Discreteness of Postcritically Finite Maps in p-Adic Moduli Space

DISCRETENESS OF POSTCRITICALLY FINITE MAPS IN p-ADIC MODULI SPACE Abstract. Let p ≥ 2 be a prime number and let Cp be the completion of an algebraic closure of the p-adic rational field Qp . Let fc (z) be a one-parameter family of rational functions of degree d ≥ 2, where the coefficients are meromorphic functions defined at all parameters c in some open disk D ⊆ Cp . Assuming an appropriate stability condition, we prove that the parameters c for which fc is postcritically finite (PCF) are isolated from one another in the p-adic disk D, except in certain trivial cases. In particular, all PCF parameters of the family fc (z) = z d + c are p-adically isolated. 1. Introduction Let K be a field with algebraic closure K, and let f (z) ∈ K(z) be a separable rational function. The degree of f is defined to be the maximum of the degrees of the numerator and denominator of f . Define f n (z) to be the n-th iterate of f under composition; that is, f 0 (z) = z, and for each n ≥ 0, f n+1 (z) = f ◦ f n (z). The forward orbit of a point x ∈ P1 (K) is the set {f n (x) : n ≥ 0}; the strict forward orbit of x is {f n (x) : n ≥ 1}. We say x is periodic under f if f n (x) = x for some n ≥ 1; equivalently, x belongs to its own strict forward orbit. Similarly, we say that x is preperiodic under f if there are integers n > m ≥ 0 such that f n (x) = f m (x); equivalently, the forward orbit of x is finite. We say x is strictly preperiodic if it is preperiodic but not periodic. We say that f is postcritically finite, or PCF, if every critical point c ∈ P1 (K) of f is preperiodic under f . Let fc (z) be a one-parameter analytic family of rational functions of degree d. That is, each coefficient of fc is an analytic function of c ∈ D, where D is some set of parameters; and for each specific parameter c ∈ D, the function fc (z) ∈ K(z) is a rational function of degree d. In this paper, we consider the set of parameters c ∈ D for which fc is PCF; we call such points PCF parameters. In particular, our focus is on the case that K = Cp , the completion of an algebraic closure of the field Qp of p-adic rationals, where p ≥ 2 is a prime number. There are a number of cases when it is easy to see that there are many PCF parameters. First, suppose that fc = h−1 c ◦g ◦hc , where g(z) ∈ K(z) is a (fixed) rational function of degree d ≥ 2, and where hc (z) is a one-parameter family of linear fractional transformations. This is the isotrivial case, and if g(z) itself is PCF, then so is fc for every parameter c. Second, suppose that Ec is a one-parameter family of elliptic curves, m ≥ 2 is an integer, and fc is the flexible Lattès map associated with the multiplication-by-m map [m] : Ec → Ec . (That is, if we write Ec in Weierstrass form, then fc ◦ xc = xc ◦ [m], where xc : Ec → P1 is the x-coordinate map. See Definition 4.1 for a slightly more general definition of flexible Lattès maps.) It is well known, and easy to check, that all Lattès maps are PCF, and hence every c is a PCF parameter for such a family. As a third, more complicated case, let K = C, and consider the quadratic polynomial family fc (z) = z 2 + c ∈ C[c, z]. Since the critical point at z = ∞ is fixed, a given map fc is PCF if and only if the critical point at z = 0 is preperiodic. The parameters c for which z = 0 is strictly preperiodic are called Misiurewicz parameters, and they are dense in the boundary ∂M of the Mandelbrot set M. (See, for example, [7, Theorem VIII.1.5ff] or [11, Proposition 2.1]. The parameters c for which z = 0 is periodic lie in the interior of M but also accumulate on ∂M.) Hence, the complex quadratic polynomial family features PCF parameters accumulating at many points, including one another. (This phenomenon is not unique to the complex case K = C. Indeed, Rivera-Letelier described similar Misiurewicz bifurcations in the p-adic setting in [14]; see Example 7.3.) By contrast, our main result, Theorem 1.1 below, states that if K = Cp and the family fc is stable in an appropriate sense, then such accumulation of PCF parameters is impossible. The underlying flavor of this statement is similar to that of Scanlon’s theorem [15, 16] on the Tate-Voloch conjecture [19], although our methods are different. See Section 2.1 for the notation D(b, R) and D(b, R) for open and closed disks in Cp , and Section 2.2 for the field Mp (D) of p-adic meromorphic functions on a disk D ⊆ Cp ; in addition, P1 (Mp (D)) of course denotes Mp (D) ∪ {∞}. Theorem 1.1. Let p ≥ 2 be a prime number, let S > 0, let Φ(c, z) ∈ Mp (D(0, S))(z), and suppose that for each c ∈ D(0, S), fc (z) := Φ(c, z)
is a rational function in Cp (z) of degree d ≥ 2. Let α1 , . . . , α2d−2 ∈ P1 Mp (D(0, S)) be meromorphic functions on D(0, S) such that for each c ∈ D(0, S), the critical points of fc , repeated according to multiplicity, are α1 (c), . . . , α2d−2 (c). For each i = 1, . . . , 2d − 2, suppose that there are integers Ni > Mi ≥ 0 and rational open disks Ui,0 , . . . , Ui,Ni ⊆ P1 (Cp ) such that for all c ∈ D(0, S), • αi (c) ∈ Ui,0 , • fc (Ui,j ) ⊆ Ui,j+1 for all j = 0, . . . , Ni − 1, and • Ui,Ni ⊆ Ui,Mi . Then either (a) for any 0 < s < S, there are only finitely many c ∈ D(0, s) such that fc is postcritically finite, (b) for every c ∈ D(0, S), fc is a flexible Lattès map, or (c) for every b, c ∈ D(0, S), fb is conjugate to fc . The stability conditions of Theorem 1.1, i.e., the bullet points concerning the disks Ui,j , happen to be satisfied by certain families of good reduction, which we now define. Recall that a non-archimedean field Kv with absolute value | · |v has ring of integers Ov := {x ∈ Kv : |x|v ≤ 1}, maximal ideal Mv := {x ∈ Kv : |x|v < 1}, and residue field kv := Ov /Mv . (For Kv = Cp , the residue field is isomorphic to Fp , the algebraic closure of the field Fp of p elements.) Definition 1.2. Let Kv be a non-archimedean field with absolute value | · |v , ring of integers Ov , and residue field kv , and let f (z) ∈ Kv (z) be a rational function of degree d ≥ 1. Write f = g/h, where g, h ∈ Ov [z], with at least one coefficient a of g or h satisfying |a|v = 1. Define g, h ∈ kv [z] by reducing each coefficient of g, h modulo v. We say f has (explicit) good reduction if its reduction, the rational function f := g/h ∈ kv (z), satisfies deg(f ) = d. The reduction map from Ov → kv induces a map P1 (Kv ) → P1 (kv ), which we also denote x 7→ x. If f ∈ Kv (z) has good reduction, then f (x) = f (x) for every x ∈ P1 (Kv ). Theorem 1.3. Let p ≥ 2 be a prime number, let S > 0, let Φ(c, z) ∈ Mp (D(0, S))(z),
1 and let α1 , . . . , α2d−2 ∈ P Mp (D(0, S)) . Suppose that for each c ∈ D(0, S), • fc (z) := Φ(c, z) is a rational function in Cp (z) of degree d ≥ 2 and of p-adic explicit good reduction; • the reduction f c ∈ Fp (z) satisfies f c = f 0 ; • the critical points of fc , repeated according to multiplicity, are α1 (c), . . . , α2d−2 (c); and • for each i = 1, . . . , 2d − 2, the reduction αi (c) ∈ P1 (Fp ) satisfies αi (c) = αi (0). Then either (a) for any 0 < s < S, there are only finitely many c ∈ D(0, s) such that fc is postcritically finite, (b) for every c ∈ D(0, S), fc is a flexible Lattès map, or (c) for every b, c ∈ D(0, S), fb is conjugate to fc . As a special case of Theorem 1.3, we can show that the unicritical family z d + c over Cp has all PCF parameters isolated. Here is a more precise statement. Theorem 1.4. Let p ≥ 2 be a prime number, and let d ≥ 2 be an integer. Define fc (z) := z d + c. If c ∈ Cp is a parameter for which fc is postcritically finite, then |c|p ≤ 1. Moreover, for any a ∈ Cp with |a|p ≤ 1 and for any radius 0 < s < 1, there are only finitely many c ∈ D(a, s) such that fc is postcritically finite. As we will see in Examples 7.1 and 7.2, Theorem 1.4 cannot be strengthened to conclude that there are only finitely many PCF parameters in an open disk D(a, 1) of radius 1. An application of Theorem 1.4, towards a result on the integrality of PCF parameters in the unicritical family, will appear in [3]. The outline of the paper is as follows. In Section 2, we set notation and recall the essential background we need, especially on non-archimedean analysis. In Sections 3 and 4, we prove technical lemmas needed to avoid degenerations of isotrivial and Lattès families. Section 5 is devoted to the statement and proof of Theorem 5.1, which is our central tool, using certain p-adic dynamical results to study the orbit of a single marked point in a one-parameter p-adic family. In Section 6, we use Theorem 5.1 and Thurston Rigidity (stated here as Theorem 6.1) to prove Theorems 1.1, 1.3, and 1.4. Finally, in Section 7, we present various examples showing that the conclusions of our theorems cannot be significantly strengthened, and that the stability conditions cannot be dropped from their hypotheses. 2. Notation and background In this section, we recall some basic facts from non-archimedean analysis, following [2, Chapters 2–3]. Throughout the rest of the paper, for ease of notation, we fix a prime number p ≥ 2, and we denote the absolute value on Cp simply as | · | rather than | · |p . 2.1. Disks and power series. Fix b ∈ Cp and R > 0. We write D(b, R) := {x ∈ Cp : |x − b| < R} and D(b, R) := {x ∈ Cp : |x − b| ≤ R} for the open and closed disks, respectively, centered at b of radius R. If R ∈ |C× p |, then we call D(b, R) a rational open disk, and D(b, R) a rational closed disk, which satisfy D(b, R) ( D(b, R). If R 6∈ |C× p |, then we call D(b, R) = D(b, R) an irrational disk. In spite of the names, all disks in Cp are open and closed topologically. In addition, for any b0 ∈ D(b, R), we have D(b0 , R) = D(b, R). Similarly, for any b0 ∈ D(b, R), we have D(b0 , R) = D(b, R). We also define a disk D in P1 (Cp ) = Cp ∪ {∞} to be either a disk in Cp or the complement P1 (Cp ) r D0 of a disk D0 ⊆ Cp . In the latter case, if D0 is rational closed (resp., rational open, irrational), we say D is rational open (resp., rational closed, irrational). Let ∞ X f (z) = an (z − b)n ∈ Cp [[z − b]] n=0 be a power series with coefficients in Cp . Then f converges on the open disk D(b, R) if and only if the sequence {|an |rn }n≥0 is bounded for every 0 < r < R; equivalently, if and only if limn→∞ |an |rn = 0 for every 0 < r < R. In that case, the Weierstrass degree of f on D(b, R) is the smallest integer m ≥ 0 such that  (2.1) |am |Rm = max |an |Rn : n ≥ 0 , or ∞ if the maximum is not attained. If the Weierstrass degree of f − a0 is an integer m ≥ 1, then f maps D(b, R) onto D(a0 , S), where S := |am |Rm ; moreover, every point of D(a0 , S) has exactly m preimages in D(b, R), counted with multiplicity. If R ∈ |C× p |, then f converges on the rational closed disk D(b, R) if and only if limn→∞ |an |Rn = 0. In that case, the Weierstrass degree of f on D(0, R) is the greatest integer m ≥ 0 satisfying equation (2.1). As with open disks, f maps D(b, R) onto D(a0 , S), where S = |am |Rm and m is the Weierstrass degree of f − a0 on D(b, R); moreover, every point of D(a0 , S) has exactly m preimages in D(b, R), counted with multiplicity. In particular, a convergent power series f can have only finitely many zeros on a rational closed disk unless f is the trivial power series, i.e., is identically zero. Since any open disk is a union of countably many nested rational closed disks, then, a nontrivial convergent power series can have only countably many zeros on an open disk. We will also occasionally need to consider power series F (z, w) ∈ Cp [[z, w]] in two variables. The main fact we will need is the following. Writing such a series as F (x, y) = P i j i,j≥0 Aij x y , suppose that F converges for all (x, y) in the bidisk D(0, r) × D(0, s), with image contained in the disk D(0, t), with r, s, t > 0. Then we have |Aij |ri sj ≤ t for all i, j ≥ 0, with |A00 | < t. 2.2. Meromorphic functions. If φ ∈ Cp (z) is a rational function of degree d ≥ 1 with no poles in a given disk D ⊆ Cp — either open or closed — then φ has a power series expansion that converges on D. Moreover, the Weierstrass degree of this power series is at most d. (The existence of such a power series holds for the exact same reasons as over C. One can add, subtract, and multiply convergent power series, and onePcan compute reciprocals of power series with no zeros using the identity 1/(1 − z) = zn for |z| < 1.) More generally, we make the following definition. Definition 2.1. Let D ⊆ Cp be a disk, either open or closed. A function f : D → P1 (Cp ) is a (rigid ) meromorphic function on D if there exist power series g, h converging on D such that f = g/h. We denote the set of all meromorphic functions on D by Mp (D). Just as for C, the set Mp (D) is a field under addition and multiplication of functions, and we may speak of zeros and poles of meromorphic functions. We will also occasionally abuse terminology and refer to the constant function ∞ as being meromorphic; that is, we consider every element of the projective line P1 (Mp (D)) = Mp (D) ∪ {∞} to be a meromorphic function on D. A meromorphic function on D that is not identically 0 or ∞ has only finitely many zeros and poles in any proper subdisk of D; in particular, it has only countably many zeros and poles in D, all of which are isolated. A meromorphic function with no poles in D is in fact given by a power series converging on D. If two meromorphic functions f, g ∈ Mp (D) agree on an uncountable subset of D, then f = g. Many of the above facts (in both Sections 2.1 and 2.2) are consequences of the Weierstrass Preparation Theorem (see, for example, [2, Theorem 14.2]). This result states that a nontrivial power series f ∈ Cp [[z − b]] which converges on a rational closed disk D = D(b, R) can be factored uniquely as f = gh, where g ∈ Cp [z] is a monic polynomial g with all of its roots in D, and where h ∈ Cp [[z − b]] converges on D with |h(z)| a nonzero constant on D. (Moreover, the degree of g is the Weierstrass degree of f on D.) We also have the following fact. Lemma 2.2. Let f (z) be a nontrivial power series converging on D(0, S), let 0 < s < S, and let y1 , . . . , ym be the zeros of f in D(0, s), where 0 ≤ m < ∞. Then for any r > 0, there exists ε > 0 so that m [ |f (x)| ≥ ε for all x ∈ D(0, s) r D(yi , r). i=1 Proof. Increasing s slightly if necessary, we may assume without loss that s ∈ |C× p |. The conclusions are then immediate from the Weierstrass Preparation Theorem.  P 2.3. Norm and distortion. Let h(z) = n≥0 an z n ∈ Cp [[z]] be a power series converging on D(0, R). Then for any radius 0 < r < R, we define khkζ(0,r) to be the sup-norm of h on D(0, r). That is,  (2.2) khkζ(0,r) := sup |h(x)| : x ∈ D(0, r) = |ad |rd , where d is the Weierstrass degree of h on D(0, r). (The notation k·kζ(0,r) is meant to evoke the fact that this norm is a point ζ(0, r) on the Berkovich disk Dan (0, R) corresponding to the disk D(0, r).) If we define a function Lh : (−∞, log R) → R by Lh (log r) := log khkζ(0,r) , then Lh is continuous and piecewise linear, with nonnegative integer slopes. Specifically, the slope of Lh just to the right of log r is the Weierstrass degree of h on D(0, r). (The graph of Lh is essentially the Newton copolygon of h.) As in [1, Section 4], we also define a quantity δ(h, ζ(0, r)), which we call the distortion of the power series h on the disk D(0, r), by (2.3) δ(h, ζ(0, r)) := log r + log kh0 kζ(0,r) − log khkζ(0,r) . Then log r 7→ δ(h, ζ(0, r)) is again a continuous, real-valued, piecewise linear function on (−∞, log R) with (possibly negative) integer slopes, since both h and h0 are power series converging on D(0, R). Indeed, if h0 and h have Weierstrass degrees ` and m, respectively, on D(0, r), then the slope of this function just to the right of log r is 1 + ` − m. 3. Isotrivial families As noted in the introduction, a family fc of rational maps is said to be isotrivial if it is conjugate to a constant map g. That is, g ∈ Cp (z) is a rational function of degree d, and fc = h−1 c ◦ g ◦ hc , where hc is a one-parameter family of linear fractional transformations. We will need the following lemma to identify isotrivial families when they arise in proving our main results. Lemma 3.1. Let S > 0, let g(z) ∈ Cp (z) be a rational function of degree d ≥ 2, and let
1 fc (z) ∈ Mp (D(0, S))(z). Let β1 , β2 , β3 ∈ P Mp (D(0, S)) be three distinct meromorphic functions on D(0, S), and let γ1 , γ2 , γ3 be three distinct points in P1 (Cp ). Suppose that for every c ∈ D(0, S), fc is a rational function in Cp (z) of degree d. Suppose also that for some uncountable subset W ⊆ D(0, S) and for every c ∈ W , there is a linear fractional transformation hc ∈ PGL(2, Cp ) such that fc = h−1 c ◦ g ◦ hc and hc (βi (c)) = γi for i = 1, 2, 3. Then fc is conjugate to g for all c ∈ D(0, S). Proof. Step 1. After a change of coordinates, we may assume without loss that γ1 = 0, γ2 = ∞, and γ3 = 1. We may similarly assume that none of β1 , β2 , β3 is identically ∞. Let Y be the set of parameters c ∈ D(0, S) for which at least two of β1 (c), β2 (c), β3 (c) have the same value. Define


z − β1 (c) β3 (c) − β2 (c)

∈ PGL 2, Mp (D(0, S)) , (3.1) h̃c (z) := z − β2 (c) β3 (c) − β1 (c) and let f˜c := h̃−1 c ◦ g ◦ h̃c ∈ Mp (D(0, S))(z). Then for each c ∈ D(0, S) r Y , we have h̃c ∈ PGL(2, Cp ). Moreover, for all c in the uncountable set W ⊆ D(0, S) r Y , we have h̃c = hc , and hence f˜c = fc for such c. Since the coefficients of both f˜c and fc are meromorphic functions on D(0, S) that agree for all c ∈ W , we in fact have f˜c = fc as elements of Mp (D(0, S))(z), and hence f˜c = fc for all c ∈ D(0, S). Thus, it suffices to show that Y is empty. Step 2. Suppose, towards a contradiction, that there is some c0 ∈ Y . Since the meromorphic functions βi − βj are nontrivial for all i 6= j, all points of Y are isolated. Translating by c0 in the c-variable, we may assume that c0 = 0. Let L := Cp ((c)), the field of formal Laurent series over Cp , which is a complete nonarchimedean field with respect to the valuation ord0 , where ord0 (α) denotes the order of b be vanishing of α ∈ L at c = 0. Let OL := Cp [[c]] be the ring of integers in L, and let L the completion of an algebraic closure of L. Then fc (z) is a rational function in L(z) of degree d, and by hypothesis, f0 (z) is a rational function in Cp (z) of the same degree d. However, f0 (z) is precisely the reduction of fc with respect to the valuation ord0 ; thus, fc ∈ L(z) has explicit good reduction with respect to ord0 , in the sense of Definition 1.2. On the other hand, g(z) ∈ L(z) also has explicit good reduction with respect to ord0 (since all its coefficients are constants in Cp ), and fc , g ∈ L(z) are conjugate via h̃c ∈ PGL(2, L). By Propositions 8.12 and 8.13 of [2], then, we must have h̃c ∈ PGL(2, OL ). (See also [13, Théorème 4]. In the language of Berkovich spaces, the explicit good reduction of fc is equivalent to saying that the Gauss point ζ(0, 1) in the Berkovich b is invariant under fc , by [2, Proposition 8.12]. However, the fact projective line over L −1 that g = h̃c ◦ fc ◦ h̃c has explicit good reduction says that h̃−1 c (ζ(0, 1)) is also invariant under fc ; but there can only be one such invariant point, by [2, Proposition 8.13]. Thus, h̃c (ζ(0, 1)) = ζ(0, 1), which is equivalent to saying h̃c ∈ PGL(2, OL ), by [2, Proposition 4.10(b)].) The fact that h̃c ∈ PGL(2, OL ), and hence that h̃−1 ∈ PGL(2, OL ), implies that c −1 −1 −1 β1 (0) = h̃0 (0), β2 (0) = h̃0 (∞), and β3 (0) = h̃0 (1) are all distinct. Thus, 0 6∈ Y , as desired.  Observe that the hypothesis that deg(fc ) = d for all c ∈ D(0, S) is essential. For example, consider the family fc (z) = cz 2 + z for c ∈ D(0, 1). Then for every c ∈ D(0, 1) r {0}, we have fc (z) = c−1 g(cz), where g(z) = z 2 + z. However, f0 (z) = z, which is not conjugate to g(z), but which also has strictly smaller degree. 4. Lattès maps There are a number of equivalent definitions of (flexible) Lattès maps in the literature. Our working definition, which is equivalent to the others by [17, Proposition 6.51] and [18, Proposition III.1.7(a)], is as follows. Definition 4.1. Let f ∈ Cp (z) be a rational function of degree d ≥ 2. We say that f is a Lattès map if there exist • an elliptic curve E defined over Cp , • a morphism ψ : E → E, and • a finite, separable morphism q : E → P1 such that the following diagram commutes ψ E −−−→   qy f E   qy P1 −−−→ P1 Furthermore, if we write E in Legendre form y 2 = x(x − 1)(x − λ) with λ ∈ Cp r {0, 1}, and if • ψ is given by ψ(P ) := [m]P + T for some integer m ≥ 2 and some 2-torsion point T ∈ E[2], and • q = h ◦ x for some linear fractional transformation h ∈ PGL(2, Cp ), where x : E → P1 is the x-coordinate map, then we say f is a flexible Lattès map. In particular, the degree d of a flexible Lattès map f is precisely d = m2 . For more on such maps, see [10] or [17, Sections 6.4–6.5]. In [10, Corollary 4.8], Milnor proved the following characterization of Lattès maps; he stated it over C, but because C and Cp are isomorphic as abstract fields, it also holds in our context. First, any Lattès map is PCF, and its strictly postcritical set consists of either three or four points. (The strictly postcritical set is, of course, the union of the strict forward orbits of the critical points.) Second, a PCF map with exactly four points in its strictly postcritical set is Lattès if and only if all of its critical points are simple — that is, they map to their images with multipliclity 2 — and none are strictly postcritical. In particular, flexible Lattès maps have this postcritical structure, with exactly four postcritical points. Specifically, the strictly postcritical set is the four-element set h(x(E[2])), where h is the linear fractional transformation of Definition 4.1. Indeed, after changing coordinates by h so that the commutative diagram of the definition becomes simply x ◦ ψ = f ◦ x, the critical points of f are precisely the 2d − 2 points of x(E[2m] r E[2]), that is, the x-coordinates of the 2m-torsion points that are not 2-torsion. The image of E[2m] under ψ is E[2], and hence the image of x(E[2m]) under f is x(E[2]). To prove Theorem 1.1, we will need the following criterion about families of maps having exactly four strictly postcritical points and satisfying Milnor’s criterion.
Lemma 4.2. Let S > 0, let fc (z) ∈ Mp (D(0, S))(z), and let α1 , . . . , α2d−2 ∈ P1 Mp (D(0, S)) . Suppose, for each c ∈ D(0, S), that fc is a rational function of degree d ≥ 2 with critical points α1 (c), . . . , α2d−2 (c), repeated according to multiplicity. Suppose also that there are uncountably many parameters c ∈ D(0, S) for which the critical points are all distinct, and for which the strictly postcritical set consists of exactly four points, none of which are critical. Then either (a) for all c ∈ D(0, S), fc is is a flexible Lattès map, or (b) for every b, c ∈ D(0, S), fb is conjugate to fc . Proof. Step 1. Let W ⊆ D(0, S) be the uncountable set of parameters c satisfying the given restrictions on the orbits of the critical points. By Milnor’s result in [10, Corollary 4.8], the map fc is Lattès for every c ∈ W . On the other hand, there are only countably many conjugacy classes of rigid (i.e., non-flexible) Lattès maps in Cp (z). Indeed, by the results of [17, Section 6.5], any such map arises from an elliptic curve E with CM (of which there are only countably many isomorphism classes), an endomorphism of E (of which there are only countably many), and a finite subgroup of Aut(E) to quotient by (of which there are only finitely many). Suppose there are an uncountable subset W 0 ⊆ W and a rigid Lattès map g(z) such that for each c ∈ W 0 , there exists hc (z) ∈ PGL(2, Cp ) satisfying fc = h−1 c ◦ g ◦ hc . Let γ1 , γ2 , γ3 ∈ P1 (Cp ) be three distinct postcritical points of g. For each i = 1, 2, 3, let βi (c) := h−1 c (γi ), each of which must be of the form
(4.1) βi (c) = fcmi αji (c) .
In particular, βi ∈ P1 Mp (D(0, S)) for each i = 1, 2, 3. (There may be different possible choices for mi and ji for different c ∈ W 0 , but only finitely many. Thus, shrinking W 0 if necessary, we may assume equations (4.1) hold for all c ∈ W 0 and all i = 1, 2, 3, with W 0 still uncountable.) The hypotheses of Lemma 3.1 therefore hold, and hence fc is conjugate to g for every c ∈ D(0, S), yielding conclusion (b). Step 2. We may assume for the remainder of the proof that fc is rigid Lattès for only countably many c; thus, shrinking W if necessary, we may assume that fc is flexible Lattès for all c in the uncountable set W . For all such c, the strictly postcritical set of fc consists of four distinct points of the form βi (c) as in equation (4.1), for i = 1, 2, 3, 4. After a change of coordinates, we may assume that no βi is identically ∞. For each c ∈ W , let Ec : y 2 = x(x − 1)(x − λ(c)) be the associated elliptic curve in Legendre form, Tc the associated 2-torsion point, m ≥ 2 the associated integer, and hc ∈ PGL(2, Cp ) the associated linear fractional transformation, as in Definition 4.1. Since hc ◦ x(Ec [2]) = {β1 (c), β2 (c), β3 (c), β4 (c)}, and x(Ec ([2])) = {∞, 0, 1, λ(c)}, we may assume (by reindexing if necessary) that (4.2) hc (λ(c)) = β1 (c), hc (0) = β2 (c), hc (1) = β3 (c), and hc (∞) = β4 (c) for uncountably many such c ∈ W . Shrink the uncountable set W if necessary so that equation (4.2) holds for all c ∈ W . For such c, then, we must have h−1 c (z) := (4.3) (z − β2 (c))(β3 (c) − β4 (c)) , (z − β4 (c))(β3 (c) − β2 (c)) and hence (4.4) λ(c) = (β1 (c) − β2 (c))(β3 (c) − β4 (c)) ∈ Mp (D(0, S)). (β1 (c) − β4 (c))(β3 (c) − β2 (c)) Note that λ cannot be identically equal to any of 0, 1, ∞, since Ec is an elliptic curve for all c ∈ W . Step 3. Let Y ⊆ D(0, S) r W be the set of parameters c ∈ D(0, S) for which at least two of the values β1 (c), β2 (c), β3 (c), β4 (c) coincide. Since the meromorphic functions βi − βj are nontrivial for all i 6= j, all points of Y are isolated. We claim that Y is empty. Suppose, towards a contradiction, that there is some c0 ∈ Y . Translating in the c-variable, we may assume c0 = 0. Reindexing β1 , . . . , β4 again if necessary, with the associated changes to Ec , Tc , and hc , we may assume that λ(0) 6= 1, ∞. Thus, ord0 (λ) ≥ 0 = ord0 (λ − 1), where ord0 denotes the order of vanishing at c = 0. To prove our claim, it suffices to show that λ(0) 6= 0, i.e., that ord0 (λ) = 0. As in the proof of Lemma 3.1, let L := Cp ((c)) with valuation ord0 , and with ring of integers OL := Cp [[c]]. Also as in that proof, the map fc ∈ L(z) has explicit good reduction with respect to ord0 . If ord0 (λ) > 0, then E0 is a singular curve. Thus, if we consider Ec as an elliptic curve over the discretely valued field L, then Ec has multiplicative reduction. (See, for example, [18, Proposition VII.5.1(b)].) In light of Exercise 10.19(a) and Proposition 8.12 of [2], then, the map fc does not have explicit good reduction. (More precisely, the Julia set of b the completion of an algebraic closure of L, fc in the Berkovich projective line over L, is a line segment, containing infinitely many points. However, by [2, Proposition 8.12], the Julia set of a map of explicit good reduction consists of only one point.) This contradiction to the previous paragraph implies that ord0 (λ) = 0, proving our claim. Step 4. By the claim of Step 3, the points β1 (c), β2 (c), β3 (c), β4 (c) are distinct for each c ∈ D(0, S). Hence, for each c ∈ D(0, S), the map hc described by equations (4.2) and (4.3) lies in PGL(2, Cp ), and λ(c) 6= 0, 1, ∞, where λ is the meromorphic function of equation (4.4). In addition, for each c ∈ W , the 2-torsion point Tc of Step 2 must be one of the four 2-torsion points of Ec , namely (λ(c), 0), (0, 0), (1, 0), or O, the point at infinity. Shrinking the uncountable set W if necessary, we may assume that Tc is always the same one of these four points, for every c ∈ W . For every c ∈ D(0, S), define T̃c to be this torsion point on Ec . For every c ∈ D(0, S), define f˜c to be the flexible Lattès map associated to the curve Ec : y 2 = x(x − 1)(x − λ(c)), morphism ψ : P 7→ [m]P + T̃c , and linear fractional transformation hc ∈ PGL(2, Cp ). Then fc , f˜c ∈ Mp (D(0, S))(z), and fc = f˜c for all c ∈ W . Since W is uncountable, we have fc = f˜c for all c ∈ D(0, S).  5. The orbit of a marked point The heart of our proof of Theorem 1.1 is the following result on the orbit of a single marked point in a family fc (z).
Theorem 5.1. Let S > 0, let fc (z) := Φ(c, z) ∈ Mp (D(0, S))(z), and let α ∈ P1 Mp (D(0, S)) . Suppose that for each c ∈ D(0, S), fc (z) is a rational function in Cp (z) of degree d ≥ 2. Let N > M ≥ 0 be integers, and let U0 , . . . , UN ⊆ P1 (Cp ) be rational open disks such that for all c ∈ D(0, S), • α(c) ∈ U0 , • fc (Uj ) ⊆ Uj+1 for all j = 0, . . . , N − 1, and • UN ⊆ UM . Then either (a) for any s ∈ (0, S), there are are only finitely many c ∈ D(0, s) such that α(c) and all critical points of fc in UM ∪ · · · ∪ UN −1 are preperiodic under fc , or (b) there are integers n > m ≥ 0 such that fcn (α(c)) = fcm (α(c)) for all c ∈ D(0, S). To prove Theorem 5.1, we will need the following lemma about dynamics in a diskshaped attracting basin. It is a variant of [4, Theorem 4.1]. Lemma 5.2. Let R > 0, let γ ∈ D(0, R), and let X h(z) = Ai z i ∈ Cp [[z]] with ` ≥ 1 and A` 6= 0 i≥` be a nontrivial power series fixing the point 0, and converging on D(0, R) with Weierstrass degree d ≥ 1. Suppose that h(D(0, R)) ⊆ D(0, R), and that  (5.1) |A` |R`−1 < min |m|m : 1 ≤ m ≤ d . Then at least one of the following three conclusions holds: (a) h(γ) = 0. (b) γ has infinite forward orbit under h. (c) h has a critical point in D(0, R) that has infinite forward orbit under h. Proof. Let r > 0 be the smallest radius such that h has a zero in the punctured disk D(0, r) r {0}, or r = R if h has no zeros in D(0, R) r {0}. Then |Ai |ri ≤ |A` |r` for all i ≥ `, as otherwise h would have a root in D(0, r) r {0}. Therefore, (5.2) |h(x)| = |A` | |x|` for all x ∈ D(0, r). Combined with condition (5.1), it follows that if 0 < |x| < r, then 0 < |h(x)| < |x|. In particular, the only preperiodic point of h in D(0, r) is the fixed point x = 0. If h(γ) ∈ D(0, r), then either conclusion (a) holds, or else h(γ) is a nonzero point in D(0, r) and hence is not preperiodic, yielding conclusion (b). Similarly, if h0 has any zeros in D(0, r) r {0}, then h has a wandering critical point, yielding conclusion (c). Hence, we assume for the remainder of the proof that |h(γ)| ≥ r, and that h0 has no roots in D(0, r) r {0}. These two assumptions imply that
h D(0, |γ|) ⊇ D(0, r) (because h(D(0, |γ|) is a disk containing both h(γ) and 0), and (5.3) |h0 (x)| = |`A` | |x|`−1 for all x ∈ D(0, r). Since the image of any disk D(0, t) is a disk D(0, u), where u is a continuous and increasing function of t, there is a unique radius S ∈ [r, |γ|] such that h maps D(0, S) onto D(0, r). Let m be the Weierstrass degree of h on D(0, S). Because 0 has ` preimages at 0 and at least one other of absolute value r, we have m ≥ ` + 1. Moreover, since D(0, S) ⊆ D(0, R), we also have m ≤ d. Define a function L : (−∞, log R) → R by L(log t) = m log t + m log kh0 kζ(0,t) − (m − 1) log khkζ(0,t) = mδ(h, ζ(0, t)) + log khkζ(0,t) , where the norm k·kζ(0,t) and distortion δ are as defined in equations (2.2) and (2.3). Then L is a continuous, piecewise linear function (with all slopes integers). Equations (5.2) and (5.3) together imply that

L(log r) = m log r + m log `A` r`−1 − (m − 1) log A` r`
= log A` r` + m log |`|. On the other hand, the distortion δ satisfies δ(h, ζ(0, S)) ≥ |m|, for example by [1, Lemma 4.2] or [4, Lemma 3.3]. Since h(ζ(0, S)) = ζ(0, r), then L(log S) = mδ(h, ζ(0, S)) + log khkζ(0,S) ≥ m log |m| + log r

> log |A` |R`−1 + log r ≥ log |A` |r`
≥ log A` r` + m log |`| = L(log r), where the second inequality is by hypothesis (5.1). Thus, there must be some ρ ∈ (r, S) such that L has positive slope at log ρ. Define M = number of roots of h0 in D(0, ρ), counted with multiplicity, a = number of roots y of h0 in D(0, ρ) with h(y) 6= 0, counted with multiplicity, and b = number of distinct roots of h in D(0, ρ). Then the slope of L at log ρ is m(M + 1) − (m − 1)m, because log kF kζ(0,t) increases with respect to log t with slope equal to the number of zeros of F in D(0, t), and because h has m roots in D(0, ρ). In addition, since a root of h of multiplicity n ≥ 2 is also a root of h0 of multiplicity n − 1, we have M − a = m − b. Furthermore, we have b ≥ 2 by our choice of r, because h has at least two distinct roots in D(0, ρ), at 0 and at some x with |x| = r. Thus, since the slope of L at log ρ is positive, we have 0 < m(M + 1) − (m − 1)m = m(m + a − b + 1) − m2 + m = m(a − b + 2) ≤ ma, and hence a > 0. That is, there is some critical point β of h in D(0, ρ) with h(β) 6= 0. Thus, h(β) ∈ D(0, r) r {0}, whence β is wandering.  Proof of Theorem 5.1. After a change of coordinates, we may assume that the open disk UM is UM = D(0, R) for some R > 0. Thus, fcM (α(c)) ∈ UM = D(0, R) for every c ∈ D(0, S), and fcN −M (z) ∈ UN ⊆ D(0, R) for every (c, z) ∈ D(0, S) × D(0, R). Define Ψ : D(0, S) × D(0, R) → D(0, R) by 

Ψc (z) = Ψ(c, z) := fcN −M z + fcM α(c) − fcM α(c) ∈ Cp [[c, z]]. Expanded as a power series, we have Ψc (z) = Ψ(c, z) = X Ai,j ci z j i,j≥0 with Ai,j ∈ Cp satisfying (5.4) |Ai,j |S i Rj ≤ R for all i, j ≥ 0, and with |A0,0 | < R. For i = 0 and j = 1, we have |A0,1 | ≤ 1. We consider two cases: that |A0,1 | < 1, and that |A0,1 | = 1. Case 1: Attracting. Suppose that |A0,1 | < 1. We will show that there is a power series β(c) ∈ Cp [[c]] such that β(c) is an attracting fixed point of Ψc for all c ∈ D(0, S). (That is, Ψc (β(c)) = β(c), and |Ψ0c (β(c))| < 1.) Then, for each s ∈ (0, S), we will construct a finite set Es ⊆ D(0, s) such that either • there exists n ≥ 1 such that for all c ∈ D(0, s), we have Ψnc (0) = β(c), or • for all c ∈ D(0, s) r Es , either 0 or a critical point of Ψc in D(0, R) has infinite forward orbit under Ψc , from which the theorem will follow. Step 1. For any radius s ∈ [0, S), define   s |A0,0 | (5.5) ts := max , , |A0,1 | < 1. S R Claim 5.2.1. For all c ∈ D(0, s) and all x, y ∈ D(0, ts R), we have Ψc (x) ∈ D(0, ts R) and Ψc (x) − Ψc (y) ≤ ts |x − y|. To prove Claim 5.2.1, observe that for any such c, x, y, and for any i, j ≥ 0, we have  i
s i j Ai,j c x ≤ tjs |Ai,j |S i Rj ≤ ti+j s R, S using inequality (5.4). Thus, |Ai,j ci xj | ≤ ts R if i + j ≥ 1; moreover, |A0,0 | ≤ ts R by definition of ts . The first conclusion of the claim is then immediate. For the second, X Ψc (x) − Ψc (y) = Ai,j ci (xj − y j ) i,j≥0 Ai,j ci (xj−1 + xj−2 y + · · · + y j−1 )  i 
s j−1 i j−1 ≤ |x − y| max ts |Ai,j |S R ≤ ts |x − y|, i≥0,j≥1 S ≤ |x − y| max  i≥0,j≥1 where the final inequality is because for i + j ≥ 2, inequality (5.4) yields  i
s tj−1 |Ai,j |S i Rj−1 ≤ ti+j−1 ≤ ts , s s S and because |A0,1 | ≤ ts for i = 0 and j = 1. Thus, we have proven Claim 5.2.1. Step 2. To prove the existence of the fixed point β(c), for each n ≥ 0, define gn (c) ∈ Cp [[c]] by gn (c) := Ψnc (0). Then each gn is a convergent power series on D(0, S), with image in D(0, R). Moreover, for each c ∈ D(0, S), by repeated application of Claim 5.2.1, we see that {gn (c)}n≥0 is a Cauchy sequence of points in D(0, R), and hence that it converges to some point β(c); furthermore, by Claim 5.2.1 again, this convergence is uniform in c ∈ D(0, s). Because each gn is a power series converging on D(0, S) with image in D(0, R), the limit function β(c) is also such a power series. By construction, we have Ψc (β(c)) = β(c), as desired. By Claim 5.2.1 yet again, for any s ∈ (0, S) and any c ∈ D(0, s), we have
(5.6) β(c) ∈ D(0, ts R), and |Ψ0c β(c) | ≤ ts < 1.
In particular, since its multiplier is Ψ0c β(c) , we see that β(c) is an attracting fixed point of Ψc . Step 3. Changing coordinates to move β(c) to the origin, define
Hc (z) = H(c, z) := Ψ c, z + β(c) − β(c) ∈ Cp [[c, z]], which converges on D(0, S)×D(0, R) with image in D(0, R) and with H(c, 0) = 0. Write XX Hc (z) = H(c, z) = Bi,j ci z j , j≥` i≥0 where ` ≥ 0 is the smallest integer such that some coefficient Bi,` is nonzero. (Such ` exists since the original function fc (z) is a nonconstant rational function of z.) In fact, we have ` ≥ 1, since H(c, 0) = 0. Furthermore, (5.7) |Bi,j |S i Rj ≤ R for all i ≥ 0 and j ≥ `, since the image of H is contained in D(0, R). Note also that for any c ∈ D(0, S), the Weierstrass degree of Hc on D(0, R) is at most deg(fcN −M ) = dN −M . For each radius s ∈ (0, S), define   X i Es := y ∈ D(0, s) : Bi,` y = 0 . i≥0 Note that Es is finite, because it is the set of zeros of a nontrivial power series on a disk D(0, s) which is strictly contained in a larger disk D(0, S) on which the series converges. For each y ∈ Es , we may choose a radius σ(y) > 0 such that X 
0< Bi,` ci < R1−` min |m|m : 1 ≤ m ≤ dN −M for all c ∈ D y, σ(y) r {y}. i≥0 Hence, for any c ∈ D(y, σ(y)) r {y}, if we set h(z) = Hc (z) and γ := −β(c), then one of the three conclusions of Lemma 5.2 holds. That is, either • Hc (−β(c)) = 0, i.e., Ψc (0) = β(c), or • z = 0 has infinite forward orbit under Ψc , or • Ψc has a critical point in D(0, R) with infinite forward orbit under Ψc . Step 4. It remains to consider parameters c in the set [  W := D(0, s) r D(y, σ(y)) . y∈Es By Lemma 2.2 and the definition of Es , there is some ε > 0 such X Cc ≥ ε for all c ∈ W, where Cc := Bi,` ci . i≥0 Let r := ts R, and recall from equation (5.6) that the power series β maps D(0, s) into D(0, r) ( D(0, R). Then for all (c, z) ∈ D(0, s) × D(0, r), we have

Hc (z) = Ψc z + β(c) − Ψc β(c) ≤ ts |z|, by Claim 5.2.1 and the fact that Ψc (β(c)) = β(c). Pick n ≥ 1 large enough that tns < εr`−1 . Then Hcn (−β(c)) ∈ D(0, εr` ) for any c ∈ W . Furthermore, we claim that (5.8) |Hc (z)| = Cc |z|` for any (c, z) ∈ W × D(0, εr` ). Indeed, for any such (c, z) with z 6= 0, and for any i ≥ 0 and j ≥ ` + 1, we have  j−1  ` |z| |z| i j i j−1 |z| Bi,j c z ≤ Bi,j S R |z| ≤ |z| ≤ ` |z|` < ε|z|` ≤ Cc |z|` , R R r using inequality (5.7). Thus, the z ` term in the power series for Hc (z) has strictly larger absolute value than all the other terms, yielding equation (5.8), as claimed. In addition, we have 0 < Cc |z|` < |z| for any such (c, z), because Cc R` ≤ R, by inequality (5.7) again. That is, by equation (5.8), we have 0 < |Hc (z)| < |z|. Thus, for any c ∈ W , since we have Hcn (−β(c)) ∈ D(0, εr` ), it follows either that Hcn (−β(c)) = 0, or else that −β(c) has infinite forward orbit under Hc . Hence, either • Ψnc (0) = β(c), or • z = 0 has infinite forward orbit under Ψc . Step 5. For any s ∈ (0, S), let Es be the finite set from Step 3, and let n ≥ 1 be the integer chosen in Step 4 (so that tns < εr` ). By the bullet lists at the end of Steps 3 and 4, for any c ∈ D(0, s) r Es , we have either (a) Ψnc (0) = β(c), or (b) z = 0 has infinite forward orbit under Ψc , or (c) Ψc has a critical point in D(0, R) with infinite forward orbit under Ψc . If statement (a) occurs for infinitely many parameters c ∈ D(0, s), then the power series Ψnc (0) − β(c) ∈ Cp [[c]] must be trivial, since it converges on the strictly larger disk D(0, S). Thus, we would have Ψnc (0) = Ψn+1 (0) = β(c) for all c ∈ D(0, S), and hence c fc(n+1)N −nM (α(c)) = fcnN −(n−1)M (α(c)) for all c ∈ D(0, S), yielding conclusion (b) of Theorem 5.1. Otherwise, the set Es0 := {c ∈ D(0, s) : Ψnc (0) = β(c)} is finite. For any c ∈ D(0, s) outside the finite set Es ∪ Es0 , either statement (b) or statement (c) above occurs. For a given such c, statement (b) implies that α(c) has infinite forward orbit under fc , and statement (c) implies that fc has a critical point in UM ∪ · · · ∪ UN −1 with infinite forward orbit. Hence, conclusion (a) of Theorem 5.1 holds, completing Case 1. Case 2: Indifferent. Assume for the remainder of the proof that |A0,1 | = 1. Then for each c ∈ D(0, S), the power series Ψc (z) ∈ Cp [[z]] has Weierstrass degree 1 and maps D(0, R) bijectively onto itself. Step 1. Because the residue field of Cp is isomorphic to Fp , there is an integer e ≥ 1 such that |Ae0,1 − 1| < 1. Replacing N by e(N − M ) + M in the statement of Theorem 5.1 and in the definition of Ψ(c, z) = Ψc (z) at the start of this proof, we may assume that e = 1, and hence that |A0,1 − 1| < 1. Given a fixed radius s ∈ (0, S), choose s̃ ∈ (s, S), and pick a radius r with   Rs̃ max |A0,0 |, < r < R. S Then |A0,0 | < r, |A0,1 − 1|r < r, and for all j ≥ 2,  r j  r 2  r  A0,j rj = A0,j Rj ≤R = r, R R R via inequality (5.4). Pick a real number t with   |A0,0 | r Rs̃ max , |A0,1 − 1|, , < t < 1. r R rS Then by the above bounds, we have (5.9) |A0,j |rj < tr for all j ≥ 2 and j = 0, with |A0,1 − 1|r < tr. In addition, for all i ≥ 1 and all j ≥ 0, inequality (5.4) yields  i   s̃ s̃ i j i j (5.10) |Ai,j |s̃ r ≤ |Ai,j |S R ≤R < tr. S S Combining inequalities (5.9) and (5.10), then, we have (5.11) Ψc (z) − z < tr for all (c, z) ∈ D(0, s̃) × D(0, r). Step 2. The iterative logarithm of Ψc , defined in [12, Définition 3.7], is the function
n (5.12) Λ(c, z) := lim p−n Ψpc (z) − z . n→∞ According to [12, Lemme 3.11(iii)], the bound (5.11) yields that
n Λ(c, z) − p−n Ψpc (z) − z ≤ Cn r|p|n for all (c, z) ∈ D(0, s̃) × D(0, r) and n ≥ 1, where, as shown in the proof of [12, Lemme 3.11(iii)],  k −1 −1/(pn (p−1)) (5.13) Cn := max |k| t|p| . k≥1 Note that for all n large enough, the expression inside the maximum on the right side of equation (5.13) approaches 0 as k → ∞, since 0 < t < 1; thus, Cn is defined and finite for all such n. Moreover, the sequence {Cn } is decreasing. Hence, the limit of equation (5.12) in fact converges uniformly on (c, z) ∈ D(0, s̃) × D(0, r). Therefore, Λ(c, z) ∈ Cp [[c, z]] is a power series converging on D(0, s̃) × D(0, r). In addition, by [12, Proposition 3.16], a point (c, z) ∈ D(0, s̃) × D(0, r) is a zero of Λ if and only if z is periodic under Ψc . Since Ψc : D(0, r) → D(0, r) is bijective, this condition is equivalent to saying that z is preperiodic under Ψc . Step 3. Define F (c) := Λ(c, 0) ∈ Cp [[c]], which is a power series converging on D(0, s̃), with zeros precisely at the parameters c for which z = 0 is preperiodic under Ψc , i.e., for which α(c) is preperiodic under fc . If F is not identically zero, then F has only finitely many zeros in the strictly smaller disk D(0, s). That is, there are only finitely many c ∈ D(0, s) for which α(c) is preperiodic under fc , implying conclusion (a) of Theorem 5.1. Otherwise, we have F = 0, and hence α(c) is preperiodic under fc for all c ∈ D(0, s̃). There are only countably many choices of integers n > m ≥ 0, and hence there must be some such choice of m and n such that uncountably many points c ∈ D(0, S) are zeros of the power series fcn (α(c)) − fcm (α(c)). A power series with uncountably many zeros must be identically zero, and hence fcn (α(c)) = fcm (α(c)) for all c ∈ D(0, S), yielding conclusion (b).  6. Proofs of main theorems Our main results are all consequences of Theorem 5.1, using Thurston’s powerful rigidity theorem, which we state below as Theorem 6.1, in the language of critical orbit relations. Fix an integer d ≥ 2, and consider a separable rational function f (z) of degree d with marked critical points α1 , . . . , α2d−2 . A set of critical orbit relations for f is a set of equations of the form f m (αi ) = f n (αj ) for integers m, n ≥ 0 and indices i, j ∈ {1, . . . , 2d − 2}, including at least one equation of the form f ni (αi ) = f mi (αi ) for each αi , with ni > mi ≥ 0. Thus, a set of critical orbit relations specifies a finite forward orbit for each critical point, and it may also include other restrictions, such as that some of the critical points coincide, or that, for example, f 3 (α1 ) = f 4 (α2 ). Theorem 6.1 (Thurston Rigidity). Fix d ≥ 2, and let C be a set of critical orbit relations. Assume that C is not consistent with a flexible Lattès map. Then, up to conjugacy by linear fractional transformations, there are only finitely many rational functions f (z) ∈ C(z) of degree d satisfying the relations of C. The original form [20] of Thurston’s result is different; see also [8, Theorem 1] for the statement and proof. An explanation of why the original implies Theorem 6.1 appears in [5, Corollary 3.7]. As with Milnor’s criterion, the fact that C and Cp are isomorphic as abstract fields shows that Theorem 6.1 also holds for Cp in place of C. Proof of Theorem 1.1. For each i = 1, . . . , 2d − 2, one of the conclusions of Theorem 5.1 holds for αi (c). If there is some such i for which conclusion (a) of Theorem 5.1 holds, then conclusion (a) of Theorem 1.1 holds, and we are done. Thus, we may assume for the remainder of the proof that conclusion (b) of Theorem 5.1 holds for every αi (c). That is, for each i = 1, . . . , 2d − 2, there exist integers ni > mi ≥ 0 such that

(6.1) fcni αi (c) = fcmi αi (c) for all c ∈ D(0, S). In addition, for each i 6= j, there may be integers ei,j , ej,i ≥ 0 so that

(6.2) fcei,j αi (c) = fcej,i αj (c) for uncountably many c ∈ D(0, S). Without loss, for each i, the integers ni > mi ≥ 0 are the smallest for which equation (6.1) holds for uncountably many c. (That is, for any other such n0i > m0i ≥ 0, we have n0i ≥ ni and m0i ≥ mi .) Similarly, for each relevant i 6= j, we may assume that the integers ei,j , ej,i ≥ 0 are the smallest for which equation (6.2) holds for uncountably many c. e e Since the meromorphic functions fcni (αi (c)) − fcmi (αi (c)) and fc i,j (αi (c)) − fc j,i (αj (c)) have uncountably many zeros, they are trivial on D(0, S). Thus, (6.3) the relations of equations (6.1) and (6.2) hold for all c ∈ D(0, S). However, for each i, and for each of the finitely many smaller choices of integers ` > k ≥ 0, there could be (at most) countably many parameters c ∈ D(0, S) for which fck (αi (c)) = fc` (αi (c)). Similarly, for each i 6= j and each of the finitely many smaller choices of k, ` ≥ 0, there could be (at most) countably many parameters c ∈ D(0, S) for which fck (αi (c)) = fc` (αj (c)). Suppose that the relations of equations (6.1) and (6.2) are consistent with a flexible Lattès map. That is, for uncountably many parameters c ∈ D(0, S), all 2d − 2 critical points of fc are distinct, and the strictly postcritical set consists of four points, none of which are critical. By Lemma 4.2, either conclusion (b) or conclusion (c) of Theorem 1.1 holds, and we are done. Otherwise, by Thurston Rigidity (Theorem 6.1), there are only finitely many rational functions, up to conjugacy, that satisfy the critical orbit relations (6.1) and (6.2). Thus, there exist some g(z) ∈ Cp (z) and an uncountable subset W ⊆ D(0, S) such that fc is conjugate to g for all c ∈ W . That is, for any c ∈ W , there exists hc ∈ PGL(2, Cp ) such that fc = h−1 c ◦ g ◦ hc . We consider two cases. Case 1. If the forward orbits of the critical points of g together contain at least three distinct points γ1 , γ2 , γ3 , then there are corresponding distinct meromorphic functions
βi (c) := fcei αji (c) for i = 1, 2, 3, for some indices ji ∈ {1, . . . , 2d − 2} and some integers ei ≥ 0, such that hc (βi (c)) = γi for each c ∈ W and each i = 1, 2, 3. Thus, by Lemma 3.1, conclusion (c) of Theorem 1.1 holds; again, we are done. Case 2. Finally, suppose that the union of the forward orbits of the critical points of g consists of at most two points. Since g must have at least two distinct critical points, this means that it has exactly two, either with each of them fixed or with each of them mapping to the other. Thus, after a change of coordinates, we may assume that g(z) = z d or that g(z) = 1/z d , with critical points at z = 0, ∞. Since fc is conjugate to g for uncountably many c, and since equations (6.1) and (6.2) are optimal for such c, then possibly after reindexing, those equations say that (6.4) α1 (c) = α2 (c) = · · · = αd−1 (c) and αd (c) = αd+1 (c) = · · · = α2d−2 (c) with either (6.5)

fc α1 (c) = α1 (c) and fc αd (c) = αd (c) if g(z) = z d , or (6.6)

fc α1 (c) = αd (c) and fc αd (c) = α1 (c) if g(z) = 1/z d . By comment (6.3), equations (6.4) hold for every c ∈ D(0, S), as do either equations (6.5) if g(z) = z d , or equations (6.6) if g(z) = 1/z d . Moreover, we must have α1 (c) 6= αd (c) for all c ∈ D(0, S); indeed, if some parameter c0 had α1 (c0 ) = αd (c0 ), then α1 (c0 ) would have more than d preimages under fc0 , counted with multiplicity. Hence, for any c ∈ D(0, S), we may change coordinates in the z-variable to move α1 (c) to 0, and to move αd (c) to ∞. Therefore, if g(z) = z d , equations (6.4) and (6.5) together imply that for every c ∈ D(0, S), the rational function fc is conjugate to z 7→ z d . Similarly, if g(z) = 1/z d , equations (6.4) and (6.6) together imply that for every c ∈ D(0, S), the rational function fc is conjugate to z 7→ 1/z d . Either way, fc is conjugate to g for every c ∈ D(0, S), yielding conclusion (c) of Theorem 1.1.  Proof of Theorem 1.3. For each point a ∈ P1 (Cp ), let W (a) be the inverse image of the point a ∈ P1 (Fp ) under the reduction map P1 (Cp ) → P1 (Fp ). That is, if |a| ≤ 1, then W (a) = D(a, 1), and if |a| > 1, then W (a) = P1 (Cp ) r D(0, 1).
For each i = 1, . . . , 2d − 2 and for every j ≥ 0, let Ui,j := W f0j (αi (0)) . By the fourth bullet point of the hypotheses, we have αi (c) ∈ Ui,0 for every c ∈ D(0, S). Furthermore, by the first and second bullet points, for every j ≥ 0 and every c ∈ D(0, S), we have       fc (Ui,j ) = fc W f0j (αi (0)) ⊆ W f c f0j (αi (0))      = W f 0 f0j (αi (0)) = W f0j+1 (αi (0)) = Ui,j+1 . In addition, because f 0 ∈ Fp (z) and αi (0) ∈ P1 (Fp ) must both be defined over some Ni Mi finite subfield of Fp , there exist some Ni > Mi ≥ 0 such that f 0 (αi (0)) = f 0 (αi (0)). Hence, Ui,Ni = Ui,Mi . We have verified that the family fc satisfies all the hypotheses of Theorem 1.1 holds. Therefore, one of the three desired conclusions follows.  Proof of Theorem 1.4. Counting with multiplicity, the map fc (z) = z d + c has d − 1 critical points at z = 0, and the other d − 1 at the (fixed) point at ∞. For any c ∈ Cp with |c| > 1, and for any x ∈ Cp with |x| > |c|1/d , we have |fc (x)| = |x|d > |x|. Since |fc (0)| = |c| > |c|1/d for such c, it follows that 0 < |fc (0)| < |fc2 (0)| < |fc3 (0)| < · · · , and hence the critical point at z = 0 is not preperiodic. Thus, fc is not PCF for |c| > 1, proving the first claim of Theorem 1.4. Given any a ∈ Cp with |a| ≤ 1, then for any c ∈ D(0, 1), define gc (z) := z d + a + c, and also define αi (c) := 0 for each i = 1, . . . , d − 1 and αi (c) := ∞ for i = d, . . . , 2d − 2. Then for each c ∈ D(0, 1), the map gc has explicit good reduction, with g c (z) = z d +a = g 0 (z), and clearly with αi (c) = αi (0). Hence, by Theorem 1.3, either every gc is flexible Lattès, or the family is isotrivial, or our desired conclusion holds. However, gc is not flexible Lattès. Indeed, for d ≥ 3, there are repeated critical points, violating Milnor’s criterion; and for d = 2, the degree of gc is not a square. That is, the second conclusion of Theorem 1.3 cannot hold. In addition, we claim that there are only finitely many c ∈ D(0, 1) for which gc is conjugate to g0 . Indeed, if a = 0, then no other gc is conjugate to g0 (z) = z d , since z d is the only map in the family with all critical points fixed. If a 6= 0, then both g0 and gc have fixed critical points at z = ∞ and non-fixed critical points at z = 0. Thus, if gc = h−1 ◦ g0 ◦ h for some h ∈ PGL(2, Cp ), then we must have h(0) = 0 and h(∞) = ∞, so that h(z) = bz for some b ∈ C× p . Therefore, z d + a + c = gc (z) = b−1 g0 (bz) = bd−1 z d + b−1 (a + c), so that b must be a (d − 1)-st root of unity. Hence there are at most d − 1 choices of c with gc conjugate to g0 , proving our claim. Therefore, the family gc is neither Lattès nor isotrivial. By Theorem 1.3, then, our desired conclusion holds.  7. Examples In this section, we present examples to justify our earlier claims that our main theorems cannot be significantly strengthened. In particular, Examples 7.1 and 7.2 illustrate that the conclusions of Theorem 1.4 cannot be significantly strengthened even for the quadratic family fc (z) = z 2 + c. Moreover, Example 7.3 illustrates the necessity of the stability hypotheses of Theorems 1.1 and 1.3. Example 7.1. Consider the family fc (z) = z 2 + c of Theorem 1.4, with d = 2 and p = 2. Then both c = 0 and c = −2 are PCF parameters of fc lying in the 2-adic disk D(0, 1), and all three of c = −1 and c = ±i are PCF parameters in the 2-adic disk D(1, 1). More generally, the roots of the polynomial gn (c) := fcn (0) + fcn−1 (0) are parameters c for which the orbit of the critical point z = 0 satisfies fcn+1 (0) = fcn (0) n−1 but no simpler critical orbit relation. However, it is easy to check that gn (c) ≡ c2 (mod 2), and therefore all roots of g n lie in the 2-adic disk D(0, 1). For any such root c, the map fc has an attracting fixed point β ∈ D(0, 1), and the critical point z = 0 lands on β after exactly n iterations. In particular, there are infinitely many PCF parameters in the disk D(0, 1). Example 7.2. Consider the family fc (z) = z 2 + c of Theorem 1.4, with d = 2, and this time with p = 3. For any parameter c in the 3-adic disk D(1, 1) ⊆ C3 , we have fc (0) ≡ 1 (mod 3) and fcn (0) ≡ −1 (mod 3) for all n ≥ 2. Changing coordinates via c = 1 + b and z = −1 + w, consider the family Fb (w) := f1+b (w − 1) + 1 = w2 − 2w + (b + 3) ∈ O3 [[b, w]], where O3 is the ring of integers in C3 . For each n ≥ 1, let In := h3, wn+1 i + bhb, win−1 , which is an ideal of the ring R := O3 [[b, w]]. The reduction of Fb modulo the ideal I1 := h3, b, w2 i is simply w 7→ w. Inspired by a similar idea in [9, Proposition 4.1], it is a simple exercise to check that if G, H ∈ R are of the form G(w) ≡ w + A mod In+1 and H(w) ≡ w + B mod In+1 with A, B ∈ In , then G ◦ H(w) ≡ w + A + B mod In+1 . Therefore, G3 (w) ≡ w mod In+1 . n By induction, it follows that Fb3 (w) ≡ w mod In+1 for every n ≥ 0, and hence that n Fb3 (−b) ≡ −b mod h3, bn+1 i. On the other hand, viewed as a polynomial in the two n n variables b, w, the highest total-degree term of Fb3 (w) is wDn , where Dn = 23 . Thus, n hn (b) := Fb3 (−b) + b ∈ O3 [b] is a monic polynomial of degree Dn whose reduction modulo 3 has a zero of order at least n + 2 at b = 0. In particular, hn has at least n + 2 roots in the disk D(0, 1) ⊆ C3 . The critical points of Fb (w) are w = 1, ∞, and we have Fb (1) = b + 2, and Fb2 (1) = Fb (b + 2) = Fb (−b) = b2 + 3b + 3. n n Thus, every root b of hn satisfies Fb3 (b + 2) = −b, and hence Fb2+3 (1) = Fb2 (1). That is, every such b is a PCF parameter of the family Fb . Since hn has at least n + 2 roots in D(0, 1), and since these roots are distinct by [6, Lemma 4], it follows that the family Fb has infinitely many PCF parameters b ∈ D(0, 1). (More precisely, Buff’s result says n that the polynomial fc1+3 (0) + fc (0) ∈ Z[c] has no repeated roots; after our change of coordinates, the same is true of hn .) Therefore, the original family fc has infinitely many PCF parameters c ∈ D(1, 1). For any such c, the map fc fixes the disk D(−1, 1), which contains periodic points of period 3n for every n ≥ 0; and the critical point z = 0 lands on one such periodic point after 2 iterations. Our final example illustrates that the stability conditions of Theorems 1.1 and 1.3 cannot be removed in general. Example 7.3. Fix a prime number p ≥ 2, and let d = p + 1. Define  
p+1 fc (z) := c + pz p+1 − (p + 1)z p + 1 , p which has critical points at z = 0, 1, ∞. Writing γ(c) := c + (p + 1)/p, we have p+1 p+1 p 2 ∞ 7−→ ∞, 1 7−→ 0 7−→ γ(c), 7−→ γ(c), p where the numbers above the arrows indicate ramification indices. Thus, fc is PCF if and only if γ(c) is preperiodic. In particular, γ(0) = (p + 1)/p is a fixed point of f0 , which in turn is therefore PCF. In addition, we have fc0 (z) = p(p + 1)γ(c)z p−1 (z − 1), and hence
1 (p + 1)p+1 f00 γ(0) = p(p + 1)γ(0) · γ(0)p−1 · = , p pp which has p-adic absolute value greater than 1. That is, γ(0) is a repelling fixed point of f0 . Since there are other choices of c for which γ(c) is not a fixed point of fc , then in the terminology of [14, Section 5], the family of rational maps fc with marked critical point z = 1 has a Misiurewicz bifurcation at c = 0. By [14, Key Lemma], then, there is an infinite sequence of distinct parameters cn ∈ Cp with cn → 0 such that z = 1 is preperiodic under fcn , and hence such that fcn is PCF, for every n. The intuitive reason such a sequence exists is because for nonzero parameters c very close to 0, the point z = γ(c) will be very close to a repelling fixed point β(c). After many iterations, fcm (γ(c)) will be pushed away from β(c), and by a careful choice of such c, we can get this orbit to land wherever we like, for example on some preimage of β(c), yielding a new PCF parameter. References [1] Robert L. Benedetto, Non-archimedean holomorphic maps and the Ahlfors Islands Theorem, Amer. J. Math., 125 (2003), 581–622. [2] Robert L. Benedetto, Dynamics in one non-archimedean variable, American Mathematical Society, Providence, 2019. [3] Robert L. Benedetto and Su-ion Ih, A finiteness property of postcritically finite maps, in preparation. [4] Robert L. Benedetto, Patrick Ingram, Rafe Jones, and Alon Levy, Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps, Duke Math. J. 163 (2014), 2325–2356. [5] Eva Brezin, Rosemary Byrne, Joshua Levy, Kevin Pilgrim, and Kelly Plummer, A census of rational maps, Conformal Geometry and Dynamics 4 (2000), 35–74. [6] Xavier Buff, On postcritically finite unicritical polynomials, New York J. Math. 24 (2018), 1111– 1122. [7] Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1997. [8] Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. [9] Jonathan Lubin, Nonarchimedean dynamical systems, Compositio Math. 94 (1994), no. 3, 321–346. [10] John Milnor, On Lattès map, in Dynamics on the Riemann sphere, European Math. Soc., Zürich, 2006, 9–43. [11] Curtis McMullen, The Mandelbrot set is universal, in The Mandelbrot Set, Theme and Variations, Tan Lei, ed., Cambridge Univ. Press, 2007, 1–18. [12] Juan Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287 (2003), 147–230. [13] Juan Rivera-Letelier, Points périodiques des fonctions rationnelles dans l’espace hyperbolique padique, Comment. Math. Helv. 80 (2005), 593-–629. [14] Juan E. Rivera-Letelier, Wild recurrent critical points, J. London Math. Soc. (2) 72 (2005), 305– 326. [15] Thomas Scanlon, p-adic distance from torsion points of semi-abelian varieties, J. Reine Angew. Math. 499 (1998), 225–236. [16] Thomas Scanlon, The conjecture of Tate and Voloch on p-adic proximity to torsion, Internat. Math. Res. Notices 1999, no. 17, 909–914. [17] Joseph H. Silverman, The Arithmetic of Dynamical Systems, Springer, New York, 2007. 22 ROBERT L. BENEDETTO AND SU-ION IH [18] Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer, New York, 2009. [19] John Tate and José Felipe Voloch, Linear forms in p-adic roots of unity, Internat. Math. Res. Notices 1996, no. 12, 589–601. [20] William Thurston, Lecture Notes, CBMS Conference, University of Minnesota at Duluth, 1983. (Benedetto) Amherst College, Amherst, MA 01002 Email address: rlbenedetto@amherst.edu (Ih) University of Colorado, Boulder, CO 80309, and, Korea Institute for Advanced Study, Seoul 02455 Email address: ih@math.colorado.edu