Commun. Korean Math. Soc. 19 (2004), No. 4, pp. 759–764
EXPANSIVE HOMEOMORPHISMS
WITH THE SHADOWING PROPERTY
ON ZERO DIMENSIONAL SPACES
Jong-Jin Park
Abstract. Let X = {a} ∪ {ai | i ∈ N} be a subspace of Euclidean
space E 2 such that limi→∞ ai = a and ai 6= aj for i 6= j. Then it
is well known that the space X has no expansive homeomorphisms
with the shadowing property. In this paper we show that the set
of all expansive homeomorphisms with the shadowing property on
the space Y is dense in the space H(Y ) of all homeomorphisms on
Y , where Y = {a, b} ∪ {ai | i ∈ Z} is a subspace of E 2 such that
limi→∞ ai = b and limi→−∞ ai = a with the following properties;
ai 6= aj for i 6= j and a 6= b.
1. Introduction
All spaces considered in this paper are assumed to be compact and
metrizable. Let Φ be a homeomorphism from a space (X, d) onto itself.
Given δ > 0, a sequence {xi | i ∈ Z} in X is called a δ-pseudo-orbit of
Φ if
d(Φ(xi ), xi+1 ) < δ
for every i ∈ Z. Given ε > 0, a sequence {xi | i ∈ Z} in X is said to be
ε-traced by a point y ∈ X if
d(Φi (y), xi ) < ε
for every i ∈ Z. We say that Φ has the shadowing property(or pseudo
orbit tracing property) if for every ε > 0 there is δ > 0 such that every
δ-pseudo orbit of Φ can be ε-traced by a point of X. Φ is called expansive
if there is c > 0 such that for every x, y ∈ X with x 6= y there is n ∈ Z
for which
d(Φn (x), Φn (y)) > c.
Received March 12, 2004.
2000 Mathematics Subject Classification: Primary 54H20; Secondary 58F99.
Key words and phrases: expansive homeomorphism, δ- pseudo-orbit, shadowing
property (pseudo orbit tracing property), Dense, Zero dimensional space.
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Jong-Jin Park
The constant c > 0 is called an expansive constant of Φ. For a space
(X, d), we denote by H(X) the space of all homeomorphisms of X with
the metric
e Ψ) = sup {d(Φ(x), Ψ(x) | x ∈ X}
d(Φ,
for Φ, Ψ ∈ H(X). Let
E(X) = {Φ ∈ H(X) | Φ is expansive}
and
P (X) = {Φ ∈ H(X) | Φ has the shadowing property}.
Given δ > 0 and a ∈ X, we denote a neighborhood Uδ (a) of a by
Uδ (a) = {b ∈ X | d(a, b) < δ}.
Aoki [1] proved that every group automorphism at the Cantor set
C has the shadowing property. Sears [5] showed that E(C)is dense in
H(C). Dateyma [2] proved that P (C) is dense in H(C) and Kimura [4]
also proved the followings.
Proposition. Let X = {a}∪{ai | i ∈ N} be a subspace of Euclidean
space E 2 such that limi→∞ ai = a and ai 6= aj for i 6= j. Then
a) the set of all expansive homeomorphisms of X is dense in H(X);
b) the set of all homeomorphisms with the shadowing property of X
is dense in H(X);
c) X has no expansive homeomorphism with the shadowing property.
A question aries naturally as to whether a zero-dimensional countable
compact space admits an expansive homeomorphism with the shadowing
property. For the question, Kato and Park [3] shows a zero-dimensional
countable compact space admits an expansive homeomorphism with the
shadowing property.
In this paper we generalize the result of Kato and Park [3] as in the
main theorem of this paper.
2. The main theorem
Theorem 2.1. Let Y = {a, b} ∪ {ai | i ∈ Z} be a subspace of Euclidean space E 2 such that limi→∞ ai = b and limi→−∞ ai = a, where
ai 6= aj for i 6= j and a 6= b. Then the set of all expansive homeomorphisms with the shadowing property on Y is dense in H(Y ).
Proof. Without loss of generality, we assume that for each i ∈ Z,
d(ai+1 , ai ) < d(ai , ai−1 ) if i ≥ 0,
Expansive homeomorphisms with the shadowing property
761
and
d(ai−1 , ai ) < d(ai , ai+1 ) if i < 0.
Let Ψ ∈ H(Y ) and ε > 0. We construct a homeomorphism Φ satisfying
e Ψ) < ε.
Φ ∈ E(Y ) ∩ P (Y ) and d(Φ,
To do this, we consider the following two cases:
Case 1. Ψ(a) = a and Ψ(b) = b.
Case 2. Ψ(a) = b and Ψ(b) = a.
Case 1: For ε > 0, we take n ∈ N such that for |i| ≥ n, i ∈ Z,
d(ai , a) < ε or d(ai , b) < ε.
we define the sets C1 , C2 , C3 by
C1 = {ai | i ≤ −n}, C2 = {ai | − n < i < n}
and
C3 = {ai | n ≤ i}.
Since Ψ is a homeomorphism, there is a q ∈ N such that
Ψ(ai ) ∈ C1 for all i < −q and Ψ(ai ) ∈ C3 for all i > q.
Put k = max{q, n}. For every i ∈ Z, we take l, l0 ∈ Z such that
l = max{i | Ψ(ai ) = aj , |j| < k}
and
l0 = min{i | Ψ(ai ) = aj , |j| < k}.
we define the sets B, B1 , B2 by
B = {ai | l0 ≤ i ≤ l} \ {ai | Ψ(ai ) = aj , |j| < k}
and
B1 = {ai | Ψ(ai ) ∈ C1 , ai ∈ B} and B2 = {ai | Ψ(ai ) ∈ C3 , ai ∈ B}.
Then we know that B = B1 ∪ B2 .
Define a mapping Φ from Y onto itself as follows:
i) Φ(a) = a and Φ(b) = b;
ii) for any ai ∈ Y ,
(
Ψ(ai ), if i ∈ {i | Ψ(ai ) = aj , |j| < k}
Φ(ai ) =
ai+1 ,
if i ≥ l + 1 or if i ≤ l0 − 2;
iii) Φ(al0 −1 ) is a point in the set
{ai | i = l0 , · · · , l} \ {ai | |i| < k};
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Jong-Jin Park
iv) Φ(ai ) ∈ {ai | l0 ≤ i ≤ −k} \ Φ(al0−1 ) if ai ∈ B1 ,
Φ(ai ) ∈ {ai | k ≤ i ≤ l + 1} \ Φ(al0−1 ) if ai ∈ B2 and
Φ(ai ) 6= Φ(aj ), ai 6= aj ∈ Bs , s = 1, 2.
Then Φ is bijective and Φ ∈ H(Y ). By the construction of Φ, it is
clear that
e Ψ) < ε.
d(Φ,
Now we show that Φ is expansive. Put
c = min{d(al+1 , al+2 ), d(al0 −1 , al0 −2 )}.
Then we have
Uc (al0 −1 ) = al0 −1 and Uc (al+1 ) = al+1 .
Let x and y be two points in Y \ {a, b} with x 6= y. If
x ∈ {ai | l0 − 1 ≤ i ≤ l + 1},
then we get d(x, y) > c. Let x be a point in the set x ∈ {ai | i >
l + 1} ∪ {ai | i < l0 − 1}. Then we can chose k, k 0 ∈ Z such that
0
f k (x) = al+1 or f k (x) = al0 −1 .
Hence Φ is an expansive homeomorphism. Now we are going to show
that Φ has the shadowing property. Let ε0 > 0 and let ε1 = min{²0 , ε}.
Take P, P 0 ∈ Z with P 0 < P satisfying the following properties:
if i ≤ P 0 then d(a, ai ) < ε1 ,
and
if i ≥ P then d(b, ai ) < ε1 .
Put
δ = min{d(ap0 −1 , ap0 −2 ), d(ap+1 , ap+2 )}.
To show that Φ has the shadowing property, it is sufficient that every δpseudo orbit of Φ can be ε1 -traced by a point of Y . Let ζ = {yi | i ∈ Z}
be a δ-pseudo orbit of Φ. we define the sets A1 , A2 , A3 by
A1 = {ai | i < P 0 }, A2 = {ai | P 0 ≤ i ≤ P }
and
A3 = {ai | i > P }.
Then we have the following three possibilities:
1) ζ ∩ Ai 6= φ, for each i = 1, 2, 3;
2) ζ ⊂ A1 ;
3) ζ ⊂ A3 .
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1): If the point y0 of the δ-pseudo orbit ζ = {yi | i ∈ Z} is in A2 ,
then ζ is ε1 -traced by the point y0 . But if y0 ∈ A1 or y0 ∈ A3 , then
we take a point p in the set {yi | i ∈ Z} ∩ A2 say p = yr . Consider the
sequence {yi | i ∈ Z} and we denote it by {yi−r | i ∈ Z}.
Then the the sequence {yi−r | i ∈ Z} is ε1 -traced by the point yr−r .
2): If ζ ⊂ A1 , then it is clear that the δ-pseudo orbit ζ = {yi ∈
A1 | i ∈ Z} is ε1 -traced by the point a.
3): If ζ ⊂ A3 , then it is also obvious that the δ-pseudo orbit ζ =
{yi ∈ A3 | i ∈ Z} is ε1 -traced by the point b.
Case 2: Let ε > 0 and l, l0 as in Case 1. Define a mapping Φ from Y
onto itself as follows:
i)
ii)
iii)
iv)
Φ(a) = b and Φ(b) = a;
for any ai ∈ Y , Φ(ai ) = Ψ(ai ), if i ∈ {i | Ψ(ai ) = aj , |j| < k};
Φ(al0 −1 ) is a point in the set {ai | i = l0 , · · · , l} \ {ai | |i| < k};
Φ(ai ) ∈ {ai | l0 ≤ i ≤ −k} \ φ(al0−1 ) if ai ∈ B1
Φ(ai ) ∈ {ai | k ≤ i ≤ l + 1} \ Φ(al0−1 ) if ai ∈ B2 and
Φ(ai ) 6= Φ(aj ), ai 6= aj ∈ Bs , s = 1, 2;
v) Φ(al+2t−1 ) = al0 −2t , t = 1, 2, · · · ,
Φ(al+2t ) = al0 −2t+1 , t = 1, 2, · · · ,
Φ(al0 −2t ) = al+2t+1 , t = 1, 2, · · · ,
Φ(al0 −2t−1 ) = al+2t , t = 1, 2, · · · .
By the same techniques as in the proof of Case 1, we know that Φ is
bijective, Φ(a) = b and Φ(b) = a. Hence Φ ∈ H(Y ). By the construction
of Φ, we have
e Ψ) < ε and Φ ∈ E(Y ) ∩ P (Y ).
d(Φ,
This completes the proof of our main theorem.
References
[1] N. Aoki, The splitting of zero-dimensional automorphisms and its application,
Colloq. Math. 49 (1985), 161–173.
[2] M. Dateyma, Homeomorphisms with the pseudo orbit tracing property of the Cantor set, Tokyo J. Math. 6 (1983), 287–290.
[3] H. Kato and J. Park, Expansive homeomorphisms of countable compacta, Topology
Appl. 95(1999), 207–216.
[4] T. Kimura, Homeomorphisms of zero-dimensional spaces, Tsukuba J. Math.
12(1988) no. 2 , 489–495.
[5] M. Sears, Expansive self-homeomorphisms of the Cantor set, Math. System Theory. 6 (1972), 129–132.
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Jong-Jin Park
Department of Mathematics
Chonbuk National University
Chonbuk 361-763, Korea
E-mail : [email protected]