallel between Theorem 1.1 and Milnor's description [7] of degenerations of quadratic maps over C, as they approach the boundary of moduli space.
In §4.2, we observe that Theorem 1.1 implies the following result, which resolves a special case of a conjecture of L.-C. Hsia ([6,Conj. 4.3]). Proposition 1.2. Let ϕ be a quadratic rational map defined over K, let J ϕ (K) ⊆ P 1 (K) be the (classical) Julia set of ϕ, and let R ϕ (K) be the closure in P 1 (K) of the set of type I repelling periodic points for ϕ. Then J ϕ (K) = R ϕ (K).
1.1. Outline of the Paper. In Section 2, we introduce notation and concepts used in the rest of the article. In particular, we give a more detailed explanation of the weights w ϕ and the conditions (W1)-(W4) under which a point can have weight. In Section 3 we relate the reduction type of the unique weighted point ξ to the multipliers at the classical fixed points. For this, we rely on two normal forms for quadratic rational maps given in [11]. In Section 4 we apply our analysis to prove Theorem 1.1, and give the application to Hsia's conjecture.
1.2. Acknowledgements. The research for this article was begun during an NSF-sponsored VIGRE research seminar on dynamics on the Berkovich line at the University of Georgia, led by the third author. We would like to thank the other members of our research group for many helpful discussions, and would especially like to thank Jacob Hicks, Allan Lacy, Marko Milosevic, and Lori Watson for their insights and contributions to this work.

Notation and Conventions
In this section we introduce terminology and notation used throughout the paper.
2.1. Berkovich Space. Formally, the Berkovich affine line A 1 K over K is the collection of equivalence classes of multiplicative seminorms on K[T ] which extend the norm | · | v on K. Berkovich showed ( [2], p.18) that each such seminorm [·] x corresponds to a decreasing sequence of discs {D(a i , r i )} in K (more precisely, to a cofinal equivalence class of such sequences) via the correspondence Here, [f ] D(a,r) = sup z∈D(a,r) |f (z)| is the sup-norm on the disc D(a, r). With this, we obtain a classification of the points of A 1 K into four types: • Points of type I correspond to nested, decreasing sequences of discs {D(a i , r i )} whose intersection is a single point in K; formally, these are the seminorms [f ] a = |f (a)| for a ∈ K. • Points of type II correspond to nested, decreasing sequences of discs whose intersection is a disc D(a, r) ⊆ K with r ∈ |K × |. In this case, we have [f ] x = sup z∈D(a,r) |f (z)|. Note that for polynomials f ∈ K[T ], the supremum is achieved at some point in D(a, r). • Points of type III correspond to nested, decreasing sequences of discs whose intersection is a disc D(a, r) ⊆ K with r ∈ |K × |; as in the case of type II points, the corresponding seminorm is the sup-norm on D(a, r). In this case, the sup is not achieved unless f is constant. • Points of type IV correspond to nested, decreasing sequences of discs {D(a i , r i )} whose intersection is empty, but for which lim i→∞ r i > 0. Such points can occur only if the field K is not spherically complete.
One often writes type I, II, and III points in terms of their corresponding discs D(a, r) using the shorthand ζ D(a,r) or simply ζ a,r . The point corresponding to the unit disc is called the Gauss point (it corresponds to the Gauss norm on polynomials), and is written ζ G = ζ D(0,1) .
The construction of P 1 K from A 1 K is similar to the construction of P 1 from A 1 , gluing two copies of A 1 K together by means of an involution of A 1 K \{0}; see Section 2.2 of [1] for details. We write H 1 K for P 1 K \P 1 (K), the 'non-classical' part of P 1 K . The Berkovich Line is typically endowed with the Berkovich-Gel'fand topology, which is the weakest topology for which the map x → [f ] x is continuous for every f ∈ K[T ]. In this topology, P 1 K is a compact Hausdorff space and is uniquely path connected. The points of type I, II and III are dense in P 1 K for this topology. In general the Berkovich-Gel'fand topology is not metrizable.
A rational map ϕ ∈ K(z) induces a continuous action on 1) . This action extends continuously to all of P 1 K , and preserves types of points. One can show that PGL 2 (K) acts transitively on type II points, and that PGL 2 (O) is the stabilizer of the Gauss point.
2.1.1. Tree Structure. The Berkovich projective line can also be viewed as a tree. The collection of points {ζ a,r } r∈[t,s] ⊆ P 1 K is naturally homeomorphic to the real segment [t, s]. The type II points are dense along such a segment, and at any type II point there are infinitely many branches away from {ζ a,r } r∈[t,s] ; indeed the branches are in 1 − 1 correspondence with the elements of P 1 (k) for the residue field k. To make this clearer, consider the type II point ζ G = ζ D(0,1) . The branches off ζ G come from equivalence classes of paths [ζ G , x] sharing a common initial segment; these classes correspond to subdiscs D(b, 1) − = {x ∈ K : |x−b| < 1} where |b| ≤ 1, and to the set P 1 (K)\D(0, 1). Identifying D(0, 1) with the valuation ring O K , the subdiscs D(b, 1) − are just the cosets b + m in k = O K /m, and P 1 (K)\D(0, 1) corresponds to ∞ ∈ P 1 (k).
A more geometric way to think of the branches is in terms of tangent directions. Formally, a tangent direction v at P is an equivalence class of paths emanating from P . The collection of tangent directions at P will be denoted by T P . For points of type II, T P is in 1 − 1 correspondence with P 1 (k) as noted above. For points of type III, T P consists of two directions, while for points of type I and IV, T P consists of the unique direction pointing into P 1 K . If P, Q are points of P 1 K with ϕ(P ) = Q, there is a canonical induced surjective map ϕ * : T P → T Q .

2.1.2.
Reduction of Rational Maps. The action of ϕ on the tangent space T P is closely related to the notion of the reduction of ϕ, which we describe here. If [F, G] is a lift of ϕ that has been scaled so that the coefficients all lie in O, and so that at least one is a unit, we call [F, G] a normalized lift, or normalized representation, of ϕ. Such a representation is unique up to scaling by a unit in O. We can reduce each coefficient of a normalized lift [F, G] modulo m. After removing common factors, we obtain a well-defined map [ F : G] on P 1 (k). This map, called the reduction of ϕ at ζ G , is denoted ϕ.
If P is an arbitrary type II point, there is a γ ∈ PGL 2 (K) for which γ(ζ G ) = P ; we define the reduction of ϕ at P to be the reduction of the conjugate ϕ γ = γ −1 • ϕ • γ: The reduction ϕ P is unique up to conjugation by an element of PGL 2 (k); in particular, the degree deg( ϕ P ) is well-defined.
It was shown by Rivera-Letelier that a type II point P ∈ H 1 K is fixed by ϕ if and only if the reduction ϕ P is non-constant (see [1], Lemma 2.17); equivalently, ϕ(P ) = P if and only if ϕ P is constant. Rivera-Letelier calls a type II point P a repelling fixed point if deg( ϕ P ) ≥ 2, and he calls P an indifferent fixed point if deg( ϕ P ) = 1. Rumely [8,Def. 2] gave a refined classification of indifferent fixed points in H 1 K : Definition 2.1. If P is a type II indifferent fixed point of ϕ, then after a change of coördinates on P 1 (k), exactly one of the following holds: • ϕ P (z) = cz for some c ∈ k × , c = 1, in which case we say P is a (Berkovich) multiplicatively indifferent fixed point for ϕ. • ϕ P (z) = z + a for some a ∈ k × , in which case we say P is an (Berkovich) additively indifferent fixed point for ϕ. • ϕ P (z) = z, in which case we say P is an id-indifferent fixed point for ϕ.
One should think of each of the above reduction types as describing the behavior of the map ϕ * acting on T P . More precisely, after conjugating ϕ by a suitable γ ∈ PGL 2 (K) we can assume that P = ζ G is fixed. Then ϕ is a well-defined non-constant map, and by making use of the identification It is common to say that a map ϕ(z) ∈ K(z) has good reduction if deg( ϕ) = deg(ϕ), and potential good reduction if deg( ϕ P ) = deg(ϕ) for some P = ζ G . However, in this paper we will not distinguish good reduction from potential good reduction. By the discussion above, a quadratic map ϕ has potential good reduction iff it has a type II repelling fixed point.

2.2.
The Crucial Set. The crucial set was constructed in [8], and arose from the study of a certain function ord Res ϕ : P 1 K → R ∪ {∞}. This function had been introduced in [9], in order to address the question of finding conjugates ϕ γ that had minimal resultant. One obtains the crucial measure and crucial set by taking the graph-theoretic Laplacian of ord Res ϕ (·), restricted to a canonical tree Γ FR ⊂ P 1 K . In this section, we briefly sketch this construction.
Using standard formulas for the resultant from [10], one sees that ord Res ϕ (ζ) is well-defined, and that ord Res ϕ (ζ) = ord Res ϕ (ζ G ) + (d 2 + d) ord det(γ) − 2d min ord(F γ ), ord(G γ ) , (The 'min' term assures we are using a normalized lift Φ γ .) It is shown in [9] that the function ord Res ϕ on type II points extends to a continuous function ord Res ϕ : P 1 K → [0, ∞], which, for any a ∈ P 1 (K), attains its minimum on the tree Γ Fix,ϕ −1 (a) spanned by the classical fixed points and the pre-images under ϕ of a. It is also shown in [8] that the tree Γ FR spanned by the classical fixed points and the repelling fixed points in H 1 K is the intersection of all the trees Γ Fix,ϕ −1 (a) : This is useful in determining Γ FR .

The Crucial
Measures. The crucial measure is obtained by taking the graph-theoretic Laplacian of ord Res ϕ (·) on (a suitable truncation 1 of) the tree Γ FR . More precisely, if µ Br is the 'branching measure' which gives each P ∈ Γ FR the weight 1 − 1 2 v(P ), where v(P ) is the valence of P in Γ FR , then Definition 2.2 (Rumely, [8, Cor. 6.5]). The crucial measure associated to ϕ is the measure ν ϕ on Γ FR defined by It is a probability measure with finite support, and its support is contained in H 1 K . The crucial measure is canonically attached to ϕ, because the function ord Res ϕ and the tree Γ FR are canonical. It is a conjugation equivariant of ϕ in H 1 K , just as the sets of classical fixed points and critical points are conjugation equivariants in P 1 (K).
Rumely gave an explicit expression for ν ϕ as a sum of weighted point masses: 1 In order to apply the theory of graph Laplacians, one must first 'prune' the tree Γ FR to remove its type I endpoints -see [8, p. 25]. We omit the details here, as they won't be necessary in this article.
where the weights w ϕ (P ) are as follows ([8, Def. 8]): Definition 2.3. For a point P ∈ H 1 K , if P fixed by ϕ, let N shearing,ϕ (P ) be the number of directions v ∈ T P that contain type I fixed points but are moved by ϕ * . Let v(P ) denote the valence of P in Γ FR (set v(P ) = 0 if P ∈ Γ FR ). Then the weight w ϕ (P ) of a point P ∈ P 1 K is as follows: (A) If P is a type II fixed point of ϕ, then w ϕ (P ) = deg( ϕ P ) − 1 + N shearing,ϕ (P ). (B) If P is a branch point of Γ Fix which is moved by ϕ (necessarily of type II), then w ϕ (P ) = v(P ) − 2. (C) Otherwise, w ϕ (P ) = 0.
The above formulas for the weights give rise to conditions (W1)-(W4) in the Introduction.

The Crucial Sets of Quadratic maps
The behavior of a rational map ϕ near a classical fixed point α ∈ P 1 (K) is governed by the multiplier at α. In this section, we explicitly describe the crucial set for quadratic rational maps in terms of the multipliers at the classical fixed points.
If α ∈ K is a fixed point for the rational map ϕ, the derivative ϕ ′ (α) is called the multiplier of ϕ at α. It is well-known that the multiplier is independent of the choice of coordinates, which means the multiplier at ϕ at ∞ ∈ P 1 (K) can be defined by changing coordinates.

Maps With Distinct Fixed Points.
We now turn to quadratic rational maps with three distinct classical fixed points. In this case, the multiplier of the third fixed point is determined by the multipliers of the other two: Let ϕ be a degree two rational map with three distinct classical fixed points. Let λ 1 and λ 2 be the multipliers of two of the fixed points. Then the third has multiplier Proof. Since ϕ has three distinct fixed points, none of the multipliers can be equal to one. Therefore, we have the well-known formula (see [10, Theorem 1.14]) Solving for λ 3 yields the desired result. (C) if ϕ has no classical repelling fixed points, then either some pair of multipliers satisfies λ i λ j = 1, or else λ 1 = λ 2 = λ 3 = 1.
Proof. Suppose that ϕ has two repelling fixed points, say α 1 and α 2 , with multipliers λ 1 and λ 2 . By (3), the multiplier λ 3 of α 3 satisfies so α 3 is attracting. This shows that ϕ cannot have three repelling fixed points, and also proves (A).
Remark. Note that in case (A)(ii), the point ξ is different from ζ G , since |λ 1 λ 2 − 1| < 1 and therefore This is the only situation where Γ FR may be strictly larger than Γ Fix .
The map ϕ γ is given by Since λ 1 , λ 2 ∈ O by assumption, all of the coefficients of ϕ γ are integers. However, as we will now see, all of the coefficients of ϕ γ lie in m, so we need a normalized representation of ϕ γ . Since |λ 3 | = 1, it follows from (3) that We also claim that |λ 2 − 1| ≤ r. Indeed, suppose to the contrary that |λ 2 − 1| > r. Then We now see that the absolute values of the coefficients of ϕ γ are as follows: so the maximum among the absolute values of the coefficients is r 2 . We therefore divide all coefficients by ρ 2 to obtain the normalized representation The resultant of the natural lift Φ γ is and therefore ϕ γ has good reduction. It follows that γ(ζ G ) = ζ D(−1,r) is a repelling fixed point for ϕ, and hence ξ = ζ D(−1,r) , as claimed.

4.1.
Proof of the Main Theorem. We are now ready to prove our main result, Theorem 1.1, which says that for quadratic maps the crucial set gives a stratification of M 2 (K) compatible with specialization to M 2 (k). Recall that M 2 ∼ = A 2 and M 2 ∼ = P 2 as schemes over Z; by abuse of notation we view the isomorphism s = (σ 1 , σ 2 ) : M 2 (K) → A 2 (K) as an embedding s = [σ 1 : σ 2 : 1] : M 2 (K) ֒→ P 2 (K) .
Finally, for a point P ∈ P 2 (K), we denote by P ∈ P 2 (k) the specialization of P modulo m.
Theorem 1.1. Let K be a complete, algebraically closed non-Archimedean field. Let ϕ be a degree two rational map over K, and let ξ denote the unique point in the crucial set of ϕ. Then Proof of Theorem 1.1. Because ξ must satisfy exactly one of (W1) -(W4), and since P 2 (k) is equal to the disjoint union it suffices to prove only the forward implications of the statements in the theorem.
First, suppose ξ satisfies (W1). If ϕ has three distinct fixed points, it follows from Proposition 3.4 that ϕ has no repelling fixed points. Thus all of the multipliers of ϕ lie in O. In particular, this means that σ 1 , σ 2 ∈ O, so If, on the other hand, ϕ has a multiple fixed point, then for a suitable ordering of the multipliers we have λ 1 = λ 2 = 1, and by Proposition 3.1 we have |λ 3 | ≤ 1. Once again, all the multipliers of ϕ lie in O, and so s([ϕ]) = [ σ 1 : σ 2 : 1] ∈ A 2 (k).
For ξ to satisfy (W2), (W3), or (W4), the map ϕ must have at least one repelling classical fixed point and one non-repelling classical fixed point. Indeed, in the case that ϕ has a multiple fixed point, this follows from Proposition 3.1; in the case that ϕ has three distinct fixed points, we know from Proposition 3.4 that ϕ must have at least one classical repelling fixed point, in which case ϕ also has a non-repelling fixed point by Lemma 3.3. Therefore, we may assume for the remainder of the proof that |λ 1 | > 1 ≥ |λ 2 |.

4.2.
An Application to Repelling Periodic Points. We now use the main theorem to prove a special case of a conjecture of Hsia. For a rational map ϕ ∈ K(z), let J ϕ (K) denote the (classical) Julia set of ϕ, let R ϕ (K) denote the set of all classical repelling periodic points for ϕ, and let R ϕ (K) be its closure in P 1 (K). It is known over the complex numbers that J φ (C) = R φ (C); the analagous result is not known when K is non-Archimedean, though it is conjectured to be true: