Generic 1-parameter pertubations of a vector field with a singular point of codimension k

TL;DR

This paper classifies generic one-parameter families of complex vector fields near a singular point of multiplicity k+1, providing normal forms, moduli space descriptions, and bifurcation diagrams.

Contribution

It introduces a comprehensive classification of germs of generic 1-parameter vector field families with a singular point of codimension k, including normal forms and bifurcation analysis.

Findings

Complete bifurcation diagram of ż = z^{k+1} - ε over CP1.

Description of the modulus space for the unfolding.

Almost unique normal forms for the vector fields.

Abstract

We describe the equivalence classes of germs of generic 1-parameter families of complex vector fields z dot = omega_epsilon(z) on C unfolding a singular point of multiplicity k+1: omega_0 = z^{k+1} + o(z^{k+1}). The equivalence is under conjugacy by holomorphic change of coordinate and parameter. We provide a description of the modulus space and (almost) unique normal forms. As a preparatory step, we present the complete bifurcation diagram of the family of vector fields z dot = z^{k+1} - epsilon, over CP1.

Equations

$$\dot{z}=\omega (z)=z^{k+1}+O(z^{k+2}).$$

$$\dot{z}=\frac{z^{k+1}+{ϵ}_{k-1}{z}^{k-1}+\cdots +{ϵ}_{1}z+{ϵ}_0}{1+a(ϵ)z^k}.$$

$$\dot{z}=(z^{k+1}-ϵ)u(z,ϵ),$$

$$\dot{z}=(z^{k+1}-ϵ)\sigma (z)$$

$$\lambda (\delta )=(k+1){\delta }^k\sigma (\delta )=(k+1){\delta }^k\left(1+\sum _{j=1}^k{a}_j({\delta }^{k+1}){\delta }^j\right),$$

$$\dot{z}=(z^{k+1}-ϵ)\sigma (z),$$

$$f_0(z)=z+z^{k+1}+O(z^{k+2}).$$

$$\dot{z}=\frac{dz}{dt}=P_{ϵ}(z)=z^{k+1}-ϵ$$

$$(z,ϵ,t)\mapsto \left(e^{\frac{\pi i}{k}}z,-e^{\frac{\pi i}{k}}ϵ,-t\right),$$

$$(z,ϵ)\mapsto (\overline{z},\overline{ϵ}).$$

$$(z,ϵ)\mapsto (-z,ϵ,-t).$$

$$(z,ϵ,t)\mapsto \left(|ϵ{|}^{-\frac{1}{k+1}}z,\frac{ϵ}{|ϵ|},|ϵ{|}^{\frac{k}{k+1}}t\right),$$

$$ϵ=e^{i\theta },\theta \in [0,2\pi ].$$

$$z_{\ell }=e^{\frac{i(\theta +2\ell \pi )}{k+1}},\ell =0,\dots ,k,$$

$$P_{ϵ}^{\prime }(z_{\ell })=(k+1)z_{\ell }^k=(k+1)e^{\frac{ik(\theta +2\ell \pi )}{k+1}}=(k+1)e^{\frac{ik\theta }{k+1}-\frac{2\ell \pi i}{k+1}}$$

$${\kappa }_{\ell }=\frac{1}{P_{ϵ}^{\prime }(z_{\ell })}=\text{Res}\left(\frac{1}{P_{ϵ}},z_{\ell }\right),$$

$$T={\int }_{\gamma }dt={\int }_{\gamma }\frac{dz}{P_{ϵ}(z)}=2\pi i\sum _{\ell \in I}{\kappa }_{\ell }\in \mathbb{R}.$$

$${\mu }_{\ell }=2\pi i{\kappa }_{\ell }=\frac{2\pi i}{P_{ϵ}^{\prime }(z_{\ell })}.$$

$$v_{\ell +\frac{1}{2}}-v_{\ell -\frac{1}{2}}=-{\mu }_{\ell }.$$

$${\theta }_j=\{\begin{array}{ll}\frac{j\pi }{k},& k\mathrm{odd}\\ \frac{\pi }{2k}+\frac{j\pi }{k},& k\mathrm{even}\end{array},$$

$$t={\int }_0^z\frac{dz}{P(ϵ)(z)}.$$

$$(z,t)\mapsto \left(z{e}^{-\frac{i\theta }{k+1}},t{e}^{-\frac{ik\theta }{k+1}}\right),$$

$$R_0=1/k{r}^k,$$

$$v_m=C_k|ϵ{|}^{-k/(k+1)}{e}^{i\frac{-2\pi m-\pi -k\theta }{k+1}},$$

$$\beta \sim \sin \beta \sim 2R_0/|v_m-v_{m^{\prime }}|$$

$$|v_m-v_{m^{\prime }}|=|ϵ{|}^{-k/(k+1)}{C}_k\left|\sin \left(\pi \frac{|m-m^{\prime }|}{k+1}\right)\right|$$

$$\alpha \sim C(k,|m-m^{\prime }|)|ϵ{|}^{k/(k+1)}$$

$$y\sim \pm C^{\prime }(k,|m-m^{\prime }|)x^{\frac{2k+1}{k+1}}$$

$$ϵ=x+iy,$$

$$E(x,y,\alpha)=\cos(k\alpha)-\frac{1}{r^{k+1}}\big{(}x\cos\alpha+y\sin\alpha\big{)}.$$

Full text

The second author is supported by NSERC in Canada. The two authors thank BIRS where this research was first initiated.

1. 1 Introduction

2. 2 Study of the polynomial vector field $\dot{z}=z^{k+1}-{\epsilon}$

3. 2.1 The phase portrait on ${\mathbb{C}\mathbb{P}}^{1}$

4. 2.2 The phase portrait on a disk $B(0,r)$

5. 2.3 Concluding remarks

6. 3 Generic one-parameter unfoldings of vector fields

7. 3.1 Principal parts

8. 3.1.1 Effect of a change of variable on the principal parts.

9. 3.2 About the position of the singularities

10. 3.3 About the eigenvalues

11. 3.4 Classification under conjugacies that fix the parameter

12. 3.5 Classification upon general conjugacies, canonical parameter

13. 3.5.1 Method 1

14. 3.5.2 Method 2

15. 3.5.3 A postponed lemma: eliminating specific coefficients of the eigenvalue function

16. 3.5.4 Auxiliary question

17. 3.6 Proof of Theorem 3.13

18. 3.7 Normal forms

19. 3.7.1 The local normal form

20. 3.7.2 The polynomial normal form

21. 3.7.3 The rational normal form

22. 3.7.4 A Kostov-type normal form

1. Introduction

In this paper we are interested in the local study of analytic vector fields $\dot{z}=\omega(z)$ over ${\mathbb{C}}$ in a neighborhood of a singular point at the origin. When $\omega^{\prime}(0)\neq 0$, the vector field is linearizable, and the local study is finished. We are interested here in the case of a singular point of multiplicity $k+1$, and hence codimension $k$:

[TABLE]

The system is not structurally stable and the multiple singular point splits into several singular points when perturbing the system. To study all possible behaviours (phase portraits) it is natural to embed the vector field in a generic $k$-parameter unfolding. This has been done by Kostov [K] who provided a simple normal form

[TABLE]

It is nearly polynomial, save for the term with $a(\epsilon)$, which cannot be removed. The Kostov normal form is obtained by a change of coordinate and multi-parameter. In [RT] it is shown that the multi-parameter is almost unique, the only degree of freedom coming from rotations in $z$ of order dividing $k$. However, in practice, it is quite common to encounter $1$-dimensional perturbations $\omega_{\epsilon}$ of (1.1). Generically, such a pertubation satisfies $\frac{\partial\omega_{\epsilon}}{\partial{\epsilon}}\neq 0$. What are the possible phase portraits and bifurcations occuring in such perturbations? This is the question we address in this paper.

We show that a change of coordinate allows bringing a generic $1$-parameter perturbation of (1.1) to the form

[TABLE]

with $u(0,0)\neq 0$. Its singular points are given by the roots of $z^{k+1}-{\epsilon}$. Coming back to the initial coordinates we immediately see that, in a generic perturbation, the singular points are approximately located at the vertices of a regular $(k+1)$-gon. If $u$ were constant, the vector field would be conjugated by a linear change of variable to $\dot{z}=z^{k+1}-{\epsilon}^{\prime}$ for some $\epsilon^{\prime}$. It is hence natural to study the family of vector fields $\dot{z}=z^{k+1}-{\epsilon}$, their phase portraits and bifurcation diagrams, as models for the behaviour of the family $\omega_{\epsilon}$. This is what we do as a first step. We first give the bifurcation diagram of this vector field over ${\mathbb{C}\mathbb{P}}^{1}$, from which we deduce the bifurcation diagram of the restriction of the vector field to a disk containing the singularities.

As a second step, we study and solve two equivalence problems:

1. (1)

Equivalence Problem 1: When are two generic perturbations $\omega_{\epsilon}$ of (1.1) conjugate under an analytic change of coordinate? 2. (2)

Equivalence Problem 2: When are two generic perturbations $\omega_{\epsilon}$ of (1.1) conjugate under an analytic change of coordinate and parameter?

In both cases, the change of coordinate is allowed to depend on ${\epsilon}$. For both equivalence problems we introduce an invariant in the form of an eigenvalue function. It is a function $\lambda$ of $\delta={\epsilon}^{1/k+1}$ which contains all the information about the eigenvalues at each singular point. More precisely, the $k+1$ eigenvalues at the singular points of $\omega_{\epsilon}$ are given by the $\lambda(\delta_{j})$, where $\delta_{1},\dots,\delta_{k+1}$ are the solutions of $\delta^{k+1}={\epsilon}$. The function $\lambda$ vanishes at $\delta=0$ with order precisely $k$ and any function with a root of order precisely $k$ at [math] arises as an eigenvalue function of some family (1.3).

For the Equivalence Problem 1, we show that two generic perturbations $\omega_{\epsilon}$ of (1.1) are conjugate under an analytic change of coordinate if and only if their eigenvalue functions are equivalent up to right-composition by a rotation of order dividing $k+1$. Our proof is geometric in the spirit of the pioneering work of Douady and Sentenac on polynomial vector fields [DS]. In particular, we use the rectifying coordinate of the vector field given by the complex time $t$. Equivalence Problem 2 is later reduced to Equivalence Problem 1 modulo a change to a canonical parameter. In addition, we show that $\omega_{\epsilon}$ is conjugate to the form

[TABLE]

by a change of coordinate preserving the parameter, for some analytic function $\sigma$ independent of ${\epsilon}$ with $\sigma(0)\neq 0$.

For the Equivalence Problem 2, it is possible to describe the action of a change of parameter on $\lambda$. In particular, we can bring the eigenvalue function to an almost unique normal form:

[TABLE]

i.e. $\sigma(0)=1$ and the power series expansion of $\sigma$ contains no terms of degree $m(k+1)$ for $m\geq 1$. The only degree of freedom comes from substituting $\nu\delta$ to $\delta$ in (1.4) with $\nu$ a $k$-th root of unity. A choice of parameter for which the eigenvalue function has this form is called canonical: it is almost unique, up to ${\epsilon}\mapsto\nu{\epsilon}$, $\nu$ a $k$-th root of unity.

If two families have equivalent eigenvalue functions, the equivalence relation being precomposition with $\delta\mapsto\nu\delta$ for $\nu$ a $k$-th root of unity, then they are conjugate by a change of coordinate and parameter. Indeed, by action of the rotation group of order $k$ on the variable and parameter on one system, we can reduce the problem to the case of equal eigenvalue functions, in which case there exists a conjugacy between the two families which preserves the parameter, obtained in Equivalence Problem 1. Moreover, any analytic function of the form (1.4) is realizable as the eigenvalue function of a germ of family. As a consequence, we get that the modulus space is exactly the space of germs of analytic functions of the form (1.4), up to this equivalence relation. A second consequence is that a generic perturbation $\omega_{\epsilon}$ of (1.1) with an eigenvalue function having normal form (1.4) is conjugate by a change of coordinate and parameter to the almost unique normal form

[TABLE]

where the function $\sigma$ does not depend on $\epsilon$, the only degree of freedom coming from $(z,{\epsilon})\mapsto(\nu z,\nu{\epsilon})$ for $\nu$ a $k$-th root of unity.

Our motivation for this work came from our interest in perturbations of parabolic points of codimension $k$ of a germ of diffeomorphism of $({\mathbb{C}},0)$

[TABLE]

The study of a generic $k$-parameter unfolding was done in [Ro]. However, understanding the dynamics of generic $1$-parameter families is crucial in many applications where the local diffeomorphisms are polynomial or rational maps.

Besides its intrisic interest, a motivation for this work is the study of the analog classification problem, but for diffeomorphisms instead of vector fields, i.e. generic $1$-parameter unfolding of parabolic germs of codimension $k$. This will be addressed in a second paper. Let us stress that one significant difficulty in deciding if two germs of families are conjugate is the change of parameter, and it is a great bonus when it is possible to identify canonical parameters. Indeed, when changing to canonical parameters, a conjugacy must preserve the parameter, thus reducing the study to the variable space.

2. Study of the polynomial vector field $\dot{z}=z^{k+1}-{\epsilon}$

2.1. The phase portrait on ${\mathbb{C}\mathbb{P}}^{1}$

This section is deeply inspired by [DS] both for the terminology and the spirit.

The polynomial vector field

[TABLE]

has a pole of order $k-1$ at infinity (a regular point if $k=1$) and $2k$ separatrices, alternately ingoing and outgoing, i.e. trajectories that tend to $\infty$ in finite positive or negative time, with asymptotical directions $\arg{z}=\frac{2\pi j}{k}$ for $j=0,\dots,2k-1$. The family is invariant under

[TABLE]

and under

[TABLE]

Note that in the particular case when $k$ is odd, (2.2) yields that the family is invariant under

[TABLE]

For nonzero ${\epsilon}$, using the rescaling

[TABLE]

we can reduce the study of the vector field to the case $|\epsilon|=1$. Hence, we now suppose that

[TABLE]

The singular points given by

[TABLE]

are the vertices of a regular $(k+1)$-gon, and their eigenvalues are

[TABLE]

using $\frac{2\pi k\ell}{k+1}\equiv-\frac{2\pi\ell}{k+1},\bmod\,2\pi$. In particular, the circular ordering of the eigenvalues is reversed as compared to that of $z_{\ell}$. Let

[TABLE]

which are ordered as the $z_{\ell}$.

When $k=1$, infinity is a regular point of the vector field on ${\mathbb{C}\mathbb{P}}^{1}$, but we will still speak of separatrices for the incoming and outgoing trajectories through $\infty$. We consider that $\infty$ is not in the phase space and stop the trajectories when they reach infinity. We will still call homoclinic loop a periodic loop on ${\mathbb{C}\mathbb{P}}^{1}$ through $\infty$ even though this denomination is not proper because $\infty$ is not singular.

When ${\epsilon}=0$, the $2k$ separatrices of $\dot{z}=z^{k+1}$ coincide with some trajectories coming from or going to the parabolic singular point [math]: they form alternating and equally spaced straight half lines from [math] to infinity. They are in this case also called the repelling (coming from [math]) or attracting (going to [math]) axes; the positive real axis is a repelling axis. The situation is symmetric by a rotation of order $k$.

When ${\epsilon}\neq 0$, the $k+1$ singular points $z_{\ell}$ fit in this picture in a way that cannot be invariant by the same symmetry. There is a pleasant geometrical way of figuring out the argument of the eigenvalues (from which one can for instance deduce if the singular points are attracting) according to the placement of these points with respect to the axes. Indeed since $P^{\prime}_{\epsilon}(z)=P_{0}^{\prime}(z)=(k+1)z^{k}$ it follows that an eigenvalue is positive and real if and only if the singularity $z_{\ell}$ of $P_{\epsilon}$ is on one of the half lines that were repelling axes for $P_{0}$. It is real negative iff $z_{\ell}$ is on an attracting axis. And it is imaginary iff $z_{\ell}$ is equidistant from a neighborhing pair of repelling/attracting axes of $P_{0}$. See Figure 2.

Generically, for all but finitely many $\theta=\arg{\epsilon}$, the separatrices of $\infty$ land at the singular points. Indeed, if a separatrix does not land, then it has to come back to infinity, thus forming a homoclinic loop $\gamma$: this loop separates a group of singular points $\{z_{\ell}\}_{\ell\in I}$ from the other singular points. It follows that $\sum_{\ell\in I}\kappa_{\ell}\in i{\mathbb{R}}$. This comes from the residue theorem. Indeed, let $T$ be the travel time along $\gamma$ (note that separatrices reach infinity in finite time, because $\infty$ is either a pole (if $k>1$) or a regular point (if $k=1$) of the vector field). Then, by the residue theorem

[TABLE]

Note that the converse does not necessarily hold and it could happen that $\sum_{\ell\in I}\kappa_{\ell}\in i{\mathbb{R}}$ without having the corresponding group of singular points separated by a (homoclinic) loop: this will be clear once we will have determined exactly when homoclinic loops occur in this family, see Theorem 2.5.

Let us call singular-gon the regular $(k+1)$-gon formed by the singularities of the vector field. The period of a singular point $z_{\ell}$ is $\mu_{\ell}$ defined by

[TABLE]

If $z_{0},\dots,z_{k}$ is the order of the vertices of the singular-gon when turning in the positive direction, then $\mu_{0},\dots\mu_{k}$ is the cyclic order of the arguments of the periods when turning in the positive direction too.

We define the period-gon, this time not using $\mu_{\ell}$ as vertices but as edge vectors, and with a minus sign. To be more precise, let us use $\frac{1}{2}+{\mathbb{Z}}/(k+1){\mathbb{Z}}$ as an indexing set for the vertices: there is a unique (convex) regular $(k+1)$-gon with vertices $v_{\ell+\frac{1}{2}}$, centered at the origin and such that

[TABLE]

Notice the minus sign in front of $\mu_{\ell}$.

The singular-gon rotates at speed $\frac{1}{k+1}$ when $\theta$ moves at speed $1$, while the periods, and thus the period-gon, turn at negative speed $-\frac{k}{k+1}$. When $\theta$ makes a full turn, the singular-gon turns by $2\pi/(k+1)$, and since it is invariant by this rotation, it returns as a set to its initial position with the indexing of its vertices having shifted by $-1$. Meanwhile, the period-gon turns by $-2\pi k/(k+1)\equiv 2\pi/(k+1)(\bmod 2\pi)$. So it is also unchanged as a set and undergoes exactly the same shift on indices as the period-gon. This is coherent with the fact that the cyclic order of the singularities and the cyclic order of the argument of their respective periods are the same.

Definition 2.1**.**

A polynomial vector field $\dot{z}=P(z)$ is generic if its singular points are simple and there are no homoclinic loops.

Douady and Sentenac [DS] classified the generic polynomial vector fields of degree $k+1$. Up to a rotation of order dividing $k$, they are completely characterized by an analytic invariant given by $k$ complex numbers with positive imaginary parts and a combinatorial invariant. We describe these now.

Definition 2.2**.**

Let $\dot{z}=P(z)$ be a unitary generic polynomial vector field in the sense of Definition 2.1. Then the $2k$ separatrices of $\infty$ land at singular points.

1. (1)

The Douady-Sentenac combinatorial invariant is the union of (see Figure 3(a)):

• •

the tree graph, an embedded graph in the oriented plane (i.e. up to an orientation preserving homeomorphism) defined as follows: for the vertices, we take the set of singular points and, whenever there is trajectory joining two singular points, we choose one of them as an edge.

• •

the information on how one separatrix of $\infty$ is attached to it (then the attachment of all other separatrices is determined). 2. (2)

The Douady-Sentenac analytic invariant is the $k$-tuple of “travel times” $\chi_{1},\dots,\chi_{k}$ along $k$ curves (the dotted curves in Figure 3(b)) disjoint from the separatrices, each one going from $\infty$ to $\infty$ and cutting one edge of the Douady-Sentenac combinatorial invariant: the direction of the trajectory is chosen so that the $\alpha_{j}$ have positive imaginary part.

In the following theorem will appear a special kind of linear ordering (i.e. a total order) on the set of singularities. A total order on a finite set $E$ of cardinal $m$ is equivalent to the data of a uniquely defined order preserving bijection $L:\{0,\ldots,m-1\}\to E$. Similarly, a circular order is defined by a bijection $L$ from the set ${\mathbb{U}}_{m}$ of $m$-th roots of unity to $E$, but this time it is unique only up to composition of $L$ by a rotation ${\mathbb{U}}_{m}\to{\mathbb{U}}_{m}$. For any ${\epsilon}\neq 0$, the set of $(k+1)$-st roots of ${\epsilon}$ is naturally circularly ordered.

Definition 2.3**.**

Let us call zig-zag ordering on the set ${\mathbb{U}}_{m}$ of $m$-th roots of unity, any linear ordering $L:\{0,\ldots,m-1\}\to{\mathbb{U}}_{m}$ for which there exists a rotation $R_{\theta}:z\mapsto e^{i\theta}z$ such that $L$, followed by $R_{\theta}$, followed by the projection $z\mapsto\operatorname{Re}z$ to the real line is strictly increasing. For a set $E$ circularly ordered via a bijection ${\mathbb{U}}_{m}\to E$, a zig-zag ordering is defined as a linear ordering that induces a zig-zag ordering of ${\mathbb{U}}_{m}$.

Consider $\theta$ (non unique) as above. If $m\geq 3$, then no points in $R_{\theta}({\mathbb{U}}_{m})$ can be on the real line. The real line cuts the unit circle into two halves. The zig-zag ordering of $R_{\theta}({\mathbb{U}}_{m})$ alternates between these two halves, because the points in $\cal U_{m}$ cuts the circle into arcs of equal angular span. Advancing two steps along the zig-zag order, one follows the points of the lower half circle in the positive circular orientation, and the points it the upper half are followed in the negative circular orientation.111There are arithmetical characterization of zig-zag orderings. For instance on ${\mathbb{Z}}/m{\mathbb{Z}}$ these orderings take the form $n\mapsto n_{0}+s\frac{1-(-1)^{n}(2n+1)}{4}(\bmod m)$ with $s=1$ or $s=-1$ and $n_{0}\in{\mathbb{Z}}/m{\mathbb{Z}}$. There are also topological characterizations: for instance “for any adjacent pair of points for the linear order, cut the circle at these points to form two arcs. Then the successors must be all on one side of and the predecessors all on the other side”. And many others.

Definition 2.4**.**

Given a finite tree graph, let us call it a trunk if it has no side branches (no vertex of valence $>2$), i.e. its topological realizations are homeomorphic to a segment.

Theorem 2.5**.**

We consider the polynomial vector field $\dot{z}=z^{k+1}-{\epsilon}$ with $\epsilon=e^{i\theta}$.

1. (1)

The Douady-Sentenac combinatorial invariant is a trunk making a zig-zag through the singular-gon: i.e. there is a zig-zag ordering of the singular points (vertices), which we also call chain of the singular points, such that there is exactly one edge from vertex numbered $n$ to vertex number $n+1$, and no other edge. 2. (2)

The homoclinic loop bifurcations occur precisely at all values of $\epsilon$ such that the imaginary axis is a symmetry axis of the period-gon.222 Recall that the period-gon was defined as the polygon centered on [math] and whose edges are given by the negated periods $-\mu_{\ell}$. This is also equivent to the singular-gon being symmetric with respect to one of the dotted axes in Figure 2. There are $2k$ such bifurcations occuring for

[TABLE]

for $j=0,\dots,2k-1$. Across each bifurcation, the Douady-Sentenac combinatorial invariant is modified as follows: consider the linear ordering of the segments and erase every other segments, i.e. preserved segments and destroyed segments alternate. There are exactly two ways of doing this, each choice corresponds to one end of the interval of structural stability in $\theta$-space with the given invariant. Then exchange the order of the two elements of each remaining segment and keep the order of the segments. This reattaches the whole chain and gives the DS invariant on the other side of the bifurcation (see Figures 4 and 5).

Proof.

The proof of (1) and (2) follows from a description of the vector field in the rectifying time coordinate

[TABLE]

This is well defined on ${\mathbb{C}}$ minus radial cuts from the singular points to $\infty$ (see Figure 6). In this coordinate, the vector field simply becomes $\dot{t}=1$, and the trajectories are horizontal lines. The vertices of the period-gon are images of $\infty$. The singular points are sent to $\infty$. The image of ${\mathbb{C}}$ minus the radial cuts is a star-shaped domain obtained by taking the filled period-gon (i.e. the convex hull of the vertices) and gluing to it $k+1$ branches: straight strips, half-infinite, perpendicular to the sides and of width given by the periods $\mu_{\ell}$. To see this, first note that this is the case for ${\epsilon}\in{\mathbb{R}}$. Then, since a change of coordinate and (complex!) time

[TABLE]

sends $\dot{z}=z^{k+1}-e^{i\theta}$ to $\dot{z}=z^{k+1}-1$, it is obvious that the descriptions for different $\theta$ just correspond to rotating the star-shaped domain.

This description is very useful. Visually, a homoclinic loop occurs when two vertices of the period-gon lie on a horizontal line. It is then clear that this can only occur when the vertical axis is a symmetry of the period-gon, in which case homoclinic bifurcation(s) occur simultaneously separating the singular points in groups of one or two points. Consider two singularities $a$, $b$ of the vector field. Each of them corresponds to a side $A$, $B$ of the period-gon. One sees that there are trajectories between $a$ and $b$ if and only if there is a non-empty intersection of the interior of the orthogonal projections of $A$ and $B$ to the imaginary axis. In the absence of homoclinic loops, this corresponds to a zig-zag ordering.

Moreover, from the rotational movement of the period-gon, we deduce the monotonic behavior of the separatrices of $\infty$ very close to the bifucations: in the $t$-coordinate, the attracting separatrices are the horizontal lines that end on a vertex of the period-gon situated on the right of the imaginary axis, and they all move in the same direction, up or down, when the argument of the parameter $\theta$ changes, while the repelling separatrices come from the vertices of the period-gon situated on the left and they all move in the other direction, thus leading to the announced change in the Douady-Sentenac combinatorial invariant. ∎

Corollary 2.6**.**

Homoclinic loops are arranged as follows.

1. (1)

For $k=2m$ even (i.e. an odd number of singularities), bifurcations of homoclinic loops occur precisely when one singular point $z_{\ell}$ has a eigenvalue in $i{\mathbb{R}}$. Then, there are $m$ simultaneous homoclinic loops separating ${\mathbb{C}}$ into $m+1$ regions containing respectively $z_{\ell}$ and the $m$ pairs of points $\{z_{\ell-j},z_{\ell+j}\}$, $j=1,\dots m$. When ${\epsilon}$ rotates in the positive direction, the next group of simultaneous bifurcations to occur is the one in which $z_{\ell-m}$ is isolated by a homoclinic loop. These occur for $\arg({\epsilon})=\frac{\pi}{2k}+\frac{n\pi}{k}$ for $n=0,\dots,2k-1$. (See Figure 8(c).) 2. (2)

For $k=2m+1$ odd (i.e. an even number of singularities), there are two kinds of possible arrangements:

• •

In the first type, two opposite singular points $z_{\ell}$ and $z_{\ell+m+1}$ (indices are modulo $k+1$) have eigenvalues in $i{\mathbb{R}}$ and there are $m+1$ simultaneous homoclinic loops separating the singular points into the $m+2$ groups $\{z_{\ell}\}$, $\{z_{\ell-1},z_{\ell+1}\}$, …, $\{z_{\ell-m},z_{\ell+m}\}$, $\{z_{\ell+m+1}\}$. These occur when $\arg({\epsilon})=\frac{\pi}{k}+\frac{2n\pi}{k}$ for $n=0,\dots,k-1$. (See Figure 9(e).)

• •

In the second type, there are $m$ homoclinic loops separating the points into $m+1$ pairs $\{z_{\ell},z_{\ell+1}\}$, $\{z_{\ell-1},z_{\ell+2}\}$ …, $\{z_{\ell-m},z_{\ell+m+1}\}$. This case of course only occurs for $k>1$, when $\arg({\epsilon})=\frac{2n\pi}{k}$ for $n=0,\dots,k-1$. (See Figure 9(b).)

When ${\epsilon}$ rotates uniformy in the positive direction, the two types of bifurcations alternate. The first type with an end group $\{z_{\ell}\}$ is followed by one of the second type with end group $\{z_{\ell},z_{\ell+1}\}$, and then by one of the first type with end group $\{z_{\ell+1}\}$, etc.

Proof.

1. (1)

Recall that the period-gon has edges given by the vectors $\mu_{\ell}=2\pi i/P^{\prime}(z_{\ell})$ and that $P^{\prime}(z_{\ell})=(k+1)e^{i\frac{k(\theta+2\pi\ell)}{k+1}}$. By the previous theorem, there is a bifurcation if and only if the imaginary axis is a symmetry axis of the period-gon. Since it has an odd number $k+1$ of sides, this happens if and only if there is a horizontal edge $\mu_{\ell}\in{\mathbb{R}}$, if and only if $P^{\prime}(z_{\ell})\in i{\mathbb{R}}$. Since $(k,k+1)=1$, then all $\beta_{j}=1/P^{\prime}(z_{j})$ are the vertices of a regular $(k+1)$-gon centered at the origin. When ${\epsilon}$ rotates uniformly in the positive direction, the $\beta_{\ell}$ rotate $k$ times faster in the negative direction. The next $\beta_{j}$ to reach $i{\mathbb{R}}$ is $\beta_{\ell+m+1}$, since $\arg(\beta_{\ell+m+1})-\arg(\beta_{\ell})=-(2m+1)\pi+\frac{\pi}{2m+1}$. 2. (2)

Again, this comes from the symmetry of the period-gon and its rotational movement.

∎

Example 2.7**.**

Let us consider the family $\dot{z}=z^{6}-{\epsilon}$.

Figure 6 represents a generic situation. It shows a branch of $t=\int\frac{dz}{z^{6}-{\epsilon}}$ defined on the following simply connected set: the complex plane minus the 6 straight slits depicted in dashed lines, radiating from the roots $z_{i}$ of $z^{6}-{\epsilon}$ and going to $\infty$. The branch is injective and its image is a star shaped domain, depicted on the right. The two parallel sides of a branch of the star on the right are the images of the two sides of a corresponding slit. To recover the complex plane (minus singularities), one has to roll and glue the branches, by glueing points separated by a period. A neighborhood of infinity on the left is cut in 6 sectors that are mapped on the right to neighborhoods of the 6 corners. The separatrices land at the singular points: indeed, no two corners of the star are at the same vertical height.

Because of the symmetries, to highlight the bifurcations it is sufficient to consider $\theta\in\left[0,\frac{\pi}{5}\right]$. The phase portraits appear in Figure 9.

2.2. The phase portrait on a disk $B(0,r)$

When considering the phase portrait on ${\mathbb{C}\mathbb{P}}^{1}$ for nonzero ${\epsilon}$ then, except when ${\epsilon}$ is on the $2k$ half-lines $e^{i\theta_{j}}{\mathbb{R}}^{+}$, where $\theta_{j}$ is defined in Equation 2.7, all separatrices of $\infty$ end at singular points, and there exist a chain of trajectories between the singular points, forming a trunk. The ${\epsilon}$-space is then decomposed as the union of:

• •

$2k$ open sectors $V_{j}=\{{\epsilon}\>:\>\arg{\epsilon}\in(\theta_{j},\theta_{j+1})\}$ where the vector field is structurally stable;

• •

the bifurcation locus of real codimension 1, which is composed of the $2k$ half-lines $e^{i\theta_{j}}{\mathbb{R}}^{+}$ where homoclinic connections occur;

• •

the limit point ${\epsilon}=0$: there, the homoclinic connections become heteroclinic connections passing through the singular point $z=0$.

What does remains of this when we consider the restriction of the phase portrait to a disk $B(0,r)$? We only study the situation where ${\epsilon}$ belongs to a disk $B(0,\rho)$ sufficiently small so that the singular points all remain in $B(0,r)$ and the separatrices of $\infty$ all enter $B(0,r)$. First, the structurally stable systems are still dense, as we now explain. Note that the separatrices of $\infty$ have no more intrinsic meaning.

The notion of separatrix is replaced by trajectories that hit the boundary of the disk $B(0,r)$, in the future or in the past. They thus come in $2k$ connected “bunches” instead of being in finite number $2k$. The notion of homoclinic loop is replaced by trajectories that cut the disk into two pieces, that we call separating trajectories. These notions are illustrated in Figure 10. Whenever separating trajectories occur, we lose the chain of trajectories between the singular points that was used to define the Douady-Sentenac invariant.

In $t$-coordinate, the boundary of $B(0,r)$ corresponds to nearly circular curves around the vertices of the star figure, which we call eyelets. By choosing $\rho$ small enough, we can assume they are as close to circles as wished, in the $C^{\infty}$ topology. Note that as ${\epsilon}\to 0$, their size remains bounded (of the order of $R_{0}=\frac{1}{kr^{k}}$), whereas their mutual distances tend to infinity. In these coordinates, a “separatrix”  corresponds to a horizontal line that hits one of the eyelets in the future (right) or in the past (left), and a separating curve to one that hits the eyelets in both directions.

The only situations that are not structurally stable are when there exists at least one horizontal line that is tangent to two eyelets. Note that two kinds of tangency can occur: the two eyelets can be on the same side or on opposite sides of the tangent line. The bifurcations from existence to non-existence of separating trajectories occur when they are on opposite sides.

Remark 2.8**.**

Denote the polar form of the parameter as ${\epsilon}=|{\epsilon}|e^{i\theta}$. If we fix $|{\epsilon}|$ and let $\theta$ vary then, because we are studying a very special situation, the set of eyelets undergoes a rigid rotation, in the opposite direction, by $k/(k+1)$ times the variation of $\theta$ (recall that the system is invariant under $(z,t,{\epsilon})\mapsto(e^{i\theta}z,e^{-ik\theta}t,e^{i(k+1)\theta}{\epsilon})$). Hence finding values of $\theta$ for which a horizontal line has a double tangency with two eyelets is equivalent to taking ${\epsilon}$ real (or any preferred argument for ${\epsilon}$), finding the non-necessarily horizontal lines that are tangent to two eyelets and determining $\theta$ from the direction of those lines.

Theorem 2.9**.**

Apart from ${\epsilon}=0$, the parameters values for which the system is not structurally stable form a finite number of real-analytic curves from the origin to the boundary of the disk, inside the pointed disk $B(0,\rho)^{*}$. In particular structural stability is dense.

The bifurcation curves are disjoint for $\rho$ sufficiently small. They are organized in groups that tend to ${\epsilon}=0$ along the $2k$ directions $\theta_{j}$ of Equation 2.7 (the $\theta_{j}$ are the directions for which the period-gon has a vertical axis of symmetry). Each group contains at least three curves, one of which is a straight ray. The other ones come in pairs on each side of the ray, with a tangency at ${\epsilon}=0$ of order $2-1/(k+1)$. See Figure 14 for an illustration.

Proof of Theorem 2.9. We will distinguish two cases:

1. (1)

The period-gon is symmetric with respect to the vertical axis. 2. (2)

It is not.

The eyelets tend, when ${\epsilon}\to 0$, to circular arcs of radius

[TABLE]

centered on the vertices of the period-gon. The direction of the tangent is monotonous on each of these circular arcs. At their endpoints, the tangent is orthogonal to the strip boundary they touch. Also, the set of eyelets share the same isometry group as the period-gon. As a consequence, in case (1), several double tangencies occur at the same time (see Figure 12). These bifurcations are very minor: on both sides there exist separating trajectories.

We now assume that we are in case (2). We start with rough estimates. Recall that the period-gon rotates regularly as ${\epsilon}=|{\epsilon}|e^{i\theta}$ rotates around [math]: in polar form, its vertices are

[TABLE]

indexed by $m\in{\mathbb{Z}}/(k+1){\mathbb{Z}}$, for some $C_{k}>0$ that depends only on $k$: more precisely $C_{k}=\frac{\pi}{k+1}\left/\sin\frac{\pi}{k+1}\right.$ but we will not need this value.

• •

A double tangency can only occur if $\arg{\epsilon}$ is close to some $\theta_{j}$. Indeed, the diameter of the eyelets is bounded, so, in case of a double tangency, the centers located at two vertices of the period-gon must have imaginary parts that differ by a bounded amount whereas their distance to [math] tends to $\infty$ as ${\epsilon}\to 0$.

• •

By case (1), a double tangency does occurs when $\theta=\theta_{j}$.

• •

Other double tangencies have to occur near $\theta_{j}$: for each pair of vertices of the period-gon that are at the same height when $\theta=\theta_{j}$, consider the two eyelets centered on each and let $\theta$ vary while fixing $|{\epsilon}|$. By the intermediate value theorem, when $\theta$ varies away from $\theta_{j}$ in one direction, a double tangency must occur between the top most point of one eyelet and the bottom most of the other one.

Another way of considering the situation is via Remark 2.8: fix $\theta=\theta_{j}$ for convenience; then there are exactly two non-horizontal curves that are tangent to the two eyelets, one with positive slope and one the opposite slope (because the vertical axis is a symmetry of the set of eyelets when $\theta=\theta_{j}$), see Figure 13. In particular there are exactly three value of $\theta$ near $\theta_{j}$ for which this particular set of two eyelets have a horizontal line tangent to both of them: $\theta_{j}$ (this is Case (1)), $\theta_{j}+\alpha$, $\theta_{j}-\alpha$, for some $\alpha>0$ close to [math]. We will justify later in this proof that $\alpha$ depends analytically on $|\epsilon|$.

Let us first estimate the order of magnitude of $\alpha$. Consider thus a situation as above, call $v_{m}$ and $v_{m^{\prime}}$ the vertices at which the two eyelets are centered, and let $a$ denote the tangency point close to $v_{m}$. The triangle whose vertices are $\frac{v_{m}+v_{m^{\prime}}}{2}$ , $v_{m}$ and $a$ is nearly rectangle in $a$: the angle tends to $\pi/2$ when $|{\epsilon}|\to 0$. Moreover the side length $|v_{m}-a|$ tends to $R_{0}$ when $|{\epsilon}|$ tends to [math] whereas the other two lengthes tend to $+\infty$. It follows that

[TABLE]

where $\beta$ denotes the angle of the triangle at its vertex $\frac{v_{m}+v_{m^{\prime}}}{2}$. From (2.10) one computes

[TABLE]

and, given the relation between $\theta$ and the rotation to be applied on the star figure: $\alpha=\frac{k+1}{k}\beta$, so

[TABLE]

for some constant $C(k,|m-m^{\prime}|)>0$. Now let ${\epsilon}e^{-i\theta_{j}}=x+iy$. Then $|{\epsilon}|\sim x$ thus Equation 2.11 occurs approximately when $y/x=\pm C^{\prime}(k,|m-m^{\prime}|)x^{\frac{k}{k+1}}$, for some constant $C^{\prime}(k,|m-m^{\prime}|)>0$. Whence the tangency order

[TABLE]

for the corresponding bifurcation set.

Note that $C^{\prime}(k,|m_{1}-m_{1}^{\prime}|)=C^{\prime}(k,|m_{2}-m_{2}^{\prime}|)\iff|v_{m_{1}}-v_{m_{1}^{\prime}}|=|v_{m_{2}}-v_{m_{2}^{\prime}}|$. Hence, any two types of double tangency near $\theta_{j}$ for which the distances between the two vertices differ (and hence the corresponding constants $C^{\prime}(k,|m-m^{\prime}|)$ differ), correspond to disjoint sets in parameter space near ${\epsilon}=0$.

Recall that for $\theta$ close to $\theta_{j}$, the pairs $(m,m^{\prime})$ for which a double tangency occurs are exactly the pairs for which the segment $[v_{m},v_{m^{\prime}}|$ is horizontal when $\theta=\theta_{j}$. The corresponding lengthes are all distinct if $k+1$ is odd. If $k+1$ is even, there may be pairs that have the same length, depending on the situation (the precise study is left to the reader, see also Figure 14). If there are two distinct horizontal segments that have the same length, they must be image one of the other by the reflection with respect to the real axis, and hence by the rotation of angle $\pi$. For $\theta$ close to $\theta_{j}$, the set of eyelets is still symmetric by the rotation of angle $\pi$ (see symmetry (2.4)), thus the two double tangencies corresponding to the two chosen segments occur for the same values of the parameters.

Analyticity: In the study above, we associated combinatorial data to a double tangency: the index $j$ of the angle $\theta_{j}$ to which $\theta$ is close, the period-gon vertex indices $\{m,m^{\prime}\}$ and the sign of $\theta-\theta_{j}$. We have also seen that the corresponding bifurcation set has precisely one intersection point (i.e. there is one value of $\theta$) with each circle $|\epsilon|=c$ for $c>0$ small enough. Let us prove that the dependence is analytic for $|\epsilon|>0$ small.

Recall that we called tangency points the points in $t$-coordinate where the boundary is tangent to the trajectories, i.e. has a horizontal tangent. The tangency points correspond in $z$-coordinate to the points on the boundary of $B(0,r)$ where the vector field is tangent to $\partial B(0,r)$. From the simple form of the vector field: $\dot{z}=z^{k+1}-{\epsilon}$, there are exactly $2k$ points of tangency of the vector field on $\partial B(0,r)$ that depend ${\mathbb{R}}$-analytically on ${\epsilon}$ and are close to be regularly spaced when ${\epsilon}$ is small. Indeed the equation on $z=re^{i\alpha}$ takes the form $\dot{z}\cdot z=0$ (where the dot denotes the Euclidean dot product of vectors) which, for

[TABLE]

is equivalent to $E(x,y,\alpha)=0$, where

[TABLE]

For $(x,y)=0$ there are exactly $2k$ solutions: $\alpha=\alpha_{j}=(\pi/2+j\pi)/k$. Let us apply the implicit function theorem, by checking that the partial derivative $\partial E/\partial\alpha$ does not vanish at these $k$ points: a computation gives $\partial E/\partial\alpha(0,0,\alpha_{j})=(-1)^{j+1}k$. Hence, for $\epsilon$ small, there are $2k$ solutions for $\alpha$, they are close to the $\alpha_{j}$ and they depend ${\mathbb{R}}$-analytically on $x$ and $y$.

There is a branch of $\int dz/(z^{k+1}-{\epsilon})$ defined on $|z|>\sqrt[k+1]{|{\epsilon}|}$: indeed $1/(z^{k+1}-{\epsilon})$ can be expanded as $z^{-k-1}\sum_{n\in{\mathbb{N}}}({\epsilon}/z^{k+1})^{n}$ which can be integrated as

[TABLE]

with $a_{n}=(n+1)(k+1)-1$. This branch has no monodromy when $z$ winds around the set of roots.

Then the image in $t$-coordinate of a point $z\in\partial B(0,r)$ is of the form $s+\xi(z)$ where $s$ denotes the appropriate vertex of the period-gon associated to the sector in which $z$ lies. Recall that these vertices are of the form $A\delta^{-k}$ where $A\neq 0$ is a complex number and $\delta$ ranges over the set of $(k+1)$-th roots of ${\epsilon}$. So the $2k$ points of tangency $t_{j}$ in $t$-coordinate are of the form

[TABLE]

for some ${\mathbb{R}}$-analytic function $h(x,y)$ defined near $(0,0)$ and an appropriate choice of $\delta=\sqrt[k+1]{\epsilon}$ that depends on $m$. This choice jumps when the tangency point crosses one of the slits in the definition of the $t$-coordinate.

We conclude by appling the implicit function theorem to the equation $\operatorname{Im}t_{m}-\operatorname{Im}t_{m^{\prime}}=0$ with the expression found above for $t_{m}$: we get that for $|{\epsilon}|=\rho^{\prime}$ small enough, there exists a unique value of $\arg{\epsilon}=\beta(\rho^{\prime})$ for which $\operatorname{Im}(t_{m}-t_{m}^{\prime})=0$, and the $\beta$ is ${\mathbb{R}}$-analytic in a neighborhood of [math] (i.e. it extends across [math]). This gives another proof, by the isolated zero theorem, that close enough to [math] the different bifurcation curves in parameter space do not cross except at [math]. $\Box$

2.3. Concluding remarks

The regions of structural stability in Theorem 2.9 are called generalized sectors. They fall hence into two classes: wide ones, that contain a sector with positive angle, and thin ones, that do not. The first class corresponds to those parameters for which there exists a trunk of trajectories joining all the singular points. Inside the small generalized sectors, the trunk does not exist anymore. The $2k$ wide generalized sectors are what remains from the open sectors $V_{j}$ we obtained at the beginning of Section 2.2 when we were considering the phase portrait on the whole of ${\mathbb{C}\mathbb{P}}^{1}$.

The complement of the $V_{j}$ form $2k$ thin sets $W_{j}$, each of which decomposes further into several thin sectors of stability separated by some bifurcation curves. Among those curves, there is a very particular one, an accident due to the high symmetry of the system we considered: it is a straight line and on it, the period-gon is symmetric with respect to the imaginary axis, and two double tangencies occurs for each pair of vertices with the same height. The bifurcation occuring when crossing these straight bifurcation curves are of a mild type: indeed there exist persistent separating trajectories, and the (broken) Douady-Sentenac invariant remains identical.

In a coming work we plan to study one-parameter unfoldings of germs of parabolic diffeomorphisms. The formal normal form of a parabolic diffeomorphism is the time-one map of a vector field. Hence, this partition of $B(0,\rho)$ into the union of the wide generalized sectors, where we have structurally stable behaviour, alternating with their thin complement $W_{j}$ will be relevant for this study: we expect all the bifurcations of parabolic diffeomorphisms to occur for parameter values in regions corresponding to the $W_{j}$.

3. Generic one-parameter unfoldings of vector fields

Notation 3.1**.**

In all this section $*$ denotes a nonzero constant.

Consider a holomorphic vector field in one complex dimension $\dot{z}=\omega_{0}(z)=*z^{k+1}+O(z^{k+2})$, with a singular point of codimension $k\geq 1$: this means that $\omega_{0}$ has a root of multiplicity $k+1$ at the origin. We call this a parabolic singularity.

Definition 3.2**.**

A germ of one-parameter analytic family of vector fields $\omega_{\epsilon}$ unfolding a parabolic germ $\omega_{0}(z)$ is generic if

[TABLE]

for333Of course it is enough to check the condition at $\epsilon=0$. small ${\epsilon}$.

We are interested in the classification of germs of generic families of vector fields unfolding a parabolic singularity.

Definition 3.3**.**

Two generic germs $\omega_{\epsilon}$ and $\widetilde{\omega}_{\tilde{{\epsilon}}}$ are conjugate if there exists $r,\rho>0$ and an analytic diffeomorphism $\Psi=(\psi,h):B(0,r)\times B(0,\rho)\rightarrow{\mathbb{C}}^{2}$ fibered in ${\epsilon}$ (i.e. $h$ depends only on $\epsilon$), such that $\Psi(0,0)=(0,0)$ and for all $\epsilon$ in $B(0,\rho)$, $\psi(\cdot,{\epsilon})$ is a conjugacy between $\omega_{\epsilon}$ and $\widetilde{\omega}_{\tilde{{\epsilon}}}$ over $B(0,r)$. The map $\Psi$ is called a change of variable and parameter. If $h(z,\epsilon)=\epsilon$, then $\Psi$ is said to preserve the parameter.

We will give a complete classification for the two equivalence relations of Definition 3.3 and identify the modulus set. As a preparatory step, we give a classification under conjugacies that preserve the parameter. It is interesting in itself, but also will be used in the general classification as follows: we will prepare the germs of families by changing the parameter to a canonical parameter; then two prepared germs of families are conjugate if and only if there exists a conjugacy preserving the parameter.

3.1. Principal parts

Consider a family $\omega_{\epsilon}$ that is generic as per Definition 3.2. This condition can be written as

[TABLE]

with $A,B\in{\mathbb{C}}^{*}$, which we call the principal part. In other words its Newton polygon is:

z$${\epsilon}

Knowledge of Newton’s polygon is not necessary to read this article: it is an aid for understanding but we will not use it directly.

Repelling axes of $\omega_{0}$ are defined as the set of complex numbers $z$ such that $Bz^{k+1}$ and $z$ have the same argument. They are the set of points for which, at first order, the vector field points away from the origin. They form $k$ half lines and should be thought of living in the tangent space to ${\mathbb{C}}$ at the origin. See Figure 15.

Lemma 3.4**.**

In a neighborhood of $(z,{\epsilon})=(0,0)$, for ${\epsilon}\neq 0$ there are exactly $k+1$ singularities of $\omega_{\epsilon}$, each of multiplicity one, they tend as a whole to [math] as ${\epsilon}\to 0$, and more precisely they are asymptotically located at the vertices $z$ of the regular $(k+1)$-gon defined by $Bz^{k+1}-A{\epsilon}=0$. We denote this by $Bz^{k+1}\sim A{\epsilon}$, the precise meaning of which is: for all $\eta>0$, there exists $\eta^{\prime}>0$ such that if $0<|{\epsilon}|<\eta^{\prime}$, then there exists a bijection between the set of singularities $z$ of $\omega_{\epsilon}$ and the set of solutions $u$ of $Bu^{k+1}-A{\epsilon}=0$, such that for corresponding pairs $(z,u)$ we have $|u-z|\leq\eta|u|$.

Proof.

By Hurwitz’s theorem, in a neighborhood of $(z,{\epsilon})=(0,0)$ there must be exactly $k+1$ singularities counted with multiplicity, and they must tend to [math]. The asymptotic analysis will show that there are at least $k+1$ distinct singularities so they must be of multiplicity one. The asymptotic analysis can be realized with an appropriate change of variable: $z=\delta w$ and ${\epsilon}=\delta^{k+1}$ with $\delta$ a complex number that will tend to [math]. Then $\omega_{\epsilon}(z)=\delta^{k+1}(Bw^{k+1}-A)+\cal O(\delta^{k+2}w^{k+2},\delta^{k+2},\delta^{2k+2})$ so $\omega_{\epsilon}(z)/\delta^{k+1}$ tends to $(Bw^{k+1}-A)$. Then we can apply Hurwitz’s theorem here too. Interpreting the result in the original coordinates gives the result. ∎

A stronger version is proved later in Corollary 3.6.

The position of a singularity with respect to the repelling axes gives us an information on its eigenvalue. Indeed $\omega^{\prime}_{\epsilon}(z)=(k+1)Bz^{k}+\cal O(z^{k+1},{\epsilon})$ so if $z$ is a singularity of $\omega_{\epsilon}$, from $Bz^{k+1}\sim A{\epsilon}$ we get $\omega^{\prime}_{\epsilon}(z)\sim(k+1)Bz^{k}$. On a repelling axis, the quantity $(k+1)Bz^{k}$ is real positive. It is real negative on the attracting axes, which are the $k$ axes equidistant to the repelling axes. And it is imaginary on the $2k$ axes equidistant from the real and attracting axes. In short: singularities closer to the repelling axes will have a tendency to be repelling and singularities close to the attracting axes to be attracting. See Figure 16.

3.1.1. Effect of a change of variable on the principal parts.

Recall that the principal part is the term $Bz^{k+1}-A{\epsilon}$ in

[TABLE]

Denote $(z,{\epsilon})=\Psi(\tilde{z},\tilde{\epsilon})$ a change of variable. The first remark is that repelling axes of the original family are obtained by applying the linear part $\tilde{z}\mapsto\beta\tilde{z}$ of the change of variable to the repelling axes of the conjugate family, with $\beta=\frac{\partial\Psi}{\partial\tilde{z}}(0,0)$.

The principal part in the new coordinate is of the form $\tilde{B}\tilde{z}^{k+1}-\tilde{A}\tilde{\epsilon}$ and the coefficients $\tilde{A}$ and $\tilde{B}$ depends only on the linear part of the change of variable and parameter, and more precisely only on the diagonal part: let

[TABLE]

denote the differential of $\Psi$ at $(z,{\epsilon})=(0,0)$. Then

[TABLE]

We have two degrees of freedom to simplify the principal part: in particular there exists precisely $k$ pairs $(\beta,\alpha)$ of non-zero complex numbers such that

[TABLE]

and they are of the form $(\beta,\beta A^{-1})$ with $\beta$ one of the $k$-th roots of $B^{-1}$. There is a natural bijective correspondence between these pairs and the choice of a repelling axis of $\omega_{0}$: indeed letting $\arg\tilde{B}=0$ is equivalent to choosing a rotation in the $z$-plane such that one repelling axis is ${\mathbb{R}}_{+}$, whereas getting $|\tilde{B}|=1$ corresponds to a unique choice of rescaling by a real factor. Once $\tilde{B}=1$ is obtained, there is only one possible value of $\alpha$ to get $\tilde{A}=1$.

A parameter fixing conjugacy has in particular $\alpha=1$. Hence in this case there is only one degree of freedom to simplify the principal part. Our choice here is to let

[TABLE]

so that the principal part is $\tilde{B}\times(z^{k+1}-{\epsilon})$, which will be natural once we have seen Corollary 3.6. There are exactly $k+1$ values of $\beta$ such that $\tilde{A}=\tilde{B}$, they are the $(k+1)$-st roots of $A/B$. To give a geometric interpretation of this choice, we need to introduce a notion that may look somewhat artificial: we call explosion axes the $k+1$ asymptotic directions in which the singularities are located when ${\epsilon}\to 0$ along the positive real axis. They are characterized by the equation $Bz^{k+1}/A\in{\mathbb{R}}_{+}$. Then letting $\arg A=\arg B$ is equivalent to choosing a rotation in the $z$-plane so that one of the explosion axes is ${\mathbb{R}}_{+}$. See Figure 17.

We now come back to general conjugacies (i.e. not necessarily fixing the parameter). We just defined explosion axes. What is the action of a conjugacy on the explosion axes? With a pure change of parameter ${\epsilon}=\alpha\tilde{\epsilon}+\cal O({\epsilon}^{2})$, the explosion axes rotate by $\arg(\alpha)/{k+1}$ because the new axes are given by $Bz^{k+1}/A\alpha\in{\mathbb{R}}_{+}$ instead of $Bz^{k+1}/A\in{\mathbb{R}}_{+}$. The general action is the composition, in any order because this commutes, of this effect and of the linear part $\tilde{z}=\beta^{-1}z$ of the change of variable. This second effect is a rotation by $-\arg\beta$. We can thus reinterpret the procedure leading to $\tilde{A}=\tilde{B}=1$ at the level of the axes as follows: choose a repelling axis, rotate the $z$-coordinates so that this axis is ${\mathbb{R}}_{+}$, then there is a unique value of $\arg\alpha$ such that ${\mathbb{R}}_{+}$ is an explosion axis for $\tilde{\epsilon}$. See Figure 18.

3.2. About the position of the singularities

We saw that the singularities are approximately on the vertices of a regular $(k+1)$-gon centered on the origin. In fact we will see in this section that we can make a change of variable placing them exactly on those vertices, for all ${\epsilon}$ sufficiently small.

We first recall a general feature of equations near a multiple root.

Proposition 3.5**.**

Consider an analytic family of equations $u_{\epsilon}(z)=0$ defined in a neighborhood of $(z,\epsilon)=0$ such that $u_{0}(z)$ has a root of order $d$ at the origin. Make the following genericity assumption: $\frac{\partial u}{\partial\epsilon}(0,0)\neq 0$. Then there exists a change of variable $z\mapsto\tilde{z}$ independent of the parameter and a restriction of $u_{\epsilon}$ in a neighborhood of $(0,0)$, such that for all $\epsilon$ the roots of $u_{\epsilon}$ are exactly the solutions of $\tilde{z}^{d}-\epsilon=0$.

Proof.

There are many ways to prove it, one goes as follows. By the implicit function theorem, the zero locus of $u$ can be locally parameterized as $\epsilon=f(z)$ for some holomorphic $f$. Hence

[TABLE]

Since $u_{\epsilon}(z)=*z^{d}+*{\epsilon}(1+O(z,{\epsilon}))+O(z^{d+1})$, it follows by substituting ${\epsilon}=\sum_{n\geq 1}a_{n}z^{n}$ in (3.5) that $f$ has a root of exact order $d$ at the origin. Then $f(z)=z^{d}h(z)=\tilde{z}^{d}$ for $\tilde{z}=g(z)=zh^{1/d}(z)$ analytic, since $h(0)\neq 0$. ∎

In the situation of the proposition above, the function $u_{\epsilon}$ factors by $\tilde{z}^{d}-{\epsilon}$ and the quotient does not vanish near $(0,0)$.

Corollary 3.6**.**

Let $\omega_{\epsilon}$ be a germ of generic one-parameter analytic family of vector fields unfolding a parabolic germ.

1. (1)

Then there exists a local change of variable $z\mapsto\tilde{z}=g(z)$ independent of the parameter bringing the family to the form

[TABLE]

where $v_{\epsilon}(\tilde{z})=K+O(\tilde{z},{\epsilon})$, with $K\neq 0$. 2. (2)

Any other change of coordinate independent of ${\epsilon}$ sending $\omega_{\epsilon}$ to some family of the form (3.6) is given by $z\mapsto\zeta{g}(z)$ for $\zeta$ satisfying $\zeta^{k+1}=1$. In particular, the change of coordinate is completey determined by ${g}^{\prime}(0)$, which can take exactly $k+1$ values.

Proof.

The first part follows from the discussion above applied to $u=\omega$. Let us deal with the second part: The map $z\mapsto\zeta{g}(z)$ for $\zeta^{k+1}=1$ also brings $\omega_{\epsilon}$ to a family of the form (3.6). If $z\mapsto\overline{z}=\overline{g}(z)$ brings $\omega_{\epsilon}$ to some family $\overline{\omega}_{\epsilon}$ of the form (3.6), then $G=\overline{g}\circ g$ sends $\widetilde{\omega}_{\epsilon}$ to $\overline{\omega}_{\epsilon}$. In particular $G$ sends $\{\tilde{z}^{k+1}={\epsilon}\}$ onto $\{\overline{z}^{k+1}={\epsilon}\}$ for all ${\epsilon}$ simultaneously. This is only possible if $G$ is linear of the form $\tilde{z}\mapsto\zeta\tilde{z}$ for $\zeta^{k+1}=1$. ∎

Note that if a family has the form (3.6) then in particular $A=B$ in the notations of Section 3.1.

Remark 3.7**.**

In the form (3.6), ${\mathbb{R}}_{+}$ is one of the explosion axes. The change of variable that brings a general $\omega_{\epsilon}$ to the form (3.6) also brings, at the level of linear parts, one of the explosion axes to ${\mathbb{R}}_{+}$. Hence selecting one of the $k+1$ changes of variables in Corollary 3.6 corresponds to selecting one of the $k+1$ explosion axes.

3.3. About the eigenvalues

The eigenvalue of the vector field $\omega$ at a singularity $z$ is just the complex number $\lambda=\omega^{\prime}(z)$. Its dynamical significance is that the flow $\Phi_{t}$ then fixes $z$ with multiplier $\Phi_{t}^{\prime}(z)=\exp(\lambda t)$.

Eigenvalues are invariant by a change of variable. Are they the only invariants of conjugacy? In general, this is not the case, but we will see that in our situation and for the notion of conjugacy we are interested in, the answer is yes (this is basically Theorem 3.13).

In the case of the family of vector fields $\omega_{\epsilon}$ that we are considering, i.e. satisfying the genericity assumption of Definition 3.2, we get for each $\epsilon\neq 0$ a collection of $k+1$ complex numbers that forms an invariant. Two natural questions are: how do they depend on $\epsilon$ and what are the collections that actually arise?

Definition 3.8**.**

Given a vector field of the form

[TABLE]

with $v_{\epsilon}(0)\neq 0$, we define the natural eigenvalue function $\lambda$ as follows: $\lambda(\delta)$ is the eigenvalue of $\omega_{\epsilon}$ at the singularity $z=\delta$ for ${\epsilon}=\delta^{k+1}$.

Remark 3.9**.**

An elementary computation yields

[TABLE]

The whole collection of eigenvalues of $\omega_{\epsilon}$ is given by the values of $\lambda$ at the $(k+1)$-st roots of ${\epsilon}$. Note here a remarkable fact: the collection is given by a single function $\lambda$.

We now extend the notion of eigenvalue function to the general case using Corollary 3.6:

Definition 3.10**.**

Given a generic vector field $\omega_{\epsilon}$ as per Definition 3.2, consider one of the $k+1$ changes of variables $z\mapsto\tilde{z}$ conjugating the family $\omega_{\epsilon}$ to a family in the form (3.7). Consider then the natural eigenvalue function $\lambda$ of the conjugated family of Definition 3.8: this function is called an eigenvalue function for $\omega_{\epsilon}$.

Corollary 3.11**.**

There are thus $k+1$ eigenvalue functions associated to a generic $\omega_{\epsilon}$ and they are all related by pre-composition with the multiplication by a $(k+1)$-st root of unity. By Remark 3.7, choosing one eigenvalue function corresponds to choosing one explosion axis.

Consider an eigenvalue function $\lambda$ of a generic vector field as we just defined. By invariance of eigenvalues, $\lambda(\delta)$ is the eigenvalue of some singularity of $\omega_{\epsilon}$ when ${\epsilon}=\delta^{k+1}$ and by the second part of Remark 3.9 the set of all eigenvalues of $\omega_{\epsilon}$ is given by the values of $\lambda$ at the $(k+1)$-st roots of ${\epsilon}$.

Proposition 3.12** (form of the eigenvalue functions).**

Let $\omega_{\epsilon}$ be a germ of generic $1$-parameter family of vector fields in the sense of Definition 3.2.

1. (1)

If $\lambda$ is an eigenvalue function for $\omega_{\epsilon}$ then $\lambda$ has a zero of order $k$ at the origin:

[TABLE]

with $\sigma(0)\neq 0$. 2. (2)

Conversely any function $\lambda$ of the form (3.8) is an eigenvalue function of some $\omega_{\epsilon}$.

Proof.

1. (1)

By definition, there exists a coordinate change, independent of the parameter for which $\omega_{\epsilon}$ takes the form $(z^{k+1}-{\epsilon})v_{\epsilon}(z)$ where $v_{\epsilon}(z)$ does not vanish in a neighborhood of $(0,0)$. By Remark 3.9: $\lambda(\delta)=(k+1)\delta^{k}v_{\delta^{k+1}}(\delta)$. This gives the first claim, letting $\sigma(\delta)=v_{\delta^{k+1}}(\delta)$. 2. (2)

For the converse, it is enough to consider $\omega_{\epsilon}:=(z^{k+1}-\epsilon)\sigma(z)$ for some $\sigma$ that does not depend on $\epsilon$. Then $\lambda(\delta)=(k+1)\delta^{k}\sigma(\delta)$.

∎

3.4. Classification under conjugacies that fix the parameter

Theorem 3.13** (Classification without change of parameters).**

Two generic families $\omega_{\epsilon}$ and $\widetilde{\omega}_{\epsilon}$ are conjugate near $(0,0)$ by a change of variable $z\mapsto\tilde{z}=\tilde{z}(z,\epsilon)$ fixing the parameter if and only if for all ${\epsilon}$ near [math], they have the same sets of eigenvalues. This occurs if and only if, given a choice of eigenvalue functions $\lambda$ and $\tilde{\lambda}$ for each family, there exists a $(k+1)$-st root of unity $\zeta$ such that $\tilde{\lambda}(\delta)=\lambda(\zeta\delta)$ for $\delta$ near [math].

We will give a proof in Section 3.6, of geometric nature. Let us comment on some aspects: in the first statement the necessity of the condition follows from the invariance of eigenvalues under a conjugacy of a vector field. For the second statement, we determined that the set of eigenvalues are respectively of the form $\{\lambda(\zeta\delta),\ \zeta^{k+1}=1\}$ and $\{\tilde{\lambda}(\zeta\delta),\ \zeta^{k+1}=1\}$. So in the second statement the sufficiency is obvious and the necessity has several elementary proofs, we present here one:

Proof of the second statement.

first we have that for all $\delta$ small, there exists a $\zeta$ with $\zeta^{k+1}=1$ and such that $\tilde{\lambda}(\delta)=\lambda(\zeta\delta)$. Secondly, for any given $\zeta$, the set of $\delta$ such that $\tilde{\lambda}(\delta)=\lambda(\zeta\delta)$ is closed. A small open disk $D$ near [math] is thus the union of these $k+1$ closed subsets. Hence one of these sets has non-empty interior. By the isolated zeroes theorem, it has to be the whole disk $D$. ∎

The sufficiency in the first statement is the hard part. We will first change variables so that the vector fields have the form $\omega_{\epsilon}(z)=(z^{k+1}-{\epsilon})v_{\epsilon}(z)$ and $\widetilde{\omega}_{\epsilon}(\tilde{z})=(\tilde{z}^{k+1}-{\epsilon})\tilde{v}_{\epsilon}(\tilde{z})$ and so that $\lambda$ and $\tilde{\lambda}$ are the associated natural eigenvalue functions (see Definition 3.8), namely $\omega_{\delta^{k+1}}^{\prime}(\delta)$ and $\widetilde{\omega}_{\delta^{k+1}}^{\prime}(\delta)$. By conjugating the first vector field under $z\mapsto\zeta z$ we are reduced to the case of identical eigenvalue functions. Then the strategy is as follows: The construction of the conjugacy between $\omega_{\epsilon}$ and $\widetilde{\omega}_{{\epsilon}}$ will be geometric for ${\epsilon}\neq 0$ and done in the rectifying coordinate $t$ and $\tilde{t}$ for the vector fields. For each ${\epsilon}$, it will be done over a region containing a fixed disk $B(0,r)$ in $z$-space. The construction will depend analytically on ${\epsilon}$. It will remain to show that the conjugacy is bounded over $B(0,r)$ for all $0<|{\epsilon}|<\rho$, thus allowing to extend it to ${\epsilon}=0$.

Corollary 3.14**.**

For any generic family $\omega_{\epsilon}$, denote by $\lambda(\delta)=(k+1)\delta^{k}\sigma(\delta)$ an associated eigenvalue function. Then there exists a conjugacy preserving the parameter between $\omega_{\epsilon}$ and $\dot{z}=\Omega_{\epsilon}=(z^{k+1}-\epsilon)\sigma(z)$.

Proof.

Let $\epsilon=\delta^{k+1}$. Then $\delta$ is a singularity of $\Omega_{\epsilon}$ and its eigenvalue is $\partial\Omega_{\epsilon}/\partial z=(k+1)\delta^{k}\sigma(\delta)$ at $z=\delta$. So $\omega_{\epsilon}$ and $\Omega_{\epsilon}$ have a common eigenvalue function and we conclude using Theorem 3.13. ∎

Theorem 3.13 also allows to determine the modulus space for the Equivalence Problem 1:

Corollary 3.15**.**

The modulus space of germs of generic parabolic unfoldings of codimension $k$ parabolic vector fields, up to conjugacy fixing the parameter, is naturally in bijection with the set of equivalence classes of holomorphic functions of the form

[TABLE]

$\sigma(0)\neq 0$, up to pre-composition by multiplication with a $(k+1)$-st root of unity.

3.5. Classification upon general conjugacies, canonical parameter

Here we study the effect of a change of parameter $\epsilon$ on the eigenvalue function $\lambda$ and provide an almost unique canonical normal form for $\lambda$. We also introduce a notion of prepared family and draw consequences in terms of classification of $\omega_{\epsilon}$ by conjugacy.

We have two approaches for presenting the classification. The first one ignores the variable $z$ and the other one focuses on the factored form $(z^{k+1}-{\epsilon})v_{\epsilon}(z)$ of Corollary 3.6. We believe both are important, which is why we included each one in the paper.

3.5.1. Method 1

Let $\omega$ be a generic family in the sense of Definition 3.2. Consider a change of variable and parameter $(z,{\epsilon})\to(\tilde{z},\tilde{\epsilon})$, with $\tilde{\epsilon}=\phi({\epsilon})$. Denote $\tilde{\omega}$ the conjugated family. Recall that the collection of eigenvalues of $\omega_{\epsilon}$ is given by the values taken by some (non-unique) function $\lambda$ at the $(k+1)$-st roots of ${\epsilon}$. Similarly, the eigenvalues of $\tilde{\omega}_{\tilde{\epsilon}}$ are the values of some $\tilde{\lambda}$ at the $(k+1)$-th roots of $\tilde{\epsilon}$. Hence we must have: for all ${\epsilon},\tilde{\epsilon}$ that correspond under $\phi$ and small enough,

[TABLE]

Note that a holomorphic change of parameter $\tilde{\epsilon}=\phi({\epsilon})$ can be reflected by a change in $\delta=\sqrt[k+1]{\epsilon}$, which is symmetric in the sense that it commutes with the rotation by $2\pi/(k+1)$: the new variable $\tilde{\delta}=\sqrt[k+1]{\tilde{\epsilon}}$ is a power series $\xi(\delta)$ with all non-zero terms having exponent belonging to $1+(k+1){\mathbb{Z}}$. In other words $\tilde{\delta}=\xi(\delta)=\delta g(\delta^{k+1})$ with $g$ holomorphic and $g(0)\neq 0$. There is a choice here too: there are exactly $k+1$ such power series for a given change of parameter ${\epsilon}\to\tilde{\epsilon}$, they all differ by multiplication by a $(k+1)$-st root of unity. Let us denote $\tilde{\delta}=\xi(\delta)$ such a choice:

[TABLE]

The constant $\zeta$ depends on the choices in $\lambda$, $\tilde{\lambda}$ and $\xi$ and can be made equal to $1$ by changing the choice of any of those three functions. Conversely, any holomorphic function $\xi$ that commutes with the rotation by $2\pi/(k+1)$ and has a non-vanishing derivative at the origin corresponds to a (unique) change of variable $\tilde{\epsilon}=\xi(\sqrt[k+1]{\epsilon})^{k+1}$.

Proposition 3.16**.**

Consider two families of vector fields $\omega$ and $\bar{\omega}$ that are generic in the sense of Definition 3.2 and choose an eigenvalue function $\lambda$ and $\bar{\lambda}$ for each of them. The families are conjugate if and only if there exists a holomorphic map $\xi$ that commutes with the rotation by $2\pi/(k+1)$ and such that the following holds near [math]:

[TABLE]

Proof.

We saw that this condition is necessary in the discussion above. For the sufficiency we will reduce to the parameter fixing classification (Theorem 3.13). Indeed, if $\lambda=\bar{\lambda}\circ\xi$ then let us perform the change of parameter $\tilde{\epsilon}=\xi(\!\sqrt[k+1]{\epsilon})^{k+1}$ on the first family $\omega$. This gives a family $\tilde{\omega}$. Now the two families $\tilde{\omega}$ and $\bar{\omega}$ have a common eigenvalue function, which is $\bar{\lambda}$. By Theorem 3.13 there is thus a further change of variable fixing the parameter that brings $\tilde{\omega}$ to $\bar{\omega}$. ∎

The classification is thus reduced to a problem of classifying germs up to composition as follows: let

• •

$\cal G_{k}$ be the set of germs of functions $\lambda$ with a zero of order exactly $k$ at the origin.

Let the set ${\mathbb{U}}_{k+1}$ of $(k+1)$-st roots of unity $\zeta$ act on $\cal G_{k}$ by pre-composition (right-composition) with multiplication by $\zeta$. Let

[TABLE]

be the quotient space. (We identified $\cal L$ in Corollary 3.15 with the modulus space for classification with fixed parameter.) Let now

• •

$\Xi$ be the group of germs $\xi$ that commute with the rotation by $2\pi/(k+1)$ and have a non-vanishing derivative at the origin.

Then $\Xi$ acts on $\cal G_{k}$ by right-composition, and this action commutes with the action of ${\mathbb{U}}_{k+1}$. This induces an action of $\Xi$ on $\cal L$ and Proposition 3.16 tells us that the modulus space for classification with non-fixed parameter is naturally identified with $\cal L/\Xi$. Now, as a matter of fact, ${\mathbb{U}}_{k+1}\subset\Xi$, hence $\cal L/\Xi$ is naturally identified with $\cal G_{k}/\Xi$: in other words, taking the quotient of the set of possible germs $\lambda$ by right-composition with a rotation in ${\mathbb{U}}_{k+1}$ and then the quotient by right-composition with the set of possible functions $\xi$ is equivalent to taking directly the quotient by the possible functions $\xi$. In this language, Proposition 3.16 can be rephrased as:

Corollary 3.17**.**

The modulus space of germs of generic parabolic unfoldings of codimension $k$ parabolic vector fields, up to conjugacy, is naturally in bijection with $\cal G_{k}/\Xi$.

We can now seek a unique representative of the orbits of $\Xi$ in $\cal G_{k}$. One natural choice is described in the theorem below:

Theorem 3.18**.**

The modulus space of germs of generic parabolic unfoldings of codimension $k$ parabolic vector fields, up to conjugacy, is in bijection with $\cal G^{\prime}_{k}/{\mathbb{U}}_{k}$ where $\cal G^{\prime}_{k}$ denotes the set of germs

[TABLE]

for which $\sigma(0)=1$ and all the terms of the power series expansion of $\sigma$ of degree $(k+1)m$ vanish for $m\geq 1$, and the group $\cal U_{k}$ of $k$-th roots444The fact that $\nu$ is of order $k$ and not $k+1$ is not a mistake. of unity $\nu$ acts by right-composition with multiplication by $\nu$.

Proof.

Part of the proof is postponed to Lemma 3.22 below. Let us prepare the function $\lambda$ so as to apply Lemma 3.22. We first study the action of linear functions $\xi(\delta)=a\delta$, $a\in{\mathbb{C}}^{*}$, they all belong to $\Xi$. Specifically look at the lower order term of the power series expansion of a general $\lambda\in\cal G_{k}$: $(k+1)\sigma(0)\delta^{k}$. Then the dominant term of $\lambda\circ\xi$ is $(k+1)\sigma(0)a^{k}\delta^{k}$. There are thus precisely $k$ choices of $a$ such that the term becomes $(k+1)\delta^{k}$. Once this is done, Lemma 3.22 ensures that there is a further $h\in\Xi$ such that $\lambda\circ\xi\circ h$ is in $\cal G^{\prime}_{k}$. So the orbit of any germ $\lambda$ intersects $\cal G^{\prime}_{k}$. Then we note that $\cal G^{\prime}_{k}$ is invariant by ${\mathbb{U}}_{k}$. There remains to check that if a function $\xi\in\Xi$ sends an element of $\cal G^{\prime}_{k}$ to an element of $\cal G^{\prime}_{k}$ then $\xi$ is linear with coefficient in ${\mathbb{U}}_{k}$. Looking at the dominant coefficient of $\lambda$, it follows that $\xi^{\prime}(0)=\nu\in{\mathbb{U}}_{k}$. Replacing $\xi$ by $\nu^{-1}\xi$ we can assume that $\xi^{\prime}(0)=1$. Then uniqueness of $h$ in Lemma 3.22 ensures that $\xi=\operatorname{id}$. ∎

3.5.2. Method 2

We will often mention coefficients $A$ and $B$ appearing in the following expansion:

[TABLE]

see Section 3.1 for more details.

Proposition 3.19**.**

Given a generic555see Definition 3.2 vector field $\omega_{\epsilon}$, there exists a change of coordinate independent of the parameter and a linear change of parameter bringing the system to the form

[TABLE]

The only linear changes of coordinate and parameter preserving this form are given by $(z,{\epsilon})\mapsto(\nu z,\nu{\epsilon})$ for $\nu$ a $k$-th root of unity. This is also the form of the diagonal of the linear part of any invertible analytic change of coordinate and parameter preserving the factorized form (3.12).

Proof.

By Corollary 3.6, using a change of coordinate in $z$ we can suppose that the system is in the form $\dot{z}=(z^{k+1}-{\epsilon})v_{\epsilon}(z)$ with $v_{\epsilon}(0)=K$. A change $(z,{\epsilon})\mapsto(\tilde{z},\tilde{{\epsilon}})=(\beta z+\gamma{\epsilon}+o(z,{\epsilon}),\alpha{\epsilon}+o({\epsilon}))$ transforms the system into

[TABLE]

from which it follows that we need to take $\beta^{k}=K$ and $\alpha=\beta^{k+1}$, which has the $k$ solutions described.666It is possible to achieve $\gamma\neq 0$: for instance consider the parameter fixing change of variable that is the time-$1$ map of the vector field for all ${\epsilon}$. The changes $(z,{\epsilon})\mapsto(\nu z,\nu{\epsilon})$ preserve the form (3.12) and by the same computations as above, a linear change $(z,{\epsilon})\mapsto(\tilde{z},\tilde{{\epsilon}})=(\beta z+c{\epsilon},\alpha{\epsilon})$ that preserves it must satisfy $\beta^{k}=1$ and $\alpha=\beta^{k+1}=\beta$. Moreover $\gamma=0$ in the linear case because for all ${\epsilon}$, $z\mapsto\beta z+\gamma{\epsilon}$ must preserve the $(k+1)$-st roots of ${\epsilon}$ thus must preserve their barycenter, which is [math]. ∎

A vector field in the form (3.12) has in particular $A=B=1$ and by Section 3.1 this implies that ${\mathbb{R}}_{+}$ is both a repelling axis and an explosion axis. The $k$ choices in Proposition 3.19 correspond to the choice of which of the $k$ repelling axes of the original vector field we want to place on ${\mathbb{R}}_{+}$ after the change of variable.

Recall that in Section 3.3, we associated to a family $\omega_{\epsilon}$ for which $(z^{k+1}-{\epsilon})$ is a factor, a notion of natural eigenvalue function: $\lambda(\delta)=\omega^{\prime}_{\delta^{k+1}}(\delta)$, see Definition 3.8. The action of the change $(z,{\epsilon})\mapsto(\nu z,\nu{\epsilon})$ on this natural eigenvalue function is to pre-compose it by multiplication by $\nu^{-1}$.

We now move to nonlinear changes of parameter. A holomorphic change of parameter in $\epsilon$ induces a change in $\delta=\sqrt[k+1]{\epsilon}$ that is symmetric in the sense that it commutes with the rotation by $2\pi/(k+1)$. Equivalenty, the new variable $\tilde{\delta}=\sqrt[k+1]{\tilde{\epsilon}}$ is a power series in $\delta$ with all non-zero terms having exponent belonging to $1+(k+1){\mathbb{Z}}$, i.e. $\tilde{\delta}=\delta\tilde{g}(\delta^{k+1})$ with $\tilde{g}$ holomorphic and $\tilde{g}(0)\neq 0$.

Theorem 3.20** (Canonical parameter).**

Let $\omega_{\epsilon}$ be a generic777see Definition 3.2 family of vector fields. For each repelling axis of $\omega_{0}$ there exists a unique analytic change of parameter, ${\epsilon}\mapsto\tilde{{\epsilon}}$, and there exists a non-unique change of variable that depends on the parameter, such that:

• •

it brings the chosen repelling axis of $\omega_{0}$ to ${\mathbb{R}}_{+}$;

• •

the conjugate $\tilde{\omega}$ of $\omega$ is in form (3.12): $\tilde{\omega}_{\tilde{\epsilon}}(\tilde{z})=(\tilde{z}^{k+1}-\tilde{\epsilon})(1+\cal O(\tilde{z},\tilde{\epsilon}))$;

• •

the natural eigenvalue function $\tilde{\lambda}$ (we recall it is defined by $\tilde{\lambda}(\tilde{\delta})=\tilde{\omega}^{\prime}_{\tilde{\delta}^{k+1}}(\tilde{\delta})$) takes the form $\tilde{\lambda}(\tilde{\delta})=(k+1)\tilde{\delta}^{k}(1+O(\tilde{\delta}))$ with $\tilde{\delta}^{k+1}=\tilde{{\epsilon}}$, where $\tilde{\lambda}(\tilde{\delta})$ contains no terms in $\tilde{\delta}^{k+m(k+1)}$ for $m\geq 1$.

The parameter $\tilde{{\epsilon}}$ will be called the canonical parameter, and the representative $\tilde{\lambda}$ above, the canonical eigenvalue function associated to the chosen repelling axis. A family whose parameter is canonical and which is in the form (3.12) will be called prepared. Changing the repelling axis changes the canonical parameter by multiplication by a $k$-th root of unity $\nu$, the effect of which is to pre-compose the canonical eigenvalue function $\tilde{\lambda}$ by multiplication by $\nu^{-1}$.

Proof.

We start by putting $\omega$ in the form (3.12) using Proposition 3.19, placing the desired repelling axis on ${\mathbb{R}}_{+}$. We get a family $\tilde{\omega}$ for which in particular $\tilde{A}=\tilde{B}=1$.

Existence of the change of variable and parameter: According to Lemma 3.22 in Section 3.5.3 below there exists a further change of parameter tangent to the identity and allowing to remove all terms in $\tilde{\delta}^{k+m(k+1)}$ for $m\geq 1$. This destroys the form (3.12) but we keep the property $\tilde{A}=\tilde{B}=1$. Applying Proposition 3.19 again, where this time the linear change of parameter is the identity, we recover the form (3.12) but this time with a natural eigenvalue that is canonical.

Uniqueness of the change of parameter: if a family $\omega$ can be put, in two different ways, in a two forms which are at the same time canonical and satisfy (3.12), for the same choice of repelling axis, then this is also the case for $\tilde{\omega}$ with moreover the two change of parameters having derivative one at the origin. To this change corresponds a change in $\delta$ with also derivative one. We can then apply the uniqueness in Lemma 3.22.

Effect of the change of axis: this follows from Proposition 3.19. ∎

By Theorem 3.20, Theorem 3.13 and the second item of Proposition 3.12:

Theorem 3.21** (Modulus space).**

The modulus space of germs of generic 1-parameter families of vector fields with a singular point of multiplicity $k+1$ (codimension $k$) at the origin is the set of equivalence classes of germs of analytic functions with first term $(k+1)\delta^{k}$ and with all terms of order $k+m(k+1)$ vanishing for $m\geq 1$, i.e.

[TABLE]

where equivalence is by pre-composition with multiplication by a $k$-th root of unity.

3.5.3. A postponed lemma: eliminating specific coefficients of the eigenvalue function

The same lemma is used in both methods, so we moved it here to avoid repetition:

Lemma 3.22**.**

For any holomorphic function $\lambda:({\mathbb{C}},0)\rightarrow({\mathbb{C}},0)$ of the form $\lambda(\delta)=(k+1)\delta^{k}\sigma(\delta)$ with $\sigma(0)=1$, there exists a unique germ of holomorphic diffeomorphism $h:({\mathbb{C}},0)\rightarrow({\mathbb{C}},0)$ of the form $h(\eta)=\eta g(\eta^{k+1})$ with $g$ analytic, such that $h$ is tangent to the identity (i.e. $g(0)=1$) and such that all terms of the power series expansion of $\tilde{\lambda}=\lambda\circ h$ of degree $k+m(k+1)$ vanish for $m\geq 1$.

Proof.

Let us group the terms in the power series expansions of $\lambda$ and $\lambda\circ h$ according to the class modulo $k+1$ of their exponent. This yields $\lambda(\delta)=(k+1)\delta^{k}\sum_{j=0}^{k}\delta^{j}a_{j}(\delta^{k+1})$, and $\lambda\circ h(\eta)=(k+1)\eta^{k}\sum_{j=0}^{k}\eta^{j}b_{j}(\eta^{k+1})$ for some holomorphic germs $a_{j}$ and $b_{j}$ with $a_{0}(0)=b_{0}(0)=1$. Substituting $h(\eta)=\eta g(\eta^{k+1})$ for $\delta$, one realizes that for all $j$, the function $b_{j}$ depends on $a_{j}$ but not on the other $a_{i}$: more precisely if we denote $\zeta=\eta^{k+1}$ then

[TABLE]

The problem is thus equivalent to finding $g$ analytic such that $g(0)=1$ and $(\eta g(\eta^{k+1}))^{k}a_{0}((\eta g(\eta^{k+1}))^{k+1})=\eta^{k}$. Let $\ell(\delta)=\delta\left(a_{0}(\delta^{k+1})\right)^{\frac{1}{k}}$ be the holomorphic germ where we take the branch of $k$-th root that maps $1$ to $1$. The function $\ell$ is analytic and invertible since $\ell^{\prime}(0)=1$. Also

[TABLE]

Hence the problem is equivalent to $\eta\nu=\ell(\eta g(\eta^{k+1}))$ for some solution of $\nu^{k}=1$. Derivating with respect to $\eta$ at $\eta=0$ yields $\nu=1$. The equation thus becomes $\eta=\ell(\eta g(\eta^{k+1}))$, i.e. $g(\eta^{k+1})=\frac{\ell^{-1}(\eta)}{\eta}$, where $\frac{\ell^{-1}(\eta)}{\eta}$ is analytic. We need to prove that $\frac{\ell^{-1}(\eta)}{\eta}$ is a function of $\eta^{k+1}$ alone. From its definition, $\ell$ has the form $\ell(\delta)=\delta b(\delta^{k+1})$ for some analytic function $b$ with $b(0)=1$, yielding that $\ell^{-1}$ has the same form. Hence, $\frac{\ell^{-1}(\eta)}{\eta}$ is a function of $\eta^{k+1}$ alone, yielding the result.

Uniqueness. Suppose that $h_{1},h_{2}$ are two solutions. Then $h_{2}\circ(h_{1})^{-1}$ sends $\tilde{\lambda}_{1}=\lambda\circ h_{1}$ to $\tilde{\lambda}_{2}=\lambda\circ h_{2}$. This is only possible if $h_{2}\circ(h_{1})^{-1}$ is linear: to prove this, denote $h_{2}\circ(h_{1})^{-1}(\eta)=\eta g(\eta)$ and use (3.14) with $j=0$. ∎

3.5.4. Auxiliary question

We may wonder when a generic family $\omega_{\epsilon}$ is conjugate near $(0,0)$ to the simple family $z^{k+1}-{\epsilon}$ studied in Section 2. The natural eigenvalue for this family is $\lambda(\delta)=(k+1)\delta^{k}$. The classification theorem implies:

Corollary 3.23**.**

Let $\lambda$ be an eigenvalue function for $\omega_{\epsilon}$. Then $\omega_{\epsilon}$ is conjugate to the family $z^{k+1}-{\epsilon}$ if and only if $\lambda(\delta)=\delta^{k}\sigma(\delta^{k+1})$ for some germ of analytic function $\sigma$ with $\sigma(0)\neq 0$.

3.6. Proof of Theorem 3.13

Let us now explain how to construct a (parameter preserving) conjugacy between $\omega_{\epsilon}$ and $\widetilde{\omega}_{\epsilon}$ having equivalent eigenvalue functions under pre-composition by a rotation of order dividing $k+1$. We can suppose that $\omega_{\epsilon}$ and $\widetilde{\omega}_{\epsilon}$ have the form (1.3). Then conjugating $\widetilde{\omega}_{\epsilon}$ with a rotation of order $k+1$, we can suppose that $\omega_{\epsilon}$ and $\widetilde{\omega}_{\epsilon}$ have the same natural eigenvalue function $\lambda$, see Section 3.3. We change to the rectifying coordinate, also called straightening coordinate, or $t$-coordinate, and which is given by the complex time associated to the vector fields: if $t=\int\frac{dz}{\omega_{\epsilon}(z)}$, then the vector field $\dot{z}=\omega_{\epsilon}(z)$ becomes $\dot{t}=1$. Similarly, for $\tilde{t}=\int\frac{d\tilde{z}}{\widetilde{\omega}_{{\epsilon}}(\tilde{z})}$, then $\dot{\tilde{z}}=\widetilde{\omega}_{{\epsilon}}$ becomes $\dot{\tilde{t}}=1$. Hence a conjugacy can only be (locally) a translation in the respective time variables. But we have to be careful since the time variables are ramified, there are periods, and the domains are not uniquely defined. To remove these problems we consider the associated translation surfaces. We will see that this amounts to quotienting by the periods.

Definition 3.24**.**

A translation surface is a Riemann surface for which the transition maps between charts are translations (instead of just holomorphic maps).

A translation surface naturally carries a flat Riemannian metric of expression $ds=|dt|$ in the charts. We will use the associated geodesic distance.

The patches of straightening coordinate of a holomorphic vector field $\omega$ on a Riemann surface $S$ define an atlas of a translation surface, on $S$ minus the singularities of $\omega$.

To prove the theorem, we start by showing that specific subsets of the translation surfaces associated to $\omega_{\epsilon}$ and $\widetilde{\omega}_{{\epsilon}}$ are isomorphic. For that purpose we introduce below an abstract translation surface $\mathcal{U}_{\epsilon}$, which we call the local $\tau$-model and we show that (subsets of) the two translation surfaces are isomorphic to the local $\tau$-model.

It is natural to expect that, since the situation is close to the vector field $\dot{z}=z^{k+1}-\epsilon$, rectifying coordinate for the family $\omega_{\epsilon}$ can be defined on domains whose image in rectifying coordinate are small deformations of the star shaped domain found in Section 2. But there are subtleties. Since we are doing a local analysis, we have to remove a neighborhood (varying but of bounded size) of the tips of the star. Also, the sum of the periods does not anymore need to be [math].

Definition 3.25**.**

The period $\mu_{j}$ associated to a singularity $z_{j}$ of the vector field $\omega$ is the integral of $1/\omega$ around a small loop enclosing $z_{j}$, so by the residue theorem, $\mu_{j}=2\pi i\operatorname{Res}\left(1/\omega,z_{j}\right)$. If the singularity is not parabolic, i.e. $\omega^{\prime}(z_{j})\neq 0$, then the period is just $\mu_{j}=2\pi i/\omega^{\prime}(z_{j})$.

As a consequence, assume we tried to proceed as follows to define the model $\cal U_{\epsilon}$: define a period-gon, i.e. a polygon whose sides are the negated periods $-\mu$, attach infinite half-strips to the sides and then remove neighborhoods of the corners. This program would fail on the first step: there is no period-gon since the sum of periods may not be zero: then the sides would not close up. Indeed, the sum of the periods is generically not zero (see Lemma 3.27 below).

Hence the approach has to be modified. We give up on the period-gon. It is important to note that, since the local study implies we are removing (bounded) neighborhoods of the tips, the sides of a strip can be shifted (by a bounded amount) and we still get an isomorphic model, see Figure 19.

Remark 3.26**.**

When we had constructed the star-shaped domain in Section 2, we could have allowed for sides that are not orthogonal to the period $\mu$, or even sides that are not straight lines, as long as they match by the translation by $\mu$. However we will stick here with orthogonality.

Lemma 3.27**.**

The sum of the periods is analytic in ${\epsilon}$ and given by $2\pi iA({\epsilon})$, where

[TABLE]

The limit of the sum of the periods as ${\epsilon}\to 0$ is equal to $2\pi i$ times the residue of $1/\omega_{0}$ at the origin. For a holomorphic germ $\lambda$ with a root of order $k$ at the origin, the sum of $1/\lambda(\delta)$, where $\delta$ ranges on the solutions of $\delta^{k+1}=\epsilon$, has a limit when $\epsilon\longrightarrow 0$, equal to $k+1$ times the constant term in the Laurent series of $1/\lambda$ at the origin.

Proof.

The first claim follows from the fact $A({\epsilon})=\frac{1}{2\pi i}\oint_{C}\frac{dz}{\omega_{\epsilon}(z)},$ where $C$ is a circle around the $k+1$ singular points. The second claim follows from the existence of a vector field with periods associated to $\lambda$ (second part of Proposition 3.12). More simply, it can also be directly checked: $1/\lambda(\delta)$ is a Laurent series in $\delta$ with exponents ranging from $-k$ to $+\infty$. The sum thus cancels out all negative exponents. ∎

Definition 3.28**.**

Given a holomorphic germ $\lambda$ vanishing with order $k$ at the origin, we construct a “local time-model” $\cal U_{\epsilon}(R)$ as follows, on which we choose the symbol $\tau$ for the time variable. We call it the $\tau$-model.

Construction of $\cal U_{\epsilon}(R)$: By Lemma 3.27, the sum of periods $2\pi iA({\epsilon})$ stays bounded: the quantity $2\pi iA({\epsilon})$ is the gap to close the polygon. Let

[TABLE]

For a given $\epsilon$, let us order the negated periods $-\mu_{0}$, …, $-\mu_{k}$ by increasing value of their argument (the construction is independent of which one is chosen first). Consider a circularly indexed chain of vertices such that the consecutive differences are given alternatively as follows: $-\mu_{0}$, $a$, $-\mu_{1}$, $a$, $-\mu_{2}$, $a$, …, $-\mu_{k}$, $a$ with $a=a({\epsilon})$. This is possible because the sum of these numbers is [math], and of course the solution is unique up to a translation. If we link consecutive vertices we obtain a polygon that may self-intersect. Each segment is alternately long and short, and the long segments nearly form a regular polygon. See Figure 20 on the right.

On each long segment, put an infinite half-strip that is orthogonal to the segment and is directed opposite to the center of the polygon (see Figure 21). Consecutive half strips may self-intersect near the short segments, which may or may not be a problem. To get rid of this messy part the idea is to remove some neighborhood of the short segments. There are several possible choices that will give different translation surfaces that are in fact isomorphic on large parts. We fix here a choice, which depends analytically on ${\epsilon}$, namely removing a disk of constant radius $R$ centered at the middle of each short segment.

Consider a short segment. From its ends draw two half lines that bound two consecutive strips. If $\epsilon$ is small enough, then they make an angle close to $2\pi/(k+1)$, so we may assume that it is at least $\pi/(k+1)$. Then the intersection point of the correspondig lines lies at a distance from the center of the short segment that is less than $\frac{1}{2}|a|\cot\frac{\pi}{2(k+1)}$. Hence, if $R>\frac{3}{4}|a(0)|\cot\frac{\pi}{2(k+1)}$, this intersection point lies inside the disk of radius $R$ centered on the middle of the short segment for all sufficiently small ${\epsilon}$. Now the idea is to remove this disk from the union of the strips and of the polygon (see Figure 21). Since the polygon is self-intersecting, to avoid technicalities, we proceed as follows: we define a path, it starts from infinity by following one of the half lines until it hits the circle. It then follows the circle, towards the inside of the strip and until it hits the other half line. Last, it follows this half line towards infinity. We now have $k+1$ disjoint simple (i.e. injective) curves going from infinity to itself, cutting the plane into $k+2$ domains. We let $U_{\epsilon}=U_{\epsilon}(R)$ be the central one.

The domain $U_{\epsilon}$ is unbounded and simply connected. We want to define an identification of the sides of the strips in $\partial U_{\epsilon}$ through the translation by the corresponding period $\mu$. It should be noted that the two removed disks at each end are, unless the sum of the periods vanishes, not image of one another by the translation by $\mu$, which means that, most likely, the identification has to omit a bounded segment on one of the sides (see Figure 21). For $\epsilon$ small enough, this segment will have a length bounded by the modulus of the sum of the periods, times a constant that depends only on $k$. We denote $\cal U_{\epsilon}=\cal U_{\epsilon}(R)$ the translation surface we have obtained and call it the $\tau$-model. The simply connected set $U_{\epsilon}(R)$ will be called the canonical chart. The chart $U_{\epsilon}$ and the translation surface $\cal U_{\epsilon}$ are endowed with the vector field $\dot{\tau}=1$.

This ends the construction. $\Box$

Note that the set $U_{\epsilon}(R)$ has only been uniquely defined up to a translation.

Remark 3.29**.**

On Figure 22 we indicated the curve obtained by following a $\tau$-coordinate along the boundary of $\cal U_{\epsilon}$. In other words we take the 6 components of the boundary and we reglue them together by the translations by the $\mu_{i}$. This cannot close up if the sum of periods is not [math]. We believe that, essentially, the isomorphism class of the model only depends on the $\mu_{i}$ and on this contour. However we will not need this.

The crucial step of the proof of Theorem 3.13 is the following.

Proposition 3.30**.**

There exist $R>0$, $r>0$ and, for all $\epsilon\neq 0$ small enough, an open neighborhood $V_{\epsilon}$ of [math] such that: $V_{\epsilon}$ is contained in the domain of $\omega_{\epsilon}$, $V_{\epsilon}$ contains the disk $B(0,r)$ and the $k+1$ singularities of $\omega_{\epsilon}$, and the translation surface defined by $\omega_{\epsilon}$ on $V_{\epsilon}$ (minus the singularities) is isomorphic to $\cal U_{\epsilon}(R)$.

In other words, $\cal U_{\epsilon}(R)$ is part of the structure and it is not too small a part.

Proof.

We can of course suppose that $\omega_{\epsilon}$ is of the form (3.6) with $v_{0}(0)=1$:

[TABLE]

By restricting the domain of $\epsilon$ and $z$ we may assume that $|v_{\epsilon}(z)-1|<\frac{1}{100}$ and $|\arg(v_{\epsilon}(z))|<\frac{1}{100(k+1)}$.

In the sequel we drop the index $\epsilon$ whenever this clutters the notation and dependence in $\epsilon$ is clear. In the process above, we restricted the domain of $z$. We are not changing it anymore and we assume that it contains a disk $B(0,r)$, where $r$ is independent of $\epsilon$. Let us now consider the following points on its boundary: $e_{i}$, $i=0,\dots,k$, that have the same arguments as the singularities, and are ordered trigonometrically; $s_{i}$, that are at mid-range from $e_{i}$ and $e_{i+1}$ (see Figure 23.)

We choose straightening coordinates $t(z)=\int_{0}^{z}\frac{dz}{\omega_{\epsilon}(z)}$ and $\overline{t}(z)=\int_{0}^{z}\frac{dz}{\Omega_{\epsilon}(z)}$ for

[TABLE]

both sending [math] to [math] on a simply connected domain, with cuts $[z_{j},e_{j}]$ as in Figure 23.

Consider then the straight segment $[0,s_{i}]$, and its two images in the straightening coordinate $t$ and $\overline{t}$. Then $|t(s_{i})-\overline{t}(s_{i})|<\frac{1}{100}|\overline{t}(s_{i})|$. The images of the $k+1$ segments $[0,s_{i}]$ by $\overline{t}$ are line segments $[0,\overline{t}(s_{i})]\subset[0,\xi_{i}]$, where $\xi_{i}$ is one tip of the star shaped domain described in Section 2, and $|\xi_{i}-\overline{t}(s_{i})|<C$, where $C$ is a constant that depends only on $r$ and $k$. Drawing the $k+1$ curves $t([0,s_{i}])$, we get a star figure whose endpoints $u_{i}=t(s_{i})$ are close to $\xi_{i}$: if $d_{\epsilon}=|\xi_{i}|$ then $|t(s_{i})-\xi_{i}|\leq|t(s_{i})-\overline{t}(s_{i})|+|\overline{t}(s_{i})-\xi_{i}|<\frac{1}{100}|\overline{t}(s_{i})|+\centering C\leq\frac{1}{100}d_{\epsilon}+\frac{101}{100}C\@add@centering$. Recall that $d_{\epsilon}\to+\infty$ as $\epsilon\to 0$.

We now follow the boundary of $B(0,r)$ from $s_{i}$ to $e_{i}$ or $e_{i+1}$. Tracing the image by $t$ starting from $u_{i}$, we get two curves, which are close to circular arcs of radius $1/r^{k}$, and together they span approximately an angle of $2\pi(1-1/(k+1))$. Hence it is natural to look for some “center” of this circle. Let us call $\gamma_{i}$ the union of these two curves, and $A_{i}$ and $A_{i}^{\prime}$ its endpoints corresponding respectively to $e_{i}$ and $e_{i+1}$. In the $z$-coordinate, the point $e_{i}$ is at the end of both the circular arc on $\partial B(0,r)$ from $s_{i}$ and the one from $s_{i-1}$. Then $A_{i}$ and $A^{\prime}_{i-1}$ differ precisely by the negated period $-\mu_{i}$: indeed with the segments and arcs we considered, we can form a closed contour from $e_{i}$ to $s_{i}$, to [math], to $s_{i-1}$, to $e_{i}$. The integral $\int\frac{dz}{\omega_{\epsilon}}$ along this contour is equal, on one hand to $A_{i}-A^{\prime}_{i-1}$, and on the other hand, to $2\pi i$ times the residue of $1/\omega_{\epsilon}$ at the unique singularity enclosed by the contour, so $A_{i}-A^{\prime}_{i-1}=-\mu_{i}$.

Since $|A_{i}^{\prime}-A_{i}|<C_{2}$ for some constant $C_{2}$ ($C_{2}=\frac{4}{kr^{k}}$ for instance), it follows that there exists some translation of the $2(k+1)$-gon of the construction of $\cal U_{\epsilon}$ following Definition 3.28 whose vertices lie at bounded distance $C_{3}$ from the $A^{\prime}_{i}$ and $A_{i}$. We can for instance put one vertex at the same place and then we get the bound $C_{3}=(k+1)(C_{2}+|a|)$.

Let us now follow the image by $t$ of the radial segment $[e_{i},z_{i}]$ from $e_{i}$ down to the corresponding singularity $z_{i}$ of $\omega_{\epsilon}$. This traces from $A_{i}$ and $A^{\prime}_{i-1}$ two parallel curves, $\zeta_{i}$ and $\zeta_{i-1}^{\prime}$, whose tangent vector deviates less than $\frac{1}{100(k+1)}$ radians from the direction of the corresponding (straight) lines which are the images of $[e_{i},z_{i}]$ by $\overline{t}$.

The union over $i$ of the curves $\zeta_{i}$ and $\zeta_{i-1}^{\prime}$ of the previous paragraph and of the curves $\gamma_{i}$ defined before, forms the boundary of an open set $W_{\epsilon}$, that serves as a global chart in $t$-coordinate of the translation surface $\cal S_{\epsilon}$ associated to $\omega_{\epsilon}$: more precisely take the union $\widehat{W}_{\epsilon}$ of $W_{\epsilon}$ with the curves $\zeta_{i}$ and $\zeta^{\prime}_{i}$, ends excluded, and quotient by identifying them with the translation by $\mu_{i}$: this gives a translation surface $\widehat{W}_{\epsilon}/\sim$ that is isomorphic to $\cal S_{\epsilon}$. We must check two things:

• •

that it contains a model $\cal U_{\epsilon}(R)$ as per Definition 3.28 for some $R$ sufficiently large;

• •

that the corresponding $V_{\epsilon}$ in $z$-coordinate contains a definite disk around the origin.

The strips we constructed have non-straight boundaries, but the boundaries can be modified to become straight and parallel to the appropriate direction (which is perpendicular to $\mu_{i}$), while preserving the initial point. Call $W^{\prime}_{\epsilon}$ the simply connected domain thus obtained. We then choose $R$ so that, for an appropriate choice of translate of the canonical chart $U_{\epsilon}(R-|a(\epsilon)|)$, the $k+1$ disks of radius $R-|a(\epsilon)|$ in its definition contain the curves $\gamma_{i}$. Since $a(\epsilon)$ is bounded, and since the curves $\gamma_{i}$ have bounded diameter, it is possible to choose a uniform $R$. Remove now from $W^{\prime}_{\epsilon}$ the disks of radius $R-|a(\epsilon)|$ having the same centers, to get a domain $W^{\prime\prime}_{\epsilon}$. The last step is to translate the two straight boundaries of each strip so that they coincide with those of $U_{\epsilon}$, in other words they must radiate from the appropriate vertex of the $2(k+1)$-gon. In this process the radius of the removed disks has to be increased to ensure that we get a subset of $W^{\prime\prime}_{\epsilon}$ after identification of the sides, and increasing this radius by $|a(\epsilon)|$ is enough: recall that the gluing may omit an initial segment on one of the sides, of length at most $|a(\epsilon)|$. We finally have proved that $\cal U_{\epsilon}(R)$ is indeed isomorphic to a subset of the translation surface $\cal S_{\epsilon}$ associated to $\omega_{\epsilon}$.

What we removed is at a bounded distance from the $\gamma_{i}$ and coming back to the $z$-coordinate, this implies that it cannot get too close to [math] for $\epsilon$ small. Indeed $\omega_{\epsilon}$ is close to $\omega_{0}$ for which a path between points $z^{\prime}$ and $z^{\prime\prime}$ will have a $t$-length at least $c|1/|z^{\prime}|^{k}-1/|z^{\prime\prime}|^{k}|$ for some $c>0$. ∎

By our choice of models, the domains $V_{\epsilon}$ have angular points on their boundaries, nevertheless they consist mostly of nearly circular arcs around [math] alternating with short curves.

It should also be noted that in the procedure above, we did not try to get a nice dependence of the isomorphism between the $z$-coordinate and the $\tau$-model, in particular the boundary of $V_{\epsilon}$ does not a priori depend continuously on $\epsilon$. When we placed a copy of the $\tau$-model in the translation surface there was some choice and it could have been slid by some translation (this amounts to applying the complex flow of $\omega_{\epsilon}$).

We now deal with the problems of getting holomorphic dependence of the model and continuous dependence of $V_{\epsilon}$. This is done as follows. In Proposition 3.30 we introduced points $s_{i}$ on $\partial B(0,R)$ and their images $u_{i}\in\partial W_{\epsilon}(R)$. Call $u(\epsilon)$ the barycenter of the $u_{i}({\epsilon})$. Call $c_{i}$ the center of the circles supporting the arcs on the boundary of $U_{\epsilon}$ (i.e. the middle points of the small segments of the $2(k+1)$-gon), and let $c(\epsilon)$ be their barycenter.

Definition 3.31**.**

The domain $U_{\epsilon}(R)$ is called balanced whenever the two barycenters $c({\epsilon})$ and $u({\epsilon})$ coincide.

The notion depends on the points $s_{i}$, and therefore on $r$. A change of $z$-coordinates would change the position of $u(\epsilon)$ and thus would slide $U_{\epsilon}(R)$ by some amount, notwithstanding the fact that it would probably change $R$.

To understand the following lemma, recall that $|a(\epsilon)|$ is the length of each of the $k+1$ small segments in the polygon involved in the construction of the $\tau$-model.

Lemma 3.32**.**

If $R^{\prime}\geq 2R+|a(\epsilon)|$, then it is possible to choose $U_{\epsilon}(R^{\prime})$ balanced.

Proof.

Finding a translated copy of $\cal U_{\epsilon}(R^{\prime})$ in $\cal U_{\epsilon}(R)$ is possible if the translation is by a vector of size at most $R^{\prime}-R-|a(\epsilon)|$. Recall that $|u_{i}({\epsilon})-c_{i}({\epsilon})|<R$, whence $|c({\epsilon})-u(\epsilon)|<R$. ∎

Recall also that ${\epsilon}\mapsto a({\epsilon})$ is bounded (it has an erasable singularity at ${\epsilon}=0$). Therefore by increasing $R$ we can assume in the sequel that $U_{\epsilon}(R)$ is balanced.

Lemma 3.33**.**

The function $u$ has the following expansion:

[TABLE]

Proof.

We begin by working with another parameter: the universal cover $\hat{{\epsilon}}\mapsto\epsilon$ of the set of $\epsilon\neq 0$. Note that the logarithm is globally well-defined as a function of $\hat{{\epsilon}}$.

Choose $k+1$ points $w_{i}$ evenly spaced on the circle of center [math] and radius $r$. Let them be independent of $\hat{{\epsilon}}$. Recall that we work in a $z$-coordinate where the singularities of $\omega_{\epsilon}$ are the solutions of $z^{k+1}=\epsilon$. For some value $\epsilon_{0}$ of $\epsilon$, each segment from [math] to $w_{i}$ passes through a midpoint between two consecutive singularities (hence for this $\epsilon$ the $w_{i}$ coincide with the moving points called $s_{i}({\epsilon})$ in the proof of Proposition 3.30). Choose one lift $\hat{{\epsilon}}_{0}$ of this $\epsilon_{0}$. Consider the quantities $t(w_{i})=\int_{[0,w_{i}]}\frac{dz}{\omega_{\epsilon_{0}}(z)}$, $i=0,\dots,k$, and let $m(\hat{{\epsilon}}_{0})=\frac{1}{k+1}\sum_{i=0}^{k}t(w_{i})$ be the average of these integrals. Note that $m(\hat{{\epsilon}}_{0})=u(\epsilon_{0})$.

Now when $\hat{{\epsilon}}$ varies away from $\hat{{\epsilon}}_{0}$, one can follow the homotopy class of the path from [math] to the unmoving $w_{i}$, in the complement of the singuarities, and the integral of $1/\omega_{\epsilon}$ from [math] to $w_{i}$ in this class is well-defined and is a function of $\hat{{\epsilon}}$. Again, we let $m(\hat{{\epsilon}})$ be their average. It is a holomorphic function of $\hat{{\epsilon}}$: indeed, locally we can choose a path independent of $\hat{{\epsilon}}$, and $\omega_{\epsilon}$ depends holomorphically on $\epsilon$.

When $\epsilon$ winds once around [math], each integral will have varied by $\mu_{j}$ for some $j$ and $m(\hat{{\epsilon}})$ will have varied by $a(\epsilon)$. We can cancel out this monodromy by considering the quantity

[TABLE]

Indeed, $f({\epsilon})$ depends uniformly on $\epsilon$, thus defining a holomorphic function on a disk minus the origin. Let us prove that the origin is a removable singularity.

When $\arg(\hat{{\epsilon}})-\arg(\hat{{\epsilon}}_{0})$ remains bounded, then $|m-u|$ is bounded, since the difference is a finite sum of $\int dz/\omega_{\epsilon}$ over bounded arcs in $\partial B(0,r)$. Recall that $\omega_{\epsilon}$ differs from $z^{k+1}-\epsilon$ by a factor of at most $\frac{1}{100}$. This implies in normalized $t$-coordinate that $u_{i}$ is at distance from [math] of order $1/|\epsilon|^{k/(k+1)}$. It follows that $u=\cal O(1/|\epsilon|^{k/(k+1)})$. Whence $f(\epsilon)=\cal O(1/|\epsilon|^{k/(k+1)})$ (the log term is much smaller). It follows by Cauchy’s formula that the singularity of $f$ at the origin is erasable. The lemma follows. ∎

Recall that we are using a balanced chart $U_{\epsilon}(R)$ as a subset of the $t$-coordinate chart $W^{\prime\prime}_{\epsilon}$ constructed in the proof of Proposition 3.30. The inverse $t\mapsto z$ of the coordinate change $z\mapsto t$ extends to $\partial U_{\epsilon}(R)$. Let $\widehat{U}_{\epsilon}(R)$ be the union of $U_{\epsilon}(R)$ and of the part of its strip boundaries that get identified by the translations by the corresponding $\mu_{i}$. Let $V^{*}_{\epsilon}=t^{-1}(\widehat{U}_{\epsilon}(R))$, and let $V_{\epsilon}$ be the union of $V^{*}_{\epsilon}$ with the singularities of $\omega_{\epsilon}$. The $t$-coordinate thus defines an isomorphism

[TABLE]

Using the inclusion $U_{\epsilon}\subset{\mathbb{C}}$ we can define a two-dimensional analytic structure over the union of $U_{\epsilon}\times\{\epsilon\}$ in ${\mathbb{C}}^{2}$. A little bit of thinking shows that it also works across the glued strip sides and defines an analytic structure on the union888A remark for the set-theoretic minded: the quotient of a subset of ${\mathbb{C}}$ by an equivalence relation is the set of equivalence classes, so it is a subset of $\cal P({\mathbb{C}})$. The union thus takes place in $\cal P({\mathbb{C}})\times{\mathbb{C}}$. of the quotients $\cal U_{\epsilon}\times\{\epsilon\}$ by these gluings: the fact that $\mu_{i}$ depends holomorphically on $\epsilon\neq 0$ has to be used.

Lemma 3.34**.**

The function $(z,\epsilon)\mapsto\phi_{\epsilon}(z)$ is analytic.

Proof.

For each fixed $\epsilon$ it is analytic as a function of $z$. By the theory of analytic functions in several variables, it is enough to check that, for each fixed $z$, it is analytic as a function of $\epsilon$.

Consider any $(z_{0},\epsilon_{0})$ such that $z_{0}$ is not a singularity and $\epsilon_{0}\neq 0$ and let us prove holomorphic dependence of $\epsilon\mapsto\phi_{\epsilon}(z)$ near $(z_{0},\epsilon_{0})$. Consider a path $\gamma:[0,1]\rightarrow B(0,r_{2})$, $\alpha\mapsto\gamma(\alpha)$ from [math] to $z_{0}$ avoiding the singularities of $\omega_{\epsilon_{0}}$. Its image in $t$-space is a path $\Gamma_{\epsilon}$, where $\Gamma_{\epsilon}(\alpha)=t(\gamma(\alpha))=\int_{\gamma[0,\alpha]}\frac{dz}{\omega_{\epsilon}(z)}$. Then for all nearby $\epsilon$, for all $\alpha\in[0,1]$, $\epsilon\mapsto\Gamma_{\epsilon}(\alpha)$ is holomorphic. If we follow this path in $\widehat{U}_{\epsilon}/\sim$, jumping by $\mu_{i}(\epsilon)$ when a boundary is crossed, we still get holomorphic dependence because $\epsilon\mapsto\mu_{i}(\epsilon)$ is holomorphic. ∎

Theorem 3.35** (Completeness of the invariant $\lambda$).**

Consider any two generic families of vector fields $\omega_{\epsilon}$ and $\widetilde{\omega}_{\epsilon}$ unfolding a parabolic germ of same codimension $k$ (see Definition 3.2). Assume that there exists $\zeta\in{\mathbb{C}}$ with $\zeta^{k+1}=1$ and such that $\lambda(\delta)=\tilde{\lambda}(\zeta\delta)$ near [math]; then there exists a parameter preserving change of variable $z\mapsto\tilde{z}=\psi_{\epsilon}(z)$ on a neighborhood of $(\epsilon,z)=(0,0)$ that sends $\omega_{\epsilon}$ to $\widetilde{\omega}_{\epsilon}$. We can moreover require that $\psi_{\epsilon}(0)=0$ for all $\epsilon$.

Proof.

Note that the two families have the same periods $\mu_{i}$ and hence the same function $a(\epsilon)=\sum\mu_{i}/(k+1)$. We carry out all the constructions done earlier in this section: this restricts the domain of $\epsilon$ to some pointed disk $B(0,\rho)\setminus\{0\}$. We can always increase either $R$ or $\tilde{R}$ so we will assume that $R=\tilde{R}$. This yields a balanced $\tau$-model $\cal U_{\epsilon}$ for $\omega_{\epsilon}$ (resp. $\tilde{\tau}$-model $\widetilde{\cal{U}}_{\epsilon}$ for $\widetilde{\omega}_{\epsilon}$). In particular, Lemma 3.34 gives analytic functions $\phi_{\epsilon}:V^{*}_{\epsilon}\to\cal U_{\epsilon}$ and $\widetilde{\phi}_{\epsilon}:\widetilde{V}^{*}_{\epsilon}\to\widetilde{\cal U}_{\epsilon}$. By Lemma 3.33, the subsets $U_{\epsilon}$ and $\widetilde{U}_{\epsilon}$ of ${\mathbb{C}}$ differ by a translation by a vector whose length is bounded as $\epsilon$ varies: call $K$ such a bound. We can use this to define an inclusion of $\cal U_{\epsilon}(R+|a(\epsilon)|+K)$ in $\widetilde{\cal U}_{\epsilon}(R)$ that is defined as the identity $(t\mapsto\tilde{t}=t)$ on a large part of $U_{\epsilon}$ containing [math], and defined elsewhere using the gluings of the strips (details left to the reader).

So the domain of the composition $\psi_{\epsilon}:=\widetilde{\phi}^{-1}_{\epsilon}\circ\phi_{\epsilon}$ contains the set $\overline{V}^{*}_{\epsilon}$ that we define as the preimage by $\phi_{\epsilon}$ of the set $\cal U_{\epsilon}(R+|a(\epsilon)|+K)$ considered above. Call $\overline{V}_{\epsilon}$ the union of $\overline{V}^{*}_{\epsilon}$ and of the singularities of $\omega_{\epsilon}$. Note that its boundary moves continuously with $\epsilon$. Note also that the sets $\overline{V}_{\epsilon}$ contain a common neighborhood $B(0,r_{2})$ of the origin for similar reasons as in Proposition 3.30.

By construction, the map $\psi_{\epsilon}$ conjugates the vector field $\omega_{\epsilon}$ to $\widetilde{\omega}_{\epsilon}$. It extends through the singularities (zeroes) of the vector field: isolated singularities (points outside the domain) of bounded holomorphic function are erasable. The extension, call it also $\psi_{\epsilon}$, is still a conjugacy of the vector fields (by continuity w.r.t. $z$).

By Lemma 3.34, for all $z\in B(0,r_{2})$, $\psi_{\epsilon}(z)$ depends analytically on $\epsilon$ whenever $z$ is not a singularity of $\omega_{\epsilon}$, i.e. whenever $\epsilon\neq z^{k+1}$ (recall also that $\epsilon\neq 0$). By the Cauchy formula in $z$-coordinate applied to a small circle centered on $z$, the analytic dependence extends to $\epsilon$ satisfying ${\epsilon}=z^{k+1}$.

Since isolated singularities of bounded holomorphic functions are erasable, the function $\epsilon\mapsto\psi_{\epsilon}(z)$ will have a holomorphic extension to $\epsilon=0$ for all $z\in B(0,r_{2})$. By continuity it sends $\omega_{0}$ to $\widetilde{\omega}_{0}$ so it cannot be constant with respect to $z$. The function $\psi_{0}$ is a non-constant limit of injective holomorphic maps, so it is injective. By the Cauchy formula in ${\epsilon}$-coordinate, the extended $\psi$ is analytic in $(z,\epsilon)$ over $B(0,r_{2})\times B(0,\rho)$. ∎

3.7. Normal forms

There are several notions of classification we can be interested in and, for each notion, there are interesting normal forms. Here we list a few, summing up what was done in the present article and adding a few more.

We still assume that $\omega_{\epsilon}$ is a generic $1$-parameter unfolding of a vector field having a parabolic singularity $\omega_{0}(z)=z^{k+1}+O(z^{k+2})$, $k\geq 1$.

For ${\epsilon}$ small enough and in a uniform neighborhood of [math] in the $z$-coordinate, $\omega_{\epsilon}$ can be turned by a change of variable $z$ into the (non-unique) form

[TABLE]

for some function $h_{\epsilon}$, i.e. we can place the singularities exactly on the solutions of $z^{k+1}={\epsilon}$. However, this is not a normal form: there are many possible changes of variables. Hence the function $h_{\epsilon}$ is highly non-unique.

3.7.1. The local normal form

The whole family (3.17) can be transformed into the local normal form

[TABLE]

by a conjugacy that preserves the parameter: this is Theorem 3.13. Its remarkable feature is that $\sigma$ does not depend on $\epsilon$. This normal form is nearly unique: we can only multiply $z$ by a rotation of order dividing $k+1$. The effect on $\sigma$ is a conjugacy by this rotation: if $\tilde{z}=\zeta z$ with $\zeta^{k+1}=1$ then the vector field reads $\dot{\tilde{z}}=(\tilde{z}^{k+1}-{\epsilon})\times\tilde{\sigma}(\tilde{z})$ with $\tilde{\sigma}(\tilde{z})=\zeta\sigma(\zeta^{-1}\tilde{z})$. Hence $\sigma$ is unique up to this action. The function $\sigma$ is related to the eigenvalue function, see Section 3.3.

However, the problem that interests us most is conjugacies that allow a change of parameter. Then the Classification Theorem 3.18 tells that $\omega_{\epsilon}$ can be turned into the same local normal form as above:

[TABLE]

where moreover

[TABLE]

i.e. the power series expansion of $\sigma(z)$ at [math] has no terms of exponent in $(k+1){\mathbb{N}}$. Three advantages of this form are that the factor $\sigma(z)$ does not depend on $\epsilon$, that the parameter is canonical and that the form is unique up to right composition of $\sigma$ with a rotation of order dividing $k$ (which corresponds to $(z,{\epsilon})\mapsto(z/\nu,{\epsilon}/\nu)$ with $\nu^{k}=1$).

Two other possibilities for which the singular points are located at the roots of $x^{k+1}-{\epsilon}$ are given by polynomial and rational vector fields on ${\mathbb{C}\mathbb{P}}^{1}$.

3.7.2. The polynomial normal form

The polynomial normal form has up to $k$ additional singular points on ${\mathbb{C}\mathbb{P}}^{1}$ and a pole at infinity:

[TABLE]

where $Q_{\epsilon}$ is a polynomial in $z$ of degree at most $k$ and:

• •

if we work with parameter preserving changes of variable, then we impose $Q_{0}(0)\neq 0$;

• •

if we allow conjugacies that change the parameter, then we impose the stronger condition $Q_{0}(0)=1$.

Similarly to Section 3.7.1, in the first case there is uniqueness up to conjugacy of $Q_{\epsilon}$ by a rotation of order dividing $k+1$, and in the second case by pre-composition of $Q_{\epsilon}$ by a rotation of order dividing $k$. To justify that we can pass from the normal form of Section 3.7.1 to this one, we can apply Theorem 3.13, which reduces the problem to ensuring that $Q_{\epsilon}(\delta)=\sigma(\delta)$ where $\epsilon=\delta^{k+1}$. A polynomial of degree at most $k$ being uniquely determined by its values at $k+1$ points, there is a unique solution for ${\epsilon}\neq 0$. If $\delta_{i}$, $i=1,\dots,k+1$ are the singular points and $w_{i}=\sigma(\delta_{i})$, such a polynomial is given by the Lagrange interpolation formula

[TABLE]

It depends holomorphically on ${\epsilon}\neq 0$ and is well-defined (single-valued) since it is invariant under permutations of the $\delta_{i}$. To prove the convergence when ${\epsilon}\to 0$ we consider the variables $\delta_{j}$ as independent, we let $\Delta=(\delta_{1},\ldots,\delta_{k+1})$, and for all $\Delta\in{\mathbb{C}}^{k+1}$ we introduce this other formula for the Lagrange interpolation polynomial

[TABLE]

Then the denominator is $\prod_{i<j}(\delta_{j}-\delta_{i})$. Each $\delta_{i}-\delta_{j}$ divides the numerator since $w_{i}-w_{j}=O(\delta_{i}-\delta_{j})$, and hence, $\delta_{i}-\delta_{j}$ divides the difference of the lines starting by $\sigma(\delta_{i})$ and $\sigma(\delta_{j})$.

With this normal form it is very easy to check if the parameter is canonical. Indeed the only terms of degree $\delta^{m(k+1)}$ in $Q_{\delta^{k+1}}(\delta)$ come from $Q_{\epsilon}(0)$. Hence the parameter is canonical if and only if $Q_{\epsilon}(0)\equiv 1$. Also, a change of coordinate and parameter in (3.18) to the canonical parameter is simply of the form $(z,{\epsilon})\mapsto(\tilde{z},\tilde{{\epsilon}})=\left(zQ_{\epsilon}^{1/k}(0),{\epsilon}Q_{\epsilon}(0)^{1+\frac{1}{k}}\right)$, which is analytic in $(z,{\epsilon})$ since $Q_{\epsilon}(0)\neq 0$.

3.7.3. The rational normal form

The rational normal form on ${\mathbb{C}\mathbb{P}}^{1}$ is given by

[TABLE]

where $R_{\epsilon}$ is again a polynomial in $z$ of degree at most $k$ and $R_{0}(0)=1$ (or $R_{0}(0)\neq 0$ if we only do parameter preserving conjugacies). The treatment is the same as above with equation $R_{{\epsilon}}(\delta)=1/\sigma(\delta)$.

In this case it is harder to have a grip on the canonical parameter, since the calculations become messy. But we could redefine the notion of canonical parameter by using $1/\sigma$ instead of $\sigma$ in the definition: with this new definition the new parameter would be canonical if and only if $R_{\epsilon}(0)\equiv 1$. Also a change of coordinate and parameter in (3.19) to the new canonical parameter is simply of the form $(z,{\epsilon})\mapsto(\tilde{z},\tilde{{\epsilon}})=\left(zR_{\epsilon}^{-1/k}(0),R_{\epsilon}^{-1-\frac{1}{k}}(0){\epsilon}\right)$, which is analytic in $(z,{\epsilon})$ since $R_{\epsilon}(0)\neq 0$.

3.7.4. A Kostov-type normal form

A fourth normal form, which we call Kostov-type normal form is given by

[TABLE]

where $A(\epsilon)$ denotes the sum of the inverses of the eigenvalues (we saw in Proposition 3.12 that it is analytic in ${\epsilon}$), $P_{\epsilon}$ is a polynomial in $z$ of degree $k+1$ with

[TABLE]

(so it is monic and centered), and $P_{0}(z)=z^{k+1}$. The fact that we can put the family in this normal form (by a parameter-preserving change of variable) is a consequence of a theorem of Kostov ([K]), as follows. We consider a $1$-parameter family of vector fields $\dot{z}=\omega_{\eta}(z)=(z^{k+1}-\eta)v_{\eta}(z)$. We enlarge it to the $k$ parameter family $\dot{z}=\omega_{\tilde{\eta}}(z)=(z^{k+1}+\eta_{k-1}z^{k-1}+\dots+\eta_{1}z-\eta)v_{\eta}(z)$ with the multi-parameter $\tilde{\eta}=(\eta_{k-1},\dots,\eta_{1},\eta)$. Kostov’s Theorem states that there exists a change of coordinate and parameters to a normal form

[TABLE]

with a new multi-parameter $a=(a_{k-1},\dots,a_{1},a_{0})$ and $\tilde{P}_{a}(\tilde{z})=\tilde{z}^{k+1}+a_{k-1}\tilde{z}^{k-1}+\dots+a_{1}\tilde{z}+a_{0}$. Since $a_{0}$ is the product of the singularities, it has the same order of magnitude as the product of the singularities before the change of coordinate and parameters, thus $\frac{\partial a_{0}}{\partial\eta}\neq 0$. Hence the restriction of the change of coordinate and parameter to $\eta_{k-1}=\dots=\eta_{1}=0$ provides the required change to the normal form.

One great advantage of this normal form is that a change of parameter $\epsilon=h(\overline{{\epsilon}})$ corresponds to substituting $f(\overline{{\epsilon}})$ for ${\epsilon}$ in the coefficients of $P_{\epsilon}$ and in the coefficient $A(\epsilon)$. Another is that when changing parameter, the $z$-coordinate of the normal form does not change.

The Kostov-type normal forms are far from unique, because any analytic change of parameter provides a normal form of the same type. Hence, it would be good to move to a canonical parameter and hope for some form of uniqueness. The canonical parameter we have defined in Section 3.5 is not adequate in practice for this normal form since an exact expression of the singular points does not exist anymore, and hence we cannot compute the eigenvalue function. Fortunately, it is possible to identify another canonical parameter. Indeed, it suffices to take ${\epsilon}=-a_{0}(0,\dots,0,\eta)$ as the new parameter, i.e. putting $b_{0}({\epsilon})\equiv-{\epsilon}$ in (3.21), as shown in the following theorem.

Theorem 3.36**.**

Let $\dot{z}=P_{\epsilon}(z)/(1+A(\epsilon)z^{k})$ and $\dot{\tilde{z}}=\widetilde{P}_{\tilde{\epsilon}}(\tilde{z})/(1+\widetilde{A}(\tilde{\epsilon})\tilde{z}^{k})$ be two $1$-parameter families of vector fields such that $P$ and $\widetilde{P}$ are monic centered of degree $k+1$, $P_{0}(z)=z^{k+1}$, $\widetilde{P}_{0}(\tilde{z})=\tilde{z}^{k+1}$, $P_{\epsilon}(0)=-{\epsilon}$, and $\widetilde{P}_{\tilde{{\epsilon}}}(0)=-\tilde{{\epsilon}}$. Suppose the two families are locally conjugate through a change of coordinate and parameter $(z,{\epsilon})\mapsto(\tilde{z},\tilde{{\epsilon}})=(\varphi(z,{\epsilon}),h({\epsilon}))$. Then there exist $m\in{\mathbb{Z}}_{k}$ and $T\in{\mathbb{C}}\{{\epsilon}\}$ such that

[TABLE]

where $\Phi_{\epsilon}^{T({\epsilon})}$ is the flow of $\dot{z}=P_{\epsilon}(z)/(1+A(\epsilon)z^{k})$ at time $T({\epsilon})$.

Moreover, $A({\epsilon})=\tilde{A}(h({\epsilon}))$ and $P_{{\epsilon}}=\tilde{P}_{h({\epsilon})}$ hold for ${\epsilon}$ near [math].

Proof.

We use a method of infinite descent as in the proof of Theorem 3.5 in [RT]. Since the proof is completely similar, we will be brief on the details. Before starting the infinite descent, we must reduce the problem.

First reduction. We first consider the case ${\epsilon}=0$, for which the theorem follows from a mere calculation. Then $\varphi_{0}^{\prime}(0)=\exp(2\pi im/k)$ for some $m$. We change $(z,{\epsilon})\mapsto\left(\exp(2\pi im/k)z,\exp(2\pi im/k){\epsilon}\right)$ in the first vector field, so as to limit ourselves to the case $\varphi_{0}^{\prime}(0)=1$.

Second reduction. It is easily checked that the flow $\Phi_{0}^{t}$ of $\dot{z}=z^{k+1}/(1+A(0)z^{k})$ at time $t$ has the form

[TABLE]

Let $\Phi_{\epsilon}^{t}$ be the flow of the first equation at time $t$, let

[TABLE]

and let

[TABLE]

We want to solve $K({\epsilon},t)=0$ by the implicit function theorem. We know that there exists $t_{0}$ such that $K(0,t_{0})=0$ because of the form of $\Phi_{0}^{t}$ in (3.23). Indeed, $K(0,t)=\phi_{0}^{(k+1)}(0)+t(k+1)!$, yielding $\frac{\partial K}{\partial t}(0,0)=(k+1)!\neq 0$. Hence there exists a unique analytic germ $T({\epsilon})$ such that $K({\epsilon},T({\epsilon}))\equiv 0$ and $T(0)=t_{0}$. We change $z\mapsto\Phi^{T({\epsilon})}z$ in the first system.

The infinite descent. After the two reductions, we can suppose that $\varphi_{0}=id$ and that $\varphi_{\epsilon}^{(k+1)}(0)\equiv 0$. We now show that $\varphi_{\epsilon}={\rm Id}$ and $h({\epsilon})={\epsilon}$. Note that $A\equiv\widetilde{A}\circ h$ since the sum of the residues at the singular points is invariant. Let

[TABLE]

where all $b_{j},c_{j},f_{j}\in{\mathbb{C}}\{{\epsilon}\}$, and we simply write $b_{j}$ instead of $b_{j}({\epsilon})$, etc. for the functions $b_{j},c_{j},f_{j}$ and $h$. We introduce the principal ideal $I=\langle{\epsilon}\rangle$ in ${\mathbb{C}}\{{\epsilon}\}$. We show by induction that $b_{j}-c_{j},h-{\epsilon},f_{j}\in I^{n}$ for all $n\in{\mathbb{N}}^{*}$, from which it will follow that they are identically zero. Note that $h-{\epsilon}=b_{0}-c_{0}$.

The conjugacy condition is

[TABLE]

which we simply write as $\sum_{j\geq 0}g_{j}z^{j}=0$. Hence all $g_{j}$ must be identically [math]. The $g_{j}$ are quite complicated but they have a very simple structure of linear terms and this is what we will exploit.

• •

From the two reductions, it is clear that $h,b_{j},c_{j},f_{j}\in I$. This is our starting point.

• •

The only linear terms in the equations $g_{j}=0$ for $j=0,\dots,k-1$, are $b_{j}-c_{j}$. Hence $b_{j}-c_{j}\in I^{2}$.

• •

The equations $g_{k+j}=0$ with $0\leq j\leq k$ yield $f_{j}\in I^{2}$, since the only linear terms are $A(c_{j}-b_{j})+(k+1-j)f_{j}=0$ when $j<k$ and $Af_{0}+f_{k}$ when $j=k$.

• •

Remember that $f_{k+1}\equiv 0$ because of the reduction.

• •

The equations $g_{\ell}=0$ with $\ell>2k+1$ yield $f_{\ell-k}\in I^{2}$, since the only linear terms in $g_{\ell}$ are $-(\ell-2k-1)(f_{\ell-k}+Af_{\ell-2k})$.

• •

Hence, all $b_{j}-c_{j},f_{j}\in I^{2}$.

• •

We now suppose that $b_{j}-c_{j},f_{j}\in I^{n}$, and we want to show that there are in $I^{n+1}$.

• •

The equations $g_{j}=0$ for $j=0,\dots k-1$, yield $b_{j}-c_{j}\in I^{n+1}$.

• •

The equations $g_{k+j}=0$ with $0\leq j\leq k$ yield $f_{j}\in I^{n+1}$.

• •

The equations $g_{\ell}=0$ with $\ell>2k+1$ yield $f_{\ell-k}\in I^{n+1}$.

• •

Hence, all $b_{j}-c_{j},f_{j}\in I^{n+1}$.

This concludes the proof. ∎

Acknowledgements

We would like to thank Wolf Jung for stimulating discussions and for suggesting a simplification in Section 2.2.