BERKOVICH DYNAMICS OF NEWTON MAPShmnie/Berkovich dynamics of Newton maps.pdf · BERKOVICH DYNAMICS...

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BERKOVICH DYNAMICS OF NEWTON MAPS HONGMING NIE Abstract. Let L be the completion of the formal Puiseux series over C, and let P 1 be the Berkovich space over L. We study the dynamics of Newton maps, defined over L, on P 1 . As an application, we give a complete description of the rescaling limits for a holomorphic family of complex Newton maps. 1. Introduction The dynamics of Newton maps for complex polynomials are well-studied in the literature, see [6, 14, 19– 22, 29, 34]. The goal of the present article is to exploit the dynamics of Newton maps in non-Archimedean settings. Our work is an attempt to describe the Berkovich dynamics of certain class of higher degree rational maps following quadratic case given by Kiwi [17] and polynomial case given by Trucco [33]. Moreover, for a suitable ground non-Archimedean field, the Berkovich dynamical systems can be regarded as the limiting of families of complex dynamical systems, see [9, 12, 18]. Applying the Berkovich dynamics we obtain, we characterize all the rescaling limits, introduced by Kiwi [18], for a holomorphic family of complex Newton maps. Now we set up the statements. Throughout this paper, we denote L the completion of the formal Puiseux series in t over C, and denote P 1 the Berkovich space over L. For more details on Berkovich spaces, we refer [2,4,15]. Let P L[z] be a degree d 2 monic polynomial with distinct roots, that is, P (z)= d Y i=1 (z - r i ), where r 1 ,...,r d are distinct elements in L. The Newton map for the polynomial P is N P (z)= z - P (z) P 0 (z) . We sometimes write N {r1,··· ,r d } , instead of N P , to indicate the roots of P . Then N P L(z) is a degree d rational map acting on the projective space P 1 L , which extends naturally to an endomorphism of P 1 . By abuse of notation, we also denote the extension by N P . In the Berkovich space P 1 , each point associates with a positive integer called local degree, see [2, Section 9.1] and [10, Section 3]. Following Faber [10,11], the ramification locus is the set of points in P 1 with local degree at least 2. A cycle in H := P 1 \ P 1 L is repelling if it intersects with the ramification locus. Otherwise, we say it is indifferent. According to multipliers, we classify cycles in P 1 L to be attracting, indifferent or repelling as in complex dynamics. The Berkovich space P 1 consists of two totally invariant sets: Berkovich Fatou set and Berkovich Julia set, see [2, Section 10.5] for definitions. A fixed Berkovich Fatou component is either the immediate basin of an attracting fixed point or a fixed Rivera domain [2, Theorem 10.76]. A notable difference between Berkovich dynamics and complex dynamics is that the Berkovich Fatou set contains all indifferent periodic points in P 1 L , see [32, Proposition 5.20]. The point is an indifferent fixed point of N P in our setting rather than a repelling fixed point as in complex dynmaics. Hence it is in a fixed Revera domain. For an open set in P 1 , we say it is an open ball if its boundary is a singleton, and we say it is an annulus if it is the intersection of two open balls whose union is P 1 . Recall a rational map is simple if its Berkovich Julia set is a singleton. Then any Berkovich Fatou component of a simple rational map is an open ball. Since any quadratic Newton map is conjugate to the square map, it is simple. Here we describe the Berkovich dynamics of N P of degree at least 3. 2010 Mathematics Subject Classification. 37P50, 37F45 (primary), and 37P40, 37P45 (secondary). Key words and phrases. Newton maps, Berkovich dynamics, wandering Fatou component, rescaling limits. 1

Transcript of BERKOVICH DYNAMICS OF NEWTON MAPShmnie/Berkovich dynamics of Newton maps.pdf · BERKOVICH DYNAMICS...

Page 1: BERKOVICH DYNAMICS OF NEWTON MAPShmnie/Berkovich dynamics of Newton maps.pdf · BERKOVICH DYNAMICS OF NEWTON MAPS HONGMING NIE Abstract. Let L be the completion of the formal Puiseux

BERKOVICH DYNAMICS OF NEWTON MAPS

HONGMING NIE

Abstract. Let L be the completion of the formal Puiseux series over C, and let P1 be the Berkovich space

over L. We study the dynamics of Newton maps, defined over L, on P1. As an application, we give a

complete description of the rescaling limits for a holomorphic family of complex Newton maps.

1. Introduction

The dynamics of Newton maps for complex polynomials are well-studied in the literature, see [6, 14, 19–22, 29, 34]. The goal of the present article is to exploit the dynamics of Newton maps in non-Archimedeansettings. Our work is an attempt to describe the Berkovich dynamics of certain class of higher degree rationalmaps following quadratic case given by Kiwi [17] and polynomial case given by Trucco [33]. Moreover, fora suitable ground non-Archimedean field, the Berkovich dynamical systems can be regarded as the limitingof families of complex dynamical systems, see [9, 12, 18]. Applying the Berkovich dynamics we obtain, wecharacterize all the rescaling limits, introduced by Kiwi [18], for a holomorphic family of complex Newtonmaps. Now we set up the statements.

Throughout this paper, we denote L the completion of the formal Puiseux series in t over C, and denoteP1 the Berkovich space over L. For more details on Berkovich spaces, we refer [2, 4, 15]. Let P ∈ L[z] be adegree d ≥ 2 monic polynomial with distinct roots, that is,

P (z) =

d∏i=1

(z − ri),

where r1, . . . , rd are distinct elements in L. The Newton map for the polynomial P is

NP (z) = z − P (z)

P ′(z).

We sometimes write N{r1,··· ,rd}, instead of NP , to indicate the roots of P . Then NP ∈ L(z) is a degree d

rational map acting on the projective space P1L, which extends naturally to an endomorphism of P1. By

abuse of notation, we also denote the extension by NP .In the Berkovich space P1, each point associates with a positive integer called local degree, see [2, Section

9.1] and [10, Section 3]. Following Faber [10, 11], the ramification locus is the set of points in P1 with localdegree at least 2. A cycle in H := P1 \ P1

L is repelling if it intersects with the ramification locus. Otherwise,we say it is indifferent. According to multipliers, we classify cycles in P1

L to be attracting, indifferent orrepelling as in complex dynamics.

The Berkovich space P1 consists of two totally invariant sets: Berkovich Fatou set and Berkovich Julia set,see [2, Section 10.5] for definitions. A fixed Berkovich Fatou component is either the immediate basin of anattracting fixed point or a fixed Rivera domain [2, Theorem 10.76]. A notable difference between Berkovichdynamics and complex dynamics is that the Berkovich Fatou set contains all indifferent periodic points inP1L, see [32, Proposition 5.20]. The point ∞ is an indifferent fixed point of NP in our setting rather than a

repelling fixed point as in complex dynmaics. Hence it is in a fixed Revera domain.For an open set in P1, we say it is an open ball if its boundary is a singleton, and we say it is an annulus

if it is the intersection of two open balls whose union is P1. Recall a rational map is simple if its BerkovichJulia set is a singleton. Then any Berkovich Fatou component of a simple rational map is an open ball. Sinceany quadratic Newton map is conjugate to the square map, it is simple. Here we describe the Berkovichdynamics of NP of degree at least 3.

2010 Mathematics Subject Classification. 37P50, 37F45 (primary), and 37P40, 37P45 (secondary).

Key words and phrases. Newton maps, Berkovich dynamics, wandering Fatou component, rescaling limits.

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Theorem 1.1. Let P ∈ L[z] be a monic polynomial of degree d ≥ 3 with distinct roots and let N := NP ∈L(z) be its Newton map. Denote J(N), F (N) and RN the Berkovich Julia set, Berkovich Fatou set andramification locus, respectively, of N . Then

(1) The following are equivalent.(a) N is simple.(b) N has exactly one repelling fixed point in P1.(c) RN is connected.

Moreover, if (a)− (c) hold, then the unique repelling fixed point is in H.(2) If N is not simple, then

(a) N has at least two repelling fixed points in H. Moreover, it has at most d − 1 repelling fixedpoints and at most d− 3 repelling cycles of periods at least 2 in H. However, the total numberof repelling cycles in H is at most d− 1.

(b) J(N) ∩H is the grand orbit of the repelling cycles in H.(c) F (N) contains at most 2d − 1 fixed components consisting of d immediate attracting basins of

the roots of P , a Rivera domain which is an open ball containing ∞ and at least one Riveradomain which is an open annulus with fixed boundaries. Moreover, periodic Berkovich Fatoucomponents of higher periods are open balls

(d) RN is disconnected with at most d− 1 components. Each component of RN contains exact onerepelling fixed point.

Besides its own significance, our above theorem serves as a meaningful example for several general results.We summarize here in the following remark.

Remark 1.2. (1) Our characterization of simplicity for Newton maps does not hold for general rationalmaps. Indeed, Faber [10, Theorem C] proved that if a rational map has a totally ramified point in P1, thenits ramification locus is connected. For example, the ramification locus of any polynomial is connected.

(2) In contrast with quadratic rational maps, see [17, Theorems 1 and 2], our result claims that a Newtonmap N ∈ L(z) always has a repelling fixed point in H, which implies that the J(N) can be not contained inP1L. Moreover, if N is not simple, F (N) always has a fixed Rivera domain which is not an open ball.

(3) Recall that the classical Julia set JI(φ) for a rational map φ ∈ L(z) is the intersection JI(φ) = J(φ)∩P1L

[2, Theorem 10.67]. The Hsia’s conjecture [13, Conjecture 4.3] asserts that the repelling periodic points inP1L are dense in JI(φ). Bezivin [5, Theorem 3] proved the Hsia’s conjecture holds if there exists at least one

repelling periodic point in JI(φ). By a result of Rivera-Letelier [2, Theorem 10.88] stating that the closureof the repelling cycles in P1 is J(φ), our result implies that a Newton map N ∈ L(z) has type I repellingcycles. Hence the Hsia’s conjecture holds for N .

(4) Rumely [30, Corollary 6.2] gave the upper bound d − 1 of repelling fixed points in P1 for generalrational maps of degree d ≥ 2. Our bounds for repelling fixed points and for repelling periodic cycles ofperiods at least 2 are sharp independently. But they achieve simultaneously if and only if d = 3, in whichcase all repelling periodic points in H are fixed points.

(5) For any 1 ≤ m ≤ d − 1, it is easy to construct a Newton map N of degree d such that RN hasexact m components. In L(z), the Newton maps optimize the general upper bound d − 1 given by Faber,see [10, Theorem A and Proposition 6.7].

Our proof of Theorem 1.1 highly depends on an elaborate analysis on the Berkovich tree generated bythe fixed points and critical points of N in P1

L. Some special branch points of this tree turn out to be therepelling fixed points. Additionally considering the iterated preimages of these fixed points, we can obtaina more clear picture of the dynamics. We exhibit detailed dynamical properties of N in Section 3.1, whichare used by the author and Pilgrim [25] to prove boundedness results of hyperbolic components in modulispace of complex Newton maps.

A Berkovich Fatou component is non-wandering if it is eventually periodic or is contained in the basinof a periodic orbit. Otherwise, we say it is wandering. Trucco [33, Corollary B] proved that each BerkovichFatou component of a polynomial of degree at least 2 in L[z] is non-wandering. Kiwi proved same result forquadratic rational maps in L(z), see [17, Theorem 1.2], and conjectured that there is no wandering BerkovichFatou component for all rational maps of degree at least 2 in L(z), see [17, Conjecture]. Our Theorem 1.1immediately implies the following result, which provides more evidence to support Kiwi’s conjecture.

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BERKOVICH DYNAMICS OF NEWTON MAPS 3

Corollary 1.3. Let N ∈ L(z) be as in Theorem 1.1. Then every Berkovich Fatou component of N isnon-wandering.

Now we apply the Berkovich dynamics of N to investigate the rescaling limits for a holomorphic familyof complex Newton maps. Following Kiwi [18], we can regard a holomorphic family of complex rationalmaps as a single rational map in L(z). Let {Nt}t∈D be a holomorphic family of Newton maps where Nt isa Newton map of a complex polynomial with distinct roots if t 6= 0. Then Nt acts on the Berkovich spaceP1. We are interested in the rescaling limits for the family {Nt}t∈D, that is, the limits M−1

t ◦Nqt ◦Mt → g,

with rescalings Mt ∈ Aut(P1C) if t 6= 0 and deg g ≥ 2, of rescaled iterates where the convergence is locally

uniform outside some finite subset of P1C, see [18, Definition 1.1].

Via coefficients, we can write a point f ∈ P2d+1C as a (degenerate) rational map, that is, a set of holes and

a rational map f of degree at most d, see [7]. We say the rational map f is the induced map of f . Identifying

the space of degree d rational maps as an open and dense subset of P2d+1C , we say a rational map N is an

induced Newton map if it is the induced map of a limit point of a sequence of complex Newton maps in

P2d+1C . If deg N = 0, then N = ∞. Otherwise, N is a Newton map for a polynomial of degree at most d

with possible multiple roots.Now we state our result of the rescaling limits for the family {Nt}t∈D.

Theorem 1.4. Let {Nt}t∈D be a holomorphic family of degree d ≥ 3 Newton maps for complex monicpolynomials with distinct roots. Then {Nt}t∈D has at most d− 1 dynamically independent rescalings. Moreprecisely,

(1) up to equivalence, {Nt}t∈D has at most d − 1 rescalings of period 1. Moreover, each such rescalingleads to a rescaling limit that is conjugate to an induced Newton map.

(2) {Nt}t∈D has at most d − 3 dynamically independent rescalings of periods at least 2. Moreover, thecorresponding rescaling limit for each such rescaling is conjugate to a polynomial of degree at most2d−3.

Roughly speaking, the equivalent rescalings of period 1 are related to the repelling fixed points of Nt inH and the dynamically independent rescalings are related to the repelling periodic cycles in H. For precisedefinition, see [18, Definitions 1.2 and 3.8] and Section 4.1.

For any given holomorphic family of degree d ≥ 2 rational maps, studying the corresponding Berkovichdynamics, Kiwi [18, Theorem 1] proved that there are at most 2d − 2 dynamically independent rescalingssuch that the corresponding rescaling limits are not postcritically finite, also see [1] for a different argument.Kiwi and Arfeux also described completely the rescaling limits for quadratic rational maps and for bicriticalrational maps, respectively. Our result is an interesting case in Kiwi’s framework. Indeed with the period1 rescaling limits, we construct a non-Hausdorff compactification of the moduli space of complex Newtonmaps, see Section 4.2. Also the upper bound of the degrees of the rescaling limits implies that the limits arenot semistable in the moduli-theoretic sense, see Section 4.3.

Outline. Section 2 covers the relevant background of degenerate Newton maps and Berkovich space. Section3 states the Berkovich dynamcs of Newton maps, which is the main part of this paper. Section 3.1 describesdetailed dynamics on several subtrees. Section 3.2 gives a characterization of type II repelling cycles ofperiods at least 2. Section 3.3 studies the tree generated by the iterated preimages of type II repellingperiodic points and shows that it contains the non type I Julia points. Then we prove Theorem 1.1 andCorollary 1.3 in Section 3.4. Section 4 discusses the rescaling limits and related results. Section 4.1 providesthe definition of rescaling limits and the proof of Theorem 1.4. With the period 1 rescaling limits, Section4.2 describes a compactification of moduli space of complex Newton maps. Section 4.3 proves that the limitsinducing the rescaling limits of periods at least 2 are not semistable.

Acknowledgement. Partial results in this work are from the author’s thesis. He would like to thank hisadvisor Kevin Pilgrim for fruitful discussions. He also thanks Jan Kiwi, Laura DeMarco and Juan Rivera-Letelier for valuable comments and thanks Matthieu Arfeux and Ken Jacobs for useful conversations.

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2. Preliminaries

This section gives the background materials used in this work. It contains two subsections. Section2.1 states the basic properties of degenerate Newton maps from analytic view. Section 2.2 introduces thedynamics from Berkovich view.

2.1. Degenerate Newton maps. For any point f ∈ P2d+1C , there exist two homogeneous polynomials Fa

and Fb in C[X,Y ], via coefficients, such that f = [Fa : Fb]. Set Hf = gcd(Fa, Fb). Then we can write

F = Hf f , where f = [Fa/Hf : Fb/Hf ] is a degree at most d rational maps. We say f is the induced map off and a zero of Hf is a hole of f . Moreover, we say the multiplicity of a point z ∈ P1

C as a zero of Hf is thedepth of z denoted by dz(f). We refer [7] for details.

Now we begin to discuss the degenerate Newton maps. For d ≥ 2, let P (z) ∈ C[z] be a degree d monicpolynomial. Let r = {r1, · · · , rd} be the roots of P (z) counted with multiplicity. Then P (z) is uniquelydetermined by r. We sometimes write Pr(z) instead of P (z) to emphasis its roots. In projective coordinates,

P ([X : Y ]) = [Ga(X,Y ) : Gb(X,Y )] =

[d∏i=1

(X − riY ) : Y d

].

Then the derivative P ′(z) of P (z) is

P ′([X : Y ]) = [Ka(X,Y ) : Kb(X,Y )] =

d∑j=1

∏i6=j

(X − riY ) : Y d−1

.Define

NP ([X : Y ]) := [XKa(X,Y )−Ga(X,Y ) : Y Ka(X,Y )] ∈ P2d+1C . (2.1)

Note that NP is a Newton map of degree d if and only if r1, · · · , rd are d distinct points in C. In this case,in affine coordinates,

NP (z) = z − P (z)

P ′(z).

For a point N ∈ P2d+1C , suppose N is not a degree d rational map. Then we say N is a degenerate Newton

map of degree d if there exists a sequence {Nn} of Newton maps of degree d such that Nn converges to N

in P2d+1C , as n → ∞. For such N , there exist an integer d ≥ m ≥ 0 and a degree d −m monic polynomial

P ∈ C[z] such that

N([X : Y ]) = Y mNP ([X : Y ]) ∈ P2d+1C (2.2)

and the degree of the induced map deg NP < d. If m 6= 0, then ∞ is a hole of N with depth d∞(N) = m.

All other holes of N are the multiple roots ri of P and the corresponding depths dri(N) are mP (ri) − 1,

where mP (ri) is the multiplicity of ri as a zero of P . Note if m = 0, then P is a degree d polynomial withmultiple roots, and the formula (2.2) coincides to formula (2.1).

In general, let r be a set of d points (not necessary distinct) in C. Suppose in r there are m ≥ 0 points

that are ∞. Let P ∈ C[z] be the monic polynomial of degree d −m whose roots coincide to the points inr ∩ C. Then we define the (degenerate) Newton map Nr of degree d by formula (2.2).

Example 2.1. Let r = {0, 0, 1,∞}. Then Nr has holes at 0 and ∞, and each hole has depth 1. Moreprecisely,

Nr([X : Y ]) = HNr (X,Y )Nr([X : Y ]) = XY [2X2 −XY : 3XY − 2Y 2].

Note here we have Nr = N{0,0,1,∞} = N{0,0,1}.

Our next result addresses the fixed points and critical points of induced Newton maps. It is well-knownand we omit the proof.

Proposition 2.2. For d ≥ 2, let r = {r1, · · · , rd} be a set of d (not necessary distinct) points in C, and letNr be the (degenerate) Newton map for the monic polynomial Pr. Then

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BERKOVICH DYNAMICS OF NEWTON MAPS 5

(1) The set of fixed points of Nr is

Fix(Nr) = {r1, · · · , rd,∞}.The point z =∞ is the unique repelling fixed point. Moreover, ri is superattracting if and only if it isa simple root of Pr. Otherwise, ri is an attracting fixed point with multiplier (mPr (ri)− 1)/d, wheremPr (ri) is the multiplicity of ri as a zero of Pr. In particular, the depth dri(Nr) = mPr (ri)− 1.

(2) The critical set Crit(Nr) is contained in the union of simple zeros of Pr and zeros of P ′′r . In particular,

if Nr is a Newton map of degree d, then Crit(Nr) coincides to the union of zeros of Pr and zeros ofP ′′r .

2.2. Berkovich space and related dynamics. We begin to summarize the definitions and main propertiesof Berkovich spaces and rational maps on Berkovich spaces used in this work. For more details, we refer[2–4,15,27,28].

Recall that the field L is the completion of the formal Puiseux series in t over C associated with thenatural non-Archimedean absolute value | · |. Then each element in L has the form

∑n≥0 ant

qn , where {qn}is a sequence of rational numbers increasing to ∞ and an ∈ C, and |a0t

q0 + · · · | = e−q0 . The ring of integersof L is OL = {z ∈ L : |z| ≤ 1} and the unique maximal idealML of OL isML = {z ∈ L : |z| < 1}. Thus, theresidue field OL/ML is canonically isomorphic to C. For more details about the fields L, we refer [16–18].

The Berkovich space P1 over L is a compact (with respect to weak topology), Hausdorff and arcwiseconnected topology space with tree structure. We denote [ξ1, ξ2] the unique arc between two distinct pointsξ1, ξ2 ∈ P1. The space P1 consists of 4 types of points. The set of type I points identifies with the projectivespace P1

L. The type II and type III points correspond to closed discs in L. A point is of type II if thecorresponding disc has radius in the value group |L×|. Otherwise, it is of type III. The type IV pointscorresponds to a decreasing sequence of closed discs in L with empty intersection. To ease notation, wesometimes write ξa,r for the point in P1 corresponding to the closed disc {z ∈ L : |z − a| ≤ r} ⊂ L. We callthe point ξ0,1 ∈ P1 the Gauss point denoted by ξg.

For a point ξ ∈ P1, we can define an equivalence relation on P1 \ {ξ}, that is, ξ1 is equivalent to ξ2 if ξ1and ξ2 are in the same connected component of P1 \{ξ}. We say such an equivalence class ~v is a direction atξ and the set TξP

1 formed by all directions at ξ is the tangent space at ξ. For ~v ∈ TξP1, denote by Bξ(~v)−

the component of P1 \ {ξ} corresponding to the direction ~v. If ξ ∈ P1 is a type I or IV point, then TξP1

consists of a single direction. If ξ ∈ P1 is a type III point, then TξP1 consists of two directions. If ξ ∈ P1 is

a type II point, the directions in TξP1 are in one-to-one correspondence with the elements in P1

C. Since theGauss point ξg is a type II point, we can identify TξgP

1 to P1C by the correspondence TξgP

1 → P1C sending

~vx to x, where ~vx is the direction at ξg such that Bξg (~vx)− contains all the type I points whose images are

x under the canonical reduction map P1L → P1

C.For a point ξ ∈ P1, denote diam(ξ) is the limit of the radii of a decreasing sequence of closed discs

corresponding to ξ ∈ P1. For any two points ξ1, ξ2 ∈ P1, there is a unique point ξ1 ∨ ξ2 ∈ P1 such that itscorresponding closed disc in L contains the discs corresponding to ξ1 and ξ2. Then diam(ξ1∨ ξ2) ≥ diam(ξ1)and diam(ξ1 ∨ ξ2) ≥ diam(ξ2). The hyperbolic space H := P1 \ P1

L carries a natural path distance as follows:for ξ1, ξ2 ∈ H, the path distance between ξ1 and ξ2 is

ρ(ξ1, ξ2) = logdiam(ξ1 ∨ ξ2)

diam(ξ1)+ log

diam(ξ1 ∨ ξ2)

diam(ξ2).

A degree d ≥ 1 rational map φ ∈ L(z) extends to a map from P1 to itself. By abuse of notation, wedenote the extension by φ as well. We refer [10, Section 2.3.2] for the extension. Write φ = P/Q ∈ L(z),where P,Q ∈ L[z] have no common zeros. We can normalize φ such that P,Q ∈ OL(z) and the maximalabsolute value of coefficients of P and Q is 1. For a normalized rational map φ = P/Q ∈ L(z), the reductionRed(φ) is defined by

Red(φ) :=

{Red(P )/Red(Q) if Q 6∈ ML[z],

∞ if Q ∈ML[z],

where Red(P ) and Red(Q) are the images of P and Q in OL[z]/ML[z], respectively. For convenience, for anarbitrary rational map φ ∈ L(z), we write Red(φ) for the reduction of the normalization of φ.

The following result, originally proved by Rivera-Letelier, gives a characterization of rational maps withnonconstant reductions.

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Lemma 2.3. [2, Corollary 9.27] Let φ(z) ∈ L(z) be a rational map. Then deg Red(φ) ≥ 1 if and only ifφ(ξg) = ξg.

Denote by degξ φ the local degree of φ at ξ, see [2, Section 9.1].

Lemma 2.4. [2, Section 9.1] Let φ(z) ∈ L(z) be a nonconstant rational map. Let ξ ∈ P1 and ~v ∈ TξP1.Then there is a positive integer m and a point ξ′ ∈ Bξ(~v)− such that

(1) degξ′′ φ = m for all ξ′′ ∈ (ξ, ξ′), and(2) ρ(φ(ξ), φ(ξ′′)) = mρ(ξ, ξ′′) for all ξ′′ ∈ (ξ, ξ′).

We say the integer m is the directional multiplicity denoted by mφ(ξ,~v).Let ξ ∈ P1. For any ~v ∈ TξP1, there is a unique ~w ∈ Tφ(ξ)P

1 such that for any ξ′ sufficiently close to ξ,

φ(ξ′) ∈ Bφ(ξ)(~w)−. Thus the rational map φ induces a tangent map

Tξφ : TξP1 → Tφ(ξ)P

1,

sending the direction ~v to the corresponding direction ~w. Then degξ φ = deg Tξφ. More precisely,

Lemma 2.5. [2, Theorem 9.22] Let φ ∈ L(z) be a nonconstant rational map. Let ξ ∈ P1. Then for eachtangent ~w ∈ Tφ(ξ)P

1, we have

degξ φ =∑

~v∈TξP1

Tξφ(~v)=~w

mφ(ξ,~v).

The following result allows us to compute the local degree of φ at a type II point with the reduction map.

Lemma 2.6. [26, Proposition 3.3] Let φ ∈ L(z) be a nonconstant rational map, and let ξ ∈ P1 be a typeII point. Set ξ′ = φ(ξ) and choose M1,M2 ∈ Aut(P1(L)) such that M1(ξg) = ξ and M2(ξg) = ξ′. Let

ψ = M−12 ◦ φ ◦M1. Then deg Red(ψ) ≥ 1 and degξ φ = deg Red(ψ).

Now let us state a fundamental result in Berkovich dynamics, which asserts the image of an open ball iseither an open ball or the whole space P1.

Proposition 2.7. [26, Lemma 2.1] Let φ : P1 → P1 be a rational map of degree at least 1. For ξ ∈ P1 and~v ∈ TξP1, the image φ(Bξ(~v)−) always contains Bφ(ξ)(Tξφ(~v))−, and either φ(Bξ(~v)−) = Bφ(ξ)(Tξφ(~v))−

or φ(Bξ(~v)−) = P1.

To end this subsection, we state a criterion to determine whether the image of an open ball Bξ(~v)− is thewhole space P1.

Lemma 2.8. [2, Theorem 9.42] Let φ ∈ L(z) be a nonconstant rational map. Then φ(Bξ(~v)−) is an openball if and only if for each ξ′ ∈ Bξ(~v)−, the local degree of φ is nonincreasing on the directed segment [ξ, ξ′].

3. Berkovich Dynamics

Fix d ≥ 3. Let r = {r1, · · · , rd} ⊂ L be a set of d distinct points. Set Pr(z) =∏di=1(z − ri) ∈ L[z] and

consider its Newton map Nr ∈ L(z). Then Nr acts on the Berkovich space P1. To ease notation, we writeN for Nr. The aim of this section is to prove Theorem 1.1. We first study detailed dynamics on severalsubtrees related to N in Section 3.1, which is summarized in Proposition 3.3. In Section 3.2, we establisha sufficient and necessary condition for the existence of a type II repelling cycle of period at least 2, seeProposition 3.15, and give a upper bound of such repelling cycles, see Corollary 3.19. In Section 3.3, westudy the tree T generated by the iterated preimages of type II repelling periodic points and prove that Tcontains the non type I Julia points, see Proposition 3.22. Finally, we prove Theorem 1.1 and Corollary 1.3in Section 3.4.

3.1. Dynamics on subtrees. Since L is a complete and algebraically closed field of characteristic zero, themap N has 2d− 2 critical points, counted with multiplicity, in P1

L. Denote Crit(N) the set of these criticalpoints. Then the set Crit(N) ∪ {∞} is a finite collection of elements in P1

L, which sits in the Berkovichspace P1. Let Hbig be convex hull of Crit(N) ∪ {∞}. As a topological space, Hbig is homeomorphic tothe underlying space of a finite tree. We may equip this undelying space with a natural graph-theoreticcombinatorial structure so that we may speak of vertices and edges. Consider the branch points as the

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BERKOVICH DYNAMICS OF NEWTON MAPS 7

vertices of Hbig. Then Hbig is a finite tree and it plays a central role in our study. Since it is the biggest ofseveral trees we will consider, we indicate this by the subscript.

In general, the vertices of Hbig come in two flavors: (1) those in the hyperbolic space H, which are calledthe internal vertices of the tree, and (2) those in P1

L, i.e. the critical points and the point∞, which are calledthe leaves of the tree. Note the fixed points of N in P1

L are r1, · · · , rd and ∞. Then the convex hull Hfix

of {r1, · · · , rd,∞} sits in Hbig. As for Hbig, we can consider vertices, internal vertices and leaves for Hfix.Then a vertex of Hfix is a vertex of Hbig. Hence Hfix is a subtree of Hbig. Let Vrep be the set of internalvertices of Hfix. We have

Vrep = {ξ ∈ Hfix : ValHfix(ξ) ≥ 3},

where ValE(ξ) is the valance of ξ ∈ E in a connected subset E ⊂ P, that is, the number of components ofE \ {ξ}. Thus each element in Vrep is an internal vertex of Hbig. As we will see that each element in Vrep isa repelling fixed point of N , we indicate this by the subscript.

To make it clear, we list the subtrees we will study.

Hbig := Hull(Crit(N) ∪ {∞}),Hfix := Hull({r1, · · · , rd,∞}),Hcrit := Hull(Crit(N)),

Hrep := Hull(Vrep),

H∞rep := Hull(Vrep ∪ {∞}),Then we have Hbig ⊃ Hfix ⊃ H∞rep ⊃ Hrep, Hbig ⊃ Hfix and Hbig ⊃ Hcrit.

To illustrate the these subtrees, following Faber [11] we first define visible point.

Definition 3.1. For a connected subset E ⊂ H and a point ξ ∈ H, we say a point v ∈ H is the visible pointfrom ξ to E if v is the unique point such that infξ′∈E ρ(v, ξ′) = 0 and ρ(ξ, v) = infξ′∈E ρ(ξ, ξ′). Denote byπE(ξ) the visible point from ξ to E.

As an motivating example to understand the above subtrees, we consider the following example.

Example 3.2. Let r1 = 0, r2 = t, r3 = 2t, r4 = 1, r5 = 1 + t and r6 = 2 be six points in L. Set r ={r1, r2, r3, r4, r5, r6} and consider the Newton map N := Nr. Then N has four non-fixed critical pointsc1, c2, c3 and c4. Figure 1 shows the convex hull Hbig for N . The hull Hbig has internal vertices v1, v2, v3 ∈ Hand leaves r1, · · · , r6, c1, · · · , c4,∞ ∈ P1

L. Then Vrep = {v1, v2, v3} and Hrep = [v1, v2] ∪ [v2, v3]. At v1, thetangent map Tv1N has degree 3. Hence there are 4 critical points of N whose visible points to Hrep are v1.At v2, the tangent map Tv2N has 1 superattracting fixed point and 2 attracting fixed points. Thus thereare at least two directions, say the ones containing c2 and c4, attracted to the attracting fixed points. Atv3, the tangent map Tv3N has degree 2.

Figure 1. The hull Hbig ⊂ P1 for N . The hull Hrep consists of the edges (v1, v2), (v2, v3)and the vertices v1, v2, v3. The points c1, c2, c3 and c4 are the non-fixed critical points ofN . At v2, the directions containing c2 and c4 are attracted to attracting fixed points of thetangent map Tv2N .

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8 HONGMING NIE

For now, we focus on combinatorial aspects. The following result summarizes the properties of thesesubtrees. The proof depends on a sequence of lemmas established later in this subsection.

Proposition 3.3. For the Newton map N ∈ L(z), consider the subtrees defined as above. Then we have

(1) The set Vrep consists of repelling fixed points of N in P1. Moreover, Vrep is a singleton if and onlyif the map N is simple.

(2) Let Fix(N) be the set of fixed points of N in P1. Then

Fix(N) = H∞rep ∪ {r1, · · · , rd}.

(3) The visible points from elements in Crit(N) to H∞rep lie in Vrep. Moreover, for each v ∈ Vrep, thereare 2 degv N − 2 points, counted with multiplicity, in Crit(N) whose visible points to H∞rep are v.

(4) For each v ∈ Vrep, the visible points from the preimages N−1(v) to Hfix lie in Vrep.(5) The set Hfix is forward invariant. Moreover, each edge of Hfix is mapped to itself. In particular,

each point in Hfix \H∞rep is attracted to some ri.(6) The ramification locus RN of N consists of the union of closed segments [v, c]s, where c ∈ Crit(N)

and v is the visible point from c to Hfix.

The remaining of this subsection is devoted to prove the above proposition. We begin with Proposition3.3 (2). Note the type I fixed points of N are r1, · · · , rd,∞. Thus Proposition 3.3 (2) is an immediateconsequence of the following lemma.

Lemma 3.4. For the map N ,

Fix(N) \ P1L = H∞rep \ {∞}.

Proof. We first show Fix(N) \P1L ⊃ H∞rep \ {∞}. Let ξ ∈ H∞rep \ {∞}. If ξ is a type II point, let M ∈ Aut(L)

be an affine map such that M(ξg) = ξ. Then M−1 ◦N ◦M ∈ L(z) is a Newton map for a polynomial in L[z].Moreover, the reduction of M−1 ◦N ◦M is an induced complex Newton map of degree at least 1. Then byLemma 2.3, the map M−1 ◦ N ◦M fixes ξg and hence N fixes ξ. If ξ is not a type II point, there exists asequence {ξn} ⊂ H∞rep of type II points such that ρ(ξn, ξ)→ 0 as n→∞. Note N is continuous. Then

ρ(ξn, N(ξ)) = ρ(N(ξn), N(ξ))→ 0.

It deduces that ξ = N(ξ) and hence ξ ∈ Fix(N).For the other direction, let ξ 6∈ H∞rep be a type II point and let M1 ∈ Aut(L) be an affine map such that

M1(ξg) = ξ. Then there exist at least d− 1 many ris such that Red(M−11 (ri)) =∞. Thus, the reduction of

M−11 ◦N ◦M1 is constant. By Lemma 2.3, it follows that ξ 6∈ Fix(N). For any type III or IV point ξ 6∈ H∞rep,

denote the visible point η := πHfix(ξ). Let ~v ∈ TηP1 be the direction such that ξ ∈ Bη(~v)−. By Lemma

2.8, we know N(Bη(~v)−) is an open ball. We first assume η ∈ Hfix \H∞rep. If ξ 6∈ Hfix \H∞rep, then η is of

type II and hence is not a fixed fixed point. So N(Bη(~v)−) and Bη(~v)− are disjoint. Hence ξ 6∈ Fix(N). Ifξ ∈ Hfix \H∞rep, then η = ξ is of type III or IV. Note degη N = 2. Then η can not be a repelling fixed pointbecause all repelling periodic points are either of type I or of type II, see [2, Lemma 10.80]. We also haveξ 6∈ Fix(N). For the remaining case that η ∈ H∞rep, we have η is a fixed point of N . Note ~v is not a fixed

point of the map TηN . It follows that N(Bη(~v)−) and Bη(~v)− are disjoint. Hence ξ 6∈ Fix(N). Therefore,Fix(N) \ P1

L ⊂ H∞rep \ {∞}. �

The above lemma implies immediately the following fundamental result for Newton maps that plays akey role in our work.

Corollary 3.5. Let ξ ∈ H∞rep be a type II point. Then via identifying TξP1 to P1

C, the tangent map TξN isconjugate to an induced Newton map.

Now we prove Proposition 3.3 (1). Recall that a q-periodic type II point ξ ∈ P1 is repelling if degξ φq ≥ 2.

Note a type I fixed point of N is either superattracting or indifferent. Thus the first statement of Proposition3.3 (1) is equivalent to the following lemma.

Lemma 3.6. The set of repelling fixed points of N in H is Vrep

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BERKOVICH DYNAMICS OF NEWTON MAPS 9

Proof. Let ξ ∈ Fix(N) be a type II point and let M ∈ Aut(L) be an affine map such that M(ξg) = ξ. Ifξ 6∈ Vrep, then Red(M−1(ri)) is either ∞ or some constant c ∈ C. Thus, the reduction M−1 ◦ N ◦M hasdegree 1. By Lemma 2.6, degξN = 1. So ξ is not a repelling fixed point. If ξ ∈ Vrep, then there exist

two distinct constants c1 and c2 in C, and exist ri and rj with i 6= j such that Red(M−1(ri)) = c1 andRed(M−1(rj)) = c2. Thus, the reduction of M−1 ◦N ◦M has degree at least 2. Again, by Lemma 2.6, wehave degξN ≥ 2. It follows that ξ is a repelling fixed point. �

If Vrep = {ξ} is a singleton, then ValHfix(ξ) = d+1. Hence degξN = d. Let M ∈ Aut(L) be an affine map

such that M(ξg) = ξ. Then degξg (M−1 ◦N ◦M) = d. By Lemma 2.6, we have deg Red(M−1 ◦N ◦M) = d.

HenceN is simple. Conversely, ifN is simple, then there exitsM ∈ Aut(L) such that deg Red(M−1◦N◦M) =d. So degξg (M−1 ◦N ◦M) = d. Letting ξ = M(ξg), we have degξN = d. Hence Vrep = {ξ} is a singleton.

This completes the proof of Proposition 3.3 (1).Now we consider the visible points from the critical points and prove Proposition 3.3 (3) by the following

result.

Lemma 3.7. If c ∈ Crit(N) is a critical point, then πH∞rep(c) ∈ Vrep. Moreover, for v ∈ Vrep,

]{a ∈ L : P ′′r (a) = 0, πH∞rep(a) = v} = 2 degv N − 2− ]{ri : πH∞rep(ri) = v},counted with multiplicity.

Proof. To ease the notations, let ξ = πH∞rep(c). Then ξ ∈ H is a type II point. Let ~v ∈ TξP1 be the vector

such that c ∈ Bξ(~v)−. Then the directional multiplicity mN (ξ,~v) ≥ 2 since c is a critical point of N . Thusby Lemma 2.5, we have

degξN ≥ mN (ξ,~v) ≥ 2.

By Lemmas 3.4 and 3.6, we have ξ ∈ Vrep.For v ∈ V , there is M ∈ Aut(L) such that M(ξg) = v since v is a point of type II. Then the reduction

Red(M−1 ◦N ◦M) has 2 degv N − 2 critical points. Note

Crit(Red(M−1 ◦N ◦M)) = {Red(M−1(a)) : a ∈ Crit(N), πH∞rep(a) = v}and

Crit(N) = {a ∈ L : P ′′r (a) = 0} ∪ {r1, · · · , rd}.Thus the conclusion holds. �

Now we prove a weak version of Proposition 3.3 (4) which allows us to characterize the fixed attractingBerkovich Fatou components of N and further to prove Proposition 3.3 (5). After that, we prove Proposition3.3 (4).

Lemma 3.8. For v ∈ Vrep, let ξ ∈ N−1(v). Then

πHrep(ξ) ∈ Vrep.

Moreover, for v1 6= v2 ∈ Vrep,

]{ξ ∈ P1 : ξ ∈ N−1(v1), πHrep(ξ) = v2} = degv2 N − 1.

Proof. Suppose η := πHrep(ξ) ∈ Hrep \ Vrep. By Lemmas 3.4 and 3.6, we have η ∈ Fix(N) and degη N = 1.

Let ~v ∈ TηP1 be the vector such that ξ ∈ Bη(~v)−. Since TηN(~v) 6= ~w, where ~w ∈ TηP1 is a direction suchthat Bη(~w)− ∩ Vrep 6= ∅, then

N(Bη(~v)−) ∩ Vrep = ∅.Hence N(ξ) 6∈ Vrep. It is a contradiction. Thus η ∈ Vrep.

At the point v2, let ~v1 ∈ Tv2P1 be the direction such that v1 ∈ Bv2(~v1)−. Then ~v1 is a fixed point ofTv2N with deg~v1 Tv2N = 1. Thus, we have

#{ξ ∈ P1 : ξ ∈ N−1(v1), πHrep(ξ) = v2} = deg Tv2N − 1 = degv2 N − 1.

Corollary 3.9. For v ∈ Vrep, if ξ ∈ N−1(v) satisfyies πHrep(ξ) = v, then ξ = v. In particular, for any

~v ∈ TvP1 with Bv(~v)− ∩ Vrep = ∅,N(Bv(~v)−) 6= P1.

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10 HONGMING NIE

Proof. Let Vrep = {v1, · · · , vn}. Note

degvi N = ValHfix(vi)− 1.

Thus

degvi N +∑j 6=i

(degvj N − 1) =

n∑j=1

degvj N − (n− 1) =

n∑j=1

(ValHfix(vj)− 1)− (n− 1)

=

n∑j=1

ValHfix(vj)− (2n− 1) = d+ 1 + 2(n− 1)− (2n− 1)

= d

Note vi totally has d preimages. Thus, if πHrep(ξi) = vi for some ξi ∈ N−1(vi), then ξi = vi.

As a consequence, for v ∈ Vrep and ~v ∈ TvP1 with Bv(~v)− ∩ Vrep = ∅, we have v 6∈ N(Bξ(~v)−). HenceN(Bv(~v)−) 6= P1. �

Applying Corollary 3.9, we describes the structure of the fixed attracting Berkovich Fatou componentsof N which will be used to prove Proposition 3.3 (5). First, recall that for a rational map φ ∈ L(z), theBerkovich Julia set J(φ) consists of the points ξ ∈ P1 such that for all (weak) neighborhood U of ξ, the set∪∞n=1φ

n(U) omits at most two points in P1. The complement of the Berkovich Julia set is the BerkovichFatou set F (φ), see [2, Section 10.5] and [3, Section 7]. If U is a periodic component of F (φ), then U iseither a Rivera domain, mapping bijectively onto itself, or an attracting component, mapping multiple toone onto itself [2, Theorem 10.76]. For the Newton map N , by Lemma 3.6, we know Vrep ⊂ J(N).

Lemma 3.10. An open set U ⊂ P1 is a fixed attracting component of F (N) if and only if U is a componentof P1 \ Vrep which contains some ri.

Proof. If U is a component of P1 \ Vrep which contains some ri, by Corollary 3.9, U is fixed by N , since Nfixes the boundary point of U and ri. Note that ri is a superattracting fixed point of N , then U is a fixedattracting component of F (N).

Now let U is a fixed attracting component of F (N). Since each ri is a superattracting fixed point of Nand N has no other attracting fixed points, then U contains some ri. Since Vrep is contained in J(N), theBerkovich Fatou component U is contained in a component W of P1 \Vrep with ri ∈W . Again by Corollary3.9, the component W is fixed and hence is a Berkovich Fatou component. Therefore U = W . �

Note each ri is a superattracting fixed point and vi := πHrep(ri) ∈ Vrep is a repelling fixed point of N .

The following lemma states that the segment [vi, ri] is invariant under N and in the path distance metric ρ,the map N pushes points in (vi, ri) away from vi to ri. It deduces Proposition 3.3(5) immediately since byLemma 3.4, we have H∞rep is fixed by N . Later, we will show N expands uniformly on [vi, ri).

Lemma 3.11. Any component L of Hfix \H∞rep is invariant by N . Moreover, for any ξ ∈ L ∩H, then

ρ(N(ξ), πHrep(ξ)) ≥ ρ(ξ, πHrep

(ξ))

Proof. Suppose there exist a component L of Hfix \H∞rep and a point ξ ∈ L such that N(ξ) 6∈ L. We assume

ri ∈ L and set v = πHrep(ξ) . Let ~v1 ∈ TvP1 and ~v2 ∈ TξP1 be the directions such that ri ∈ Bξ(~v1)− ⊂

Bv(~v2)−. Let ξ′ = N(ξ) and ~w = TξN(~v) ∈ Tξ′P1. By Corollary 3.9,

N(Bξ(~v1)−) = Bξ′(~w)−.

Note N(ri) = ri. It follows that [ri, v] ⊂ Bξ′(~w)−. Thus N−1(v) ∩ Bv(~v2)− 6= ∅. It contradicts withCorollary 3.9. So N(L) ⊂ L. Note N fixes the endpoints of L. We have N(L) = L.

By Lemma 3.10, the open ball Bv(~v2)− is a fixed attracting component of F (N). Thus

ρ(ξ, ri) > ρ(N(ξ), ri).

Note πHrep(N(ξ)) = πHrep

(ξ). Thus we have

ρ(ξ, πHrep(ξ)) < ρ(N(ξ), πHrep(N(ξ))) = d(N(ξ), πHrep(ξ)).

Now we can prove Proposition 3.3 (4).

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BERKOVICH DYNAMICS OF NEWTON MAPS 11

Corollary 3.12. For v ∈ V , let ξ ∈ N−1(v). Then πHfix(ξ) ∈ Vrep.

Proof. By Lemma 3.8, it is sufficient to show πHfix(ξ) 6∈ Hfix \Hrep. Suppose that η := πHfix

(ξ) ∈ Hfix \Hrep.By Corollary 3.9, the open ball with boundary η containing ξ maps to an open ball B with boundary N(η)and πHrep

(η) 6∈ B. By Lemma 3.11, we know

ρ(N(η), πHrep(η)) > ρ(η, πHrep(η)).

Thus B ∩ Vrep = ∅. It is a contradiction. �

Now we are in the stage to prove Proposition 3.3 (6) about the structure of ramification locus RN of N .For a rational map over L of degree at least 2, Faber [10, Corollary 7.13] proved that its ramification locus iscontained in the convex hull of its critical points. Thus, RN ⊂ Hcrit. The following lemma gives the preciselocations of RN and implies Proposition 3.3 (6).

Lemma 3.13. For the Newton map N , the ramification locus is

RN = (Hcrit \Hrep) ∪ Vrep.

Proof. For any point ξ ∈ Hrep \ Vrep, by Lemmas 3.4 and 3.6, degξN = 1. Thus ξ 6∈ RN . Hence, RN ⊂(Hcrit \Hrep) ∪ Vrep.

Conversely, by Lemma 3.6, Vrep ⊂ RN . Now pick ξ ∈ Hcrit \ Hrep. By Corollary 3.9, there exists avector ~v ∈ TξP1 such that Bξ(~v)− ∩ Crit(N) 6= ∅ and N(Bξ(~v)−) 6= P1. Thus, the directional multiplicitymN (ξ,~v) ≥ 2. By Lemma 2.5,

degξN ≥ mN (ξ,~v) ≥ 2.

It follows that ξ ∈ RN . �

To end this section, we show the map N expands uniformly on each segment [vi, ri), where vi = πHrep(ri).

Lemma 3.14. Let vi = πHrep(ri). Then for any ξ1, ξ2 ∈ [vi, ri), then

ρ(N(ξ1), N(ξ2)) ≥ 2ρ(ξ1, ξ2).

Proof. By Lemma 3.13, for each point ξ ∈ [vi, ri], we have degξN ≥ 2. In fact, the directional multiplicity

mN (ξ,~v) ≥ 2, where ~v ∈ TξP1 is the direction at ξ such that ri ∈ Bξ(~v)−. Then the conclusion follows fromLemma 2.4. �

3.2. Type II repelling cycles of periods at least 2. We already figured out the repelling fixed points ofN in previous subsection. Now we work on repelling periodic points of periods at least 2 in H. We begin withthe following result which establish a sufficient and necessary condition for the existence of such periodicpoints.

Proposition 3.15. The Newton map N has a type II repelling cycle of period q ≥ 2 if and only if thereexist a point v ∈ Vrep, a critical point ~v ∈ TvP1 of TvN , a critical point c ∈ Crit(N) in the open ball Bv(~v)−

and an integer 1 ≤ ` ≤ q − 1 such that the following holds.

(1) Let ~w := TvN`(~v) ∈ TvP1. The open ball Bv(~w)− contains a repelling fixed point of N ;

(2) Nq(c) ∈ Bv(~v)−;(3) Let η := c ∨Nq(c). For k ≥ 1, there exists ξk ∈ N−kq(v) ∩ (v, η) such that

ρ(η, v) ≥ ρ(ξ∞, v),

where ξ∞ = limk→∞

ξk.

In particular, if (1) − (3) holds, the point ξ∞ is periodic of period q. Moreover, for any point ξ ∈ (v, ξ∞),ρ(Nq(ξ), ξ∞) > ρ(ξ, ξ∞), and hence Nq(ξ) 6= ξ.

Proof. We first show the “only if” part. If statement (1) does not hold, then all critical points of N arecontained in the Berkovich Fatou set F (N). Moreover, for any c ∈ Crit(N), the segment [c, πHrep

(c)) is inF (N). Note the repelling cycles of type II points intersect with the ramification locus RN . By Lemmas 3.4and 3.13, we know the type II repelling cycles are contained in Vrep. Thus, all of them have period 1. It isa contradiction. If statement (2) does not hold, then for any c ∈ Crit(N), the segment [c, πHrep

(c)) disjointswith the periodic cycles. Thus, again the type II repelling cycles are contained in Vrep. It is a contradiction.Now we show statement (3) holds. From statements (1), (2) and the existence of a periodic cycle of period

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12 HONGMING NIE

at least 2, we know for k ≥ 1, there exists unique ξk ∈ (N)−kq(v) ∩ (v, η). By Lemma 3.13, we know (v, η)is contained in the ramification locus RN . Then

ρ(ξk, ξk+1) ≥ 2ρ(ξk+1, ξk+2)

for all k ≥ 0. So ρ(ξk, ξk+1) → 0, as k → ∞. Hence the sequence {ξk} converges. Therefore, ξ∞ exists.Moreover, we have Nq(ξ∞) = ξ∞. Note for any point ξ ∈ (v, ξ∞), we have

ρ(Nq(ξ), ξ∞) > ρ(ξ, ξ∞).

Hence there is no q-periodic point of N in the segment (v, ξ∞). Thus statement (3) holds.Now we prove the “if” part. Note the statements (1) − (3) imply that the point ξ∞ is a fixed point of

Nq. Moreover, by Lemma 3.13, we know ξ∞ ∈ RN . Thus ξ∞ is a repelling fixed point of Nq. In particular,ρ(ξk, ξk+1) ≥ 2ρ(ξk+1, ξk+2). Then ξ∞ ∈ H is a repelling periodic point, and hence ξ∞ is a type II point,since all repelling periodic points in H are of type II, see [2, Lemma 10.80] �

Now we give the following example to illustrate Proposition 3.15.

Example 3.16. Consider the quartic Newton maps N := Nr with r = {r1, r2, r3, r4} ⊂ L,where

r1 = −1−√

3i+ (5 +40√

3

9i)t2,

r2 = −1 +√

3i+ (5− 40√

3

9i)t2,

r3 = 2 +

√30

3t− 5t2,

r4 = 2−√

30

3t− 5t2.

Note Vrep = {ξg, ξ2,|t|}. Set v := ξg and v1 := ξ2,|t|. Let q = 2, ` = 1. The visible points from bothnon-fixed critical points c1 and c2 to Hrep are v, that is, πHrep(c1) = πHrep(c2) = v. Let c1 be the critical

point such that πHrep(N(c1)) = v1 and let ~v ∈ TvP

1 be such that c1 ∈ Bv(~v)−. Then TvN(~v) ∈ TvP1

and v1 ∈ Bv(TvN(~v))−. Let ~w ∈ Tv1P1 be such that N(c1) ∈ Bv1(~w)−. Then Tv1N(~w) ∈ Tv1P

1 andv ∈ Bv1(Tv1N(~w))−. Moreover, we have N2(c1) ∈ Bv(~v)−. Considering ξk and ξ∞ as in Proposition 3.15,we have ξ∞ is a 2-periodic point for N and η = ξ∞, as it is shown in Figure 2. Moreover, we have

ρ(v, ξ∞) = 2ρ(v, ξ1) = 2ρ(v, v1) = 2.

Figure 2. The 2-periodic type II cycle for Example 3.16.

Now we associate each critical point at most one type II repelling periodic point, which implies that Nhas only finitely many type II repelling cycles. We will give the sharp upper bound later.

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BERKOVICH DYNAMICS OF NEWTON MAPS 13

Proposition 3.17. For each c ∈ Crit(N), the segment (c, πHrep(c)) contains at most one type II repellingperiodic point.

Proof. By Lemma 3.11, there is no type II repelling periodic point in (c, πHrep(c)) if πHfix(c) /∈ Vrep. Now we

consider a critical point c ∈ Crit(N) with πHfix(c) ∈ Vrep. To ease notation, let Lc := (c, πHrep(c)). Suppose

Lc contains a type II repelling periodic point ξ∞ of period q ≥ 2. Set ξ = πHrep(ξ∞). By Proposition 3.15, we

can choose ξ∞ such that there is no repelling periodic point in (ξ, ξ∞). Suppose ξ′ ∈ Lc \ (ξ, ξ∞] is anothertype II repelling periodic point. Let ~v ∈ Tξ∞P1 be the direction such that ξ′ ∈ Bξ∞(~v)−. By Corollary 3.9,for 1 ≤ ` ≤ q, we know N `(Bξ∞(~v)−) is an open ball with boundary at N `(ξ∞). Thus ξ′ has period kq forsome integer k ≥ 1. Now consider the segment [ξ∞, ξ

′]. For 1 ≤ ` ≤ kq, again by Corollary 3.9, we knowN `([ξ∞, ξ

′]) = [N `(ξ∞), N `(ξ′)] is a segment. Thus

Nkq([ξ∞, ξ′]) = [Nkq(ξ∞), Nkq(ξ′)] = [ξ∞, ξ

′].

However, by Lemma 2.4, we know

ρ(Nkq(ξ∞), Nkq(ξ′)) ≥ 2ρ(ξ∞, ξ′).

It is a contradiction. Thus ξ∞ is the only type II repelling periodic point in Lc. �

Let us agree that a critical point c ∈ Crit(N) is totally free if at the point v := πHrep(c), the vector ~vc is

not in the immediate basin of any (super)attracting fixed point of TvN , where ~vc ∈ TξP1 with c ∈ Bξ(~vc)−.

Lemma 3.18. The Newton map N has at most d− 1− ]Vrep totally free critical points.

Proof. At a point v ∈ Vrep, let Cv be the set of totally free critical points c of N with πHrep(c) = v and let

Av be the set of attracting fixed points of TvN . Then at v, we have

]Cv ≤ 2 degv N − 2− ]{ri : πHrep(ri) = v} −#Av.

Note that if a direction ~v ∈ TvP1 at v is such that B(~v)−∩Hrep 6= ∅ and∞ 6∈ B(~v)−, then ~v is an attractingfixed point of TvN . Thus∑

v∈Vrep

]Av =∑v∈Vrep

ValHrep(v)− (]Vrep − 1) = 2(]Vrep − 1)− (]Vrep − 1) = ]Vrep − 1.

It follows that ∑v∈Vrep

]Cv ≤∑v∈Vrep

(2 degv N − 2)−∑v∈Vrep

]{ri : πHrep(ri) = v} −

∑v∈Vrep

]Av

= 2d− 2− d− (]Vrep − 1) = d− 1− ]Vrep.

If a critical point c ∈ Crit(N) is not totally free, then the segment (c, πHrep(c)) is in the Berkovich Fatou set

F (N). Hence, in this case, there is no type II repelling periodic point in (c, πHrep(c)). Thus, by Proposition

3.17, the number of totally free critical points gives an upper bound of the the number of type II repellingcycles of periods at least 2 for N .

Corollary 3.19. The Newton map N has at most d− 1− ]Vrep type II repelling cycles of periods at least 2.

To end this subsection, we show that if a type II repelling periodic cycle exists, then the correspondingtangent map is conjugate to a polynomial whose degree is bounded and independent of the period.

Proposition 3.20. Let ξ ∈ H be a type II repelling periodic point of period q ≥ 2 of N . Then the tangentmap TξN

q is conjugate to a polynomial of degree at most 2d−1−]Vrep .

Proof. Let ξ ∈ H be a repelling periodic point of period q ≥ 2. By Proposition 3.15, there is a uniquedirection ~v ∈ TξP1 such that Vrep ⊂ Bξ(~v)−. By Lemma 3.8 and Corollary 3.9, the direction ~v is totallyinvariant under TξN

q at ξ. Thus the map TξNq is conjugate to a polynomial.

To determine the degree, let

I = {i ∈ {0, · · · , q − 1} : degNi(ξ)N ≥ 2}.

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14 HONGMING NIE

Then by Lemma 2.5,

deg TξNq =

∏i∈I

degNi(ξ)N.

If c ∈ Crit(N) is not a totally free critical point, then the segment (c, πHrep(c)) is contained in the BerkovichFatou set F (B). Thus there is no repelling periodic point in (c, πHrep(c)). Hence by Lemma 3.18, we have

]I ≤ d− 1− ]Vrep. It is easy to check that deg TξNq ≤ 2d−1−]Vrep . Indeed, if there are multiple totally free

critical points, then the number of the totally free critical points drops. �

3.3. Iterated preimages of repelling periodic points. Let T0 be the convex hull of type II repellingperiodic points of N . Then by Proposition 3.3 (1), Hrep ⊂ T0. Moreover, T0 = Hrep if and only if N has notype II repelling periodic points of periods at least 2. In particular, by Proposition 3.3 (2) and Proposition3.15, we have N(T0) = T0.

For n ≥ 0, define Tn = N−n(T0).

Lemma 3.21. For n ≥ 0, Tn is connected and Tn ⊂ Tn+1.

Proof. By Proposition 3.3 (2), Hrep ⊂ Tn. Suppose Tn is disconnected. Let U1 be the connected componentof Tn containing Hrep and let U2 be a component of Tn disjointing with U1. Let a1 ∈ U1 be the visiblepoint from any point in U2 to U1 and let a2 ∈ U2 be the visible point from any point in U1 to U2. Then thesegment (a1, a2) ⊂ P1 \ (U1 ∪ U2) and U1 ∪ U2 ∪ [a1, a2] is connected. Moreover, [a1, a2] in not contained inTn, for otherwise U1 = U2. By Proposition 3.15, the point a1 is eventually mapped to some periodic pointξ in T0. Since U2 is a component of Tn, then in both cases, under Ta1N

n for some n ≥ 0, the direction~v ∈ Ta1P1 at a1 containing U2 is mapped to a direction at ξ containing a point in Vrep. Thus there is a pointa′1 ∈ Ba1(~v) close to a1 such that [a1, a

′1] is contained in U1. It contradicts to the choice of a1. Therefore,

Tn is connected.By Lemma 3.8, Corollary 3.9 and Proposition 3.15, we have T0 ⊂ T1. It follows immediately that

Tn ⊂ Tn+1. �

Now setT =

⋃n≥0

Tn.

Then T ⊂ H is connected and N−1(T ) = T . Moreover, T carries a natural tree structure with vertices atthe iterated preimages of type II repelling periodic points which makes it to be an infinite tree. Next resultclaims that T is large enough to contain the non type I points in Julia set J(N).

Proposition 3.22. Let (ξ1, ξ2) be an edge of T such that ξi is not an iterated preimage of a type II repellingperiodic point of period at least 2 for i = 1, 2. Then there exists an edge (v1, v2) in Hrep such that (ξ1, ξ2) iseventually mapped to (v1, v2). In particular, any component of P1 \ T is contained in the Berkovich Fatouset F (N).

Proof. For i = 1, 2, let ni be the smallest nonnegative integers such that Nni(ξi) ∈ Vrep and set vi = Nni(ξi).Now we show (v1, v2) is an edge in Hrep. We may assume n2 ≥ n1. Let ξ := Nn1(ξ2). Then Nn1 maps(ξ1, ξ2) onto (v1, ξ). If ξ = v2, that is, n1 = n2. We are done. If ξ 6= v2, then the visible point πHrep(ξ) = v1.For otherwise, (ξ1, ξ2) contains an iterated preimage of a point in Vrep. It follows immediately that (v1, v2)is an edge in Hrep. For otherwise, (v1, ξ) contains a preimage of a point v ∈ Vrep under Nn2−n1 and hence(ξ1, ξ2) contains a preimage of v ∈ Vrep under Nn2 , which is a contradiction.

If U is a component of P1 \ T , let a be the visible point from any point in U to T . If a is not a vertexof T , then there exists an edge (ξ1, ξ2) of T such that (ξ1, ξ2) contains a and is eventually mapped onto anedge (v1, v2) of Hrep. Observe at any point ξ′ ∈ (v1, v2), the tangent map Tξ′N has degree 1 and has fixedpoints at ~v1 and ~v2, where ~vi ∈ Tξ′P1 is the direction such that vi ∈ Bξ′(~vi)

−. By Lemma 3.4 and Corollary3.9, the component of P1 \Hrep containing (v1, v2) is a fixed Berkovich Fatou component. Thus in this case,U is contained in F (N). Now we assume a is a vertex of T . Let v be a type II repelling periodic point suchthat Nn(a) = v for some n ≥ 0. Then by Corollary 3.9, it follows that Nn(U) is a component of P1 \ {v}disjointing with Vrep. If v /∈ Vrep, then v is a repelling periodic point of period at least 2. By Proposition3.20, Nn(U) and hence U is a Berkovich Fatou component. If v ∈ Vrep, then Nn+m(U) disjoints with Vrep

for any m ≥ 0. For otherwise, Nn(U) and hence U intersect with T . Thus by Corollary 3.9, Nn(U) andhence U are Berkovich Fatou components. �

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BERKOVICH DYNAMICS OF NEWTON MAPS 15

3.4. Proof of Theorem 1.1. We are ready to prove Theorem 1.1.

Proof of Theorem 1.1. We first prove the equivalence of simplicity of Newton maps. By Proposition 3.3 (1),N is simple if and only if N has only one repelling fixed point in P1. If Vrep is a singleton, then Hrep = Vrep.Thus, by Lemma 3.13, RN = Hcrit and hence RN is connected. Conversely, note Hrep is contained in thesubstree of Hcrit generated by the vertices of valence at least 3. To show Vrep is a singleton, we argue bycontradiction. Suppose Vrep contains at least two points. Then Hrep contains a segment. Hence by Lemma3.13, RN is contained in Hcrit removed a least one edge connecting two inner vertices. Hence RN is notconnected. It is a contradiction. Thus if RN is connected, then Vrep is a singleton.

From now on, we consider the case that N is not simple. Since N is not simple, by Proposition 3.3 (1),N has at least two repelling fixed points in P1. Note N has no repelling fixed points of type I. Thus all therepelling fixed points are in H. Moreover, the number of repelling fixed points in P1 is ]Vrep. Since Hfix isthe convex hull of d+ 1 points, the set Vrep contains at most d− 1 elements. It follows that N has at mostd− 1 repelling fixed points. For the type II repelling cycles of period at least 2, by Corollary 3.19, N has atmost d − 1 − ]Vrep such cycles. Since ]Vrep ≥ 2, the N has at most d − 3 such cycles. Moreover, the totalnumber of type II repelling cycles is at most ]Vrep + d− 1− ]Vrep = d− 1. Thus statement (a) holds

For the Berkovich Julia set J(N), obviously the grand orbits of repelling cycles in H are contained inJ(N)∩H. For the other direction, by Proposition 3.22, J(N)∩H is contained in the tree T . Now for a pointξ ∈ T that is not in the grand orbit of a repelling cycle in H. Then ξ is contained in an edge of T , which,again by Proposition 3.22, is eventually mapped to a point in an edge of Hrep. Thus ξ is eventually mappedinto a fixed Fatou component and hence it is contained in Berkovich Fatou set. Therefore, statement (b)holds.

Now consider the Berkovich Fatou set F (N). By Lemma 3.10, F (N) contains exact d attracting fixedBerkovich Fatou components, which are the immediate attracting basins of ris. Since ∞ is an indifferentfixed point, then there is a fixed Rivera domain containing ∞. Moreover, any component of P1 \ Vrep

containing an edge of Hrep is a fixed Rivera domain which is an annulus. By statement (b), Propositions3.20 and 3.22, there is no other fixed Berkovich Fatou component. Thus the total number of fixed BerkovichFatou components is d+ 1 + ]Vrep − 1 ≤ 2d− 1. Let U be a periodic Berkovich Fatou component of periodat least 2. Again by statement (b) and Proposition 3.22, the boundary of U is a repelling periodic point,and hence U is an open ball. Thus statement (c) holds.

Now we prove statement (d) about the ramification locus RN . By Lemma 3.13, each component of RNcontains exact one point in Vrep. Thus the number of components of RN is at most ]V ≤ d− 1. �

Proof of Corollary 1.3. By Theorem 1.1 (2b) and Proposition 3.22, we only need to consider the BerkovichFatou components U that are open balls with periodic boundaries. Suppose U is not eventually periodic.Let ξ be the boundary of U that is a q-periodic point for q ≥ 1. Then by Corollary 3.9 and Proposition 3.20,{Nkq(U)}k≥0 is a set of infinite open balls with same boundary at ξ. Thus U is in the basin of the cyclecontaining ξ. Thus N has no wandering Berkovich Fatou domain. �

4. Rescaling Limits

This section provides proofs of Theorem 1.4 and related results. From Kiwi’s general results [18], Theorem1.4 is a consequence of the Berkovich dynamics in Section 3. We briefly introduces Kiwi’s results and proveTheorem 1.4 in Section 4.1. Then in Section 4.2, considering the period 1 rescaling limits, we construct a non-Hausdorff compactification of the moduli space of complex Newton maps. Finally, we show the instabilityof the degenerate rational maps that induces the rescaling limits of periods at least 2 in Section 4.3,

4.1. Definitions and proof of Theorem 1.4. We start by introducing the definition of holomorphic familyand state a relation between convergences. For t ∈ D, a collection {ft} ⊂ P2d+1 is a holomorphic family of

degree d ≥ 1 rational maps if the map D → P2d+1C , sending t to ft, is a holomorphic map such that ft is a

degree d rational map if t 6= 0. We say a holomorphic family of degree 1 rational maps is a moving frame.

Lemma 4.1. [18, Lemma 3.2 ] Let {ft} be a holomorphic family of rational maps. Suppose that ft converges

to f = Hf f in P2d+1C . Then ft converges to f locally uniformly on P1

C \ {h : Hf (h) = 0}.

Now we consider the rescaling limits for a holomorphic family.

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16 HONGMING NIE

Definition 4.2. [18, Definition 1.1] Let {ft} be a holomorphic family of degree d ≥ 2 rational maps. Wesay a moving frame {Mt} is a rescaling for {ft} of period q ≥ 1 if there exist a degree e ≥ 2 rational map gand a finite subset S ⊂ P1 such that as t → 0, M−1

t ◦ fqt ◦Mt converges to g locally uniformly on P1C \ S.

We say g is a rescaling limit on P1 \ S and the minimal such q ≥ 1 is the period of the rescaling {Mt}.

The holomorphic families {ft} and {Mt}, regarded as maps in L, act on the Berkovich space P1. Therescaling limits are closely related to the type II repelling cycles in P1.

Proposition 4.3. [18, Proposition 3.4] Let {ft} be a holomorphic family of degree d ≥ 2 rational maps andlet {Mt} be a moving frame. Then for e ≥ 2, the following are equivalent.

(1) {Mt} is a rescaling for {ft} of period ` ≥ 1 with rescaling limit g of degree e.(2) In P1, set ξ = Mt(ξg). Then f `t (ξ) = ξ and degξ f

`t = e.

In the case that (1) and (2) hold, via identifying TξP1 to P1

C, the tangent map Tξf`t : TξP

1 → TξP1 is

conjugate to g : P1C → P1

C.

To count the number of rescalings, we needs the following two equivalence relations.

Definition 4.4. [18, Definitions 1.2 and 3.8 and Lemmas 3.6 and 3.7] Let {ft} be a holomorphic family ofdegree d ≥ 2 rational maps and let {Mt} and {Lt} are two rescalings for {ft}.

(1) We say {Mt} and {Lt} are equivalent if Mt(ξg) = Lt(ξg).(2) We say {Mt} and {Lt} are dynamical dependent if Mt(ξg) and Lt(ξg) are in the same cycle of ft.

Otherwise, we say {Mt} and {Lt} are dynamical independent.

Let {Nt} be a holomorphic family of Newton maps. Applying the Berkovich dynamics of Newton maps,we now can prove Theorem 1.4.

Proof of Theorem 1.4. By Theorem 1.1 and Proposition 4.3, the family {Nt} has at most d − 1 dynamicalindependent rescalings. Again by Proposition 4.3, the equivalent rescalings of period 1 are related to thetype II repelling fixed points. Thus statement (1) holds immediately from Theorem 1.1 and Corollary 3.5.For statement (2), if Nt has only one repelling fixed point in H, then by Theorem 1.1 (1), Nt is simple andhas no type II repelling cycle of period at least 2. In this case, the family {Nt} has no rescaling of period atleast 2. Thus if the family {Nt} has a rescaling of period at least 2, then the map Nt has at least 2 repellingfixed points in H. Then by Theorem 1.1 (2a) and Corollary 3.20, statement (2) holds. �

For a holomorphic family {Nt} of quartic Newton maps, by Theorem 1.4, {Nt} has at most 1 rescaling ofperiod at least 2 and the corresponding rescaling limit is a quadratic polynomial. To end this subsection, weuse a perturbation of the map in Example 3.2 to show that any quadratic polynomial is a possible rescalinglimit for a holomorphic family of quartic Newton maps. Consider r(t) = {r1(t), r2(t), r3(t), r4(t)}, where

r1(t) = −1−√

3i+ (5 +40√

3

9i)t2,

r2(t) = −1 +√

3i+ (5− 40√

3

9i)t2,

r3(t) = 2 +

√30

3t− 5t2 + at3,

r4(t) = 2−√

30

3t− 5t2 − at3.

Then by computation, we have {Mt(z) = 5t2z/18} is a period 2 rescaling for {Nr(t)} and the correspondingrescaling limit is

fa(z) = z2 + 62 +144a

√30

25.

4.2. Compactification of moduli space via rescaling limits. Denote NMd be the space of degree d ≥ 2complex Newton maps and let NMd be the compactification of NMd in P2d+1

C . Since∞ is the unique repellingfixed points of Newton maps, we define the moduli space of degree d ≥ 2 Newton maps is by

nmd := NMd/Aut(C),

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BERKOVICH DYNAMICS OF NEWTON MAPS 17

modulo the action by conjugation of affine maps. There are known compactification of nmd. For examplewe refer [8, 23, 31] for distinct compactifications of moduli spaces of rational maps. It turns out that theydeduce same compactification of nmd, see [24]. Now we use the period 1 rescaling limits to construct anon-Hausdorff compactification of nmd that is definitively different from the previous one.

We first define a relation ∼ on P2d+1C ×P2d+1

C . For f = Hf f , g = Hg g ∈ P2d+1C , we say f ∼ g if there exists

M ∈ Aut(P1C) such that M−1 ◦ f ◦M = g. Denote by [f ]∼ for the equivalence class of f . Let NM

{0,1}d be

the subset of NMd consisting of degree d ≥ 2 Newton maps Nr, where r = {0, 1, r3, · · · , rd} ⊂ C is a set of d

distinct points. Let NM{0,1}d be the compactification of NM

{0,1}d in P2d+1

C . Then for any f = Hf f ∈ NM{0,1}d ,

we have deg f ≥ 2. Moreover, for any point g = Hg g ∈ NMd with deg g ≥ 2, there exists f ∈ NM{0,1}d such

that f ∼ g. Thus, we have

Lemma 4.5. For d ≥ 3,

NM{0,1}d / ∼= NMd ∩ {Hf f : deg f ≥ 2}/ ∼ .

To relate NM{0,1}d / ∼ to the rescaling limits, we now need to characterize the possible period 1 rescaling

limits for holomorphic families of Newton maps.

Lemma 4.6. For any Hf f ∈ NMd with deg f ≥ 2, there exist a holomorphic family {Nt} of degree d ≥ 2

Newton maps and a period 1 rescaling {Mt} for {Nt} such that the corresponding rescaling limit s f .

Proof. Since Hf f ∈ NMd, there exists a holomorphic family {Nt} of degree d Newton maps such that Ntconverges to Hf f , as t → 0, in P2d+1

C . Set Mt(z) = z. Then {Mt} is a period 1 rescaling for {Nt} with

rescaling limit f since deg f ≥ 2. �

Fix d ≥ 3 and denote by RL(NMd, 1) the set of all period 1 rescaling limits for holomorphic families ofdegree d Newton maps. Then by Theorem 1.4 (1) and Lemma 4.6,

RL(NMd, 1) = NMd ∩ {Hf f : deg f ≥ 2}.Consider the quotient space nmRL

d := RL(NMd, 1)/ ∼ associated with the quotient topology. Then

nmRLd = NM

{0,1}d / ∼ .

Since NM{0,1}d is compact, then the quotient space NM

{0,1}d / ∼ and hence nmRL

d are compact. Note nmRLd

contains nmd as a dense subset. We say nmRLd is the compactification of the moduli space nmd via rescaling

limits.

Proposition 4.7. For d ≥ 3, the space nmRLd is not Hausdorff.

Proof. To show nmRLd is not Hausdorff, it is sufficient to show the relation ∼ is not closed on NM

{0,1}d ×

NM{0,1}d . Set r(t) = {0, 1, t, r4(t), · · · , rd(t)} and r′(t) = {0, 1/t, 1, r4(t)/t, · · · , rd(t)/t}, where ri(t)s are

holomorphic functions on t with ri(t) 6= rj(t) if i 6= j and |ri(t)| = o(|t|), as t → 0, for i = 4, · · · , d. LetMt(z) = tz. Then we have if t 6= 0,

M−1t ◦Nr(t) ◦Mt = Nr′(t).

Hence [Nr(t)]∼ = [Nr′(t)]∼.

However, as t→ 0, we have Nr(t) converges to N{0,1,0,··· ,0} and Nr′(t) converges to N{0,∞,1,0,··· ,0} in P2d+1C .

NoteN{0,1,0,··· ,0} 6∼ N{0,∞,1,0,··· ,0}.

Thus ∼ is not closed on NM{0,1}d ×NM

{0,1}d . �

For cubic Newton maps, we have

nmRL3 = nm3 ∪ {[N{0,1,∞}]∼, [N{0,0,1}]∼},

and there are no open sets can separate [N{0,1,∞}] and [N{0,0,1}]. As another example, we state explicitly

the elements in the space nmRL4 . Note NM

{0,1}4 = {N{0,1,r3,r4} : r3, r4 ∈ C}. Thus

nmRL4 \ nm4 = {[N{0,1,r,∞}]∼, [N{0,1,∞,∞}]∼, [N{0,0,1,r}]∼, [N{0,0,0,1}]∼, N{0,0,1,1}]∼ : r ∈ C \ {0, 1}}.

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18 HONGMING NIE

4.3. Unstable limits. Following Silverman [31], denote Ratssd ⊂ P2d+1C the semistable locus. The following

criterion of semistablity is due to DeMarco.

Lemma 4.8. [8, Section 3] Let f = Hf f ∈ P2d+1C . Then f = Hf f ∈ Ratssd if and only if the depth of each

hole is ≤ (d+ 1)/2 and if dh(f) ≥ d/2, then f(h) 6= h.

Our next result asserts that rescalings of period at least 2 lead to non-semistable limits.

Proposition 4.9. Let {Nt} be a holomorphic family of degree d ≥ 3 Newton maps, and let {Mt} be a

rescaling of period q ≥ 2 for {Nt}. Suppose that M−1t ◦Nq

t ◦Mt converges to f in P2dq+1C . Then f 6∈ Ratssdq .

Proof. We write f = Hf f . By Lemma 4.1 and Theorem 1.4 (2), we can assume that f is a polynomial. ByProposition 3.15, the point ∞ is the unique hole of f . To show f 6∈ Ratssdq , by Lemma 4.8, it is sufficient toshow the hole ∞ has depth d∞(f) ≥ dq/2.

If q > d− 3, then by Theorem 1.4 (2), we have

deg f

dq≤ 2d−3

dq<

1

2.

In this case, d∞(f) ≥ dq − deg f ≥ dq/2 and we are done.If 2 ≤ q ≤ d−3, let {ξ0, ξ1, · · · , ξq−1} be the corresponding type II repelling cycle. Suppose {ξ0, ξ1, · · · , ξq−1}∩

RNt = {ξi0 , · · · , ξi`−1} for some 1 ≤ ` ≤ q. Then

deg f =

`−1∏k=0

degξikNt.

At a point ξik , let ~vk ∈ TξikP1 be the direction such that ∞ ∈ Bξik(~vk)−. For each 0 ≤ k ≤ `− 1, suppose

there are nk critical points of Nt, counted with multiplicity, that are not in Bξik(~vk)−. By Lemma 3.18, we

haven0 + · · ·+ n`−1 ≤ d− 3.

Now we claim degξikNt = nk + 1 for 0 ≤ k ≤ ` − 1. By Lemma 2.4, there exists ξ′k ∈ Bξik

(~vk)− such

that the directional multiplicity mNt(ξik , ~vk) = degξ′′k Nt for any ξ′′k ∈ (ξik , ξ′k). By Lemma 3.13, we know

degξ′′k Nt ≥ degξikNt. It follows that mNt(ξik , ~vk) ≥ degξik

Nt. Hence

mNt(ξik , ~vk) = degξikNt.

Let Mk, Lk ∈ Aut(P1L) be affine maps such that Mk(ξg) = ξik and Lk(ξg) = Nt(ξik). Then at ξik , the

direction TξikM−1k (~vk) is a fixed point of Tξg (L−1

k ◦Nt ◦Mk), and

degTξikM−1k (~vk) Tξg (L−1

k ◦Nt ◦Mk) = deg Tξg (L−1k ◦Nt ◦Mk).

Thus, ~vk is a critical point of TξikNt with multiplicity deg TξikNt − 1. It follows that

nk + deg TξikNt − 1 = 2 deg TξikNt − 2.

Thus nk = deg TξikNt − 1. Therefore, we have

deg f =

`−1∏k=0

(nk + 1).

Note(n0 + 1) + · · ·+ (n`−1 + 1) ≤ d− 3 + `.

It follows that

deg f ≤ (d− 3 + `

`)`.

Since d− 3 + ` < `d, we have

deg f

dq≤ 1

dq−`(d− 3 + `

`d)` ≤ 1

dq−`(d− 3 + `

`d).

If 1 ≤ ` < q, then deg f/dq ≤ 1/dq−` ≤ 1/d ≤ 1/2. If ` = q, then ` ≥ 2 and deg f/dq ≤ (d−3+`)/(`d) ≤ 1/2

Therefore, d∞(f) = dq − deg f ≥ dq/2. �

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BERKOVICH DYNAMICS OF NEWTON MAPS 19

References

[1] Matthieu Arfeux. Dynamics on trees of spheres. Journal of the London Mathematical Society, 95(1):177–202, 2017.

[2] Matthew Baker and Robert Rumely. Potential theory and dynamics on the Berkovich projective line, volume 159 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2010.

[3] Robert L. Benedetto. Non-archimedean dynamics in dimension one, lecture notes. Preprints,http://math.arizona.edu/ swc/aws.

[4] Vladimir G. Berkovich. Spectral theory and analytic geometry over non-Archimedean fields, volume 33 of Mathematical

Surveys and Monographs. American Mathematical Society, Providence, RI, 1990.[5] Jean-Paul Bezivin. Sur les points periodiques des applications rationnelles en dynamique ultrametrique. Acta Arith.,

100(1):63–74, 2001.

[6] Professor Cayley. Desiderata and Suggestions: No. 3.The Newton-Fourier Imaginary Problem. Amer. J. Math., 2(1):97,1879.

[7] Laura DeMarco. Iteration at the boundary of the space of rational maps. Duke Math. J., 130(1):169–197, 2005.

[8] Laura DeMarco. The moduli space of quadratic rational maps. J. Amer. Math. Soc., 20(2):321–355, 2007.[9] Laura DeMarco and Xander Faber. Degenerations of complex dynamical systems II: analytic and algebraic stability. Math.

Ann., 365(3-4):1669–1699, 2016. With an appendix by Jan Kiwi.

[10] Xander Faber. Topology and geometry of the Berkovich ramification locus for rational functions, I. Manuscripta Math.,142(3-4):439–474, 2013.

[11] Xander Faber. Topology and geometry of the Berkovich ramification locus for rational functions, II. Math. Ann., 356(3):819–844, 2013.

[12] Charles Favre and Thomas Gauthier. Continuity of the Green function in meromorphic families of polynomials. Algebra

Number Theory, 12(6):1471–1487, 2018.[13] Liang-Chung Hsia. Closure of periodic points over a non-Archimedean field. J. London Math. Soc. (2), 62(3):685–700,

2000.

[14] John Hubbard, Dierk Schleicher, and Scott Sutherland. How to find all roots of complex polynomials by Newton’s method.Invent. Math., 146(1):1–33, 2001.

[15] Mattias Jonsson. Dynamics of Berkovich spaces in low dimensions. In Berkovich spaces and applications, volume 2119 of

Lecture Notes in Math., pages 205–366. Springer, Cham, 2015.[16] Jan Kiwi. Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier (Grenoble),

56(5):1337–1404, 2006.

[17] Jan Kiwi. Puiseux series dynamics of quadratic rational maps. Israel J. Math., 201(2):631–700, 2014.[18] Jan Kiwi. Rescaling limits of complex rational maps. Duke Math. J., 164(7):1437–1470, 2015.

[19] R. Lodge, Y. Mikulich, and D. Schleicher. A classification of postcritically finite Newton maps. ArXiv e-prints, October2015.

[20] R. Lodge, Y. Mikulich, and D. Schleicher. Combinatorial properties of Newton maps. ArXiv e-prints, October 2015.

[21] Curt McMullen. Families of rational maps and iterative root-finding algorithms. Ann. of Math. (2), 125(3):467–493, 1987.[22] Y Mikulich. Newton’s method as a dynamical system. PhD thesis, Jacobs University Bremen, 2011.

[23] John Milnor. Geometry and dynamics of quadratic rational maps. Experiment. Math., 2(1):37–83, 1993. With an appendix

by the author and Lei Tan.[24] H. Nie. Compactifications of the moduli spaces of Newton maps. arXiv:1803.08391, Mar 2018.

[25] H. Nie and K. M. Pilgrim. Boundedness of Hyperbolic Components of Newton Maps. arXiv:1809.02722, September 2018.

[26] Juan Rivera-Letelier. Dynamique des fonctions rationnelles sur des corps locaux. Asterisque, (287):xv, 147–230, 2003.Geometric methods in dynamics. II.

[27] Juan Rivera-Letelier. Espace hyperbolique p-adique et dynamique des fonctions rationnelles. Compositio Math., 138(2):199–

231, 2003.[28] Juan Rivera-Letelier. Points periodiques des fonctions rationnelles dans l’espace hyperbolique p-adique. Comment. Math.

Helv., 80(3):593–629, 2005.

[29] P. Roesch. On local connectivity for the Julia set of rational maps: Newton’s famous example. Ann. of Math. (2),168(1):127–174, 2008.

[30] Robert Rumely. A new equivariant in nonarchimedean dynamics. Algebra Number Theory, 11(4):841–884, 2017.[31] Joseph H. Silverman. The space of rational maps on P1. Duke Math. J., 94(1):41–77, 1998.

[32] Joseph H. Silverman. The arithmetic of dynamical systems, volume 241 of Graduate Texts in Mathematics. Springer, NewYork, 2007.

[33] Eugenio Trucco. Wandering Fatou components and algebraic Julia sets. Bull. Soc. Math. France, 142(3):411–464, 2014.[34] F. von Haeseler and H.-O. Peitgen. Newton’s method and complex dynamical systems. Acta Appl. Math., 13(1-2):3–58,

1988.

Einstein Institute of Mathematics, The Hebrew University of JerusalemE-mail address: [email protected]