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 Open Access
Configuration of the crucial set for a quadratic rational map
 John R. Doyle^{1},
 Kenneth Jacobs^{2}Email author and
 Robert Rumely^{2}
 Received: 4 November 2015
 Accepted: 13 April 2016
 Published: 5 July 2016
Abstract
Let K be a complete, algebraically closed nonArchimedean 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 classical nonArchimedean Julia set.
Keywords
 Crucial set
 Quadratic map
 Moduli space
 Potential good reduction
 Stratification
Mathematics Subject Classification
 Primary 37P50
 11S82
 Secondary 37P05
 11Y40
1 Background
Let K be an algebraically closed field, complete with respect to a nonArchimedean absolute value \(\cdot _v\). Let \(\mathcal {O}= \mathcal {O}_K\) denote its ring of integers and \(\mathfrak {m}=\mathfrak {m}_K\subseteq \mathcal {O}\) its maximal ideal. Let \(k = \mathcal {O}/\mathfrak {m}\) denote the residue field. We assume that \(\cdot _v\) and the logarithm \(\log _v\) are normalized so that \({{\mathrm{ord}}}_{\mathfrak {m}} (x) = \log _v x_v\). We will typically drop the dependence on v and \(\mathfrak {m}\) in the notation and simply write \(\cdot \) and \({{\mathrm{ord}}}\). Let \(\mathbf {P}^1_K\) denote the Berkovich projective line over K; it is a compact, uniquely path connected Hausdorff space which contains \({\mathbb P}^1(K)\) as a dense subset.
Let \(\varphi (z) \in K(z)\) be a rational map of degree \(d \ge 2\). In [9], the third author showed there is a canonical way to assign nonnegative integer weights \(w_{\varphi }(P)\) to points in \(\mathbf {P}^1_K\), for which the sum of the weights is \(d1\). The set of points which receive weight is called the crucial set of \(\varphi \), and the probability measure \(\nu _{\varphi } := \frac{1}{d  1} \sum _{P \in \, \mathbf {P}^1_K} w_{\varphi }(P) \delta _P\) is called the crucial measure of \(\varphi \). When \(\varphi \) has potential good reduction, the crucial set consists of the single point where \(\varphi \) has good reduction. Otherwise, the crucial set appears to classify the type of bad reduction that \(\varphi \) has; this paper provides quantitative support for that idea when \(\varphi \) is a quadratic rational map.
The points in \(\mathbf {P}^1_K\) are classified into four types, customarily labeled as type I through type IV. We will define these types in Sect. 2, but we mention here that the type I points are the “classical” points belonging to \({\mathbb P}^1(K)\), while the type II points are the nonclassical points P at which \(\varphi \) has a meaningful reduction \({\widetilde{\varphi }}_P \in k(z)\). The crucial set is contained in the set of type II points ([9, Proposition 6.1]); in particular, it lies in \(\mathbf {P}^1_K\backslash {\mathbb P}^1(K)\), and it is appropriate to talk about the reduction of \(\varphi \) at the points of the crucial set.

\(\varphi \) has potential good reduction if the reduction of \(\varphi \) at \(\xi \) is again a degree 2 map.

\(\varphi \) has potential additive reduction if the reduction of \(\varphi \) at \(\xi \) is a degree 1 map conjugate to \(z\mapsto z+\widetilde{a}\) for some \(\widetilde{a}\in k{\setminus } \{\widetilde{0}\}\), and \(\xi \) is contained in \(\Gamma _{\mathrm {Fix}}\).

\(\varphi \) has potential multiplicative reduction if the reduction of \(\varphi \) at \(\xi \) is a degree 1 map conjugate to \(z\mapsto \widetilde{\lambda } z\) for some \(\widetilde{\lambda }\in k{\setminus }\{\widetilde{0},\widetilde{1}\}\), and \(\xi \) is a branch point of \(\Gamma _{\mathrm {Fix}}\).

\(\varphi \) has potential constant reduction if the reduction of \(\varphi \) at \(\xi \) is a constant map, and \(\xi \) is a branch point of \(\Gamma _{\text {Fix}}\).
Our main result shows there is a relation between the reduction type of \(\varphi \) at \(\xi \) and the image of \(\varphi \) in \(\mathcal {M}_2\), the moduli space of degree 2 rational maps. Using geometric invariant theory, Silverman [13] constructed \(\mathcal {M}_d\) as a scheme over \({\mathbb Z}\) for all \(d\ge 2\), and for \(d=2\) showed there is a canonical isomorphism \(\mathbf {s}: \mathcal {M}_2 \rightarrow \mathbb {A}^2\). (Milnor had shown this earlier over \(\mathbb {C}\); see [8, Lemma 3.1].) This leads to a natural compactification of \(\mathcal {M}_2\) as \(\overline{\mathcal {M}}_2 \cong \mathbb {P}^2\).
For a quadratic map \(\varphi (z) \in K(z)\), let \([\varphi ] \in \mathcal {M}_2(K)\) denote the point corresponding to the equivalence class of \(\varphi \). We will basechange to \({\mathcal O}\) and regard \(\mathcal {M}_2\) and \(\overline{\mathcal {M}}_2\) as schemes over \({\mathcal O}\). The isomorphism \(\mathbf {s}: \mathcal {M}_2 \rightarrow \mathbb {A}^2\) is given by \(\mathbf {s}([\varphi ]) = (\sigma _1(\varphi ),\sigma _2(\varphi ))\) where \(\sigma _1, \sigma _2\) are the first and second symmetric functions in the multipliers at the fixed points of \(\varphi \). We identify \(\mathbb {A}^2(K)\) with \(\{\,[x:y:1] \} \subset {\mathbb P}^2(K)\). Given a point \(P \in \mathbb {P}^2(K)\), we write \(\widetilde{P} \in \mathbb {P}^2(k)\) for the specialization of P modulo \(\mathfrak {m}\).
For arbitrary \(d \ge 2\), the connection between the crucial set and \(\mathcal {M}_d\) was first noted in [9], where it was shown that points in the barycenter of \(\nu _\varphi \) correspond to conjugates of \(\varphi \) having semistable 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 \(\mathcal {M}_2(K)\) compatible with specialization of \([\varphi ]\) to \(\overline{\mathcal {M}}_2(k)\):
Theorem 1.1
 (A)
\(\varphi \) has potential good reduction if and only if \(\widetilde{\mathbf {s}([\varphi ])} \in \mathbb {A}^2(k)\).
 (B)
\(\varphi \) has potential additive reduction if and only if \(\widetilde{\mathbf {s}([\varphi ])} = [\widetilde{1} : \widetilde{2} : \widetilde{0}]\).
 (C)
\(\varphi \) has potential multiplicative reduction if and only if \(\widetilde{\mathbf {s}([\varphi ])} = [\widetilde{1} : \widetilde{x} : \widetilde{0}]\) for some \(\widetilde{x} \in k\) with \(\widetilde{x} \ne \widetilde{2}\). In this case, \(\widetilde{x} = \widetilde{\lambda } + \widetilde{\lambda }^{1}\), where \(\lambda \) is the multiplier of an indifferent fixed point for \(\varphi \).
 (D)
\(\varphi \) has potential constant reduction if and only if \(\widetilde{\mathbf {s}([\varphi ])} = [\widetilde{0} : \widetilde{1} : \widetilde{0}]\).
Theorem 1.1 is proved by explicitly determining \(\xi \) and the reduction type of \(\varphi \) at \(\xi \) for a canonical representative of the equivalence class \([\varphi ]\), and correlating these with the behavior of the multipliers of \(\varphi \). If one is given an arbitrary quadratic rational map \(\varphi \), it is possible to determine the unique point \(\xi \) in the crucial set of \(\varphi \) (as well as the reduction type of \(\varphi \) at \(\xi \)) by first conjugating to this canonical representative of \([\varphi ]\) and then applying the results in Sect. 3. The fact that \(\widetilde{\mathbf {s}([\varphi ])} \in \mathbb {A}^2(k)\) if and only if \(\varphi \) has potential good reduction had previously been shown by D. Yap in her thesis [14, Theorem 3.0.3]; our theorem may be considered a refinement of Yap’s result. One also notes the parallel between Theorem 1.1 and Milnor’s description [8] of degenerations of quadratic maps over \({\mathbb C}\) as they approach the boundary of moduli space.
In Sect. 4.2, we also observe that part (A) of Theorem 1.1—and hence also the result of D. Yap cited above—implies the following result, which resolves a special case of a conjecture of L.C. Hsia ([7, Conjecture 4.3]).
Proposition 1.2
Let \(\varphi \) be a quadratic rational map defined over K, let \(\mathcal {J}_\varphi (K) \subseteq \mathbb {P}^1(K)\) be the (classical) Julia set of \(\varphi \), and let \(\overline{\mathcal {R}_\varphi (K)}\) be the closure in \({\mathbb P}^1(K)\) of the set of type I repelling periodic points for \(\varphi \). Then \(\mathcal {J}_\varphi (K) = \overline{\mathcal {R}_\varphi (K)}\).
Our method of proof for Proposition 1.2 is special to quadratic rational maps, and unfortunately cannot be extended to maps of arbitrary degree.
There is reason to believe that the connection between the configuration of the crucial set of a map \(\varphi \) and the location of \([\varphi ]\) in moduli space should hold even in higher degrees. For example, Harvey, Milosevic, Rumely and Watson have given a classification theorem similar to Theorem 1.1 for cubic polynomials [6].
A significant difficulty which arises in higher degrees is that there are many more possible configurations for the crucial set: for a degree d map, there can be between 1 and \(d1\) points in the crucial set, and each point will exhibit one of four reduction types. One must also consider refinements to the reduction type that capture the geometric action of \(\varphi \) near a given point in the crucial set, and the relative locations of the different points in the crucial set. The number of configurations for a given d is clearly finite, but the problem of explicitly classifying the possible configurations for large d seems quite challenging. Nonetheless, the explicit connections in the case of quadratic rational maps and cubic polynomials suggest the possibility of a general theory linking the crucial set to the moduli space \(\mathcal {M}_d\) for all d.
1.1 Outline of the paper
In Sect. 2, we introduce notation and concepts used in the rest of the article. In particular, we give a more detailed explanation of the weights \(w_\varphi \) for maps of arbitrary degree and the conditions under which a point can have weight. In Sect. 3 we relate the reduction type of the unique weighted point \(\xi \) to the multipliers at the classical fixed points. For this, we rely on two normal forms for quadratic rational maps given in [12]. In Sect. 4 we apply our analysis to prove Theorem 1.1, and give the application to Hsia’s conjecture.
2 Notation and conventions
In this section we introduce terminology and notation used throughout the paper.
2.1 Berkovich space

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 \(a\in K\).

Points of type II correspond to nested, decreasing sequences of discs whose intersection is a disc \(D(a,r)\subseteq K\) with \(r\in K^\times \). For polynomials \(f\in K[T]\), we have \([f]_x = \sup _{z\in D(a,r)} f(z)\), and 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 \(D(a,r) \subseteq K\) with \(r\not \in K^\times \); as in the case of type II points, the corresponding seminorm is the supnorm on D(a, r). In this case, the supremum is not achieved in general.

Points of type IV correspond to nested, decreasing sequences of discs \(\{D(a_i, r_i)\}\) whose intersection is empty, but for which \(\lim _{i\rightarrow \infty } r_i >0\). Such points can occur only if the field K is not spherically complete.
The construction of \(\mathbf {P}^1_K\) from \(\mathbf {A}^1_K\) is similar to the construction of \(\mathbb {P}^1\) from \(\mathbb {A}^1\), gluing two copies of \(\mathbf {A}^1_K\) together by means of an involution of \(\mathbf {A}^1_K{\setminus } \{0\}\); see Sect. 2.2 of [1] for details. We write \(\mathbf {H}^1_K\) for \(\mathbf {P}^1_K\backslash {\mathbb P}^1(K)\), the ‘nonclassical’ part of \(\mathbf {P}^1_K\).
The Berkovich projective line is typically endowed with the BerkovichGel’fand topology, which is the weakest topology for which the map \(x\mapsto [f]_x\) is continuous for every \(f\in K[T]\). In this topology, \(\mathbf {P}^1_K\) is a compact Hausdorff space and is uniquely path connected. The points of type I are dense in \(\mathbf {P}^1_K\) for this topology; so are the points of type II. In general the BerkovichGel’fand topology is not metrizable.
A rational map \(\varphi \in K(z)\) induces a continuous action on \(\mathbb {P}^1(K)\) by means of a lift \(\Phi = [F,G]\), where \(F,G\in K[X,Y]\) are homogeneous polynomials of degree \(d = \deg (\varphi )\) such that \(\varphi (z) = \frac{F(z,1)}{G(z,1)}\). This action extends continuously to all of \(\mathbf {P}^1_K\), and preserves types of points. One can show that \({{\mathrm{PGL}}}_2(K)\) acts transitively on type II points, and that \({{\mathrm{PGL}}}_2(\mathcal {O})\) is the stabilizer of the Gauss point.
2.1.1 Tree structure
The Berkovich projective line has a canonical tree structure. The collection of points \(\{\zeta _{a,r}\}_{r\in [t,s]}\subseteq \mathbf {P}^1_K\) 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 \(\{\zeta _{a,r}\}_{r\in [t,s]}\); indeed the branches are in \(11\) correspondence with the elements of \({\mathbb P}^1(k)\) for the residue field k. To illustrate this, consider the type II point \(\zeta _G= \zeta _{D(0,1)}\). The branches off \(\zeta _G\) come from equivalence classes of paths \([\zeta _G,x]\) sharing a common initial segment; these classes correspond to subdiscs \(D(b,1)^ = \{x \in K : xb < 1\}\) where \(b \le 1\), and to the set \({\mathbb P}^1(K) \backslash D(0,1)\). Identifying D(0, 1) with the valuation ring \({\mathcal O}\), the subdiscs \(D(b,1)^ \) are just the cosets \(b + \mathfrak {m}\) in \(k = {\mathcal O}/\mathfrak {m}\), and \({\mathbb P}^1(K) \backslash D(0,1)\) corresponds to \(\widetilde{\infty } \in {\mathbb P}^1(k)\).
A more geometric way to think of the branches is in terms of tangent directions. Formally, a tangent direction \(\mathbf {v}\) 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 \(11\) correspondence with \(\mathbb {P}^1(k)\) 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 \(\mathbf {P}^1_K\). If P, Q are points of \(\mathbf {P}^1_K\) with \(\varphi (P) = Q\), there is a canonical induced surjective map \(\varphi _* : T_P \rightarrow T_Q\).
2.1.2 Reduction of rational maps
The action of \(\varphi \) on the tangent space \(T_P\) is closely related to the notion of the reduction of \(\varphi \), which we now describe. If [F, G] is a lift of \(\varphi \) that has been scaled so that the coefficients all lie in \(\mathcal {O}\), and so that at least one is a unit, we call [F, G] a normalized lift, or normalized representation, of \(\varphi \). Such a representation is unique up to scaling by a unit in \({\mathcal O}\). We can reduce each coefficient of a normalized lift [F, G] modulo \(\mathfrak {m}\). After removing common factors, we obtain a welldefined map \([\widetilde{F}: \widetilde{G}]\) on \(\mathbb {P}^1(k)\). This map, called the reduction of \(\varphi \) at \(\zeta _G\), is denoted \(\widetilde{\varphi }\).
It was shown by RiveraLetelier that a type II point \(P\in \mathbf {H}^1_K\) is fixed by \(\varphi \) if and only if the reduction \(\widetilde{\varphi }_P\) is nonconstant (see [1, Lemma 2.17]). RiveraLetelier calls a type II point P a repelling fixed point if \(\deg (\widetilde{\varphi }_P) \ge 2\), and he calls P an indifferent fixed point if \(\deg (\widetilde{\varphi }_P) = 1\). The third author [9, Definition 2] gave a refined classification of indifferent fixed points in \(\mathbf {H}^1_K\):
Definition 2.1

\(\widetilde{\varphi }_P(z) = \widetilde{\lambda }z\) for some \(\widetilde{\lambda }\in k{\setminus }\{\widetilde{0}, \widetilde{1}\}\), in which case we say P is a (Berkovich) multiplicatively indifferent fixed point for \(\varphi \).

\(\widetilde{\varphi }_P(z) = z+\widetilde{a}\) for some \(\widetilde{a}\in k{\setminus }\{\widetilde{0}\}\), in which case we say P is a (Berkovich) additively indifferent fixed point for \(\varphi \).

\(\widetilde{\varphi }_P(z) = z\), in which case we say P is an idindifferent fixed point for \(\varphi \).
One should think of each of the above reduction types as describing the behavior of the map \(\varphi _*\) acting on \(T_P\). More precisely, after conjugating \(\varphi \) by a suitable \(\gamma \in {{\mathrm{PGL}}}_2(K)\) we can assume that \(P = \zeta _G\) is fixed. Then \(\widetilde{\varphi }\) is a welldefined nonconstant map, and by making use of the identification \(T_P \cong \mathbb {P}^1(k)\), if \(\mathbf {v}_{\widetilde{a}}\in T_P\) corresponds to the point \(\widetilde{a}\in \mathbb {P}^1(k)\), then \(\varphi _*(\mathbf {v}_{\widetilde{a}}) = \mathbf {v}_{\widetilde{\varphi }(\widetilde{a})}\).
2.2 The crucial set
The crucial set was constructed in [9] and arose from the study of a certain function \({{\mathrm{ordRes}}}_\varphi : \mathbf {P}^1_K\rightarrow \mathbb {R}\cup \{\infty \}\). The function \({{\mathrm{ordRes}}}_\varphi (\cdot )\) had been introduced in [10] to address the problem of finding conjugates \(\varphi ^\gamma \) with minimal resultant. One obtains the crucial measure and crucial set by taking the graphtheoretic Laplacian of \({{\mathrm{ordRes}}}_\varphi (\cdot )\), restricted to a canonical tree \(\Gamma _{\mathrm {FR}}\subset \mathbf {P}^1_K\).
In this section, we briefly recall this construction.
2.2.1 The function \({{\mathrm{ordRes}}}_{\varphi }(x)\)
2.2.2 The crucial measures
The crucial measure is obtained by taking the graphtheoretic Laplacian of \({{\mathrm{ordRes}}}_\varphi (\cdot )\) on (a suitable truncation^{1} of) the tree \(\Gamma _{\mathrm {FR}}\). More precisely, if \(\mu _{Br}\) is the ‘branching measure’ which gives each \(P \in \Gamma _{\mathrm {FR}}\) the weight \(1\frac{1}{2} v(P)\), where v(P) is the valence of P in \(\Gamma _{\mathrm {FR}}\), then the crucial measure is defined as follows:
Definition 2.2
The crucial measure is canonically attached to \(\varphi \), because the function \({{\mathrm{ordRes}}}_\varphi (\cdot )\) and the tree \(\Gamma _{\mathrm {FR}}\) are canonical. It is a conjugation equivariant of \(\varphi \) in \(\mathbf {H}^1_K\), just as the sets of classical fixed points and critical points are conjugation equivariants in \({\mathbb P}^1(K)\).
Definition 2.3
 (A)
If P is a type II fixed point of \(\varphi \), then \(w_{\varphi }(P)= \deg (\widetilde{\varphi }_P)  1 + N_{\text {shearing}, \varphi }(P)\).
 (B)
If P is a branch point of \(\Gamma _\mathrm{{Fix}}\) which is moved by \(\varphi \) (necessarily of type II), then \(w_{\varphi }(P) = v(P)2\).
 (C)
Otherwise, \(w_{\varphi }(P) = 0\).
The fact that \(\nu _\varphi \) is a probability measure is equivalent to the following formula:
Theorem 2.4

\(\widetilde{\varphi }_\xi \) has degree 2: Here, \(\xi \) is a repelling fixed point for \(\varphi \), and \(w_\varphi (\xi ) > 0\) by part (A) of Definition 2.3. In this case, \(\varphi \) has potential good reduction.

\(\widetilde{\varphi }_\xi \) is conjugate to \(z\mapsto z+\widetilde{a}\) for some \(\widetilde{a}\in k{\setminus } \{\widetilde{0}\}\), and \(\xi \) is contained in \(\Gamma _{\mathrm {Fix}}\): Here, \(\xi \) is an additively indifferent fixed point with at least two tangent directions containing type I fixed points. Since \(\varphi _*\) fixes only one tangent direction, \(\xi \) must have a shearing direction, hence \(w_\varphi (\xi ) > 0\) by Definition 2.3 (A). In this case, \(\varphi \) has potential additive reduction. (Note that since \(w_\varphi (\xi ) = 1\), \(\xi \) can have only one shearing direction; thus \(\xi \) cannot be a branch point of \(\Gamma _{\mathrm {Fix}}\).)

\(\widetilde{\varphi }_\xi \) is conjugate to \(z\mapsto \widetilde{\lambda } z\) for some \(\widetilde{\lambda }\in k{\setminus } \{\widetilde{0}, \widetilde{1}\}\), and \(\xi \) is a branch point of \(\Gamma _{\mathrm {Fix}}\): Here, \(\xi \) is a multiplicatively indifferent fixed point, but now \(\xi \) has three tangent directions containing type I fixed points. Since \(\varphi _*\) fixes only two tangent directions, \(\xi \) must have a shearing direction, so \(w_\varphi (\xi ) > 0\)—once again, by part (A) of Definition 2.3. In this case, \(\varphi \) has potential multiplicative reduction

\(\widetilde{\varphi }_\xi \) is constant, and \(\xi \) is a branch point of \(\Gamma _{\mathrm {Fix}}\): That \(w_\varphi (\xi ) > 0\) follows immediately from part (B) of Definition 2.3. In this case, \(\varphi \) has potential constant reduction.

Let \(\varphi (z) = z(zF_1)(zF_2) + z\), where \(1< F_2\) and \(1/F_2< F_1<F_2\). Then the points \(\zeta _{D\left( 0, F_1\right) }\) and \(\zeta _{D\left( 0, F_2\right) }\) are branch points of the tree \(\Gamma _{\text {Fix}}\) with valence 3 which are moved by \(\varphi \). Thus, \(w_\varphi (\zeta _{D\left( 0, F_1\right) })= w_\varphi (\zeta _{D\left( 0, F_2\right) }) = 32 = 1\). By the weight formula (2), these are the only points in the crucial set.

Let \(\varphi (z) = z(zF_1)(zF_2)+z\) be as above, where \(1<F_2\) but now \(F_1 \le 1/F_2\). Then \(\zeta _{D\left( 0, F_2\right) }\) and \(\zeta _{D\left( 0, 1/F_2\right) }\) are the points in the crucial set, each of weight 1: The point \(\zeta _{D\left( 0, F_2\right) }\) is a branch point of \(\Gamma _{\text {Fix}}\) with valence 3 that is moved by \(\varphi \), hence its weight is \(w_\varphi (\zeta _{D\left( 0, F_2\right) }) = 32 = 1\). The point \(\zeta _{D\left( 0, 1/F_2\right) }\), however, is fixed by \(\varphi \), but has no shearing—here, the reduction of \(\varphi \) at \(\zeta _{D\left( 0, 1/F_2\right) }\) is a quadratic polynomial. Thus \(w_\varphi \left( \zeta _{D\left( 0, 1/F_2\right) }\right) = 1\) as well.
2.3 The moduli space of quadratic rational maps
Using geometric invariant theory, Silverman [13, Theorem 1.1] constructed the moduli space \(\mathcal {M}_d\) for rational maps of degree \(d \ge 2\) as a scheme over \(\mathbb {Z}\).
If \(\alpha \in K\) is a fixed point for the rational map \(\varphi \), the derivative \(\varphi ^{\prime }(\alpha )\) is called the multiplier of \(\varphi \) at \(\alpha \). It is wellknown that the multiplier is independent of the choice of coordinates, which means the multiplier at \(\varphi \) at \(\infty \in {\mathbb P}^1(K)\) can be defined by changing coordinates.
Silverman showed [13, Theorem 5.1] there is a natural isomorphism \(\mathbf {s}: \mathcal {M}_2 \rightarrow \mathbb {A}^2\) as schemes over \(\mathbb {Z}\). More precisely, he showed that the first and second elementary symmetric functions \(\sigma _1\), \(\sigma _2\) of the multipliers at the fixed points give coordinates on \(\mathcal {M}_2\).
3 The crucial sets of quadratic maps
The behavior of a rational map \(\varphi \) near a classical fixed point \(\alpha \in {\mathbb P}^1(K)\) is governed by the multiplier at \(\alpha \). In this section, we explicitly describe the crucial set for quadratic rational maps in terms of the multipliers at the classical fixed points.
3.1 Maps with a multiple fixed point
Proposition 3.1
 (A)
If \(\lambda _3 \le 1\), then \(\xi =\zeta _G\) and \(\varphi \) has (potential) good reduction.
 (B)
If \(\lambda _3 > 1\), then \(\xi =\zeta _{D\left( 0, \sqrt{\lambda _3}\right) }\) and \(\varphi \) has potential additive reduction.
Proof
3.2 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:
Lemma 3.2
Proof
Lemma 3.3
 (A)
if \(\varphi \) has two classical repelling fixed points, then the third is attracting;
 (B)
if \(\varphi \) has only one classical repelling fixed point, then the other two are indifferent;
 (C)
if \(\varphi \) has no classical repelling fixed points, then either some pair of multipliers satisfies \(\widetilde{\lambda _i\lambda _j} \ne \widetilde{1}\), or else \(\widetilde{\lambda _1} = \widetilde{\lambda _2} = \widetilde{\lambda _3} = \widetilde{1}\).
Proof
To show (C), suppose that \(\lambda _1, \lambda _2, \lambda _3 \le 1\). If some pair of multipliers satisfies \(\widetilde{\lambda _i\lambda _j} \ne \widetilde{1}\), we are done. Otherwise \(\widetilde{\lambda _1\lambda _2} = \widetilde{\lambda _1\lambda _3} = \widetilde{\lambda _2\lambda _3} = \widetilde{1}\). Considering these equalities in pairs, we conclude there is a \(\widetilde{c} \in k\) such that \(\widetilde{\lambda }_1 = \widetilde{\lambda }_2 = \widetilde{\lambda }_3 = \widetilde{c}\). In particular, we have \(\widetilde{c}^2 = \widetilde{1}\), so \(\widetilde{c} \in \{\pm \widetilde{1}\}\). If \(\widetilde{c} = \widetilde{1}\) (which is necessarily true if \(\mathrm{{char}}(k) = 2\)), then we are done, so assume that \(\mathrm{{char}}(k) \ne 2\) and \(\widetilde{c} = \widetilde{1}\). In this case, reducing (4) modulo \(\mathfrak {m}\) yields the equation \(\widetilde{\left( 3/2\right) } = \widetilde{1}\), which implies that \(\widetilde{2} = \widetilde{3}\), a contradiction. Hence \(\widetilde{\lambda }_1 = \widetilde{\lambda }_2 = \widetilde{\lambda }_3 = \widetilde{1}\). \(\square \)
Furthermore, if \(\varphi \) has no repelling classical fixed points, i.e., if \(\lambda _1, \lambda _2, \lambda _3 \le 1\), then by Lemma 3.3 either \(\widetilde{\lambda }_1 = \widetilde{\lambda }_2 = \widetilde{\lambda }_3 = \widetilde{1}\) or, by permuting the fixed points via an additional conjugation if necessary, \(\widetilde{\lambda _1\lambda _2} \ne \widetilde{1}\). On the other hand, if \(\varphi \) has a repelling classical fixed point, then by Lemma 3.3 it also has a nonrepelling classical fixed point; permuting the fixed points via a conjugation of \(\varphi \) if necessary, we can assume that \(\lambda _1 > 1 \ge \lambda _2\).
Proposition 3.4
 (A)If \(\varphi \) has no repelling classical fixed points, then \(\varphi \) has (potential) good reduction. Permuting the indices via an additional conjugation of \(\varphi \) if necessary, we can assume that either \(\widetilde{\lambda _1 \lambda _2} \ne 1\), or that \(\widetilde{\lambda }_1 = \widetilde{\lambda }_2 = \widetilde{\lambda }_3 = \widetilde{1}\). With this ordering of the indices, we have
 (i)
if \(\widetilde{\lambda _1\lambda _2} \ne \widetilde{1}\), then \(\xi = \zeta _G;\)
 (ii)
if \(\widetilde{\lambda }_1 = \widetilde{\lambda }_2 = \widetilde{\lambda }_3 = \widetilde{1}\), then \(\xi = \zeta _{D\left( 1,\sqrt{\lambda _1\lambda _2  1}\right) }\).
 (i)
 (B)Suppose \(\varphi \) has at least one repelling classical fixed point, hence also a nonrepelling fixed point by Lemma 3.3. Permuting the indices via an additional conjugation of \(\varphi \) if necessary, we can assume that \(\lambda _1 > 1 \ge \lambda _2\). With this ordering of the indices, we find that \(\xi = \zeta _{D(0,\lambda _1)}\). Moreover,
 (i)
if \(\widetilde{\lambda _2} \not \in \{\widetilde{0},\widetilde{1}\}\), then \(\varphi \) has potential multiplicative reduction.
 (ii)
if \(\widetilde{\lambda _2} = \widetilde{1}\), then \(\varphi \) has potential additive reduction.
 (iii)
if \(\widetilde{\lambda _2} = \widetilde{0}\), then \(\varphi \) has potential constant reduction.
 (i)
Remark
Proof
First, assume that \(\varphi \) has no classical repelling fixed points, which means that each of the multipliers lies in \(\mathcal {O}\). In particular, this implies that the expression for \(\varphi \) given in the proposition is already normalized. Since \({{\mathrm{Res}}}(\Phi ) = 1  \lambda _1\lambda _2\), we see that if \(\widetilde{\lambda _1\lambda _2} \ne \widetilde{1}\), then \({{\mathrm{Res}}}(\Phi ) = 1  \lambda _1\lambda _2 = 1\), so \(\varphi \) has good reduction at \(\zeta _G\), proving (A)(i).
Now suppose that \(\widetilde{\lambda }_1 = \widetilde{\lambda }_2 = \widetilde{\lambda }_3 = \widetilde{1}\). This means that \(\lambda _1\lambda _2  1 < 1\). Set \(\rho := \sqrt{\lambda _1\lambda _2  1}\), and let \(r := \rho  < 1\). Put \(\gamma (z) := \rho z  1\), so that \(\gamma (\zeta _G) = \zeta _{D(1,r)}\). To prove (A)(ii), it suffices to show that \(\varphi ^{\gamma }\) has good reduction.

If \(\widetilde{\lambda _2} \not \in \{\widetilde{0},\widetilde{1}\}\), then \(\widetilde{\varphi ^{\gamma }}\) is conjugate to the map \((\widetilde{1}/\widetilde{\lambda _2})z\) via \(z \mapsto z + \widetilde{1}/(\widetilde{\lambda _2}  \widetilde{1})\); hence \(\zeta _{D(0,\lambda _1)}\) is a multiplicatively indifferent fixed point for \(\varphi \). Moreover,so \(\zeta _{D(0,\lambda _1)} = \zeta _{D(0,\alpha _3)}\) is a branch point of \(\Gamma _{\mathrm {Fix}}\). We conclude that \(\varphi \) has potential multiplicative reduction, thereby proving (B)(i).$$\begin{aligned} \alpha _3 = \frac{\lambda _1  1}{\lambda _2  1} = \lambda _1, \end{aligned}$$

If \(\widetilde{\lambda _2} = \widetilde{1}\), then \(\widetilde{\varphi ^{\gamma }}(z) = z + \widetilde{1}\), thus \(\zeta _{D(0,\lambda _1)}\) is an additively indifferent fixed point for \(\varphi \). Moreover, since\(\zeta _{D(0, \lambda _1)}\) lies on the segment \([0,\zeta _{D\left( 0,\alpha _3\right) }] \subset [0,\alpha _3] \subset \Gamma _{\mathrm {Fix}}\). Therefore \(\varphi \) has potential additive reduction, proving (B)(ii).$$\begin{aligned} \alpha _3 = \frac{\lambda _1  1}{\lambda _2 1} > \lambda _1, \end{aligned}$$

Finally, if \(\widetilde{\lambda _2} = \widetilde{0}\), then \(\widetilde{\varphi ^{\gamma }}\) is the constant map \(\widetilde{\infty }\). This means that \(\zeta _{D(0,\lambda _1)}\) is not a fixed point under \(\varphi \). Arguing just as we did for (B)(i), we see that \(\zeta _{D(0,\lambda _1)}\) is the branch point for \(\Gamma _{\mathrm {Fix}}\). Therefore \(\varphi \) has potential constant reduction, which proves (B)(iii). \(\square \)
4 Applications to moduli space
4.1 Proof of the main theorem
Proof of Theorem 1.1
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 \(\varphi \in K(z)\), let \(\mathcal {J}_\varphi (K)\) denote the (classical) Julia set of \(\varphi \), which is the complement of the equicontinuity locus (in \(\mathbb {P}^1(K)\)) of the family of iterates \(\{\varphi ^n : n \in \mathbb {N}\}\) ([11, §5.4]). Let \(\mathcal {R}_\varphi (K)\) denote the set of all classical repelling periodic points for \(\varphi \), and let \(\overline{\mathcal {R}_\varphi (K)}\) be its closure in \({\mathbb P}^1(K)\).
It is known over the complex numbers that \(\mathcal {J}_\varphi ({\mathbb C}) = \overline{\mathcal {R}_\varphi ({\mathbb C})}\); the analagous result is not known when K is nonArchimedean, though it is conjectured to be true:
Conjecture 4.1
(Hsia [7, Conjecture 4.3]) Let \(\varphi \) be a rational function defined over a nonArchimedean field with \(\deg \varphi \ge 2\). Then \(\mathcal {J}_\varphi (K) = \overline{\mathcal {R}_\varphi (K)}\).
This conjecture has not yet been resolved in general. Using part (A) of Theorem 1.1—which we recall is Theorem 3.0.3 in Yap’s thesis [14]—we show that Hsia’s conjecture holds for a quadratic rational map over a complete, algebraically closed nonArchimedean field:
Proposition 1.2
Let \(\varphi \) be a quadratic rational map defined over K. Then \(\mathcal {J}_\varphi (K) = \overline{\mathcal {R}_\varphi (K)}\).
Proof
We separate the proof based on whether or not \(\varphi \) has potential good reduction.
If \(\varphi \) has potential good reduction, the Berkovich Julia set is a single point in \(\mathbf {H}^1_K\) (see [4, Proposition 0.1]); thus \(\varphi \) has no type I repelling periodic points (such would necessarily be Julia), hence \(\mathcal {J}_\varphi (K)= \emptyset = \overline{\mathcal {R}_\varphi (K)}\) as desired.
If \(\varphi \) does not have potential good reduction, then by Theorem 1.1 the image \(\mathbf s ([\varphi ]) \in {\mathbb A}^2(K) \subset {\mathbb P}^2(K)\) cannot specialize to \(\mathbb {A}^2(k)\). It follows that \(\varphi \) must have a type I repelling fixed point, for if all of its type I fixed points were nonrepelling then the symmetric functions in their multipliers would lie in \(\mathcal {O}\). By a theorem of Bézivin ([3, Théorème 3]), this implies that \(\mathcal {J}_\varphi (K) = \overline{\mathcal {R}_\varphi (K)}\). \(\square \)
The argument above shows that for quadratic maps, if \(\varphi \) does not have potential good reduction, then \(\varphi \) must have a classical repelling fixed point. Such an argument cannot be used to prove Hsia’s conjecture for \(\varphi \) of arbitrary degree, since there are examples (due to Favre and RiveraLetelier [5]) of functions \(\varphi \) whose classical Julia set is empty, but whose Berkovich Julia set is a segment contained in \(\mathbf {H}^1_K\) (for instance, Lattès maps associated to Tate elliptic curves; see [1, Example 10.124]). Such \(\varphi \) do not have good reduction, but neither they nor their iterates can have classical repelling fixed points, since such points would belong to the Julia set.
Declarations
Open Access
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Acknowledgements
The research for this article was begun during an NSFsponsored VIGRE research seminar on dynamics on the Berkovich line at the University of Georgia, led by the third author. Funding was provided by NSF Grant DMS0738586. 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. We also thank the anonymous referees for suggestions concerning exposition.
Authors’ Affiliations
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