J-Stability in non-archimedean dynamics
aa r X i v : . [ m a t h . D S ] F e b J-STABILITY IN NON-ARCHIMEDEAN DYNAMICS
ROBERT L. BENEDETTO AND JUNGHUN LEE
Abstract.
Let C v be a complete, algebraically closed non-archimedean field, and let f ∈ C v ( z ) be a rational function of degree d ≥
2. If f satisfies a bounded contrac-tion condition on its Julia set, we prove that small perturbations of f have dynamicsconjugate to those of f on their Julia sets. Introduction
Fix the following notation throughout this paper. C v an algebraically closed field of characteristic zero. | · | a nontrivial non-Archimedean absolute value on C v ,with respect to which C v is complete. N the set { , , , . . . } of positive integers. N N ∪ { } .The Berkovich projective line P is a natural compactification of the classical pro-jective line P ( C v ) = C v ∪ {∞} , which we describe in greater detail in Section 2.3. Weconsider the dynamics of a rational function f ∈ C v ( z ) on P ( C v ) and on P . Thatis, writing f ( z ) = z and f n +1 = f ◦ f n for all n ∈ N , we consider the action of theiterates f n on P ( C v ) and P . See [BR10, Chapter 10], [B19], or [Sil07, Chapter 5]for more thorough treatments of such non-archimedean dynamics. We will be especiallyinterested in the case that two such maps f, g ∈ C v ( z ) are conjugate on a subset of P ;more precisely, there is some invertible map h : V → V such that h ◦ f | W = g ◦ h | W ,where W = f − ( V ) ⊆ V ⊆ P .A rational function f ∈ C v ( z ) may be written as f = F/G for relatively primepolynomials
F, G ∈ C v [ z ]. We define the degree of f to be deg f := max { deg F, deg G } .Every point of P ( C v ) has deg f preimages under f , counted with multiplicity. For anyinteger d ≥
2, we define Rat d ( C v ) := (cid:8) f ∈ C v ( z ) (cid:12)(cid:12) deg f = d (cid:9) to be the set of rational functions of degree d , defined over C v , with the topology inducedfrom the natural inclusion of Rat d ( C v ) in P d +1 ( C v ), which maps f to the (2 d + 2)-tupleof its coefficients.The main result of this paper is motivated by Ma˜n´e, Sad, and Sullivan’s result [MSS83]in complex dynamics. They introduced the notion of J -stability of a rational map f ∈ C ( z ), a property which, roughly speaking, means that the dynamics of all maps g in some neighborhood of f in Rat d ( C ) are conjugate on their Julia sets. In particular,they showed that a rational map is J -stable if it is expanding on its Julia set. McMullenand Sullivan [MS98, Sections 7–8] extended these conjugacies to neighborhoods of the Date : February 9, 2021.
Key words and phrases.
Berkovich space, expanding map, dynamical moduli space.
Julia set if the conjugating function is allowed to be quasiconformal. See also [McM94]for more details on J -stability in complex dynamics.Motivated by the results of [MSS83], T. Silverman [Sil17] proved a non-archimedeanstability result for one-parameter families via a condition on the Berkovich analytifica-tion of the appropriate moduli space. In a different direction, the second author [L19]investigated non-archimedean rational functions f ∈ C v ( z ) acting on P ( C v ), provingthat f is J -stable if it is expanding in a sense parallel to that in complex dynamics.Specifically, as in [L19, Definition 1.1], the map f ∈ C v is expanding on its (type I) Juliaset J f := J an ,f ∩ P ( C v ) if J f is nonempty and there exist real constants c > λ > (cid:0) f n (cid:1) ♮ ( z ) ≥ cλ n for every z ∈ J an ,f ∩ P ( C v ) and n ∈ N , where g ♮ denotes the spherical derivative of g ∈ C v ( z ), defined in Section 3. (See alsoRemark 7.3.) In a different context [B01], the first author had previously studied aslightly weaker version of this condition for the case that C v = C p and f is defined overa locally compact subfield K of C v . (Specifically, such a map f is hyperbolic if for eachfinite extension L/K , there exist c = c L > λ = λ L > z ∈ J f ∩ P ( L ).) However, besides the fact that the results of [L19] applyonly to the type I Julia set J f , both the expanding and the hyperbolic hypotheses areunnecessarily restrictive, as we illustrate in Section 7.In this paper, we strengthen the main result of [L19] both by generalizing the ex-panding hypothesis of equation (1.1) and by extending the resulting conjugacy fromthe classical Julia set in P ( C v ) to the Berkovich Julia set J an ,f ⊆ P of the map f .(See Section 2.4 for more on the Berkovich Julia set.) Moreover, in analogy with thequasiconformal conjugacy of [MS98], we show that our conjugacy can be extended toan appropriate neighborhood of J an ,f . As in [L19], our statement involves the sphericalderivative f ♮ of the rational function f , but extended to the Berkovich space P , asdescribed in Section 3, and with a less restrictive hypothesis. Our extension to P alsoallows us to avoid the assumption that J an ,f ∩ P ( C v ) = ∅ required in both [L19] and[Sil17]. Theorem 1.1.
Let f ∈ C v ( z ) be a rational function of degree d ≥ with BerkovichJulia set J an ,f . Suppose there exists δ > such that (cid:0) f n (cid:1) ♮ ( ζ ) ≥ δ for all ζ ∈ J an ,f and n ∈ N . Then there exist a neighborhood W ⊆ Rat d ( C v ) of f and an open set U ⊆ P containing J an ,f ∩ P ( C v ) with the following properties. For each g ∈ W , there is a homeomorphism h : P → P for which (a) h is an isometry on the set P ( C v ) of type I points , (b) h is the identity map on P r U , and (c) h ◦ f ( ζ ) = g ◦ h ( ζ ) for all ζ ∈ U ∪ J an ,f . Note in particular that the points of J an ,f r U are fixed by the map h of Theorem 1.1.Hence, we have J an ,f r U = J an ,g r U , and moreover f ( ζ ) = g ( ζ ) for all ζ ∈ J an ,f r U .The outline of this paper is as follows. We recall some essentials from non-archimedeananalysis and dynamics in Section 2, and we describe the spherical derivative on P inSection 3. Next, we present several necessary lemmas in Sections 4 and 5. Section 6 -STABILITY 3 is devoted to the proof of Theorem 1.1. Finally, in Section 7, we present examples ofrational maps which satisfy the hypotheses of Theorem 1.1 but which are not expandingin the sense of [L19]. 2. Preliminaries
In this section, we recall some relevant facts about dynamics on P ( C v ) and on P .Here and in the rest of the paper, we set the following notation for disks in C v . D ( a, r ) for a ∈ C v and r >
0, the open disk { x ∈ C v | | x − a | < r } . D ( a, r ) for a ∈ C v and r >
0, the closed disk { x ∈ C v | | x − a | ≤ r } . O the ring of integers D (0 ,
1) = { z ∈ C v | | z | ≤ } of C v .2.1. The chordal metric.
The chordal metric is the distance function ρ on P ( C v )given in homogeneous coordinates by ρ (cid:0) [ z : z ] , [ w : w ] (cid:1) := | z w − z w | max {| z | , | w |} max {| z | , | w |} . Equivalently, in affine coordinates we have ρ (cid:0) z, w (cid:1) = | z − w | max { , | z |} max { , | w |} = | z − w | if z, w ∈ O , (cid:12)(cid:12)(cid:12)(cid:12) z − w (cid:12)(cid:12)(cid:12)(cid:12) if z, w ∈ C v r O , h ∈ PGL(2 , O ) is an isometry with respect to the chordal metric. See [Sil07,Section 2.1] or [B19, Section 5.1]for more on the chordal metric.2.2. Weierstrass degrees of power series.
Let a ∈ C v and r >
0. A power series F ( z ) = ∞ X i =0 c i ( z − a ) i ∈ C v [[ z − a ]]converges on D ( a, r ) if and only iflim n →∞ | c n | s n = 0 for all 0 < s < r .If F converges on D ( a, r ), then the derivative of FF ′ ( z ) = ∞ X i =1 ic i ( z − a ) i − ∈ C v [[ z − a ]]also converges on D ( a, r ). In particular, F ( a ) = c and F ′ ( a ) = c .The Weierstrass degree of F on D ( a, r ) is defined to be the smallest n ∈ N such that | c n | r n = sup {| c i | r i | i ∈ N } , or ∞ if this supremum is never attained. If n ∈ N is the Weierstrass degree of F − c on D ( a, r ), then F maps D ( a, r ) onto the disk D ( c , | c n | r n ), and every point of thelatter disk has n preimages in the former, counted with multiplicity. In particular, F isinjective on D ( a, r ) if and only if n = 1, in which case F ( D ( a, r )) = D ( F ( a ) , | c | r ), and | F ( x ) − F ( y ) | = | F ′ ( a ) || x − y | for all x, y ∈ D ( a, r ) . ROBERT L. BENEDETTO AND JUNGHUN LEE If F is injective on D ( a, r ), then F ′ has no zeros in D ( a, r ). However, the converse isnot necessarily true if C v has positive residue characteristic, although Lemma 4.2 showsthat F is injective on a smaller disk in that case.If f ∈ Rat d ( C v ) has no poles in D ( a, r ), then there exists a convergent power series F ∈ C v [[ z − a ]] on D ( a, r ) such that F ( x ) = f ( x ) for all x ∈ D ( a, r ). Thus, the image f ( D ( a, r )) is a disk of the form D ( b, s ), where b = f ( a ). Note that the Weierstrassdegree of F − b on D ( a, r ) is at most d .We refer the reader to [B19, Chapters 3,14] or [Rob00, Chapter 6] for more details onpower series over non-archimedean fields.2.3. The Berkovich projective line.
It has become clear that although a significantamount of non-archimedean dynamics can be done on the classical projective line P ( C v ),the appropriate setting is the Berkovich projective line P . In this section we summarizesome relevant facts about P and its associated dynamics. For more details, see [BR10,Chapters 1,2,9,10] or [B19, Chapters 6–8].The Berkovich affine line A is the set of all multiplicative seminorms on C v [ z ] thatextend the absolute value on C v . That is, ζ = k · k ζ is a function from C v [ z ] to [0 , ∞ )satisfying k f g k ζ = k f k ζ k g k ζ , k f + g k ζ ≤ max {k f k ζ , k g k ζ } , and k a k ζ = | a | for all f, g ∈ C v [ z ] and a ∈ C v . We will generally write an element of A as ζ when we thinkof it as a point, and as k · k ζ when we think of it as a seminorm.There are four types of points in A . Type I points correspond to the points of C v ,with k f k x := | f ( x ) | for x ∈ C v . Points of type II and III correspond to closed disks D ( a, r ), with a ∈ C v and r >
0, where r ∈ | C × v | gives a point of type II, and r
6∈ | C × v | gives a point of type III. In both cases, the corresponding point ζ ( a, r ) ∈ A is thesup-norm on the disk D ( a, r ). Finally, type IV points correspond to descending chainsof disks D ) D ) · · · with empty intersection. We denote by H := A r C v thesubset of points not of type I.We equip A with the Gel’fand topology, i.e., the weakest topology such that forevery f ∈ C v [ z ], the function ζ
7→ k f k ζ maps A continuously to R . The projective line P may be formed either by taking the one-point compactification P = A ∪ {∞} orby glueing two copies of A via ζ /ζ . (The new point ∞ is of type I.) Then P isa compact Hausdorff space which contains P ( C v ), the set of type I points, as a densesubspace.For a ∈ C v and r >
0, the sets D an ( a, r ) := { ζ ∈ A | k z − a k ζ < r } and D an ( a, r ) := { ζ ∈ A | k z − a k ζ ≤ r } are called open and closed Berkovich disks, respectively. A type I point x ∈ C v lies in D an ( a, r ) if and only if x ∈ D ( a, r ), and it lies in D an ( a, r ) if and only if x ∈ D ( a, r ). Atype II or III point ζ = ζ ( b, s ) lies in D an ( a, r ) if and only if D ( b, s ) ⊆ D ( a, r ); and it liesin D an ( a, r ) if and only if D ( b, s ) ⊆ D ( a, r ). (The one exception to the last rule is thata type III point ζ ( a, r ) itself does not lie in D an ( a, r ), even though D ( a, r ) = D ( a, r ) for r
6∈ | C × v | .) As is the case for disks in C v , if two Berkovich disks intersect, then one diskcontains the other.If ζ lies in the Berkovich disk D an ( a, r ), we will sometimes abuse notation and write D an ( a, r ) as D an ( ζ , r ), and even D ( a, r ) as D ( ζ , r ). We will similarly write D an ( a, r ) = D an ( ζ , r ) and D ( a, r ) = D ( ζ , r ) if ζ ∈ D an ( a, r ). -STABILITY 5 More generally, an open connected affinoid is either P with finitely many closedBerkovich disks removed, or else an open Berkovich disk with finitely many closedBerkovich disks removed. A closed connected affinoid is defined similarly, with theroles of “open” and “closed” reversed. The open connected affinoids form a basis for theGel’fand topology on P . If U is either an open or closed connected affinoid, then boththe set of type I points of U and the set of type II points of U are dense in U .2.4. Dynamics on the Berkovich line.
Any seminorm ζ ∈ A extends from C v [ z ]to C v ( z ) by defining k F/G k ζ := k F k ζ / k G k ζ , where we understand ∞ to be a legal valuefor this expression, in case k G k ζ = 0. Any rational function f ∈ C v ( z ) then defines acontinuous function f : P → P , given by k h k f ( ζ ) := k h ◦ f k ζ , which extends the usual action of f on the type I points of P ( C v ).Moreover, if f is a convergent power series on D ( a, r ), then f similarly induces acontinuous function f : D an ( a, r ) → A . For any open disks D ( a, r ) , D ( b, s ) ⊆ C v , wehave f (cid:0) D ( a, r ) (cid:1) = D ( b, s ) ⇐⇒ f (cid:0) D an ( a, r ) (cid:1) = D an ( b, s ) . Furthermore, in that case, the following are equivalent: • f ( z ) − b has Weierstrass degree 1 on D ( a, r ). • f : D ( a, r ) → D ( b, s ) is a bijective function. • f : D an ( a, r ) → D an ( b, s ) is a bijective function. • f has an inverse function f − : D ( b, s ) → D ( a, r ) also given by a convergentpower series.(The fact that bijectivity implies Weierstrass degree 1 uses our assumption that C v hascharacteristic zero; that implication fails in positive characteristic for totally inseparablemaps.)The (Berkovich) Fatou set of a rational function f ∈ C v ( z ) of degree d ≥ ζ ∈ P having a neighborhood U such that S n ∈ N f n ( U ) omits infinitely manypoints of P . The complement P r F an ,f is the (Berkovich) Julia set J an ,f of f . Bothsets are nonempty (see [B19, Corollaries 5.15 and 12.6]), and both are invariant under f , meaning that f − ( J an ,f ) = f ( J an ,f ) = J an ,f and f − ( F an ,f ) = f ( F an ,f ) = F an ,f . The Fatou set is open in P , and the Julia set is closed (and hence compact).The type I Fatou set F an ,f ∩ P ( C v ) consists of those points of P ( C v ) having a neigh-borhood on which the sequence of iterates { f n } ∞ n =0 is equicontinuous with respect tothe chordal metric ρ . If a type I point x ∈ C v is periodic, i.e., if f n ( x ) = x for some(minimal) positive integer n ∈ N , then the multiplier of x is ( f n ) ′ ( x ). If the multiplier λ of x satisfies | λ | >
1, then x is said to be repelling , and we have x ∈ J an ,f . Otherwise,i.e. if | λ | ≤
1, then x is said to be nonrepelling , and x ∈ F an ,f .3. The spherical derivative
The spherical kernel is a natural extension to P of the chordal metric ρ on P ( C v ).We recall its definition and some of its properties from [BR10, Section 4.3]. ROBERT L. BENEDETTO AND JUNGHUN LEE
Definition 3.1.
The spherical kernel is the unique function ||· , ·|| : P × P → R suchthat • || x, y || = ρ ( x, y ) for any x, y ∈ P ( C v ), • ||· , ·|| is continuous on P × P r { ( ζ , ζ ) | ζ ∈ H } , and • for any ζ , ξ ∈ P , || ζ , ξ || = lim sup ( x,y ) → ( ζ,ξ ) ρ ( x, y )where the lim sup is over ( x, y ) ∈ P ( C v ) × P ( C v ).See [BR10, Equation (4.21)] for an explicit construction of the spherical kernel. Al-though it is not a metric, the spherical kernel has the following related properties; see[BR10, Proposition 4.7]. Proposition 3.2.
The spherical kernel is symmetric and takes values in [0 , . More-over, it is continuous in each variable separately, and it is upper semicontinuous as afunction of two variables. The spherical kernel is discontinuous on the diagonal in H × H , but it is precisely onthis diagonal that we are most interested in it, as illustrated by the next two definitions. Definition 3.3.
The spherical diameter sphdiam( · ) : P → [0 ,
1] is defined bysphdiam( ζ ) := || ζ , ζ || for any ζ ∈ P .In [B19, Section 6.1.2], the diameter of ζ ∈ A is defined asdiam( ζ ) := inf {k z − a k ζ : a ∈ C v } Defining | ζ | := k z k ζ , we have the identitysphdiam( ζ ) = diam( ζ )max { , | ζ | } for all ζ ∈ A , with sphdiam( ∞ ) = 0. Definition 3.4.
Let f ∈ C v ( z ) be a rational function. The spherical derivative of f on P is f ♮ ( ζ ) := lim ζ ′ → ζ || f ( ζ ) , f ( ζ ′ ) |||| ζ , ζ ′ || where the convergence ζ ′ → ζ is with respect to the Gel’fand topology on P .Our next result shows how to compute the spherical derivative in practice. Proposition 3.5.
Let f ∈ C v ( z ) be a nonconstant rational function, and let ζ ∈ P . (a) If ζ = x ∈ P ( C v ) , then f ♮ ( x ) = f ( x ) is the classical spherical derivative on P ( C v ) , given by f ♮ ( x ) = f ( x ) := lim y → x ρ ( f ( x ) , f ( y )) ρ ( x, y ) ∈ R ≥ , where y → x in P ( C v ) . In particular, if x, f ( x ) ∈ C v , then f ♮ ( x ) = | f ′ ( x ) | · max { , | x | } max { , | f ( x ) | } . -STABILITY 7 (b) If ζ ∈ H , then f ♮ ( ζ ) = sphdiam( f ( ζ ))sphdiam( ζ ) ∈ R > . Proof.
By Proposition 3.2, the maps || ζ , ·|| : P → R and || f ( ζ ) , ·|| : P → R arecontinuous. Since f : P → P is also continuous, we havelim ζ ′ → ζ || f ( ζ ) , f ( ζ ′ ) || = || f ( ζ ) , f ( ζ ) || = sphdiam( f ( ζ )) , and lim ζ ′ → ζ || ζ , ζ ′ || = || ζ , ζ || = sphdiam( ζ ) . If ζ ∈ H , then sphdiam( ζ ) >
0. Because f is nonconstant, we have f ( ζ ) ∈ H aswell, and hence also sphdiam( f ( ζ )) >
0. Therefore, f ♮ ( ζ ) = sphdiam( f ( ζ ))sphdiam( ζ ) > . Otherwise, we have ζ = x ∈ P ( C v ), so that f ( ζ ) = f ( x ) ∈ P ( C v ), and hence f ♮ ( ζ ) = lim ζ ′ → ζ || f ( ζ ) , f ( ζ ′ ) |||| ζ , ζ ′ || = lim y → x || f ( x ) , f ( y ) |||| x, y || = lim y → x ρ ( f ( x ) , f ( y )) ρ ( x, y ) = f ( x ) ≥ , where the second and third limits are for y → x = ζ in P ( C v ), and the second equalityfollows from the density of P ( C v ) in P . (cid:3) Recall that the chordal metric ρ is invariant under the action of PGL(2 , O ). Therefore,by the third bullet point of Definition 3.1, we have(3.1) h ♮ ( ζ ) = 1 for all h ∈ PGL(2 , O ) and ζ ∈ P . The spherical derivative also satisfies the following chain rule.
Proposition 3.6.
For any rational functions f, g ∈ C v ( z ) and any ζ ∈ P , we have ( f ◦ g ) ♮ ( ζ ) = f ♮ ( g ( ζ )) · g ♮ ( ζ ) Proof.
By continuity, we have( f ◦ g ) ♮ ( ζ ) = lim ζ ′ → ζ || f ( g ( ζ )) , f ( g ( ζ ′ )) |||| ζ , ζ ′ || = lim ζ ′ → ζ || f ( g ( ζ )) , f ( g ( ζ ′ )) |||| g ( ζ ) , g ( ζ ′ ) || · lim ζ ′ → ζ || g ( ζ ) , g ( ζ ′ ) |||| ζ , ζ ′ || = f ♮ ( g ( ζ )) · g ♮ ( ζ ) (cid:3) We close this section with the following lemma concerning a disk on which the rationalfunction f has Weierstrass degree 1. Lemma 3.7.
Let f ∈ C v ( z ) , let a, b ∈ C v with | a | , | b | ≤ , and < r, s ≤ . Suppose f maps D ( a, r ) bijectively onto D ( b, s ) . Then f ♮ ( ζ ) = s/r for any ζ ∈ D an ( a, r ) . Inparticular, we have sphdiam( f ( ζ )) = sr sphdiam( ζ ) ROBERT L. BENEDETTO AND JUNGHUN LEE
Proof.
By [B19, Proposition 3.20], we have(3.2) | f ( x ) − f ( y ) | = sr | x − y | for all x, y ∈ D ( a, r ) . Recall that ρ ( x, y ) = | x − y | for any x, y ∈ O . Because D ( a, r ) , D ( b, s ) ⊆ O , it followsthat (cid:13)(cid:13) f ( x ) , f ( y ) k = sr k x, y k for all x, y ∈ D ( a, r ) . Therefore, by the third bullet point of Definition 3.1, we have (cid:13)(cid:13) f ( ζ ) , f ( ζ ) (cid:13)(cid:13) = sr k ζ , ζ k for all ζ ∈ D an ( a, r ) , which is the desired conclusion for ζ not of type I, by definition of the spherical di-ameter. Finally, the conclusion for ζ of type I is immediate from equation (3.2) andProposition 3.5(a). (cid:3) Basic Lemmas
Lemma 4.1.
Let f ∈ C v ( z ) . Suppose there exists δ > such that (cid:0) f n (cid:1) ♮ ( ζ ) ≥ δ for all ζ ∈ J an ,f and n ∈ N . Then there exist δ ′ > and h ∈ PGL(2 , C v ) such that the map g := h ◦ f ◦ h − satisfies: • | g ( ζ ) | > for all ζ ∈ P with | ζ | > , • J an ,g ⊆ D an (0 , , and • (cid:0) g n (cid:1) ♮ ( ζ ) ≥ δ ′ for all ζ ∈ J an ,g and n ∈ N .Proof. By [B19, Proposition 4.2], there is a type I point α ∈ P ( C v ) that is a nonrepellingfixed point of f . Let h ∈ PGL(2 , O ) be a M¨obius transformation satisfying h ( α ) = ∞ .By [B19, Proposition 4.3(c)], there is some R > g := h ◦ f ◦ h − satisfies | g ( x ) | > R for all x ∈ P ( C v ) with | x | > R ; and by [B19, Theorem 4.18], wehave | g ( x ) | ≥ | x | for all such x .Choose b ∈ C × v with | b | ≥ R , and define h ∈ PGL(2 , C v ) by h ( z ) := z/b . Define h := h ◦ h and g := h ◦ f ◦ h − . Then | g ( x ) | ≥ | x | for all x ∈ C v with | x | > F an ,g = h ( F an ,f ) of g must contain P r D an (0 , J an ,g = h ( J an ,f ) is contained in D an (0 , h ♮ ( ζ ) , (cid:0) h − (cid:1) ♮ ( ζ ) ≥ min (cid:8) | b | , | b | − (cid:9) for all ζ ∈ P . Therefore, by equation (3.1) and Proposition 3.6, we also have h ♮ ( ζ ) , (cid:0) h − (cid:1) ♮ ( ζ ) ≥ min (cid:8) | b | , | b | − (cid:9) for all ζ ∈ P . Define δ ′ := δ min {| b | , | b | − } >
0. For any ζ ∈ J an ,g , again by Proposition 3.6, we have (cid:0) g n (cid:1) ♮ ( ζ ) = h ♮ (cid:0) f n ( h − ( ζ )) (cid:1) · (cid:0) f n (cid:1) ♮ (cid:0) h − ( ζ ) (cid:1) · (cid:0) h − (cid:1) ♮ ( ζ ) ≥ min (cid:8) | b | , | b | − (cid:9) · (cid:0) f n (cid:1) ♮ (cid:0) h − ( ζ ) (cid:1) · min (cid:8) | b | , | b | − (cid:9) ≥ δ ′ . (cid:3) -STABILITY 9 Define the real number κ := ( | p | / ( p − if p > , p = 0 , where p is the residue characteristic of C v . Note that 0 < κ ≤
1, since C v itself hascharacteristic zero.If a convergent power series F ∈ C v [[ z − a ]] on a disk D ( a, r ) has no critical points, itis still possible that F may not be injective on D ( a, r ). However, the next result showsthat F is injective on the smaller disk D ( a, r ), scaling distances by a factor of | F ′ ( a ) | . Lemma 4.2.
Fix a ∈ C v and r > . Let F ( z ) = ∞ X i =0 c i ( z − a ) i ∈ C v [[ z − a ]] be a power series converging on D ( a, r ) . If F has no critical points in D ( a, r ) , then F maps D ( a, κr ) bijectively onto D ( c , | c | κr ) .Proof. Because F has no critical points in D ( a, r ), the power series F ′ has Weierstrassdegree zero on this disk, and hence | nc n | r n − ≤ | c | for all n ∈ N . In addition, by definition of κ , we have κ n − ≤ | n | , and hence | c n | ( κr ) n ≤ | nc n | κr n ≤ | c | ( κr ) for all n ∈ N . Therefore, F − c has Weierstrass degree 1 and hence is injective on D ( a, κr ). By [B19,Theorem 3.15], F maps D ( a, κr ) bijectively onto D ( c , | c | κr ). (cid:3) Lemma 4.3.
Let f ∈ C v ( z ) be a nonconstant rational map. Suppose that all poles and(type I) critical points of f lie in the Fatou set F an ,f . Then there exists ǫ > suchthat for any point a ∈ C v for which D an ( a, ǫ ) ∩ J an ,f = ∅ , we have that f maps D ( a, r ) bijectively onto D ( f ( a ) , | f ′ ( a ) | r ) for any radius r with < r ≤ ǫ .Proof. Denote by CP( f ) the set of poles and (type I) critical points of f . Since each c ∈ CP( f ) lies in F an ,f , there is an associated radius δ c > D an ( c, δ c ) ⊆ F an ,f .Because CP( f ) is finite, we may define ǫ := min { δ c | c ∈ CP( f ) } > ǫ := κǫ > . For any a ∈ C v for which D an ( a, ǫ ) intersects J an ,f , the larger disk D an ( a, ǫ ) alsointersects J an ,f , and hence cannot contain any points of CP( f ). After all, if c ∈ CP( f )lies in D an ( a, ǫ ), then D an ( a, ǫ ) = D an ( c, ǫ ) is contained in F an ,f , a contradiction.For such a ∈ C v , since f has no poles in D ( a, ǫ ), we may write f as a power series f ( z ) = ∞ X i =0 c i ( z − a ) i ∈ C v [[ z − a ]]converging on D ( a, ǫ ), with c = f ( a ) and c = f ′ ( a ). By Lemma 4.2, for 0 < r ≤ ǫ , wehave that f maps D ( a, r ) bijectively onto D ( f ( a ) , | f ′ ( a ) | r ). (cid:3) Technical Lemmas
To prepare for the proof itself, we need to set some notation and present severaltechnical lemmas. Throughout this section, we assume f ∈ C v ( z ) is as in Theorem 1.1.By Lemma 4.1, we may assume that J an ,f ⊆ D an (0 , | f ( x ) | > | x | >
1, andsuch that ( f n ) ♮ ( ζ ) ≥ δ for all ζ ∈ J an ,f and n ∈ N . Choose ǫ > f is injective on D ( a, ǫ ) for any a ∈ C v for which D an ( a, ǫ ) ∩ J an ,f = ∅ . Without loss,assume that δ, ǫ < ζ ∈ J an ,f , define the real quantities σ ( ζ ) := inf (cid:8) ( f n ) ♮ ( ζ ) (cid:12)(cid:12) n ∈ N (cid:9) , ν ( ζ ) := δ ǫσ ( ζ ) , and for each n ∈ N , µ n ( ζ ) := ν (cid:0) f n ( ζ ) (cid:1) ( f n ) ♮ ( ζ ) ν ( ζ ) = σ ( ζ )( f n ) ♮ ( ζ ) σ (cid:0) f n ( ζ ) (cid:1) . The function σ : J an ,f → R will serve as a local scaling factor with respect to which f will be everywhere expanding on J an ,f (see Lemma 5.1.(b) below).We also partition J an ,f into two pieces: J +an ,f := { ζ ∈ J an ,f | sphdiam( ζ ) ≥ ν ( ζ ) } , and J ,f := { ζ ∈ J an ,f | sphdiam( ζ ) < ν ( ζ ) } . Moreover, for each n ∈ N , define J n an ,f := f − n ( J ,f ).Finally, we cover J ,f with open disks, by setting Ω := [ ζ ∈J ,f D an (cid:0) ζ , ν ( ζ ) (cid:1) We will multiply the radii ν ( ζ ) by the contracting factors µ n ( ζ ) to produce even smallerneighborhoods of J n an ,f . Lemma 5.1.
For any ζ ∈ J an ,f , the following statements hold. (a) δ ≤ σ ( ζ ) ≤ and δ ǫ ≤ ν ( ζ ) ≤ δǫ . (b) f ♮ ( ζ ) σ (cid:0) f ( ζ ) (cid:1) ≥ σ ( ζ ) .Proof. (a) . For any ζ ∈ J an ,f , choosing n = 0 in the definition of σ ( ζ ) yields the upperbound σ ( ζ ) ≤
1. The lower bound follows from the hypothesis that ( f n ) ♮ ( ζ ) ≥ δ for all n ∈ N . The bounds on ν follow immediately. (b) . For any n ∈ N , we have f ♮ ( ζ ) · ( f n ) ♮ (cid:0) f ( ζ ) (cid:1) = (cid:0) f n +1 (cid:1) ♮ ( ζ ) , by Proposition 3.6. Taking the infimum over all n ∈ N , we have f ♮ ( ζ ) σ (cid:0) f ( ζ ) (cid:1) = inf n(cid:0) f n (cid:1) ♮ ( ζ ) (cid:12)(cid:12)(cid:12) n ∈ N o ≥ inf n(cid:0) f n (cid:1) ♮ ( ζ ) (cid:12)(cid:12)(cid:12) n ∈ N o = σ ( ζ ) . (cid:3) Lemma 5.2.
For each ζ ∈ J an ,f , we have -STABILITY 11 (a) µ n ( ζ ) = n − Y i =0 µ (cid:0) f i ( ζ ) (cid:1) for all n ∈ N . (b) 1 = µ ( ζ ) ≥ µ ( ζ ) ≥ µ ( ζ ) ≥ · · · . (c) If ζ ∈ J +an ,f , then f ( ζ ) ∈ J +an ,f .Proof. (a) . Given ζ ∈ J an ,f and n ∈ N , Proposition 3.6 yields n − Y i =0 µ (cid:0) f i ( ζ ) (cid:1) = n − Y i =0 σ (cid:0) f i ( ζ ) (cid:1) f ♮ (cid:0) f i ( z ) (cid:1) σ ( f i +1 ( z )) = (cid:18) n − Y i =0 σ (cid:0) f i ( ζ ) (cid:1) σ ( f i +1 ( ζ )) (cid:19) · (cid:18) n − Y i =0 f ♮ (cid:0) f i ( z ) (cid:1) (cid:19) = σ ( ζ ) σ (cid:0) f n ( ζ ) (cid:1) · f n ) ♮ ( ζ ) = µ n ( ζ ) . (b) . For any ζ ∈ J an ,f , clearly µ ( ζ ) = 1. Observe that µ (cid:0) f i ( ζ ) (cid:1) ≤ i ∈ N , by Lemma 5.1(b) applied to f i ( ζ ). Thus, part (a) of the current lemma immediatelyimplies part (b). (c) . For ζ ∈ J +an ,f , we havesphdiam (cid:0) f ( ζ ) (cid:1) = f ♮ ( ζ ) sphdiam( ζ ) ≥ f ♮ ( ζ ) ν ( ζ )= f ♮ ( ζ ) σ ( ζ ) · δ ǫ ≥ δ ǫσ (cid:0) f ( ζ ) (cid:1) = ν (cid:0) f ( ζ ) (cid:1) , where the first equality is by definition of f ♮ , the second and third equalities are bydefinition of ν , the first inequality is because ζ ∈ J +an ,f , and the second inequality is byLemma 5.1(b). (cid:3) It is immediate from Lemma 5.2(c) that J ,f ⊇ J ,f ⊇ J ,f ⊇ · · · . Lemma 5.3.
For any n ∈ N , f − n ( Ω ) = [ ζ ∈J n an ,f D an (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1) . Moreover, we have Ω ⊇ f − ( Ω ) ⊇ f − ( Ω ) ⊇ · · · .Proof. We prove the equality by induction on n . It is trivial for n = 0. Assume it holdsfor some n = m ∈ N ; we will prove it for m + 1.For the forward inclusion, given ξ ∈ f − ( m +1) ( Ω ), there exists ζ ∈ J m an ,f such that f ( ξ ) ∈ D an ( ζ , µ m ( ζ ) ν ( ζ )). Write f − ( ζ ) = { θ , . . . , θ d } ⊆ J m +1an ,f . For each i = 1 , . . . , d , Lemma 5.1(a) yields µ m +1 ( θ i ) ν ( θ i ) = ν ( f m +1 ( θ i ))( f m +1 ) ♮ ( θ i ) ≤ δǫδ = ǫ. Therefore, by Lemma 4.3, f is injective on each disk D an ( θ i , µ m +1 ( θ i ) ν ( θ i )), scalingdistances by a factor of f ♮ ( θ i ). Hence, the points θ , . . . , θ d are indeed distinct, and f (cid:16) D an (cid:0) θ i , µ m +1 ( θ i ) ν ( θ i ) (cid:1)(cid:17) = D an (cid:0) f ( θ i ) , f ♮ ( θ i ) µ m +1 ( θ i ) ν ( θ i ) (cid:1) = D an (cid:16) f ( θ i ) , µ m (cid:0) f ( θ i ) (cid:1) ν (cid:0) f ( θ i ) (cid:1)(cid:17) = D an (cid:0) ζ , µ m ( ζ ) ν ( ζ ) (cid:1) . Since deg f = d , we have accounted for all preimages of D an ( ζ , µ m ( ζ ) ν ( ζ )). Thus, thereis some j ∈ { , . . . , d } such that ξ ∈ D an (cid:0) θ j , µ m +1 ( θ j ) ν ( θ j ) (cid:1) , completing our proof of the forward inclusion.Conversely, given ζ ∈ J m +1an ,f and ξ ∈ D an ( ζ , µ m +1 ( ζ ) ν ( ζ )), we have f ( ξ ) ∈ f (cid:16) D an (cid:0) ζ , µ m +1 ( ζ ) ν ( ζ ) (cid:1)(cid:17) = D an (cid:0) f ( ζ ) , f ♮ ( ζ ) µ m +1 ( ζ ) ν ( ζ ) (cid:1) = D an (cid:16) f ( ζ ) , µ m (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1)(cid:17) ⊆ f − m ( Ω ) , verifying the reverse inclusion.Finally, for any n ∈ N , we have µ n +1 ( ζ ) ≤ µ n ( ζ ) for all ζ ∈ J n +1an , by Lemma 5.2(b).Since J n +1an ⊆ J n an , it follows immediately that f − n − ( Ω ) ⊆ f − n ( Ω ). (cid:3) Lemma 5.4.
We have \ n ∈ N f − n ( Ω ) = \ n ∈ N J n an ,f = J an ,f ∩ C v . Moreover, for any ζ ∈ J an ,f ∩ C v , we have lim n →∞ µ n ( ζ ) = 0 .Proof. The inclusion ( ⊇ ) in the first equality is immediate from the definitions of Ω and J n an ,f , and the inclusion ( ⊇ ) in the second equality is because sphdiam( f n ( ζ )) = 0 forevery point ζ of type I and every n ∈ N . Thus, to show these two equalities, it sufficesto show that the first set is contained in the third.Given ξ ∈ T ∞ n =0 f − n ( Ω ), by Lemma 5.3, there is a sequence of points { ζ n } ∞ n =0 suchthat for every n ∈ N , we have(5.1) ζ n ∈ J n an ,f and ξ ∈ D an (cid:0) ζ n , µ n ( ζ n ) ν ( ζ n ) (cid:1) Define t := inf (cid:8) µ n ( ζ n ) ν ( ζ n ) (cid:12)(cid:12) n ∈ N (cid:9) ≥ . We claim that t = 0. If not, i.e., if t >
0, then there is some n ∈ N such that t > δ / µ n ( ζ n ) ν ( ζ n ). There must be some m ∈ N such that f m ( D an ( ζ n , µ n ( ζ n ) ν ( ζ n ))) isnot contained in an open disk of radius ǫ , or else f i ( D an ( ζ n , µ n ( ζ n ) ν ( ζ n ))) ⊆ D an (0 , i ∈ N , contradicting the fact that ζ n ∈ J an ,f . Let m be the smallest such integer.Since f i ( D an ( ζ n , µ n ( ζ n ) ν ( ζ n ))) is contained in D an ( ζ n , ǫ ) for every 0 ≤ i < m , repeatedapplication of Lemma 4.3 shows that f m maps D an ( ζ n , µ n ( ζ n ) ν ( ζ n )) bijectively onto adisk of radius greater than ǫ . -STABILITY 13 Choose a point θ ∈ P as follows. If diam( ξ ) ≥ δ / t , then choose θ := ξ ; otherwise,choose θ to be the unique boundary point of the disk D an ( ξ, δ / t ). Then for every i ∈ N , we have θ ∈ D an ( ζ i , µ i ( ζ i ) ν ( ζ i )) , and sphdiam( θ ) = diam( θ ) ≥ δ / t > δµ i ( ζ i ) ν ( ζ i ) . Because θ lies in the disk D an ( ζ n , µ n ( ζ n ) ν ( ζ n )), Lemma 3.7 applied to f m implies that(5.2) sphdiam (cid:0) f m ( θ ) (cid:1) > ǫµ n ( ζ n ) ν ( ζ n ) · diam( θ ) > δǫ ≥ ν (cid:0) f m ( ζ m ) (cid:1) , where the last inequality is by Lemma 5.1(a). However, since θ also lies in the disk D an ( ζ m , µ m ( ζ m ) ν ( ζ m )), we have f m ( θ ) ∈ f m (cid:16) D an (cid:0) ζ m , µ m ( ζ m ) ν ( ζ m ) (cid:1)(cid:17) = D an (cid:16) f m ( ζ m ) , ν (cid:0) f m ( ζ m ) (cid:1)(cid:17) . Therefore, sphdiam( f m ( θ )) = diam( f m ( θ )) ≤ ν ( f m ( ζ m )), contradicting inequality (5.2).Our claim follows; we must have t = 0.The point ξ is therefore contained in disks D an ( ζ n , µ n ( ζ n ) ν ( ζ n )) of arbitrarily smallpositive radius. Thus, diam( ξ ) = 0, implying that ξ ∈ C v . The points ζ n ∈ J an ,f accumulate at ξ , and hence ξ ∈ J an ,f ∩ C v , as desired.Finally, given ξ ∈ J an ,f ∩ C v , we may choose the sequence { ζ n } ∞ n =0 of (5.1) to be theconstant sequence ζ n := ξ . Since the sequence { µ n ( ξ ) } ∞ n =0 is decreasing by Lemma 5.2(b),the above claim immediately yields µ n ( ξ ) → (cid:3) Lemma 5.5.
For any γ > , there is an open subset W of Rat d ( C v ) containing f suchthat for any g ∈ W , we have (a) | g ( x ) | > for any x ∈ C v with | x | > , and (b) | g ( x ) − f ( x ) | < γ for all x ∈ f − ( D (0 , .Proof. Write f = F/G for relatively prime polynomials
F, G ∈ C v [ z ], with F ( z ) = a d z d + · · · + a and G ( z ) = b d z d + · · · + b , and write an arbitrary g ∈ Rat d ( C v ) as ˜ F / ˜ G , with ˜ F , ˜ G ∈ C v [ z ] given by(5.3) ˜ F ( z ) = A d z d + · · · + A and ˜ G ( z ) = B d z d + · · · + B . As we assumed at the start of this section, we have | f ( x ) | > x ∈ C v with | x | >
1. Therefore, a d = 0, and | a i | , | b i | ≤ | a d | for all i = 0 , . . . , d . Let W be the subsetof Rat d ( C v ) defined by the (open) conditions | A i − a i | < | a d | and | B i − b i | < | a d | for all i = 0 , . . . , d. Then any g ∈ W has | g ( x ) | > x ∈ C v with | x | > y , . . . , y ℓ denote the distinct poles of f in D (0 , < r < f ( D ( y i , r )) ⊆ P ( C v ) r D (0 ,
1) for each i . Then | G ( x ) | ≥ C for all x ∈ D (0 , r (cid:0) D ( y , r ) ∪ · · · ∪ D ( y ℓ , r ) (cid:1) , where C := min {k G k ζ ( y ,r ) , . . . , k G k ζ ( y ℓ ,r ) } > ℓ ≥ , or C := | G (0) | = k G k ζ (0 , if ℓ = 0.Because | a i | , | b i | ≤ | a d | for all i = 0 , . . . , d , we have | F ( x ) | ≤ | a d | and | G ( x ) | ≤ | a d | forall x ∈ D (0 , With notation as in equations (5.3), define W to be the open neighborhood of f inRat d ( C v ) given by the conditions | A i − a i | < C γ | a d | and | B i − b i | < min (cid:26) C γ | a d | , C (cid:27) for each i. Then any g = ˜ F / ˜ G ∈ W satisfies (cid:12)(cid:12) ˜ F ( x ) − F ( x ) (cid:12)(cid:12) < C γ | a d | and (cid:12)(cid:12) ˜ G ( x ) − G ( x ) (cid:12)(cid:12) < min (cid:26) C γ | a d | , C (cid:27) for all x ∈ D (0 , . Therefore, for any g = ˜ F / ˜ G ∈ W and x ∈ C v such that | f ( x ) | ≤
1, we have (cid:12)(cid:12) ˜ G ( x ) − G ( x ) (cid:12)(cid:12) < C ≤ | G ( x ) | , and hence (cid:12)(cid:12) ˜ G ( x ) (cid:12)(cid:12) = | G ( x ) | ≥ C. Thus, (cid:12)(cid:12) g ( x ) − f ( x ) (cid:12)(cid:12) = (cid:12)(cid:12) G ( x ) (cid:0) ˜ F ( x ) − F ( x ) (cid:1) − F ( x ) (cid:0) ˜ G ( x ) − G ( x ) (cid:1)(cid:12)(cid:12) | G ( x ) | · | ˜ G ( x ) |≤ C max (cid:8) | a d || ˜ F ( x ) − F ( x ) | , | a d || ˜ G ( x ) − G ( x ) | (cid:9) < C max n C γ, C γ (cid:9) = γ. Finally, defining W := W ∩ W , we are done. (cid:3) Lemma 5.6.
Let W ⊆ Rat d ( C v ) be the open neighborhood of f from Lemma 5.5 forsome γ with < γ < δ ǫ . Then for any g ∈ W and any ζ ∈ J ,f , we have f ♮ ( ζ ) = g ♮ ( ζ ) ,and g maps D (cid:0) ζ , µ ( ζ ) ν ( ζ ) (cid:1) bijectively onto D (cid:16) f ( ζ ) , ν (cid:0) f ( ζ ) (cid:1)(cid:17) . Moreover, g − ( Ω ) = f − ( Ω ) .Proof. Given g and ζ as specified, let r := µ ( ζ ) ν ( ζ ), so thatdiam( ζ ) = sphdiam( ζ ) < r ≤ δǫ, by Lemmas 5.1(a) and 5.2(b). Choose x ∈ C v with k z − x k ζ < r , so that ζ ∈ D an ( x, r ) ⊆ Ω . Recall that f has no poles in Ω , and hence neither does g , by the defining propertyof W in Lemma 5.5(b). Thus, we may expand both f and g as power series f ( z ) = ∞ X i =0 a i ( z − x ) i and g ( z ) = ∞ X i =0 b i ( z − x ) i convergent on D ( x, r ). Because r ≤ ǫ , Lemma 4.3 implies that f − a is injective on D ( x, r ) and hence has Weierstrass degree 1. That is, | a i | r i ≤ | a | r for all i ≥
1. We willshow the analogous statement for g − b .By the defining property of W , the power series g ( z ) − f ( z ) = ∞ X i =0 ( b i − a i )( z − x ) i satisfies | g ( y ) − f ( y ) | < γ for y ∈ D ( x, r ), and hence(5.4) (cid:12)(cid:12) b i − a i (cid:12)(cid:12) r i ≤ γ for all i ∈ N . -STABILITY 15 On the other hand, we have | a | = | f ′ ( x ) | = f ♮ ( ζ ), and therefore | a | r = f ♮ ( ζ ) µ ( ζ ) ν ( ζ ) = ν (cid:0) f ( ζ ) (cid:1) ≥ δ ǫ > γ. Combined with (5.4) for i = 1, it follows that | b − a | < | a | , and hence(5.5) g ♮ ( ζ ) = | g ′ ( x ) | = | b | = | a | = f ♮ ( ζ ) . Furthermore, applying (5.4) for i ≥
1, we have | b i | r i ≤ max (cid:8)(cid:12)(cid:12) b i − a i (cid:12)(cid:12) r i , | a i | r i (cid:9) ≤ max (cid:8) γ, | a | r (cid:9) = | b | r for all i ≥ . That is, g − b has Weierstrass degree 1 on D ( x, r ).Thus, g maps D ( ζ , r ) = D ( x, r ) bijectively onto D ( g ( x ) , | g ′ ( x ) | r ). However, since | b | = | a | by equation (5.5), we have | g ′ ( x ) | r = | b | r = | a | r = ν (cid:0) f ( ζ ) (cid:1) . Hence, | g ( y ) − f ( y ) | < γ ≤ δ ǫ ≤ ν ( f ( ζ )) = (cid:12)(cid:12) g ′ ( x ) (cid:12)(cid:12) r for all y ∈ D ( x, r ) . Therefore, the image of D ( ζ , r ) under g is D (cid:0) g ( x ) , | g ′ ( x ) | r (cid:1) = D (cid:16) f ( x ) , ν (cid:0) f ( ζ ) (cid:1)(cid:17) = D (cid:16) f ( ζ ) , ν (cid:0) f ( ζ ) (cid:1)(cid:17) . Lastly, we must show that g − ( Ω ) = f − ( Ω ). For any ζ ∈ J ,f , let θ , . . . , θ d ∈ J ,f be the d preimages of ζ under f , which we know to be distinct as in the proof ofLemma 5.3. By the first part of the current lemma, we also know that g maps each disk D ( θ i , µ ( θ i ) ν ( θ i )) bijectively onto D ( ζ , ν ( ζ )), accounting for all d preimages of D ( ζ , ν ( ζ ))under g . Thus, g − (cid:16) D an (cid:0) ζ , ν ( ζ ) (cid:1)(cid:17) = D an (cid:0) θ , µ ( θ ) ν ( θ ) (cid:1) ∪ · · · ∪ D an (cid:0) θ d , µ ( θ d ) ν ( θ d ) (cid:1) = f − (cid:16) D an (cid:0) ζ , ν ( ζ ) (cid:1)(cid:17) . Taking the union across all ζ ∈ J ,f , we have g − ( Ω ) = f − ( Ω ). (cid:3) Proof of Theorem 1.1
We are now prepared to prove our main result, as follows. In Step 1, we define asequence { h n } ∞ n =0 of maps from subsets of Ω to Ω , and we investigate properties of thissequence in Step 2. Then, in Step 3, we glue the maps h n to produce the desired map h : P → P that is a conjugacy on f − ( Ω ) ∪ J an ,f . Finally, in Steps 4 and 5, we showthat h has all the claimed properties. Proof of Theorem 1.1.
Step 1 . Let W ⊆ Rat d ( C v ) be the neighborhoood W of f givenby Lemma 5.5 for γ = δ ǫ/
2. For the rest of this proof, consider an arbitrary map g ∈ W .By Lemma 5.6, for each ζ ∈ J ,f , the map g is injective on D ( ζ , µ ( ζ ) ν ( ζ )), withimage D ( f ( ζ ) , ν ( f ( ζ ))). Thus, there exists a map G ζ : D (cid:16) f ( ζ ) , ν (cid:0) f ( ζ ) (cid:1)(cid:17) → D (cid:0) ζ , µ ( ζ ) ν ( ζ ) (cid:1) which is an inverse to g given by a power series convergent on D ( f ( ζ ) , ν ( f ( ζ ))). Notethat if ξ ∈ J ,f lies in the same disk D ( ζ , µ ( ζ ) ν ( ζ )), then the power series G ζ and G ξ agree, since g is injective on both D ( ζ , µ ( ζ ) ν ( ζ )) and D ( ξ, µ ( ξ ) ν ( ξ )). As usual, thepower series defining G ζ extends via continuity to(6.1) G ζ : D an (cid:16) f ( ζ ) , ν (cid:0) f ( ζ ) (cid:1)(cid:17) → D an (cid:0) ζ , µ ( ζ ) ν ( ζ ) (cid:1) We now define a sequence { h n } ∞ n =0 of functions, with h : P → P and h n : f − n ( Ω ) → Ω for n ≥ h : P → P by h ( ζ ) := ζ . For each n ∈ N ,having already defined h n − , we define h n as follows. For each ζ ∈ J n an ,f , define h n on D an ( ζ , µ n ( ζ ) ν ( ζ )) by h n := G ζ ◦ h n − ◦ f, where G ζ is the local inverse of g defined in (6.1). Step 2 . We will now show that for each n ∈ N , • h n is a well-defined function mapping f − n ( Ω ) bijectively onto g − n ( Ω ), given by aconvergent power series on each disk D an ( ζ , µ n ( ζ ) ν ( ζ )) for ζ ∈ J n an ,f , • h n is an isometry on f − n ( Ω ) ∩ C v , and • for every ζ ∈ J n an ,f and x ∈ D ( ζ , µ n ( ζ ) ν ( ζ )), we have (cid:12)(cid:12) h n ( x ) − x (cid:12)(cid:12) < µ ( ζ ) ν ( ζ ),with(6.2) (cid:12)(cid:12) h n ( x ) − h n − ( x ) (cid:12)(cid:12) < µ n ( ζ ) ν ( ζ ) if n ≥ . We proceed by induction. For n = 0, all three properties hold trivially.For n ≥
1, assume the three bullet points hold for n −
1. Then h n is well-definedbecause if ξ lies in both D an ( ζ , µ ( ζ ) ν ( ζ )) and D an ( ζ ′ , µ ( ζ ′ ) ν ( ζ ′ )), then as we noted inStep 1, the power series G ζ and G ζ ′ agree. That is, the value of h n ( ξ ) is independent ofwhich point ζ is chosen as the center of the disk.For any ζ ∈ J n an ,f , it is immediate from Proposition 3.6 and the definition of µ n that f ♮ ( ζ ) µ n ( ζ ) ν ( ζ ) = ν (cid:0) f n ( ζ ) (cid:1) ( f n − ) ♮ (cid:0) f ( ζ ) (cid:1) = µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1) , and hence, by Lemmas 3.7 and 4.3, f is a convergent power series on the disk U n := D ( ζ , µ n ( ζ ) ν ( ζ )), mapping D (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1) bijectively onto D (cid:16) f ( ζ ) , µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1)(cid:17) , and multiplying all distances by a factor of f ♮ ( ζ ). By our inductive assumptions, h n − acts as a power series mapping D (cid:16) f ( ζ ) , µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1)(cid:17) isometrically onto V n − := D (cid:16) h n − (cid:0) f ( ζ ) (cid:1) , µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1)(cid:17) ⊆ D (cid:16) f ( ζ ) , ν (cid:0) f ( ζ ) (cid:1)(cid:17) , where the inclusion is because of the inductive assumption that (cid:12)(cid:12) h n − (cid:0) f ( x ) (cid:1) − f ( x ) (cid:12)(cid:12) < µ (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1) ≤ ν (cid:0) f ( ζ ) (cid:1) for all x ∈ V n − . Thus, G ζ is defined as an injective power series on the disk V n − = h n − ( f ( U n )), mul-tiplying all distances by ( g ♮ ( ζ )) − = ( f ♮ ( ζ )) − , where this equality is by Lemma 5.6.Therefore, h n = G ζ ◦ h n − ◦ f is a power series on U n , mapping(6.3) D (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1) isometrically onto D (cid:0) h n ( ζ ) , µ n ( ζ ) ν ( ζ ) (cid:1) . -STABILITY 17 We will prove h n is an isometry on all of f − n ( Ω ) ∩ C v shortly, but first we prove thethird bullet point for our given n ≥
1. Given ζ ∈ J n an ,f and x ∈ D ( ζ , µ n ( ζ ) ν ( ζ )), we firstclaim that(6.4) (cid:12)(cid:12) h n − (cid:0) f ( x ) (cid:1) − g (cid:0) h n − ( x ) (cid:1)(cid:12)(cid:12) < µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1) . Indeed, if n = 1, we have | f ( x ) − g ( x ) | < δ ǫ/ < ν ( f ( ζ )) by Lemmas 5.1 and 5.5, yield-ing (6.4). If n ≥
2, we have g ( h n − ( x )) = h n − ( f ( x )), and by our inductive assumptionfor f ( x ), we also have (cid:12)(cid:12) h n − (cid:0) f ( x ) (cid:1) − h n − (cid:0) f ( x ) (cid:1)(cid:12)(cid:12) < µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1) , proving (6.4). Moreover, h n − ( f ( x )) and h n − ( f ( x )) both lie in D ( f ( ζ ) , ν ( f ( ζ ))), andhence we may apply G ζ . Recalling that G ζ scales distances by ( f ♮ ( ζ )) − , we have(6.5) (cid:12)(cid:12) h n ( x ) − h n − ( x ) (cid:12)(cid:12) < ( f ♮ ( ζ )) − µ n − (cid:0) f ( ζ ) (cid:1) ν (cid:0) f ( ζ ) (cid:1) = µ n ( ζ ) ν ( ζ ) , giving inequality (6.2). The first part of the third bullet point then follows from thisbound together with the inductive assumption, because (cid:12)(cid:12) h n ( x ) − x (cid:12)(cid:12) ≤ max (cid:8)(cid:12)(cid:12) h n ( x ) − h n − ( x ) (cid:12)(cid:12) , (cid:12)(cid:12) h n − ( x ) − x (cid:12)(cid:12)(cid:9) < max (cid:8) µ n ( ζ ) ν ( ζ ) , µ ( ζ ) ν ( ζ ) (cid:9) = µ ( ζ ) ν ( ζ ) , where the final equality is by Lemma 5.2(b).As for the second bullet point, that h n is an isometry on f − n ( Ω ) ∩ C v , consider arbitrary x, y ∈ f − n ( Ω ) ∩ C v . Then there exist ζ , ξ ∈ J n an ,f such that x ∈ D ( ζ , µ n ( ζ ) ν ( ζ )) and y ∈ D ( ξ, µ n ( ξ ) ν ( ξ )), by Lemma 5.3. Without loss, µ n ( ζ ) ν ( ζ ) ≥ µ n ( ξ ) ν ( ξ ).If | x − y | < µ n ( ζ ) ν ( ζ ), then we have | h n ( x ) − h n ( y ) | = | x − y | by (6.3). Otherwise, (cid:12)(cid:12) h n ( x ) − h n − ( x ) (cid:12)(cid:12) < µ n ( ζ ) ν ( ζ ) ≤ | x − y | = (cid:12)(cid:12) h n − ( x ) − h n − ( y ) (cid:12)(cid:12) , and similarly for (cid:12)(cid:12) h n ( y ) − h n − ( y ) (cid:12)(cid:12) , where the first inequality is by (6.5), and the equalityis by our inductive assumption. Thus, (cid:12)(cid:12) h n ( x ) − h n ( y ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) h n ( x ) − h n − ( x ) (cid:1) − (cid:0) h n ( y ) − h n − ( y ) (cid:1) + h n − ( x ) − h n − ( y ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) h n − ( x ) − h n − ( y ) (cid:12)(cid:12) = | x − y | , as desired.It remains to show that h n maps f − n ( Ω ) bijectively onto g − n ( Ω ). Because h n is givenlocally by power series, it suffices to show that h n maps f − n ( Ω ) ∩ C v bijectively onto g − n ( Ω ) ∩ C v .Since h n is an isometry, we already know it is injective on f − n ( Ω ) ∩ C v . In addition,for any x ∈ f − n ( Ω ) ∩ C v , we have f ( x ) ∈ f − ( n − ( Ω ), and therefore by our inductiveassumption, we also have h n − (cid:0) f ( x ) (cid:1) ∈ g − ( n − ( Ω ) . Since each map G ζ is a local inverse of g , it follows that h n ( x ) ∈ g − n ( Ω ).Finally, given y ∈ g − n ( Ω ) ∩ C v , we have g ( y ) ∈ g − ( n − ( Ω ), and hence there is some˜ x ∈ f − ( n − ( Ω ) such that h n − (˜ x ) = g ( y ), by our inductive assumption. By Lemma 5.3,there is some ζ ∈ J n − ,f such that ˜ x ∈ D ( ζ , µ n − ( ζ ) ν ( ζ )). Writing f − ( ζ ) = { θ , . . . , θ d } ,each disk D (cid:0) θ i , µ ( θ i ) ν ( θ i ) (cid:1) maps bijectively onto D (cid:0) ζ , ν ( ζ ) (cid:1) under both f and g , by Lemmas 4.3 and 5.6. Moreover, because (cid:12)(cid:12) g ( y ) − ˜ x (cid:12)(cid:12) = (cid:12)(cid:12) h n − (˜ x ) − ˜ x (cid:12)(cid:12) < µ ( ζ ) ν ( ζ )by our inductive assumption, we have g ( y ) ∈ D ( ζ , µ ( ζ ) ν ( ζ )). Therefore, there is some j ∈ { , . . . , d } such that y ∈ D ( θ j , ν ( θ j )), and there is some x ∈ D ( θ j , ν ( θ j )) such that f ( x ) = ˜ x . Since ˜ x ∈ f − ( n − ( Ω ), we have x ∈ f − n ( Ω ). Writing θ := θ j , we have G θ ( g ( y )) = y , and hence h n ( x ) = y . Thus, h n does indeed map f − n ( Ω ) ∩ C v bijectivelyonto g − n ( Ω ) ∩ C v , completing our induction. Step 3 . For each n ∈ N , define H n : P → P by the following inductive procedure.Let H = h , and for n ∈ N and ζ ∈ P , let H n ( ζ ) := ( H n − ( ζ ) if ζ ∈ P r f − n ( Ω ) ,h n ( ζ ) if ζ ∈ f − n ( Ω ) . Define h : P → P by h ( ζ ) := lim n →∞ H n ( ζ ) , or equivalently h ( ζ ) = ζ if ζ ∈ P r f − ( Ω ) ,h n ( ζ ) if ζ ∈ f − n ( Ω ) r f − ( n +1) ( Ω ) for n ∈ N , lim n →∞ h n ( ζ ) if ζ ∈ T n ∈ N f − n ( Ω ) . For the third case, recall from Lemma 5.4 that T n ∈ N f − n ( Ω ) = J an ,f ∩ C v , and thatlim n →∞ µ n ( ζ ) = 0 for such ζ . Thus, by the third bullet point of Step 2, the sequence { h n ( ζ ) } ∞ n =0 is Cauchy and hence converges to h ( ζ ) ∈ Ω ∩ C v . Together with Lemmas 5.3and 5.6, as well as the first bullet point of Step 2, it follows that h is indeed a functionfrom P to itself. Moreover, by the second bullet point of Step 2, h maps f − n ( Ω ) ∩ C v bijectively onto g − n ( Ω ) ∩ C v for each n ∈ N .We claim that h is an isometry on C v . To see this, given x, y ∈ C v , we consider severalcases. First, if x, y ∈ J an ,f , then | h ( x ) − h ( y ) | = (cid:12)(cid:12)(cid:12) lim n →∞ h n ( x ) − h n ( y ) (cid:12)(cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12) h n ( x ) − h n ( y ) (cid:12)(cid:12) = lim n →∞ | x − y | = | x − y | , where the third equality is because h n is an isometry on f − n ( Ω ) ∩ C v , by Step 2. Second,if x, y ∈ f − n ( Ω ) r f − ( n +1) ( Ω ) for some n ∈ N , or if x, y ∈ P r f − ( Ω ) with n = 0, then | h ( x ) − h ( y ) | = (cid:12)(cid:12) h n ( x ) − h n ( y ) (cid:12)(cid:12) = | x − y | . Finally, suppose there is some n ∈ N such that(6.6) x ∈ ( P r f − ( Ω ) if n = 0 ,f − n ( Ω ) r f − ( n +1) ( Ω ) if n ≥ , and y ∈ f − ( n +1) ( Ω ). Then for every m > n for which y ∈ f − m ( Ω ), there is some ζ m ∈ J m an ,f such that y ∈ D ( ζ m , µ m ( ζ m ) ν ( ζ m )). For any integer ℓ with n < ℓ ≤ m , wehave ζ m ∈ J ℓ an ,f . By Lemma 5.2(b), we also have y ∈ D ( ζ m , µ ℓ ( ζ m ) ν ( ζ m )). Thus, itfollows from the third bullet point of Step 2 that | h ℓ ( y ) − h ℓ − ( y ) | < µ ℓ ( ζ m ) ν ( ζ m ) -STABILITY 19 On the other hand, it follows from our assumption (6.6) that | x − y | ≥ µ n +1 ( ζ m ) ν ( ζ m ).Therefore, | h m ( y ) − h n ( y ) | ≤ max (cid:8)(cid:12)(cid:12) h ℓ ( y ) − h ℓ − ( y ) (cid:12)(cid:12)(cid:9) < max { µ ℓ ( ζ m ) ν ( ζ m ) } = µ n +1 ( ζ m ) ν ( ζ m ) ≤ | x − y | = | h n ( x ) − h n ( y ) | where the two maxima are over ℓ ∈ { n + 1 , . . . , m } , and where the first equality is byLemma 5.2(b). Hence,(6.7) (cid:12)(cid:12) H m ( x ) − H m ( y ) (cid:12)(cid:12) = (cid:12)(cid:12) h n ( x ) − h m ( y ) (cid:12)(cid:12) = (cid:12)(cid:12) h n ( x ) − h n ( y ) (cid:12)(cid:12) = | x − y | . If y ∈ J an ,f , we obtain | h ( x ) − h ( y ) | = | x − y | by taking the limit as m → ∞ in (6.7).Otherwise, we obtain | h ( x ) − h ( y ) | = | x − y | by choosing m in (6.7) to be the largestinteger for which y ∈ f − m ( Ω ).Next, we claim that(6.8) h (cid:0) f ( ζ ) (cid:1) = g (cid:0) h ( ζ ) (cid:1) for all ζ ∈ f − ( Ω ) . To see this, suppose first that ζ ∈ f − n ( Ω ) r f − ( n +1) ( Ω ) for some n ∈ N . Then h ( ζ ) = h n ( ζ ), and h ( f ( ζ )) = h n − ( f ( ζ )). Hence, by the construction of h n in Step 1, we have g (cid:0) h ( ζ ) (cid:1) = g (cid:0) h n ( ζ ) (cid:1) = h n − (cid:0) f ( ζ ) (cid:1) = h (cid:0) f ( ζ ) (cid:1) . The only other possibility is that ζ ∈ J an ,f ∩ C v , in which case ζ , f ( ζ ) ∈ f − n ( Ω ) for all n ∈ N . Therefore, g (cid:0) h ( ζ ) (cid:1) = g (cid:16) lim n →∞ h n ( ζ ) (cid:17) = lim n →∞ g (cid:0) h n ( ζ ) (cid:1) = lim n →∞ h n − (cid:0) f ( ζ ) (cid:1) = h (cid:0) f ( ζ ) (cid:1) , proving our claim. Step 4 . Our goal in this step is to show that h : P → P is a homeomorphism. Wealready know that h fixes every point of P r f − ( Ω ) and maps f − ( Ω ) bijectively ontoitself. It follows that h : P → P is bijective. Since P is a compact Hausdorff space,it suffices to show that h − is continuous.To that end, we first recall that for every n ∈ N and every ζ ∈ J n an ,f , both h n − and h n are power series convergent on D an ( ζ , µ n ( ζ ) ν ( ζ )) with Weierstrass degree 1. Therefore,it is immediate from inequality (6.2), along with the fact that h n is an isometry on thetype I points, that h n − (cid:16) D an (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1)(cid:17) = h n (cid:16) D an (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1)(cid:17) for all ζ ∈ J n an ,f By the definition of H n : P → P from Step 3, it follows that(6.9) H n − (cid:16) D an (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1)(cid:17) = H n (cid:16) D an (cid:0) ζ , µ n ( ζ ) ν ( ζ ) (cid:1)(cid:17) for all ζ ∈ J n an ,f . Second, we claim that for every a ∈ C v , every r >
0, and every n ∈ N , we have(6.10) H n (cid:0) D an ( a, r ) (cid:1) = D an (cid:0) H n ( a ) , r (cid:1) and H n (cid:0) D an ( a, r ) (cid:1) = D an (cid:0) H n ( a ) , r (cid:1) . We prove equation (6.10) by induction on n ; it is clearly true for n = 0, since H is theidentity map. For n ∈ N , assuming equation (6.10) holds for H n − , we now show it for H n . Let X be the disk D an ( a, r ) or D an ( a, r ). If X does not intersect D an ( ζ , µ n ( ζ ) ν ( ζ ))for any ζ ∈ J n an ,f , then X ∩ f − n ( Ω ) = ∅ by Lemma 5.3, so that H n ( X ) = H n − ( X )Similarly, if there are any points ζ ∈ J n an ,f for which D an ( ζ , µ n ( ζ ) ν ( ζ )) ⊆ X , then byequation (6.9), we again have H n ( X ) = H n − ( X ). In either case, equation (6.10) follows immediately. The only remaining case is that X ⊆ D an ( ζ , µ n ( ζ ) ν ( ζ )) for some ζ ∈ J n an ,f .In that case, H n | X = h n | X is a power series convergent on the disk X which is anisometry on the type I points, and hence equation (6.10) holds, proving our claim.Third, we make the same claim for h : that for every a ∈ C v and r >
0, we have(6.11) h (cid:0) D an ( a, r ) (cid:1) = D an (cid:0) h ( a ) , r (cid:1) and h (cid:0) D an ( a, r ) (cid:1) = D an (cid:0) h ( a ) , r (cid:1) . Let X be D an ( a, r ) or D an ( a, r ), and let Y be D an ( h ( a ) , r ) or D an ( h ( a ) , r ), respectively.If there is some n ∈ N such that X ∩ f − n ( Ω ) = ∅ , then h ( X ) = H n ( X ), and we aredone by equation (6.10). Otherwise, by Lemma 5.3, for each n ∈ N , there is some ζ n ∈ J n an ,f such that X ∩ D an ( ζ n , µ n ( ζ n ) ν ( ζ n )) = ∅ . If X ⊆ D an ( ζ n , µ n ( ζ n ) ν ( ζ n )) foreach n , then X ⊆ C v by Lemma 5.4, contradicting the fact that the Berkovich disk X contains points of type II, for example.Thus, there must be some m ∈ N such that X ⊇ D an ( ζ m , µ m ( ζ m ) ν ( ζ m )). By equa-tion (6.10) again, we have that H n ( X ) = Y for every n ≥ m . To prove the currentclaim, then, it suffices to show, for every ξ ∈ P , that ξ ∈ X if and only if there is some j ≥ m such that h ( ξ ) ∈ H j ( X ).Consider arbitrary ξ ∈ P . If there is some j ∈ N such that ξ f − j ( Ω ), then h ( ξ ) = H j ( ξ ) ∈ H j ( X ) by the definitions of h and H j . Otherwise, ξ ∈ T n ∈ N f − n ( Ω ).Therefore, by Lemma 5.4, we have ξ ∈ J an ,f ∩ C v , with lim n →∞ µ n ( ξ ) = 0. Hence,there is some j ≥ m such that µ j ( ξ ) ν ( ξ ) < r . By equation (6.2) and the fact that h ( ξ ) = lim n →∞ h n ( ξ ), we have (cid:12)(cid:12) h ( ξ ) − H j ( ξ ) (cid:12)(cid:12) = (cid:12)(cid:12) h ( ξ ) − h j ( ξ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∞ X n = j (cid:0) h n +1 ( ξ ) − h n ( ξ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) < r. Therefore, ξ ∈ X if and only if h ( ξ ) ∈ H j ( X ), completing our proof our claimed equa-tion (6.11).We are now prepared to show that h − is continuous, and hence that h is a homeo-morphism. For any connected open affinoid V ⊆ P , it suffices to show that h ( V ) isalso open in P . Write V = P r (cid:0) D an ( a , r ) ∪ · · · ∪ D an ( a ℓ , r ℓ ) (cid:1) or V = D an ( b, s ) r (cid:0) D an ( a , r ) ∪ · · · ∪ D an ( a ℓ , r ℓ ) (cid:1) . By equation (6.11) and the fact that h is bijective, we have h ( V ) = P r (cid:16) D an (cid:0) h ( a ) , r (cid:1) ∪ · · · ∪ D an (cid:0) h ( a ℓ ) , r ℓ (cid:1)(cid:17) or h ( V ) = D an (cid:0) h ( b ) , s (cid:1) r (cid:16) D an (cid:0) h ( a ) , r (cid:1) ∪ · · · ∪ D an (cid:0) h ( a ℓ ) , r ℓ (cid:1)(cid:17) , respectively. Either way, h ( V ) is a connected open affinoid, completing our proof that h is a homeomorphism. Step 5 . We have shown that h : P → P is a homeomorphism, mappng C v bijectively and isometrically onto itself, and satisfying the conjugacy formula (6.8) on f − ( Ω ). Moreover, f − ( Ω ) = g − (Ω) by the final statement of Lemma 5.6. It remainsto extend the conjugacy to f − ( Ω ) ∪ J an ,f , and to show that h ( J an ,f ) = J an ,g . -STABILITY 21 To this end, we first claim that(6.12) f ( ζ ) = g ( ζ ) for all ζ ∈ f − (cid:0) D an (0 , (cid:1) with diam (cid:0) f ( ζ ) (cid:1) > δ ǫ . To see this, consider an arbitrary such point ζ . The subset f − ( D (0 , f − ( D an (0 , { x i } ∞ i =0 ⊆ f − ( D (0 , i →∞ x i = ζ . For each such type I point x i , we have | g ( x i ) − f ( x i ) | < δ ǫ/
2, byLemma 5.5(b). Therefore, (cid:13)(cid:13) g − f (cid:13)(cid:13) ζ = lim i →∞ (cid:13)(cid:13) g − f (cid:13)(cid:13) x i = lim i →∞ (cid:12)(cid:12) g ( x i ) − f ( x i ) (cid:12)(cid:12) ≤ δ ǫ < diam (cid:0) f ( ζ ) (cid:1) . Thus, for any a ∈ C v , we have (cid:13)(cid:13) f ( z ) − a (cid:13)(cid:13) ζ ≥ diam (cid:0) f ( ζ ) (cid:1) > (cid:13)(cid:13) g − f (cid:13)(cid:13) ζ , and hence k z − a k g ( ζ ) = (cid:13)(cid:13) g ( z ) − a (cid:13)(cid:13) ζ = (cid:13)(cid:13)(cid:0) g ( z ) − f ( z ) (cid:1) + (cid:0) f ( z ) − a (cid:1)(cid:13)(cid:13) ζ = (cid:13)(cid:13) f ( z ) − a k ζ = k z − a k f ( ζ ) . Since this is true for all a ∈ C v , we have f ( ζ ) = g ( ζ ) by [B19, Lemma 15.2(d)], provingour claim.Consider arbitrary ζ ∈ J an ,f r f − ( Ω ). Then f ( ζ ) ∈ J an ,f r Ω , and in particular f ( ζ ) ∈ J +an ,f . Hence,diam (cid:0) f ( ζ ) (cid:1) = sphdiam (cid:0) f ( ζ ) (cid:1) ≥ ν (cid:0) f ( ζ ) (cid:1) > δ ǫ , where the equality is because J an ,f ⊆ D an (0 , f ( ζ ) ∈J +an ,f , and the second is by Lemma 5.1(a). Therefore, by the claim of (6.12), we have(6.13) f ( ζ ) = g ( ζ ) for all ζ ∈ J an ,f r f − ( Ω ) . Finally, recall that J an ,f is a nonempty compact set, and hence so is its homeomorphicimage h ( J an ,f ). In addition, the functions h ◦ f and g ◦ h coincide on J an ,f , whence thefunctions h ◦ f ◦ h − and g coincide on h ( J an ,f ). Thus, g − (cid:0) h ( J an ,f ) (cid:1) = h (cid:0) f − ( J an ,f ) (cid:1) = h ( J an ,f ) . Therefore, by [B19, Theorem 8.15(d)], it follows that h ( J an ,f ) ⊇ J an ,g , since h ( J an ,f ) isclosed in P . Furthermore, because of this inclusion, we have h − ◦ g = f ◦ h − on J an ,g ,and hence we may apply the same argument to the image of the compact set J an ,g underthe homeomorphism h − , to obtain h − ( J an ,g ) ⊇ J an ,f , or equivalently, h ( J an ,f ) ⊆ J an ,g .Combining these two inclusions yields the desired equality h ( J an ,f ) = J an ,g . (cid:3) Examples
We now present examples of rational functions satisfying the hypotheses of Theo-rem 1.1 but which are not expanding in the sense of equation (1.1).
Example 7.1.
Assume the residue characteristic of C v is 0, and fix c ∈ C v with 0 < | c | <
1. Define f ( z ) := ( z + c )( z + 1) z = z + ( c + 1) + cz ∈ C v ( z ) , which is a rational function of degree d = 2. A straightforward calculation shows that (cid:12)(cid:12) f ( x ) − ( x + 1) (cid:12)(cid:12) < x ∈ C v with | x | ≥ , and therefore(7.1) f maps D an ( x,
1) bijectively onto D an ( x + 1 ,
1) for all x ∈ C v with | x | ≥ . It follows that P r D an (0 , ⊆ F an ,f , and that D an ( n, ⊆ F an ,f for every positiveinteger n ∈ N . Further simple calculations show that f (cid:0) D an (0 , | c | ) (cid:1) ⊆ P r D an (0 ,
1) and f (cid:0) D an (0 , r D an (0 , | c | ) (cid:1) ⊆ D an (1 , . Combining these facts, it follows that(7.2) J an ,f ⊆ (cid:8) ζ ∈ P (cid:12)(cid:12) | ζ | = 1 or | ζ | = | c | (cid:9) . Conversely, J an ,f is nonempty, and by [B19, Theorem 7.34], we have f ( D an (0 , P .Therefore, by (7.1), a simple induction shows D an ( − n, ∩ J an ,f = ∅ for all n ∈ N . Thus, f is not expanding in the sense of equation (1.1), since for any n ∈ N , there issome ζ ∈ D an ( − n, ∩ J an , but equation (7.1) together with Lemma 3.7 shows that( f i ) ♮ ( ζ ) = 1 for all 0 ≤ i ≤ n .On the other hand, we have f ( cw ) = w − + 1 + c ( w + 1), and hence f maps D an ( x, | c | ) bijectively onto D an (cid:18) cx + 1 , (cid:19) for all x ∈ C v with | x | = | c | , whence f ♮ ( ζ ) = | c | − for all ζ ∈ P with | ζ | = | c | . Combining this fact with equa-tions (7.1) and (7.2), as well as Lemma 3.7 again, shows that (cid:0) f n (cid:1) ♮ ( ζ ) ≥ ζ ∈ J an ,f and n ∈ N . That is, even though f is not expanding, it satisfies the hypotheses of Theorem 1.1 andhence is J -stable in the moduli space Rat . Example 7.2.
Choose an integer m ≥ | m | = 1, i.e., such that m is notdivisible by the residue characteristic of C v . Fix c ∈ C v with 0 < | c | <
1. Define f ( z ) := cz m +1 + z m = z m ( cz + 1) ∈ C v [ z ] , which is a polynomial of degree d = m + 1 ≥
3. Then | f ( x ) | = | c || x | m +1 > | x | for all x ∈ C v with | x | > | c | − and | f ( x ) | = | x | m < | x | for all x ∈ C v with | x | < . It follows that f maps both P r D an (0 , | c | − ) and D an (0 ,
1) into themselves, and hence D an (0 , ∪ (cid:0) P r D an (0 , | c | − ) (cid:1) ⊆ F an ,f . Furthermore, it is not difficult to check that(7.3) f − (cid:0) D (0 , | c | − ) (cid:1) ⊆ D (0 , | c | − /m ) ∪ D ( − c − , | c | m − )by writing f ( z ) = cz m ( z + c − ). (In fact, we have equality in (7.3).)Therefore, we have J an ,f ⊆ X ∪ Y , where X := (cid:8) ζ ∈ P (cid:12)(cid:12) ≤ | ζ | ≤ | c | − /m (cid:9) and Y := D an ( − c − , | c | m − ) . -STABILITY 23 For any x ∈ X ∩ C v , writing f as a power series centered at x , it is straightforward tocheck that f maps D ( x, | x | ) bijectively onto D ( x m , | x | m ). Thus, (the proof of) Lemma 3.7shows that for any ζ ∈ X , we have f ♮ ( ζ ) = | ζ | m | ζ | · max { , | f ( ζ ) | } max { , | ζ | } ≥ | f ( ζ ) | | ζ | . Similarly, because f maps the disk Y (of diameter | c | m − ) bijectively onto D an (0 , | c | − ),Lemma 3.7 shows that for any ζ ∈ Y with f ( ζ ) ∈ X ∪ Y , we have f ♮ ( ζ ) = | c | − | c | m − · max { , | f ( ζ ) | } max { , | ζ | } = | c | − m | f ( ζ ) | | ζ | ≥ | f ( ζ ) | | ζ | . Combining these two bounds, and using the fact that J an ,f ⊆ X ∪ Y , we have (cid:0) f n (cid:1) ♮ ( ζ ) ≥ | f n ( ζ ) | | ζ | ≥ | c | for all ζ ∈ J an ,f and n ∈ N . Thus, f satisfies the hypotheses of Theorem 1.1 and hence is J -stable in the modulispace Rat m +1 .On the other hand, the Newton polygon of the equation f ( z ) − z = 0 reveals that f has a fixed point a ∈ C v with | a | = | c | − . By inclusion (7.3), we must have a ∈ D ( − c − , | c | m − ). Since f ′ ( z ) = ( m + 1) cz m + mz m − = z m − (cid:0) ( m + 1) cz + m (cid:1) , we have | f ′ ( a ) | = | c | − m >
1, and hence a is repelling and therefore lies in J an ,f .For each b ∈ C v with 1 < | b | < | c | − , the Newton polygon of the equation f ( z ) − b =0 shows that b has m preimages α , . . . , α m with | α i | = | b | /m . Applying this factinductively starting with b = a , and choosing only one such preimage each time, thereis an infinite sequence { a n } ∞ n =0 in C v with | a n | = | c | − /m n and f ( a n ) = a n − for all n ∈ N . Each point a n eventually maps to a and hence lies in J an ,f , with f ♮ ( a n ) = | f ′ ( a n ) | · max { , | f ( a n ) | } max { , | a n | } = | a n | m − · | a n | m | a n | = | c | − m − /m n . Thus, (cid:0) f n (cid:1) ♮ ( a n ) = n − Y i =0 f ♮ ( a n ) = | c | − e , where e = 3( m − m n + 3( m − m n − + · · · + 3( m − m = 3 (cid:18) − m n (cid:19) < . Hence, ( f n ) ♮ ( a n ) < | c | − for every n ≥
1, and as in Example 7.1, f is not expandingin the sense of equation (1.1). Remark 7.3.
Motivated by condition (1.1), let us call a rational function f : P → P uniformly expanding on its Julia set if there exist c > λ > n ∈ N and ζ ∈ J an ,f , ( f n ) ♮ ( ζ ) ≥ cλ n . Any uniformly expanding rational functions clearly satisfies the assumption of Theo-rem 1.1 and hence is J -stable in the moduli space Rat d . However, although this condition is appropriate in complex dynamics, the above examples show that uniform expansionis too restrictive a condition in the non-archimedean setting.In fact, any uniformly expanding rational function has Julia set consisting only oftype I points, as we now prove. Suppose there is ζ ∈ J an ,f ∩ H . Then by Proposition 3.5,we have sphdiam( f n ( ζ ))sphdiam( ζ ) = ( f n ) ♮ ( ζ ) ≥ cλ n for any n ∈ N . Therefore,lim n →∞ sphdiam( f n ( ζ )) ≥ sphdiam( ζ ) · lim n →∞ cλ n = ∞ , contradicting the fact that sphdiam( ξ ) ∈ [0 ,
1] for any ξ ∈ P . Acknowledgments . The first author gratefully acknowledges the support of NSF grantDMS-150176. The second author was supported by JSPS KAKENHI Grant Number16J01139 and the JSPS Postdoctoral Fellowship for Foreign Researchers. This researchstarted during the second author’s stay in Amherst College. The authors thank LauraDeMarco for helpful discussions.
References [B01] R. L. Benedetto,
Hyperbolic maps in p -adic dynamics , Ergodic Theory Dynam. Systems (2001),1–11.[B19] R.L. Benedetto, Dynamics in One Non-Archimedean Variable , American Mathematical Society,Providence, 2019.[BR10] M. Baker and R. Rumely,
Potential Theory and Dynamics on the Berkovich Projective Line ,American[L19] J. Lee, J -stability of expanding maps in non-Archimedean dynamics , Ergodic Theory Dynam.Systems (2019), 1002–1019.[MSS83] R. Ma˜n´e, P. Sad and D. Sullivan, On the dynamics of rational maps , Ann. Sci. ´Ecole Norm.Sup. (4) (1983), 193–217.[McM94] C. McMullen, Complex Dynamics and Renormalization , Princeton University Press, Prince-ton, 1994.[MS98] C. T. McMullen and D. P. Sullivan,
Quasiconformal homeomorphisms and dynamics. III. TheTeichm¨uller space of a holomorphic dynamical system , Adv. Math. (1998), 351–395.[Rob00] A. M. Robert,
A Course in p -adic Analysis , Springer-Verlag, New York, 2000.[Sil07] J. H. Silverman, The Arithmetic of Dynamical Systems , Springer, New York, 2007.[Sil17] T. Silverman,
A non-archimedean λ -lemma , preprint 2017. Available at arXiv:1712.01372 Amherst College, Amherst, MA 01002, USA
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