Period functions for C⁰- and C¹-flows
Let F:M×R→M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let ξ:V→R be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all peri...
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irk-123456789-1661832020-02-19T01:26:22Z Period functions for C⁰- and C¹-flows Maksymenko, S.I. Статті Let F:M×R→M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let ξ:V→R be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C1-flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows. Нехай F:M×R→M — неперервний потік на многовиді M, V⊂M — відкрита підмножина ξ:V→R - неперервна функція. Назвемо ξ функцією періодів, якщо F(x,ξ(x))=x для всіх x∈V. Нещодавно для кожної відкритої зв'язної множини V⊂M автором було описано структуру множини P(V) всіх функцій періодів на V. Припустимо, що F є топологічно спряженим до деякого потоку класу C1. У даній роботі показано, що тоді функції періоду F задовольняють додаткові умови, які, взагалі кажучи, не виконуються для загальних неперервних потоків. 2010 Article Period functions for C⁰- and C¹-flows / S.I. Maksymenko // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 954–967. — Бібліогр.: 19 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166183 515.145+515.146 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Maksymenko, S.I. Period functions for C⁰- and C¹-flows Український математичний журнал |
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Let F:M×R→M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let ξ:V→R be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C1-flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows. |
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Maksymenko, S.I. |
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Maksymenko, S.I. |
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Maksymenko, S.I. |
| title |
Period functions for C⁰- and C¹-flows |
| title_short |
Period functions for C⁰- and C¹-flows |
| title_full |
Period functions for C⁰- and C¹-flows |
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Period functions for C⁰- and C¹-flows |
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Period functions for C⁰- and C¹-flows |
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period functions for c⁰- and c¹-flows |
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Інститут математики НАН України |
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2010 |
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Period functions for C⁰- and C¹-flows / S.I. Maksymenko // Український математичний журнал. — 2010. — Т. 62, № 7. — С. 954–967. — Бібліогр.: 19 назв. — англ. |
| series |
Український математичний журнал |
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AT maksymenkosi periodfunctionsforc0andc1flows |
| first_indexed |
2025-07-14T20:56:26Z |
| last_indexed |
2025-07-14T20:56:26Z |
| _version_ |
1837657320425783296 |
| fulltext |
UDC 515.145 + 515.146
S. I. Maksymenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
PERIOD FUNCTIONS FOR C0- AND C1-FLOWS
ФУНКЦIЇ ПЕРIОДIВ ДЛЯ C0- ТА C1-ПОТОКIВ
Let F : M ×R→M be a continuous flow on a manifold M, V ⊂M be an open subset, and let ξ : V → R
be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any
open connected subset V ⊂M, the author has described the structure of the set P (V ) of all period functions
on V. Assume that F is topologically conjugate to some C1-flow. It is shown in this paper that, in this case, the
period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general
continuous flows.
Нехай F : M × R → M — неперервний потiк на многовидi M, V ⊂ M — вiдкрита пiдмножина i
ξ : V → R — неперервна функцiя. Назвемо ξ функцiєю перiодiв, якщо F(x, ξ(x)) = x для всiх x ∈ V.
Нещодавно для кожної вiдкритої зв’язної множини V ⊂ M автором було описано структуру множини
P (V ) всiх функцiй перiодiв на V. Припустимо, що F є топологiчно спряженим до деякого потоку класу
C1. У данiй роботi показано, що тодi функцiї перiоду F задовольняють додатковi умови, якi, взагалi
кажучи, не виконуються для загальних неперервних потокiв.
1. Introduction. Let F : M × R → M be a continuous flow on a topological finite-
dimensional connected manifold M. Let also Σ be the set of fixed points of F. For
x ∈M we will denote by ox the orbit of x. If x is periodic, then Per(x) will denote the
period of x.
Definition 1.1. Let V ⊂M be a subset and ξ : V → R be a continuous function.
We will say that ξ is a period function or simply a P -function (with respect to F) if
F(x, ξ(x)) = x for all x ∈ V.
The set of all P -functions on V with respect to a flow F will be denoted by P (F, V ),
or simply by P (V ). The following easy lemma explains the term P -function.
Lemma 1.1 [1]. 1. For any subset V ⊂M the set P (V ) is a group with respect to
the pointwise addition.
2. Let x ∈ V and ξ ∈ P (V ). Then ξ is locally constant on ox ∩V. In particular, if x
is nonperiodic, then ξ|ox∩V = 0. Suppose x is periodic, and let ω be a path component
of ox ∩ V. Then ξ = nω Per(ox) for some nω ∈ Z depending on ω.
The following theorem, describing P (V ) for open connected subsets V ⊂ M, is a
particular case of results obtained in [1]. It also extends [2] (Theorem 12) to the case of
continuous flows.
Theorem 1.1 [1, 2]. Let M be a finite-dimensional topological manifold possibly
noncompact and with or without boundary, F : M × R → M be a flow, and V ⊂ M
be an open, connected set. Suppose that Int(Σ) ∩ V = ∅. Then one of the following
possibilities is realized: either P (V ) = {0} or P (V ) = {nθ}n∈Z for some continuous
function θ : V → R having the following properties:
(1) θ > 0 on V \ Σ, so this set consists of periodic points only.
(2) There exists an open and everywhere dense subset Q ⊂ V such that θ(x) =
= Per(x) for all x ∈ Q.
(3) θ is constant on ox ∩ V for each x ∈ V.
(4) Put U = F(V × R). Then θ extends to a P -function on U and there is a circle
action G : U × S1 → U defined by G(x, t) = F(x, tθ(x)), x ∈ U, t ∈ S1 = R/Z. The
orbits of this action coincides with the ones of F.
c© S. I. MAKSYMENKO, 2010
954 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
PERIOD FUNCTIONS FOR C0- AND C1-FLOWS 955
The aim of this paper is to show that if F is conjugate to some C1-flow, then P -
functions of F have additional properties which may fail for P -functions of general
continuous flows (Theorems 1.2 and 1.3).
Before formulating these results let us discuss the behaviour of P -functions under
conjugations of flows.
1.1. Conjugation of flows. Let M be Cr, r ≥ 1, manifold and F : M × R → M
be a Cr-flow. Then in general, F is generated by a Cr−1-vector field
F (x) =
∂F
∂t
(x, t)|t=0.
Nonetheless, it is proved by D. Hart [3] that every Cr-flow F is Cr-conjugate to a Cr-
flow generated by a Cr-vector field. Thus in order to study P -functions for Cr-flows one
can assume that these flows are generated by Cr-vector fields.
Now let h : M → M be a homeomorphism and G : M × R→ M be the conjugate
flow:
Gt(x) = h ◦ Ft ◦ h−1(x) = h ◦ F(h−1(x), t)).
Let also V ⊂M be an open set and θ : V → R be a continuous P -function for F. Then
θ ◦ h : h−1(V )→ R is a P -function for G. Indeed, if x ∈ V and y = h−1(x), then
G(y, θ ◦ h−1(y)) = h ◦ F(h−1(x), θ ◦ h−1(x))) = h ◦ h−1(x) = x.
In other words P (G, h−1V ) = P (F, V ) ◦ h. In particular, the groups P (G, h−1V ) and
P (F, V ) are isomorphic. Thus the structure of the set of P -functions of the flow F
depends only on its conjugate class.
Let Ek be the unit (k × k)-matrix, C be a square (k × k)-matrix, and a, b ∈ R.
Define the following matrices:
Jp(C) =
C 0 . . . 0
Ek C . . . 0
. . . . . . . . . . . .
0 . . . Ek C
, R(a, b) =
(
a b
−b a
)
,
Jp(a± ib) = Jp(R(a, b)).
For square matrices B,C it is also convenient to put B ⊕ C =
(
B 0
0 C
)
.
Theorem 1.2. Let F be a continuous flow on M and V ⊂M be a connected open
subset such that P (V ) = {nθ}n∈Z for some nonnegative P -function θ : V → [0,+∞)
being strictly positive on M \ Σ, see (1) of Theorem 1.1. If F is conjugate to a C1-flow,
then, in fact, θ > 0 on all of M.
Suppose, in addition, that F is generated by a C1-vector field F. Then for every
z ∈ Σ there are local coordinates in which the linear part j1F (z) of F at z is given by
the following matrix:(
0 β1
−β1 0
)
⊕ . . .⊕
(
0 βk
−βk 0
)
⊕ 0⊕ . . .⊕ 0 (1.1)
for some k ≥ 1 and βj ∈ R \ 0.
The proof of this theorem will be given in Section 7.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
956 S. I. MAKSYMENKO
Remark 1.1. Under assumptions of Theorem 1.2 suppose that θ is C1 and θ(z) 6= 0
for some z ∈ Σ. In this case existence of (1.1) at z is easy to prove, c.f. [4].
Indeed, define the flow G : M × R → M by G(x, t) = F(x, t θ(x)). Then G is
generated by the C1-vector field G = θF and satisfies G1 = idM . Hence G yields a
C1-differentiable R/Z = S1-action on M.
Moreover, Gt(z) = z and so G yields a linear S1-action TzGt on the tangent
space TzM. Now it follows from standard results about S1 representations in GL(R, n)
that the linear part of G at z in some local coordinates is given by (1.1). It remains
to note that j1F (z) = j1G(z)/θ(z). Notice that these arguments do not prove that the
matrix (1.1) is non-zero.
Example 1.1. Let F : C× R→ C be a continuous flow on C defined by
F(z, t) =
e
2πi t/|z|2 z, z 6= 0,
0, z = 0.
The orbits of F are the origin 0 ∈ C and the concentric circles centered at 0. Then
P (C) = {nθ}n∈Z, where θ(z) = |z|2. Also notice that θ(0) = 0 and θ > 0 on C \ 0.
This shows that a non-zero P -function of a continuous flow F may vanish at its fixed
points. It also implies that F in not conjugate to a C1-flow.
Our second result shows that if F is conjugate to a C1-flow, then discontinuity of
P -functions at points of Σ is almost always the result of unboundedness of periods of
points near Σ.
Theorem 1.3. Let F be a C1-flow generated by a C1-vector field, V ⊂ M be an
open connected subset such that P (V ) = 0, while P (V \ Σ) = {nθ}n∈Z for a certain
non-zero P -function θ : V \ Σ → R, so θ can not be continuously extended to all of V.
Suppose that there exists a point z ∈ V ∩Σ in which j1F (z), the linear part of F at z, is
not similar to a matrix of the form (1.1). Then there exists a sequence {xi}i∈N ⊂ V \Σ
(consisting of periodic points) which converges to z and satisfies
lim
i→∞
Per(xi) = lim
i→∞
θ(xi) = +∞.
Example 1.2. Let F : C× R→ C be a C∞-flow on C defined by
F(z, t) =
e
2πi t |z|2 z, z 6= 0,
0, z = 0,
where 0 is the origin. Then θ =
1
|z|2
is a C∞ P -function on C \ 0 and P (C \ 0) =
= {nθ}n∈Z. On the other hand, lim
z→0
θ(z) = +∞, whence θ can not be extended even
to a continuous function on C, so P (C) = {0}.
1.2. Structure of the paper. In next section we discuss applications of P -functions
to reparametrizations of flows to circle actions and also describe the relationships of
Theorem 1.3 to the results of other authors.
Section 3 presents a variant of results of M. Newman, A. Dress, D. Hoffman, and
L. N. Mann about lower bounds for diameters of orbits of Zp-actions on manifolds.
Sections 4 and 5 give sufficient conditions for unboundedness of periods of C0- and
C1-flows near singular points. The last two Sections 8 and 7 contain the proofs of
Theorems 1.3 and 1.2 respectively.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
PERIOD FUNCTIONS FOR C0- AND C1-FLOWS 957
2. P -functions and circle actions. In this section we discuss relation of our results
to the following problem:
Problem 2.1. Let F be a continuous flow on M. Do there exist a continuous
S1-action on M whose orbits coincides with the ones of F?
The following simple statement shows applications of P -functions to Problem 2.1.
Lemma 2.1. Let θ : M → R be a P -function on all of M. Then the following
map G : M × R → M defined by G(x, t) = F(x, t · θ(x)) is a flow on M such that
G1 = idM , so G factors to a circle action.
Proof. Indeed, G1(x) = F(x, θ(x)) = x.
An evident necessary (but not sufficient) condition for Problem 2.1 is that every
orbit of F is either periodic or fixed. Moreover, due to the well-known theorem of
M. Newman the set Σ should be nowhere dense, see [5 – 8].
Suppose now that Σ = ∅ and all points are periodic for F. It will be convenient to
call such a flow F a P -flow. Then we have a well-defined function
λ : M → (0,+∞), λ(x) = Per(x).
This function was studied by many authors. It can be shown that λ is lower semicon-
tinuous and the set B of its continuity points is open in M, see e.g. D. Montgomery [9]
and D. B. A. Epstein [10] (§ 5). Thus in the sense of Definition 1.1 λ is a P -function
on B.
There are certain typical situations in which λ is discontinuous.
For instance, if λ is locally unbounded, then it can not be continuously extended to
all M. Say that a P -flow F has property PB (resp. property PU ) if λ is locally bounded
(resp. locally unbounded). Equivalently, if F is at least C1, then instead of periods one
can consider lengths of orbits with respect to some Riemannian metric on M.
The first example of a PU -flow was constructed by G. Reeb [11]. He produced a
C∞ PU -flow on a noncompact manifold. Further D. B. A. Epstein [10] constructed a
real analytic PU -flow on a noncompact 3-manifold, D. Sullivan [12] a C∞ PU -flow on
a compact 5-manifold S3 × S1 × S1, and D. B. A. Epstein and E. Vogt [13] a PU -flow
on a compact 4-manifold defined by polynomial equations, with the vector field defining
the flow given by polynomials, see also E. Vogt [14].
On the other hand, the following well-known example of Seifert fibrations shows
that even if λ is discontinuous, then in some cases it can be continuously extended to
all of M so that the obtained function is a P -function.
Example 2.1. Let D2 ⊂ C be the closed unit 2-disk centered at the origin, S1 =
= ∂D2 be the unit circle, and T = D2×S1 be the solid torus. Fix k ≥ 2 and define the
following flow on T :
F : T × R→ T, F(z, τ, t) = (ze2πit/k, τe2πit),
for (z, τ, t) ∈ D2×S1×R. It is easy to see that every (z, τ) ∈ T is periodic. Moreover,
Per(z, τ) = k if z 6= 0 ∈ D2, while Per(0, τ) = 1. Thus the function Per: T 2 → R is
discontinuous on the central orbit 0× S1, but it becomes even smooth if we redefine it
on 0×S1 by the value k instead of 1. This new constant function θ ≡ k is a P -function
and P (T ) = {nk}n∈Z.
Notice that in this example F is a suspension flow of a periodic homeomorphism
h : D2 → D2 being a rotation by 2π/k.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
958 S. I. MAKSYMENKO
More generally, let h : M → M be a homeomorphism of a connected manifold M
such that the corresponding suspension flow F of h on M ×S1 is a P -flow. This is pos-
sible if and only if all the points of M are periodic with respect to h. D. Montgomery [9]
shown that such a homeomorphism is periodic itself. Let k be the period of h. Then the
periods of orbits of F are bounded with k, so F is a PB-flow. Moreover, similarly to
Example 2.1, it can be shown that P (M) = {nk}n∈Z.
D. B. A. Epstein [10] also proved that if M is a compact orientable 3-manifold then
any Cr P -flow with (1 ≤ r ≤ ω) has property BP and even there exists a Cr-circle
action with the same orbits. In fact he shown that the structure of one-dimensional Cr-
foliations (1 ≤ r ≤ ∞) on compact orientable 3-manifolds, possibly with boundary, is
similar to Seifert fibrations described in Example 2.1.
The problem of bounded periods has its counterpart for foliations with all leaves
compact. The question is whether the volumes of leaves are locally bounded with re-
spect to some Riemannian metric, see e.g. [15 – 17]. For instance the mentioned above
statements for flows can be adopted for foliations.
The results of the present paper describe the behaviour of period functions near fixed
poins of C1-flows.
3. Diameters and lengths of orbits. 3.1. Effective Zp-actions. We recall here
results of A. Dress [8] (Lemma 3) and D. Hoffman and L. N. Mann [18] (Theorem 1)
about diameters of orbits of effective Zp-actions.
For x, y ∈ Rn denote by d(x, y) the usual Euclidean distance, and by Br(x), r > 0,
the open ball of radius r centered at x.
Let W be an open subset of the half-space Rn+ = {xn ≥ 0} and x ∈ W. Define the
radius rx of convexity of W at x as follows. If x ∈ Int(Rn+) ∩W, then
rx = sup
{
r > 0: Br(x) ⊂W
}
.
Otherwise, x ∈ ∂Rn+ ∩W and we put
rx = sup
{
r > 0: (Br(x) ∩ Rn+) ⊂W
}
.
Lemma 3.1 ([8], Lemma 3). Let U ⊂ Rn be an open, relatively compact and con-
nected subset, p be a prime, and h : U → U be a homeomorphism which induces a
nontrivial Zp-action, that is h 6= idU but hp = idU . Define two numbers:
D(U) = max
{
min{d(x, y) : y ∈ U} : x ∈ U
}
,
C(U) = max
{
d(x, ha(x)) : a = 0, . . . , p− 1, x ∈ U \ Int(U)
}
.
Then D(U) < C(U).
The next Lemma 3.2 is a variant of [18] (Theorem 1). It seems that in the proof of
[18] (Theorem 1) the condition of connectedness of the set U, see (3.1) below, is missed,
c.f. paragraph after the assumption (H) on [18, p. 345]. Therefore we recall the proof
which is also applicable to manifolds with boundary.
Lemma 3.2 (c.f. [18], Theorem 1). Let W ⊂ Rn+ = {xn ≥ 0} be an open subset,
p be a prime, and h : W → W be a homeomorphism which induces a nontrivial Zp-
action. Suppose h(z) = z for some z ∈ W. Let also rz be the radius of convexity of W
at z and r ∈ (0, rz).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
PERIOD FUNCTIONS FOR C0- AND C1-FLOWS 959
If z ∈ Int(Rn+), then there exist x ∈ ∂Br/2(z) and a = 1, . . . , p− 1 such that
d(z, x) ≤ 2d(x, ha(x)).
If z ∈ ∂Rn+, then there exist x ∈ ∂(B2r/3(z) ∩W ) and a = 1, . . . , p− 1 such that
d(z, x) ≤ 4d(x, ha(x)).
Proof. For simplicity denote Br(z) by Br.
1. First suppose that z ∈ Int(Rn+). For each s ∈ (0, rz) put
Us = Bs ∪ h(Bs) ∪ . . . ∪ hp−1(Bs). (3.1)
Then Us is open, relatively compact, and h yields a nontrivial Zp-action on U.Moreover,
by assumption h(z) = z, therefore Us is connected. Then by Lemma 3.1
D(Us) ≤ C(Us).
Notice that Bs ⊂ Us, whence s ≤ Ds.
On the other hand, suppose
d(y, ha(y)) < r − s, for all y ∈ ∂Us and a = 1, . . . , p− 1. (3.2)
Then in particular, C(Us) < r − s and thus
s ≤ D(Us) ≤ C(Us) < r − s,
whence s < r/2.
Thus if s = r/2, then (3.2) fails, whence there exist y ∈ ∂Ur/2 and b ∈ {0, . . . , p−
− 1} such that d(y, hb(y)) ≥ r − r/2 = r/2. However
∂Ur/2 ⊂
p−1
∪
i=0
hi(∂Br/2),
whence y = hc(x) for some x ∈ ∂Br/2. Therefore at least one of the distances d(x, y) or
d(x, hb(y)) is not less than r/4. In other words, d(x, ha(x)) ≥ r/4 for some x ∈ ∂Br/2
and a ∈ {1, . . . , p− 1}. Then
d(x, z) =
r
2
= 2
r
4
≤ 2d(x, ha(x)).
2. Let z ∈ ∂Rn+. For each s ∈ (0, rz) let As = Bs ∩ Int(∂Rn+) be the open upper
half-disk centered at z, and
U ′s = As ∪ h(As) ∪ . . . ∪ hp−1(As).
Then U ′s is open, relatively compact, and h yields a nontrivial Zp-action on U ′s. More-
over, it is easy to see that U ′s is connected, whence by Lemma 3.1
D(U ′s) ≤ C(U ′s).
Moreover, Bs/2 ⊂ U ′s. Therefore s/2 ≤ D(U ′s). Hence if we suppose that C(U ′s) <
< r − s, then s/2 < r − s and thus s < 2r/3.
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960 S. I. MAKSYMENKO
Put s = 2r/3. Then there exists y ∈ ∂U ′2r/3 and b ∈ {1, . . . , p − 1} such that
d(y, hb(y)) > r − 2r/3 = r/3.
Again ∂U ′s ⊂
p−1⋃
i=0
hi(∂As), whence we can find x ∈ ∂A2r/3 such that d(x, ha(x)) >
> r/6 for some a ∈ {1, . . . , p− 1}. Then
d(x, z) ≤ 2r
3
= 4 · r
6
≤ 4 · d(x, ha(x)).
The lemma is proved.
3.2. Periodic orbits. Let F be a C1-vector field on a manifold M and d be any
Riemannian metric on M. Then for any periodic point x of F the length l(x) of its orbit
can be calculated as follows:
l(x) =
Per(x)∫
0
∥∥F (F(x, t))
∥∥dt.
Hence
l(x) ≤ Per(x) sup
t∈[0,Per(x)]
∥∥F (F(x, t))
∥∥. (3.3)
Also notice that
2 diam(ox) ≤ l(x). (3.4)
Indeed, let y, z ∈ ox be points for which d(y, z) = diam(ox). These points divide ox
into two arcs each of which has the length ≥ d(y, z). This implies (3.4).
4. P -functions on the set of nonfixed points. Let V ⊂ M be an open subset,
ξ : V \Σ→ R be a P -function, and α ∈ R. Define the following map hα : V →M by:
hα(x) =
F(x, α ξ(x)), x ∈ V \ Σ,
x, x ∈ Σ ∩ V.
(4.1)
Then hα is continuous on V \ Σ but in general it is discontinuous at points of Σ ∩ V.
The aim of this section is to establish implications between the following five con-
ditions:
(A) The periods of periodic points in V \ ξ−1(0) are uniformly bounded above with
some constant C > 0, that is for each x ∈ V with ξ(x) 6= 0 we have that Per(x) < C.
(B) Every z ∈ Σ∩V has a neighbourhood W ⊂ V such that ξ is regular on W \Σ,
that is for every y ∈W \ Σ the restriction of ξ to oy ∩W is constant.
(C)α The map hα is continuous on all of V.
Let z ∈ Σ ∩ V.
(D)α Suppose α = q/p ∈ Q, where q ∈ Z and p ∈ N. There exists a neighbourhood
W ⊂ V of z such that hα(W ) = W, the restriction hα : W →W is a homeomorphism,
and hpα = idW .
(E) There exist T > 0, a Euclidean metric d on some neighbourhood W of z, and
a sequence {xi}i∈N ⊂ V \ Σ converging to z such that ξ(xi) 6= 0 and
d(z, xi) < Tdiam(oxi ∩W ).
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Remark 4.1. By a Euclidean metric on W in (E) we mean a metric induced
by some embedding W ⊂ Rn. In fact, this condition can be formulated for arbitrary
Riemannian metrics, but for technical reasons (see especially Lemma 5.1) we restrict
ourselves to Euclidean ones.
Remark 4.2. If z ∈ Σ ∩ V has a neighbourhood W ⊂ V such that W \ Σ is
connected, then by (3) of Theorem 1.1 condition (B) holds for z.
Lemma 4.1. The following implications hold true:
(A)⇒ (B) ∨ (C)α.
If α ∈ Q, then for every z ∈ Σ ∩ V
(B) ∨ (C)α ⇒ (D)α.
If h is not the identity on some neighbourhood of z ∈ Σ ∩ V, then
(D)α ⇒ (E),
and we can take T = 4 in (E).
Proof. (A) ⇒ (B). Since z ∈ Σ ∩ V, there exists a neighbourhood W of z such
that F(W × [0, c]) ⊂ V. Then W satisfies (B). Indeed, let y ∈W \Σ. We have to show
that ξ|oy∩W is constant.
If ξ = 0 on oy∩W there is nothing to prove. Therefore we can assume that ξ(y) 6= 0.
Then, by (A), Per(y) < C, whence
oy = F(y × [0,Per(y)]) = F(y × [0,C]) ⊂ F(W × [0,C]) ⊂ V.
Thus oy ∩ V = oy is connected. Then, by Lemma 1.1, ξ is constant along oy and
therefore on oy ∩W.
(A) ⇒ (C)α. It suffices to show that hα is continuous at each z ∈ Σ ∩ V. Let
V ′ ⊂ V be any neighbourhood of z and W be another neighbourhood of z such that
F(W × [0, c]) ⊂ V ′. We claim that hα(W ) ⊂ V ′. This will imply continuity of hα at z.
Let x ∈ W. If x ∈ Σ ∩W or ξ(x) = 0, then hα(x) = x ∈ W ⊂ V ′. Otherwise,
ξ(x) 6= 0 and x is periodic. Hence hα(x) = F(x, τ) for some τ ∈ [0,Per(x)] ⊂ [0, c].
Therefore hα(x) ∈ F(W × [0, c]) ⊂ V ′.
(B)∨(C)α ⇒ (D)α. We have that α = q/p, where q ∈ Z and p ∈ N. For simplicity
denote hα by h. Since h(z) = z and h is continuous, there exists a neighbourhood B of
z such that hi(B) ⊂ V for all i = 0, . . . , p. Denote
W = B ∪ h(B) ∪ . . . ∪ hp−1(B).
We claim that W satisfies (D).
Indeed, let x ∈ V and suppose that h(x) ∈ V as well. Since h(x) belongs to the
orbit of x, then, by (B), ξ(x) = ξ(h(x)). Hence
h2(x) = F
(
h(x), αξ(h(x))
)
= F
(
h(x), αξ(x)
)
=
= F
(
F(x, αξ(x)), α · ξ(x)
)
= F
(
x, 2αξ(x)
)
. (4.2)
By induction we will get that if hi(x) ∈ V for all i = 0, . . . , j − 1, then
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962 S. I. MAKSYMENKO
hj(x) = F
(
x, jαξ(x)
)
.
In particular, hkp(x) = F(x, kqξ(x)) = x for any k ∈ Z. By (C)α we have that h is
continuous on V, whence h yields a homeomorphism of W onto itself and hp|W = idW .
(D)α ⇒ (E). Again denote hα by h. Since h is not the identity on W, we can
assume that p is a prime and thus the action of h is effective. This can be done by
replacing h with hn for some n ∈ N such that p/n is a prime.
Decreasing W we can assume that W is an open subset of the half-space Rn+ =
= {xn ≥ 0}. Let d be the corresponding Euclidean metric on W and rz be the radius
of convexity of W at z. Then by Lemma 3.2 for each r ∈ (0, rz) there exist xr ∈ W
and ar ∈ {1, . . . , p− 1} such that
d(z, xr) ≤ Sr ≤ Td(xr, h
ar (xr)) ≤ Tdiam(oxr ),
for some S,T > 0. In fact, S =
1
2
and T = 2 if z ∈ Int(M), and S =
2
3
and T = 4
if z ∈ ∂M. Notice that ar may take only finitely many values. Therefore we can find
a ∈ {1, . . . , p − 1} and a sequence {xri}i∈N such that lim
i→∞
ri = 0 and ari = a for all
i ∈ N.
The lemma is proved.
5. Condition (E) for C1-flows. Condition (E) defined in the previous section
gives some lower bound for diameters of orbits of a sequence {xi}i∈N of periodic
points converging to a fixed point z. In this section it is shown that for a C1-flow that
condition allows to estimate periods of xi.
Let M be a Cr, r ≥ 1, connected, m-dimensional manifold possibly noncompact
and with or without boundary. Let also F be a Cr-vector field on M tangent to ∂M and
generating a Cr-flow F : M ×R→M. Again by Σ we denote the set of fixed points of
F which coincides with the set of zeros (or singular points) of F.
Proposition 5.1. Let V ⊂M be an open subset, ξ : V \ Σ→ R be a P -function,
z ∈ Σ ∩ V, and {xi}i∈N ⊂ V \ Σ be a sequence of periodic points converging to z and
satisfying (E). Thus ξ(xi) 6= 0, and there exists T > 0 and a Euclidean metric on some
neighbourhood of z such that d(z, xi) < Tdiam(oxi). If the periods of xi are bounded
above with some C > 0 (in particular, condition (A) holds true), then
(e1) there exists ε > 0 such that |ξ(xi)| ≥ Per(xi) > ε for all i ∈ N, so the periods
are bounded below as well, and
(e2) j1F (z) 6= 0.
For the proof we need some statements. The proofs are straightforward, and we left
them for the reader.
Claim 5.1. Let X be a topological space, K be a compact space, and g : X×K →
→ R be a continuous function. Then the following function γ : X → R defined by
γ(x) = sup
y∈K
g(x, y) is continuous.
Claim 5.2. Let K be a compact manifold and
Q = (Q1, . . . , Qn) : Rn ×K → Rn
be a continuous map satisfying the following conditions:
(a) Q(0×K) = 0.
(b) For each k ∈ K the map Qk = Q(·, k) : Rn → Rn is C1.
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(c) For every i, j = 1, . . . , n the partial derivative
∂Qi
∂xj
: Rn ×K → R of the i-th
coordinate function Qi of Q in xj is continuous.
In particular, conditions (b) and (c) hold if K is a manifold and Q is a C1 map.
Then there exists a continuous function α : Rn → R such that
‖Q(x, k)‖ ≤ ‖x‖α(x), (x, k) ∈ Rn ×K.
If j1Qk(0) = 0 for all k ∈ K, then α(0) = 0.
Lemma 5.1. Let F be a C1-vector field in Rn such that F (0) = 0 and (Ft) be
the local flow of F. Then for every C > 0 there exist a neighbourhood W of the origin
0 ∈ Rn and a continuous function γ : W → R such that
‖F (F(x, t))‖ ≤ ‖x‖γ(x)
for all (x, t) ∈W × [−C,C]. If j1F (0) = 0, then γ(0) = 0.
If F is C2, then we have a usual estimation ‖F (F(x, t))‖ ≤ A‖x‖2 for some A > 0.
Proof. Since F(0, t) = 0 for all t ∈ R, there exists a neighbourhood W of z such
that for each (x, t) ∈ W × [−C,C] the point F(x, t) is well-defined and belongs to V,
so F(W × [−C,C]) ⊂ V.
Moreover, F is C1 and therefore it satisfies assumptions (a) – (c) of Claim 5.2 with
K = [−C,C]. Hence there exists a continuous function α : W → R such that∥∥F(x, t)
∥∥ ≤ ‖x‖α(x), (x, k) ∈W × [−C,C].
Moreover, F is also C1 and F (0) = 0, whence again by Claim 5.2 (for K = ∅)
there exists a continuous function β : W → R such that ‖F (x)‖ ≤ ‖x‖β(x). Define
γ : W → R by
γ(x) = sup
t∈[−C,C]
α(x)β(F(x, t)).
Then by Claim 5.1 γ is continuous and
‖F (F(x, t))‖ ≤ ‖F(x, t))‖β(F(x, t)) ≤ ‖x‖α(x)β(F(x, t)) ≤ ‖x‖γ(x).
Moreover, if j1F (0) = 0, then β(0) = 0. Since in addition F(0, t) = 0, we obtain that
γ(0) = sup
t∈[−C,C]
α(0)β(0) = 0 as well.
The lemma is proved.
Proof of Proposition 5.1. We have to show that violating either of assumptions (e1)
or (e2) leads to a contradiction.
By Lemma 5.1 for any C > 0 there exist a neighbourhood W of z and a continuous
function γ : W → R such that
‖F (F(x, t))‖ ≤ d(x, z)γ(x), (x, t) ∈W × [−C,C]. (5.1)
Then
d(z, xi) < Tdiam(oxi)
(3.4)
≤ T
2
l(xi)
(3.3)∨(5.1)
≤ T
2
Per(xi)d(z, xi)γ(xi).
Therefore
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964 S. I. MAKSYMENKO
0 <
2
T
< Per(xi)γ(xi) ≤ |ξ(xi)|γ(xi).
Hence if (e1) is violated, i.e., lim
i→∞
Per(xi) = 0, then lim
i→∞
γ(xi) = +∞, which contra-
dicts to continuity of γ near z.
Suppose j1F (z) = 0. Then by Lemma 5.1 γ(z) = 0, whence lim
i→∞
γ(xi) = γ(z) =
0. Therefore lim
i→∞
Per(xi) = lim
i→∞
|ξ(xi)| = +∞, which contradicts to boundedness of
periods of xi.
The proposition is proved.
6. Unboundedness of periods. Now let F be a C1-vector field in Rn such that
F (0) = 0. We can regard F as a C1 map F = (F1, . . . , Fn) : Rn → Rn. Let
A =
(
∂Fi
∂xj
(0)
)
i,j=1,...,n
be the Jacobi matrix of F at 0. This matrix also called the linear part of F at 0. By the
real Jordan’s normal form theorem A is similar to the matrix of the following form:
s
⊕
σ=1
Jqσ (aσ ± ibσ))⊕
r
⊕
τ=1
Jpτ (λτ ), (6.1)
where λσ ∈ R and aτ ± ibτ ∈ C are all the eigen values of A.
Theorem 1.3 is a direct consequence of the following theorem.
Theorem 6.1. Suppose one of the following conditions holds:
(1) A has an eigen value λ such that <(λ) 6= 0.
(2) The matrix (6.1) has either a block Jq(±ib) or Jq(0) with q ≥ 2.
(3) A = 0 and there exists an open neighbourhood V of 0 in Rn and a continuous
P -function ξ : V \ Σ → R which takes non-zero values arbitrary close to 0, that is
0 ∈ V \ ξ−1(0).
Then there exists a sequence {xi}i∈N ⊂ V \ Σ which converges to 0 and such that
either every xi is nonperiodic, or every xi is periodic and lim
i→∞
Per(xi) = +∞.
Proof. (1) In this case by Hadamard – Perron’s theorem, e.g. [19], we can find a
nonperiodic orbit o of F such that 0 ∈ o \ o. This means that there exists a sequence
{xi}i∈N ⊂ o converging to 0.
(2) Consider two cases.
(a) If (6.1) has a block Jq(0) with q ≥ 2, then it can be assumed that
A =
0 0 . . . 0
1 0 . . . 0
. . . . . . . . . . . .
. . . . . . . . . . . .
.
(b) Suppose (6.1) has a block Jq(±ib) with q ≥ 2. Then we can regard Rn as
C2 ⊕ Rn−4, so the first two coordinates x1 and x2 are complex. Therefore it can be
supposed that
A =
ib 0 . . . 0
1 ib . . . 0
. . . . . . . . . . . .
. . . . . . . . . . . .
.
In both cases denote by p1 the projection to the first (either real or complex) coordi-
nate, i.e., p1(x1, . . . , xn) = x1.
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Lemma 6.1. Let Sn−1 be the unit sphere in Rn centered at the origin 0, ε > 0,
Yε =
{
(x1, . . . , xn) ∈ Sn−1 : |x1| ≥ ε
}
,
and Cε be the cone over Yε with vertex at 0.
Then for each L > 0 there exists a neighbourhood W = WL,ε of 0 such that every
x ∈ (W ∩ Cε) \ 0
is either nonperiodic or periodic with period Per(x) > L.
Proof. Let (Ft) be the local flow of F. Then in general, F is defined only on some
open neighbourhood of Rn×0 in Rn×R. Nevertheless, since F(0, t) = 0 for all t ∈ R,
it follows that for each L > 0 there exists a neighbourhood V of 0 such that F is defined
on V × [−2L, 2L].
We claim that in both cases there exists c > 0 such that
‖eAtx− x‖ ≥ c|tx1| = c|tp1(x)|. (6.2)
(a) In this case eAtx = (x1, tx1 + x2, . . .) and we can put c = 1:
‖eAtx− x‖ = ‖(0, tx1, . . .)‖ ≥ |tx1| = |tp1(x)|.
(b) Now eAtx = (eibtx1, e
ibt(tx1 + x2), . . .). Notice that we can write eibt =
= 1 + tγ(t) for some smooth function γ : R → C \ {0}. Denote c = min
t∈[−2L,2L]
|γ(t)|.
Then c > 0 and
‖eAtx− x‖ =
∥∥(eibtx1 − x1, . . .)
∥∥ ≥ |tγ(t)x1| = c|tp1(x)|.
Since F is a C1 map and F(0, t) = 0 for all t ∈ R, it follows from Claim 5.2 that
there exists a continuous function α : V → [0,+∞) such that
(1) α(0) = 0,
(2) ‖F(x, t)− eAtx‖ ≤ ‖x‖α(x) for all (x, t) ∈ V × [−2L, 2L].
Hence
‖F(x, t)− x‖ ≥ ‖eAtx− x‖−‖F(x, t)− eAtx‖ ≥ c|tp1(x)|−‖x‖α(x)
for (x, t) ∈ Rn × [−2L, 2L]. Moreover, if p1(x) 6= 0, then
‖F(x, t)− x‖ ≥ c|tp1(x)| − ‖x‖α(x) = c|p1(x)|
(
|t| − ‖x‖α(x)
c|p1(x)|
)
.
Since α(0) = 0, there exists a neighbourhood W ⊂ V of 0 such that
α(x) < cεL, x ∈W.
Now let x ∈ (W ∩ Cε) \ 0. Then |x| ≤ 1 and |p1(x)| ≥ ε, whence
‖x‖α(x)
c|p1(x)|
<
1cεL
cε
= L.
Therefore ‖F(x, t) − x‖ ≥ cε(|t| − L) for t ∈ [−2L, 2L]. In particular, F(x, t) 6= x for
t ∈ [L, 2L]. It follows that x is either nonperiodic, or periodic with the period being
greater than 2L− L = L.
The lemma is proved.
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966 S. I. MAKSYMENKO
(3) Suppose that there exists a neighbourhood V of 0 such that all points in V \ Σ
are periodic. If the periods of points in V \ ξ−1(0) are bounded above, i.e., condition
(A) holds true, then by (e2) of Proposition 5.1 j1F (0) 6= 0.
Theorem 6.1 is proved.
The following statement extends [2] (Proposition 10).
Proposition 6.1 ([2], Proposition 10). Let F be a C1-vector field on a manifold M,
z ∈ Σ \ Int(Σ), V ⊂ M be a neighbourhood of z, and ξ ∈ P (V ) be a P -function. Let
also W = ∪λ∈ΛWλ be the union of those connected components of V \ Σ for which
z ∈Wλ. Then each of the following conditions implies that ξ = 0 on W ∩ V :
(a) j1F (z) has an eigen value λ such that <(λ) 6= 0;
(b) a real normal Jordan form of j1F (z) has either a block Jq(±ib) or Jq(0) with
q ≥ 2;
(c) j1F (z) = 0;
(d) z ∈ Int(ΣF ) \ Int(Σ);
(e) ξ(z) = 0.
Proof. Suppose that ξ takes non-zero values on periodic points arbitrary close to z.
First we prove that every of the assumptions (a) – (d) implies (e), and then show that (e)
gives rise to a contradiction.
(a)∨(b)∨(c) ⇒ (e) Suppose j1F (z) satisfies either of the conditions (a), (b) or (c).
Then by assumption on ξ and Theorem 6.1 there exists a sequence {xi}i∈N ⊂ V \Σ con-
verging to z and such that every xi is either nonperiodic, or periodic but lim
i→∞
Per(xi) =
= +∞.
If every xi is nonperiodic, then by Lemma 1.1 ξ(xi) = 0, whence by continuity of
ξ we obtain ξ(z) = 0 as well.
Suppose every xi is periodic. Then ξ(xi) = ni Per(xi) for some ni ∈ Z. Since
lim
i→∞
Per(xi) = +∞ and ξ is continuous, it follows that lim
i→∞
ni = 0, that is ni = 0 for
all sufficiently large i, whence ξ(xi) = 0 which implies ξ(z) = 0.
(d) ⇒ (c) The assumption z ∈ Int(Σ) \ Int(Σ) means that there is a sequence
{zi}i∈N ⊂ Int(Σ) converging to z. But then j1F (zi) = 0 for all i. Since F is C1, we
obtain j1F (z) = 0 as well.
(e) Suppose that ξ(z) = 0. Let U be a neighbourhood of z with compact closure
U ⊂ V, and C = sup
x∈U
|ξ(x)|. Then the periods of points in U \ ξ−1(0) are bounded
above with C, that is ξ satisfies condition (A), and therefore by Lemma 4.1 condition
(E). Let {xi}i∈N ⊂ U \ Σ be a sequence converging to z and satisfying (E). Then by
Proposition 5.1 there exists ε > 0 such that |ξ(xi)| > ε. Since ξ is continuous, we get
|ξ(z)| ≥ ε > 0, which contradicts to the assumption ξ(z) = 0.
The proposition is proved.
7. Proof of Theorem 1.2. Let F be a flow conjugate to a C1-flow. Then by [3],
F is conjugate to a flow generated by a C1-vector field F. As noted in Subsection 1.1,
conjugation does not change the structure of the set of P -functions, therefore we can
assume that F itself is generated by C1-vector field F.
Let θ ∈ P (M) be a nonnegative generator of P (M). Put Y = θ−1(0). Then Y is
closed.
We claim that Y is also open in M. Indeed, if x is a non-fixed point of F, then by [2]
(Corollary 8) θ = 0 on some neighbourhood of x. Suppose x ∈ Σ. Since Σ is nowhere
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PERIOD FUNCTIONS FOR C0- AND C1-FLOWS 967
dense in M, it follows from (1) of Proposition 6.1 that θ = 0 on some neighbourhood
of x as well.
As M is connected, we obtain that either Y = ∅ or Y = M. By Theorem 1.1 θ > 0
on M \ Σ, whence Y 6= M and therefore Y = ∅, so θ > 0 on all of M.
Let z ∈ Σ. To establish (1.1) it suffices to prove that
(a) j1F (z) has no eigen values λ with <(λ) 6= 0;
(b) a real normal Jordan form of j1F (z) has neither a block Jq(±ib) nor Jq(0) with
q ≥ 2;
(c) j1F (z) 6= 0.
But if either of these conditions were violated, then it would follow from Proposi-
tion 6.1 that θ(z) = 0. This contradiction completes Theorem 1.2.
8. Proof of Theorem 1.3. Let z ∈ Σ be such that j1F (z) is not similar to a matrix
of the form (1.1). Then for this point z one of the conditions (1) – (3) of Theorem 6.1
holds true. Since every x ∈ V \ Σ is periodic, it follows from Theorem 6.1 that there
exists a sequence {xi}i∈N ⊂ V \ Σ converging to z and such lim
i→∞
Per(xi) = +∞.
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Received 22.02.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 7
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