Bull. Korean Math. Soc. 51 (2014), No. 2, pp. 387–400
http://dx.doi.org/10.4134/BKMS.2014.51.2.387
SPECTRAL DECOMPOSITION OF k-TYPE
NONWANDERING SETS FOR Z2 -ACTIONS
Daejung Kim and Seunghee Lee
Abstract. We prove that the set of k-type nonwandering points of a
Z2 -action T can be decomposed into a disjoint union of closed and T invariant sets B1 , . . . , Bl such that T |Bi is topologically k-type transitive
for each i = 1, 2, . . . , l, if T is expansive and has the shadowing property.
1. Introduction
We consider the notions of k-type limit set, k-type nonwandering sets and ktype chain recurrent sets (k = 1, 2, 3, 4) of Z2 -actions on a compact metric space
are introduced and studied as generalizations of those of classical dynamical
systems; that is, Z-actions or R-actions.
The shadowing property (or pseudo orbit tracing property) and expansivity
play important roles in the qualitative theory of classical dynamical systems,
and quite well developed in the theory (for more details, see [1, 2, 4, 5, 8, 12]).
One of the most important results in the qualitative theory of dynamical systems is the Spectral Decomposition Theorem, first obtained by Smale (proof
for the smooth case) and then extended by Bowen, which says that the set of
nonwandering points of an expansive homeomorphism f with the shadowing
property on a compact metric space can be decomposed into a disjoint union
of closed and f -invariant sets Ω1 , Ω2 , . . . , Ωk such that f |Ωi is topologically
transitive for each i = 1, 2, . . . , k (see [1]). Very recently, T. Das et al. [3] generalized the Smale’s spectral decomposition theorem to topologically Anosov
homeomorphisms on first countable, locally compact, paracompact, Hausdorff
spaces.
It is known that the multidimensional time dynamical system case significantly differ from situations known for homeomorphisms (see [6, 7, 9, 10, 11]).
In fact, there is a Z2 -action without periodic points, but it is expansive and
has the shadowing property (for more details, see [10]). As we know, such
Received June 21, 2012; Revised June 14, 2013.
2010 Mathematics Subject Classification. 37B10, 37C85, 54H20.
Key words and phrases. spectral decomposition theorem, k-type nonwandering sets, expansive, shadowing property.
c
2014
Korean Mathematical Society
387
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D. KIM AND S. LEE
a situation is impossible for expansive homeomorphisms with the shadowing
property.
In Section 2, we recall the basic definitions in Z2 -actions that are necessary
in other sections, and introduce an example which is a general version of onedimensional shift spaces.
In Section 3, we introduce the notions of k-type limit set, k-type nonwandering set (k = 1, 2, 3, 4) of a Z2 -action, and study their dynamic properties.
In Section 4, we introduce the notion of k-type chain recurrent set of a Z2 action, and obtain the following facts: If a Z2 -action T on a compact metric
space X has the shadowing property, then T |CRk (T ) has the shadowing property
for each k = 1, 2, 3, 4, where CRk (T ) is the set of all k-type chain recurrent
points of T .
In Section 5, we prove that the set of k-type nonwandering points of a Z2 action T can be decomposed into a disjoint union of closed and T -invariant
sets B1 , . . . , Bl such that T |Bi is topologically k-type transitive for each i =
1, 2, . . . , l if T is expansive and has the shadowing property.
2. Preliminaries
We give basic notions and properties on Z2 -actions which are necessary in
other sections. Let (X, ρ) be a compact metric space and let H(X) be the set
of homeomorphisms on X with the C 0 metric d0 ; that is, for any f, g ∈ H(X),
d0 (f, g) = supx∈X {d(f (x), g(x)), d(f −1 (x), g −1 (x))}.
Then we can see that the space H(X) with the metric d0 is a Banach space.
Throughout this section, we will denote by e1 , e2 vectors of the standard
canonical basis of R2 .
Definition 2.1. Let k ∈ {1, 2, 3, 4}, and let k b represent k−1 in the 2-positional
P2
binary system; that is, k b ∈ {0, 1}2 , k = 1+ i=1 kib ·2i−1 . We define a function
mt : {1, 2, 3, 4} → Z2 by
2
X
b
(−1)ki ei .
mt(k) =
i=1
Remark 2.1. The above function mt plays the role of a 2-dimensional quarters
enumerator.
Definition 2.2. Let k ∈ {1, 2, 3, 4} and let x, y ∈ R2 . We will write x k y if
b
b
(−1)ki xi ≥ (−1)ki yi for i = 1, 2. If all inequalities are strong, then we write
x ≻k y.
Definition 2.3. A Z2 -action on X is a continuous map T : Z2 × X → X such
that
(1) T (0, x) = x for 0 = (0, 0) ∈ Z2 and any x ∈ X,
(2) T (n, T (m, x)) = T (n + m, x) for any n, m ∈ Z2 and x ∈ X.
SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS
389
Remark 2.2. Let T : Z2 × X → X be a Z2 -action on X. Then it is easy to
show that for each n ∈ Z2 , the map T n : X → X defined by T n (x) = T (n, x)
is a homeomorphism.
Moreover we can see that each T n can be expressed as a finite composition
of T e1 and T e2 ; that is, T n = T ei1 ◦ T ei2 ◦ · · · ◦ T eik for i1 , i2 , . . . , ik ∈ {1, 2}.
A point x ∈ X is said to be periodic if the set {T n (x) : n ∈ Z2 } is finite.
The set of all periodic points is denoted by P er(T ). A Z2 -action T on X is
said to be expansive if there is a constant e > 0, called an expansive constant of
T , such that if x, y ∈ X and x 6= y, then ρ(T n (x), T n (y)) ≥ e for some n ∈ Z2 .
Let δ > 0. A sequence ξ = {xn }n∈Z2 in X is said to be a δ-pseudo orbit of T
(or δ-chain) if ρ(T ei (xn ), xn+ei ) < δ for every n ∈ Z2 and i ∈ {1, 2}. For any
ε > 0, we say that a δ-pseudo orbit ξ = {xn }n∈Z2 is said to be ε-shadowed by
y ∈ X if ρ(T n (y), xn ) < ε for any n ∈ Z2 . We say that a Z2 -action T on X
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 T is ε-shadowed by some point
of X.
3. k-type limit and nonwandering
We study the k-type limit sets and k-type nonwandering sets of Z2 -actions.
Definition 3.1. Let T be a Z2 -action on a compact metric space (X, ρ) and
let k ∈ {1, 2, 3, 4}. By a k-type limit set of a point x in X we mean the set
Lk (x) = {y ∈ X | there exists a sequence {ts }s∈N ⊂ Z2 such that
ts+1 ≻k ts ,
lim T ts (x) = y}.
s→+∞
The k-type limit set of T is defined by Lk (T ) =
S
x∈X
Lk (x).
Definition 3.2. Let T be a Z2 -action on a compact metric space (X, ρ) and
let k ∈ {1, 2, 3, 4}. The set
J k (x) = {y ∈ X | ∃ a sequence {xs }s∈N ⊂ X and a sequence {ts }s∈N ⊂ Z2
such that
s+1
≻k ts , lim xs = x,
s→+∞
lim T ts (xs ) = y}
s→+∞
is said to be the k-type limit prolongation of point x, where x ∈ X.
Theorem 3.1. Let T be a Z2 -action on a compact metric space (X, ρ) and let
x ∈ X. We have the following properties:
(1) Lk (x) ⊂ J k (x),
(2) Lk (x) is nonempty closed subset of X,
(3) Lk (x) and J k (x) are T -invariant sets in X.
Proof. (1) For any y ∈ Lk (x), there exists a sequence {ts }s∈N ⊂ Z2 such
that ts+1 ≻k ts , lims→∞ T ts (x) = y. Also we can choose a sequence {xs }s∈N
such that xs = x for any s ∈ N. Then we have lims→∞ xs = x. And
lims→+∞ T ts (xs ) = lims→+∞ T ts (x) = y. Thus we get y ∈ J k (x).
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D. KIM AND S. LEE
(2) Let {yj }j∈N be a sequence in Lk (x) converge to y ∈ X. Since yj ∈ Lk (x)
for each j ∈ N, there exists a sequence {tjs }s∈N ⊂ Z2 such that
tjs+1 ≻k tjs and lim T tjs (x) = yj .
s→∞
For any ε > 0, take N ∈ N such that if j, s ≥ N , then
ε
ε
ρ(yj , y) < and ρ(T tjs (xs ), yj ) < .
2
2
Hence we have
ρ(T tjs (x), y) < ε
for sufficiently large j and s. This implies that y ∈ Lk (x).
(3) For any y ∈ Lk (x), there exists a sequence {ts }s∈N ⊂ Z2 such that
ts+1 ≻k ts , lim T ts (x) = y.
s→∞
So,
T ei (y) = T ei ( lim T ts (x)) = lim T ts +ei (x).
s→∞
s→∞
This implies that T ei (y) ∈ Lk (x). Thus Lk (x) is T -invariant.
For any y ∈ J k (x), there exists a sequence {xs }s∈N ⊂ X and {ts }s∈N ⊂ Z2
such that
ts+1 ≻k ts , lim (xs ) = x and lim T ts (x) = y.
s→∞
s→∞
So,
T ei (y) = T ei ( lim T ts (x)) = lim T ts +ei (x).
s→∞
s→∞
ei
k
k
This implies that T (y) ∈ J (x). Thus J (x) is T -invariant.
Definition 3.3. A point x ∈ X is said to be a k-type nonwandering point of
T if x ∈ J k (x). The set of k-type nonwandering points of T will be denoted by
Ωk (T ).
Example 3.1. Consider a Z2 -action T : Z2 × R2 → R2 defined by
T ((n1 , n2 ), (x1 , x2 )) = (
1
1
x1 , n2 x2 ).
2n1
2
Then the origin 0 ∈ R2 is 1-type nonwandering point of T , but it is not a 2-type
nonwandering point of T .
Theorem 3.2. Let T be a Z2 -action defined on a compact metric space (X, ρ).
A point x ∈ X is k-type nonwandering point of T if and only if for any δ > 0
and m ∈ Z2 , there exists n ∈ Z2 such that n ≻k m and T n (Uδ (x)) ∩ Uδ (x) 6= ∅.
Proof. Let x ∈ X be a k-type nonwandering point of T . Then there exist two
sequences {xs }s∈N ⊂ X and {ts }s∈N ⊂ Z2 such that
ts+1 ≻k ts , lim xs = x and lim T ts (xs ) = x.
s→∞
s→∞
SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS
391
For any δ > 0, let Uδ (x) be a δ-neighborhood of x. Since lims→∞ xs = x, there
exists N ∈ N and ε > 0 such that for any p > N ,
xp ∈ Uδ (x) and Uε (xp ) ⊂ Uδ (x).
Since lims→∞ T (xs ) = x, for any m ∈ Z2 , there exists n ∈ Z2 such that
ts
n ≻k m and T n (Uε (xp )) ∩ Uδ (x) 6= ∅.
Thus we have T n (Uδ (x)) ∩ Uδ (x) 6= ∅.
Conversely, let x ∈ X be a point such that for any δ > 0 and m ∈ Z2 , there
exists n ∈ Z2 such that
n k m and T n (Uδ (x)) ∩ Uδ (x) 6= ∅.
For each s ∈ N, we have ts ∈ Z2 such that ||ts || < 1s . By the assumption, for
any s ∈ N, there exists ts ∈ Z2 such that
ts+1 ≻k ts and T ts (U s1 (x)) ∩ U s1 (x) 6= ∅.
Choose xs ∈ T s (U s1 (x)) ∩ U s1 (x) 6= ∅ for each s ∈ N. Then we have
xs → x and T ts (xs ) → x
as n → ∞. This implies that x is a nonwandering point of T .
2
Theorem 3.3. Let T be a Z -action on a compact metric space (X, ρ) and let
k, s ∈ {1, 2, 3, 4}. If k + s = 5, then Ωk (T ) = Ωs (T ).
Proof. Let us take x ∈ Ωk (T ). For any δ > 0 and m ∈ Z2 , there exists n ∈ Z2
such that n ≻k m and T n (Uδ (x)) ∩ Uδ (x) 6= ∅. Since k + s = 5, n ≻k m and
n ∈ Z2 , we have −n >s m and −n ∈ Z2 . Since T n (Uδ (x)) ∩ Uδ (x) 6= ∅, we have
Uδ (x) ∩ T −n (Uδ (x)) = T −n+n (Uδ (x)) ∩ T −n (Uδ (x))
= T −n (T n (Uδ (x)) ∩ T −n (Uδ (x)))
6= ∅.
s
This means that x ∈ Ω (T ).
Theorem 3.4. Let T be a Z2 -action defined on a compact metric space (X, ρ).
We have the following properties:
(1) Ωk (T ) is nonempty, closed and T -invariant,
(2) P er(T ) ⊂ Lk (T ) ⊂ Ωk (T ).
Proof. (1) Take a sequence {xj }i∈N ⊂ Ωk (T ) converging to x ∈ X. Then for
any δ > 0, we can choose ε > 0 and l ∈ N such that Uε (xl ) ⊂ Uδ (x). Since
xl ∈ Ωk (T ), there exists n ≻k m such that
T n (Uε (xl )) ∩ Uε (xl ) 6= ∅.
Since T n (Uε (xl )) ∩ Uε (xl ) ⊂ T n (Uδ (x)) ∩ Uδ (x), we have then x ∈ Ωk (T ). Thus
Ωk (T ) is closed in X.
Let x ∈ Ωk (T ), let δ > 0 and m ∈ Z2 be arbitrary. For any i ∈ Z2 , since
i
T is continuous, there exists ε > 0 such that T i (Uε (x)) ⊂ Uδ (U i (x)). Since
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D. KIM AND S. LEE
x ∈ Ωk (T ), there exists n ≻k m such that T n (Uε (x)) ∩ Uε (x) 6= ∅. Then we
have
T n (Uδ (T i (x))) ∩ Uδ (T i (x)) ⊃ T n (T i (Uε (x))) ∩ T i (Uε (x))
= T i (T n (Uε (x)) ∩ Uε (x)) 6= ∅.
Therefore T i (x) ∈ Ωk (T ). So, Ωk (T ) is T -invariant.
(2) For any x ∈ P er(T ), there is n ∈ Z2 such that T n (x) = x (n k 0). And
there is {ts }s∈N ⊂ Z2 such that ts+1 k ts , ts = sn. Hence lims→∞ T ts (x) =
lims→∞ T sn (x) = x. This means that x ∈ Lk (x).
For any y ∈ Lk (T ), there is a sequence {ts }s∈N ⊂ Z2 such that
ts+1 ≻k ts , lim T ts (x) = y.
s→∞
For any δ > 0, let Uδ (y) is a δ-neighborhood of y. Since lims→∞ T ts (x) = y,
there is n ∈ Z2 such that for any p ∈ Z2 with p ≻k n, T p (x) ∈ Uδ (y).
Let T p (x) = z. Then there exists ε > 0 such that Uε (z) ⊂ Uδ (y). Since
T n (z) ∈ Uδ (y), we get T n (Uδ (z)) ∩ Uδ (y) 6= ∅. Hence T n (Uδ (y)) ∩ Uδ (y) 6= ∅.
This implies that y ∈ Ωk (T ).
The following two examples show that P er(T ) 6= Lk (T ) and Lk (T ) 6= Ωk (T ).
Example 3.2. Define a Z2 -action T : Z2 × S 1 → S 1 by
T ((n1 , n2 ), θ) = 2n1 +n2 θ.
If θ ∈ P er(T ), then there exists n ≻k 0 such that T n (θ) = θ. For any n =
(n1 , n2 ) ∈ Z2 , T ((n1 , n2 ), θ) = 2n1 +n2 θ = θ + 2kπ and so, θ = 2n12kπ
+n2 −1
√
for k ∈ N. Hence 23 π ∈
/ P er(T ), but
P er(T ) 6= Lk (T ) for k ∈ {1, 2, 3, 4}.
√
3
2 π
∈ Lk (T ). This implies that
Example 3.3. Define a map f : [0, 1] → [0, 1] by f (x) = x −
Consider a Z2 -action T : Z2 × [0, 1] → [0, 1] defined by
1
10
sin 2πx.
T ((n1 , n2 ), x) = f n1 +n2 (x).
Let 0 < ε < δ where δ be the number with Theorem 3.2. Then 1 − 2ε ∈
/ L1 (T )
ε
1
k
k
but 1 − 2 ∈ Ω (T ). This implies that L (T ) 6= Ω (T ) for k ∈ {1, 2, 3, 4}.
Definition 3.4. Let T be a Z2 -action on a compact metric space (X, ρ) and
let k ∈ {1, 2, 3, 4}. We say that T is topologically k-type transitive if for any
nonempty open sets U, V ⊂ X there exists n ∈ Z2 such that n ≻k 0 and
T n (U ) ∩ V 6= ∅.
Theorem 3.5. A Z2 -action T is k-type transitive if and only if there exists
x ∈ X such that {T ts (x)|ts ∈ Z2 , ts+1 ≻k ts } = X.
Proof. Suppose there is a points x ∈ X such that
{T ts (x)|ts ∈ Z2 , ts+1 ≻k ts } = X.
SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS
393
Then for any nonempty open sets U, V ⊂ X, there exist m, n ∈ Z2 with m k n
such that
T n (x) ∈ U and T m (x) ∈ V.
Let m − n = l k 0. Then T n+l (x) ∈ T l (U ) and T m (x) ∈ V . Hence T l (U ) ∩
V 6= ∅ for some l ∈ Z2 . This means that T is k-type transitive.
Conversely, we assume that T is k-type transitive.
Let {T ts (x) | ts ∈ Z2 , ts+1 ≻k ts } = Ax and {Ui | i ≥ 1} be a countable basis
for X. It is enough to show that Ax = X for some x ∈ X. Suppose not. Then
for any x ∈ X, we have
Ax 6= X ⇔ Ax ∩ Ui = ∅ for some i ≥ 1
⇔ T m (x) ∈ X\Ui for everym ∈ Z2 and some i ≥ 1
\
⇔ x∈
T −m (X\Ui ) for some i ≥ 1
⇔ x∈
Since
S
m∈Z2
∞ \
[
T −m (X\Ui ).
i=1 m∈Z2
m∈Z2
T
−m
(Ui ) is dense in X by the assumption,
[
\
X\(
T −m (Ui )) =
(X\T −m(Ui ))
m∈Z2
m∈Z2
is nowhere
S∞dense.
S∞ T
T
Thus i=1 m∈Z2 T −m (X\Ui ) = i=1 m∈Z2 (X\T −m(Ui )) is the set of first
category. Since X is compact, {x ∈ X | Ax 6= X} is a set of first category. Theorem 3.6. Let T be a Z2 -action defined on a compact metric space (X, ρ).
If T has the shadowing property, then P er(T ) is dense in Ωk (T ) for any k ∈
{1, 2, 3, 4}.
Proof. Let x ∈ Ωk (T ) for any k ∈ {1, 2, 3, 4}, and let ε > 0 be arbitrary. Since
T has the shadowing property, we can take δ > 0 such that every δ-pseudo
orbit of T is ε-shadowed by a point in X. Since x ∈ Ωk (T ), we can select a
δ-pseudo orbit ξ of T in Ωk (T ) as follows;
ξ = {x, T t1 (x), T t2 (x), . . . , T tk (x), x}.
Then there is a periodic point y ∈ X which ε-shadows ξ. This means that
Bε (x) ∩ P er(T ) 6= ∅ for any ε > 0, and so x ∈ P er(T ).
4. k-type chain recurrence
We study the notion of k-type chain recurrence in Z2 -actions.
Definition 4.1. Let T be a Z2 -action on a compact metric space (X, ρ). For
x, y ∈ X and δ > 0, x is k-type δ-related to y (written x ∼k(δ) y) if there exists
a δ-pseudo orbits ξ1 , ξ2 of T from x to y; that is, ξ1 = {x0 = x, . . . , xn = y}
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D. KIM AND S. LEE
and ξ2 = {y0 = y, . . . , ym = x} for some m, n ∈ Z2 (m, n ≻k 0). If x ∼k(δ) y for
any δ > 0, then x is k-type related to y (written x ∼k y).
Definition 4.2. Let T be a Z2 -action on a compact metric space (X, ρ). Then
CRk (T ) = {x ∈ X | x ∼k x} is said to be the k-type chain recurrent set of T .
Remark 4.1. The k-type relation x ∼k y is equivalence relation on the set
CRk (T ). Each equivalence class is called a k-type chain component of T .
Let T be a Z2 -action on a compact metric space (X, ρ). By the definition
of k-type nonwandering, we know that for α > 0, every k-type nonwandering
point x of T is k-type α-related to T ei (x) for i ∈ {1, 2}. And let a sequence
{xs }s∈N ⊂ Ωk (T ) with limx→∞ xs = x. If xs is α-related to xs+1 for s ∈ N,
then xs is α-related to x for all s ∈ N.
Lemma 4.1. Let T be a Z2 -action on a compact metric space (X, ρ). If T has
the shadowing property, then CRk (T ) = Ωk (T ) for each k ∈ {1, 2, 3, 4}.
Proof. By the definition, it is clear that Ωk (T ) ⊂ CRk (T ) for each k ∈ {1, 2, 3, 4}.
For any ε > 0, choose δ > 0 satisfying the shadowing property of T . Let
x ∈ CRk (T ), and let ξ = {xts }ns=0 be a δ-pseudo orbit of T from x to xj ; that
is, xt0 = x = xtn . Then there exists y ∈ X such that
ρ(T ts (y), xts ) < ε
for all s = 0, 1, . . . , n. In particular, we have T tn (y) ∈ Uε (x). This means that
x ∈ Ωk (T ).
Suppose a Z2 -action T on a compact metric space (X, ρ) has the shadowing
property. For any ε > 0, let δ = δ(ε) be a number taken by the shadowing
property of T . Then we can split Ωk (T ) into S
k-type chain components Aλ
under the k-type δ-relation; that is, Ωk (T ) = λ Aλ . If we show that each
k-type chain component Aλ is open, then Ωk (T ) is decomposed by a finite
Sk
number of k-type chain components; that is, Ωk (T ) = i=1 Ai . The following
lemma shows that each k-type chain component is open in Ωk (T ) if T has the
shadowing property.
Lemma 4.2. Each k-type chain component Aλ is open in Ωk (T ) if T has
shadowing property.
Proof. Take x ∈ Aλ . For every y ∈ Aλ , there is a δ-pseudo orbit {xt0 =
x, . . . , xtp = y} in Ωk (T ) where {tn }n∈{0,1,2,...,p} ⊂ Z2 . We write Uα (x) = {z ∈
Ωk (T ) | ρ(z, x) < α}. Choose γ with 0 < γ < δ3 such that
T ei (Uγ (xt0 )) ⊂ Uδ (xt1 ) for i ∈ {1, 2}.
Then for every x′t0 ∈ Uγ (xt0 ), {x′t0 , . . . , xtp } is a δ-pseudo orbit of T in Ωk (T ).
On the other hand, let ξ = {ys0 = y, . . . , ysl = x} be a δ-pseudo orbit of T
SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS
395
/ Uγ (xt0 ) ∩ Ωk (T ), then
in Ωk (T ) where {sn }n∈{0,1,2,...,l} ⊂ Z2 . If T ei (ysl−1 ) ∈
there is z ∈ Uγ (xt0 ) ∩ Ωk (T ) with
ρ(T ei (ysl−1 ), Uγ (x0 ) ∩ Ωk (T )) = ρ(T ei (ysl−1 ), z) < δ.
And so ρ(x′0 , z) ≤ 2γ. Since z ∈ Ωk (T ), there exists a γ-pseudo orbit {zu0 =
z, . . . , zub+1 = z} in Ωk (T ) where {un }n∈{0,1,2,...,b+1} ⊂ Z2 . Since
ρ(T ei (zub ), x′t0 ) ≤ ρ(T ei (zub ), z) + ρ(z, x′t0 ) ≤ 3γ < δ,
we have that
(ξ\{ysl }) ∪ {zu0 , . . . , zub+1 = z} = {ys0 , . . . , ysl−1 , zu0 , . . . , zub , x′t0 }
is a δ-pseudo orbit of T from ys0 to x′t0 . Thus x′t0 ∈ Aλ and so Uγ (xt0 ) ⊂
Aλ .
Theorem 4.1. Let T be a Z2 -action defined on a compact metric space (X, ρ).
If T has the shadowing property, then T |Ωk (T ) has the shadowing property.
Proof. For any ε > 0, let δ > 0 be as in the definition of shadowing property.
Since each k-type chain component Ai is open and closed, we have
ρ(Ai , Aj ) = inf{ρ(a, b) | a ∈ Ai , b ∈ Aj } > 0
for i 6= j. Put δ1 = min{ρ(Ai , Aj ) | i 6= j}. For any α > 0 with α < min{δ, δ1 },
let {xn }n∈Z2 is an α-pseudo orbit of T in Ωk (T ). It will be enough to prove that
a ε-shadowing point of {xn }n∈Z2 is chosen in Ωk (T ). By Lemma 4.1, we see
that {xn }n∈Z2 is contained in some Aj . Take xta , xtb ∈ {xn }n∈Z2 with ta ≺k tb .
Then we get xta ∼k(δ) xtb , so that there are m1 , m2 ≻k 0 and (m1 + m2 )-cyclic
δ-pseudo orbit {zn }n∈Z2 such that
xta = z(m1 +m2 )i and xtb = zm1 +(m1 +m2 )i
for all i ∈ Z. Put m = m1 + m2 . Since T has the shadowing property, there is
yta ,tb ∈ X such that
ρ(T n (yta ,tb ), zn ) < ε
for n ∈ Z2 and so,
ρ(T mi+j (yta ,tb ), zj ) < ε
for i ∈ Z, 0 k j ≺k m.
If D = {T mi (yta ,tb ) | i ∈ Z} is discrete, then there is l ≻k 0 such that
l
T (yta ,tb ) = yta ,tb . Hence yta ,tb ∈ Ωk (T ).
′
′
If D is not discrete, then there is a subsequence {T mi (yta ,tb )} with T mi (yta ,tb )
→ y ′ ta ,tb as i′ → ∞. Obviously we have
ρ(y ′ ta ,tb , xa ) ≤ ε and ρ(T j (yta ,tb ), zj ) ≤ ε
for j ∈ Z2 . We shall see that y ′ ta ,tb ∈ Ωk (T ). For α > 0, we can take N > 0
such that
′
ρ(T mj (yta ,tb ), yt′ a ,tb ) ≤ α′ and ρ(T mj
′
+ei
(yta ,tb ), T ei (yt′ a ,tb )) ≤ α′
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D. KIM AND S. LEE
′
for j ′ > N and i ∈ {1, 2}. From this we see that yt′ a ,tb ∼k(α ) yt′ a ,tb for α′ > 0.
By Lemma 4.3, we have yt′ a ,tb ∈ Ωk (T ). By Lemma 4.2, if a subsequence {yt′ a ,tb }
converges to y as a → −∞ and b → ∞, then y ∈ Ωk (T ) and ρ(T ei (y), xei ) ≤ ε
for i ∈ I. Thus T |Ωk (T ) has the shadowing property
By our construction, we see that for a Z2 -action T defined on a compact
metric space (X, ρ), CRk (T ) is a nonempty closed and T -invariant subset of
X. And if T has the shadowing property, then T |CRk (T ) also has the shadowing
property.
5. Spectral decomposition in Z2 -actions
One of the most important results in the theory of topological dynamics
is the Spectral Decomposition Theorem, first obtained by Smale (proof for
the smooth case) and then extended by Bowen, which says that the set of
nonwandering points of an expansive homeomorphism f with the shadowing
property on a compact metric space can be decomposed into a disjoint union
of closed and f -invariant sets Ω1 , Ω2 , . . . , Ωk such that f |Ωi is topologically
transitive for any i = 1, . . . , k. In this section, we obtain a general version of
the Spectral Decomposition Theorem for the set of k-type nonwandering points
of Z2 -actions.
Definition 5.1. T : Z2 × X → X be a Z2 -action on a compact metric space
(X, ρ). Let ε > 0 and x ∈ X. We define a k-type local stable set Wεs (x, k) and
k-type local unstable set Wεu (x, k) by
Wεs (x, k) = {y ∈ X | ρ(T n (x), T n (y)) ≤ ε, ∀n k 0},
Wεu (x, k) = {y ∈ X | ρ(T n (x), T n (y)) ≤ ε, ∀n k 0}, respectively.
And also we define a k-type stable set W s (x, k) and k-type unstable set
u
W (x, k) by
W s (x, k) = {y ∈ X | lim (T ts (x), T ts (y)) = 0
s→∞
for a sequence {ts }s∈N in Z2 with ts+1 ≻k ts },
W u (x, k) = {y ∈ X | lim (T −ts (x), T −ts (y)) = 0
s→∞
for a sequence {ts }s∈N in Z2 with ts+1 ≻k ts }, respectively.
First we prepare the following two lemmas to prove the Spectral Decomposition Theorem for the k-type nonwandering set Ωk (T ).
Lemma 5.1. Let a Z2 -action T on X be expansive with an expansive constant
e. For γ > 0, there exists nγ ∈ Z2 such that for all n k nγ ,
T n (Wes (x, k)) ⊂ Wγs (T n (x), k) and T n (Weu (x, k)) ⊂ Wγu (T n (x), k).
Proof. Suppose that there exists γ > 0 such that for any nγ ∈ Z2 (nγ ≻k 0),
there is xn ∈ X such that
T n (Wes (xn , k)) * Wγs (T n (xn ), k).
SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS
397
Let {nt }t∈N be a sequence in Z2 such that nt1 ≻k nt2 (t1 > t2 ). By our standing
assumptions, there exists ynt ∈ Wes (xnt , k) such that
/ Wγs (T nt (xnt ), k)
T nt (ynt ) ∈
for each t ∈ N. So, there exists i ∈ Z2 (i k 0) such that
ρ(T i (T n (xn1 )), T i (T n (yn1 ))) > γ.
Let i + n = m1 . Continuing this process, we can find mn ≻k 0, xtn and
ysn ∈ X such that
(1) ynt ∈ Wes (xnt , k),
(2) ρ(T mn (xtn ), T mn (ytn )) > γ,
(3) for a, b ∈ N, if a > b, then ma ≻k mb and limn→∞ kmn k = ∞.
It follows from ynt ∈ Wes (xnt , k) that for each i ≥k −mn ,
ρ(T i+mn (xnt ), T i+mn (ynt )) ≤ e, ∀i ∈ Z2 .
If T mn (xnt ) converge to x and T mn (ynt ) converge to y as t → ∞, then
ρ(T i (x), T i (y)) ≤ e, ∀i ∈ Z2 .
Since e is an expansive constant for T , we have x = y. This is contradicting
for
ρ(x, y) = lim ρ(T i+mn (xnt ), T i+mn (ynt )) > γ.
n→∞
Thus T n (Wes (x, k)) ⊂ Wγs (T n (x), k). Similarly, we can show that
T n (Weu (x, k)) ⊂ Wγu (T n (x), k).
Lemma 5.2. Let a Z2 -action T on X be expansive with an expansive constant
e, and let 0 < ε < e and x ∈ X. Then we have
[
[
W s (x, k) =
T −n (Wεs (T n (x), k)) and W u (x, k) =
T n (Wεu (T −n (x), k)).
nk 0
nk 0
Proof. For any y ∈ W s (x, k), there exists N ≻k 0 such that for each n k N ,
ρ(T n (x), T n (y)) ≤ ε.
Thus we have
ρ(T i (T N (x)), T i (T N (y))) ≤ ε for all i k 0.
This implies that T N (y) ∈ Wεs (T N (x), k). Therefore we get
[
y ∈ T −N (Wεs (T n (x), k)) ⊂
T −n (Wεs (T n (x), k)).
nk 0
Let y ∈
S
nk 0
T −n (Wεs (T n (x), k)). For some n k 0, we have
y ∈ T −n (Wεs (T n (x), k));
that is,
T n (y) ∈ Wεs (T n (x), k).
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D. KIM AND S. LEE
By Lemma 5.1, for every γ > 0 there exists Nγ ≻k 0 such that for each x ∈ X
and m k Nγ we obtain
T m+n (y) ∈ T m (Wεs (T n (x), k)) ⊂ Wγs (T m+n (x), k).
Consequently for each n k N, we have
ρ(T m+n (x), T m+n (y)) ≤ γ.
Thus y ∈ W s (x, k).
Now we introduce a general version of the Spectral Decomposition Theorem
for the set of k-type nonwandering points of Z2 -actions.
Theorem 5.1 (Spectral Decomposition Theorem). Let T be an expansive Z2 action on compact metric space. If T has the shadowing property, then there
exist closed pairwise disjoint and invariant sets B1 , . . . , Bl ⊂ Ωk (T ) which additionally fulfill the following conditions:
Sl
(1) Ωk (T ) = i=1 Bi ,
(2) T |Bi is topologically k-type transitive for each i = 1, . . . , l.
Proof. (1) Since T has the shadowing property, we have P er(T ) = Ωk (T ) =
CRk (T ) by Theorem 3.6 and Lemma 4.3. Thus Ωk (T ) splits into the equivalence classes Bλ under the k-type relation ∼k . By Lemmas 4.1 and 4.2, we know
that each k-type chain component Bλ is closed and T -invariant. Moreover we
know that T |Ωk (T ) has the shadowing property by Theorem 4.1.
Claim: Each Bλ is open in Ωk (T ).
Let e be an expansive constant of T . For any ε with 0 < ε < e, there exists
δ > 0 such that for any δ-pseudo orbit of T in Ωk (T ) is ε-shadowed by some
point of Ωk (T ). Denote
Uδ (Bλ ) = {y ∈ Ωk (T ) | ρ(y, Bλ ) < δ}.
Then for p ∈ Uδ (Bλ ) ∩ P er(T ), there is y ∈ Bλ such that ρ(y, p) < δ. By
Lemma 5.2, we have
[
[
W s (x, k) =
T −i (Wεs (T i (x), k)) and W u (x, k) =
T i (Wεu (T −i (x), k))
ik 0
ik 0
for x ∈ X and i ∈ Z2 . Since T has the shadowing property on Ωk (T ), we get
W u (p, k) ∩ W s (y, k) 6= ∅ and W s (p, k) ∩ W u (y, k) 6= ∅.
Therefore there is y0 ∈ Bλ with y0 ∼k p, that is, p ∈ Bλ . Thus Uδ (Bλ ) ∩
P er(T ) ⊂ Bλ . Since Bλ is closed in Ωk (T ), we have
Bλ = Bλ ⊃ Uδ (Bλ ) ∩ P er(T ).
Let z ∈ Uδ (Bλ ) ∩ P er(T ) and ε′ > 0 be given. Since z ∈ Uδ (Bλ ) and Bλ is
closed, 0 ≤ ρ(z, Bλ ) < δ. Put γ = min{δ − ρ(z, Bλ ), ε′ }. Since z ∈ P er(T ),
there exists z̃ ∈ P er(T ) such that ρ(z, z̃) < γ. Then ρ(z̃, Bλ ) < δ. This means
SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS
399
that z̃ ∈ Bε′ (z) ∩ [Uδ (Bλ ) ∩ P er(T )] 6= ∅. Therefore z ∈ Uδ (Bλ ) ∩ P er(T ).
Thus we have
Uδ (Bλ ) ∩ P er(T ) ⊃ Uδ (Bλ ) ∩ P er(T ).
And so,
Bλ ⊃ Uδ (Bλ ) ∩ P er(T ) ⊃ Uδ (Bλ ) ∩ P er(T ) = Uδ (Bλ ).
Thus each Bλ is open in Ωk (T ). By compactness of Ωk (T ) and openness of Bλ ,
Ωk (T ) is expressed as a union of a finite set of {Bλ }.
e ∩Bi , V = Ve ∩Bi
(2) Let U and V be nonempty open sets in Bi where U = U
e
e
for some open sets U , V in X. Let x ∈ U and y ∈ V , take ε > 0 such that
Bε (x) ∩ Bi ⊂ U and Bε (y) ∩ Bi ⊂ V.
Claim: T |Bi has the shadowing property.
Choose ε > 0 such that ρ(Bi , Bj ) > ε for all i, j ∈ {1, . . . , l} i 6= j. For ε > 0,
choose δ with 0 < δ < ε, for any δ-pseudo orbit of T in Ωk (T ) is ε-shadowed
by a point in Ωk (T ). Let ξ = {xn }n∈Z2 be a δ-pseudo orbit of T in Bi . Then
there exists a point y in Ωk (T ) such that ρ(T n (y), xn ) < ε for any n ∈ Z2 .
Since xn ∈ Bi , T n (y) ∈ Bi for i ∈ {1, . . . , l}.
Since T |Bi has the shadowing property, there exists δ > 0 such that for any
δ-pseudo orbit of T in Bi is ε-shadowed by a point in Bi . Since x and y are
k-type δ-related, there is a δ-pseudo orbit ξ = {x0 = x, xt1 , xt2 , . . . , xtn = y}
of T from x to y, where {tn }n∈N ⊂ Z2 . Then there exists z ∈ Bi such that
ρ(z, x) < ε and ρ(T ti (z), xti ) < ε for all i ∈ N. In particular, ρ(z, x) < ε and
ρ(T tn (z), xtn ) < ε. Since z ∈ Bε (x) ∩ Bi ⊂ U , we have U ∩ T −tn (V ) 6= ∅. Thus
T |Bi is a topologically k-type transitive for any i ∈ {1, . . . , l}.
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Daejung Kim
Department of Mathematics
Chungnam National University
Daejeon 305-764, Korea
E-mail address: [email protected]
Seunghee Lee
Department of Mathematics
Chungnam National University
Daejeon 305-764, Korea
E-mail address: [email protected]