Symmetries of a center singularity of a plane vector fields
Let D² ⊂ R² be a closed unit 2-disk centered at the origin O ∈ R², and F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a „center” singularity. Let D⁺(F) be the group of all diffe...
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Maksymenko, S.I. 2021-02-19T07:28:44Z 2021-02-19T07:28:44Z 2009 Symmetries of a center singularity of a plane vector fields / S.I. Maksymenko // Нелінійні коливання. — 2009. — Т. 12, № 4. — С. 507-526. — Бібліогр.: 18 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178419 515.145+515.146 Let D² ⊂ R² be a closed unit 2-disk centered at the origin O ∈ R², and F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a „center” singularity. Let D⁺(F) be the group of all diffeomorphisms of D² which preserve orientation and orbits of F. Recently the author described the homotopy type of D⁺(F) under the assumption that the 1-jet j¹ F(O) of F at O is non-degenerate. In this paper degenerate case j¹ F(O) is considered. Under additional ” nondegeneracy assumptions” on F the path components of D⁺(F) with respect to distinct weak topologies are described. These conditions imply that for each h ∈ D⁺(F) its path component in D⁺(F) is uniquely determined by the 1-jet of h at O. Нехай D² ⊂ R² — замкнений одиничний двовимiрний диск з центром у початку координат O ∈ R² та F — гладке векторне поле таке, що O є єдиною особливою точкою F, а всi iншi орбiти — простими замкненими кривими, що огортають O один раз. Таким чином, топологiчно O є особливiстю типу центр. Нехай D⁺(F) — група всiх дифеоморфiзмiв D², що зберiгають орiєнтацiю та орбiти поля F. Нещодавно автором було описано гомотопiчний тип D⁺(F) за умови, що 1-струмiнь j¹F(O) поля F в O є невиродженим. У цiй статтi розглядається вироджений випадок j¹F(O). За додаткової умови невиродженостi на F описано компоненти лiнiйної зв’язностi простору D⁺(F) вiдносно рiзних слабких топологiй. З цих умов випливає, що для кожного h ∈ D⁺(F) його компонента лiнiйної зв’язностi в D⁺(F) єдиним чином визначається 1-струменем h в O. This research is partially supported by grant of Ministry of Science and Education of Ukraine, № M/150-2009. en Інститут математики НАН України Нелінійні коливання Symmetries of a center singularity of a plane vector fields Симетрії центральної сингулярності векторних полів на площині Симметрии центральной сингулярности векторных полей на плоскости Article published earlier |
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Symmetries of a center singularity of a plane vector fields |
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Symmetries of a center singularity of a plane vector fields Maksymenko, S.I. |
| title_short |
Symmetries of a center singularity of a plane vector fields |
| title_full |
Symmetries of a center singularity of a plane vector fields |
| title_fullStr |
Symmetries of a center singularity of a plane vector fields |
| title_full_unstemmed |
Symmetries of a center singularity of a plane vector fields |
| title_sort |
symmetries of a center singularity of a plane vector fields |
| author |
Maksymenko, S.I. |
| author_facet |
Maksymenko, S.I. |
| publishDate |
2009 |
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English |
| container_title |
Нелінійні коливання |
| publisher |
Інститут математики НАН України |
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Article |
| title_alt |
Симетрії центральної сингулярності векторних полів на площині Симметрии центральной сингулярности векторных полей на плоскости |
| description |
Let D² ⊂ R² be a closed unit 2-disk centered at the origin O ∈ R², and F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a „center” singularity. Let D⁺(F) be the group of all diffeomorphisms of D² which preserve orientation and orbits of F. Recently the author described the homotopy type of D⁺(F) under the assumption that the 1-jet j¹ F(O) of F at O is non-degenerate. In this paper degenerate case j¹ F(O) is considered. Under additional ” nondegeneracy assumptions” on F the path components of D⁺(F) with respect to distinct weak topologies are described. These conditions imply that for each h ∈ D⁺(F) its path component in D⁺(F) is uniquely determined by the 1-jet of h at O.
Нехай D² ⊂ R² — замкнений одиничний двовимiрний диск з центром у початку координат
O ∈ R² та F — гладке векторне поле таке, що O є єдиною особливою точкою F, а всi iншi орбiти — простими замкненими кривими, що огортають O один раз. Таким чином, топологiчно
O є особливiстю типу центр. Нехай D⁺(F) — група всiх дифеоморфiзмiв D², що зберiгають
орiєнтацiю та орбiти поля F.
Нещодавно автором було описано гомотопiчний тип D⁺(F) за умови, що 1-струмiнь j¹F(O)
поля F в O є невиродженим. У цiй статтi розглядається вироджений випадок j¹F(O). За додаткової умови невиродженостi на F описано компоненти лiнiйної зв’язностi простору D⁺(F)
вiдносно рiзних слабких топологiй. З цих умов випливає, що для кожного h ∈ D⁺(F) його компонента лiнiйної зв’язностi в D⁺(F) єдиним чином визначається 1-струменем h в O.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/178419 |
| citation_txt |
Symmetries of a center singularity of a plane vector fields / S.I. Maksymenko // Нелінійні коливання. — 2009. — Т. 12, № 4. — С. 507-526. — Бібліогр.: 18 назв. — англ. |
| work_keys_str_mv |
AT maksymenkosi symmetriesofacentersingularityofaplanevectorfields AT maksymenkosi simetríícentralʹnoísingulârnostívektornihpolívnaploŝiní AT maksymenkosi simmetriicentralʹnoisingulârnostivektornyhpoleinaploskosti |
| first_indexed |
2025-11-26T11:58:49Z |
| last_indexed |
2025-11-26T11:58:49Z |
| _version_ |
1850620639522586624 |
| fulltext |
UDC 515.145+515.146
SYMMETRIES OF DEGENERATE CENTER SINGULARITIES
OF PLANE VECTOR FIELDS*
СИМЕТРIЇ ВИРОДЖЕНИХ ОСОБЛИВОСТЕЙ ТИПУ ЦЕНТР
ВЕКТОРНИХ ПОЛIВ НА ПЛОЩИНI
S. I. Maksymenko
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivska St. 3, Kyiv, 01601, Ukraine
e-mail: maks@imath.kiev.ua
Let D2 ⊂ R2 be a closed unit 2-disk centered at the origin O ∈ R2, and F be a smooth vector field such
that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once
around O. Thus topologically O is a „center” singularity. Let D+(F ) be the group of all diffeomorphisms
of D2 which preserve orientation and orbits of F.
Recently the author described the homotopy type ofD+(F ) under the assumption that the 1-jet j1 F (O)
of F at O is non-degenerate. In this paper degenerate case j1 F (O) is considered. Under additional ” non-
degeneracy assumptions” on F the path components of D+(F ) with respect to distinct weak topologies
are described. These conditions imply that for each h ∈ D+(F ) its path component in D+(F ) is uniquely
determined by the 1-jet of h at O.
Нехай D2 ⊂ R2 — замкнений одиничний двовимiрний диск з центром у початку координат
O ∈ R2 та F — гладке векторне поле таке, що O є єдиною особливою точкою F, а всi iншi ор-
бiти — простими замкненими кривими, що огортають O один раз. Таким чином, топологiчно
O є особливiстю типу центр. Нехай D+(F ) — група всiх дифеоморфiзмiв D2, що зберiгають
орiєнтацiю та орбiти поля F.
Нещодавно автором було описано гомотопiчний типD+(F ) за умови, що 1-струмiнь j1F (O)
поля F в O є невиродженим. У цiй статтi розглядається вироджений випадок j1F (O). За до-
даткової умови невиродженостi на F описано компоненти лiнiйної зв’язностi простору D+(F )
вiдносно рiзних слабких топологiй. З цих умов випливає, що для кожного h ∈ D+(F ) його ком-
понента лiнiйної зв’язностi в D+(F ) єдиним чином визначається 1-струменем h в O.
1. Introduction. Let D2 = {x2 + y2 ≤ 1} ⊂ R2 be a closed unit 2-disk centered at the origin
O ∈ R2, V ⊂ R2 be a closed subset diffeomorphic to D2, z ∈ IntV, and
F = F1
∂
∂x
+ F2
∂
∂y
be a C∞ vector field on V. We will say that F is a TC vector field on V with topological center
at z if it satisfies the following conditions:
(T1) z is a unique singular point of F,
(T2) F is tangent to ∂V, so ∂V is an orbit of F, and
(T3) all other orbits of F are closed.
∗ This research is partially supported by grant of Ministry of Science and Education of Ukraine, № M/150-2009.
c© S. I. Maksymenko, 2009
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4 507
508 S. I. MAKSYMENKO
Let F be a TC vector field on V. Then it easily follows from Poincaré – Bendixson theorem
[1] that there exists a homeomorphism
h = (h1, h2) : V → D2 (1.1)
Fig. 1
such that h(z) = O and for every other orbit o of F its image h(o) is a circle of some radius
c ∈ (0, 1] centered at the origin, see Fig. 1. This motivates the term TC which we use, see [2].
Moreover, since the first recurrent map of closed orbits is smooth, see [1], it can be assumed
that the restriction h : V \z → D2\O is a C∞ diffeomorphism.
Let F be a TC vector field on D2. In this case we will always assume that F (O) = 0.
Consider the following matrix
∇F =
∂F1
∂x
(O)
∂F1
∂y
(O)
∂F2
∂x
(O)
∂F2
∂y
(O)
.
We will call ∇F the linear part or the linearization of F at O.
Suppose ∇F is non-degenerate. Then it can be shown that there are local coordinates at O
in which
j1F (O) = −ay ∂
∂x
+ ay
∂
∂y
for some a 6= 0, so the 1-jet of F at O is a “rotation” . This class of singularities is well-studied
from many points of view, see e.g. [3 – 8]. In particular, in [5] normal forms of such vector fields
are obtained.
Denote by D(F ) the group of C∞ diffeomorphisms h of D2 such that h(o) = o for each
orbit o of F . Let also D+(F ) be the subgroup of D(F ) consisting of all orientation preserving
diffeomorphisms, andD∂(F ) be the subgroup ofD+(F ) consisting of all diffeomorphisms fixed
of ∂D2. We endowD(F ),D+(F ), andD∂(F ) with the weak W∞-topologies, see Section 6.
The main result of [2] describes the homotopy types of D∂(F ) and D+(F ) for a TC vector
field with non-degenerate ∇F, see (i) of Proposition 2.1.
Moreover, in [9] the author calculated the homotopy type ofD+(F ) for F being a “reduced”
Hamiltonian vector field of a real homogeneous polynomial f : R2 → R in two variables having
a local extremum at O, see Example 2.1. Notice that in almost all the cases of f, we have that
∇F = 0.
In the present paper we study D∂(F ) and D+(F ) for TC vector fields with degenerate ∇F
but satisfying certain additional “non-degeneracy” assumptions, see Theorems 2.1, 2.2, and 2.3.
The obtained results are not so complete as in the non-degenerate case due to the variety
of normal forms. We are able to prove that the inclusion D∂(F ) ⊂ D+(F ) induces isomor-
phisms of the homotopy groups πk for k ≥ 1 and also describe path components of D∂(F ) and
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 509
D+(F ) with respect to distinct weak topologies. It turns out that for each h ∈ D+(F ) its path
component in D+(F ) is uniquely determined by the 1-jet of h at O.
These results agree with the ones of [9] and they are essentially new for the case when ∇F
is degenerate but is not zero.
The principal difference of the presented technique from [2] that we do not require conti-
nuity of the inverse of the so-called shift-map of F, see § 2.4.
In [10] for each compact surface M the author calculated the homotopy types of the stabili-
zers and orbits of Morse functions on M with respect to the right action of the diffeomorphism
groupD(M) ofM.The results of the present paper as well as of [2] will be used in another paper
to extend calculations of [10] to a large class of smooth functions with degenerate singularities
on surfaces.
2. Formulation of results. Let F be a TC vector field on D2. Denote by E(F ) the subset of
C∞(D2, D2) which consists of all maps h : D2 → D2 satisfying the following conditions:
(i) h(o) = o for every orbit o of F ; in particular, h(O) = O;
(ii) h is a local diffeomorphism at O, though it can be non-bijective and even degenerate
outside O.
Evidently, E(F ) is a subsemigroup of C∞(D2, D2) with respect to the usual composition of
maps. Consider the map
j : E(F ) → GL (2,R), j (h) = J(h,O),
associating to every h ∈ E(F ) its Jacobi matrix J(h,O) at O. Let
L(F ) := j(E(F ))
be the image of j. Then a priori L(F ) is a subsemigroup of GL(2,R).
Let E+(F ) = j−1(GL+(2,R)) be the subset of E(F ) consisting of all maps h with positive
Jacobian at O.
Let also E∂(F ) ⊂ E+(F ) be the subsemigroup consisting of all maps h fixed on ∂D2, i.e.,
h(x) = x for all x ∈ ∂D2. Evidently,
D(F ) ⊂ E(F ), D+(F ) ⊂ E+(F ), D∂(F ) ⊂ E∂(F ). (2.1)
For r = 0, 1, . . . ,∞ denote by E(F )r (E+(F )r, etc.) the space E(F ), (E+(F ), etc.) endowed
with the weak Wr-topology, see § 6. Let also Eid(F )r (E+
id(F )r etc.) be the path component of
the identity map idD2 in E(F ), (E+(F ), etc.) with respect to the Wr-topology. Evidently, each
h ∈ E(F )\E+(F ) (if it exists) changes the orientation of D2, whence E+(F )r consists of full
path components of E(F )r. In particular,
E+
id(F )r = Eid(F )r, r = 0, 1, . . . ,∞.
It turns out that it is more convenient to work with E(F ) instead of D(F ). Moreover,
the following theorem shows that such a replacement does not loose the information about
homotopy types.
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
510 S. I. MAKSYMENKO
We will assume throughout that the identity map idD2 is a base point and therefore omit it
from the notation. For instance, we denote the n-th homotopy group πn(E(F )r, idD2) simply by
πnE(F )r and so on.
Theorem 2.1. Let F be a TC vector field on D2. Let D denotes one of the groups D(F ),
D+(F ), or D∂(F ), and E be the corresponding semigroup E(F ), E+(F ), or E∂(F ). By Dr (resp.
Er) we denote the topological space D (resp. E) endowed with the Wr-topology. Then
(1) the inclusion Dr ⊂ Er is a weak homotopy equivalence1 for r ≥ 1;
(2) in the W0-topology, the induced map π0D0 → π0E0 is a surjection;
(3) for each r ≥ 0 the semigroup π0Er is a group and any two path components of Er are
homeomorphic to each other.
Remark 2.1. In general, a topological semigroup may have path components which are non
homeomorphic to each other. For instance, this is often so for the semigroup of continuous
maps C(X,X) of a topological space X with non-trivial homotopy groups, see e.g. [11].
The next result describes the relative homotopy groups of the pair (E+(F ), E∂(F )).
Theorem 2.2. Let F be a TC vector field on D2. Then for each r ≥ 0,
πn(E+(F )r, E∂(F )r) =
{
Z, n = 1,
0, otherwise.
Hence the inclusion E∂(F ) ⊂ E+(F ) yields isomorphisms,
πnE∂(F )r → πnE+(F )r, n ≥ 2, (2.2)
and we also have the following exact sequence:
0 → π1E∂(F )r → π1E+(F )r → Z → π0E∂(F )r → π0E+(F )r → 0. (2.3)
Our next aim (see Theorem 2.3 below) is to obtain some information about the homotopy
groups of E(F ), E+(F ) and E∂(F ). First we recall necessary definitions and some preliminary
results.
Shift map. Let F : D2 × R → D2 be the flow generated by F and
ϕ : C∞(D2,R) → C∞(D2, D2)
be the map defined by
ϕ(α)(z) = F(z, α(z))
for α ∈ C∞(D2,R) and z ∈ D2. We will call ϕ the shift map along orbits of F and denote its
image in C∞(D2, D2) by Sh(F ),
Sh(F ) := ϕ(C∞(D2,R)) ⊂ C∞(D2, D2).
1 Recall that a map i : D → E is a weak homotopy equivalence if for each n ≥ 0 the induced map in :
πn(D, x) → πn(E , x) of homotopy sets (groups for n ≥ 1) is a bijection for each x ∈ D.
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 511
Lemma 2.1. The following inclusions hold true:
Sh(F ) ⊂ Eid(F )∞ ⊂ . . . ⊂ Eid(F )1 ⊂ Eid(F )0. (2.4)
If Sh(F ) = Eid(F )r for some r = 0, 1, . . . ,∞, then
Did(F )∞ = . . . = Did(F )r, (2.5)
whence the identity maps id : D(F )∞ → D(F )s and id : E(F )∞ → E(F )s for s ≥ r yield the
following bijections:
π0D(F )∞ ≈ . . . ≈ π0D(F )r, π0E(F )∞ ≈ . . . ≈ π0E(F )r.
Proof. The first inclusion in (2.4) follows from [12] (Corollary 21) and the others are evident.
The fact that (2.5) is implied by the assumption Sh(F ) = Eid(F )r is proved in [9].
The lemma is proved.
The following Proposition 2.1 and Example 2.1 describe some results about ker j, Sh(F ),
and Eid(F )r for TC vector fields. The most complete information is given for the cases when∇F
is non-degenerate and when F is a “reduced” Hamiltonian vector field of some homogeneous
polynomial on R2.
Proposition 2.1. Let F be a TC vector field on D2.
(1) If ∇F = 0, then Sh(F ) ⊂ ker j.
(2) Suppose that ∇F is degenerate but is not zero. Then there are local coordinates at O in
which ∇F =
(
0 a
0 0
)
for some a ∈ R\{0}. Define the following subsets of GL+(2,R) :
A++ =
{(
1 d
0 1
)
, d ∈ R
}
, A−− =
{(
−1 d
0 −1
)
, d ∈ R
}
,
A+− =
{(
1 d
0 −1
)
, d ∈ R
}
, A−+ =
{(
−1 d
0 +1
)
, d ∈ R
}
, (2.6)
A = A++ ∪ A−− ∪ A+− ∪ A−+.
Then
j(Sh(F )) = A++, j(E+(F )) ⊂ A++ ∪ A−−, j(E(F )) ⊂ A. (2.7)
(3) If ∇F is non-degenerate, then there are local coordinates at O in which F is given by
F (x, y) = α(x, y)
(
−y ∂
∂x
+ x
∂
∂y
)
+X
∂
∂x
+ Y
∂
∂y
, (2.8)
where α is a C∞-function such that α(O) 6= 0, and X, Y are flat at O. Moreover,
j−1(SO(2)) = Sh(F ) = Eid(F )∞ = . . . = Eid(F )0 = E+(F ),
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
512 S. I. MAKSYMENKO
Did(F )∞ = . . . = Did(F )0,
the inclusions D∂(F ) ⊂ E∂(F ) and D+(F ) ⊂ E+(F ) are homotopy equivalences with respect to
the W∞-topologies, D∂(F ) is contractible, and D+(F ) is homotopy equivalent to a circle.
(4) Let θ : D2\O → (0,+∞) be the function associating to each z ∈ D2\O its period θ(z)
with respect to F. Then θ is C∞ on D2\O and we will call it the period function for F.
In the cases (1) and (2), i.e., when ∇F is degenerate, lim
z→O
θ(z) = +∞ and thus θ can not
be even continuously extended to all of D2. On the other hand in the case (3) θ extends to a
C∞-function on all of D2 such that θ(O) 6= 0.
Proof. Statement (1) is a particular case of [13] (Lemma 5.3). (2) and (4) are established
in [2].
(3) Representation (2.8) is due to F. Takens [5], and all other statements are proved in [2].
Actually, F. Takens has shown that except for (2.8) there is also an infinite series of normal
forms for vector fields with a “rotation as 1-jet” , however the orbits of these vector fields are
non-closed, and so they are not TC.
The proposition is proved.
Example 2.1. Let f : R2 → R be a real homogeneous polynomial in two variables such that
O ∈ R2 is a unique critical point of f being its global minimum. Then we can write
f(x, y) =
k∏
j=1
Q
βj
j (x, y), (2.9)
where every Qj is a positive definite quadratic form, βj ≥ 1, and
Qj
Qj′
6= const for j 6= j′.
Then it is easy to see that D =
k∏
j=1
Q
βj−1
j is the greatest common divisor of partial derivatives
f ′x and f ′y. Let G = −f ′y
∂
∂x
+ f ′x
∂
∂y
be the Hamiltonian vector field of f and
F = −(f ′y/D)
∂
∂x
+ (f ′x/D)
∂
∂y
.
Then the coordinate function of F are relatively prime in the ring R[x, y]. We will call F the
reduced Hamiltonian vector field for f.
Fix ε > 0 and put V = f−1[0, ε]. Then F is a TC vector field on V with singularity at O.
If k = 1, then ∇F is non-degenerate and a description of π0E+(F )∞ and Eid(F )∞ is given
by (3) of Proposition 2.1.
If k ≥ 2, then ∇F = 0. In this case, see [9, 14],
ker j = Sh(F ) = Eid(F )∞ = . . . = Eid(F )1 6= Eid(F )0 = E+(F ),
Eid(F )∞ is contractible with respect to the W∞-topology, and π0E+(F )∞ ≈ Z2n for some
n ≥ 1.
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 513
Now we can formulate our last result, Theorem 2.3. It gives some information about weak
homotopy types of E(F ) and E∂(F ) under certain restrictions on F. The main assumption is the
following one:
ker j ⊂ Sh(F ). (2.10)
It means that for every h ∈ E(F ),whose 1-jet atO is the identity, there exists aC∞ shift function
on all of D2.
Theorem 2.3. Let F be a TC vector field onD2 such that∇F is degenerate and ker j ⊂ Sh(F ).
Let also r ≥ 1. Then the following statements hold true.
(1) If ∇F = 0, then Sh(F ) = ker j. Let id ∈ GL (2,R) be the unit matrix. If in addition the
path component of id in the image L(F ) = j(E(F )) of j coincides with {id} (e.g. when L(F ) is
discrete), then Sh(F ) = Eid(F )1, and therefore j induces the isomorphisms
π0E+(F )r ≈ L(F ) ∩GL+(2,R), π0E(F )r ≈ L(F ). (2.11)
(2) If ∇F =
(
0 a
0 0
)
for some a 6= 0, then
Sh(F ) = Eid(F )∞ = . . . = Eid(F )1 = j−1(A++), (2.12)
whence j yields a monomorphism, see (2.6),
π0E(F )∞ −→ π0A ≈ Z2 ⊕ Z2.
(3) The inclusion E∂id(F )r ⊂ E+
id(F )r between the identity path components is a weak homotopy
equivalence, whence from Theorem 2.2 we have the isomorphisms
πnE∂(F )r ≈ πnE+(F )r, n ≥ 1,
and the following exact sequence:
0 → Z → π0E∂(F )r → π0E+(F )r → 0. (2.13)
(4) Suppose that the image L(F ) of j is finite. Then π0E∂(F )r ≈ Z, π0E+(F )r ≈ Zn for some
n ≥ 0, and (2.13) has the following form:
0 → Z ·n−−−→ Z mod n−−−−−−→ Zn → 0.
If E(F ) 6= E+(F ), then π0E(F )r ≈ Dn, the dihedral group.
The proof of Theorems 2.1, 2.2, and 2.3 will be given in Sections 9 – 11. All of them are based
on results of [14] described in Section 7 about existence and uniqueness of shift functions for
deformations in E+(F ), see also Proposition 8.1.
3. The inclusion D(F ) ⊂ E(F ). Let F be a TC vector field of D2. The aim of this section
is to prove Lemma 3.1 which allows to change elements of E(F ) outside some neighbourhood
of O to produce diffeomorphisms.
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514 S. I. MAKSYMENKO
Definition 3.1. A continuous function f : D2 → [0, 1] will be called a first strong intergal for
F if
(i) f is C∞ on D2\O and has no critical points in D2\O,
(ii) f−1(0) = O, f−1(1) = ∂D2, and for c ∈ [0, 1] the set f−1(c) is an orbit of F.
Notice that we do not require that f be C∞ at O. It also follows from the definition that f
takes distinct values on distinct orbits.
Fig. 2
A first strong integral for F always exists. For instance let h = (h1, h2) : D2 → D2 be a
homeomorphism which maps orbits of F onto concentric circles around O, see (1.1). If h is C∞
on D2\O, then function f = h2
1 + h2
2 is the first strong integral for F.
For every c ∈ (0, 1] put Uc = f−1[0, c]. Then Uc is invariant with respect to F.
Lemma 3.1. Let h ∈ E(F ). Then there exists g ∈ D(F ) such that h = g on some nei-
ghbourhood of O.
Proof. By definition, h ∈ E(F ) is a diffeomorphism at O, whence there exists ε ∈ (0, 1/2)
such that h : U2ε → U2ε is a diffeomorphism. Fix any C∞-diffeomorphism µ : [0, 2ε] → [0, 1]
such that µ = id on [0, ε], see Fig. 2.
We will now construct a diffeomorphism ψ : U2ε → D2 fixed on Uε and such that f ◦ ψ =
= µ ◦ f, i.e., it makes the following diagram commutative:
U2ε
ψ−−−−→ D2
f
y yf
[0, 2ε]
µ−−−−→ [0, 1]
It follows that if c ∈ [0, 2ε] and o = f−1(c) is an orbit of F, then ψ(o) = f−1(µ(c)) is also an
orbit of F. Then we can define a diffeomorphism g : D2 → D2 by
g = ψ ◦ h|U2ε ◦ ψ−1 : D2 ψ−1
−→ U2ε
h−→ U2ε
ψ−→ D2.
Then g ∈ D(F ) and since ψ is fixed on Uε, it follows that g = h on Uε.
The construction of ψ is standard, see e.g. [15], (Ch. 1, § 3). Consider the gradient vector
field ∇f of f defined on D2\O, and let (Φt) be the local flow of ∇f. Let z ∈ U2ε and γ be the
orbit of z with respect to Φ. Then γ intersects the level-set f−1(µ(f(z))) at a unique point ψ(z),
see Fig. 3. Similarly to [16] (Lemma 5.1.3) it can be shown that the correspondence z 7→ ψ(z) is
a diffeomorphism of U2ε → D2 if and only if so is µ.
The lemma is proved.
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 515
Fig. 3
4. Shift functions. Let M be a smooth (C∞) manifold, F be a C∞ vector field on M genera-
ting a flow F : M × R → M, and ϕ : C∞(M,R) → C∞(M,M) be the shift map along orbits
of F defined by ϕ(α)(z) = F(z, α(z)).
If a subset V ⊂ M, a function α : V → R, and a map h : M → M are such that
h(z) = F(z, α(z)), then we will say that α is a shift function for h on V, and that the restriction
h|V is in turn a shift along orbits of F via α.
For a C∞-function α : M → R we will denote by F (α) the Lie derivative of α along F.
Lemma 4.1 ([12], Theorem 19). Let V ⊂ M be an open subset, α : V → R a C∞-function,
and h : V → M be a map defined by h(z) = F(z, α(z)). Then h is a local diffeomorphism at
some z0 ∈ M if and only if F (α)(z0) 6= −1.
Lemma 4.2 [12]. Let αg, αh, αk : M → R be C∞-functions and
g = ϕ(αg), h = ϕ(αh), k = ϕ(αk)
be the corresponding shifts. Suppose also that k is a diffeomorphism. Then the functions
αg◦h = αg ◦ h+ αh, αk−1 = −αk ◦ k−1,
αg◦k−1 = (αg − αk) ◦ k−1
are C∞ shift functions for g ◦ h, k−1, and g ◦ k−1, respectively.
Proof. The formulae for αg◦h and αk−1 coincide with [12] (Equations (8), (9)). They also
imply the formula for αg◦k−1 .
5. Shift functions for E+(F ). Let B = {(φ, r) ∈ R2 : 0 ≤ r ≤ 1} be a closed strip,
B̆ = {(φ, r) ∈ R2 : 0 < r ≤ 1} = B \ {r = 0}
be a half-closed strip in R2, and P : B → D2 be the map given by
P (r, φ) = (r cosφ, r sinφ).
Then P (B̆) = D2\O and the restriction P : B̆ → D2\O is a Z-covering map such that the
corresponding group of covering transformations is generated by the following map:
η : B → B, η(φ, r) = (φ+ 2π, r).
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516 S. I. MAKSYMENKO
It follows that every C∞ map h : D2\O → D2\O lifts to a P -equivariant (i.e., commuting
with η) map h̃ : B̆ → B̆ such that
P ◦ h̃ = h ◦ P.
Such h̃ is not unique and can be replaced with h̃ ◦ ηn = ηn ◦ h̃ for any n ∈ Z.
Remark 5.1. It is well-known that if h : D2 → D2 is a C∞-map being a local diffeomor-
phism at O, and such that h−1(O) = O, then h̃ extends to a C∞-map h̃ : B → B being a
diffeomorphism near the φ-axis {r = 0}. We will not use this fact in the present paper.
Let F be a TC vector field on D2. Since F is non-singular on D2\O, F lifts to a unique
vector field G on B̆ such that F ◦ P = TP ◦G, where TP : TB → TD2 is the tangent map.
It is easy to see that every orbit õ of G is non-closed, its image o = P (õ) is an orbit of F,
and the map P : õ → o is a Z-covering map.
Let G : B̆ × R → B̆ be the flow generated by G, then we have the following commutative
diagram:
B̆ × R G−−−−→ B̆
P×idR
y yP
(D2\O)× R F−−−−→ D2\O
(5.1)
In other words, Ft ◦ P (z̃) = P ◦Gt(z̃) for all z̃ ∈ B̆ and t ∈ R.
In particular, if α : D2 → R is a C∞-function and h = ϕ(α), i.e., h(z) = F(z, α(z)), then
the map h̃ : B → B given by G(z̃, α ◦ P (z̃)) is a lifting of h. Indeed,
h ◦ P (z̃) = F(P (z̃), α ◦ P (z̃)) = P ◦G(z̃, α ◦ P (z̃)) = P ◦ h̃(z̃). (5.2)
Lemma 5.1. Let h ∈ E+(F ). Then there exists a C∞ shift function β : D2\O → R for h on
D2\O, i.e., h(z) = F(z, β(z)) for z ∈ D2\O. Moreover, the set {β + nθ : n ∈ Z} is the set of all
C∞ shift functions for h on D2\O, where θ : D2\O → (0,∞) is the period function for F, see
(4) of Proposition 2.1.
If ∇F is degenerate, then any h ∈ E+(F ) has at most one C∞ shift function defined on all
of D2.
Proof. By the definition, h−1(O) = O and h is a local diffeomorphism at O. Then, as noted
above, there exists a C∞ lifting h̃ : B̆ → B̆ of h such that P ◦ h̃ = h ◦P. Moreover, h preserves
orbits of F, whence h(õ) = õ for each orbit õ of G.
Since the orbits of G are non-closed, there exists a unique C∞ shift function β̃ : B̆ → R for
h̃, i.e., h̃(z̃) = G(z̃, α̃(z̃)) for all z̃ ∈ B̆. Also notice that Gt and h̃ are Z-equivariant. This easily
implies that β̃ is Z-invariant, whence it defines a unique C∞-function β : D2\O → R such that
β̃ = β ◦ P. Then it follows from (5.2) that β is a shift function for h with respect to F.
Suppose that α : D2\O → R is another C∞ shift function for h on D2\O. Then h(z) =
= F(z, α(z)) = F(z, β(z)) for z 6= 0, whence the difference α(z) − β(z) is a certain integer
multiple of the period θ(z) of z. Since θ and α− β are C∞ on D2\O, it follows that α− β = nθ
for some n ∈ Z.
Conversely, for each n ∈ Z and z 6= O we have that
F(z, β(z) + nθ(z)) = F(F(z, nθ(z), β(z)) = F(z, β(z)) = h(z).
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 517
Thus β + nθ is a shift function for h on D2\O.
Finally, suppose that ∇F is degenerate, and α, β : D2 → R are two C∞ shift function for
h defined on all of D2. Then they also shift functions for h on D2\O, whence α − β = nθ for
some n ∈ Z. But, by Proposition 2.1, lim
z→O
θ(z) = +∞, while α − β is C∞ on all of D2. Hence
n = 0, i.e., α = β.
The lemma is proved.
6. (K, r)-deformations. LetA andB be smooth manifolds. Then the spaceC∞(A,B) admits
a series {Wr}∞r=0 of weak topologies, see [17]. The W0-topology coincides with the compact
open one. Let Jr(A,B), r < ∞, be the manifold of r-jets of maps A → B. Then there is a
natural inclusion ir : C∞(A,B) ⊂ C∞(A, Jr(A,B)) associating to each f : A → B its r-jet
prolongation jr(f) : A → Jr(A,B). Endow C∞(A, Jr(A,B)) with the W0-topology. Then the
topology on C∞(A,B) induced by ir is called the Wr-topology. Finally, the W∞-topology is
generated by all Wr for 0 ≤ r < ∞.
Let X ⊂ C∞(A,B) be a subset, K be a Hausdorff, locally compact topological space, and
ω : K → X be a map. Then ω induces the following mapping Ω : K × A → B defined by
Ω(a, k) = ω(k)(a). Conversely, every map Ω : K× A → B such that Ω(k, ·) : A → B belongs
to X induces a map ω : K → X .
Endow X with the induced W0-topology. Then it is well known, e.g. [18] (§ 44.IV), that ω is
continuous if and only if Ω is so.
Definition 6.1. Let r = 0, . . . ,∞. Then the map Ω : K × A → B will be called a (K, r)-
deformation in X if Ωk ∈ X for all k ∈ K and the induced map ω : K → X is continuous
whenever X is endowed with the Wr-topology. In other words, the map jr : K×A → Jr(A,B)
associating to each (k, a) ∈ K×A the r-jet prolongation jrΩk(a) of Ωk at a is continuous.
If K = [0, 1] then the (K, r)-deformation will be called an r-homotopy.
7. Shift functions for (K, r)-deformations. Let K be a Hausdorff, locally compact, and path
connected topological space, F be a TC vector field on D2,
ω : K → E+(F )r
be a continuous map into some Wr-topology of E+(F ), and
Ω : K×D2 → D2, Ω(k, z) = ω(k)(z) (7.1)
be the corresponding (K, r)-deformation in E+(F ), so Ωk ∈ E+(F ) for all k ∈ K.
Then by Lemma 5.1 for each k ∈ K the map Ωk has a (not unique) C∞ shift function Λk
defined on D2\O. Thus we can define a map Λ : K× (D2\O) → R by Λ(k, z) = Λk(z) which,
in general, is not even continuous, though it is C∞ for each k.
Lemma 7.1 below is a particular case of results of [14], see also [12] (Theorem 25). It shows
that Λk can always be chosen so that Λ becomes continuous in (k, z).
Definition 7.1. A (K, r)-deformation Λ : K× (D2\O) → R satisfying
Ω(k, z) = F(z,Λ(k, z)) ∀(k, z) ∈ K× (D2\O) (7.2)
will be called a shift function for the (K, r)-deformation Ω.
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518 S. I. MAKSYMENKO
Lemma 7.1 [14]. Let k0 ∈ K and Λk0 be any C∞ shift function for Ωk0 . Then there exists at
most one shift function Λ : K× (D2\O) → R for Ω such that Λ(k0, z) = Λk0(z).
Moreover, if K is simply connected, i.e., π1K = 0, then any shift function Λk0 for Ωk0 uniquely
extends to a shift function
Λ : K× (D2\O) → R
for Ω.
This lemma will be used in the proofs of Theorems 2.1 and 2.2. For the proof of Theorem 2.3
we will also need the following Lemmas 7.2 and 7.3.
Suppose now that ω(K) ⊂ Sh(F ), that is, for each k ∈ K the map Ωk has a C∞ shift
function defined on all of D2. Let Λ : K× (D2\O) → R be a shift function for Ω such that Λk0
for some k0 ∈ K smoothly extends to all ofD2. The following lemma gives sufficient conditions
when any other shift function Λk = Λ(k, ·) smoothly extends to all ofD2.Again it is a particular
case of results of [14].
Lemma 7.2 [14]. Let Ω : K ×D2 → D2 be a (K, r)-deformation admitting a shift function
Λ : K× (D2\O) → R. Suppose that
(i) ker j ⊂ Sh(F ),
(ii) Ω0 = idD2 for some k0 ∈ K, and
(iii) jΩk = id, i.e., Ωk ∈ ker j ⊂ Sh(F ), for all k ∈ K.
Then for each k ∈ K the function Λk : D2\O → R extends to a C∞-function on all of D2,
though the induced function Λ : K×D2 → R is not necessarily continuous.
Finally, we present a sufficient condition when a map into Sh(F ) can be deformed into ker j.
Lemma 7.3. Let K be path connected and simply connected, r ≥ 1, and ω : K → Sh(F ) be a
continuous map into the Wr-topology of Sh(F ). Suppose ∇F is degenerate and ker j ⊂ Sh(F ).
Then there exists a homotopy B : I × K → Sh(F ) such that B0 = ω, B1(K) ⊂ ker j, and
Bt(k) = ω(k) for all k such that ω(k) ∈ ker j.
Proof. If ∇F = 0, then Sh(F ) = ker j and there is nothing to prove.
Suppose that∇F =
(
0 a
0 0
)
for some a 6= 0. Let Ω : K×D2 → D2 be the corresponding
(K, r)-deformation in Sh(F ). Then, by (2.7),
j (Ωk) =
(
1 a τ(k)
0 1
)
, k ∈ I,
for some τ(k) ∈ R. Since Ω is an r-homotopy with r ≥ 1, it follows that the function τ : K → R
is continuous. Moreover, τ(k) = 0 if and only if Ωk ∈ ker j.
Define the homotopy B : I ×K → Sh(F ) by
B(t, k)(z) = F (Ω(k, z),−tτ(k)).
Then it is easy to see that B satisfies the statement of our lemma.
8. Deformations in E+(F ). In this section we prove the key Proposition 8.1 that will imply
Theorems 2.1, 2.2, and 2.3.
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 519
Let K be a Hausdorff, locally compact topological space and
ω : K → E+(F )r
be a continuous map into some Wr-topology of E+(F ). Our aim is to show that under certain
mild assumptions ω is homotopic to a map into D∂(F ) = D+(F )∩E∂(F ) so that the intersecti-
ons of ω(K) with D∂(F ), D+(F ) and E∂(F ) remain in the corresponding spaces during the
homotopy. More precisely the following result holds true:
Proposition 8.1. Suppose that either
(i) K is a point and r ≥ 0, or
(ii) K is compact, path connected, and simply connected, and r ≥ 1.
Let L ⊂ K be a (possibly empty) subset such that ω(L) ⊂ D+(F ), and P ⊂ K be a connected
subset such that ω(P) ⊂ E∂(F ). Thus we can regard ω as a map of triples,
ω : (K;L,P′) → (E+(F )r;D+(F )r, E∂(F )r).
Then there exists a homotopy of triples,
At : (K;L,P′) → (E+(F )r;D+(F )r, E∂(F )r), t ∈ I,
such that
A0 = ω and A1(K) ⊂ D∂(F ). (8.1)
The phrase homotopy of triples means that
At(L) ⊂ D+(F ), At(P) ⊂ E∂(F ), (8.2)
and therefore At(L ∩P) ⊂ D+(F ) ∩ E∂(F ) = D∂(F ) for all t ∈ I.
The proof will be given at the end of this section. Let
Ω : K×D2 → D2, Ω(k, z) = ω(k)(z)
be the corresponding (K, r)-deformation in E+(F ). Then by Lemma 7.1 there exists a shift
function Λ : K × (D2\O) → R for Ω. The deformation of Ωk we will be produced via a
deformation of Λk.
Let a, b ∈ (0, 1) be such that a < b, f : D2 → [0, 1] be the first strong integral for F, see
Definition 3.1, and ν : [0, 1] → [0, 1] be a C∞-function such that ν[0, a] = 1 and ν[b, 1] = 0.
Define a function α : K× (D2\O) → R by
α(k, z) = ν ◦ f(z) · Λ(k, z), (k, z) ∈ K× (D2\O), (8.3)
the map Ω′ : K×D2 → D2 by
Ω′(k, z) =
F (z, α(k, z)), z 6= O,
O, z = O,
(8.4)
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520 S. I. MAKSYMENKO
and a homotopy A : I ×K×D2 → D2 by
A(t, k, z) =
F (z, (1− t)α(k, z) + tΛ(k, z)), z 6= O,
O, z = O.
(8.5)
Lemma 8.1. For (t, k) ∈ I×K denote At = A(t, ·, ·) : K×D2 → D2 and At,k = A(t, k, ·) :
D2 → D2. Then
(a) A0 = Ω, A1 = Ω′, and At is a (K, r)-deformation in E(F ) for each t ∈ I.
(b) Ω′
k is fixed on D2 \ Ub for all k ∈ K. In particular, A1 = Ω′ is a deformation in E∂(F ).
(c) If for some (k, z) ∈ K×D2 the map Ωk is a local diffeomorphism at z, then so is At,k for
each t ∈ I.
(d) Denote Z = Λ−1(0) ⊂ K× (D2\O). Thus Ω(k, z) = z for all (k, z) ∈ Z. Then
A(t, k, z) = z ∀ t ∈ I, (k, z) ∈ Z.
(e) Let P ⊂ K be a connected subset such that Ωk is fixed on ∂D2 for each k ∈ P and
Λk0 |∂D2 = 0 for some k0 ∈ P. Then At,k is also fixed on ∂D2 for all (t, k) ∈ I ×P.
Thus A induces a homotopy
At : K → E+(F )r, At(k)(z) = A(t, k, z) (8.6)
such that A0 = ω, and A1(K) ⊂ E∂(F ).
Proof. Statements (a) and (b) follow from (8.3) – (8.5).
(c) Denote
βt,k(z) = (1− t)α(k, z) + tΛ(k, z) = ((1− t)ν ◦ f(z) + t) · Λk(z). (8.7)
Then by (8.5) βt,k is a shift function for At,k on D2\O.
The assumption that Ωk is a local diffeomorphism at z means that
F (Λk)(z) > −1, (8.8)
see Lemma 4.1. Therefore by that lemma it suffices to verify that F (βt,k)(z) > −1 for all t ∈ I.
Notice that
F (((1− t)ν ◦ f + t) · Λk) = (1− t)F (ν ◦ f) + ((1− t)ν ◦ f + t)F (Λk).
The first summand is zero since f and, therefore, ν ◦f are constant along orbits of F. Moreover,
0 ≤ ν(z) ≤ 1,whence we get from (8.8) that the second summand is> −1. Hence F (βt,k)(z) >
> −1 for all k ∈ K.
(d) If Λ(k, z) = 0 for some (k, z) ∈ K × (D2\O), then by (8.7) βt,k(z) = 0, whence
A(t, k, z) = F (z, βt,k(z)) = F (z, 0) = z.
(e) If Ωk is fixed on ∂D2 for some k ∈ K, then Λk takes on ∂D2 a constant value,
Λk(∂D2) = nk · θ(∂D2),
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 521
for some nk ∈ Z. Since P is connected, Λ is continuous on P × ∂D2, and the set of possible
values of Λk on ∂D2 is discrete, it follows that Λ is constant on P×∂D2. In particular, Λ|P×∂D2 =
= Λk0 |P×∂D2 = 0. Then by (d) At,k is fixed on ∂D2 for all (k, t) ∈ I ×P.
The lemma is proved.
Proof of Proposition 8.1. We will find a, b ∈ (0, 1) and a shift function Λ : K×(D2\O) → R
for Ω such that the corresponding homotopy At constructed in Lemma 8.1 will satisfy (8.1) and
(8.2).
Choice of Λ. Let Λ′ : K × (D2\O) → R be any shift function for Ω. Since Ωk0 is fixed on
∂D2 for some k0 ∈ P, we have that Λ′k0 |∂D2 = nθ(∂D2) for some n ∈ Z.
Define another function Λ : K× (D2\O) → R by
Λ(k, z) = Λ′(k, z)− nθ(∂D2).
Then Λ is also a shift function for Ω in the sense of (7.2) and satisfies
Λk0 |∂D2 = 0. (8.9)
Choice of a, b ∈ (0, 1). Notice that Ωk(Ub) = Ub for all k ∈ K and b ∈ (0, 1]. We claim that
there exists b ∈ (0, 1) such that the map Ωk : Ub → Ub is a diffeomorphism for all k ∈ I . Indeed,
by the definition of E+(F ), the map Ωk is a diffeomorphism at O for each k ∈ K. This implies
existence of b in the case (i), i.e., when K is a point. In the case (ii) the assumption r ≥ 1 means
that the partial derivatives of Ωk are continuous functions on K×D2. Then existence of b now
follows from compactness of K×D2.
Take arbitrary a ∈ (0, b) and let At be a homotopy constructed in Lemma 8.1 for Λ and a, b.
We claim that A satisfies (8.1) and (8.2).
By (a) of Lemma 8.1 A0 = ω.
Let us prove that A1(K) ⊂ D∂(F ), i.e., for each k ∈ K the map A1,k = Ω′
k is a di-
ffeomorphism of D2 fixed on ∂D2. By (b) of Lemma 8.1 Ω′
k is fixed even on D2\Ub. Moreover,
by the assumption on b, we have that Ωk : Ub → Ub is a diffeomorphism, whence, by (c) of
Lemma 8.1, At,k = At(k) is also a self-diffeomorphism of Ub and therefore of all D2.
To show that At(L) ⊂ D+(F ) notice that by the assumption Ωl : D2 → D2 is a di-
ffeomorphism for all l ∈ L. Then again, by (c) of Lemma 8.1, At,l = At(l) is also a self-
diffeomorphism D2 for all l ∈ L, i.e., At(L) ⊂ D+(F ).
Finally, the inclusion At(P) ⊂ E∂(F ) follows from (8.9) and (e) of Lemma 8.1.
9. Proof of Theorem 2.1. First we prove (1) and (2) for the inclusions D+(F ) ⊂ E+(F )
and D∂(F ) ⊂ E∂(F ). Then we establish (3) and deduce from it (1) and (2) for the inclusion
D(F ) ⊂ E(F ).
(1) We have to show that πn(Er,Dr) = 0 for all n ≥ 0 if r ≥ 1. Then the result will follow
from the exact homotopy sequence for the pair (Er,Dr).
Let ω : (In, ∂In) → (Er,Dr) be a continuous map representing some element of the relati-
ve homotopy set πn(Er,Dr). Our aim is to show that ω is homotopic as a map of pairs to a map
into D, i.e., ω = 0 in πn(Er,Dr), whence we will get πn(Er,Dr) = 0.
Inclusion D+(F ) ⊂ E+(F ). If r ≥ 1, then applying Proposition 8.1 to the case K = In,
L = ∂In we obtain that ω is homotopic as a map of pairs to a map into D+(F ).
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522 S. I. MAKSYMENKO
Inclusion D∂(F ) ⊂ E∂(F ). Since
(E∂(F ),D∂(F )) ⊂ (E+(F ),D+(F )),
we see that ω is also an element of πn(E+(F )r,D+(F )r), which, as just shown, is trivial. Then
Proposition 8.1 can be applied to the case K = P = In and L = ∂In, and we obtain that ω is
homotopic as a map of pairs (K,L) → (E∂(F )r,D∂(F )r) to a map into D∂(F )r. Hence ω = 0
in πn(E∂(F )r,D∂(F )r).
(2) We have to show that the map π0D0 → π0E0 is surjective for all r ≥ 0. Let h ∈ E+(F ).
It can be regarded as a map from the set K consisting of a unique point into E(F ),
ω : K → E , ω(K) = h.
Then applying (i) of Proposition 8.1 we obtain that ω is C∞-homotopic to a map into D∂(F ),
whence the inclusion D∂(F ) ⊂ E+(F ) yields a surjecitve map π0D∂(F )r → π0E+(F )r for all
r ≥ 0. Therefore in the following diagram induced by inclusions all arrows are surjective:
π0D∂(F )r −−−−→ π0E∂(F )ry y
π0D+(F )r −−−−→ π0E+(F )r
(9.1)
The proof of the surjectivity π0D(F )r → π0E(F )r is the same as in (1).
(3) It is well known and is easy to prove that for a topological semigroup E the set π0E of
path components of E admits a semigroup structure such that the natural projections E → π0E is
a semigroup homomorphism. If E is a group, then so is π0E .
If D ⊂ E is a subsemigroup, then the induced map π0D → π0E is a semigroup homomor-
phism.
In our case Er is a topological semigroup andDr is a topological group. From (1) we get that
for r ≥ 1 the homomorphism π0D → π0E is a bijection, whence it is a semigroup isomorphism.
But π0D is a group, whence so is π0E .
Let us prove that all path components of E are homeomorphic to each other. By (1) and
(2) the map i0 : π0D → π0E is surjective for each of Wr-topologies, r ≥ 0. In particular, this
implies that each path component of E contains an invertible element. Now the result is implied
by the following statement.
Claim 9.1. Let E be a topological semigroup such that each path component of E contains an
invertible element. Then all path components of E are homeomorphic each other.
Moreover, let D be the subgroup consisting of all invertible elements. Then for any two path
components E1 and E2 there exists a homeomorphismQ : E1 → E2 such thatQ(E1∩D) = E2∩D.
Proof. Let h1 ∈ E1 and h2 ∈ E2 be any invertible elements. Then we can define a homeomor-
phism Q : E1 → E2 by Q(h) = h2 · h−1
1 · h. Evidently it is continuous, its inverse is given by
Q−1(g) = h1 · h−1
2 · g, and Q(E1 ∩ D) = E2 ∩ D.
(1) and (2) for the inclusion D(F ) ⊂ E(F ). Put
D′ = D(F )\D+(F ), E ′ = E(F )\E+(F ).
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 523
Then E ′ consists of full path components of E(F ) with respect each of Wr-topologies. Hence
we have to prove our statement for the inclusion D′ ⊂ E ′. We can also assume that E ′ 6= ∅.
Then it follows from Lemma 3.1 thatD′ 6= ∅ as well. Let g ∈ D′. Then we can define a mapQ :
E ′ → E+(F ) by Q(h) = g−1 ◦h for h ∈ E ′. Evidently, Q is a homeomorphism onto with respect
to any of Wr-topologies. Moreover, Q(D′) = D+(F ). Hence, πn(E ′,D′) = πn(E+(F ),D+(F )).
It remains to note that by (1) and (2) πn(E+(F ),D+(F )) = 0 if either r ≥ 1 and n ≥ 0, or
r = 0 and n = 0.
10. Proof of Theorem 2.2. The proof is similar to the one given in Section 9. Let
ω : (In, ∂In) → (E+(F )r, E+(F )r)
be a continuous map being a representative of some element in the relative homotopy set
πn(E+(F )r, E∂(F )r).
We have to show that ω is homotopic as a map of pairs to a map into E∂(F ). Again we will
apply Proposition 8.1 but now the situation is more complicated.
For n 6= 1 denote K = In and P = ∂In. Then P is path connected and, by Proposition 8.1,
ω is homotopic as a map of pairs,
(K,P) → (E+(F )r, E∂(F )r),
to a map into E∂(F ). In this case we can take b arbitrary, and therefore the arguments hold for
the case r = 0 as well. This implies πn(E+(F )r, E∂(F )r) = 0 for all n 6= 1 and r ≥ 0.
Suppose n = 1. Then I1 = [0, 1] and ∂I1 = {0, 1} is not connected, so Proposition 8.1 can
be applied only to each of the path components {0} and {1} of ∂I1. Actually this is the reason
why
π1(E+(F )r, E∂(F )r, idD2) ≈ Z. (10.1)
To prove (10.1), use 0 ∈ I1 and idD2 ∈ E+(F ) as base points, and thus assume that ω(0) = idD2 .
Consider the (I1, r)-deformation in E+(F )r corresponding to ω,
Ω : I1 ×D2 → D2, Ω(k, z) = ω(k)(z).
Then Ω0 = idD2 and therefore the zero function Λ0 = 0 is a shift function for Ω. By Lemma 7.1,
Λ0 extends to a unique (I1, r)-deformation
Λ : I1 × (D2\O) → R
being a shift function for Ω on D2\O in the sense of that lemma. In particular the last function
Λ1 is a shift function for Ω1 ∈ E∂(F ) which is fixed on ∂D2. Then by Proposition 8.1 applied to
P = {1}, we get that Λ1 takes constant value on ∂D2 being an integer multiple of the period of
orbit ∂D2 with respect to F . Thus
Λ(∂D2) = ρω · θ(∂D2)
for some ρω ∈ Z. Evidently, ρω counts the number of “full rotations” of ∂D2 during the
homotopy Ω. We claim that the correspondence ρ : ω 7→ ρω yields an isomorphism (10.1).
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524 S. I. MAKSYMENKO
It is easy to see that ρ induces a surjective homomorphism
R : π1(E+(F )r, E∂(F )r) → Z.
To show that R is a monomorphism suppose that ρ(ω) = 0, so Λ0(∂D2) = Λ1(∂D2) = 0.
Then it follows from (d) of Lemma 8.1 that
A(t, 0, z) = A(t, 1, z) = z
for all t ∈ I and z ∈ ∂D2. In other words, At(0),At(1) ∈ E∂(F )r for all t ∈ I.
Thus ω is homotopic to a map into E∂(F ) via a homotopy relatively to ∂I1, and therefore it
represents a trivial element of π1(E+(F )r, E∂(F )r). This implies that R is an isomorphism.
11. Proof Theorem 2.3. Let F be a TC vector field on D2 such that ∇F is degenerate and
ker j ⊂ Sh(F ).
(1) If ∇F = 0, then the relation Sh(F ) = ker j follows from the assumption Sh(F ) ⊃ ker j
and (1) of Proposition 2.1.
Suppose that {id} is the path component of id in L(F ). Since j is continuous from Wr-
topology of E(F ) for r ≥ 1, it follows that Eid(F )r ⊂ ker j = Sh(F ) ⊂ Eid(F )r. This also
implies (2.11).
(2) Suppose that ∇F =
(
0 a
0 0
)
for some a 6= 0. We have to show that Sh(F ) =
= Eid(F )1 = j−1(A++).
It follows from the definition, see (2.6), that A is a group, A++ is its unity component in
GL+(2,R), andA−−, A+−, A−+ are another path components ofA. Since j is continuous in the
Wr-topology of E+(F ) for r ≥ 1, it follows that the inverse images of these path components
are open-closed in E(F ). On the other hand, Sh(F ) is path connected in all Wr-topologies, as
a continuous image of a path connected space C∞(M,R), whence
Sh(F ) ⊂ Eid(F )1 ⊂ j−1(A++, ).
Conversely, let h ∈ j−1(A++, ), so j (h) =
(
1 aτ
0 1
)
for some τ ∈ R. We have to show
that h ∈ Sh(F ).
Evidently j (h) coincides with
exp(τ · ∇F ) = exp
(
0 aτ
0 0
)
.
Consider the flow (Ft) of F. Then j (Ft) = exp
(
0 at
0 0
)
for all t ∈ R. Hence j (Fτ ) = j (h),
Define the map g : D2 → D2 by g(z) = F(h(z),−τ) = F−τ ◦ h. Then j (g) =
(
1 0
0 1
)
,
i.e., g ∈ ker j ⊂ Sh(F ).
In other words, g(z) = F(z, α(z)) for some α ∈ C∞(D2,R). Put β(z) = α(z) + τ. Then
h(z) = F (z, β(z)), i.e., h ∈ Sh(F ).
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SYMMETRIES OF DEGENERATE CENTER SINGULARITIES OF PLANE VECTOR FIELDS 525
(3) Due to (2.2) we have only to show that the mapping
i1 : π1E∂(F )r → π1E+(F )r
induced by the inclusion is an isomorphism. Moreover, by exactness of the sequence (2.3) it
remains to show that i1 is surjective.
Let ω : I → E+(F )r be a continuous map representing a loop in E+(F )r, i.e.,
ω(0) = ω(1) = idD2 . (11.1)
We have to show that ω is r-homotopic relatively to ∂I to a map into E∂(F )r.
It follows from (11.1) that ω(I) is contained in E+
id(F )r which, by (1) and (2), coincides with
Sh(F ). Thus ω(I) ⊂ Sh(F ). Moreover, ω(∂I) ⊂ ker j. Then, by Lemma 7.3, ω is homotopic to
a map into ker j relatively to ∂I. Hence we can assume that ω is a loop in ker j.
Consider the (I, r)-deformation corresponding to ω,
Ω : I ×D2 → D2, Ω(t, z) = ω(t)(z).
Then Ω0 = Ω1 = idD2 and Ωk ∈ ker j ⊂ Sh(F ) for all k ∈ I.
In particular, every Ωk has a C∞ shift function Λk : D2 → R defined on all ofD2. Since∇F
is degenerate, we have by Lemma 5.1 that such Λk is unique. In particular, Λ0 = Λ1 = 0.
Then it follows from Lemma 7.2 that the map Λ : I × (D2\O) → R defined by Λ(k, z) =
= Λk(z) is a (I, r)-deformation being a shift function for Ω.
Take any a, b ∈ (0, 1) such that a < b and consider the homotopy At of ω into E∂(F ) defined
by (8.5). Since Ω(0, z) = Ω(1, z) = z for all z ∈ D2, we obtain from (d) of Proposition 8.1 that
At(0, z) = At(1, z) = z for all t ∈ I . In other words At is a homotopy relatively ∂I .
(4) This statement follows from (1) – (3) and the well known fact that any finite subgroup of
GL+(2,R) is cyclic.
Theorem 2.3 is proved.
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Received 16.07.09
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