On the linearization ofnon-Archimedean holomorphic functions

near an indifferent fixed point

Acta Wexionensia No 127/2007

Mathematics

On the linearization ofnon-Archimedean holomorphic functions

near an indifferent fixed point

Karl-Olof Lindahl

Växjö University Press

On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point. Thesis for the degree of Doctor of Philosophy, Växjö University, Sweden 2007.

Series editor: Kerstin Brodén ISSN: 1404-4307 ISBN: 978-91-7636-574-8 Printed by: Intellecta Docusys, Göteborg 2007

Abstract

Lindahl, Karl-Olof, 2007. On the linearization of non-Archimedean holo-morphic functions near an indifferent fixed point, Acta Wexionensia No127/2007. ISSN: 1404-4307, ISBN: 978-91-7636-574-8. Written in English.

It is known since the work of Herman and Yoccoz in 1981, that Siegel’slinearization theorem is true also for non-Archimedean fields. As a conse-quence, in characteristic zero, linearization is always possible near a non-resonant, indifferent fixed point. However, they also showed that the con-dition in Siegel’s theorem is ‘usually’ not satisfied over fields of prime char-acteristic. Indeed, we prove that there exist power series f , such that theassociated conjugacy function diverges. On the other hand, we also provethat if the degrees of the monomials of f are co-prime to p, then f is an-alytically linearizable. We find a lower, sometimes the best, bound of thesize of the corresponding linearization disc. In the cases where we find theexact size of the linearization disc, we show, using the Weierstrass degreeof the conjugacy, that f has an indifferent periodic point on the boundary.

In the p-adic case we give lower bounds for the size of linearization discs inCp. For quadratic maps, and certain power series containing a ‘sufficientlylarge’ quadratic term, we find the exact size of the linearization disc.

Moreover, we show that analytic linearization has applications to ergod-icity. In particular, we prove that a power series f over a non-Archimedeanfield K, is uniquely ergodic on all spheres inside a linearization disc abouta fixed point in K if and only if K is isomorphic to Qp for some prime p,and the multiplier is a generator of the group of units modulo p2.

Mathematics Subject Classification (2000): primary 37F50, 32P05,32H50; secondary 37B05, 37A50, 11S99.

Key words: dynamical system, linearization, non-Archimedean field.

v

Acknowledgements

I would like to thank my supervisor Prof. Andrei Yu. Khrennikov for intro-duction to the theory of p-adic dynamical systems, the formulation of theproblem on conjugation of p-adic dynamical systems, and for fruitful dis-cussions. I thank Robert L. Benedetto and Juan Rivera-Letelier for fruitfuldiscussions and consultancy on non-Archimedean analysis and its dynamics.Special thanks goes to Robert L. Benedetto for his hospitality during myvisit in Amherst spring 2003.

I am very grateful to my colleagues at the School of Mathematics andSystems Engineering at Vaxjo University. In particular, I thank the othermembers in the dynamical systems group at Vaxjo University, i.e. MarcusNilsson, Robert Nyqvist, and Per-Anders Svensson for many interestingdiscussions. Special thanks goes to Per-Anders Svensson for comments andcorrections of the manuscript.

Finally, I would like to thank my family and friends, for valuable supportthroughout the years.

vi

Preface

This thesis is organized in the following way. First, an introduction to thesubject is presented, together with a summary of the papers that this thesisis based upon. Secondly, the papers are given in fulltext, and they are:

I. On Siegel’s linearization theorem for fields of prime characteristicK.-O. LindahlNonlinearity, Vol. 17, No. 3, 745–763, 2004.

II. Divergence and convergence of solutions to the Schroder functionalequation in fields of prime characteristicK.-O. LindahlRevised version of preprint 04015, MSI, Vaxjo University, 2004.

III. On ergodic behavior of p-adic dynamical systemsM. Gundlach, A. Khrennikov, and K.-O. LindahlInf. Dim. Anal. Quantum Prob. Rel. Top., Vol. 4, No. 4, 569–577, 2001.

IV. Estimates of linearization discs in p-adic dynamics with application toergodicityK.-O. LindahlRevised version of preprint 04098, MSI, Vaxjo University, 2004.

vii

Contents

1 Introduction 12 Background on non-Archimedean fields 3

p-adic numbers Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Locally compact fields . . . . . . . . . . . . . . . . . . . . . . . . . 6Finite extensions of locally compact fields . . . . . . . . . . . . . . 7p-adic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Complex p-adic numbers Cp . . . . . . . . . . . . . . . . . . . . . . 8Non-Archimedean discs . . . . . . . . . . . . . . . . . . . . . . . . 9Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Background on dynamical systems 13Linearization discs and periodic points . . . . . . . . . . . . . . . . 14Fatou components of rational functions . . . . . . . . . . . . . . . 16

4 Summary of papers 19Paper I: On Siegel’s linearization theorem for fields of prime char-

acteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Paper II: Divergence and convergence of solutions to the Schroder

functional equation in fields of prime characteristic . . . . . . 20Paper III: On ergodic behavior of p-adic dynamical systems . . . . 22Paper IV: Estimates of linearization discs in p-adic dynamics with

application to ergodicity . . . . . . . . . . . . . . . . . . . . . 23Bibliography 26

ix

1 Introduction

The Archimedean theory of complex dynamics, i.e. iteration of holomor-phic functions of a complex variable, has been intensively studied over thelast hundred years, see e.g. [7, 24, 31, 46] and references therein. Over thelast decades, there has been an increasing interest in non-Archimedean dy-namics and its relationship, similarities, and differences with respect to theArchimedean theory of complex dynamics, see for example the early workof Herman and Yoccoz [32] and the work in [2,3,11–15,17–19,33,34,36,38,39,41–43,45,50,57,58,62,65,66]. Non-Archimedean fields and their dynam-ics have many applications; e.g. the counting of periodic points over globalfields using local non-Archimedean fields [45, 47, 48, 53, 54, 68, 70], codingtheory and computer oriented theories [21–23, 49, 63, 69], biochemistry andphysics [4, 5, 35, 55, 56], and cognitive sciences [37]. Non-Archimedean dy-namical systems can also be considered as a part of an extended domain ofresearch in number theory and dynamical systems, see e.g. [6, 48].

Recall that a non-Archimedean field is a field K equipped with a non-trivial absolute value | · |, satisfying the following ultrametric triangle in-equality:

|x + y| ≤ max{|x|, |y|}, for all x, y ∈ K.

Standard examples of non-Archimedean fields include the p-adics, and var-ious function fields. The p-adics include the p-adic numbers and their ex-tensions. These fields are all of characteristic zero. The most importantfunction fields include fields of formal Laurent series over various fields.These can be of zero or prime characteristic.

As in complex dynamics, the study of non-Archimedean dynamics be-gins with the description of the local behavior near fixed points. We areconcerned with the existence of a linearization at indifferent fixed points.Recall that a fixed point is said to be indifferent if the absolute value of thederivative at the fixed point is equal to one. Let f ∈ K[[x−x0]] be a powerseries over K of the form

f(x) = x0 + λ(x − x0) + a2(x − x0)2 + a3(x − x0)3 + . . . , |λ| = 1.

We say that f is locally linearizable at the fixed point x = x0, if there is aconvergent power series g, such that g ◦ f ◦ g−1(x) = λx, that is, such that

g(f(x)) = λg(x), (1.1)

on some non-empty disc about x0. The function g is called the conjugacyand plays an important role in the dynamics of f . The idea of analyticconjugation was introduced by Ernst Schroder in 1971 to study iterationof complex rational functions. Equation (1.1) and its variants are thereforeoften referred to as the Schroder functional equation.

Note that in the special case that f is a power function xλ, then the con-jugacy g is the logarithm. Loosely speaking, a solution g, of the Schroderfunctional equation, can then be viewed as a ‘logarithm’ of the function f .

1

In fact, the functional equation (1.1) plays a central role in local arithmeticgeometry, see e.g. Lubin [44] for a discussion on the relationship betweenthe construction of conjugacies in p-adic dynamics and well-established tech-niques of local arithmetic geometry. Similarly, in fields of prime character-istic, the construction of conjugacies is related to the construction of socalled Drinfeld Modules which play a central role in the arithmetic of func-tion fields, see e.g. Goss [29] and references therein.

Provided that λ is not a root of unity, there always exist a formal solutionthe Schroder functional equation, as shown by Herman and Yoccoz [32]. Bythe ansatz of a power series we can then obtain a recursive formula for thecoefficients of g. Explicit formulas for the conjugacy are known in a fewcases, e.g. in characteristic zero for power functions. In addition, we obtainan explicit formula for the conjugacy of λx + apx

p in characteristic p.From a dynamical point of view it is interesting to study the convergence

properties of the conjugacy and its inverse. Indeed, by definition

g ◦ fn ◦ g−1(x) = λnx

on the disc where g converges and is one-to-one. The maximal such discwill be referred to as the linearization disc. Hence, there is a one-to-onecorespondence between iterations of f and repeated multiplication by λinside the linearization disc. In other words, the local dynamics under fcan be determined solely in terms of λ.

In this thesis we use majorant series to estimate the radius of convergenceof the conjugacy. This technique was used by Siegel [61] in the complex fieldcase. We then use a generalization, by Benedetto [15], of the Weierstrasspreparation theorem to estimate the maximal disc on which the conjugacyis one-to-one. As manifest in this thesis, although explicit formulas for theconjugacy are rare to find, it may still be possible to find the exact radiusof the linearization disc, or at least a ‘decent’ estimate.

Finally, as noted in the last paper of this thesis, ergodicity of f is preservedunder analytic conjugation. Applying our results on conjugation, we thenshow that f is (uniquely) ergodic on an entire sphere, inside a linearizationdisc, if and only if the field K is isomorphic to a field of p-adic numbers andthe multiplier is a generator of the group of units modulo p2.

2

2 Background on non-Archimedean fields

2 Background on non-Archimedean fields

Definition 2.1. Let K be a field. By an absolute value (or valuation)| · | on K we mean a function

K � x �→ |x| ∈ R+ ∪ {0}such that

(i) |x| = 0 if and only if x = 0,

(ii) |xy| = |x| · |y| for every x, y ∈ K, and

(iii) there is a real number C ≥ 1 such that |x + y| ≤ C · max{|x|, |y|} forevery x, y ∈ K.

An absolute value is said to be non-trivial if it is not identically 1 onK∗ = K \ {0}. If an absolute value is defined on K, then K is said to be avalued field.

Note that if K = R and | · | is the ‘usual’ absolute value, then we maychoose C = 2. If it is possible to choose C = 1, then K is said to benon-Archimedean:

Definition 2.2. A valued field K is said to be non-Archimedean if it isequipped with a non-trivial absolute value | · |, which in addition, satisfiesthe ultrametric triangle inequality:

|x + y| ≤ max{|x|, |y|}, for all x, y ∈ K. (2.1)

From now on, K will be assumed to be non-Archimedean. One usefulconsequence of ultrametricity is that for any x, y ∈ K with |x| = |y|, theinequality (2.1) becomes an equality. In other words, if x, y ∈ K and wehave |x| < |y|, then |x + y| = |y|. As a consequence, all integers n ∈ Zhave absolute value less than or equal to 1 in K. Hence, the ArchimedeanProperty (given x, y ∈ K, x = 0, then there exists a positive integer n suchthat |nx| > |y|) does not hold in K.

By ultrametricity, it is easy to see that the closed unit disc

O = {x ∈ K : |x| ≤ 1}is a subring of K. The elements of O are called the integers of K. More-over, the open unit disc

M = {x ∈ K : |x| < 1}is an ideal of O. An element x ∈ K is called a unit if |x| = 1. Hence, Mconsists exactly of the non-units of O, and thereby M has to be maximal.In fact, M is the only maximal ideal of O. Since M is maximal, the factorring

k = O/Mis a field — the residue class field of K.

3

Note that if K has positive characteristic p, then also char k = p; butif char K = p, then k could have characteristic 0 or p. Note also that ifx, y ∈ O reduce to residue classes x, y ∈ k, then |x − y| is 1 if x = y, andit is strictly less than 1 otherwise.

For a non-Archimedean field K with absolute value | · | we define thevalue group as the image

|K∗| = {|x| : x ∈ K∗}. (2.2)

Note that, since |·| is multiplicative, |K∗| is a multiplicative subgroup of thepositive real numbers. We will also consider the full image |K| = |K∗|∪{0}.The absolute value | · | is said to be discrete if the value group is cyclic,that is if there is a uniformizer1 π ∈ K such that |K∗| = {|π|n : n ∈ Z}.

Standard examples of non-Archimedean fields include the p-adics, andvarious function fields defined below. The p-adics include the p-adic num-bers and their extensions. These fields are all of characteristic zero. Themost important function fields include fields of formal Laurent series overvarious fields. These can be of any characteristic.

There is a vast literature on the theory of non-Archimedean fields ofwhich we mention only a few in the subsequent sections. For instance,a nice introduction to p-adic numbers can be found in Gouvea [30]. InCassels [25], the topic is treated from an algebraic point of view, while forexample Schikhof [60] have a more analytic approach to the concept.

p-adic numbers Qp

The primary examples of K with char K = 0 and char k = p > 0, sometimesrefered to as the mixed characteristic case, are the p-adic numbers. Thep-adic numbers were invented by Kurt Hensel in the last decade of thenineteenth century. His purpose was to provide a treatment of algebraicnumbers. In order to do this, Hensel used an arithmetical analogue of KarlWeierstrass’ method of developing analytical functions of a complex variableinto power series. Combined with general field theory, this later on led tothe theory of valuations, introduced by Jozef Kurschak in 1912. Then in1918, Alexander Ostrowski determined the set of all valuations on the fieldQ of rational numbers.

Just like the real numbers, the p-adic numbers Qp are obtained as acompletion of the rational numbers Q with respect to an absolute value.More precisely, let p be a prime number and let n be an integer. The p-adicabsolute value of n is defined by

|n|p = p−ν(n),

where ν(n) denotes the order of p in n. If n = 0, then we set ν(0) = ∞. Inthe same way, the p-adic absolute value of a rational number a/b is givenby

|a/b|p = p−ν(a)+ν(b).

1 Some authors use the terminology prime element or local coordinate.

4

2 Background on non-Archimedean fields

The p-adic numbers Qp are obtained as the completion of the rational num-bers with respect to | · |p.Theorem 2.1 (Ostrowski). Any non-trivial absolute value on Q is equiv-alent to one of the absolute values | · |p for some prime number p, or to theordinary absolute value on Q.

In other words, there is basically only two kinds of analysis on Q: real andp-adic. These have quite distinctive properties and the essential differencebetween R and Qp, from a geometrical point of view, is that the p-adicnumbers are non-Archimedean.

We shall often use the shorter notation | · | instead of | · |p. Given a primep, a p-adic number x can be expressed in base p as

x =∞∑

i=ν

xipi, xi ∈ {0, 1, . . . , p − 1},

for some integer ν. The absolute value of x is given by |x| = p−ν , providedthat xν = 0. If x is an integer, its p-adic expansion contains no negativepowers of p and hence |x| ≤ 1. Note that the positive integers always havea finite p-adic expansion whereas for example

−1 =∞∑

i=0

(p − 1)pi.

The set Zp = {x ∈ Qp : |x| ≤ 1} is called the p-adic integers and consistsof elements whose p-adic expansion contains only nonnegative powers of p.The set of integers Z is dense in Zp.

Note that the p-adic absolute value is discrete and that the value group|Q∗

p| consists of all integer powers of p.

Function fields

The case char K = char k is sometimes refered to as the equal charac-teristic case. The main type of such K are function fields as describedbelow.Example 2.1 (char K = char k = p). Let Fp be a field of characteristicp > 0, e.g. Fp could be the finite field Fpn of pn elements, an infinite field likethe algebraic closure of Fpn , or the infinite field of rational functions overFpn . Let K = Fp((T )) be the field of all formal Laurent series in variableT , with coefficients in the field Fp. An element x ∈ K is of the form

x =∑i≥i0

xiTi, xi0 = 0, xi ∈ F, (2.3)

for some integer i0 ∈ Z. Given 0 < ε < 1, we define an absolute value | · |on K such that |T | = ε and ∣∣∣∣∣∣

∑i≥i0

xiTi

∣∣∣∣∣∣ = εi0 . (2.4)

5

Hence, π = T is a uniformizer of K. Furthermore, K is complete withrespect to | · | and, analogously to the p-adic numbers, can be viewed as thecompletion of the field of rational functions Fp(T ) over Fp with respect tothe absolute value defined by (2.4). Note that i0 is the order of the zero(or if negative, the order of the pole) of x at T = 0. Note also that in thiscase, K and its residue field k = Fp have the same characteristic, namelychar Fp = p. Moreover, | · | is the trivial absolute value on Fp, the subfieldof K consisting of all constant series in K. As for the p-adic numbers, wecan construct a completion K of an algebraic closure of K with respectto an extension of | · |. Then K is a complete, algebraically closed non-Archimedean field, and its residue field k is an algebraic closure of k = Fp.It follows that k has to be infinite. The value group |K∗|, that is the set ofreal numbers which are actually absolute values of non-zero elements of K,will consist of all rational powers of ε, rather than just integer powers of εas in |K∗|. In particular, the absolute value is discrete on K but not on K.

Example 2.2 (char K = char k = 0). Let F be a field of characteristiczero, e.g. F could be either Q, Qp, R or C. Then define K = F ((T )) and| · | as in the previous example. One major difference is that in this case theresidue field k = F must be infinite since F is of characteristic zero.

For information on the arithmetic of function fields, the reader may con-sult Goss [29].

Locally compact fields

Recall that a topological space is said to be locally compact if each ofits points has a neighborhood with compact closure. We note the followingtwo results. The proofs can be found in Cassels [25].

Proposition 2.1. Let K be a non-Archimedean field with absolute value| · |. Then K is locally compact (w.r.t. | · |) if and only if all three of thefollowing conditions are satisfied: (i) K is complete, (ii) | · | is discrete, and(iii) the residue field k is finite.

In a locally compact non-Archimedean field, every element has a uniquerepresentation as a Laurent series in π with coefficients in the residue fieldk.

Proposition 2.2. Suppose K is locally compact and let π be a uniformizer.Let R be a complete system of representatives of the residue field k. Thenevery x ∈ O is uniquely of the form

x =∞∑

i=0

xiπi, xi ∈ R. (2.5)

Conversely, the right hand side always converges to give an x ∈ O.

Proof. Let x ∈ O be given. There is precisely one x0 ∈ k, namely x0 = x,such that |x − x0| < 1 and then x = x0 + πy1 for some y1 ∈ O. There is

6

2 Background on non-Archimedean fields

precisely one x1 ∈ k such that |y1 − x1| < 1, namely x1 = y1, and theny1 = x1 + πy2 for som y2 ∈ O. Continuing in this way we get for everypositive integer N

x = x0 + x1π + . . . + xNπN + yN+1πN+1

with xi ∈ k and yN+1 ∈ O. Now |yN+1πN+1| → 0, and so (2.5) holds.

As noted above, O is certainly a ring. If K is locally compact withuniformizer π, then πO is a unique maximal ideal in K so that the residueclass field

k = O/πO.

Moreover, the set

(O/πnO)∗ = {a : a = a0 + a1π + . . . an−1πn−1, ai ∈ k, a0 = 0} (2.6)

is a group under multiplication modulo πn, for each integer n ≥ 1. We willrefer to this group as the group of units modulo πn.

Finite extensions of locally compact fields

It is easy to see that no non-Archimedean locally compact field is alge-braically closed; if K is locally compact and π is a uniformizer in K, thenx2 − π ∈ K[x] is irreducible since |√x| = |π|1/2 /∈ |K|. Thus there arenon-trivial finite extensions of any non-Archimedean locally compact field.

Let K be a locally compact non-Archimedean field. If L is a finite exten-sion of K, it is possible to prolong the absolute value | · | on K to an absolutevalue | · |L on L, in such a way that L will also be locally compact. In fact,there is only one possible way to do this, see for example Gouvea [30] for thep-adic case. In what follows, we will use the notation | · | for the valuationon K, as well as for its prolongation to any finite extension of K.

We define the corresponding residue field l of L in the same manner aswhat was done for K. Then, k may be considered as a subfield of l. Thedegree f = [l : k], of the field extension l/k is finite whenever [L : K] is,and is called the residue class degree of the extension L/K. We writef = f(L/K).

Turning to the valuation groups |K∗| and |L∗|, we see that |K∗| is asubgroup of |L∗|. Let πK and πL be uniformizers in K and L, respectively.Then, there is an integer e such that |πL|e = |πK |. The ramification indexe is defined as the least integer with this property. We write e = e(L/K).The notation of e and f for the ramification index and residue class degree,respectively, are standard in this context.

Proposition 2.3. Let L be a finite extension of a locally compact non-Archimedean field K. Then

n = e · f,

where n = [L : K], e = e(L/K), and f = f(L/K).

7

Sketch of proof. Choose α1, α2, . . . , αf ∈ OL such that the correspondingresidue classes α1, α2, . . . , αf ∈ l becomes a basis of the vector space lover k. Let πL be a uniformizer in L. Then, neither of the e elements|πL|0, |πL|1, . . . , |πL|e−1 ∈ |L∗| belong to the same coset of the subgroup|K∗|. The ef elements in the set

{αiπjL : 1 ≤ i ≤ f, 0 ≤ j ≤ e − 1}

can now be shown to be a basis of the vector space L over K. See Cassels [25]for the details.

Of special interest are those extensions of which the ramification indexe equals one. A finite extension L/K of degree n is called unramified, ife(L/K) = 1 (or equivalently, f(L/K) = n), and ramified, if e(L/K) > 1(or equivalently, f(L/K) < n).

p-adic fields

Definition 2.3. Let K be a complete non-Archimedean field with absolutevalue | · |. We say that K is a p-adic field if K has characteristic zero, | · |is discrete, and the residue class field k is finite.

Cassels [25] give two alternative characterizations:

Proposition 2.4. A non-Archimedean field K of characteristic zero is ap-adic field if and only if it is locally compact.

Proposition 2.5. A non-Archimedean field K is a p-adic field if and onlyif it is isomorphic (as a valued field) to a finite extension of Qp for some p.

As noted above, to any finite field extension K of Qp there is an associatedramification index e = e(K/Qp) such that

|K∗| = {pn/e : n ∈ Z}. (2.7)

For example, by adjoining√

p to Qp we get a ramified extension with ram-ification index e = 2.

Complex p-adic numbers Cp

Let Qap be the algebraic closure of Qp and let the complex p-adic numbers

Cp be the completion of Qap with respect to the p-adic metric. Let us note

that whereas the complex numbers are obtained by adjoining the single root√−1 to the real numbers, Qap is an infinite algebraic extension of Qp. For

Cp, the value group is|C∗

p| = {pr : r ∈ Q}.In other words, the p-adic absolute value is extended from all integer powersof the prime p to all rational powers of p. In particular, the absolute valueon Cp is not discrete.

8

2 Background on non-Archimedean fields

Non-Archimedean discs

Given an element x ∈ K and a real number r > 0 we denote by Dr(x)the open disc of radius r about x, by Dr(x) the closed disc, and by Sr(x)the sphere of radius r about x. If r ∈ |K∗| (that is if r is actually theabsolute value of some nonzero element of K), we say that Dr(x), Dr(x),and Sr(x) are rational. Note that Sr(x) is non-empty if and only if it isrational. If r /∈ |K∗|, then we will call Dr(x) = Dr(x) an irrational disc.In particular, if a ∈ K and r = |a|s for some rational number s ∈ Q, thenDr(x) and Dr(x) are rational considered as discs in K. However, they maybe irrational considered as discs in K. Note that all discs are both openand closed as topological sets, because of ultrametricity. However, as wewill see in Section 2 below, power series distinguish between rational open,rational closed, and irrational discs.

Again by ultrametricity, any point of a disc can be considered as its center.In other words, if b ∈ Dr(a), then Dr(a) = Dr(b); the analogous statementis also true for closed discs. In particular, if two discs have nonemptyintersection, then they are concentric, and therefore one must contain theother.

Holomorphic functions

Much of the study of mapping properties of complex holomorphic functionsfocuses on functions defined on a simply connected domain; by the Riemannmapping theorem, any such domain is bi-holomorphic either to a disc or toC. Over a non-Archimedean field K, however, the notion of “simply con-nected” cannot be defined naively, because K itself is a totally disconnectedtopological space. Fortunately, it does make sense to discuss power serieswhich converges on a disc; in fact such power series are fundamental objectsof rigid analysis, see e.g. [20, 27, 28]. As stated below, the image of a discunder a power series is either a disc or all af K. In light of the Riemannmapping theorem, it is then reasonable to define a simply connected domainin a non-Archimedean field K to be either a disc or all of K.

Let K be a complete non-Archimedean field with absolute value | · |. Letf be a power series over K of the form

f(x) =∞∑

i=0

ai(x − α)i, ai ∈ K.

Then f converges on the open disc DRf(α) of radius

Rf =1

lim sup |ai|1/i, (2.8)

and diverges outside the closed disc DRf(α) in K. The power series f

converges on the sphere SRf(α) if and only if

limi→∞

|ai|Rif = 0.

9

Notice that, in contrast to the complex field case, what happens at the pointson the ‘boundary’ of the region of convergence is rather simple; either theseries is convergent at all such points or at none of them. Over the complexfield C, the set of points on the boundary for which the series converges canbe rather complicated.

Definition 2.4. Let U ⊂ K be a disc, let α ∈ U and let f : U → K. Wesay that f is holomorphic on U if we can write f as a power series

f(x) =∞∑

i=0

ai(x − α)i ∈ K[[x − α]]

which converges for all x ∈ U .

Holomorphicity is well-defined since, contrary to the complex field case, itdoes not matter which α ∈ U we choose in the definition of holomorphicity.

Proposition 2.6. Let U be a disc in K and let f : U → K be holomor-phic. Then, for each β ∈ U , there exist b0, b1, . . . ∈ K such that the seriesexpansion f(x) =

∑bi(x − β)i holds for all x ∈ U .

Proof. By assumption, f(x) =∑

ai(x − α)i for all x ∈ U . Now

(x − α)i = (x − β + β − α)i =i∑

j=0

(i

j

)(x − β)j(β − α)i−j .

Set tij = ai

(ij

)(x− β)j(β −α)i−j if j ≤ i and tij = 0 otherwise. Recall that

by ultrametricity, |(ij

)| ≤ 1, and |x − β| ≤ max{|x − α|, |α − β|}. Then, forall i, j we have |tij | ≤ |ai|max{|x − α|, |β − α|}i. Since both

∑ai(x − α)i

and∑

ai(β − α)i converges, it follows that limi→∞ tij = 0 uniformly in j.By definition, limj→∞ tij = 0 for all i. We conclude that

f(x) =∞∑

i=0

∞∑j=0

tij =∞∑

j=0

∞∑i=0

tij =∞∑

j=0

bj(x − β)j

where bj =∑∞

i=0 ai

(ij

)(β − α)i−j . The theorem follows.

A peculiar consequence of this theorem is that the recipe of analyticcontinuation often used in complex function theory, to extend the domainof a holomorphic function, does not work in the non-Archimedean case. Infact, suppose we have a holomorphic function

f(x) =∞∑

i=0

aixi

on a disc U , where U is the region of convergence of the power series∑

aixi.

If we choose any β ∈ U and develop f in a neighborhood V ⊂ U of β intoa power series in x − β, then by Proposition 2.6, the region of convergence

10

2 Background on non-Archimedean fields

of our new power series is again U . So by this procedure “we never get outof U”.

The basic mapping properties of non-Archimedean power series on discsare given by the following theorem.

Proposition 2.7 (Weierstrass Preparation Theorem). Let K be alge-braically closed. Let f be a non-zero holomorphic function on D1(0). Thenthere are finitely many points α1, . . . , αd ∈ D1(0) (not necessarily distinct)and a function u such that both u and 1/u are holomorphic on D1(0), and

f(x) = u(x) ·d∏

j=1

(x − αj).

Furthermore, if we write f(x) =∑∞

i=0 aixi and s = max{|ai|}, then

d = max{i ≥ 0 : |ai| = s}.Proof. See [20] or [27]. An alternative proof, using Newton polygons, canbe found in [40].

It follows that d measures the degree of f , counting multiplicity, and wewill refer to d as the Weierstrass degree defined below. The WeierstrassPreparation Theorem says that non-Archimedean power series on rationalclosed discs are essentially polynomials. After all, the condition that u and1/u are both holomorphic is equivalent to saying that u has no zeros inthe disc. Although the result is stated for rational closed discs, it has thefollowing generalization to arbitrary discs.

Proposition 2.8 (Benedetto [15] Lemma 2.2). Let K be algebraicallyclosed. Let f(x) =

∑∞i=0 ai(x−α)i be a nonzero power series over K which

converges on a rational closed disc U = DR(α), and let 0 < r ≤ R. LetV = Dr(α) and V ′ = Dr(α). Then

s = max{|ai|ri : i ≥ 0},d = max{i ≥ 0 : |ai|ri = s}, and

d′ = min{i ≥ 0 : |ai|ri = s}are all attained and finite. Furthermore,

a. s ≥ |f ′(x0)| · r.b. if 0 ∈ f(V ), then f maps V onto Ds(0) exactly d-to-1 (counting multi-

plicity).

c. if 0 ∈ f(V ′), then f maps V ′ onto Ds(0) exactly d′-to-1 (counting mul-tiplicity).

Definition 2.5. Let U ⊂ K be a rational closed disc, and let f be a holo-morphic function on U . For any disc V ⊆ U , the Weierstrass degree (orsimply the degree) degV f of f on V is the number d (if V is closed) or d′

(if V is open) in Proposition 2.8.

11

Note that if 0 ∈ f(V ), then the Weierstrass degree is the same as thenotion of degree as ‘the number of pre-images of a given point, countingmultiplicity’. If 0 /∈ f(V ), then the Weierstrass degree of f is 0, eventhough points in the image of f may have many pre-images.

A power series f over a complete non-Archimedean field K is always one-to-one and hence isometric on some non-empty disc about an indifferentfixed point x0 ∈ K. This is a consequence of the local invertibility theorem,see [60]. As noted in paper IV, the maximal such disc is given by thefollowing proposition.

Proposition 2.9. Let K be algebraically closed. Let f ∈ K[[x]] be conver-gent on some non-empty disc about x0 ∈ K. Suppose that f(x0) = x0 and|f ′(x0)| = 1, and write

f(x) = x0 + λ(x − x0) +∑i≥2

ai(x − x0)i, a = supi≥2

|ai|1/(i−1).

Let M be the largest disc, with x0 ∈ M , such that f : M → M is bijectiveand isometric. Then M = D1/a(x0) if either maxi≥2 |ai|1/(i−1) is attained(as for polynomials) or f diverges on S1/a(x0). Otherwise, M = D1/a(x0).

If f ∈ K(x) is a non-polynomial rational function, then f may be one-to-one on a possibly larger, open connected affinoid. For example, the rationalfunction

R(x) =1 + xp+1

xp(1 + (px)p+1)∈ Cp(x),

has indifferent fixed points on the sphere {|z| =√

p} and is one-to-one onthe annulus C = {1 < |z| < p}, see Rivera-Letelier [57].

Affinoids are regions consisting of a disc from which a finite number ofdiscs are removed. More precisely, let K be algebraically closed and definethe projective line P(K) = K ∪ {∞}. A P(K)-disc is either a disc in Kor the complement in P(K) of a disc in K. A connected affinoid U is anynon-empty subset of P(K) which may be written as

U = P(K) \ (D1 ∪ · · · ∪ Dn),

where the Di’s are pairwise disjoint P(K)-discs. If all the Di’s are open(resp., closed, rational, irrational), we say that U is open (resp., closed,rational, irrational). Affinoids arise naturally in the theory of rigid analysis,see e.g. [28].

Although connected affinoinds are topologically disconnected, they be-have like connected sets under the application of rational functions, ormore generally, under the application of (appropriately defined) meromor-phic functions, i.e. the quotient of two holomorphic functions.

For more information on non-Archimedean power series the reader canconsult [26, 59, 60]. From a dynamical point of view, the paper [15] con-tains many useful results on non-Archimedean analogues of complex ana-lytic mapping theorems relevant for dynamics.

12

3 Background on dynamical systems

3 Background on dynamical systems

Let K be a field, and let R ∈ K(x) be a rational function defined overK. Denote by Rn the n-fold composition of R with itself. Thus, R0 isthe indentity map, R1 = R, R2 = R ◦ R, and so on. We will consider thedynamics of R meaning the action of {Rn}n≥0 on the projective line P(K).

The first objects of study in such a dynamical system are the periodicpoints of R. Given a point x ∈ P(K), we say that x is periodic of periodn ≥ 1 if Rn(x) = x; the smallest positive period n of x is called the minimalperiod of x. If x has minimal period 1, we say that x is a fixed pointof R. More generally, if a point x ∈ P(K) has the property that Rm(x)is periodic for some m ≥ 0, we say that x is preperiodic. Note that allperiodic points are preperiodic.

For any x ∈ P(K), the set {Rn}n≥1 is called the (forward) orbit of x.If x is periodic, the forward orbit of x is called a (periodic) cycle. Clearly,a point is preperiodic if and only if its forward orbit is finite.

If x ∈ K is a periodic point of R of minimal period n ≥ 1, we define themultiplier of x to be λ = (Rn)′(x). Note that the multiplier is the samefor all points in a given periodic cycle.

The character of a periodic point x is determined by the multiplier λ. If|λ| > 1, we say that x is repelling. If |λ| < 1, we say that x is atttracting.Finally, if |λ| = 1 we say that x is indifferent. This last case can bedivided into two subcases: if λ is a root of unity, we say that x is resonant;otherwise we say that x is non-resonant.

To analyze the various cases, we assume that x0 is a fixed point of R withmultiplier λ = R′(x0). Because R is rational, we know that its Taylor series

f(x) = x0 + λ(x − x0) +∞∑

i=2

ai(x − x0)i,

at x0 converges on a disk U about x0. The map f is said to be (analyt-ically) linearizable at x0 if there is a convergent power series solutiong, with g(x0) = 0 and g′(x0) = 1, to the following form of the Schroderfunctional equation

g ◦ f(x) = λg(x). (3.1)

There always exist a unique formal power series solution g of (3.1), see[32] for the non-Archimedean case. The coefficients of g can be obtainedrecursively by the formula

bk =1

λ(1 − λk−1)

k−1∑l=1

bl(∑ l!

α1! · ... · αk!aα11 · ... · aαk

k ) (3.2)

where α1, α2, . . . , αk are nonnegative integer solutions of⎧⎨⎩α1 + ... + αk = l,α1 + 2α2... + kαk = k,1 ≤ l ≤ k − 1.

(3.3)

13

We will refer to the formal solution g as the conjugacy (function). Inthe attracting and repelling cases, the formal conjugacy always convergeson some non-empty neighborhood of the fixed point, both in the complexand the non-Archimedean case. If λ is a root of unity, and if deg R ≥ 2,then R cannot be locally conjugate to multiplication by λ. The remainingcase, that the corresponding fixed point is indifferent and non-resonant, hasa different outcome depending on the field K. In fact, it is not completelysolved yet.

It was proven in [43] that in characteristic p > 0, like in complex dy-namics, the conjugacy function may diverge, due to the presence of smalldivisors, i.e. small divisor terms 1 − λn in the denominator of the coeffi-cients of the conjugacy g. The problem is that |1− λn| may be small if themultiplier λ is ‘too close’ to a nth root of unity. In this way the problem isrelated to the topological and arithmetic structure of the roots of unity inK as described in [43]. In 1942 Siegel proved in his celebrated paper [61]that the condition

|1 − λn| ≥ Cn−β for some real numbers C, β > 0, (3.4)

is sufficient for convergence in the complex field case.In 1981 Herman and Yoccoz [32] proved that Siegel’s theorem is true also

for non-Archimedean fields:Theorem 3.1 (Non-Archimedean Siegel Theorem [32]). Let K be acomplete non-Archimedean field, and let f be a power series over K of theform f(x) = λx + O(x2), convergent on some non-empty disc about 0. If λsatisfies Siegel’s condition (3.4), then f is analytically linearizable at x = 0.If 0 is either attracting or repelling, then Siegel’s condition (3.4) is alwayssatisfied. Moreover, if char K = 0, every λ not a root of unity satisfies theSiegel condition (3.4).

Their general theorem applies to any finite dimension (with the modifi-cation that in dimension d > 1 there exist examples in characteristic zerowhere the corresponding Siegel condition is violated and the associated for-mal conjugacy diverges).

In what follows, we will focus on non-resonant indifferent fixed points.

Linearization discs and periodic points

We will use the following definition of a linearization disc.Definition 3.1. Let K be a complete algebraically closed non-Archimedeanfield, and let f be a power series over K. Suppose that f is analyticallylinearizable at an indifferent fixed point x0 ∈ K. Then, the corresponding(indifferent) linearization disc of f about x0, denoted by Δf (x0), isdefined as the largest disc U ⊂ K, with x0 ∈ U , such that the semi-conjugacy

g(f(x)) = λg(x),

holds for all x ∈ U , and g converges and is one-to-one on U . We will referto the maximal disk of semi-conjugacy as the (indifferent) semi-disc.

14

3 Background on dynamical systems

Remark 3.1. This notion of a linearization disc is well-defined since therealways exist a largest disc on which g is one-to-one (provided that g isconvergent). Also note that f must certainly be one-to-one on a linearizationdisc Δf (x0); if f fails to be one-to-one, so does g.

We will sometimes discuss linearization discs over a field L which is notalgebraically closed. In this case we will refer to

Δf (x0, L) := Δf (x0) ∩ L

as the linearization disc in L.As noted in paper IV, both f and the conjugacy g are one-to-one and

isometric on a non-Archimedean linearization disc.

Lemma 3.1. Let K be algebraically closed. Suppose that f ∈ K[[x]] hasa linearization disc Δf (x0) about x0 ∈ K. Let g, such that g(x0) = 0 andg′(x0) = 1, be the corresponding conjugacy function. Then, the two map-pings g : Δf (x0) → g(Δf (x0)) and f : Δf (x0) → Δf (x0) are bijective andisometric. In particular, if x0 = 0, then g(Δf (x0)) = Δf (x0). Furthermore,Δf (x0) ⊆ M ⊆ D1/a(x0), where M and a are defined as in Proposition 2.9.

As a consequence, the radius of a linearization disc Δf (x0) is equal to tothat of g(Δf (x0)). In particular, the radius of a linearization disc is inde-pendent of the location of the fixed point x0. Therefore, we may, withoutloss of generality, assume that x0 = 0.

Given K, we will estimate the radius of the linearization disc of powerseries in the two-parameter family

Fλ,a(K) :={

λx +∑

aixi ∈ K[[x]] : |λ| = 1, a = sup

i≥2|ai|1/(i−1)

}. (3.5)

As noted above, for all f ∈ Fλ,a(K), the corresponding linearization discsatisfies

Δf (0) ⊆ D1/a(0).

The linearization disc and the semi-disc do not necessarily coincide; thelinearization disc may be strictly contained in the semi-disc as manifest inour results in characteristic p, as well as for p-adic power functions, as shownby Arrowsmith and Vivaldi [3]. The reason for this is that the semi-discmay contain other periodic points, breaking the full conjugacy; if x belongsto a semi-disc about the origin and x = 0 is periodic of period n, then bythe semi-conjugacy relation g ◦ fn(x) = λng(x) and the fact that λ is nota root of unity, the conjugacy g cannot be one-to-one at the point x, andconsequently x cannot belong to the linearization disc. The semi-disc mayin fact contain several linearization discs. In particular, Arrowsmith andVivaldi showed that power functions over K = Cp (for which the conjugacyis the logarithm), have periodic points on the boundary of the linearizationdisc. We consider this example in more detail in section 6 on power functionsin paper IV of this thesis.

15

In fact, in view of the the results in paper II, it seems that the presence ofperiodic points on the boundary of an indifferent linearization disc is typicalin the general setting when f is a power series defined over an algebraicallyclosed non-Archimedean field K; if the linearization disc Δ is rational openand the radius of the semi-disc of f is strictly greater than that of Δ, sothat there is a change of the Weierstrass degree of the conjugacy on theboundary of Δ, then f has an indifferent periodic point on the boundary.

The local and global dynamics on the semi-disc and the larger domainof quasi-periodicity (i.e. the interior of the set of points in P(K) that arerecurrent by R), containing D1/a(x0) was classified for rational functionsover K = Cp by Rivera-Letelier in his thesis [57]. We will mention some ofhis results in the next section.

Fatou components of rational functions

Let K be an algebraically closed field and let R ∈ K(x) be a rationalfunction over K. As in the complex field case we define the Fatou set ofR to be

F = {x ∈ P(K) : {Rn} is equicontinous on some neighboorhood of x}.Here, equicontinuity is defined using the spherical metric on the projectiveline P(K). Recall that a family F of functions from a metric space X to ametric space Y is called equicontinous at x0 ∈ X, if for every ε > 0, thereis a δ > 0 such that dY (f(x), f(x0)) < ε for all f ∈ F and for all x ∈ Xsuch that dX(x, x0) < δ. The key point is that ε is chosen independently off . Intuitively, the Fatou set is the region of order, where the iterates {Rn}are well behaved; the Julia set J = P(K) \F is the region of chaos, wheresmall errors become huge after many iterations of R.

Remark 3.2. In the complex dynamics litterature, the Fatou set is usuallydefined to be the set of points on the Riemann sphere P(C) having a neigh-boorhood on which the family {Rn} is normal (with respect to the sphericalmetric). By the Arzela-Ascoli Theorem, the above definition is equivalent.However, for an algebraically closed non-Archimedean field K, the Arzela-Ascoli Theorem does not apply, because K is not locally compact.

The Fatou set, being an open subset of P(C), may be subdivided intoits connected components, and it is not difficult to show that if U is aconnected Fatou component of FR, then R(U) is also. Thus, R induces adiscrete dynamical system on the set of connected components of F . Inparticular, components may be fixed, periodic, or preperiodic; or if a com-ponent is not preperiodic, then it is called a wandering domain. By afamous theorem of Sullivan [64], F has no wandering domains in complexdynamics. Sullivan’s work completed a program of fully classifying all pos-sible dynamics on complex Fatou components. The classification theoremthen states that: if R ∈ C(z) is a rational function with Fatou set F , andU is a periodic component of F , then U is either an attracting component

16

3 Background on dynamical systems

(containing an attracting fixed point), a parabolic component (containinga resonant indifferent periodic point), a linearization disc (containing anindifferent non-resonant periodic point), or a Herman ring (containing nopreperiodic point).

Because non-Archimedean fields are totally disconnected, it would bepointless to discuss the connected components of the Fatou set. However,there are several other definitions of ‘components’ of the Fatou set which arebetter analogues of complex Fatou components [9,10,14,57]. The dynamicsof non-Archimedean periodic components seems to have much more varietythan that of Archimedean components. In particular, the constructions byBenedetto [16] of wandering domains over any algebraically closed completenon-Archimedean field suggest that Sullivan’s proof has no analogue for K.

However, we will only discuss what Rivera-Letelier [57] refers to as a fixedanalytical component of R: the union of all connected affinoids U (openor closed) containing x0, on which R maps U into U and is one-to-one.These components are non-Archimedean analogues of Siegel discs and Her-man rings. Rivera-Letelier gave a classification of analytical components.He showed that every such component C is one of two general types:

Quasi-periodic: Every point of C is recurrent (i.e. contained in the clo-sure of its own forward orbit), and C contains infinitely many indifferentperiodic points, which form a discrete subset of C. Every closed affinoid Dcontained in C contains at most finitely many periodic points. Moreover,C is an open connected affinoid (which is a big deal since the infinite unionof intersecting connected affinoids may be a very complicated non-affinoidspace. Nevertheless, such a space is called a connected analytic space,because it still behaves very much like a connected set). Conversely, givenan open connected affinoid A, then there is a rational function R, such thatA is a connected component of the domain of quasi-periodicity of R.

Attracting: C contains a unique attracting fixed point x0, and all theother points of C are attracted to x0 under iteration. C is either a rationalopen disk or else a ‘space of Cantor type’ (a certain complicated type ofconnected analytic space).

Example 3.1 (Rivera-Letelier [57]). Let R be the rational function

R(x) =1 + xp+1

xp(1 + (px)p+1)∈ Cp(x).

Let x0 ∈ {|z| =√

p} be a fixed point of R. Then x0 is indifferent and thecorresponding fixed analytical component is C(x0) = {1 < |z| < p}.Example 3.2. If f ∈ Fλ,a(Cp), where Fλ,a(K) is defined by (3.5), is apolynomial, then the corresponding analytical component C(0) = D1/a(0).

17

In the complex field case, a rational function R is locally linearizable nearan indifferent fixed point z0 if and only if the connected component U , of theFatou set, containing z0 is conformally isomorphic to the unit disc under anisomorphism which conjugates R on U to multiplication by λ on the disc,see e.g. Milnor [46, p. 116]. A complex linearization disc is then usuallyrefered to as “a Fatou component on which f is conformally conjugate tomultiplication by λ”. This definition would not make sense in the non-Archimedean case; the component containing an indifferent fixed point x0,may contain infinitely many periodic points of R, breaking the conjugacy.In fact, the sometimes even smaller set M , the maximal topological disc onwhich R is one-to-one, may contain infinitely many periodic points, differentfrom x0. In other words, in contrast to the complex field case, the non-Archimedean linearization disc is in general strictly smaller than the ‘Siegeldisc’ (i.e. the fixed analytical component).

18

4 Summary of papers

4 Summary of papers

Paper I: On Siegel’s linearization theorem for fields ofprime characteristic

In this paper we consider the problem of linearization of a power seriesof the form f(x) = λx + O(x2), |λ| = 1, defined over a field K of primecharacteristics p. We describe the problem and its relation to the problem ofsmall divisors, arithmetic of the multiplier λ, geometry of the unit sphere,and the roots of unity in K. A brief review of related results in non-Archimedean and complex dynamics is also given.

We prove that the conjugacy may diverge due to the presence of smalldivisor terms in the coefficients of the conjugacy. When small divisor termsare present this brings about a problem of a combinatorial nature, where theconvergence of the conjugacy is determined in terms of the characteristicsof the state space K and the powers of the monomials of f , rather than interms of the diophantine properties of the multiplier λ, as in the complexfield case.

We prove that quadratic polynomials are always linearizable if the char-acteristics char K = 2. We obtain an explicit formula for the coefficientsof the conjugacy and find the exact size of the corresponding linearizationdisc, and show the existence of a periodic point on the boundary. As notedin paper II, this result has the following generalization for arbitrary p:Theorem 4.1. Let char K = p, and let f ∈ K[x] be polynomial of the form

f(x) = λx + apxp, ap = 0. (4.1)

Then, the conjugacy function is given by

g(x) = x +∞∑

j=1

bpj xpj

,

where

bpj =a(pj−1)/(p−1)p

λj(1 − λpj−1)(1 − λpj−1−1) . . . (1 − λp−1). (4.2)

Moreover, g converges on the open disc of radius ρp = 1/a, where the num-ber a = |ap|1/(p−1), and diverges on the sphere Sρp

(0). Let

σp =pm′−1

√|1 − λm|a

,

where m′ = 1 if m = 1, and otherwise

m′ = min{n ∈ Z : n ≥ 1, pn ≡ 1 mod m}.Then, the linearization disc of f about the origin is the disc Dσp

(0). More-over in the algebraic closure K we have deg(g, Dσp

(0)) = pm′. Furthermore,

f has an indifferent periodic point of period κ ≤ pm′on the sphere Sσp(0)

in K, with multiplier λκ.

19

In the case char K = p > 2 we give a sufficient condition for divergenceof the conjugacy for quadratic maps as well as for certain class of powerseries containing a quadratic term.

These results unveils a connection between convergence of the conjugacyand the mutual relation between the powers of the monomials of f and thecharacteristic p. ‘Good’ powers for convergence being those co-prime to p,‘bad’ powers being those prime to p. However, the blend of prime and co-prime powers may sometimes yield convergence, at least for power series,as shown by the following example.Example 4.1. Let char K = 3 and let g(x) = x + x2, then

g−1(x) = x − x2 + 2x3 + x4 + . . . .

Moreover, g is the conjugacy for the power series f defined by the equationf(x) = g−1(λg(x)). Hence f is a power series of the form

f(x) = λx + (λ − λ2)x2 + 2(λ3 − λ2)x3 + (λ4 − λ2)x4 + . . . ,

which is certainly linearizable.This brings about a problem of a combinatorial nature of seemingly great

complexity and a complete description seems difficult. However, there mightbe a sharp distinction for polynomials.Open problem 4.1. Let K be of characteristic p > 0. Is there a polynomialof the form f(x) = λx + O(x2) ∈ K[x], with λ not a root of unity satisfying|1−λn| < 1 for some n ≥ 1, and containing a monomial of degree prime top, such that the formal conjugacy g converges?

Paper II: Divergence and convergence of solutions to theSchroder functional equation in fields of prime charac-teristic

In this paper we develop further the ideas in paper I and prove some moregeneral results. First we give a class of polynomials containing a monomialof degree prime to p, such that the conjugacy diverges:Theorem 4.2. Let char K = p > 0 and let λ ∈ K, |λ| = 1. Then,polynomials of the form

f(x) = λx + ap+1xp+1 ∈ K[x], ap+1 = 0,

are not analytically linearizable at the fixed point at the origin if |1−λ| < 1.Then we prove that if the degrees of the monomials of a power series f

are co-prime to p, then f is analytically linearizable:Theorem 4.3. Let char K = p > 0 and let λ ∈ K, |λ| = 1, but not a rootof unity. Then, convergent power series of the form

f(x) = λx +∑p|i

aixi ∈ K[[x]],

are linearizable at the origin.

20

4 Summary of papers

As noted in paper I, for the same multiplier λ, the the blend of mono-mials of degree prime and co-prime to char K = p, may sometimes yielddivergence sometimes not, and a complete description seems difficult.

The rest of the paper is devoted to estimates of semi-discs and lineariza-tion discs for power series that satisfy the convergence condition of Theorem4.3.

Let K be a field of prime characteristic p. Let λ ∈ K, not a root of unity,be such that the integer

m = m(λ) = min{n ∈ Z : n ≥ 1, |1 − λn| < 1}, (4.3)

exists. Given λ and hence m, the integer k′ is defined by

k′ = k′(λ) = min{k ∈ Z : k ≥ 1, p|k, m|k − 1}. (4.4)

Let a > 0 be a real number. We will associate with the pair (λ, a), thefamily of power series

Fpλ,a(K) =

⎧⎨⎩λx +∑p|i

aixi ∈ K[[x]] : a = sup

i≥2|ai|1/(i−1)

⎫⎬⎭ , (4.5)

and the real numbers

ρ = ρ(λ, a) =|1 − λm| 1

mp

a, (4.6)

and

σ = σ(λ, a) =|1 − λm| 1

k′−1

a, (4.7)

respectively. Our most general estimate on the semi-disc and linearizationdisc for f ∈ Fp

λ,a(K) can then be stated in the following way.

Theorem 4.4. Let f ∈ Fpλ,a(K). Then, the semi-conjugacy holds on the

open disc Dρ(0). The full conjugacy g ◦ f ◦ g−1(x) = λx holds on Dσ(0).

Under further assumptions on f , the linearization disc may contain thelarger disc Dρ(0).

Theorem 4.5. Let f ∈ Fpλ,a(K) be of the form

f(x) = λx +∑i≥i0

aixi,

for some integer i0 > k′. Then, the full conjugacy g ◦ f ◦ g−1(x) = λx holdson a disc larger than or equal to Dρ(0) or the closed disc Dρ(0), dependingon whether g converges on Dρ(0) or not.

Note that the estimate of the linearization disc in Theorem 4.3 is max-imal in the sense that in K, the completion of the algebraic closure of K,quadratic polynomials have a fixed point on the sphere Sσ(0) if m(λ) = 1,breaking the conjugacy there. In fact, the estimate is maximal in a broadersense, according to the following theorem.

21

Theorem 4.6. Let f ∈ Fpλ,a(K). Suppose a = |ak′ |1/(k′−1) and |ai| < ai−1

for all i < k′. Then, Dσ(0) is the linearization disc of f about the origin. InK we have deg(g, Dσ(0)) = k′. Furthermore, f has an indifferent periodicpoint in K on the sphere Sσ(0) of period κ ≤ k′, with multiplier λκ.

The existence of a a periodic point on the boundary follows from Lemma6.1 in this paper which states that if there is a shift of the value of theWeierstrass degree from 1 to d > 1, of the conjugacy function on a sphereS, then there is an indifferent periodic point of period κ ≤ d, on the sphereS. This is true in any non-Archimedean field K. In other words, we havethe following theorem.

Theorem 4.7. Let K be a complete algebraically closed non-Archimedeanfield. Let f ∈ K[[x]] have a linearization disc Δ about an indifferent fixedpoint. Suppose that Δ is rational open, and that the radius of the corre-sponding semi-disc of f is strictly greater than that of Δ, then f has anindifferent periodic point on the boundary of Δ.

Paper III: On ergodic behavior of p-adic dynamical sys-tems

This paper is a joint work with Matthias Gundlach and Andrei Khrennikov.We study the ergodic properties of p-adic monomial maps of the form

f(x) = xs,

defined on spheres about the fixed point x = 1. It is shown that f is minimal(every forward orbit is dense) on the sphere Sp−l(1), l ≥ 1, if and only if sis a generator of the group of units (Z/p2Z)∗.

Allthough not explicitly stated, the proof makes extensive use of the factthat the p-adic logarithm conjugates xs to the multiplier map s �→ sx nearx = 1. The minimality of xs is then shown to be equivalent to its uniqueergodicity. Recall that a continuous map T : X → X is uniquely ergodic ifthere exists only one probability measure μ, defined on the Borel σ-algebraB(X) of X, such that T is measure-preserving with respect to μ. In our casethe unique invariant measure is the normalized Haar measure (the measureof a disc is equal to its radius).

The paper ends with the formulation of the following problem.

Problem 4.1 (Lindahl, Khrennikov, and Gundlach). Is a polynomialof the form f(x) = xs + q(x), with q ‘small’, ergodic if and only if x �→ xs

is?

The answer is affirmative if we add the assumption that q contains nolinear term. A solution was obtained by Anashin [1] for polynomials andcertain locally analytic functions. In paper IV we obtain a generalization ofthis result, using analytic conjugation. See Theorem 4.12 below.

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4 Summary of papers

Paper IV: Estimates of linearization discs in p-adic dy-namics with application to ergodicity

We study linearization discs in p-adic dynamics, i.e. over the p-adic numbersQp, finite extensions of Qp, as well as over the algebraic closure Cp. Asstated above, it is known since the work of Herman and Yoccoz, that p-adic power series are always linearizable near an indifferent non-resonantperiodic point.

A mainly unresolved issue which we consider in this paper concerns theradius of the corresponding linearization disc. Estimates were previously ob-tained for quadratic polynomials over Qp by Ben-Menahem [8], and Thiran,Verstegen, and Weyers [67]. Pettigrew, Roberts, and Vivaldi [52], obtainedestimates for power series over the p-adic integers, Zp, with maximal mul-tipliers. Some related estimates of small divisors in Cp were also obtainedby Khrennikov [36].

We generalize these results and give lower bounds for the size of lineariza-tion discs for power series over Cp. The expressions for these estimates takea more complicated form, compared to that of the characteristic p casestudied in paper II. This stems from the fact that the geometry of the rootsof unity is more complex in Cp.

Given λ, not a root of unity, and a real number a, let Fλ,a(Cp) be thefamily

Fλ,a(Cp) :={

λ(x − x0) +∑

ai(x − x0)i ∈ Cp[[x]] : a = supi≥2

|ai|1/(i−1)

},

and let

R(t) :=

{0, if t = 0,

p− 1

pt−1(p−1) , if t ≥ 1.(4.8)

Then, there is an integer m

m = m(λ) = min{n ∈ Z : n ≥ 1, |1 − λn| < 1}, (4.9)

and a unique nonnegative integer s = s(λ) such that

R(s) ≤ |1 − λm| < R(s + 1),

and a root of unity α = α(λ) in Cp such that

|α − λm| ≤ |γ − λm|for every root of unity γ ∈ Cp.

Theorem 4.8 (General estimate). Let f ∈ Fλ,a(Cp). Then, the lin-earization disc Δf (x0) ⊇ Dσ(x0) where

σ = σ(λ, a) := a−1R(s + 1)1m |1− λm| 1

m (1+ p−1p s)

( |α − λm||1 − λm|

)1/mps

. (4.10)

Moreover, if the conjugacy function converges on the closed disc Dσ(x0),then Δf (x0) ⊇ Dσ(x0).

23

Note that the estimate σ = σ(λ, a) depends only on λ and the real numbera. To find the exact size of the linearization disc we do in general needmore information about the coefficients of f . However, for a large class ofquadratic polynomials, and certain power series containing a ‘sufficientlylarge’ quadratic term, we prove that

τ = |1 − λ|−1/pσ(λ, a) (4.11)

is the exact radius of the linearization disc. More precisely, our main resultcan be stated in the following way.Theorem 4.9 (Quadratic case). Let p be an odd prime and let λ ∈ Cp,not a root of unity, belong to the annulus {z : p−1 < |1 − z| < 1}. Let f bea quadratic polynomial of the form f(x) = λx + a2x

2 ∈ Cp[x]. Then, thecoefficients of the conjugacy g satisfy

|bk| =|a2|k−1|1 − λ|� k−1

p �∏k−1n=1 |1 − λn|

. (4.12)

Moreover, the linearization disc about the origin, Δf (0) = Dτ (0), whereτ = |1 − λ|−1/pσ(λ, a).

This result is then extended to power series containing a ‘sufficiently large’quadratic term. We also give sufficient conditions, there being a fixed pointon the ‘boundary’ of the linearization disc, i.e. the sphere Sτ (x0) about x0

of radius τ .Note that τ(λ, a) < 1/a. Hence, at least in this case, the linearization

disc cannot contain the maximal disc D1/a(x0) on which f is one-to-one.Theorem 4.10 (Asymptotic behavior of σ). Let |α − λm| be fixed.Then, the estimate σ of the radius of the linearization disc goes to 1/a asm or s goes to infinity. If s and m are fixed, then σ → 0 as |α − λm| → 0.

For finite extensions of Qp, we give a sufficient condition on the multiplierunder which the corresponding linearization disc is maximal, i.e. its radiuscoincides with that of the maximal open disc D1/a(0) in Cp on which f isone-to-one.Theorem 4.11 (Maximal linearization discs in extensions of Qp).Let K be a finite extension of Qp of degree n, with ramification index e,residue field k of degree [k : Fp] = n/e, and uniformizer π. Let f ∈Fλ,a(Cp) ∩ K[[x]] be a power series. Suppose that λ has a maximal cyclemodulo π2 and

logp e ≤ (pn/e − 3)p/(p − 1) − ν

(α − λpn/e−1

1 − λpn/e−1

)+ logp(p − 1),

where ν(x) is the order p in x. Then, the corresponding linearizationdisc Δf (x0, K) = Δf (x0) ∩ K is maximal in the sense that Δf (x0, K)is either the open or closed disc of radius 1/a. In particular, if eithermaxi≥2 |ai|1/(i−1) is attained (as for polynomials) or f diverges on S1/a(x0),then Δf (x0, K) = D1/a(x0, K).

24

4 Summary of papers

Moreover, we also show that, for any complete non-Archimedean field,transitivity is preserved under analytic conjugation into a linearization disc.

Using a result by Oxtoby [51], we note that minimality is equivalentto the unique ergodicity of f on compact subsets of a non-Archimedeanlinearization disc. In particular, a power series f over Qp is minimal, henceuniquely ergodic, on all spheres inside a linearization disc about a fixed pointif and only if the multiplier is a generator of the group of units (Z/p2Z)∗. Wealso note that in finite extensions of Qp, a power series cannot be ergodic onan entire sphere, that is contained in a linearization disc, and centered at thecorresponding fixed point. The same is true for fields of prime characteristic.Hence, ergodicity on spheres inside a non-Archimedean linearization disc isonly possible for fields of p-adic numbers.

Theorem 4.12 (Ergodic non-Archimedean linearization spheres).Let K be a complete non-Archimedean field and let f be holomorphic on adisc U in K. Suppose that f has a linearization disc Δ ⊂ U and S ⊂ Δis a sphere about the corresponding fixed point x0 ∈ K. Then f : S → Sis ergodic if and only if K is isomorphic to Qp and the multiplier is agenerator of the group of units (Z/p2Z)∗. Furthermore, if K = Qp and λis a generator of the group of units (Z/p2Z)∗, then the radius of Δ is 1/a(considered as a disc in Qp).

25

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