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@ Appl. Gen. Topol. 17, no. 2(2016), 129-137doi:10.4995/agt.2016.4593
c© AGT, UPV, 2016
Homeomorphisms on compact metric spaces
with finite derived length
V. Kannan a and Sharan Gopal b
a
School of Mathematics and Statistics, University of Hyderabad, Hyderabad, India. (vksm@uohyd.ernet.in)
b
Department of Mathematics, BITS-Pilani, Hyderabad campus, Hyderabad, India. (sharanraghu@gmail.com)
Abstract
The sets of periodic points of self homeomorphisms on an ordinal of
finite derived length are characterised, thus characterising the same for
homeomorphisms on compact metric spaces with finite derived length.
A partition of ordinal is introduced to study this problem which is also
used to solve two more problems: one about an equivalence relation
and the other about a group action, both on an ordinal of finite derived
length.
2010 MSC: 06A99; 54H20.
Keywords: ordinal; homeomorphism; periodic point.
1. Introduction
Periodicity is one of the important properties that are well studied in Dy-
namical systems. The study of sets of periodic points for a class of dynamical
systems has been an interesting one in the literature (See [1], [3]). Some recent
results characterise these sets for toral automorphisms and solenoids (See [12],
[11]). In this paper, we do this for all self-homeomorphisms on compact met-
ric spaces with finite derived length. It can be proved that a compact metric
space with finite derived length is countable and a countable compact metric
space is homeomorphic to a countable ordinal (See [6]). So the sets of periodic
points for self homeomorphisms on an ordinal with finite derived length are
characterised here.
Received 27 January 2016 – Accepted 12 July 2016
http://dx.doi.org/10.4995/agt.2016.4593
V. Kannan and S. Gopal
Another main result is on the group actions. Given a topological space
X, the study of action of group of homeomorphisms on it is well studied in
the literature (See [5]). Here we consider the actions of a particular kind of
subgroups of this group of homeomorphisms on metric spaces. It is proved that
the separable metric spaces with finite derived length have finitely many orbits
under the action of all these subgroups.
An account of some preliminaries and notations that will be used in the
paper is given in this and the next paragraph. By a dynamical system (X,f),
we mean a topological space X together with a continuous self map f on X.
The set {fl(x) : l ∈ N0} is called the orbit of x ∈ X. A subset A ⊂ X is said to
be forward f−invariant if f(A) ⊂ A and backward f−invariant if f−1(A) ⊂ A.
A is said to be f−invariant (or just invariant, when the context is clear) if A
is both forward f−invariant and backward f−invariant.
If A and B are two ordered sets, then we define a relation, saying that A ∼ B
if there is an order preserving bijection from A to B. This is an equivalence
relation and each equivalence class (in fact a representative of this class) is
called an ordinal number. Hereafter, by an ordinal number, we actually refer
to a representative of the corresponding class. This makes it easy to convey
the idea of order among the ordinals. If α and β are two ordinal numbers
such that there is an order preserving bijection from α to a subset of β, then
define α ≤ β. This relation “≤” is an order on the set of ordinal numbers
according to which, this is well ordered. Note that any non-negative integer
can be regarded as an ordinal and such ordinals are called as finite ordinals
and the least infinite ordinal is denoted by ω (which is equivalent to the set of
all non-negative integers).
On an ordered set X, with more than one element, let B be the collection
of all sets of the following types: (i) all open intervals (a,b) in X, (ii) all
intervals of the form [a0,b), where a0 is the smallest element (if any) of X and
(iii) all intervals of the form (a,b0], where b0 is the largest element (if any) of
X. The collection B is a basis for a topology on X, which is called the order
topology. In this paper, we consider the order topology on ordinal spaces.
The set of limit points of an ordinal α is precisely the set of limit ordinals
less than α. Zero and the successor ordinals less than α are isolated points in
α. It can be seen that any ordinal of finite derived length can be written as
ωn · mn + ω
n−1 · mn−1 + ... + ω · m1 + m0, mi ∈ N0. Hereafter, we use the
notation O(n) = ωn ·mn +ω
n−1 ·mn−1 + ...+ω ·m1 +m0. The notions of limit
ordinal and arithmetic of ordinals can be seen in [10]. We also use the notation
|W | to denote the cardinality of a set W .
2. Partition of O(n)
Here, a partition is induced on the ordinal O(n). The definition of this
partition involves only elementary operations: intersection, complementation
and formation of derived set. This partition is done with respect to a given
subset of O(n) i.e., every subset S ⊂ O(n) gives rise to a partition PS of O
(n).
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 130
Homeomorphisms on compact metric spaces
This is finer than the partition of O(n) in to the different levels of limit points.
To define the partition PS, the following notations are introduced.
1. Let X be a topological space. For a subset Z ⊂ X, D(Z) denotes the set
of limit points of Z in X and for every n ∈ N, define inductively Dn+1(Z) =
D(Dn(Z)). Then, for k ∈ N, denote by Xk, the set D
k(X) \ Dk+1(X) and let
X0 denote the set of isolated points of X. Xk is called the set of k
th level limit
points of X in X for every k ∈ N0.
2. Define a sequence J (l), l ∈ N0 inductively as follows :
J (0) = {a,b}, J (l+1) = (P(J (l)) \ {∅}) × J (0) (P(X) stands for the set of all
subsets of X).
Remark 2.1.
(1) Here, a and b can be any two distinct symbols. What is actually needed
to define the partition is a set containing two elements, which we denote
by J (0).
(2) The letters P and P are used in three different contexts. PS denotes
the partition associated with the set S, P(X) stands for the collection
of all subsets of X and P(h) will be used to denote the set of periodic
points of h.
The partition PS: Let S ⊂ O
(n), n ∈ N. For every V ∈
⋃n
l=0 J
(l), we
associate a subset SV of O
(n) in the following way. Note that these SV ’s are
defined recursively.
Define Sa = (O
(n) \ S) ∩ (O(n))0 and Sb = S ∩ (O
(n))0. If V ∈ J
(k+1) for some
0 ≤ k ≤ n − 1 and say V = (W,i) (i ∈ {a,b}) for some W ⊂ J (k), then define
SV = [(
⋂
A∈W SA) \ (
⋃
A∈J (k)\W SA)] ∩ S
i ∩ (O(n))k+1,
where Si =
{
O(n) \ S, i = a
S, i = b
.
Remark 2.2. In the above partition, O(n) is partitioned in to (O(n))k’s and each
(O(n))k is partitioned into S ∩ (O
(n))k and (O
(n) \ S) ∩ (O(n))k. This partition
is further refined in such a way that the common limit points of some and only
those partition classes in (O(n))k−1 constitute the partition classes of (O
(n))k;
these are denoted by SV , V ∈
⋃n
l=0 J
(l). In other words, for every partition
class SV ⊂ (O
(n))k, there exist partition classes SV1, SV2, . . . SVm (m depends
on V ) in (O(n))k−1 such that SV is precisely the set ∩
m
i=1D(SVi).
Thus every subset S ⊂ O(n) gives rise to a partition PS of O
(n). This is finer
than the partition of O(n) in to the different levels of limit points. PS helps to
prove the following four different results.
• A subset S ⊂ O(n) arises as the set of periodic points of a homeo-
morphism on O(n) if and only if every finite partition class in PS is
contained in S.
• Using this partition PS, it is proved that in a separable metric space
X with finite derived length, there are finitely many orbits under the
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 131
V. Kannan and S. Gopal
action of the group GS = {h : X → X : h is a homeomorphism such
that h(S) = S}, for any S ⊂ X.
• PS is the smallest partition of ω
n such that
(i)S is a union of partition classes and
(ii)if A ⊂ ωn is a union of partition classes, then D(A) is also a union
of partition classes.
PS is the smallest partition in the sense that any finer refinement of
PS will not satisfy the above conditions (i) and (ii).
• Starting from a subset S ⊂ ωn, go on forming many other sets using the
operation of derived set, complement and union i.e, S, ωn\S, S, ωn \ S
and so on. In other words, if n sets are already formed, the (n + 1)th
set can be the derived set of any of the n sets, or the complement of
any of the n sets, or the union of any two of these n sets. Among
the distinct subsets that can be formed in this way, the minimal ones
are precisely the partition classes in PS i.e., in the collection of sets
formed as above, there is no set which is strictly contained in any of
the partition classes of the partition PS.
The first two results form the main part of this paper and the last two follow
from these. The last result is similar to, and is a natural sequel to the closure-
complementation problem described in [7] and further investigated in [4].
3. Sets of periodic points
3.1. Invariance of partition classes.
Proposition 3.1. If X is a topological space with finite derived length n and
h is a homeomorphism on X, then Xk is h−invariant for every 0 ≤ k ≤ n.
Proof. Since, being an isolated point is a topological property, X0 is h−invariant
for any homeomorphism h on X. For any 1 ≤ m < n, by the definition of
Xm+1, there is no sequence in ∪
n
i=m+1Xi that converges to a point in Xm+1;
because, the limit of such a sequence would be a limit point of a level larger than
m + 1. Thus the points of Xm+1 consists of the isolated points of X \ ∪
m
i=0Xi
i.e., Xm+1 = (X \ ∪
m
i=0Xi)0. In particular, X1 = (X \ X0)0. Since X0 is
h−invariant, h is a self homeomorphism on X \ X0. Thus by the above argu-
ment, it follows that X1 is h−invariant. Now let 1 ≤ k < n. Suppose that
Xl is h−invariant for every 0 ≤ l ≤ k. Then, h is a self homeomorphism on
X \
⋃k
i=0 Xi and since Xk+1 = (X \ ∪
k
i=0Xi)0, Xk+1 is h−invariant. Hence Xk
is h−invariant for every 0 ≤ k ≤ n. �
Theorem 3.2. If S ⊂ O(n) is h−invariant for some homeomorphism h on
O(n), then SV is h−invariant ∀V ∈
⋃n
k=0 J
(k).
Proof. From the above proposition, it is clear that Sa and Sb are h−invariant.
We now prove that if SW is h−invariant ∀W ∈ J
(k) for some 0 ≤ k ≤ n − 1,
then SV is h−invariant ∀V ∈ J
(k+1).
Consider a partition class SV , where V ∈ J
(k+1). Then V = (W,i) for some
W ⊂ J (k) and i ∈ {a,b}. By definition, SV = (
⋂
A∈W SA \
⋃
A∈J (k)\W SA) ∩
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 132
Homeomorphisms on compact metric spaces
Si ∩ (O(n))k+1. Since h is a homeomorphism, observe that for any A ∈ J
(k),
h(SA) = h(SA) = SA. Then h(
⋂
A∈W SA) =
⋂
A∈W SA and h(
⋃
A∈J (k)\W SA) =
⋃
A∈J (k)\W SA. Hence SV is h−invariant. �
3.2. Periodic points.
Lemma 3.3. Let n ∈ N and S ⊂ ωn+1. If 1 ≤ k ≤ n and (ωn+1 \ S) ∩ SA is
either empty or infinite for every A ∈ J (k−1) ∪ J (k) then any bijection f on
(ωn+1)k such that
(1) f has no orbit of even length
(2) SV is f−invariant ∀V ∈ J
(k) and
(3) P(f) = S ∩ (ωn+1)k
can be extended to a homeomorphism h on (ωn+1)k−1 ∪ (ω
n+1)k such that
P(h) = S ∩ ((ωn+1)k−1 ∪ (ω
n+1)k).
Proof. f being a bijection on (ωn+1)k with no orbits of even length, defines
three types of orbits in (ωn+1)k :
(1) an infinite orbit ({xl : l ∈ Z and f(xi) = xi+1}),
(2) a periodic orbit with odd length greater than 1 ({xl : −m ≤ l ≤ m for
some fixed m ∈ N; f(xi) = xi+1 ∀i < m and f(xm) = x−m}) and
(3) a singleton orbit ({x : f(x) = x}) i.e., a fixed point.
Since P(f) = S ∩ (ωn+1)k, (ω
n+1)k \ S is a union of infinite orbits and
S ∩ (ωn+1)k is a union of finite orbits.
We now define a homeomorphism h on (ωn+1)k−1 ∪ (ω
n+1)k with P(h) =
S ∩ ((ωn+1)k−1 ∪ (ω
n+1)k) in such a way that h = f on (ω
n+1)k and moreover
the orbit of each point in (ωn+1)k−1 falls in to one of the three types. To cover
the general case, we assume that each point in (ωn+1)k is in both S as well as
(ωn+1)k−1 \ S; and the same method works for other cases also.
We first define maps on some clopen neighbourhoods of points of (ωn+1)k.
This is done in one of the three different ways depending on the type of orbit
in which the point lies.
Type I : {xi : i ∈ Z and f(xi) = xi+1}
Here, xi ∈ SV ∀i ∈ Z for some V ∈ J
(k). Let Bi and B
′
i be two deleted clopen
neighborhoods of xi (i.e., xi /∈ Bi ∪ B
′
i and Bi ∪ {xi}, B
′
i ∪ {xi} are clopen
neighborhoods of xi) such that Bi ⊂ S and B
′
i ⊂ (ω
n+1 \ S). Choose Bi’s
and B′i’s in such a way that Bi ∩ SW and B
′
i ∩ SW are either empty or infinite
∀W ∈ J (k−1). Since all the xi’s are in the same SV , we can assume that for
any W ∈ J (k−1), |Bi ∩ SW | = |Bj ∩SW | and |B
′
i ∩ SW | = |B
′
j ∩ SW |, ∀i,j ∈ Z.
Say Bi ∩ (ω
n+1)k−1 = {xij : j ∈ N} and B
′
i ∩ (ω
n+1)k−1 = {x
′
ij : j ∈ N}.
Define a bijection hI on (
⋃
i∈Z(Bi ∪ B
′
i)) ∩ (ω
n+1)k−1 as
hI(xij) =
x(i+1)j if − j ≤ i < j
x(−j)j if i = j
xij if |i| > j
and hI(x
′
ij) = x
′
(i+1)j
.
Extend hI to (
⋃
i∈Z Bi)
⋃
(
⋃
i∈Z B
′
i) ∪ {xi : i ∈ Z} by defining hI(xi) = f(xi).
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 133
V. Kannan and S. Gopal
Type II : {xi : −m ≤ i ≤ m for some fixed m ∈ N and f(xi) = xi+1 ∀i < m
and f(xm) = x−m}
Let Bi and B
′
i be two deleted clopen neighborhoods of xi such that Bi ⊂ S and
B′i ⊂ (ω
n+1 \S). Choose Bi’s and B
′
i’s in such a way that Bi∩SW and B
′
i∩SW
are either empty or infinite ∀W ∈ J (k−1). Here again, xi ∈ SV ∀ − m ≤ i ≤ m
for some V ∈ J (k). Also, |Bi ∩ SW | = |Bj ∩ SW | and |B
′
i ∩ SW | = |B
′
j ∩ SW |,
∀−m ≤ i,j ≤ m and for any W ∈ J (k−1). Say Bi ∩(ω
n+1)k−1 = {xij : j ∈ N}
and B′i ∩ (ω
n+1)k−1 = {x
′
ij : j ∈ N}.
Define a bijection hII on [
⋃
−m≤i≤m(Bi ∪B
′
i)]∩(ω
n+1)k−1 as hII(xij) = xψ(i)j
and
hII(x
′
ij) = x
′
ψ(i)φ(j)
, where ψ is a bijection on {−m,−m + 1, ..,0,1, ...,m}
defined as
ψ(i) =
{
i + 1 ∀i < m
−m if i = m
and φ : N → N is defined as φ(2i − 1) = 2i + 1 ∀i ∈ N, φ(2i) = 2i − 2 ∀i ≥ 2
and φ(2) = 1.
Extend hII to (
⋃
−m≤i≤m Bi)
⋃
(
⋃
−m≤i≤m B
′
i)∪{xi : −m ≤ i ≤ m} by defining
hII(xi) = f(xi).
Type III : {x : f(x) = x} (i.e., x is a fixed point).
Let B and B′ be two deleted clopen neighborhoods of x such that B ⊂ S and
B′ ⊂ (ωn+1 \ S). Say B = {xi : i ∈ N} and B
′ = {x′i : i ∈ N}. Define hIII on
(B ∪B′)∩(ωn+1)k−1 as hIII(xi) = xi and hIII(x
′
i) = φ(x
′
i), where φ is defined
as above. Extend hIII to B ∪ B
′ ∪ {x} by defining hIII(x) = x.
The neighborhoods considered above will form a partition of a clopen set,
say X ⊂ (ωn+1)k−1 and we can assume that for every W ∈ J
(k−1), SW ⊂ X
or SW ⊂ (ω
n+1)k−1 \ X. Thus (ω
n+1)k−1 \ X is a union of SU’s for some
U ∈ J (k−1) such that D(SU) = ∅. Thus any bijection on (ω
n+1)k−1 \ X will
be a homeomorphism on it. So, we define a bijection on (ωn+1)k−1 \ X so that
SU is a single infinite orbit (type I) if SU ⊂ (ω
n \ S) or otherwise each point
of SU is fixed. Since the neighborhoods defined above for the three types form
a partition of X,, the domains of hI, hII and hIII are mutually disjoint and
further each of them is a clopen set. Hence hI, hII and hIII can be pasted
to get a homeomorphism on X and by pasting this homeomorphism and the
bijection on (ωn+1)k−1 \X, we get a homeomorphism h on (ω
n+1)k∪(ω
n+1)k−1
such that P(h) = S ∩ ((ωn+1)k−1 ∪ (ω
n+1)k). �
Remark 3.4. Similar to the bijection f, the extended homeomorphism h satisfies
the following :
(1) h has no orbit of even length
(2) SV is h−invariant ∀V ∈ J
(k) ∪ J (k−1) and
(3) P(h) = S ∩ ((ωn+1)k−1 ∪ (ω
n+1)k).
Theorem 3.5. Let S ⊂ ωn for some n ∈ N. The following are equivalent:
(1) S = P(h) for some self-homeomorphism h on ωn.
(2) SV ∩ (ω
n \ S) is either empty or infinite for every V ∈
⋃n−1
k=0 J
(k).
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 134
Homeomorphisms on compact metric spaces
Proof. If S = P(h), then S is h−invariant and thus by Theorem 3.2, SV is
h−invariant for every V . Also, either SV ⊂ S or SV ⊂ (ω
n \ S). Since a
finite non-empty invariant set certainly contains a periodic point, it follows
that SV ∩ (ω
n \ S) is either empty or infinite for every V .
For the converse, let S ⊂ ωn such that SV ∩ (ω
n \ S) is either empty or
infinite for every V ∈
⋃n−1
k=0 J
(k).
Suppose V1, V2, ...,Vm ∈ J
(n−1) such that (ωn)n−1\S =
⋃m
j=1 SVj and SVj 6= ∅
∀j ∈ {1,2, ...,m}. Say SVj = {xjl : l ∈ Z}.
Define f : (ωn)n−1 → (ω
n)n−1 as f(x) =
{
x if x ∈ S
xj(l+1) if x = xjl ∈ (ω
n)n−1 \ S
.
If m does not exist i.e., if there is no V ∈ J (n−1) such that SV ⊂ (ω
n \ S),
then define f(x) = x ∀x ∈ (ωn)n−1. Then f is a bijection on (ω
n)n−1.
If n = 1, then f is a homeomorphism on ω such that P(f) = S. Otherwise,
using the above Lemma, this f can be extended to a homeomorphism hn−2 on
(ωn)n−1 ∪ (ω
n)n−2 such that P(hn−2) = S ∩ ((ω
n)n−1 ∪ (ω
n)n−2).
hn−2 defines a bijection on (ω
n)n−2 which can be further extended to a
homeomorphism hn−3 on (ω
n)n−2 ∪ (ω
n)n−3 such that P(h) = S ∩ ((ω
n)n−2 ∪
(ωn)n−3). By pasting hn−2 and hn−3, we get a homeomorphism on (ω
n)n−1 ∪
(ωn)n−2 ∪(ω
n)n−3 with S∩((ω
n)n−1 ∪(ω
n)n−2 ∪(ω
n)n−3) as the set of periodic
points. Continuing this way, we get a homeomorphism h on ωn with P(h) =
S. �
Now, we have the main theorem:
Theorem 3.6. The following are equivalent for a subset S ⊂ O(n):
(1) S = P(h) for some self-homeomorphism h on O(n).
(2) SV ∩ (O
(n) \ S) is either empty or infinite for every V ∈
⋃n
k=0 J
(k).
Proof. The first part of the proof is same as that in the above proof. For the
converse, recall that O(n) = ωn · mn + ω
n−1 · mn−1 + ... + ω · m1 + m0 where
mi ∈ N0. Here, the highest level of limit points is n and since (O
(n))n is a finite
set, (O(n))n ⊂ S. The rest of the proof follows by taking f to be the identity
map on (O(n))n in Lemma 3.3 and also using the ideas of Theorem 3.5. �
4. An equivalence class and a group action
Definition 4.1. Let Z be a topological space and S ⊂ Z. Let x, y ∈ Z.
x is said to be topologically same as y in Z with respect to S if there exists a
homeomorphism h on Z such that h(S) = S and h(x) = y.
Given S ⊂ Z, this definition induces an equivalence relation RS on Z as :
x RS y if x is topologically same as y with respect to S. It is easy to see that
this is an equivalence relation on Z and the following theorem describes the
equivalence classes of O(n).
Theorem 4.2. The family {SV : V ∈
⋃n
l=0 J
(l), SV 6= ∅} is the set of
equivalence classes of O(n) with respect to RS.
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 135
V. Kannan and S. Gopal
Proof. From the proof of Theorem 3.2, it follows that SV is invariant for every
V ∈
⋃n
l=0 J
(l), under any homeomorphism h on O(n) such that h(S) = S. So
if V 6= V ′, then x ∈ SV and y ∈ SV ′ are not related.
Now, let x,y ∈ SV for V ∈ J
(k) for some k ∈ {0,1, ...,n}. Define a bijection h
on (On)k such that h(z) =
z if z /∈ {x,y}
y if z = x
x if z = y
.
Using the ideas of the proof of Lemma 3.3, this map can be extended to a
homeomorphism h on
⋃k
i=0(O
(n))i such that
h(S ∩ (
⋃k
i=0(O
(n))i)) = S ∩ (
⋃k
i=0(O
(n))i),
which can be further extended to O(n) by defining
h(z) = z, ∀z ∈ O(n) \
⋃k
i=0(O
(n))i.
Hence the family {SV : V ∈
⋃n
k=0 J
(k), SV 6= ∅} is the set of equivalence
classes. �
If a group G is acting on a set X and x ∈ X, then the set Gx = {gx : g ∈ G}
is called the G−orbit of x. When a finite group acts on a finite set, one natural
problem is to count the number of orbits. Burnside’s lemma (see [2]) and
Polya’s theorem (see [9]) are in this direction. Occasionally, there are some
infinite groups acting on infinite sets, but having only finitely many orbits.
The problem of the present section has such background.
To every topological space X, we can associate a natural group, namely the
group of self homeomorphisms on it. Here, we consider a subgroup of this
group. Given a subset S of a topological space X, let GS = {f : X → X : f
is a homeomorphism on X such that f(S) = S} , i.e., the collection of self-
homeomorphisms on X under which S is invariant. It can be easily seen that
GS is a group.
The following theorem gives a neat description of the GS−orbits in O
(n) in
terms of SV ’s. Its proof follows from Theorem 4.
Theorem 4.3. The family {SV : V ∈
⋃n
k=0 J
(k), SV 6= ∅} is the collection
of GS−orbits in O
(n) for any subset S ⊂ O(n).
It follows from the above theorem that a compact metric space X with finite
derived length has finitely many GS−orbits for any subset S ⊂ X. In fact, any
separable metric space with finite derived length has this property. This is
proved in the following theorem.
Theorem 4.4. If X is a separable metric space with finite derived length, then
X has finitely many GS-orbits for every subset S ⊂ X.
Proof. Let X be a separable metric space of finite derived length. If X is
compact, then the result follows from Theorem 4.3.
Now, suppose X is not compact. Since X is separable, it is known that X
is homeomorphic to a subspace of ωn for some n (See [8]). So, it is enough
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 136
Homeomorphisms on compact metric spaces
to consider the number of GS−orbits in X, where S ⊂ X ⊂ ω
n. Let H =
{h : ωn → ωn : h(X) = X and h(S) = S}. Then H is isomorphic to a
subgroup of GS. Thus the number of GS−orbits in X cannot exceed the
number of H−orbits in ωn. It follows from Theorem 4.3 and the proof of
Theorem 4.2, that the non-empty members of the family {S ∩ XV : V ∈
⋃n
k=0 J
(k)} ∪ {(ωn \ S) ∩ XV : V ∈
⋃n
k=0 J
(k)} are precisely the collection
of H−orbits in ωn. So, the number of H−orbits is atmost twice the number
of XV ’s, which are finite in number. Thus there are finitely many H−orbits in
ωn and hence finitely many GS−orbits in X.
�
5. Conclusion
We conclude this article by posing a question. The problem of characterising
the sets of periodic points of homeomorphisms on ωn leads to a very natural
and interesting question of characterising the same sets for continuous maps on
ωn. In another direction, the same problem can be extended to the problem
of characterising the sets of periodic points of homeomorphisms on ordinals of
infinite derived length. It is hoped that the ideas in this paper will be useful
in discussing the later question.
Acknowledgements. We thank the referee for his suggestions.
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