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A SHARPER ESTIMATION OF ANALYTIC CONJUGACIES IN THENON-ARCHIMEDEAN SETTING

ADRIAN JENKINS, HAROLD BLUM, AND HANK DITTON

Abstract. We look at the conjugacy of mappings of the form f(x) = x +∑

anxn. The

goal of this paper is to further develop an approach of Jenkins and Spallone [7] whichis both elementary and effective in constructing convergent power series conjugating twosuch functions (recall that two such maps f and g are conjugate if there is an h so thath ◦ f ◦ h−1 = g). Using this approach, we are also able to construct groups of germs. Wetake the time to explain all concepts, and knowledge of power series is the only prerequisiterequired for reading. This work arose out of research performed during the Kansas StateUniversity Research Experience for Undergraduates.

1. Introduction

We fix a field K, complete with respect to any nontrivial norm. For example, K mightbe the field of complex (or even real) numbers, or the field of p-adic numbers. We considerpower series f , convergent in some open set S ⊆ K (such as a ball) with coefficients in K.

In studying such a function, it is often useful to understand the action of f on a pointx0 ∈ S, as well as on “nearby” points (this generally consists of some ball of positive,though often unspecified, radius centered at x0). We will refer to this study as the “localdynamics” of f at the point x0. In this paper, we will consider the local dynamics of aspecific class of power series. Since our interest is local, it is mostly irrelevant what theactual set S is. Because S is open, we know that for any point x0 ∈ S, f is defined onsome open ball BR(x0) of positive radius R centered at x0. When studying the dynamicsof f near x0, we will freely shrink the radius of this ball as necessary.

One of the most interesting points at which to study local dynamics is a fixed point. Wewill therefore restrict our study to points x0 ∈ S so that f(x0) = x0. In either the euclideanor non-archimedean setting, such fixed points are isolated, and so by shrinking R, we mayassume that x0 is the only fixed point within the ball BR(x0). Moreover, by applying the

simple change of variable T (x) = x− x0, we can consider the function f(x) = T ◦ f ◦ T−1.Note that our new function f satisfies f(0) = 0. Obviously, the dynamics of the function fnear 0 are identical to the dynamics of the function f near x0 (because the nth composition

f ◦n = T ◦f ◦n ◦T−1), and therefore, for the purpose of local dynamics, we may assume thatour function f has the form

(1.1) f(x) =∞∑n=1

anxn.

2000 Mathematics Subject Classification. Primary 30D05, 32P05.Key words and phrases. formal conjugacy, analytic conjugacy, formal normal form, non-archimedean

analysis, p-adic dynamics.1

2 ADRIAN JENKINS, HAROLD BLUM, AND HANK DITTON

One of the important techniques in determining the local dynamics of an arbitrary powerseries f is to find some change of variable which reduces f to a simpler form, with whichit is easier to work. Such a reduced form might have fewer terms than the original powerseries, or be more easily iterated, etc. We have already demonstrated this in assuming thatthat the fixed point is 0; let us now make this more general. Given two power series f andg fixing 0, we say that they are formally conjugate (or formally equivalent) if there is somepower series h fixing 0 which satisfies h ◦ f ◦h−1 = g. We say that f and g are analyticallyconjugate if there is a power series h fixing 0 and converging in some ball Br(0) of positiveradius which satisfies h◦f ◦h−1 = g. Obviously, when studying analytic conjugacy, we willassume that the power series f and g have positive radii of convergence. If f and g areanalytically conjugate, then they are clearly formally conjugate, so formal equivalence isweaker than its analytic counterpart. However, it is often the case that formal power seriesare easily constructed (often by completely arithmetical means), and thus provide a wealthof possible test series for the analytic case. Finally, we note again that if an analytic powerseries h exists which satisfies h(0) = 0 and h◦ f ◦h−1 = g, then knowledge of the dynamicsof one function provides knowledge of the other. This is clear, since for any n ∈ N we have

(1.2) h ◦ f ◦n ◦ h−1 = g◦n.

This of course begs the question: what is a suitable “simpler” form g to which we canreduce our given f? At this point, the study of local dynamics breaks into multiple paths.In this work, we will assume that a1 = 1. The reader interested in some of the othercases can browse through the survey paper of Abate [1] (for complex cases), or considerthe fundamental work of Herman and Yoccoz [5] (for non-archimedean cases).

Thus, here and what follows, we will assume that f is defined and analytic in a neigh-borhood of 0, and that f has the form

(1.3) f(x) = x+∞∑n=m

anxn

where am 6= 0.Provided that the field K has characteristic 0 and that the number am possesses an

(m−1)-root (for example, K = C), it is well known that any such map f may be conjugatedto the polynomial form

(1.4) fm,µ(x) = x+ xm + µx2m−1

by means of a formal power series h fixing 0. In fact, the numbers m and µ provide acomplete set of formal invariants for the map F , and thus fm,µ is the “formal normal form”of f (for the argument, see, e.g., [7], among others). Unfortunately, in our situation (i.e. afield K complete with respect to a non-archimedean norm), it is not the case that K willalways possess such a root. One possible fix to to consider instead a finite extension fieldof K, and perform the normalization there (this is the tactic of [7]). Another possibility isto allow for other coefficients to appear within the formal normal form (this is the tacticof the current paper). We will return to these questions in Section 3. For the remainderof this introduction, this subtlety is unnecessary - we will simply assume that K possesseswhatever is necessary (e.g. we may assume that m = 2).

ESTIMATION OF CONJUGACIES 3

In conjugating f to this formal normal form (or indeed, to any other power series g), theintertwining series h is clearly not unique. If g is any element of the formal centralizer of f ,then the map h◦g will also formally conjugate f to g. This centralizer is always nontrivial;e.g. if g = f ◦n for any n, then g centralizes f . Thus, in determining whether f and g areanalytically conjugate, we must consider any power series which conjugates f and g. Thiscan create problems. Indeed, when the field K = C, the question of analytic equivalencewas studied for many years (going as far back as Fatou [4]), and after many years, a fullanalytic classification was determined (see any of Ecalle [3], Voronin [10], Ilyashenko [6]).However, this classification depends on functional invariants, most of which are impossibleto compute for precise examples (Ahern and Rosay [2]). More to the point, the classificationcannot answer the following question: given that f and g are formally conjugate, are theyanalytically conjugate? And if so, which of the formal power series h conjugating f to gconverge?

Thankfully, the question is simpler when K is a complete field with respect to a nontrivialnon-archimedean norm. In these cases, it turns out that if f is formally conjugate to g,then f is analytically conjugate to g, and any formal map h conjugating f to g will in factconverge in a neighborhood of 0. For the field K = Cp (see Section 2 for the definition),this result is stated in a 2003 paper of Rivera-Letelier [8]. Later, Jenkins and Spallone[7] demonstrated the result within any non-archimedean field via the estimation of formalpower series.

It is this estimation that brings us to the current problem. In [7], it was shown that, fora particular formal conjugating series h between f and fm,µ, the growth of the coefficientsof h was estimated by a relatively simple function (we will recall this function in section 3).This allowed the authors not only to prove that formal conjugacy was the same as analyticconjugacy, but also to give a lower bound on the radius of convergence of the particularpower series h which was estimated.

This lower bound is by no means the best possible; as we will show, when the formalinvariant µ = 0, much better bounds are present. Moreover, it can be shown that theestimating functions of Jenkins and Spallone uncover hidden structure among the analyticgerms fixing 0 which are tangent to the identity. In particular, these functions (and otherslike them) define groups of analytic germs. Finally, while Jenkins and Spallone freely passedto extension fields within their work, there is no real need for this; one can construct both aformal and analytic theory within the given field K. These issues guide the work presentedhere.

For this paper, we will consider an analytic function f(x) = x +∑

n≥m anxn fixing

the origin whose coefficients lie in a non-archimedean field K. By formal manipulations(which are explained in detail in Section 3), we may assume that our map takes the formf(x) = x+Axm +

∑∞n=k anx

n, where A ∈ K with |A| ≥ 1, and k ≥ 2m− 1. This number kis a key components in improving the estimates of [7]. We first prove the following theorem:

Theorem 1.1. Let K be any field of characteristic 0, complete with respect to a non-archimedean norm | · |. Let

(1.5) f(x) = x+ Axm +∞∑n=k

anxn


be an analytic function fixing the origin with coefficients an ∈ K for n ≥ k ≥ 2m− 1 andn = m, with |A| ≥ 1.

Then, there are a B ∈ K, an element q ∈ K with 0 < |q| ≤ 1, and a convergent powerseries H(x) = x+O(xm+1) conjugating f to a polynomial normal form

(1.6) fm,A,B(x) = x+ Axm +Bx2m−1.

where the coefficients Cn of H satisfy the estimate

(1.7) |(n−m)!qηm,k(n)Cn| ≤ 1

for the function

ηm,k(n) = n− 1 +m

[n− 1

k −m

].

Here, and in what follows [x] denotes the greatest integer less than or equal to [x]. Wenote again that there is no real loss of generality in assuming that |A| ≥ 1 (for if |A| < 1,we could simply conjugate f by the linear map LC(x) = Cx with C ∈ K having sufficientlysmall norm). This choice is merely for convenience of proof.

We remark that if k > 2m − 1, then B = 0. Moreover, the series H converges on the

disc Br(0) where r = |q|2k−mk−m . The element q in Theorem 1.1 is related to the radius of

convergence ρ of f , but in general, |q| < ρ (see Section 3 for more details). For any fixedm, when k > 2m − 1, the estimates governing the coefficients of H are better than thosefound by Jenkins and Spallone, improving as k → ∞ (see again Section 3 for a summaryof the estimates of [7]).

One of the most interesting aspects about the functions ηm,k is their “dynamical” nature.The following theorem demonstrates this.

Theorem 1.2. Fix m ≥ 2 and k ≥ 2m− 1, and write

(1.8) Gη,n! = Gηm,k,n! = {h(x) = x+∞∑

n=m+1

cnxn | (n−m)!qηm,k(n)cn ∈ ∆ ∀n ≥ k}.

Then, Gη,n! is a group.

As the subscript might hint, the groups Gη,n! are not the only ones we can construct;they are simply the most useful in our estimations. We give many other examples of groupsof power series governed by coefficient estimates in Section 4. This provides an elementaryalgorithm for constructing subgroups of analytic germs which are tangent to the identitypossessing good estimates. As far as we know, these are among the first ways to easilyconstruct such groups.

The organization of this paper is as follows: Section 2 is devoted to a brief recollectionof some basic facts regarding non-archimedean norms. Section 3 recalls some previousresults, in both the formal and analytic setting, for the local dynamics of non-archimedeanfunctions tangent to the identity. We use this section to fix notation, as well as give aversion of the formal theory suitable to our purposes. We also give a brief review of thework of Jenkins and Spallone on which this paper builds. Section 4 discusses the functionswhich we will use in estimation of the formal conjugating power series, as well as the groupstructure they impose on the set of analytic germs. We give a complete proof of Theorem1.2 here, and the set of properties of our functions are one of the essential ingredients in


understanding the estimates which follow. Section 5 is devoted to those estimates necessaryin the proof of Theorem 1.1. We leave the proof of Theorem 1.1 to Section 6, and give afew observations leading to new research directions. Finally, Section 7 is devoted to theproofs of some lemmas and claims which are very similar to those found in [7]; we providethese proofs for the reader’s convenience.

This research was performed during the Research Experience for Undergraduates atKansas State University, under NSF grant DMS-1004336. The authors would like to ex-press their thanks to the Kansas State Graduate School of Mathematics for its generoussupport; in particular, would like to thank the Principle Investigators of the REU, Mari-anne Korten and David Yetter. Finally, the authors would like to thank Steven Spallonefor his visitation and consultation, and in particularly, for directing us to Theorem 1.2, aswell as the possibility of eliminating any need for field extensions.

2. Preliminaries

Before moving on, we note that in Section 1 and in what follows, we have used thebold notation for those fields complete with respect to standard norms (R and C are thereal and complex numbers, resp.), while reserving blackboard bold notation for those fieldscomplete with respect to non-archimedean norms (e.g. Qp is the set of p-adic numbers).We will always write the natural numbers N, the integers Z and the rational numbers Q inblackboard-bold style.

This short section is devoted to a brief explanation of the non-archimedean setting inwhich we work. As our results require only elementary means, we list only the barestessentials here, so as not to burden the reader with many (unnecessary) facets of thetheory. The reader interested in a more full description of the theory should consult one ofthe standard texts, such as [9].

Definition 2.1. Let K be a field. A non-archimedean norm on K is a map | · | : K → Rsatisfying the following rules, for all x, y ∈ K:

i) |x| ≥ 0, |x| = 0 if and only if x = 0.ii) |xy| = |x||y|.

iii) |x+ y| ≤ max{|x|, |y|}.The pair (K, | · |) is called a non-archimedean valued field, or simply a non-archimedean

field.

As can be seen from part (iii) in the definition above, the significant difference in a non-archimedean norm is the so-called “strong triangle inequality”. When the norm is implicit,we will simply write K. Finally, the “constant” (or “trivial”) norm, where |x| = 1 if x 6= 0,will not be considered in this work.

We note here a couple of simple facts about any non-archimedean norms, for the conve-nience of the reader:

• Let n be any integer. From (ii), |1| = 1, and so from (iii), |n| ≤ 1.• Let F be a non-archimedean field, let a, b ∈ F , and suppose |b| < |a|. Then,|a+ b| ≤ max{|a|, |b|} ≤ |a| and |a| = |a+ b− b| ≤ max{|a+ b|, |b|} ≤ |a+ b| (usingthat |a| > |b|). Thus, we have |a+ b| = |a|.


One of the most important examples of non-archimedean fields are the so-called “p-adic”fields. To begin, let K = Q and choose a prime p ∈ Z. Consider the map

(2.1)∣∣∣mn

∣∣∣p,α

=

(1

p

)ordp(m)−ordp(n)

,

where ordp(n) is the exponent of p in the prime factorization of n. Then | · |p is a non-archimedean norm on Q. The completion of Q with respect to this norm is denoted Qp,and is called the set of p-adic numbers. In fact, given any real number 0 < α < 1, we candefine a non-archimedean norm on Q by

(2.2)∣∣∣mn

∣∣∣p

= αordp(m)−ordp(n).

Moreover, by a theorem of Ostrowski, these are the only non-trivial non-archimedean normson Q.

The field Qp is never algebraically complete (e.g., there is no√p). In fact, the alge-

braic completion Qp is an infinite-degree field extension, which unfortunately fails to be

topologically complete. We thus define the field Cp as the topological completion of Qp.Within a non-archimedean field, balls are defined in the usual way:

Definition 2.2. Given a positive number r ∈ R, and x ∈ K, define Br(x) = {y ∈ K :|x− y| ≤ r}.

The topology on K is then generated by the basis {Br(x) : r ∈ R, x ∈ K}. We will,however, have no need for any topological facts in this paper.

We will write the unit disc ∆ = B1(0). Note that if K is any non-archimedean field,then ∆ is a sub-ring (which is often called the ring of integers of K). Within any non-archimedean field of characteristic 0, we have Z ⊂ ∆. Finally, for all n ≥ 2, the ballsBp−n(0) are ideals of ∆.

Throughout the rest of this paper, we will take K to be a complete, non-archimedeanfield of characteristic 0. Because of this, K contains a copy of Q. Within any such field K,we have a simple lower bound on |n!|.Proposition 2.3. If the norm of K restricts trivially to Q then |n!| = 1. Otherwise,|n!| = |n!|p,α ≥ αn.

Proof. If the norm is trivial on Q, equality is obvious. For inequality, it is well-known thatordp(n!) = n−Sn

p−1 , where Sn is the sum of the digits of n in base p. Therefore ordp(n!) ≤ n,

and the result follows. �

Finally, we remark that since our norms are nontrivial, within any field K, there is anelement π ∈ K with 0 < |π| < 1. Moreover, for every ε > 0 there is a k ∈ N so that ifq = πk, then |q| < ε.

3. Previous Results

As stated earlier, our interest in this paper will be in convergent power series of the form

(3.1) f(x) = x+∞∑n=m

anxn,


where an ∈ K for a field K complete with respect to some given norm. As usual, a seriesf is convergent if its radius of convergence is strictly positive. Recall that the radius ofconvergence ρ of a power series f is defined by

(3.2) ρ =

(lim supn→∞

n√|an|)−1

However, it is convenient to consider first a formal power series f of the form (3.1). Thus,we will put no restrictions on the growth of the coefficients an appearing in the expansionof f . For example, if K = Qp, a non-convergent formal power series is given by

g(x) =∑n

1

pn2 xn

Returning to f in (3.1), note that since the linear term a1 = 1 6= 0, the formal series fis formally invertible. Indeed, it is a relatively simple exercise to see that each coefficientof the inverse f−1 is uniquely determined. It is worth noting that if the series f is actuallyanalytic, then its inverse will also have a positive radius of convergence. (However, wewill not need this fact in our analysis, as the inverses which we encounter will certainly beanalytic, by Theorem 1.2).

So, assume that f is a formal power series of the form (3.1), whose coefficients lie in somefield K of characteristic 0. Here and throughout the rest of the paper, we write f ◦ g tobe the composition of f and g, and write fg to mean the standard formal, multiplicativeproduct. Also, given n ∈ Z, we will write f ◦n to be the nth iterate of f , while fn will bethe nth formal multiplicative power of f . We add some additional notation here: given aseries f , we define [f ]n to be the coefficient of xn. In other words, if f is of the form (3.1),then [f ]n = an, and we may write any series in the form f(x) =

∑[f ]nx

n.Within the field C, the formal theory of maps f tangent to the identity is well known.

We present a slightly different form of this theory, suitable for our purposes. We provideboth a polynomial reduction for any map f of the form (3.1), as well as give an algorithmfor constructing a conjugating power series. We will refer to this constantly in our work.

Proposition 3.1. Let f be a formal power series of the form

(3.3) f(x) = x+∞∑j=m

ajxj,

whose coefficients aj lie in a field K of characteristic 0. Then, there exists 0 6= A ∈ K,B ∈ K and a formal power series H(x) = x+ · · · so that

(3.4) H ◦ f ◦H−1(x) = fm,A,B(x) = x+ Axm +Bx2m−1.

Proof. We can eliminate each term of f individually by a different polynomial map. Tothat end, let hn(x) = x + cnx

n, where cn is to be determined for n ≥ 2. We defineinductively H2(x) = h2(x), and Hn(x) = hn ◦ Hn−1(x) for n > 2. Finally, we will defineFn = Hn ◦ F ◦H−1n .

The proof is given by induction. Letting am = A, we write fm,A,B(x) = x+Axm+Bx2m−1,where B is to be determined. The cn above are defined so that Fn and g agree through


order m+ n− 1. For n = 2, we define

(3.5) c2 =am+1

(m− 2)A=

[f ]m+1

(m− 2)A.

Note that we have used the fact that K has characteristic 0 in assuming that m − 2 6= 0for any m > 2. Note also that if m = 2, then 2m− 1 = 3, and so we may take B = a3 andc2 = 0, which clearly satisfies the hypothesis (see below for more details on the choice ofcm for m > 2).

We now assume cl has been chosen for 2 ≤ l ≤ n− 1 so that Fl and g agree up to orderl +m− 1, and will define cn so that

(3.6) hn ◦ Fn−1 ◦ h−1n (x) = g(x) mod xn+m.

We can now define cn as:

(3.7) cn =[Fn−1]n+m−1(m− n)A

.

We remark here that for n = m, the formula (3.7) is not defined, and in fact, it is of nouse. An easy check shows that the coefficient cm will have no effect on the 2m−1 coefficientof Fm−1. Thus, we define B = [Fm−1]2m−1. Further, we may define cm to be any elementof K, so for the remainder of this paper, we take cm = 0.

This completes our inductive definition of the polynomials hl and Hl, l ≥ 2. Finally, theformal map H is defined to be H = limn→∞Hn, taken in the formal sense. Since the nthcoefficient of Hl is stable for all Hl with l > n, we see that each coefficient in the formalseries H depends algebraically on a finite number of terms, and thus is well-defined. �

One of the important corollaries of Proposition 3.1 is that any series f of the form (3.3)

is polynomially-conjugate to the reduced series f(x) = fm,A,B(x) +O(x2m). Obviously, thenumbers A and B are not formal invariants of f . However, we can determine which triples(m,A,B) allow for further formal normalization. The key is the following proposition.

Proposition 3.2. Let f(x) = x + Axm + Bx2m−1 + O(x2m) and g(x) = x + A′xm′

+B′x2m

′−1 + O(x2m′) be formally-equivalent power series. Then, m = m′ and there exists

c ∈ K, c 6= 0 so that cm−1A′ = A and c2m−2B′ = B.

In particular, B′ =

(A′

A

)2

B.

Proof. Let h(x) = cx +∑

n≥2 cnxn be any formal power series conjugating f to g, with

c 6= 0. Since h ◦ f = g ◦ h, we have

(3.8) h(x+ Axm +O(xm+1)) = h(x) + A′(h(x))m′+O(xm

′+1),

which is easily reduced to the form

(3.9) h(x) + cAxm +O(xm+1) = h(x) + A′cm′xm′+O(xm

′+1).

This forces m = m′, and then it follows that cm−1A′ = A.Now, Proposition 3.1 guarantees that f is conjugate to the polynomial fm,A,B(x) =

x + Axm + Bx2m−1. We therefore take f to have this form. Similarly, we write g(x) =


x+A′xm+B′x2m−1. Suppose there is a formal power series h(x) = cx+c`x`+· · · conjugating

f to g, with c` 6= 0. Formal manipulations yield the equation

h(x) + cAxm + cBx2m−1 + `c`x`+m−1 +O(x`+m)

= h(x) + A′(cx)m + A′mc`cm−1x`+m−1 +B′(cx)2m−1.

(3.10)

From this, we see that ` = m, and so c2m−2B′ = B. �

The above proposition tells us that further formal normalization among the polynomialsfm,A,B is therefore reduced to the determination of a linear term c, together with theaccompanying centralizers of these polynomials. In fact, we can even determine a set offormal normal forms from these polynomials. However, in the interest of the elementarynature of this paper, we will not pursue this any further. Rather, we are content withnoting the following: any mapping f of the form (3.3) can be taken to the form

(3.11) f(x) = x+ Axm +Bx2m−1 +O(x2m)

with A = am and B ∈ K by a polynomial change of variable of degree m − 1. Moreover,this polynomial change of variable is unique if chosen so that it is tangent to the identity.Moreover, Proposition 3.2 allows us to choose A ∈ K so that |A| ≥ 1. We will refer to thisreduction as the “prenormal” form of f , and henceforth, we will assume that f always hasthis form (this validates the form of f in the statement of Theorem 1.1). Note finally thatif in Proposition 3.1 the map f has the form (3.11), then the formal map h conjugatingf to fm,A,B is unique if chosen to satisfy h(x) = x + O(xm+1). This completes the formaltheory.

3.1. More Notation. We recall the last bit of notation needed for our presentation here.

Definition 3.3. Given a finite sequence i = (i1, . . . , i`), write |i| = i1 + · · ·+ i`. Also write`(i) = `, the “length” of i.

3.2. Estimating a Formal Power Series H. This paragraph will recall the main theo-rem of the work of Jenkins and Spallone [7]. We first give a short summary.

From Proposition 3.1, we see that any analytic map of the form f(x) = x+xm+µx2m−1+∑∞n=2m anx

n can be formally conjugated to the polynomial fm,1,µ(x) = x + xm + µx2m−1.Moreover, the series H conjugating f to fm,1,µ is unique provided that H(x) = x+O(xm+1).The obvious question is whether or not this series is actually analytic.

The idea in proving analyticity is, quite simply, to estimate each coefficient appearing inH. Of course, one has to use the analyticity of f in a non-trivial way. To that end, let ρbe the radius of convergence of the function f . Since ρ is positive, the sequence {1/ n

√an}

must be bounded below by some ε > 0. Thus, we may choose an element q ∈ K with0 < |q| < ε so that bn = anq

n ∈ ∆. This can be done in any field.However, for a non-archimedean norm, if we have two fractions bj/q

j and bk/qk, then

both their product AND sum have the form bl/ql, where |bl| ≤ 1. This simple fact allows

us to understand the size of a particular coefficient cn of the formal series H by estimatingthe decay of a “denominator” of cn in terms of the norm of |q|, thus obtaining a radiusof convergence for the formal conjugating series. This idea will be explained more fully inSection 5.

Using this idea, Jenkins and Spallone obtained the following theorem:


Theorem 3.4. Let

(3.12) f(x) = x+ xm +∞∑

n=2m−1

bnqnxn,

where bn ∈ ∆ and 0 < |q| < 1 for some q ∈ K. Write µ = b2m−1/q2m−1, and let H be the

unique formal power series of the form H(x) = x+O(xm+1) conjugating f to its polynomialnormal form fm,µ(x) = x + xm + µx2m−1. The coefficients [H]n of H satisfy the estimateq4n[H]n ∈ ∆. In particular, H is analytic in the disc centered at 0 of radius |q|4.

While the theorem is easily stated, its proof, unfortunately, is not. The function 4n inthe power of q is simple enough to write, but it is not the function which is used to estimatethe coefficients, due to difficulties in the proof.

Instead, Jenkins and Spallone invented another set of functions, indexed by the invariantm, to handle the estimation. While these functions are more complicated than 4n, theybehave very well under composition, which is a key ingredient in their proof of Theorem3.4 as well as our own proof Theorem 1.1 (this will be seen in Section 5). The functionswere given by

(3.13) σm(n) = (n− 1) +m

[n− 2

m− 1

],

where, as before, [x] denotes the greatest integer less than or equal to x.It was then proven that (n −m)!qσm(n)[H]n ∈ ∆. From Proposition 2.3, together with

the simple fact that σm(n) ≤ 3n, Theorem 3.4 follows.We can now see some of the improvements of our current paper. The new functions

governing the growth of the formal conjugating power series are more refined than theircounterparts, and in many cases, they yield strictly larger radii of convergence for theconjugating series H. In addition, from Theorem 1.2, the coefficients of the inverse H−1

will also be governed by these functions. Thus, we obtain not only a lower bound on theradius of convergence of H, but also on its inverse H−1 by very elementary means.

4. The η-Functions

We begin the heart of this paper with a discussion of the estimating functions we shalluse, and the algebraic structures they create within the set of analytic germs. In the courseof this study, we will prove Theorem 1.2, as well as construct other groups of germs usingestimating functions (e.g. the “sigma” function defined in (3.13).

The motivation of the functions we create is the following: as in [7], we consider analyticpower series f of the form

(4.1) f(x) = x+ amxm +

∞∑n=k

anxn,

with m, k ∈ N, m ≥ 2 and k ≥ 2m−1. By Propositions 3.1, there is a unique formal powerseries H(x) = cx+O(xm+1) that will conjugate f to the form fm,A,B.

Let us now elaborate on the choice of q from the previous section. Since the radius ofconvergence of the function is positive, the sequence {1/ n

√|an|} is bounded below by some


ε > 0. We pick q ∈ K with 0 < |q| ≤ ε and bn = anqn ∈ ∆ for all n ≥ k and n = m. Thus,

we may write f in the following form

(4.2) f(x) = x+bmqmxm +

∞∑n=k

bnqnxn.

The objective is as follows: we wish to show that the function H conjugating f to itspolynomial form f − m,A,B can be estimated in terms of |q|. To do this, we constructan estimating function which governs the coefficients of h, but which behaves well undervarious operations (including composition with f , inversion, etc.).

We recall the functions from Theorem 1.1 which we will use to estimate the coefficientsof our conjugating power series H. We will refer to these functions as η-functions:

Definition 4.1. Fix m ≥ 2 and k ≥ 2m−1. For all n ∈ N satisfying n ≥ m+1, we definethe η-function

(4.3) ηm,k(n) = (n− 1) +m

[n− 1

k −m

].

The η-functions have the following useful properties.

Lemma 4.2. For fixed m and k, the following properties hold for ηk,m:

i) ηm,k is strictly increasing, integer-valued function of n and ηm,k(n + k − m) =ηm,k(n) + k;

ii) If a, b, n ∈ N, and b− a ≥ n(k −m), then ηm,k(b)− ηm,k(a) ≥ (b− a) + nm;iii) Let i = (i1, . . . , i`) be an l-tuple of positive integers and let n = |i| − `+ 1. Then,

(4.4)∑j=1

ηm,k(ij) ≤ ηm,k(n).

Proof. It is clear that ηm,k is integer-valued and strictly increasing. Since[n+k−mk−m

]=[

nk−m

]+ 1, it follows that ηm,k(n+ (k −m)) = ηm,k(n) + k. For property (ii), note that

(4.5) ηm,k(b)− ηm,k(a) = b− a+m

([b− 1

k −m

]−[a− 1

k −m

]).

By assumption, (b − a)/(k −m) ≥ n. Since the floor functions in the right hand side ofEquation (4.5) differ by a value of at least the natural number n.

(4.6) ηm,k(b)− ηm,k(a) ≥ (b− a) + nm.

For property (iii), we want to show∑`

j=1 ηm,k(ij) ≤ ηm,k(n). By basic arithmeticalcomputation, this reduces to showing:

(4.7)∑j=1

[ij − 1

k −m

]≤[n− 1

k −m

].


Since n = |i|− `+1, we have∑`

j=1(ij−1) = n−1. Our proof now follows from the generalfact that for any ` integers a1, . . . a`, and positive integer N, one has

(4.8)∑j=1

[ajN

]≤

[∑`j=1 aj

N

].

�

When m and k are clear, we will suppress the subscripts and write η = ηm,k. Beforebeginning with the actual estimation of power series by the η-function, we wish to focus onproperty (iii), which is arguably its most interesting aspect. To hint at this interest, we lista couple of technical algebraic lemmas. Let i = (i1, . . . , i`) be a finite sequence of positiveintegers, with ` and |i| given by Definition 3.3. For a set of field elements (ai1 , . . . , ai`), wewill write

ai =∏j=1

aij .

Lemma 4.3. Let f(x) = x +∑

n≥2 anxn, and let f−1(x) = g(x) = x +

∑n≥2 bnx

n be itsformal inverse. Then, for all n ≥ 2, we can write

bn =∑i

αiai,

where αi is an integer (possibly 0), and the sum is taken over those i satisfying n =|i| − `(i) + 1.

Lemma 4.4. Consider formal maps f(x) = x +∑

n≥2 anxn and g(x) = x +

∑n≥2 bnx

n.Write f ◦ g(x) = x+

∑n≥2 cnx

n. Then, we have

cn =∑k,i

αkiakbi,

where αki is an integer (possibly 0), and the sum is taken over those k and i satisfyingn = (|i|+ k)− (`(i) + 1) + 1.

Like their statements, the proofs of these lemmas are rather technical; we defer themuntil Section 7. The point of these lemmas is as follows: while it might be difficult (if notimpossible) to compute the individual terms of the compositions of two functions f andg precisely, we can at least relate the degree of each term of f ◦ g to the indices of thecoefficients of f and g appearing in the formula of [f ◦ g]n. Note that if we call i′ = (i, k),then the conclusion of Lemma 4.4 is the same as that of Lemma 4.3.

We now have everything we need to prove Theorem 1.2.

Proof. For some fixed m ≥ 2 and k ≥ 2m − 1, consider the η-function ηm,k. We drop them and k for convenience. Recall the corresponding set Gη,n! as defined in Theorem 1.2,(4.9)

Gη,n! = {h(x) = x+∞∑

n=m+1

cnxn : There is q ∈ K with 0 < |q| ≤ 1 so that (n−m)!qη(n)cn ∈ ∆}.


We show that Gη,n! is a group. Let h1 and h2 be elements of Gη,n!, and write h1(x) =x+∑

n≥k anxn and h2(x) = x+

∑n≥k bnx

n. Write h1 ◦h2(x) = x+∑

n≥k cnxn. Fix N ≥ K.

By Lemma 4.4, the coefficient cN can be written as a sum of terms of the form

(4.10) α

(ak∏j=1

bij

),

where α is an integer, and (i1 + · · · + i` + k) − (` + 1) + 1 = N . Since h1, h2 ∈ Gη,n!, wehave the following estimate:

(4.11) (k −m)!(i1 −m)! · · · (i` −m)!qη(k)+η(i1)+···+η(i`)cN ∈ ∆.

What we want to show is that

(4.12) |(N −m)!qη(N)cN | ≤ 1.

Note that this estimate will hold if the following two facts hold:

(4.13) η(k) +∑j=1

η(ij) ≤ η(N);

(4.14) |(N −m)!| ≤

∣∣∣∣∣(k −m)!∏j=1

(ij −m)!

∣∣∣∣∣ .Note that inequality (4.13) is simply a restatement of (4.4) in Lemma 4.2. On the otherhand, for the second inequality (4.14), it is well known that, given the finite sequence ofnatural numbers i = (k, i1, . . . , i`) of length `+ 1, the multinomial coefficient

(4.15)

(k + i1 + i2 + · · ·+ i` −m(`+ 1)

k −m, i1 −m, i2 −m, . . . , i` −m

)=

(|i| − (`+ 1)m)!

(k −m)!(i1 −m)! . . . (i` −m)!

is an integer. Since our norm is non-archimedean, we can conclude

(4.16) |(|i| − (`+ 1)m)!| ≤ |(k −m)!|∏j=1

|(ij −m)!|.

The inequality (4.14) is therefore proven if we can show

(4.17) |(N −m)!| ≤ |(i− (`+ 1)m)!|,which we will prove by showing that N −m ≥ (k + i1 + · · ·+ i`)−m(` + 1).This reducesto proving that `(m− 1) ≥ 0, which follows immediately since m ≥ 2 and ` ≥ 0.

The argument for inverses is almost identical, but we provide (most of) it anyway. Let

(4.18) h(x) = x+∞∑

n=m+1

anxn

be an element of Gη,n!, and consider its inverse h−1. From Lemma 4.3, we can write h−1 as

(4.19) h−1(x) = x+∞∑

n=m+1

bnxn,


where, for any N ≥ 2, we can write bN as a sum of terms of the form

(4.20) α∏j=1

aij

where α is an integer, and the `-tuple i = (i1, . . . , i`) satisfies |i|−`+1 = N . By assumption,we have the estimate

(4.21) (i1 −m)! · · · (i` −m)!qη(i1)+···+η(i`)bN ∈ ∆.

We would like to show that (N−m)!qη(N)bN ∈ ∆. As before, we can reduce this to showingthe following two inequalities:

(4.22)∑j=1

η(ij) ≤ η(N);

(4.23) (N −m)! ≤ (i1 −m)! · · · (i` −m)!.

But again, the first inequality follows immediately from (4.4), while the second (after asimilar chase through multinomial coefficients) follows from the fact that (m−1)(`−1) ≥ 0for m ≥ 2 and ` ≥ 1. This completes the proof. �

As the reader probably noted, the argument in the previous proof treated estimationof factorials independently from the estimation of the η-function. This can be exploitedto construct a variety of groups in a similar way. The proofs of the following theoremsare based strongly on the ideas in the proof of Theorem 1.2, and are left to the interestedreader.

Theorem 4.5. Let K be a nontrivial, non-archimedean field. Fix m ≥ 2, k ≥ 2m− 1 andrecall the function ηm,k defined in (4.3). Fix some q ∈ K with 0 < |q| ≤ 1, and define theset

(4.24) Gη = {h(x) = x+∞∑

n=m+1

anxn : qηm,k(n)an ∈ ∆}.

Then, Gη is a group.

Theorem 4.6. Let K be a nontrivial, non-archimedean field. Fix m ≥ 2, and recall thefunction σm defined in (3.13). Fix some q ∈ K with 0 < |q| ≤ 1 and define the set

(4.25) Gσ = {h(x) = x+∞∑

n=m+1

anxn : qσm(n)an ∈ ∆}.

Then, Gσ is a group.

In a similar fashion, one can compute other groups, as well as subgroups, etc. All ofthese proofs are made possible because of (4.4) (which is also true for σm for all m).


5. Estimation of the Conjugating Maps

We now provide a proof of Theorem 1.1. Let us summarize our situation. Let K be afield of characteristic 0 complete with respect to a non-trivial, non-archimedean norm. Weassume that f is an analytic function which is tangent to the identity. From the results ofSection 3, we may assume that f takes the form (3.11), that is, we can write f as

f(x) = x+ Axm +Bx2m−1 +∞∑

k=2m

anxn,

where A,B ∈ K with |A| ≥ 1. Moreover, Proposition 3.1 guarantees the existence of aunique formal map H(x) = x + O(xm+1) conjugating f with the polynomial fm,A,B(x) =x+ Axm +Bx2m−1. The formal map H is a (formal) limit of polynomials

(5.1) Hj(x) = hj ◦ · · · ◦ hm+1(x) = x+∑

n≥m+1

Γjnxj,

where for each k = m + 1, · · · , j, we have hj(x) = x + cjxj. From Theorem 1.2, we know

that if, for all k = m+ 1, . . . , j, we have an estimate of the form (k−m)!qη(k)ck ∈ ∆, thenwe can conclude that for all n ≥ m+ 1,

(5.2) (n−m)!qη(n)Γjn ∈ ∆.

From this, we can conclude the same estimate for the coefficients of the formal map H.We will thus prove the following.

Proposition 5.1. Fix q ∈ K with 0 < |q| ≤ 1, and suppose that

(5.3) f(x) = x+bmqmxm +

∞∑n=k

bnqn

is an analytic function, where k ≥ 2m − 1, 0 < |bm| ≤ 1 and |bn| ≤ 1 for n ≥ k. For alln ≥ k + 1 − m, let cn, hn and Hn be defined as in the proof of Proposition 3.1, and letη = ηm,k be defined as in (4.3). Then, we have

(5.4) (n−m)!qη(n)cn ∈ ∆.

Proof. We induct on n. For n = k + 1 − m, we have ck+1−m = −bk/Aqk. Note thatη(k + 1 −m) = k, and since |A| ≥ 1, our claim is true (and if k = 2m − 1, then we havechosen cm = 0, and so the statement is trivially satisfied). Now, suppose that n ≥ k+1−m,and that

(5.5) (n−m)!qη(n)cn ∈ ∆.

We must show that

(n−m+ 1)!qη(n+1)cn+1 ∈ ∆.

From the induction hypothesis, together with (5.2), we can write the polynomial Hn in theform

Hn(x) = x+∑

b≥k+1−m

Γnb xb


where (b − m)!qη(b)Γnb ∈ ∆ for all b ≥ m + 1. For the remainder of the proof, we willsuppress the superscript n. Writing

(5.6) Hn+1 = hn+1 ◦Hn = Hn + cn+1Hn+1n ,

Proposition 3.1 says that up to order O(xn+m+1), we must have Hn+1 ◦ f = fm,A,B ◦Hn+1.Since B = b2m−1/q

2m−1, up to this order we must have

Hn ◦ f + cn+1(Hn ◦ f)n+1 =Hn + cn+1Hn+1n + A(Hn + cn+1H

n+1n )m

+b2m−1q2m−1

(Hn + cn+1Hn+1n )2m−1.

We consider the (n + m)-degree coefficient of each side, and look at the individual terms.Since (Hn ◦ f)n+1 = x+ Axm +O(xm+1), we have that:[

(Hn ◦ f)n+1]n+m

=[(x+ Axm +O

(xm+1

))n+1]n+m

=[xn+1 + A(n+ 1)xn+m +O(xn+m+1)

]n+m

= A(n+ 1)

Additionally:[(Hn + cn+1H

n+1n )m

]n+m

=[Hmn +mcn+1H

n+mn +O(xn+m+1)

]n+m

=[Hmn +mcn+1

(xn+m +O

(xn+m+1

))+O

(xn+m+1

)]n+m

=[Hmn +mcn+1x

n+m +O(xn+m+1)]n+m

= [Hmn ]n+m +mcn+1

Similarly: [(Hn + cn+1H

n+1n )2m−1

]n+m

=[H2m−1n

]n+m

Finally:

[Hn+1n

]n+m

=

(x+∑

b≥m+1

Γbxb

)n+1n+m

=[xn+1 + (n+ 1)Γm+1x

n+m+1 + . . .]n+m

= 0

Combining all of the above results, our induction reduces to:

(5.7) (n−m+ 1)Acn+1 = [Hn −Hn ◦ f ]n+m + A [Hmn ]n+m +

b2m−1q2m−1

[H2m−1n

]n+m

.

We show that the terms on the right-hand side of (5.7) each lie in

Aq−η(n+1)

(n−m)!∆.

Cancelling A from both sides, this will complete the proof. Note that since |A| ≥ 1, wehave

q−η(n+1)

(n−m)!∆ ⊆ A

q−η(n+1)

(n−m)!∆,


and so it is enough to show that any term of the right-hand side of (5.7) lies in the left-handset (however, this is not always the case, and sometimes, we will need the larger set). Thisfact rests on three claims regarding the individual terms on the right-hand side of (5.7).The proofs of these claims are very similar to those found in [7]. Nonetheless, for the sakeof completion, we provide these proofs in Section 7.

Claim 1:

(n−m)!qη(n+1) b2m−1q2m−1

[H2m−1n

]n+m∈ ∆

Claim 2:A−1(n−m)!qη(n+1)

(A [Hm

n ]n+m −mAΓn+1

)∈ ∆

Claim 3:

A−1(n−m)!qη(n+1)([Hn −Hn ◦ f ]n+m + A(n+ 1)Γn+1) ∈ ∆

We take a moment to explain our tactic. The ultimate goal is to obtain the estimation ofthe right-hand side of (5.7). Because our norm is non-archimedean, we can accomplish thisby estimating each term individually. Unfortunately, we are not able to obtain the estimatefor every term appearing in [Hn −Hn ◦ f ]n+m and [Hm

n ]n+m. So, we simply estimate whatwe can, and subtract away the offending terms. Thankfully, these castoff terms will cometogether conveniently, as follows: we may rewrite the right-hand side of (5.7) as follows:

([Hn −Hn ◦ f ]n+m + A(n+ 1)Γn+1) + ([Hmn ]n+m −mAΓn+1)

+b2m−1q2m−1

[H2m−1n

]n+m− A(n−m+ 1)Γn+1.

(5.8)

Note that the first three terms in the sum (5.8) are estimated by each of the three claims,while the fourth term is simply the sum of the castoff terms. From (5.2), we have

(5.9) (n−m+ 1)!qη(n+1)(Γn+1) = −A−1(n−m)!qη(n+1)(−A(n−m+ 1)Γn+1) ∈ ∆.

This is exactly what we want. Putting it all together, we see that

(n−m+ 1)!qη(n+1)cn+1 ∈ ∆.

This completes the proof of Proposition 5.4.�

6. Proof of Theorem 1.1 and Concluding Remarks

Let H be the unique formal series H(x) = x + O(xm+1) which conjugates f to thepolynomial fm,A,B. From Proposition 5.1 together with (5.2), we have for all n ≥ m+ 1,

(6.1) (n−m)!qη(n) [H]n ∈ ∆


This completes the proof of Theorem 1.1.We now give the stated lower bound on the radius of convergence of H. By choosing q

so that |q| is sufficiently small, Proposition 2.3, together with the fact that m ≥ 2, yields:

(6.2) |(n−m)!| ≥ qn−m > qn

Also,

η(n) = n− 1 +m

[n− 1

k −m

]≤ n− 1 +m

n− 1

k −m

=(k −m)(n− 1) +m(n− 1)

k −m

=k(n− 1)

k −m

≤ kn

k −m.

Putting this all together, we may conclude

(6.3)∣∣qη(n)∣∣ ≥ |q| kn

k−m

Substituting the inequalities (6.2) and (6.3) into (6.1), we have

1 ≥∣∣(n−m)!qη(n) [H]n

∣∣ ≥ ∣∣∣qnq knk−m [H]n

∣∣∣ ,and so

|q|−n(2k−m)

k−m ≥ | [H]n |The radius of convergence of H, ρ is given by:

ρ =

(lim supn→∞

n

√|[H]n|

)−1≥

(lim supn→∞

n

√q−

n(2k−m)k−m

)−1=

(lim supn→∞

q−2k−mk−m

)−1= |q|

2k−mk−m

Therefore the radius of convergence of H has a lower bound of |q|2k−mk−m �.

We now conclude this section with an open problem and some future research directions.Let f(x) = x + Axm +

∑n≥k anx

n. When k > 2m − 1 (i.e. when B = 0), the estimatespresented here are strictly better than the counterparts in the work of Jenkins and Spallone,and only improve as k →∞. However, it is still unknown whether these estimates are thebest available. By using Maple, we have “tested” our estimations on a few well-chosenfunctions (e.g., f(x) = x + x3 +

∑n≥k q

−nxn, where k ≥ 5), and these tests suggest thatour η-functions are sharp. Nonetheless, we are currently unable to prove the existence of


a function f for which the unique conjugating series H(x) = x + O(xm+1) have preciselythese estimates.

Theorem 1.2 opens interesting possibilities for the study of groups of analytic germs. Asof now, we have almost no understanding of the structure of the groups we have constructed.

Finally, the same questions can be asked in higher dimensions. For example, let Fbe a locally-invertible analytic function fixing 0 which is formally conjugate to G. Is Fanalytically conjugate to G? Can the formal conjugating maps H be estimated? Do suchestimating functions produce groups within the set of formal (or analytic) mappings?

7. Other Proofs

We devote this section to the proof of the lemmas needed in Section 4, as well as thethree claims necessary for the proof of Proposition 5.4.

7.1. Proof of Lemma 4.3.

Proof. Since g is the formal inverse of f , we have g ◦ f(x) = x. Thus, we have therelationship

0 =∞∑n=2

anxn +

∞∑n=2

bn(x+∞∑j=2

ajxj)n.

We use induction. Since b2 = −a2, it is clear that the lemma holds for n = 2. We assumethe lemma is true for n ≤ N , and prove the statement for N + 1. We consider the termbN+1. We have

(7.1) bN+1 = −aN+1 −

[N∑n=2

bn

(x+

k∑j=2

ajxj

)n]N+1

.

Note that −aN+1 satisfies our conclusion, by assumption on f . We now consider the term

(7.2) −

[N∑n=2

bn

(x+

k∑n=2

ajxj

)n].

Here, the value of k is irrelevant for our purposes (k is obviously bounded by N + 1). Forsome value of n ≥ 2, we consider a term of the outer sum of the form

(7.3) bn

[(x+

k∑j=3

ajxj

)n]N+1

where, as before [f ]N+1 is the coefficient of the term of degree N + 1 in the expansion off . By our induction hypothesis, we know that

(7.4) bn =∑i′

αi′ai′

where αi′ is an integer (possibly 0), and the sum is taken over those sequences i′ satisfyingn = |i′| − `(i′) + 1. We now focus on the terms of degree N + 1 in the power of the sumappearing in (7.3). The term of degree N + 1 is a sum of terms of the form

(7.5)[xe1(a2x

2)e2 · · · (akxk)ek]N+1

= (a2)e2 · · · (ak)ek .


Here, n =∑k

i=1 ei and N + 1 =∑k

i=1 iei. Note that the right-hand side of (7.5) takes theform ai′′ for a finite sequence of natural numbers i′′ satisfying

(7.6) |i′′|+ e1 = N + 1 `(i′′) + e1 = n.

(This sequence is very simple to describe, but unwieldy to write - it is the sequence whosefirst e2 terms are identically 2, and whose next e3 terms are identically 3, etc.) Putting thetwo equations of (7.6) together, we obtain

(7.7) |i′′| − `(i′′) = N + 1− n.Now, we put it all together. We can write the typical term of degree N + 1 in (7.3) as asum of products of the form ai′ai′′ , where i′ and i′′ satisfy the requirements of (7.4) and(7.7). Consider the sequence i formed by concatenating the two sequences i′ and i′′. Notethat |i| = |i′| + |i′′| and `(i) = `(i′) + `(i′′). Thus, by direct substitution, we see that|i| − `(i) + 1 = N + 1. Hence, the lemma is proved by induction. �

7.2. Proof of Lemma 4.4.

Proof. The idea here is very similar to the proof of Lemma 4.3. We consider functionsf(x) = x+

∑n≥2 anx

n and g(x) = x+∑

n≥2 bnxn. Then, we may write the composition

(7.8) f ◦ g(x) = x+∞∑n=2

bnxn +

∞∑n=2

an

(x+

∞∑j=2

bjxj

)n

= x+∞∑n=2

cnxn.

We analyze each term individually. For c2 = a2 + b2, it is clear that the lemma holds. Letus consider cN for n ≥ 3. We write

(7.9) cN = bN +

[∞∑n=2

an

(x+

∞∑j=2

bjxj

)n]N

.

Obviously, bN satisfies the conclusion of our lemma, so we focus on the second part of (7.9).Fix n ≥ 2, and consider the single term

(7.10)

[an

(x+

∞∑j=2

bjxj

)n]N

.

If we can prove that the lemma holds for this term, then we will have proven the lemma,since the coefficient cN will be the sum of these terms over all n. As in the proof of Lemma4.3, we can write this term as a sum of terms of the form

(7.11) anxe1(b2x

2)e2 · · · (bkxk)ek.Since this term has degree N , we have the following identities:

(7.12)k∑

n=1

ei = nk∑

n=1

iei = N.

Thus, we can write this term as anbi′ , where i′ is a finite sequence of natural numberssatisfying |i′| − `(i′) = N − n (see the description of this sequence in the proof of Lemma4.3). Writing i as the singleton (n) (note that |i| = n and `(i) = 1, we obtain |i| + |i′| −`(i)− `(i′) + 1 = N , completing the lemma. �


We now turn to proving the three claims on which Proposition 5.4 in Section 5 rests.

7.3. Proof of Claim 1.

Proof. Note that if k > 2m − 1, then B = b2m−1 = 0 and the claim holds trivially. Next,consider the case where k = 2m−1 and b2m−1 6= 0. The coefficient [H2m−1

n ]n+m comes froma sum of terms of the form:

(7.13) xk0∏t=1

(Γstxst)it

where we have the sums

(7.14) k0 + i1 + · · ·+ i` = 2m− 1

and

(7.15) k0 + i1s1 + · · ·+ i`s` = n+m.

Let us call the coefficient of this (n + m)-degree term Bn+m. From (5.2), this coefficientsatisfies the estimate

(7.16)∏t=1

((st −m)!)itqitη(st)Bn+m ∈ ∆

To complete the claim, we would like to replace the product of factorials and the powers ofq appearing in (7.16) with (n−m)! and η(n+1), respectively. We consider the multinomialcoefficient (

i1(s1 −m) + · · ·+ i`(s` −m)

s1 −m, . . . , s1 −m, . . . , s` −m, . . . , s` −m

)where sk −m appears ik times for k = 1, 2, . . . , `. Since this is an integer, we may replacethe factorials appearing in (7.16) with (n−m)!, provided that:

(7.17) i1(s1 −m) + · · ·+ i`(s` −m) ≤ n−m.From (7.15), this inequality is true when

(7.18) 2m ≤ k0 + (i1 + · · ·+ i`)m.

This inequality certainly holds if i1 + · · ·+ i` ≥ 2. This is only possibly false when ` = 1and i1 = 1. But then by (7.14), we have k0 = 2m− 2. Then (7.18) follows since m ≥ 2.

In order to estimate the power of q appearing in (7.13), we will show that

(7.19)∑t=1

itηk,m(st) ≤ ηk,m(n+ 1)− (2m− 1)

(The slightly smaller decay is necessary, since ultimately, we multiply [H2m−1n ]n+m by b2m−1

q2m−1 )

Denote by s the (i1 + · · ·+ i`)-tuple of integers consisting of s1 in the first i1 components,s2 in the next i2 components, etc. Then |s| = i1s1 + · · ·+ i`s` and `(s) = i1 + · · ·+ i`.

From (7.14) and (7.15), we see

|s| = n+m− k0 = n+m− ((2m− 1) + `(s)).

Therefore, we haven+ 1 = (|s| − `(s) + 1) + (m− 1).


Note that in this case k −m = m− 1. Therefore, from part (i) of Lemma 4.2,

ηk,m(n+ 1) = ηk,m(|s| − `(s) +m− 1) + 2m− 1.

Finally, part (iii) of Lemma 4.2 yields the inequality (7.19).This completes the proof of Claim 1. �


Proof. Note that after canceling A we are reduced to showing;

(n−m)!qη(n+1)([Hm

n ]n+m −mΓn+1

)∈ ∆

The coefficient [Hmn ]n+m comes from a sum of terms of the form:

(7.20) xk0∏t=1

(Γstxst)it

where we have the following sums:

k0 + i1 + · · ·+ i` = m(7.21)

k0 + i1s1 + · · ·+ i`s` = n+m(7.22)

Again, call the coefficient of this (n + m)-degree term Bn+m. From 5.2), this coefficientsatisfies the estimate;

(7.23)∏t=1

((st −m)!)it qitηk,m(st)Bn+m ∈ ∆

We would like to replace the eta functions appearing there, but first we must show that:

(7.24)∑t=1

itηk,m(st) ≤ ηk,m(n+ 1)

Define s to be the (i1+ · · ·+i`)-tuple with s1 in the first i1 components, etc. By subtractingEquation (7.21) from (7.22), we have that:

|s| − `(s) + 1 = n+ 1

Thus we can apply part (iii) of Lemma 4.2 to Equation (7.24) to see it is true.We now deal with the factorials of ((7.23)). Consider the multinomial coefficient(

i1(s1 −m) + · · ·+ i`(s` −m)

s1 −m, . . . , s1 −m, . . . , s` −m, . . . , s` −m

)where sk −m appears ik times, for k = 1, 2, . . . , ` This is an integer, therefore:

i1(s1 −m) + · · ·+ i`(s` −m)! ≥∏t=1

((st −m)!)it

Thus we can replace the factorials appearing in ((7.23)) by (n−m)! if we show that

(7.25) n−m ≥ i1(s1 −m) + · · ·+ i`(s` −m)

Distributing and using Equation (7.22), this inequality is true exactly when

(7.26) 2m ≤ k0 + (i1 + · · ·+ i`)m.


This inequality certainly holds if i1 + · · ·+ il ≥ 2. This is only possibly false when ` = 1and i1 = 1. But then by (7.21), we have k0 = m − 1. Then (7.26) follows since m ≥ 2.This finishes the proof for Claim 2. �


Proof. We study the above expression for n ≥ k. The computation of [Hn − Hn ◦ f ]n+mreduces to the study of the (n+m)-degree coefficient of

Hn(x)−Hn ◦ f(x) =n+m∑j=m+1

Γj(xj − f(x)j) (mod xn+m+1).

Let us write gj(x) = xj − f(x)j. If j ≥ n + 1, then gj(x) = O(xn+m+1), so those termsmay be discarded. Therefore, we may assume that j ≤ n + 1. If j = n + 1, the only termappearing is

[Γn+1gn+1(x)]n+2 = −A(n+ 1)Γn+1.

This exception is why we subtract this term from [Hn −Hn ◦ f ]n+m in the claim. We moveon to consider the case where j ≤ n.

We will consider each coefficient [Γj(f(x))j]n+m individually. Because the norm is non-archimedean, proving the necessary estimate of these terms will give us the result for thecomplete (n + m)-degree coefficient. We first expand the jth power of f . Each term willbe a sum of terms of the form

(7.27) Γjxe1

(bmx

m

qm

)em (bkxkqk

)ek· · ·(b`x

`

q`

)e`,

with

(7.28) e1 + em +∑s=k

es = j.

A term will have degree n+m when

(7.29) e1 +mem +∑s=k

ses = n+m.

Now we define t = ek + · · ·+ e`. We will prove that

(7.30) η(n+ 1)− η(j) ≥ mem +∑s=k

ses

unless em = t. If t = 0, (7.30) reduces to η(n + 1)− η(j) ≥ 0. The inequality holds, sincethe eta function is strictly increasing and we are analyzing only the cases when j ≤ n.Subtracting (7.28) from (7.29) gives us

(7.31) n+m− j = (m− 1)em +∑s=k

(s− 1)es.


We would like to rewrite the expression n+ 1− j, so that we may apply part (ii) of Lemma4.2. From ((7.31)), we find that

(7.32) n+ 1− j = (1−m) + (m− 1)em +∑s=k

(s− 1)es

Since (m− 1)em ≥ 0, we write

n+ 1− j ≥ (1−m) +∑s=k

(s− 1)es(7.33)

Now, we further simplify our inequality by noting that s− 1 ≥ (k− 1) for all s in the sum.

n+ 1− j ≥ (1−m) + (k − 1)t

= (1−m) + (k − 1) + (k − 1)(t− 1)

= (k −m) + (k − 1)(t− 1)

Since k − 1 ≥ k −m, we conclude our analysis to find that

(7.34) n+ 1− j ≥ t(k −m).

We now apply part (ii) of Lemma 4.2 to conclude.

(7.35) η(n+ 1)− η(j) ≥ (1−m) + (m− 1)em +∑s=k

(s− 1)es + tm

Now, we consider two cases: (i) em 6= t, or (ii) em = t and en = 0 for all n ≥ k. Consider

case (i). Since∑`

s=k es ≤ t,

(7.36) η(n+ 1)− η(j) ≥ (1−m) + (m− 1)em +∑s=k

ses + t(m− 1).

To prove that (7.30) holds, it is sufficient to show that

(7.37) (1−m)− em + t(m− 1) ≥ 0.

Since t > em, where both em and t are integers, t ≥ em + 1. Thus,

(1−m)− em + t(m− 1) ≥ (1−m)− em + (em + 1)(m− 1)

≥ (m− 1)em − em≥ 0.

Therefore, (7.30) holds for case (i).Next, consider case (ii). Since em = t and en = 0 for all n ≥ k, it follows from inequality

(7.35) that

(7.38) η(n+ 1)− η(j) ≥ (1−m) + (m− 1)em +mem ≥ mem

and that (7.30) holds.This finishes the proof of Claim 3. �


References

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[2] P. Ahern and J.-P. Rosay, Entire functions, in the classification of differentiable germs tangent to theidentity, in one or two variables, Trans. Amer. Math. Soc., 310 (1995), 543-572.

[3] J. Ecalle Sur les functions resurgentes, I, II, Publ. Math. d’Orsay, Universite de Paris-Sud, Orsay,1981.

[4] P. Fatou Sur les equations fonctionelles, Bull. Soc. Math. France, 47 (1919), 161-271.[5] M. Herman and J.-C. Yoccoz Generalization of some theorems of small divisors to non-archimedean

fields, in Geometric Dynamics (Rio de Janeiro, 1981). Springer, Berlin, 408-447 (1983).[6] Y. S. Il’yashenko, Nonlinear Stokes Phenomena. Adv. in Soviet Math., vol. 14, Amer. Math. Soc.,

Providence, RI, 1993[7] A. Jenkins and S. Spallone, A p-adic approach to local dynamics: analytic flows and analytic maps

tangent to the identity, Ann. Fac. Sci. Toulouse Math., Vol. XVIII, no. 3 (2009), 611-634.[8] J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Asterisque 287 (2003),

147-230.[9] W. H. Schikhof, Ultrametric Calculus: an Introduction to p-adic Analysis, Cambridge Studies

in Advanced Mathematics, 4, Cambridge University Press, Cambridge, 1984.[10] S. M. Voronin, Analytic classification of germs of conformal maps (C, 0)→ (C, 0) with identical linear

part, Func. Anal. Appl. 15 (1981), 1-17. .

Department of Mathematics, Kansas State University, Manhattan, KS, 66506E-mail address: [email protected]

Department of Mathematics, Swarthmore College, Swarthmore, PA, 19081E-mail address: [email protected]

Department of Mathematics, University of Northern Colorado, Greeley, CO, 80639E-mail address: [email protected]

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