Wildly ramified power series with large multiplicity

In this paper we consider wildly ramified power series, \emph{i.e.}, power series defined over a field of positive characteristic, fixing the origin, where it is tangent to the identity. In this setting we introduce a new invariant under change of coordinates called the \emph{second residue fixed point index}, and provide a closed formula for it. As the name suggests this invariant is closely related to the residue fixed point index, and they coincide in the case that the power series have small multiplicity. Finally, we characterize power series with large multiplicity having the smallest possible multiplicity at the origin under iteration, in terms of this new invariant.


INTRODUCTION
Let be a field. We recall that a fixed point 0 of a function ∶ → is said to be parabolic if ′ ( 0 ) is a root of unity. In this paper we are interested in formal power series having a parabolic fixed point at the origin, such that ′ (0) = 1.
We say that two elements , ∈ (1+ [[ ]]) are formally conjugated if there exists a formal power series ℎ of the form ℎ(0) = 0, ℎ ′ (0) ≠ 0 such that = ℎ −1 • •ℎ. Clearly, conjugation is an equivalence relation on (1 + [[ ]]). Henceforth, we assume that is not the identity. A property that is invariant under conjugation by ℎ is the multiplicity of , that is the unique integer in the expansion The multiplicity of (at the origin) is denoted by mult( ), and we put mult( ) ∶= ∞. Another property of that is invariant under conjugation is the residue fixed point index which is defined as the coefficient of 1 in the Laurent expansion about 0 of 1 − ( ) , and denoted by index( ), see e.g. [Mil06,§12] for a description of the residue fixed point index. i Let ∶=Ĉ = ℂ ∪{∞}, then for a rational map ∈ ( ), not the identity, of the form (0) = 0 and ′ (0) = 1, classification under formal conjugation is described in terms of the multiplicity and the residue fixed point index (see e.g., [É81a,É81b,Vor81]). One of the main results of this paper is that this statement is false if we replace by a field of positive characteristic, and that the classification depends on at least one other invariant which we describe in detail in this paper.
Let be a field of positive characteristic. A power series with coefficients in is said to be wildly ramified if and only if (0) = 0, and ′ (0) = 1. The group under composition formed by all wildly ramified power series is called the Nottingham group of , and is denoted by  ( ). Moreover, every wildly ramified power series has an associated sequence of integers called the lower ramification numbers. It encodes the multiplicity of the origin for the iterates of the power series. Henceforth, we say that has large multiplicity if mult( ) > char( ) + 1, and we study the lower ramification numbers of power series with large multiplicity at the origin, and give a characterization of those power series having the smallest possible lower ramification numbers. Wildly ramified power series and its lower ramification numbers have been studied in many papers, see e.g., [Sen69,Kea92,LS98,Win04] for i We further note that the residue fixed point index may be defined in a similar manner for any fixed point of . classic results and background on wildly ramified power series, and [KK16,LMS02,LN18,LRL16b,LRL16a,Nor17,RL03,NR19] for results on lower ramification numbers related to this paper. For a background on the Nottingham group and result on the subject relating to this paper we refer to [Joh88,Cam00,Kea05] and references therein. Further, as previously mentioned, one of the results of this paper concerns formal classification of parabolic maps in positive characteristic. Similar studies have been made in the case of superattracting germs in [Spe11] and this was later further investigated in [Rug15].
Endowing a field of positive characteristic with an absolute value will necessarily make it nonarchimedean. Studying the iterates of a wildly ramified power series with coefficients in can be thought of as a discrete dynamical system, in which we are particularly interested in the fixed points of and its iterates. The topic of discrete dynamical systems over non-archimedean fields is a well-studied subject see e.g. the monographs [Sil07, AK09, Ben19] and references therein.
In the next section we describe our results in detail.

MAIN RESULTS
The main results are naturally divided into two parts. First we introduce a 'new' invariant under local change of coordinates in positive characteristic. The second part concerns lower ramification numbers and in particular we provide a characterization in terms of this new invariant of wildly ramified power series with large multiplicity having the smallest possible lower ramification numbers.
2.1. A positive characteristic phenomena. In order to state the main result of the first part we define the key concept of this section, and in order to state said concept we need to introduce the following notation. Given a field and an integer ≥ 1, denote by Λ( , ) the ordered set of integers corresponding to the set The th element of Λ( , ) is denoted by , and to be precise we have the indexing 1 ≤ 2 ≤ … . In particular, in the case that = 1 then we say that ( ) is the second residue fixed point index.
Remark 2.4. In the complex case for ∈Ĉ( ) such that (0) = 0, converging in a neighbourhood of the origin, the residue fixed point index is typically described by the contour integral where is a small, closed, simple curve in positive direction around the origin. The formal definition in Definition 2.2 using the theory of abstract residues (cf. [Sil07, Exercise 5.10]) is equivalent in the case that is convergent in a neighbourhood of the origin.
In §6 we give a closed formula to compute ( ) in terms of its first significant coefficients. For a wildly ramified power series the residue fixed point index is invariant under local change of coordinates. This statement is also true for 1 ( ) as manifested by the following result.
In fact, we can say more, and the proof of Proposition 2.5 follows by the following generalization.
Proposition 2.6. Let be a field, ≥ 1 an integer, and a power series, with coefficients in , of multiplicity + 1. If it exists, denote by the smallest integer such that ( ) ≠ 0. In that case, for every power series ℎ with coefficients in such that ℎ(0) = 0 and ℎ ′ (0) ≠ 0, the power serieŝ ∶= ℎ• •ℎ −1 satisfies Écalle [É81a, É81b] and Voronin [Vor81] independently proved that analytic classification of parabolic maps inĈ( ) depends on an infinite number of parameters. However, as previously mentioned, the formal classification soley depends on the multiplicity and the residue fixed point index of the fixed point. Hence, Proposition 2.5 is an obstruction for the same situation to be true for formal equivalence in positive characteristic, and as a consequence we obtain the following corollary.
Corollary A. In positive characteristic, the moduli space of formal classification of parabolic maps with large multiplicity is at least three dimensional.
2.2. Lower ramification numbers. The second part of our results concerns lower ramification numbers of wildly ramified power series. Let ≥ 0 be an integer then the lower ramification numbers of a wildly ramified power series are defined as ( ) ∶= mult − 1. Let ∶= mult( ) − 1 and denote by the smallest nonnegative integer such that = in . Then we have (2.3) ( ) ≥ (1 + + ⋯ + −1 ) + , see Proposition 5.1, in §5. Our primary result in this section is a characterization of power series with large multiplicity, having lower ramification numbers satisfying equality in (2.3), i.e. the smallest possible sequence of lower ramification numbers. We divide the results into two parts: the generic and subgeneric case.
2.2.1. Generic case. The following result is a characterization of power series with large multiplicity + 1 having the smallest possible lower ramification numbers. In particular, among power series with multiplicity + 1 these form a generic set. The proof of Theorem A is given in §5.2. In a famous paper by Sen [Sen69] we have that if | 0 ( ) = then ( ) = , which clearly satisfies equality in (2.3). Using this fact together with Theorem A and [NR19, Theorem 2] we obtain the following corollary, extending [NR19, Corollary 1] to include power series with large multiplicity.

Corollary B. Let be a field of odd characteristic. Then, among wildly ramified power series with coefficients in , those which have the smallest possible lower ramification numbers are generic.
Corollary B together with (2.3) answers [NR19, Question 1], which asks for what the lower ramification numbers for a generic power series with large multiplicity are.
2.2.2. Subgeneric case. The following result is a characterization of power series with large multiplicity + 1 which does not necessarily attain the smallest possible lower ramification numbers.
Theorem B. Let be a field of characteristic . Further, suppose that ≥ + 1 is an integer not divisible by , the smallest nonnegative integer such that = in . Finally, let be a positive integer such that ∈ {1, 2, … , ⌈ ∕ ⌉ − 1}, and put ∶= + ( − 1) . Then for every power series ∈ [[ ]] such that mult( ) = + 1, and every integer ≥ 1, the lower ramification numbers of are of the form if and only if is the smallest integer such that ( ) ≠ 0.
We note that Theorem A follows from Theorem B. However, as Theorem A is the generic case we consider it of value to be highlighted. Also the proofs of Theorem A and Theorem B differ, since Theorem A follows by the invariance of 1 ( ) together with [NR19, Main Lemma], and is thus selfcontained. However, Theorem B does not follow as a consequence by the aforementioned lemma, and we give a direct proof depending on a known result by Laubie and Saïne [LS98, Corollary 1]. The proof of Theorem B is given in §5.3.
We conclude this section by defining the concept of -ramification and state our results regarding these matters.
we say that is -ramified.
Hence, partly as a consequence of Theorem B we obtain the following corollary.
Corollary C. Let be a field of odd characteristic , an integer not divisible by , such that ≥ +1, and put ∶= ⌈ ∕ ⌉. Furthermore, let ∈ [[ ]] be a wildly ramified power series of multiplicity +1.

Then is -ramified if and only if
and for every positive integer < we have ( ) = 0.
2.3. Organization of the paper. In §3 we discuss some applications of the main results related to one-dimensional dynamical systems over non-archimedean fields. In §4 we prove the invariance of the second residue fixed point index and its generalization ( ), and we also discuss some properties of these invariants. Further, in §5 we give a proof of Theorem A, Theorem B and Corollary C. In §6 we give a closed formula for ( ) in terms of the first significant coefficients of the power series . Finally, in Appendix A we give a proof of the results presented in §3.
ii The left hand side is known as the résidu itératif, or iterative residue and is introduced in §4.

APPLICATIONS
As an application of our results we compute the exact minimal distance between periodic points in the open unit disk of a convergent wildly ramified power series with large multiplicity and the origin. To state this result, we introduce some notation. Given an non-archimedean field ( , | ⋅ |), denote by  its closed unit disk and by its open unit disk respectively, i.e.
Theorem 3.1. Let be a prime number, let ≥ + 1 be an integer not divisble by , and let ( , | ⋅ |) be an non-archimedean field of characteristic . Furthermore, let be a power series with coefficients in  of the form Then, for every fixed point 0 of in  that is different from 0 we have | 0 | ≥ | |, and for every periodic point 0 of in  that is not a fixed point, we have Again, we stress that we have a closed formula for 1 ( ) given in §6. We also note that the proof of Theorem 3.1 is almost verbatim the same proof as the proof of [NR19, Theorem 3] with résit( ) replaced by 1 ( ). However, even though this proof contains little to no new information, for completion, the proof is given in Appendix A. As a consequence of Theorem 3.1, thus covering the case of large multiplicity, we obtain the following strengthening of [LRL16a, Main Theorem] and [NR19, Corollary 4].

Corollary D.
Let be a prime number and fix an integer ≥ 2, such that ≢ 1 (mod ). Then, over a field of characteristic , a generic fixed point of multiplicity is isolated as a periodic point.

THE SECOND RESIDUE FIXED POINT INDEX AND ITS GENERALIZATIONS
This section is divided into two parts. In the first part we give a proof that the second residue fixed point index invariant under local change of coordinates, and in the case that 1 ( ) = 0 the first ( ) which is nonvansihing is also invariant. In the second part we discuss some properties of the second residue fixed point index and its generalizations. Before proceeding we give some definitions that will be used throughout the paper.
Given a ring and elements 1 , … , of , denote by ⟨ 1 , … , ⟩ the ideal generated by 1 , … , . 4.1. Proof of Proposition 2.5 and 2.6. This section is devoted to prove Proposition 2.5 and 2.6. As noted before the former follows from the latter as a special case and the proof of Proposition 2.6 is given after the following lemma.  Proof. To prove the first assertion, we assume that is not divisible by and note that which is well-defined in . The coefficient of 1 in the Laurent expansion about 0 of (4.1) corresponds to the coefficient of 1 +1 in the Laurent expansion about 0 of (4.2) which is clearly equal to 0 since | . This completes the proof of the first assertion of the lemma. Put , and ∶= .
Proof of Proposition 2.6. Assume that is the smallest integer for which ( ) ≠ 0 and put ∶= + ( − 1) , where is the smallest integer such that = in . In the case that = the proof follows from the fact that the residue fixed point index is invariant under change of coordinates. Thus, we may assume that < . Further, we also put ∶= ℎ ′ (0), and recall that ( ) is the coefficient of 1 in the Laurent expansion of  Clearly, ord (Δ) = ord (Δ) = + 1 since the multiplicity is preserved under conjugation by ℎ. Thus, using (4.3) we obtain Hence, we conclude that (̂ ) is equal to the coefficient of 1 in the Laurent series expansion about 0 of We have We  The proof of Proposition 4.2 is given at the end of this section. The following lemma will be useful.

Lemma 4.4. Let be a field of positive characteristic . Furthermore, let ≥ 1 be an integer not divisible by , and denote by the smallest integer such that ( ) ≠ 0. Furthermore, suppose that is a power series with coefficients in , of the form
Proof. Recall that ( ) is the coefficient of 1 in the Laurent expansion of Clearly, the coefficient of 1 in (4.5) is 2 , which proves the proposition.

Lemma 4.5. Let be a prime number and a field of characteristic . Moreover, let be a power series with coefficients in such that
and let be the smallest integer such that ( ) ≠ 0. Then, is conjugated to a power series with coefficients in , of the form Proof. We utilize Lemma 4.3 repeatedly to eliminate each term of degrees in the set  = { + 2, + 3, … , + +1 } ⧵ { + 1 + 1, + 2 + 1, … , + + 1}.
Before we give the proof of Proposition 4.2 we make the following elementary observation.
Observation 4.6. For any power series having the origin as a fixed point of multiplicity + 1 we have for every integer ≥ 0, Proof of Proposition 4.2. Let ∈ × , ∈ , and put ∕ 2 ∶= 1 ( ), ∶= mult( ) − 1, and denote by the smallest nonnegative integer such that = in . We may assume by Lemma 4.5 that is of the form Hence, we have Thus, by Lemma 4.4 and (4.6) we have This completes the proof of the proposition.

LOWER RAMIFICATION NUMBERS FOR POWER SERIES WITH LARGE MULTIPLICITY
In this section we first prove the lower bound for the sequence of lower ramification numbers for power series in which the origin as a fixed point have multiplicity which is larger than the ground characteristic of the field. In the second part, we give a proof of Theorem A and in the last part we prove Theorem B and Corollary C.

Lower bound of lower ramification numbers.
This section is devoted to prove the following proposition. For completion, we acknowledge that it is possible to deduce the proposition using [Cam00, Theorem 6]. In the proof of Proposition 5.1 we make several times use of [Sen69, Theorem 1] referred to as Sen's theorem, which states that if is a wildly ramified power series, and ( ) is finite for every integer ≥ 0, then for every integer ≥ 1 we have Further in this and proceeding sections we make several times use of the 'Δ'-operators for a wildly ramified power series , introduced in [RL03, §3.2], and [LRL16b], of defining recursively for every integer ≥ 0 the power series Δ , by Δ 0 ( ) ∶= and for ≥ 1, by In particular we make several times use of the fact that for a prime we have The proof of Proposition 5.1 follows after the following lemma. Then ord (Δ ) − ord (Δ −1 ) ≥ ord (Δ 1 ) − 1.

Proof of Theorem A.
This section is devoted to the proof of Theorem A, which essentially depends on two results, one which is a special case of [NR19, Main Lemma], stated below as Lemma 5.3. The other is the computation of the first significant terms of -power iterates of wildly ramified power series with large multiplicity, stated as Proposition 5.4. Hence, we have = 1 ( ) . Note further that the conjugation mapping to is a composition of coordinates ℎ eliminating terms of degree ≥ + + 2. Thus, let ℎ be the composition of all ℎ , i.e., the coordinate taking to then mult(ℎ) ≥ + 2. Further, put

Lemma 5.3 (Main Lemma, [LN17]). Let be an odd prime number, and consider the rings
.
Repeating the same argument as for the case = 1 and specializinĝ to , then by Lemma 5.3 usinĝ =̂ we obtain (5.12) Finally, we recall that the coordinate ℎ that maps to satisfies mult(ℎ) ≥ + 2, and thus we have Hence, (5.12) proves the desired induction assumption (5.11), which proves the proposition.
Proof of Theorem A. By our hypothesis that ≥ +1 the proof follows immediately by Lemma 4.5 and Proposition 5.4 since the former implies that is conjugated to a power series of the form (5.8).

Proof of Theorem B and Corollary C.
In this section we give a proof of Theorem B, and its corollary. We make use of the following result. The proof of Theorem B is given after the following proposition in which we compute the first significant terms of the th iteration of a power series with large multiplicity at the origin.
The proof of Proposition 5.6 is given at the end of this section.
This completes the proof of the corollary.

CLOSED FORMULA FOR ( )
For a wildly ramified power series we can, from the definitions of the second residue fixed point index and its generalizations ( ), compute a closed formula for these invariants in terms of the first significant coefficients of . In order to state this result, denote by ℕ the set of nonnegative integers and for an integer ≥ 1 and ( 0 , … , ) in ℕ +1 , define Thus, ( ) is equal to the coefficient of in the sum in (6.3). We see that for ≥ + + 1 the coefficient makes no contributions to the coefficient of in the sum in (6.3). Furthermore, if > the corresponding term in the sum (6.3) also makes no contributions. Hence, we may replace the sum by