Ωk such that f |Ωi is topologically transitive for each i = 1, 2

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, . . . , Blsuch that T |^B_iis 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 k- type chain recurrent sets (k = 1, 2, 3, 4) ofZ²-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 sys- tems 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] gen- eralized 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 signifi- cantly 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, ex- pansive, shadowing property.

c

2014 Korean Mathematical Society 387

a situation is impossible for expansive homeomorphisms with the shadowing property.

In Section 2, we recall the basic definitions inZ²-actions that are necessary in other sections, and introduce an example which is a general version of one- dimensional shift spaces.

In Section 3, we introduce the notions of k-type limit set, k-type nonwan- dering 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 |_CR^k(T )has the shadowing property for each k = 1, 2, 3, 4, where CR^k(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 C⁰metric d0; that is, for any f, g ∈ H(X),

d0(f, g) = sup_x∈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^brepresent k−1 in the 2-positional binary system; that is, k^b∈ {0, 1}², k = 1+P2

i=1k_i^b·2ⁱ⁻¹. We define a function mt : {1, 2, 3, 4} →Z²by

mt(k) = X2 i=1

(−1)^kbiei.

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 (−1)^kbixi ≥ (−1)^kbiyi for i = 1, 2. If all inequalities are strong, then we write x ≻^ky.

Definition 2.3. AZ²-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.

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 Tⁿ: X → X defined by Tⁿ(x) = T (n, x) is a homeomorphism.

Moreover we can see that each Tⁿ can be expressed as a finite composition of T^e1 and T^e2; that is, Tⁿ= T^ei1◦ T^ei2◦ · · · ◦ T^eik for i1, i2, . . . , ik ∈ {1, 2}.

A point x ∈ X is said to be periodic if the set {Tⁿ(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(Tⁿ(x), Tⁿ(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(Tⁿ(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 ofZ²-actions.

Definition 3.1. Let T be aZ²-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

L^k(x) = {y ∈ X | there exists a sequence {ts}s∈N⊂Z² such that ts+1≻^kts, lim

s→+∞T^ts(x) = y}.

The k-type limit set of T is defined by L^k(T ) =S

x∈XL^k(x).

Definition 3.2. Let T be aZ²-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 thats+1≻^kts, lim

s→+∞xs= x, lim

s→+∞T^ts(xs) = y}

is said to be the k-type limit prolongation of point x, where x ∈ X.

Theorem 3.1. LetT be a Z²-action on a compact metric space(X, ρ) and let x ∈ X. We have the following properties:

(1) L^k(x) ⊂ J^k(x),

(2) L^k(x) is nonempty closed subset of X, (3) L^k(x) and J^k(x) are T -invariant sets in X.

Proof. (1) For any y ∈ L^k(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).

(2) Let {yj}j∈Nbe a sequence in L^k(x) converge to y ∈ X. Since yj∈ L^k(x) for each j ∈N, there exists a sequence {tjs}s∈N⊂Z²such that

tjs+1 ≻^k tjs and lim

s→∞T^tjs(x) = yj. For any ε > 0, take N ∈N such that if j, s ≥ N , then

ρ(yj, y) < ε

2 and ρ(T^tjs(xs), yj) < ε 2. Hence we have

ρ(T^tjs(x), y) < ε

for sufficiently large j and s. This implies that y ∈ L^k(x).

(3) For any y ∈ L^k(x), there exists a sequence {ts}s∈N⊂Z² such that ts+1≻^kts, lim

s→∞T^ts(x) = y.

So,

T^ei(y) = T^ei( lim

s→∞T^ts(x)) = lim

s→∞T^ts+ei(x).

This implies that T^ei(y) ∈ L^k(x). Thus L^k(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≻^kts, lim

s→∞(xs) = x and lim

s→∞T^ts(x) = y.

So,

T^ei(y) = T^ei( lim

s→∞T^ts(x)) = lim

s→∞T^ts+ei(x).

This implies that T^ei(y) ∈ J^k(x). Thus J^k(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 Ω^k(T ).

Example 3.1. Consider aZ²-action T :Z²×R²→R²defined by T ((n1, n2), (x1, x2)) = ( 1

2ⁿ¹x1, 1 2ⁿ²x2).

Then the origin 0 ∈R²is 1-type nonwandering point of T , but it is not a 2-type nonwandering point of T .

Theorem 3.2. LetT be aZ²-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 andm ∈Z², there existsn ∈Z² such thatn ≻^k m and Tⁿ(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 {t_s}_s∈N⊂Z² such that

ts+1≻^k ts, lim

s→∞xs= x and lim

s→∞T^ts(xs) = x.

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∈ U_δ(x) and Uε(xp) ⊂ Uδ(x).

Since lim_s→∞T^ts(xs) = x, for any m ∈Z², there exists n ∈Z² such that n ≻^km and Tⁿ(Uε(xp)) ∩ Uδ(x) 6= ∅.

Thus we have Tⁿ(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 Tⁿ(Uδ(x)) ∩ Uδ(x) 6= ∅.

For each s ∈ N, we have ts∈Z² such that ||ts|| < ¹_s. By the assumption, for any s ∈N, there exists ts∈Z² such that

ts+1≻^ktsand T^ts(U¹

s(x)) ∩ U¹

s(x) 6= ∅.

Choose xs∈ T^s(U¹

s(x)) ∩ U¹

s(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 .  Theorem 3.3. LetT be a Z²-action on a compact metric space(X, ρ) and let k, s ∈ {1, 2, 3, 4}. If k + s = 5, then Ω^k(T ) = Ω^s(T ).

Proof. Let us take x ∈ Ω^k(T ). For any δ > 0 and m ∈Z², there exists n ∈Z² such that n ≻^k m and Tⁿ(Uδ(x)) ∩ Uδ(x) 6= ∅. Since k + s = 5, n ≻^k m and n ∈Z², we have −n >^sm and −n ∈Z². Since Tⁿ(Uδ(x)) ∩ Uδ(x) 6= ∅, we have

Uδ(x) ∩ T⁻ⁿ(Uδ(x)) = T⁻ⁿ⁺ⁿ(Uδ(x)) ∩ T⁻ⁿ(Uδ(x))

= T⁻ⁿ(Tⁿ(Uδ(x)) ∩ T⁻ⁿ(Uδ(x))) 6= ∅.

This means that x ∈ Ω^s(T ). 

Theorem 3.4. LetT be aZ²-action defined on a compact metric space(X, ρ).

We have the following properties:

(1) Ω^k(T ) is nonempty, closed and T -invariant, (2) P er(T ) ⊂ L^k(T ) ⊂ Ω^k(T ).

Proof. (1) Take a sequence {xj}i∈N⊂ Ω^k(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∈ Ω^k(T ), there exists n ≻^km such that

Tⁿ(Uε(xl)) ∩ Uε(xl) 6= ∅.

Since Tⁿ(Uε(xl)) ∩ Uε(xl) ⊂ Tⁿ(Uδ(x)) ∩ Uδ(x), we have then x ∈ Ω^k(T ). Thus Ω^k(T ) is closed in X.

Let x ∈ Ω^k(T ), let δ > 0 and m ∈Z² be arbitrary. For any i ∈Z², since Tⁱ is continuous, there exists ε > 0 such that Tⁱ(Uε(x)) ⊂ Uδ(Uⁱ(x)). Since

x ∈ Ω^k(T ), there exists n ≻^k m such that Tⁿ(Uε(x)) ∩ Uε(x) 6= ∅. Then we have

Tⁿ(Uδ(Tⁱ(x))) ∩ Uδ(Tⁱ(x)) ⊃ Tⁿ(Tⁱ(Uε(x))) ∩ Tⁱ(Uε(x))

= Tⁱ(Tⁿ(Uε(x)) ∩ Uε(x)) 6= ∅.

Therefore Tⁱ(x) ∈ Ω^k(T ). So, Ω^k(T ) is T -invariant.

(2) For any x ∈ P er(T ), there is n ∈Z²such that Tⁿ(x) = x (n ^k0). And there is {ts}_s∈N⊂Z² such that ts+1 ^k ts, ts = sn. Hence lim_s→∞T^ts(x) = lim_s→∞T^sn(x) = x. This means that x ∈ L^k(x).

For any y ∈ L^k(T ), there is a sequence {ts}_s∈N⊂Z² such that ts+1≻^kts, lim

s→∞T^ts(x) = y.

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 Tⁿ(z) ∈ Uδ(y), we get Tⁿ(Uδ(z)) ∩ Uδ(y) 6= ∅. Hence Tⁿ(Uδ(y)) ∩ Uδ(y) 6= ∅.

This implies that y ∈ Ω^k(T ). 

The following two examples show that P er(T ) 6= L^k(T ) and L^k(T ) 6= Ω^k(T ).

Example 3.2. Define aZ²-action T :Z²× S¹→ S¹ by T ((n1, n2), θ) = 2ⁿ¹⁺ⁿ²θ.

If θ ∈ P er(T ), then there exists n ≻^k 0 such that Tⁿ(θ) = θ. For any n = (n1, n2) ∈ Z², T ((n1, n2), θ) = 2ⁿ¹⁺ⁿ²θ = θ + 2kπ and so, θ = _2n1+n2^2kπ₋₁ for k ∈ N. Hence ^√₂³π /∈ P er(T ), but ^√₂³π ∈ L^k(T ). This implies that P er(T ) 6= L^k(T ) for k ∈ {1, 2, 3, 4}.

Example 3.3. Define a map f : [0, 1] → [0, 1] by f (x) = x − ₁₀¹ sin 2πx.

Consider aZ²-action T :Z²× [0, 1] → [0, 1] defined by T ((n1, n2), x) = fⁿ¹⁺ⁿ²(x).

Let 0 < ε < δ where δ be the number with Theorem 3.2. Then 1 − ^ε₂ ∈ L/ ¹(T ) but 1 −₂^ε∈ Ω¹(T ). This implies that L^k(T ) 6= Ω^k(T ) for k ∈ {1, 2, 3, 4}.

Definition 3.4. Let T be aZ²-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 Tⁿ(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≻^kts} = X.

Proof. Suppose there is a points x ∈ X such that {T^ts(x)|ts∈Z², ts+1≻^kts} = X.

Then for any nonempty open sets U, V ⊂ X, there exist m, n ∈Z²with m ^k n such that

Tⁿ(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

Ax6= X ⇔ A_x∩ U_i= ∅ for some i ≥ 1

⇔ T^m(x) ∈ X\Ui for everym ∈Z² and some i ≥ 1

⇔ x ∈ \

m∈Z²

T^−m(X\Ui) for some i ≥ 1

⇔ x ∈ [∞ i=1

\

m∈Z²

T^−m(X\Ui).

SinceS

m∈Z²T^−m(Ui) is dense in X by the assumption, X\( [

m∈Z²

T^−m(Ui)) = \

m∈Z²

(X\T^−m(Ui)) is nowhere dense.

ThusS_∞

i=1

T

m∈Z²T^−m(X\Ui) =S_∞

i=1

T

m∈Z²(X\T^−m(Ui)) is the set of first category. Since X is compact, {x ∈ X | Ax6= X} is a set of first category.  Theorem 3.6. LetT be aZ²-action defined on a compact metric space(X, ρ).

If T has the shadowing property, then P er(T ) is dense in Ω^k(T ) for any k ∈ {1, 2, 3, 4}.

Proof. Let x ∈ Ω^k(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 ∈ Ω^k(T ), we can select a δ-pseudo orbit ξ of T in Ω^k(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 aZ²-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}

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 ∼^ky).

Definition 4.2. Let T be aZ²-action on a compact metric space (X, ρ). Then CR^k(T ) = {x ∈ X | x ∼^kx} 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 CR^k(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 ⊂ Ω^k(T ) with limx→∞xs = x. If xs is α-related to xs+1 for s ∈ N, then xsis α-related to x for all s ∈N.

Lemma 4.1. Let T be aZ²-action on a compact metric space (X, ρ). If T has the shadowing property, then CR^k(T ) = Ω^k(T ) for each k ∈ {1, 2, 3, 4}.

Proof. By the definition, it is clear that Ω^k(T ) ⊂ CR^k(T ) for each k ∈ {1, 2, 3, 4}.

For any ε > 0, choose δ > 0 satisfying the shadowing property of T . Let x ∈ CR^k(T ), and let ξ = {xts}ⁿ_s=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 ∈ Ω^k(T ). 

Suppose aZ²-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 Ω^k(T ) into k-type chain components Aλ

under the k-type δ-relation; that is, Ω^k(T ) = S

λAλ. If we show that each k-type chain component Aλ is open, then Ω^k(T ) is decomposed by a finite number of k-type chain components; that is, Ω^k(T ) =Sk

i=1Ai. The following lemma shows that each k-type chain component is open in Ω^k(T ) if T has the shadowing property.

Lemma 4.2. Each k-type chain component Aλ is open in Ω^k(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 Ω^k(T ) where {tn}_n∈{0,1,2,...,p}⊂Z². We write Uα(x) = {z ∈ Ω^k(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 Ω^k(T ).

On the other hand, let ξ = {ys0 = y, . . . , ysl = x} be a δ-pseudo orbit of T

in Ω^k(T ) where {sn}_n∈{0,1,2,...,l}⊂Z². If T^ei(ys_l−1) /∈ Uγ(xt0) ∩ Ω^k(T ), then there is z ∈ Uγ(xt0) ∩ Ω^k(T ) with

ρ(T^ei(ys_l−1), Uγ(x0) ∩ Ω^k(T )) = ρ(T^ei(ys_l−1), z) < δ.

And so ρ(x^′₀, z) ≤ 2γ. Since z ∈ Ω^k(T ), there exists a γ-pseudo orbit {zu0 = z, . . . , zub+1= z} in Ω^k(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, . . . , ys_l−1, zu0, . . . , zub, x^′_t0} is a δ-pseudo orbit of T from ys0 to x^′_t0. Thus x^′_t0 ∈ A_λ and so Uγ(xt0) ⊂

Aλ. 

Theorem 4.1. LetT be aZ²-action defined on a compact metric space(X, ρ).

If T has the shadowing property, then T |Ω^k(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∈Z²is an α-pseudo orbit of T in Ω^k(T ). It will be enough to prove that a ε-shadowing point of {xn}_n∈Z² is chosen in Ω^k(T ). By Lemma 4.1, we see that {xn}_n∈Z²is contained in some Aj. Take xta, xtb ∈ {xn}_n∈Z²with ta≺^ktb. Then we get xta∼^k(δ) xtb, so that there are m1, m2≻^k 0and (m1+ m2)-cyclic δ-pseudo orbit {zn}_n∈Z² such that

xta= z(m1+m2)iand 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(Tⁿ(yta,tb), zn) < ε for n ∈Z² and so,

ρ(T^mi+j(yta,tb), zj) < ε for i ∈Z, 0 ^kj ≺^km.

If D = {T^mi(yta,tb) | i ∈Z} is discrete, then there is l ≻^k 0 such that T^l(yta,tb) = yta,tb. Hence yta,tb ∈ Ω^k(T ).

If D is not discrete, then there is a subsequence {T^m′i(yta,tb)} with T^m′i(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 ∈ Ω^k(T ). For α > 0, we can take N > 0 such that

ρ(T^mj′(yta,tb), y_t^′_a,tb) ≤ α^′ and ρ(T^mj′+ei(yta,tb), T^ei(y^′_ta,tb)) ≤ α^′

for j^′> N and i ∈ {1, 2}. From this we see that y^′_ta,tb ∼^k(α′)y_t^′_a,tb for α^′ > 0.

By Lemma 4.3, we have y^′_ta,tb∈ Ω^k(T ). By Lemma 4.2, if a subsequence {y_t^′_a,tb} converges to y as a → −∞ and b → ∞, then y ∈ Ω^k(T ) and ρ(T^ei(y), xei) ≤ ε for i ∈ I. Thus T |_Ω^k_{(T )} has the shadowing property  By our construction, we see that for a Z²-action T defined on a compact metric space (X, ρ), CR^k(T ) is a nonempty closed and T -invariant subset of X. And if T has the shadowing property, then T |CR^k(T )also has the shadowing property.

5. Spectral decomposition inZ²-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 ofZ²-actions.

Definition 5.1. T :Z²× X → X be aZ²-action on a compact metric space (X, ρ). Let ε > 0 and x ∈ X. We define a k-type local stable set W_ε^s(x, k) and k-type local unstable set W_ε^u(x, k) by

W_ε^s(x, k) = {y ∈ X | ρ(Tⁿ(x), Tⁿ(y)) ≤ ε, ∀n ^k 0},

W_ε^u(x, k) = {y ∈ X | ρ(Tⁿ(x), Tⁿ(y)) ≤ ε, ∀n ^k0}, respectively.

And also we define a k-type stable set W^s(x, k) and k-type unstable set W^u(x, k) by

W^s(x, k) = {y ∈ X | lim

s→∞(T^ts(x), T^ts(y)) = 0

for a sequence {ts}s∈N inZ² with ts+1≻^kts}, W^u(x, k) = {y ∈ X | lim

s→∞(T^−ts(x), T^−ts(y)) = 0

for a sequence {ts}_s∈Nin Z²with ts+1 ≻^k ts}, respectively.

First we prepare the following two lemmas to prove the Spectral Decompo- sition Theorem for the k-type nonwandering set Ω^k(T ).

Lemma 5.1. Let aZ²-actionT on X be expansive with an expansive constant e. For γ > 0, there exists nγ ∈Z² such that for all n ^k nγ,

Tⁿ(W_e^s(x, k)) ⊂ W_γ^s(Tⁿ(x), k) and Tⁿ(W_e^u(x, k)) ⊂ W_γ^u(Tⁿ(x), k).

Proof. Suppose that there exists γ > 0 such that for any nγ ∈Z² (nγ ≻^k 0), there is xn ∈ X such that

Tⁿ(W_e^s(xn, k))* W_γ^s(Tⁿ(xn), k).

Let {nt}t∈Nbe a sequence inZ²such that nt1 ≻^k nt2(t1> t2). By our standing assumptions, there exists ynt ∈ W_e^s(xnt, k) such that

T^nt(ynt) /∈ W_γ^s(T^nt(xnt), k) for each t ∈N. So, there exists i ∈ Z² (i ^k 0) such that

ρ(Tⁱ(Tⁿ(xn1)), Tⁱ(Tⁿ(yn1))) > γ.

Let i + n = m1. Continuing this process, we can find mn ≻^k 0, xtn and ysn ∈ X such that

(1) ynt ∈ W_e^s(xnt, k),

(2) ρ(T^mn(xtn), T^mn(ytn)) > γ,

(3) for a, b ∈N, if a > b, then ma≻^kmb and lim_n→∞km_nk = ∞.

It follows from ynt ∈ W_e^s(xnt, k) that for each i ≥^k −m_n, 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(Tⁱ(x), Tⁱ(y)) ≤ e, ∀i ∈Z².

Since e is an expansive constant for T , we have x = y. This is contradicting for

ρ(x, y) = lim

n→∞ρ(T^i+mn(xnt), T^i+mn(ynt)) > γ.

Thus Tⁿ(W_e^s(x, k)) ⊂ W_γ^s(Tⁿ(x), k). Similarly, we can show that

Tⁿ(W_e^u(x, k)) ⊂ W_γ^u(Tⁿ(x), k).  Lemma 5.2. Let aZ²-actionT on X be expansive with an expansive constant e, and let 0 < ε < e and x ∈ X. Then we have

W^s(x, k) = [

n^k0

T⁻ⁿ(W_ε^s(Tⁿ(x), k)) and W^u(x, k) = [

n^k0

Tⁿ(W_ε^u(T⁻ⁿ(x), k)).

Proof. For any y ∈ W^s(x, k), there exists N ≻^k 0such that for each n ^kN , ρ(Tⁿ(x), Tⁿ(y)) ≤ ε.

Thus we have

ρ(Tⁱ(T^N(x)), Tⁱ(T^N(y))) ≤ ε for all i ^k 0.

This implies that T^N(y) ∈ W_ε^s(T^N(x), k). Therefore we get y ∈ T^−N(W_ε^s(Tⁿ(x), k)) ⊂ [

n^k0

T⁻ⁿ(W_ε^s(Tⁿ(x), k)).

Let y ∈S

n^k0T⁻ⁿ(W_ε^s(Tⁿ(x), k)). For some n ^k 0, we have y ∈ T⁻ⁿ(W_ε^s(Tⁿ(x), k));

that is,

Tⁿ(y) ∈ W_ε^s(Tⁿ(x), k).

By Lemma 5.1, for every γ > 0 there exists Nγ ≻^k 0such that for each x ∈ X and m ^k Nγ we obtain

T^m+n(y) ∈ T^m(W_ε^s(Tⁿ(x), k)) ⊂ W_γ^s(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 expansiveZ²- action on compact metric space. If T has the shadowing property, then there exist closed pairwise disjoint and invariant sets B1, . . . , Bl⊂ Ω^k(T ) which ad- ditionally fulfill the following conditions:

(1) Ω^k(T ) =Sl i=1Bi,

(2) T |Biis topologically k-type transitive for each i = 1, . . . , l.

Proof. (1) Since T has the shadowing property, we have P er(T ) = Ω^k(T ) = CR^k(T ) by Theorem 3.6 and Lemma 4.3. Thus Ω^k(T ) splits into the equiva- lence classes Bλunder 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 |Ω^k(T ) has the shadowing property by Theorem 4.1.

Claim: Each Bλ is open in Ω^k(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 Ω^k(T ) is ε-shadowed by some point of Ω^k(T ). Denote

Uδ(Bλ) = {y ∈ Ω^k(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) = [

i^k0

T⁻ⁱ(W_ε^s(Tⁱ(x), k)) and W^u(x, k) = [

i^k0

Tⁱ(W_ε^u(T⁻ⁱ(x), k))

for x ∈ X and i ∈Z². Since T has the shadowing property on Ω^k(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 Ω^k(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 ˜z ∈ P er(T ) such that ρ(z, ˜z) < γ. Then ρ(˜z, Bλ) < δ. This means

that ˜z ∈ 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 Bλis open in Ω^k(T ). By compactness of Ω^k(T ) and openness of Bλ, Ω^k(T ) is expressed as a union of a finite set of {Bλ}.

(2) Let U and V be nonempty open sets in Biwhere U = eU ∩Bi, V = eV ∩Bi

for some open sets eU , eV 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 Ω^k(T ) is ε-shadowed by a point in Ω^k(T ). Let ξ = {xn}_n∈Z² be a δ-pseudo orbit of T in Bi. Then there exists a point y in Ω^k(T ) such that ρ(Tⁿ(y), xn) < ε for any n ∈ Z². Since xn∈ Bi, Tⁿ(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 |Biis 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: kimdj623@hanmail.net Seunghee Lee

Department of Mathematics Chungnam National University Daejeon 305-764, Korea E-mail address: shlee@cnu.ac.kr