Abstract
In this paper we study topological and analytical conditions on the orbits of a germ of diffeomorphism in the complex plane in order to obtain periodicity. In particular, we give a simple proof of a finiteness criteria for groups of analytic diffeomorphisms, stated in Brochero Martínez 2003. As an application, we derive some consequences about the integrability of complex vector fields in dimension three in a neighborhood of a singular point.
Key words
Complex germs of diffeomorphims; singular holomorphic foliations; integrability of vector fields; closed orbits
1 - INTRODUCTION
The relationship between periodic subgroups of and the integrability of germs of vector fields at was established in Mattei and Moussu 1980. There the authors show that the topological condition of finiteness of the orbits is sufficient to ensure the periodicity of a finitely generated sugbroup at . As a consequence, they obtain that the topological condition of closeness of the orbits of a germ of vector field is equivalent to the existence of a first integral, i.e., a germ of map whose level sets contain the orbits of . The link between these two objects is obtained in terms of the so called holonomy group, introduced by Charles Ehresmann.
In this paper we show that for a completely distinct phenomena occur. In fact we present and explicit example (Example 2.1 2.1 - Preliminaries Let Diff(ℂn,0) denote the group of germs of diffeomorphisms of ℂn fixing the origin. Let G∈Diff(ℂ2,0) and V be a neighborhood of the origin, where a representative (also denoted by G) of the germ G is defined. Then we denote by 𝒪 V + ( G , x ) = { G ∘ ( n ) ( x ) : G ∘ ( j ) ( x ) ∈ V , j = 0 , … , n } the so-called positive semiorbit of x∈V by G. Analogously, the negative semiorbit of x∈V by G is the set 𝒪V-(G,x):=𝒪V+(G-1,x). The orbit of x∈V by G is the set 𝒪V(G,x)=𝒪V+(G,x)∪𝒪V-(G,x). The cardinality of 𝒪V(G,x) is denoted by |𝒪V(G,x)|. A diffeomorphism f∈Diff(ℂn,0) is said to be tangent to the identity if it can be expanded in series of homogeneous polynomials as f=(f1,…,fn) with fj(z)=zj+fj,νj(z)+⋯, where fj,νj≢0 and ord(fj,k)=k whenever fj,k≢0. Then one says that ν(f)=min{ν1,…,νn} is the order of f. The set of germs of diffeomorphisms tangent to the identity is denoted by Diff1(ℂn,0). Let f∈Diff1(ℂ2,0), then we say that φ:Ω⟶ℂn is a parabolic curve for f at the origin if it is an injective holomorphic map satisfying the following properties: Ω⊂ℂ is a simply connected domain with 0∈ ∂Ω; φ is continuous at the origin and φ(0)=0∈ℂn; φ(Ω) is invariant under f and (f|φ(Ω))∘(n)→0∈ℂn as n→∞. Let f(x)=x+fν(x)+O(∥x∥ν+1), where fν(x)=(f1,ν(x),f2,ν(x))≢0 with fj,ν being homogeneous polynomial of degree ν. Then we say that f is dicritical if x2f1,ν(x)-x1f2,ν(x)≡0 and non-dicritical otherwise. The relationship between parabolic curves and dicritical fixed points is given by the following result. Theorem 2.1 (Abate 2001). Let f∈Diff1(C2,0) be a dicritical germ of holomorphic map tangent to the identity, then f admits infinitely many parabolic curves. ) showing that the finiteness of the orbits of a cyclic subgroup of is not enough to ensure its periodicity. From this example we construct a vector field at (Example 3.1) whose leaves are closed but are not contained in the level sets of a map-germ .
In the sequel we show that the periodicity of a germ of diffeomorphism at is achieved if we add one further topological condition, i.e., the finiteness of the orbits together with the Lyapunov stability condition. In terms of vector fields this may be translated by saying that admits a first integral if and only if its orbits are closed off the origin and transversely stable (in the sense of Lyapunov and Reeb) with respect to a distinguished smooth separatrix.
In the final part of the paper we show that these new topological aspects may be reinterpreted in terms of flags of foliations (Theorem 3.4). Finally we want to mention that we also present in this paper a simple proof of a finiteness criteria for groups of analytic diffeomorphisms (Theorem 2.2), stated in Brochero Martínez 2003, since it is one of our main ingredients.
2 - Germs of diffeomorphisms with finite orbits
2.1 - Preliminaries
Let denote the group of germs of diffeomorphisms of fixing the origin. Let and be a neighborhood of the origin, where a representative (also denoted by ) of the germ is defined. Then we denote by
the so-called positive semiorbit of by . Analogously, the negative semiorbit of by is the set . The orbit of by is the set . The cardinality of is denoted by .
A diffeomorphism is said to be tangent to the identity if it can be expanded in series of homogeneous polynomials as with , where and whenever . Then one says that is the order of . The set of germs of diffeomorphisms tangent to the identity is denoted by .
Let , then we say that is a parabolic curve for at the origin if it is an injective holomorphic map satisfying the following properties:
-
is a simply connected domain with ;
-
is continuous at the origin and ;
-
is invariant under and as .
Let , where with being homogeneous polynomial of degree . Then we say that is dicritical if and non-dicritical otherwise. The relationship between parabolic curves and dicritical fixed points is given by the following result.
Theorem 2.1 (Abate 2001).
Let be a dicritical germ of holomorphic map tangent to the identity, then admits infinitely many parabolic curves.
2.2 - Infinitely many invariant sets
In Brochero Martínez 2003 the following result is announced as Theorem 3.1, but unfortunately its proof seems to be incomplete.
Theorem 2.2.
Let , then the group generated by is finite if and only if there exists a neighborhood of the origin such that for all and preserves infinitely many analytic invariant curves at .
Here we shall present a simple and consistent proof for it. Since the necessary part of Theorem 2.2 is trivial, then we only prove its sufficient part in the following series of claims. For this sake, we need to recall some terminology from complex dynamics.
A germ of curve is a separatrix for if as germs of curves at the origin. Such a separatrix is called periodic of period if for each . The curve is not necessarily irreducible and may have several branches. The map takes a branch into a branch and may interchange these branches. Nevertheless, since has only a finite number of branches, we conclude that for each branch there is such that . Then clearly we have , for all , where . Thus, we have:
Claim 1.
For each separatrix of there is such that each branch of is invariant with respect to .
Remark 2.1.
It is well-known that a map germ with finite orbits is necessarily periodic (Mattei and Moussu 1980). Let now be an irreducible separatrix of . Put . We claim that if has finite orbits then is periodic. Indeed, in case is (irreducible and) smooth this is immediate. In the general (irreducible) case we take a Newton-Puiseux parametrization . This is a holomorphic injective map so that we may consider the “lift”, i.e., the holomorphic map that satisfies . Then also has finite orbits and thus, is periodic. Therefore, the same holds for .
Combining Claim 1 and Remark 2.1 we promptly obtain:
Claim 2.
For any separatrix of there exists such that is periodic of period , i.e., .
In view of the above result, from now on we suppose that admits infinitely many periodic separatrices.
We say that a set of separatrices for is in general position if their first tangent cone intersect the exceptional divisor in at least three distinct points.
Claim 3.
Let and be distinct periodic separatrices for , then after a finite number of blowing-ups on there appears a local map-germ admitting an infinite set of irreducible periodic separatrices in general position.
Proof.
It is immediate that after blowing-up a finite number of times at least three distinct curves will intersect the exceptional divisor in three distinct points in such a way that one of these points intersects infinitely many separatrices. ∎
Claim 4.
Suppose has a set of irreducible periodic separatrices in general position. Then there exists such that is tangent to the identity.
Proof.
Let , , be three distinct periodic separatrices in general position of orders respectively . Let , then has three distinct irreducible separatrices in general position. Now, let be the strict transform of by a blow-up . If is the exceptional divisor of this blow-up, then it follows immediately from the hypothesis that . Let us prove that the first jet of is given by a diagonal matrix. Let , where . After one blowing-up we have
Since , then and . Therefore, Now, suppose the three irreducible invariant curves are smooth (or resolved after one blowing-up), then we may suppose without loss of generality that is an invariant periodic curve for . In particular, its strict transform is periodic with respect to . Since , then is a root of unity. The result then follows. On the other hand, suppose is a singular irreducible curve invariant by (not necessarily periodic) and recall Remark 2.1. Let be the Newton-Puiseux parametrization for and be given by . If , then
thus and ; i.e., is a root of unity. The result then follows. ∎
Claim 5.
Suppose admits an infinite number of separatrices, then along its resolution there appears a dicritical map-germ tangent to the identity in a neighborhood of .
Proof.
Let be the strict transform of by a resolution , then to each invariant curve there corresponds a fixed point for along . Therefore, at least one of the projective spaces composing the exceptional divisor admits infinitely many separatrices for transversal to it. The result then follows. ∎
Therefore, in order to prove Theorem 2.2 we just have to combine the above claims with Abate’s theorem.
2.3 - Invariant functions
A finitely generated subgroup is finite provided that it has finite pseudo-orbits (Mattei and Moussu 1980). Contrasting with the one dimensional case, in greater dimensions the finiteness of the orbits is not enough to ensure the periodicity of the group.
Example 2.1.
Consider the map . The orbits of are confined in the level sets of and are clearly finite. Notice that as , thus is not periodic nor linearizable. Furthermore, the orbits are far from being stable, since in each line the map acts as a translation.
Blowing up this diffeomorphism (cf. Abate 2001) at the origin one has
whose orbits are finite and confined in the level sets of . Further, acts in these level sets of in some sort of translation whose orbits increase in cardinality as .
Therefore in order to obtain periodicity we need to ask some further conditions. We say that two germs of holomorphic functions are generically transverse if is not identically zero.
Theorem 2.3.
Let be generically transverse germs and be a complex map germ having finite orbits and preserving the level sets of both and . Then is periodic.
Proof.
The idea of the proof is the following: Since and are generically transverse, then one can find a pure meromorphic function whose level sets are preserved by . Hence, the infinitely many curves , with , intersect the origin and are invariant by . Thus Theorem 2.2 ensures that is periodic. Now let us construct . If is already pure meromorphic, then it is enough to pick . Otherwise one has , where , and is a germ of holomorphic function not divisible by . Clearly, is -invariant, thus if it has an irreducible component distinct from the irreducible components of , then must be a -invariant pure meromorphic function. Suppose that the decomposition in irreducible components of and are of the form and . Since is not divisible by , then there must be such that . If there is also such that , then is a pure meromorphic -invariant function. From now on we suppose that for all with at least one such that . If there is such that , then after reordering the indexes (if necessary) we may suppose that: (i) for all ; (ii) for all ; for some . Then is a -invariant germ of holomorphic function. Now, let (where denotes the integer part of ), then a straightforward calculation shows that is a pure meromorphic function. Hereafter we suppose that for all . Recall that the Euclid’s algorithm of a pair of positive integers , , is the sequence of pairs of positive integers given by: (1) ; (2) , where and ; (3) ; and (4) and . This is called the Euclid’s sequence of the pair . For simplicity, suppose that and have only two irreducible components, say and , and let and be the Euclid’s sequence of and , respectively. If , then and . If , then is a -invariant germ of pure meromorphic function, otherwise and is a -invariant germ of pure meromorphic function. Arguing inductively along the Euclid’s sequences of and one can always construct a -invariant pure meromorphic function unless for all . But this means that and for some . Therefore, , , and are powers of the same holomorphic function , thus and cannot be generically transverse. A contradiction! The reasoning in the case of many irreducible factors is analogous, being in fact a consequence of the above reasoning. ∎
In particular, Theorem 2.3 ensures that the map defined in Example 2.1 does not preserve the level sets of a couple of generically transverse functions .
2.4 - Invariant foliations
Let be a germ of holomorphic 1-form at . Assume that Sing and let be the germ of foliation at given by the Pfaff equation . We denote by the subgroup of given by those preserving , i.e., such that .
Given a map , we have for a (unique) formal vector field of order at least two (cf. Brochero Martínez et al. 2008, Câmara and Scárdua 2012) called the infinitesimal generator of . Let be a germ of foliation at having a dicritical component (i.e., admitting infinitely many separatrices through the origin). We shall say that is adapted to if there is a resolution of such that has infinitely many curves transverse to (this happens precisely when we blow-up a dicritical component of along the resolution of ).
Lemma 2.4.
Let be a germ of foliation at having a dicritical component and adapted to . Then having finite orbits if and only if is the identity.
Proof.
For simplicity we shall write . Let be the resolution of introduced in Abate M. 2001, the strict transform of via , and the lifting of . Since , then . If is a dicritical component of defined in a neighborhood of the irreducible component , then it is given in appropriate coordinate systems by a fibration transversal to , up to a finite number of singular leaves or smooth leaves tangent to . More precisely, there is an open set such that can be seen as a family of germs of automorphisms of with parameters in (see Figure ). Let be given by for some , then the classical Leau-Fatou flower theorem (Bracci 2004) says that has a parabolic fixed point at the origin, unless it is the identity. The result then follows by analytic continuation. ∎
2.5 - Stability
A germ of map is said to be positively semistable if for any representative , where is an open neighborhood of the origin, and any open set there is an open subset such that for all , i.e., all positive iterates of starting in remain in . Notice that any hyperbolic attractor satisfy the previous condition, but if we add the hypothesis of finiteness of the positive orbits, then the map must be periodic.
Lemma 2.5.
Let be represented by the map , where are open neighborhoods of the origin with compact closure. Suppose that:
-
is positively semistable.
-
has finite orbits.
Then is periodic, i.e., there is such that .
Proof.
By the positive semistability, there are and as above with and satisfying for all and . Now consider the analytic set , where . Then is a closed set without interior points. Since the orbits of are all periodic we have . From Baire’s category theorem, some must have an interior point and therefore by the Identity Principle is periodic. ∎
3 - Applications to foliations
3.1 - Integrability, first integrals, and closed orbits
The problem of deciding whether a vector field or, more generally, an ordinary differential equation can be integrated by studying its number of non-transcendent solutions goes back to H. Poincaré, Dulac (cf. Dulac 1912) and other authors. The classical theorem of G. Darboux (cf. Jouanolou 1979) states that a polynomial vector field in the complex plane admits a rational first integral provided that it admits infinitely many algebraic solutions. Our natural framework is the class of analytic equations. With the arrival of the Theory of Foliations the use of geometrical/topological methods has given an important contribution to the comprehension of the problem as well as some important results. Indeed, a holomorphic vector field defined in a neighborhood , , of the origin , with an isolated singularity at the origin, defines a germ of one-dimensional holomorphic foliation (with a singularity at the origin), and vice-versa.
In dimension two (or codimension one) a classical result (cf. Mattei and Moussu 1980) states that a germ of holomorphic vector field at the origin of admits a holomorphic first integral if and only if : (i) the leaves are closed off the origin; (ii) only finitely many of them are separatrices, i.e., adhere to the origin.
Condition (ii) is usually known as non-dicriticalness of the (germ of) foliation induced by the (germ of) vector field (cf. Camacho and Sad 1982). A foliation germ admitting a pure meromorphic first integral is necessarily dicritical. An example of Suzuki shows that there is no such topological criteria for existence of a meromorphic first integral (cf. Suzuki 1977, Klughertz 1992). Also interesting is the point of view adopted in Alexander and Verjovsky 1988, where the authors prove the existence of a holomorphic first integral for a germ of singular holomorphic vector field in dimension , under the hypothesis of existence of a uniform bound for the volume of the orbits of the vector field and some additional condition that restricts the “dicritical case”.
Our goal is to investigate topological conditions assuring the existence of holomorphic first integrals for vector field germs in dimension . This is done in Theorem 3.4. In few words, our result shows, for a generic class of singularities, an equivalence between the existence of a holomorphic first integral and the fact that the orbits are closed off the origin, plus the existence of a suitable stable separatrix, or the existence of a suitable flag, i.e., a codimension one foliation containing the orbits of the vector field. Our result may be seen as a kind of Reeb stability theorem for singularities of complex vector fields.
Let us introduce the notation we use, already used in Câmara and Scárdua 2009. Denote the ring of germs of holomorphic functions at by and its maximal ideal by . By we denote the -module of germs at the origin of holomorphic vector fields. Given a germ , we denote by the germ of one-dimensional holomorphic foliation at induced by . Let us make clear the notions we use:
Definition 3.1 (holomorphic first integral).
Let be a germ of one-dimensional holomorphic foliation at . A germ of holomorphic map is a holomorphic first integral for if:
-
is a submersion off a codimension analytic subset.
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The leaves of are contained in the level curves of .
A meromorphic function germ at is called -invariant if the leaves of are contained in the level sets of . This can be written as .
3.2 - Non-degenerate generic vector field germs
Next we describe the class of vector field germs we shall work with. Agerm is non-degenerate if its linear part is non-singular. Generically (in terms of the Krull topology for the coefficients of ) the map has distinct eigenvalues, is diagonalizable, and has an isolated singularity at the origin. From Poincaré-Dulac, Siegel, and Brjuno linearization theorems and from Camacho, Kuiper, and Palis 1978, generically (i.e., for a full measure subset of the set of the set of germs of holomorphic vector fields), up to a change of coordinates, the vector field leaves invariant the coordinate hyperplanes. We then introduce the following definition:
Definition 3.2 (Non-degenerate generic vector field germs).
We shall say that is non-degenerate generic if is non-singular, diagonalizable and, after some suitable change of coordinates, leaves invariant the coordinate planes.
We shall denote by the set of germs of non-degenerate generic vector fields at . Let and be a separatrix of , i.e., an analytic curve germ invariant by intersecting the origin. From the Newton-Puiseux parametrization theorem one knows that has the topology of a punctured disc . Then we denote by the cyclic holonomy of with respect to evaluated at a section transverse to , where is a single point. We can choose to be biholomorphic to a disc in with center corresponding to . With this identification the group is analytically conjugate to a subgroup of the group .
An -invariant germ is called adapted to if it can be written locally in theform , where are relatively prime, , where and denote the zero sets of and respectively, and is pure meromorphic for a generic transverse section to . Given we have if and only if for some nonvanishing holomorphic function germ we have . In this case we say that and are tangent. Any vector field germ admitting a holomorphic first integral must satisfy the following condition (cf. Câmara and Scárdua 2009).
Definition 3.3 (condition ).
A germ of generic vector field satisfies condition if there is a real line through the origin containing all the eigenvalues of and such that not all the eigenvalues belong to the same connected component of .
There is therefore one isolated eigenvalue of . The above condition holds for if and only if holds for any vector field such that and are tangent. Condition implies that is in the Siegel domain, but is stronger than this last. Denote by the isolated eigenvalue of and by its corresponding invariant manifold (the existence is granted by the classical invariant manifold theorem). We call the distinguished axis of . We shall say that is transversely stable with respect to if for any representative of the germ , defined in an open neighborhood of the origin, any open section transverse to with , and any open set there is an open subset such that all orbits of through intersect only in .
In this paper we prove various equivalent conditions for the integrability of a generic germ of complex vector field singularity in dimension three (cf. Theorem 3.4). This full statement involves the notion of flag (Corrêa and Soares 2013, Mol 2011) and Kupka singularities (Calvo-Andrade 1994, Kupka 1964), to be developed later in this paper. For the moment we state the following topological criteria.
Theorem 3.1.
For any the following conditions are equivalent:
-
has a holomorphic first integral.
-
satisfies condition and the leaves of are closed off the origin and transversely stable with respect to .
From this result we conclude the invariance of the existence of a holomorphic first integral for generic germs in dimension three under topological equivalence.
Corollary 3.2.
Let be generic germs of holomorphic vector fields, both satisfying condition . Assume that and are topologically equivalent. Then has a holomorphic first integral if and only if admits a holomorphic first integral.
We stress that Theorem 3.1 above can be completed (cf. Theorem 3.4) by weakening the topological hypothesis on the orbits, replacing the transverse stability by the existence of a suitable flag, i.e., a codimension one foliation tangent to .
3.3 - Closed leaves and first integrals
We show that the closing of the leaves is not sufficient to ensure the existence of first integrals for with . We first remark (cf. also Câmara and Scárdua 2009, Proposition 1, Section 2.3) that for a generic vector field germ the local holonomy group is generated by a resonant map preserving two smooth curves crossing transversely. In particular, one cannot expect a map like the map in Example 2.1 appearing as the (generator of the) holonomy of some with respect to . Thus, we blow up such map and look to a neighborhood of the point determined by the exceptional divisor and the strict transform of . Let be given by
where , , and . Now consider the closed loop given by and let be its lifting along the leaves of starting at . In particular, the map given by is a generator of . Since belongs to a leaf of , then
From this vector equation one has , thus . Furthermore
Example 3.1.
Let , then is invariant by and the holonomy of with respect to evaluated at has the form with , where and satisfy respectively equations (1) and (2) above. Now if we let , then (2) is written in the form
More precisely , thus for some . Since , then and otherwise. Therefore and . On the other hand, (1) is written in the form
Analogously, for all . Since , then and for all . Finally . Now recall that the solution to the Cauchy problem
is given by
In particular, . Thus, if we set , then and .
Completing the above example we obtain:
Proposition 3.3.
There is a vector field which satisfies condition and has all leaves closed but does not admit a holomorphic first integral.
Proof.
We consider the vector field . Then . After one blow up along the -axis one has
which has an isolated singularity at the origin, and whose holonomy with respect to the -axis is precisely the map in Example 2.1. Thus it satisfies condition and has all leaves closed but, from Proposition 2.3 and from what we have observed in Example 3.1 above, does not admit a holomorphic first integral. ∎
3.4 - Stability, flags, and first integrals
Proposition 3.3 shows that the statements of Theorems 1.2 and 1.3 in Câmara and Scárdua 2009 are incomplete. The correct statement involves a natural adaptation of a classical notion for regular smooth foliations.
3.4.1 Stability
We consider a germ satisfying condition .
Definition 3.4 (stability).
The germ is transversely stable with respect to if for any representative of the germ , defined in an open neighborhood of the origin, any open section transverse to with , and any open set there is an open subset such that all orbits of through intersect only in .
As above mentioned, Definition 3.4 is a natural adaptation to our singular framework of the classical notion of stability due to Lyapunov (Lyapunov 1892) and rediscovered by Reeb (cf. Godbillon 1991).
3.4.2 Flags, dicritical components, and Kupka singularities
Let us first recall some basic notions from singularities of foliations in dimension two. Let be a germ of singular foliation at the origin , then Seidenberg’s theorem (Seidenberg 1968) gives a reduction of the singularities of by the blow-up method. This is also called desingularization or resolution of . We say that has a dicritical component if its resolution contains a non-invariant projective line. This is equivalent to say that has infinitely many separatrices, i.e., infinitely many analytic leaves intersecting the origin (Camacho and Sad 1982).
Given , we denote its corresponding foliation by . By a flag containing, we mean a germ of codimension one holomorphic foliation at giving by an integrable holomorphic -form with singular set of codimension containing the origin and with the property that (for some representatives of each foliation defined in a common domain containing the origin) each leaf of is contained in some leaf of . This last property is enclosed in the formula . The notion of flag is detailed in Mol 2011. A codimension two irreducible component is a Kupka type component if does not vanish along . According to Kupka’s theorem (Calvo-Andrade 1994, Kupka 1964), for a representative of in an open neighborhood , where is given by an integrable holomorphic 1-form , and a representative of the component , there is a germ of foliation at such that for each point there is a holomorphic submersion , with the property that and . The foliation is then called the Kupka transverse type of along the Kupka component . One says that the Kupka component is dicritical if the corresponding transverse type has a dicritical singularity at the origin , in the above sense.
Example 3.2.
A particular case of a dicritical Kupka component is the one induced by the codimension one foliation , where and . In fact it represents the product foliation with .
We shall call a dicritical Kupka component of radial type if in suitable coordinates is given by a -form as above.
Given a flag containing the foliation , consider its restriction to a transverse section as above. Since is transverse to , it is also transverse to off the singular set and therefore one may identify the germ induced by at the point with the germ of foliation at the origin . Then one says that is dicritical if this corresponding germ in dimension two is dicritical.
Definition 3.5 (Adapted flag).
Let now with linear part given by , then the local holonomy generator of with respect to the distinguished axis is periodic with linear part given by . In particular, is tangent to the identity. Therefore, this map can be written locally in the form , where is its infinitesimal generator. Then one says that is an adapted flag if is a flag such that is a germ of foliation having a dicritical component adapted to .
Notice that the last definitions are of finite determinacy character. Furthermore, if is dicritical, then is automatically an adapted flag. Using this terminology, one may complete the statements in Câmara and Scárdua 2009 as follows.
Theorem 3.4.
Suppose that satisfies condition and let be the distinguished axis of . Then the following conditions are equivalent:
-
The leaves of are closed off the origin and transversely stable with respect to ;
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has finite orbits and is (topologically) stable;
-
is periodic;
-
has a holomorphic first integral.
Moreover, in terms of flags of foliations, the above conditions are also equivalent to each of the following conditions:
-
The leaves of are closed off the origin and there is an adapted flag ;
-
The leaves of are closed off the origin and there is a flag such that is a Kupka component of radial type.
Proof of the first part of Theorem 3.4.
It follows immediately from the definition of transverse stability of germs of vector fields and from (topological) stability of maps that (1) implies (2). It comes from Lemma 2.5 that (2) implies (3). Now let us prove that (3) implies (4). Since satisfies condition and is linearizable, then Elizarov and Ilyashenko 1984 ensures that is linearizable. Therefore, one may suppose without loss of generality that , where . Since is periodic, one may suppose without loss of generality that . The result then follows from Lemma 2.3 in Câmara and Scárdua 2009. Finally let us verify that (4) implies (1). The existence of a first integral for ensures that the leaves of are closed off . Furthermore, admits a couple of generically transverse -invariant holomorphic functions whose restrictions to have the level sets preserved by . Thus Theorem 2.3 ensures that is periodic and, in particular, topologically stable. Hence the leaves of are transversely stable with respect to . This proves the first four equivalences in Theorem 3.4. ∎
As a straightforward consequence (cf. Theorem 2 in Câmara and Scárdua 2009), one has the following topological criterion for the existence of invariant meromorphic functions for elements in .
Theorem 3.5.
Let satisfy condition with distinguished axis . Suppose that has closed leaves off the origin and is transversely stable with respect to . Then there is an -invariant meromorphic function adapted to .
Now we study the topological invariance of the existence of a holomorphic first integral for a generic germ of holomorphic vector field, as a consequence of our preceding results. We recall that two germs of holomorphic vector fields and at the origin are topologically equivalent if there is a homeomorphism , where are neighborhoods of the origin and are vector fields representing respectively, such that takes orbits of into orbits of . Such a map takes separatrices of into separatrices of : indeed, a separatrix of is an orbit which is closed off the origin, and the same holds for its image under . Assume that the vector field is generic satisfying condition and admits a holomorphic first integral. In this case one has:
Claim 6.
The vector field is analytically linearizable, say with in suitable local coordinates . In particular, admits a unique separatrix off the dicritical plane , and this separatrix corresponds to the distinguished separatrix .
Proof.
Indeed, the analytic linearization of is a straightforward consequence of the first part of Theorem 3.4 (or, since by hypothesis there is a holomorphic first integral, in view of Lemma 2.5 and Elizarov and Ilyashenko 1984). In this normal form
the “dicritical plane” is the plane and the distinguished separatrix is the -axis. The orbit of through the point is given by
Thus, if accumulates at the origin, either or and . For instance, if , then the orbit is contained in the hypersurface , which does not accumulate at the origin. ∎
Lemma 3.6.
A topological equivalence takes the distinguished axis of into the distinguished axis of .
Proof.
Indeed, as we have seen above, the image is some separatrix of . If this is not the distinguished axis of , then the distinguished axis of is taken by into a separatrix other than the distinguished axis of . Therefore, according to Claim 6, must be a separatrix of the “dicritical” part of , i.e., in the coordinates above, where , we have . Nevertheless, any invariant neighborhood of a leaf contained in a dicritical separatrix of off the origin intersects infinitely many separatrices (namely, those contained in the intersection of this neighborhood with the dicritical plane ). On the other hand, this same phenomena does not occur for arbitrarily small invariant neighborhoods of a leaf contained in the distinguished axis of . Therefore, necessarily is the distinguished axes of . ∎
From the above considerations we immediately obtain Corollary 3.2 from Theorem 3.4.
3.4.3 Flags and integrability
In this section we prove the remaining part of Theorem 3.4. We address therefore the following problem. Given a germ of foliation by curves induced by a germ of vector field of the form
with , what are the consequences of the existence of a codimension germ of holomorphic foliation tangent to , which is transversely dicritical with respect to ?
We begin by studying the consequences of the existence of a flag foliation with a dicritical transverse type for a vector filed .
Lemma 3.7.
Let be a germ of foliation by curves at , an invariant curve of through the origin, and a codimension one foliation satisfying the following conditions:
-
is tangent to ;
-
There is a section transverse to such that is dicritical.
Then is transversely dicritical with respect to
Proof.
Since the orbits of are contained in the leaves of , then these leaves are invariant by the flow of . Therefore, if is another section transversal to and is an element of the holonomy pseudogroup of with respect to , then it is a diffeomorphism taking the leaves of onto the leaves of . ∎
We are now in a position to finish the proof of Theorem 3.4. For this sake, let us first recall some facts proved along this work and introduce some terminology. First notice that any admitting a holomorphic first integral must satisfy condition in Definition 3.3 (cf. Câmara and Scárdua 2009). Assume the curve is the -axis, let be a section transverse to , and be the holonomy of with respect to evaluated at .
End of the proof of Theorem 3.4.
First suppose all the leaves of are closed off the origin and that there is an adapted flag . Given a germ of leaf of it follows that the closure is a germ of analytic subset of pure dimension one (Gunning and Rossi 1965) at . Since this leaf is transverse to , one concludes that is a finite set. On the other hand, given a point , its orbit in the holonomy group is also contained in , so that it is a finite set. Thus the orbits of the generator of are finite. By hypothesis, for any the foliation has a dicritical component. Now consider a simple loop around the origin inside the -axis starting from . Pick a leaf of and consider the liftings of starting at points of , along the trajectories of . Then these liftings form a three dimensional real variety, say , whose intersection with is given by and (see Figure ). In particular, if is the generator of , then . For the -form , one has that is tangent to and is tangent to the induced foliation . Thus, is a leaf of . Since has a dicritical component and is a diffeomorphism with resonant linear part having finite orbits, then Lemma 2.4 ensures that is periodic (in particular linearizable and finite). Since has linearizable periodic holonomy, then it follows from Elizarov and Ilyashenko 1984 that the foliation is also analytically linearizable. Therefore, one may suppose without loss of generality that . This vector field has a holomorphic first integral. From the above linearization, it is easy to see that the flag foliation containing must have a linear dicritical Kupka transverse type along the -axis. In particular, is of radial type. This proves that (5) implies (1)-(4) and also (6). Since the converse is immediate, this proves that the first four conditions in Theorem 3.4 are equivalent to conditions (5) and (6). ∎
Remark 3.1 (Parabolic curves and smooth sets of fixed points cf. Abate 2001).
In our previous paper Câmara and Scárdua 2009 it is stated an integrability result mentioning only the fact that the leaves of are closed off . Nevertheless, as we saw above, this result is not correct. Indeed, there are such kind of vector fields without holomorphic first integral (cf. Example 3.3).
Let us identify precisely the missing point in Câmara and Scárdua 2009. This justifies the further topological conditions introduced above in order to correct the statements of the main theorems therein (Theorems 1.2 and 1.3 in Câmara and Scárdua 2009). Along these lines we shall keep all the notations introduced in Câmara and Scárdua 2009. In Theorem 3.6 of Câmara and Scárdua 2009 we have stated that every non trivial complex map germ fixing the origin admits a parabolic curve. Javier Ribon draw our attention to the fact that this is not true with the following example:
Let with and , then the orbits of the map are confined in the level sets of the first integral to the vector field . Therefore, has no orbit attracting to the origin, thus it does not admit any parabolic curve at the origin.
Some time after that Marco Abate communicated us the same fact showing that Theorem 3.6 in Câmara and Scárdua 2009 contradicts Proposition 2.1, p. 185, in Abate 2001. As a matter of fact, Lemma 3.5 (and thus Theorem 3.6) is not correct. This is due to the authors misinterpretation of the proof of Corollary 3.1 in Abate 2001 wrongly stated as Theorem 3.2 in Câmara and Scárdua 2009. Indeed, the correct statement is the following: Let and suppose that is a smooth curve through the origin such that . Then admits parabolic curves.
More precisely, one can check that this would be the appropriate hypothesis looking to the proof of Theorem 3.1 in Abate 2001. Now one can check that the diffeomorphism in the proof of Lemma 3.5 in Câmara and Scárdua 2009 does not satisfy the conditions of the above theorem.
We finish with an immediate consequence of the proof of Theorem 2.3.
Corollary 3.8.
Let and be the distinguished axis of . Suppose that admits a pure meromorphic first integral, then the holonomy group is periodic.
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Publication Dates
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Publication in this collection
Dec 2017
History
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Received
10 Apr 2017 -
Accepted
01 June 2017