On iteration stability of the Birkhoff center with respect to power 2
It is proved that the Birkhoff center of a homeomorphism on an arbitrary metric space coincides with the Birkhoff center of its power 2.
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| Дата: | 2006 |
|---|---|
| Автори: | , , , |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509586876792832 |
|---|---|
| author | Polulyakh, E. O. Vlasenko, I. Yu. Полулях, Є. О. Власенко, І. Ю. |
| author_facet | Polulyakh, E. O. Vlasenko, I. Yu. Полулях, Є. О. Власенко, І. Ю. |
| author_sort | Polulyakh, E. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:42Z |
| description | It is proved that the Birkhoff center of a homeomorphism on an arbitrary metric space coincides with the Birkhoff center of its power 2. |
| first_indexed | 2026-03-24T02:43:28Z |
| format | Article |
| fulltext |
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UDC 513.83
I. Yu. Vlasenko, E. O. Polulyakh (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ON ITERATION STABILITY OF THE BIRKHOFF
CENTER WITH RESPECT TO POWER 2
*
ITERACIJNA STIJKIST| CENTRA BIRKHOFA
VIDNOSNO STEPENQ 2
It is proved that the Birkhoff center of a homeomorphism on an arbitrary metric space coincides with the
Birkhoff center of its power 2.
Dovedeno, wo centr Birkhofa homeomorfizmu na dovil\nomu metryçnomu prostori zbiha[t\sq z
centrom Birkhofa joho stepenq 2.
It is known that nonwandering set of a homeomorphism can change when we take a
power of the homeomorphism. Such examples are shown in [1, 2]. On the other hand,
the sets of recurrent points [3], chain recurrent points [4], and the limit set do not
change when we take a power. A natural question arises: What can happen with the
Birkhoff center of a dynamical system when we take its power? Here, we prove that
the Birkhoff center of dynamical systems on arbitrary metric spaces coincides with the
Birkhoff center of their power 2.
Let M be a metric space and let f, g : M → M be homeomorphisms of M. Denote
by Ω ( f ) the set of nonwandering points of f. Iterate the construction of the
nonwandering set. Let Ω 1 ( f ) = Ω ( f ). Define by induction Ω n + 1 ( f ) = Ω Ωf
n f( )( ) .
Denote by Ω ω ( f ) the intersection of the obtained sequence of embedded closed
invariant sets. This process can be continued using transfinite induction. According to
the Zorn lemma, the process will stop at an ordinal number α for which Ω α ( f ) =
= Ω Ωf fα ( )( ) . The obtained closed invariant set is called the Birkhoff center and is
denoted by B C ( f ).
Lemma 1. Ω ( g n ) ⊆ Ω ( g ).
Proof. Let x ∈ M \ Ω ( g ) be a wandering point of g. By definition, there exists a
neighborhood U of x such that g k ( U ) ∩ U = ∅. Then gn k ( U ) ∩ U = f k ( U ) ∩ U =
= ∅, k ∈Z, and x is a wandering point of f. Thus, M \ Ω ( g ) ⊆ M \ Ω ( gn ) and
Ω ( g ) ⊇ Ω ( gn ).
Definition 1. A point ξ is called tied with a point µ by g if for all
neighborhoods U (µ ) and V (ξ ) of points µ and ξ , correspondingly, there
exists N ∈ Z such that U ( µ ) ∩ g VN ( )ξ( ) ≠ ∅.
Definition 2. If one can choose N ∈ Z
– ( N ∈ Z
+
) in the previous definition,
then the point ξ is called α -tied ( ω-tied) with the point µ . The point α-
and ω-tied with the point µ is called bi-tied with the point µ.
*
Partially supported by NFBD of Ukraine (grant 01.07 / 00132).
© I. YU. VLASENKO, E. O. POLULYAKH, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5 705
706 I. YU. VLASENKO, E. O. POLULYAKH
Definition 3. Points ξ0 , … , ξn – 1 are called cyclically tied with a cycle length
n if there exists k ∈ 1, … , n – 1 such that for each i = 1, … , n – 1, the point ξi
is ω- (α-)tied with the point ξ m
, m = k + i ( mod n ).
Lemma 2. Let ξ ∈ Ω ( g ) \ Ω ( gn ). Then the trajectories O g
g
i
n ( )ξ( ) , i = 1, …
… , n – 1, related to g n on which the trajectory Og
( ξ ) is split are wandering
cyclically tied ones.
Proof. Let ξ ∈ Ω ( g ) \ Ω ( gn ). Since ξ ∈ Ω ( g ), we have ∀ U (ξ ) ∃ m :
g m ( U (ξ ) ) ∩ U (ξ ) ≠ ∅. Further, since ξ ∉ Ω ( gn ), ∀ l ∈ Z : g l n ( U (ξ ) ) ∩ U (ξ ) ≠ ∅.
We obtain that m ≠ 0 mod n and, hence, ξ and g m ( ξ ) belong to different trajectories
related to g n. Choose U (ξ ) to be the set B1
/
p
(ξ ) of 1 / p-neighborhoods of the point
ξ, p ∈ N. Since n is finite, but the set of values of p is infinite, there exists k ≠ 0
mod n such that m (p ) = k mod n for infinite number of values of p in the previous
formulae. Show that ξ is tied with g k ( ξ ). Let V1
( ξ ) and V2
( g k (ξ ) ) be arbitrary
neighborhoods of points ξ and g k ( ξ ). Choose p such that m (p ) = t n + k to be large
enough for inclusions B 1
/
p
(ξ ) ⊂ V 1
( ξ ) and g k ( B1
/
p
(ξ ) ) ⊂ V 2
( g k (ξ ) ). Then
g t n + k ( B1
/
p
(ξ ) ) ∩ B1
/
p
(ξ ) ≠ ∅ and, hence, V1
( ξ ) ∩ g V gtn k
2 ( )ξ( )( ) ≠ ∅.
Cyclicity of tying follows from the fact that g maps tied points into tied ones.
The lemma is proved.
Theorem 1. A nonwandering set coinciding with the whole space is iteration
stable with respect to power 2.
Proof. Suppose on the contrary that M = Ω ( f ) ≠ Ω ( f 2 ).
Let x ∈ Ω ( f ) \ Ω ( f 2 ). Then, according to Lemma 2, x and y = f ( x ) are tied:
∀ n ≥ 1 ∃ an ∈ B1
/
n
( x ) ∃ An : f aA
n
n ( ) ∈ B1
/
n
( y ). Assume for definiteness that x
and y are ω-tied (including the bi-tied case too). Then An > 0. Otherwise, x and y
are α-tied, and this case reduces to previous one but with f – 1 instead of f, because
by definition the nonwandering sets of f and f – 1 coincide. Note that we can assume
all an to be wandering points of f 2 because the set of wandering points is open.
Denote bn = f f aA
n
n− ( )1 ( ) . The sequence f aA
n
n ( ) tends to y. The sequence
( bn
) tends to x by continuity. Denote with Bε
( x ) the open ball of radius ε around
the point x. According to Lemma 2, b n and f aA
n
n ( ) are tied. Then ∀ n ≥ 1
∀ m ≥ ≥ 1 ∃ cm n ∈ B1
/
m
( bn
) ∃ Km n : f cK
mn
mn ( ) ∈ B f am
A
n
n
1/ ( )( ) . Similarly, we
can assume cm n to be wandering points of f 2. Since bn and f aA
n
n ( ) are tied, there
exists a subsequence such that either bn and f aA
n
n ( ) are ω-tied or bn and f aA
n
n ( )
are α-tied. Switching to such subsequence, we can assume that all bn and f aA
n
n ( )
are ω-tied (or α-tied).
Assume that they are ω-tied. Then Km n > 0. Consider the sequence ( cn
n
). By
construction, ( cn
n
) tends to x and f cA
nn
n−( )( ) tends to y. But f cK
nn
nn ( )( ) also
tends to y. Since An > 0 and Km n > 0, Kn n ≠ – An . We obtain that the point y is self-
tied and, hence, is nonwandering.
Assume that they are α-tied. Consider the sequence f – 1 ( cn
n
). By continuity, it
tends to y. According to Lemma 2, f – 1 ( cm
n
) and ( cm
n
) are tied. The further proof
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, #5
ON ITERATION STABILITY OF THE BIRKHOFF CENTER WITH RESPECT … 707
is reduced to the considered cases. If we can choose a subsequence from f – 1 ( cn
n
)
which consists of points ω-tied to their images, then, repeating the reasons for ω-tied
bn and f aA
n
n ( ) , we obtain that x is nonwandering point. Otherwise, bn and
f aA
n
n ( ) are α-tied and the sequences f – 1 ( cm
n
) and ( cm
n
) which tie them are also
α-tied. This case reduces by substituting f with f – 1 to already considered one when
points are ω-tied and the sequences which tie them are also ω-tied.
The theorem is proved.
Lemma 3. B C ( g n ) ⊆ B C ( g ).
Proof. Consider the family of nonwandering sets in the definition of the Birkhoff
center Ω ( f ) = Ω
1
( f ) ⊇ Ω
2
( f ) ⊇ … ⊇ Ω
ω
( f ) ⊇ Ω
ω
+
1
( f ) ⊇ … indexed with ordinal
numbers. The family is ordered by inclusion. At that relations, Ω
α
( f ) ⊇ Ω
β
( f ) and
α ≤ β are equivalent. Consider the same family for f n.
Prove that Ω
λ
( f n ) ⊆ Ω
λ
( f ) for each ordinal λ using transfinite induction. When
λ = 1, it is Lemma 1. Let it be true for each α < λ. Show that it is true for λ. If λ
is limit ordinal, then by construction, Ω
λ
( f ) = β λ β<∩ Ω ( )f ⊇ β λ β<∩ Ω ( )f n =
= Ω
λ
( f n ). Otherwise, λ has previous ordinal λ̂ . According to inductive proposition,
Ω ˆ ( )
λ
f ⊇ Ω ˆ ( )
λ
f n .
Denote f
λ̂
= f fΩ ˆ ( )
λ
: Ω ˆ ( )
λ
f → Ω ˆ ( )
λ
f and ′f
λ̂
= f n
f nΩ ˆ ( )
λ
: Ω ˆ ( )
λ
f →
→ Ω ˆ ( )
λ
f . Note that Ω f
λ̂( ) ⊇ Ω f
n
λ̂( )( ) according to Lemma 1 and ′f
λ̂
=
= f
n
f nˆ ( )ˆλ λ
( ) Ω . Hence, Ω
λ
( f ) = Ω Ωf fˆ ( )
λ
( ) = Ω f
λ̂( ) ⊇ Ω f
n
λ̂( )( ) ⊇
⊇ Ω Ωf
n
f nˆ ( )ˆλ λ
( )
= Ω ′( )f
λ̂
= Ω Ωf n
f n
ˆ ( )
λ
= Ω
λ
( f n ). The second inclusion
follows from the fact that the nonwandering set of a map cannot be smaller that the
nonwandering set of its restriction.
Let λ, λ ′ be depths of centers B C ( f ) and B C ( f n ), correspondingly. Denote
β = max ( λ, λ ′ ). Then B C ( f ) = Ω
β
( f ) ⊇ Ω
β
( f n ) = B C ( f n ).
The lemma is proved.
It follows that the iteration stability of Birkhoff center is equivalent to the iteration
stability of nonwandering set coinciding with the whole space. As a consequence, we
have the following theorem:
Theorem 2. The Birkhoff center is iteration stable with respect to power 2.
1. Coven E., Nitecki Z. Nonwandering sets of the powers of maps of the interval // Ergod. Theory and
Dynam. Systems. – 1981. – 1. – P. 9 – 31.
2. Savada K. On the iterations of diffeomorphisms without C
0
-Ω-explosions: an example // Proc.
Amer. Math. Soc. – 1980. – 79, # 1. – P. 110 – 112.
3. Gottschalk W. H. Powers of homeomorphisms with almost periodic properties // Bull. Amer. Math.
Soc. – 1944. – 50. – P. 222 – 227.
4. Conley C. Isolated invariant sets and the Morse index // CBMS Reg. Reg. Conf. Ser. Math. –
Providence: Amer. Math. Soc., 1978. – 38.
Received 30.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
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| language | English |
| last_indexed | 2026-03-24T02:43:28Z |
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| spelling | umjimathkievua-article-34872020-03-18T19:55:42Z On iteration stability of the Birkhoff center with respect to power 2 Ітераційна стійкість центра Біркгофа відносно степеня 2 Polulyakh, E. O. Vlasenko, I. Yu. Полулях, Є. О. Власенко, І. Ю. It is proved that the Birkhoff center of a homeomorphism on an arbitrary metric space coincides with the Birkhoff center of its power 2. Доведено, що центр Біркгофа гомеоморфізму на довільному метричному просторі збігається з центром Біркгофа його степеня 2. Institute of Mathematics, NAS of Ukraine 2006-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3487 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 5 (2006); 705–707 Український математичний журнал; Том 58 № 5 (2006); 705–707 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3487/3711 https://umj.imath.kiev.ua/index.php/umj/article/view/3487/3712 Copyright (c) 2006 Polulyakh E. O.; Vlasenko I. Yu. |
| spellingShingle | Polulyakh, E. O. Vlasenko, I. Yu. Полулях, Є. О. Власенко, І. Ю. On iteration stability of the Birkhoff center with respect to power 2 |
| title | On iteration stability of the Birkhoff center with respect to power 2 |
| title_alt | Ітераційна стійкість центра Біркгофа відносно степеня 2 |
| title_full | On iteration stability of the Birkhoff center with respect to power 2 |
| title_fullStr | On iteration stability of the Birkhoff center with respect to power 2 |
| title_full_unstemmed | On iteration stability of the Birkhoff center with respect to power 2 |
| title_short | On iteration stability of the Birkhoff center with respect to power 2 |
| title_sort | on iteration stability of the birkhoff center with respect to power 2 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3487 |
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