Kernel of a map of a shift along the orbits of continuous flows
Let F:M × R → M be a continuous flow on a topological manifold M. For every subset V⊂M, we denote by P(V) the set of all continuous functions ξ:V→R such that F(x,ξ(x))=x for all x∈V. These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples...
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irk-123456789-1661512020-02-19T01:26:10Z Kernel of a map of a shift along the orbits of continuous flows Maksymenko, S.I. Статті Let F:M × R → M be a continuous flow on a topological manifold M. For every subset V⊂M, we denote by P(V) the set of all continuous functions ξ:V→R such that F(x,ξ(x))=x for all x∈V. These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the corresponding periods (in general, not minimal). In this paper, the structure of P(V) is described for an arbitrary connected open subset V⊂M. Нехай F:M×R→M — неперервний потік на топологічному многовиді M. Для кожної підмножини V⊂M позначимо через P(V) множину всіх неперервних функцій ξ:V→R , що задовольняють умову F(x,ξ(x))=x для всіх x∈V. Такі функції набувають нульового значення в неперіодичних точках потоку, а в періодичних точках їх значення є цілими кратними відповідних періодіб (в загальному не мінімальними). В статті описано структуру P(V) для довільної відкритої зв'язної підмножини V⊂M. 2010 Article Kernel of a map of a shift along the orbits of continuous flows / S.I. Maksymenko // Український математичний журнал. — 2010. — Т. 62, № 5. — С. 651–659. — Бібліогр.: 11 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166151 515.145+515.146 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Maksymenko, S.I. Kernel of a map of a shift along the orbits of continuous flows Український математичний журнал |
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Let F:M × R → M be a continuous flow on a topological manifold M. For every subset V⊂M, we denote by P(V) the set of all continuous functions ξ:V→R such that F(x,ξ(x))=x for all x∈V. These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the corresponding periods (in general, not minimal). In this paper, the structure of P(V) is described for an arbitrary connected open subset V⊂M. |
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Maksymenko, S.I. |
| author_facet |
Maksymenko, S.I. |
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Maksymenko, S.I. |
| title |
Kernel of a map of a shift along the orbits of continuous flows |
| title_short |
Kernel of a map of a shift along the orbits of continuous flows |
| title_full |
Kernel of a map of a shift along the orbits of continuous flows |
| title_fullStr |
Kernel of a map of a shift along the orbits of continuous flows |
| title_full_unstemmed |
Kernel of a map of a shift along the orbits of continuous flows |
| title_sort |
kernel of a map of a shift along the orbits of continuous flows |
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Інститут математики НАН України |
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2010 |
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Статті |
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| citation_txt |
Kernel of a map of a shift along the orbits of continuous flows / S.I. Maksymenko // Український математичний журнал. — 2010. — Т. 62, № 5. — С. 651–659. — Бібліогр.: 11 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT maksymenkosi kernelofamapofashiftalongtheorbitsofcontinuousflows |
| first_indexed |
2025-07-14T20:48:55Z |
| last_indexed |
2025-07-14T20:48:55Z |
| _version_ |
1837656845584433152 |
| fulltext |
UDC 515.145 + 515.146
S. I. Maksymenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
KERNEL OF MAP OF A SHIFT ALONG THE ORBITS
OF CONTINUOUS FLOWS*
ЯДРО ВIДОБРАЖЕННЯ ЗСУВУ ВЗДОВЖ ОРБIТ
НЕПЕРЕРВНИХ ПОТОКIВ
Let F : M × R → M be a continuous flow on a topological manifold M. For every subset V ⊂ M we
denote by P (V ) the set of all continuous functions ξ : V → R such that F(x, ξ(x)) = x for all x ∈ V. These
functions vanish at non-periodic points of the flow, while their values at periodic points are integer multiples
of the corresponding periods (in general not minimal). In this paper, the structure of P (V ) is described for
arbitrary connected open subset V ⊂ M.
Нехай F : M × R → M — неперервний потiк на топологiчному многовидi M. Для кожної пiдмножини
V ⊂ M позначимо через P (V ) множину всiх неперервних функцiй ξ : V → R, що задовольняють умо-
ву F(x, ξ(x)) = x для всiх x ∈ V. Такi функцiї набувають нульового значення в неперiодичних точках
потоку, а в перiодичних точках їх значення є цiлими кратними вiдповiдних перiодiв (в загальному не
мiнiмальними). В статтi описано структуру P (V ) для довiльної вiдкритої зв’язної пiдмножини V ⊂ M.
1. Introduction. Let F : M × R → M be a continuous flow on a topological finite-
dimensional manifoldM. For x ∈ M we will denote by ox the orbit of x. If x is periodic,
then Per(x) is the period of x. The set of fixed points of F will be denoted by Σ.
For each subset V ⊂ M define the following map
ϕV : C(M,R) → C(V,M), ϕV (α)(x) = F(x, α(x)),
for α ∈ C(V,R) and x ∈ V. We will call ϕV the shift map along the orbits of F. It was
used by the author for study of homotopy types of certain infinite-dimensional functional
spaces, see, e.g., [1 – 6].
Let iV : V ⊂ M be the inclusion map. Then the following set
P (V ) = ϕ−1
V (iV )
will be called the kernel of ϕV .
Thus a continuous function ξ : V → R belongs to P (V ) iff
F(x, ξ(x)) = x ∀x ∈ V. (1.1)
In this case we will say that ξ is a period function or simply a P -function for F on V.
The aim of this paper is to give a description of P (V ) for open connected subsets
V ⊂ M with respect to a continuous flow on a topological manifold M (Theorem 1.1).
Such a description was given in [1] (Theorem 12) for C∞ flows. It turns out that
both descriptions almost coincide. Our methods are based on well-known theorems of
M. Newman about actions of finite groups.
The following easy lemma explains the term P -function. The proof is the same as in
[1] (Lemmas 5 and 7) and we leave it for the reader.
*This research is done within the program of National Academy of Sciences of Ukraine “Modern methods
of investigation of mathematical models in the problems of natural sciences”, research No. 0107U002333.
c© S. I. MAKSYMENKO, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5 651
652 S. I. MAKSYMENKO
Lemma 1.1 [1] (Lemmas 5 and 7). For any subset V ⊂ M the set P (V ) is a group
with respect to the point-wise addition.
Let x ∈ V and ξ ∈ P (V ). Then ξ is locally constant on ox ∩ V. In particular, if
x is non-periodic, then ξ|ox∩V = 0. Suppose x is periodic, and let ω be some path
component of ox ∩ V. Then ξ = nω · Per(ox) for some nω ∈ Z depending on ω.
It is not true that any P -function on any subset V ⊂ M is constant on all of ox ∩ V
for each x ∈ V, see Example 1.1 below. Therefore we give the following definition.
Definition 1.1. A P -function ξ : V → R will be called regular if ξ is constant on
ox ∩ V for each x ∈ V.
Denote by RP (V ) the set of all regular P -functions on V. Then RP (V ) is a subgroup
of P (V ).
Remark 1.1. If for any periodic orbit o the intersection o ∩ V is either empty or
connected, e.g. in the case when V is F-invariant, then any P -function on V is regular.
The following theorem extends [1] (Theorem 12) for continuous case.
Theorem 1.1. Let M be a finite-dimensional topological manifold possibly non-
compact and with or without boundary, F : M × R → M be a flow, and V ⊂ M be an
open, connected set.
(A) If Int(Σ) ∩ V 6= ∅, then
P (V ) =
{
ξ ∈ C(V,R) : ξ|V \Int(Σ) = 0
}
.
(B) Suppose 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 a regular P -function.
(4) Denote 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 coincide with the ones of F.
In particular, in all the cases RP (V ) = P (V ).
Theorem 1.1 will be proved in Section 3.
The following simple example illustrates necessity of conditions of Theorem 1.1.
It shows that on non-open or disconnected sets V ⊂ M there may exist non-regular
P -functions and that P -functions for continuous flows may vanish at fixed points.
Example 1.1. Let F : C × R → C be a continuous flow on the complex plane C
defined by
F(z, t) =
e2πi 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
θ = |z|2 is a P -function on C and
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
KERNEL OF MAP OF A SHIFT ALONG THE ORBITS OF CONTINUOUS FLOWS 653
RP (C) = P (C) = {nθ}n∈Z.
Also notice that θ(0) = 0 and θ > 0 on C \ 0. This agrees with (1) of Theorem 1.1
and shows that non-zero P -functions may vanish at fixed points of flows.
Let Vi, i = 1, 2, 3, be the corresponding subset in C shown in Fig. 1.1. Thus V1 is
an open segment, say (−1, 1), on the real axis, V2 is a union of two closed triangles
with common vertex at the origin 0, and V3 is union of a triangle with a segment (−1, 0]
of the real axis intersecting at the origin. In particular, Int(V1) = ∅, Int(V2) is not
Fig. 1.1
connected, and Int(V3) is not dense in V. For any pair m,n ∈ Z define the function
ξm,n : Vi → R by
ξm,n(z) =
−m|z|, ℜ(z) ≤ 0,
n|z|, ℜ(z) > 0.
Evidently, P (Vi) = {ξm,n}m,n∈Z, while RP (Vi) = {ξm,m}m∈Z. Thus not every P -
function is regular.
Structure of the paper. In next section we describe certain properties of P -function
for continuous flows: local uniqueness, local regularity, and continuity of extensions of
regularP -functions. We also deduce from well-known M. Newman’s theorem a sufficient
condition for divisibility of regular P -functions by integers in P (V ). These results will
be used in Section 3 for the proof of Theorem 1.1.
2. Properties of P -functions.
Lemma 2.1. Let z ∈ M. Suppose there exists a sequence of periodic points
{xi}i∈N converging to z and such that lim
i→∞
Per(xi) = 0. Then z ∈ Σ.
Proof. Suppose z 6∈ Σ, so there exists τ > 0 such that z 6= Fτ (z). Let U be a
neighbourhood of z such that
U ∩ Fτ (U) = ∅. (2.1)
Since F(z, 0) = z, there exists ε > 0 and a neighbourhood W of z such that F(W ×
× [0, ε]) ⊂ U. Then we can find xi ∈ W with Per(xi) < ε. Hence
Fτ (xi) ∈ Fτ (U).
On the other hand,
Fτ (xi) ∈ oxi
= F(xi, [0,Per(xi)]) ⊂ F(W × [0, ε]) ⊂ U,
which contradicts to (2.1).
Lemma 2.2 (Local uniqueness of P -functions, c.f. [1], Corollary 8). Let V ⊂ M
be any subset, z ∈ V \Σ and ξ ∈ P (V ). If ξ(z) = 0, then ξ = 0 on some neighbourhood
of z in V.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
654 S. I. MAKSYMENKO
Proof. Suppose ξ is not identically zero on any neighbourhood of z in V \Σ. Then
there exists a sequence {xi}i∈N ⊂ V \Σ converging to z and such that ξ(xi) 6= 0. Hence
every xi is periodic and ξ(xi) = niPer(xi) for some ni ∈ Z \ {0}. By continuity of ξ
we get
0 = lim
i→∞
ξ(xi) = lim
i→∞
niPer(xi).
Since |ni| ≥ 1, it follows that lim
i→∞
Per(xi) = 0, whence by Lemma 2.1 z ∈ Σ, which
contradicts to the assumption.
Lemma 2.3 (Local regularity of P -functions on open sets). Let V ⊂ M be an open
subset and ξ ∈ P (V ). Then for each z ∈ V there exists a neighbourhood W ⊂ V such
that the restriction ξ|W is regular.
Proof. Suppose ξ is not regular on arbitrary small neighbourhood of z. Then we can
find two sequences {xi}i∈N and {yi}i∈N converging to z such that yi = F(xi, τi) for
some τi ∈ R and ξ(xi) < ξ(yi) for all i ∈ N.
It follows that xi and yi are periodic. Otherwise, by Lemma 1.1, we would have
ξ(xi) = ξ(yi). Hence 0 < ξ(yi)− ξ(xi) = niPer(xi) for some ni ∈ Z \ {0}.
We claim that lim
i→∞
Per(xi) = 0. Indeed, take any ε > 0. Then there is a nei-
ghbourhood W of z such that |ξ(y) − ξ(x)| < ε for all x, y ∈ W. Let N > 0 be such
that xi, yi ∈ W for i > N,
Per(xi) ≤ niPer(xi) = ξ(yi)− ξ(xi) < ε, i > N.
This implies lim
i→∞
Per(xi) = 0, whence, by Lemma 2.1, z ∈ Σ. But in this case there
exists a neighbourhood W1 of z and ε > 0 such that F(W1 × [0, ε]) ⊂ V. Take xi ∈ W1
such that Per(xi) < ε, then
oxi
= F(xi × [0,Per(xi)) ⊂ F(W1 × [0, ε]) ⊂ V.
In other words oxi
∩ V = oxi
is connected, whence by Lemma 1.1 ξ is constant on oxi
.
Therefore ξ(xi) = ξ(yi) which contradicts to the assumption.
Lemma 2.4 (Continuity of extensions of regular P -functions). Let V ⊂ M be an
open subset and ξ ∈ RP (V ) be a regular P -function on V. Put U = F(V × R). Then
ξ extends to a P -function ξ̃ on all of U.
If M is a Cr manifold, F is Cr on V × R, and ξ is Cr on V, then ξ̃ is Cr on U.
Proof. The definition of ξ̃ is evident: if y ∈ U, so y = F(x, τ) for some (x, τ) ∈
∈ V × R, then we put ξ̃(y) = ξ(x). Since ξ is regular, this definition does not depend
on a particular choice of such (x, τ).
It remains to prove continuity of ξ̃ on U. Let y = F(x, t) ∈ U for some (x, t) ∈
∈ V × R. Since V is open, there exists a neighbourhood W of y in U such that
F−t(W ) ⊂ V. Then ξ̃ can be defined on W by ξ̃(z) = ξ ◦ F−t(z) for all z ∈ W. This
shows continuity of ξ̃ on W.
Moreover, if M is a Cr manifold, ξ and F are Cr, then so is ξ̃.
In order to formulate the last preliminary result we recall the following well-known
theorem of M. Newman:
Theorem 2.1 (M. Newman [7], see also [8 – 10]). If a compact Lie group effecti-
vely acts on a connected manifold M, then the set Σ of fixed points of this action is
nowhere dense in M and, by [9], it does not separate M.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
KERNEL OF MAP OF A SHIFT ALONG THE ORBITS OF CONTINUOUS FLOWS 655
Lemma 2.5 (Condition of divisibility by integers). Let V ⊂ M be a connected
open subset and ξ : V → R be a regular P -function. Suppose that there exist an
integer p ≥ 2 and a non-empty open subset W ⊂ V such that F(x, ξ(x)/p) = x for all
x ∈ W, so the restriction of ξ/p to W is a P -function. Then ξ/p is also a P -function on
all of V.
Proof. By Lemma 2.4 we can assume that V is F-invariant. Moreover, it suffices to
consider the case when p is a prime. Define the following map h : V → V by h(x) =
= F(x, ξ(x)/p). Since ξ is constant along orbits of F, it follows that ξ(h(x)) = ξ(x),
whence
h ◦ h(x) = F
(
h(x), ξ(h(x))/p
)
= F(F
(
x, ξ(x)/p, ξ(x)/p
)
= F
(
x, 2ξ(x)/p
)
.
Similarly,
hk(x) = F
(
x, kξ(x)/p
)
, k ∈ N.
In particular, we obtain that hp = idV , and thus h yields a Zp-action on V. But by
assumption this action is trivial on the non-empty open set W. Then by M. Newman’s
Theorem 2.1 the action is trivial on all of V, so ξ/p is a P -function on V.
Corollary 2.1. Let ξ be a regular P -function on a connected open subset V ⊂ M.
(i) If V ∩ Int(Σ) 6= ∅, then ξ = 0 on V \ Int(Σ).
(ii) If V ∩ Int(Σ) = ∅ and ξ = 0 on some open non-empty subset W ⊂ V, then
ξ = 0 on all of V.
Proof. Evidently, it suffices to show that in both cases ξ = 0 on V \ Σ.
In the case (i) put W = V ∩ Int(Σ).
Let p be any prime. Then in both cases F(y, ξ(y)/p) = y for all y ∈ W, where W
is a non-empty open set. Hence by Lemma 2.5 F(y, ξ(y)/p) = y for all y ∈ V, that is
ξ/p is a P -function on V. Thus if ξ(x) = nPer(x) 6= 0 for some x ∈ V \Σ and n ∈ Z,
then n is divided by p. Since p is arbitrary, we get n = 0.
3. Proof of Theorem 1.1. (A). Suppose Int(Σ) ∩ V 6= ∅. We should prove that the
following set
P ′ =
{
ξ ∈ C(V,R) : ξ|V \Int(Ξ) = 0
}
coincides with P (V ). Evidently, P ′ ⊂ P (V ).
Conversely, let ξ ∈ P (V ). We claim that for every connected component T of
V \ Int(Σ) there exists z ∈ T such that ξ(z) = 0. By Lemma 2.2 this will imply that
ξ|T = 0. Since T is arbitrary we will get that ξ = 0 on all of V \ Int(Σ) and, in
particular, that ξ is a regular P -function.
As V is connected, the following set is non-empty, see Fig. 2.1:
B := T ∩ V ∩
(
Int(Σ) \ Int(Σ)
)
6= ∅.
Let x ∈ B ⊂ V = Int(V ). Then by Lemma 2.3 there exists an open connected
neighbourhood W such that ξ|W is a regular P -function. Then we have that W ∩
∩ Int(Σ) 6= ∅ and W ∩ T 6= ∅ as well. Since ξ is regular on W, it follows from (i) of
Corollary 2.1 that ξ = 0 on W \ Int(Σ) and, in particular, on W ∩ T.
(B). Suppose that Int(Σ) ∩ V = ∅ and P (V ) 6= {0}, so there exists ξ ∈ P (V )
which is not identically zero on V. We have to show that P (V ) = {nθ}n∈Z for some
P -function θ : V → R satisfying (1) – (4).
Denote by Y the subset of V consisting of all points x having one of the following
two properties:
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
656 S. I. MAKSYMENKO
Fig. 2.1
(L1) x ∈ V \ Σ and ξ(x) = 0;
(L2) x ∈ V ∩ Σ and there exists a sequence {xi}i∈N ⊂ V \ Σ converging to x and
such that ξ(xi) = 0 for all i ∈ N.
Evidently, ξ = 0 on Y.
Lemma 3.1. Y is open and closed in V. Hence if V is connected and ξ(x) = 0
for some x ∈ V \ Σ, then ξ = 0 on all of V.
Proof. Y is open. Let x ∈ Y. We will show that there exists an open neighbourhood
W of x such that W ⊂ Y.
If x ∈ V \Σ, then, by Lemma 2.2, ξ = 0 on some neighbourhood W ⊂ V \Σ of x.
Hence, by (L1), W ⊂ Y.
Suppose x ∈ Σ ∩ V ⊂ V = Int(V ). Then by Lemma 2.3 there exists an open
connected neighbourhood Wx of x such that ξ|Wx
is regular. We claim that Wx ⊂ Y.
First we show that ξ = 0 on Wx. Indeed, by (L2) there exists a sequence {xi}i∈N ⊂
⊂ V \ Σ converging to x and such that ξ(xi) = 0 for all i ∈ N. In particular, xi ∈ Wx
for some i ∈ N. Let C be the connected component of Wx \ Σ containing xi. Then
ξ = 0 on an open set C ⊂ Wx, whence, by (ii) of Corollary 2.1, ξ = 0 on Wx.
Therefore Wx \ Σ ⊂ Y. Let y ∈ Wx ∩ Σ. Since Wx ∩ Σ is nowhere dense in Wx,
there exists a sequence {yi}i∈N ⊂ Wx \Σ converging to y. But then ξ(yi) = 0, whence,
by (L2), y ∈ Y as well.
Y is closed. Let {xi}i∈N ⊂ Y be a sequence converging to some x ∈ V. We have to
show that x ∈ Y. Since ξ(xi) = 0, we have ξ(x) = 0 as well.
If x ∈ V \ Σ, then by (L1) x ∈ Y.
Suppose x ∈ V ∩Σ. Then we can assume that either {xi}i∈N ⊂ V \Σ or {xi}i∈N ⊂
⊂ V ∩Σ. In the first case x ∈ Y by (L2).
Suppose {xi}i∈N ⊂ V ∩ Σ. Since xi ∈ Y, it follows from (L2) for xi that there
exists a sequence {yji }j∈N ⊂ V \Σ converging to xi and such that ξ(yji ) = 0. Then for
each i ∈ N we can find n(i) ∈ N such that the diagonal sequence {y
n(i)
i }i∈N ⊂ V \ Σ
converges to x, and satisfies ξ(y
n(i)
i ) = 0. Hence, by (L2), x ∈ Y.
The lemma is proved.
Thus we can assume that ξ 6= 0 on V \ Σ. In particular, all points in V \ Σ are
periodic.
Take any x ∈ V \ Σ and consider the following homomorphism
ex : P (V ) → Z, ex(ν) = ν(x)/Per(x),
for ν ∈ P (V ). If ν(x) = 0, then, as noted above, ν = 0 on all of V, whence ex is
a monomorphism. Moreover, ex(ξ) = ξ(x) 6= 0, whence ex yields an isomorphism of
P (V ) onto a non-zero subgroup kZ of Z for some k ∈ N. Put θ = e−1
x (k). Then
P (V ) = {nθ}n∈Z.
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KERNEL OF MAP OF A SHIFT ALONG THE ORBITS OF CONTINUOUS FLOWS 657
It remains to verify properties of θ.
(2) ⇒ (1). We have that θ(x) = Per(x) > 0 on an open and everywhere dense subset
Q ⊂ V, whence θ ≥ 0 on V. On the other hand, by Lemma 3.1, θ 6= 0 on V \Σ, whence
θ > 0 on V \ Σ.
(2) ⇒ (3). We have to show that θ is regular, that is
θ(x) = θ(Fτ (x))
for any x ∈ V \ Σ and τ ∈ R such that Fτ (x) ∈ V.
First notice that for any open subsets A,B ⊂ M we have that
A ∩B = A ∩B = A ∩B. (3.1)
Since Q is open and everywhere dense in V, it follows that
Fτ (x) ∈ V ∩ Fτ (V ) ⊂ Q ∩ Fτ (V )
(3.1)
=
(3.1)
= Q ∩ Fτ (V ) = Q ∩Fτ (Q)
(3.1)
= Q ∩Fτ (Q).
In other words, there exists a sequence {xi}i∈N ⊂ Q converging to x and such that
{Fτ (xi)}i∈N ⊂ Q. Then θ(Fτ (xi)) = θ(xi) = Per(xi). Whence
θ(Fτ (x)) = lim
i→∞
θ(Fτ (xi)) = lim
i→∞
θ(xi) = θ(x).
(3) ⇒ (4). See Lemma 2.4.
(2) The proof consists of the following three statements.
Claim 3.1. Let x ∈ V \ Σ. Then there exist an open connected neighbourhood
Wx of x in V, a regular P -function θx ∈ P (Wx), a number mx ∈ Z \ {0}, and an open
and everywhere dense subset Qx ⊂ Wx consisting of periodic points such that
(a) P (Wx) = {mθx}m∈Z,
(b) θ = mxθx on Wx,
(c) θx(y) = Per(y) for all y ∈ Qx.
Proof. By Lemma 2.3 there exists an open connected neighbourhood Wx of x such
that Wx ⊂ V \Σ and θ|Wx
is regular. Notice that if we decrease Wx, then the restriction
of θ to Wx remains regular. Therefore we can additionally assume that there exists
ε ∈ (0,Per(x)) such that
(i) θ(y) < θ(x) + ε for all y ∈ Wx;
(ii) Per(x) < Per(y) + ε for all y ∈ Wx;
(iii) there is N > 0 such that ny := θ(y)/Per(y) < N for all y ∈ Wx.
Indeed, (i) follows from continuity of θ, and (ii) from lower semicontinuity of Per,
c.f. [11].
More precisely, suppose (ii) fails. Then there exists a sequence {xi}i∈N ⊂ V \ Σ
converging to x and such that Per(x) ≥ Per(xi) + ε. In particular, periods of xi are
bounded above and we can assume that lim
i→∞
Per(xi) = τ < ∞ for some τ. Then
Per(x) ≥ τ + ε > τ. (3.2)
But F(x, τ) = lim
i→∞
F(xi,Per(xi)) = x, so τ = nPer(x) ≥ Per(x) for some n ∈ N,
which contradicts to (3.2). This proves (ii).
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658 S. I. MAKSYMENKO
To establish (iii) notice that it follows from (i) and (ii) that
ny(Per(x)− ε) < nyPer(y) = θ(y) < θ(x) + ε,
whence
N :=
θ(x) + ε
Per(x)− ε
> ny.
This proves (iii).
Consider the group P (Wx). As Wx is open and connected, we have that P (Wx) =
= {mθx}m∈Z for some θx ∈ C(W,R). By assumption, θ is a P -function on Wx,
whence θ|Wx
= mxθx for some mx ∈ Z \ {0}.
To construct Qx notice that for each y ∈ Wx \ Σ there exists a unique ny ∈ Z such
that θx(y) = nyPer(y). For every n ∈ N denote by Tn the subset of Wx consisting of
all y such that ny is divided by n. Since the values ny are bounded above, it follows
that Tn is non-empty only for finitely many n. Also notice that
Wx \ Σ =
N⋃
n=1
Tn.
We claim that Tn is nowhere dense for n ≥ 2. Indeed, suppose Int(Tn) 6= ∅. Then
θx/n is a regular P -function on Int(Tn) and therefore, by Lemma 2.5, on all of Wx.
However this is possible only for n = 1 as θx generates P (Wx). Thus the subset
Qx := Int(T1) ∩ Wx is open and everywhere dense in W and θ(y) = Per(y) for all
y ∈ Qx.
Claim 3.1 is proved.
Claim 3.2. Let x, y ∈ V \ Σ. Then θx = θy on Wx ∩Wy and mx = my.
Proof. Indeed, since Qx (Qy) is open and everywhere dense in Wx (Wy), it follows
that Qx∩Qy is open and everywhere dense in Wx∩Wy . Moreover, for each z ∈ Qx∩Qy
we have that θx(z) = θy(z) = Per(z). Then by continuity θx = θy on Wx ∩Wy .
In particular, if z ∈ Qx ∩ Qy, then θ(z) = mxPer(z) = myPer(z), whence mx =
= my.
Claim 3.2 is proved.
Let T be a connected component of V \ Σ. Then by Claim 3.2 mx is the same for
all x ∈ T and we denote their common value by mT . It also follows that the functions
{θx}x∈T define a continuous function θT : T → R such that θ|T = mT θT . Thus if we
put QT = ∪
x∈T
Qx, then QT is open and everywhere dense in T and θT (y) = Per(y)
for all y ∈ QT .
Claim 3.3. Let S and T be any connected components of V \Σ such that S∩T 6=
6= ∅. Then mS = mT .
Proof. We can assume that T 6= S. Let x ∈ S ∩ T ⊂ V ∩ Σ and Wx be an open,
connected neighbourhood of x in V such that θ|Wx
is a regular P -function on Wx.
Notice that θS = θ/mS is a regular P -function on the non-empty open set Wx ∩ S,
whence, by Lemma 2.5, θ/mS is a P -function on all of Wx.
If x ∈ QT ∩Wx, then θ(x) = mT θT (x) = mTPer(x), therefore mT is divided by
mS . By symmetry mS is divided by mT as well, whence mS = mT .
Claim 3.3 is proved.
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KERNEL OF MAP OF A SHIFT ALONG THE ORBITS OF CONTINUOUS FLOWS 659
Since V is connected, it follows from Claim 3.3 that the number mT is the same
for all connected components T of V \ Σ. Denote the common value of these numbers
by m. Then θ/m is continuous on V and F(x, θ(x)/m) = x for all x ∈ V. Since θ
generates P (V ), we obtain that m = 1.
Let Q be the union of all QT , where T runs over the set of all connected components
of V \ Σ. Since for every such component T we have that θ = mθT = θT on T, it
follows that θ(x) = Per(x) for all x ∈ QT .
Theorem 1.1 is proved.
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Received 22.02.10
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