TOPOLOGICAL STABILITY AND SHADOWING OF ZERO-DIMENSIONAL DYNAMICAL SYSTEMS

. In this paper, we examine the notion of topological stability and its relation to the shadowing properties in zero-dimensional spaces. Several counter-examples on the topological stability and the shadowing properties are given. Also, we prove that any topologically stable (in a modiﬁed sense) homeomorphism of a Cantor space exhibits only simple typical dynamics.


1.
Introduction. Stability is one of the most important notions in the qualitative study of dynamical systems, and the relationship with it is a basic subject of the theory of shadowing properties. The topological stability introduced by Walters in [22] is a kind of structural stability defined for all homeomorphisms of compact metric spaces. Among the papers dealing with the notion, there are [9,18,23] (see also [24]), and some recent attempts were made to explore the variants of topological stability [4,15,16]. Although there are a number of studies on the topological stability, the main attention seems to have been focused on the homeomorphisms (or diffeomorphisms) of topological (or differentiable) manifolds. In this paper, we examine the notion in zero-dimensional spaces, mainly in relation with the shadowing properties, and observe some singular behaviors possibly different from when the spaces are manifolds. Especially, we give several counter-examples on the topological stability and the shadowing properties. Moreover, we prove that all topologically stable (in a modified sense) homeomorphisms of Cantor spaces exhibit only simple typical dynamics.
Let us begin with the definition of the topological stability. Throughout this paper, (X, d) denotes a compact metric space X endowed with a metric d. We denote by C(X) the set of continuous self-maps of X and by H(X) the set of homeomorphisms of X. We define metrics d C 0 on C(X) and D on H(X) by for f, g ∈ C(X), and D(f, g) = max{d C 0 (f, g), d C 0 (f −1 , g −1 )} for f, g ∈ H(X). As H(X) ⊂ C(X), d C 0 gives another metric on H(X), but we know that d C 0 and D are equivalent metrics on H(X). We give a proof of this fact in Section 2 for the sake of completeness. Then, for f ∈ H(X), we say that f is 2744 NORIAKI KAWAGUCHI topologically stable if for any > 0, there is δ > 0 such that for every g ∈ H(X) with D(f, g) < δ, there is h ∈ C(X) with d C 0 (h, id X ) < and h • g = f • h.
The most part of this paper is concerned with the relation between the topological stability and the shadowing properties, so we shall recall the definitions of shadowing properties dealt with in this paper. Given a map f : X → X, a finite sequence (x i ) k i=0 of points in X, where k is a positive integer, is called a δ-chain of f if We say that f has the shadowing property if for any > 0, there is δ > 0 such that every δ-pseudo orbit of f is -shadowed by some point of X. The following definition is not so standard as the shadowing property. We say that f has the strict periodic shadowing property if for any > 0, there is δ > 0 such that for any δ-cycle (x i ) m i=0 of f , where m is a positive integer, there is p ∈ X such that f m (p) = p and d(x i , f i (p)) ≤ for all 0 ≤ i ≤ m. Let P er(f ) denote the set of periodic points for f . A δ-pseudo orbit (x i ) i≥0 of f is said to be periodic when there is an integer m > 0 such that x i = x i+m for every i ≥ 0. Then, we say that f has the periodic shadowing property if for any > 0, there is δ > 0 such that every periodic δ-pseudo orbit of f is -shadowed by some point of P er(f ) (such a definition is given in, for example, [25]). It is obvious from the definitions that if f has the strict periodic shadowing property, then f has the periodic shadowing property. A point x ∈ X is said to be a chain recurrent point for f if for any We denote by CR(f ) the set of chain recurrent points for f . Note that we have CR(f ) = P er(f ) when f has the (strict) periodic shadowing property.
There is a property of compact metric spaces derived from [21,Lemma 4], which relates the topological stability to the shadowing properties. In [22], by using the fact that the closed differentiable manifolds of dimension grater than 1 have such a property [22,Lemma 10] (see also [19,Lemma 13]), Walters proved that all topologically stable homeomorphisms of those spaces satisfy the shadowing property. The property is called generalized homogeneity in [1] and also in some other papers. Its definition is given as follows. For n ≥ 1, an n-tuple (x 1 , x 2 , . . . , x n ) ∈ X n is said to be proper if x 1 , x 2 , . . . , x n are pairwise distinct, that is, x i = x j for all 1 ≤ i < j ≤ n. For n ≥ 1, we define a metric d n on X n by . , x n ), η = (y 1 , y 2 , . . . , y n ) ∈ X n . For any map f : X → X and n ≥ 1, the n-fold product f (n) : . . , f (x n )) for ζ = (x 1 , x 2 , . . . , x n ) ∈ X n . Then, we say that a compact metric space (X, d) is generalized homogeneous when for any > 0, there is δ > 0 such that the following condition holds: Given any integer n ≥ 1 and any pair of proper n-tuples ζ, η ∈ X n , if d n (ζ, η) < δ, then there is φ ∈ H(X) such that D(φ, id X ) < and φ (n) (ζ) = η.
A compact metric space (X, d) is said to be a Cantor space if it is perfect, that is, it has no isolated point, and its topological dimension (denoted by dim X) is zero, or equivalently, it is totally disconnected. Every Cantor space is homeomorphic to the Cantor ternary set in the unit interval. We let the following lemma be a base of the study in this paper. Under the assumption that the space is perfect and generalized homogeneous, we prove that a topologically stable homeomorphism satisfies not only the shadowing property but also the strict periodic shadowing property. Theorem 1.1. Let (X, d) be a compact metric space which is perfect and generalized homogeneous. For any f ∈ H(X), if f is topologically stable, then f has the shadowing property and the strict periodic shadowing property, especially f satisfies CR(f ) = P er(f ).
When the generalized homogeneity is absent, the implication as in Theorem 1.1 does not hold in general. For example in [7], Cook gave an example of a nondegenerate continuum C with H(C) = {id C }. For such C, id C is trivially topologically stable, but we easily see that it has neither the shadowing property nor the strict periodic shadowing property.
It is known that there exists a circle homeomorphism which satisfies the shadowing property but is not topologically stable [23]. The continuous shadowing property introduced by Lee in [14] is a stronger property than the shadowing property (its precise definition is given in Section 2). The following statement is a consequence of [14,Theorem 2.5]: If f ∈ H(M ) has the continuous shadowing property, then f is topologically stable. There seems to be an implicit assumption in [14,Theorem 2.5] that the space M is a closed differentiable manifold. In contrast, by using Theorem 1.1, we prove the following corollary. Corollary 1.1. Let (X, d) be a Cantor space. Then, there exists f ∈ H(X) which has the continuous shadowing property but is not topologically stable.
Indeed, Corollary 1.1 is an immediate consequence of the fact that the odometers satisfy the continuous shadowing property, but by Theorem 1.1, they are not topologically stable. This corollary clarifies that the continuous shadowing property does not necessarily imply the topological stability unless there are proper assumptions on the space. The next result concerns the notion of equicontinuity. For f ∈ H(X), we say that f is equicontinuous if for any > 0, there is δ > 0 such that d(x, y) ≤ δ implies sup i∈Z d(f i (x), f i (y)) ≤ for all x, y ∈ X. Theorem 1.2. Let (X, d) be a compact metric space and let f ∈ H(X) be an equicontinuous homeomorphism. If f has the strict periodic shadowing property, then f is topologically stable, and dim X = 0.
This theorem gives a converse of Theorem 1.1 for equicontinuous homeomorphisms. As a direct consequence of Lemma 1.1, Theorem 1.1, and Theorem 1.2, we obtain the following corollary. As proved in Section 5 (see Lemma 5.1), for any equicontinuous homeomorphism f ∈ H(X), if dim X = 0 and X = P er(f ), then f has the periodic shadowing property. Then, it is natural to expect that the same conditions still imply the strict periodic shadowing property. However, this is not the case. In Section 5, by modifying an odometer, we give an example of a homeomorphism f of a Cantor space (X, d) with the following properties.
(3) f has the periodic shadowing property.
(4) f does not have the strict periodic shadowing property.
(5) f 3 has the strict periodic shadowing property. As a consequence, this example shows that the periodic shadowing property is not equivalent to the strict periodic shadowing property in general. It also shows that even if f n has the strict periodic shadowing property for some n > 0, f does not necessarily have the same property. Moreover, by the properties (2), (4), (5), and Corollary 1.2, we obtain the following corollary. Corollary 1.3. Let (X, d) be a Cantor space. Then, there exists f ∈ H(X) such that f 3 is topologically stable, but f is not topologically stable.
It seems to be a natural attempt to seek a kind of shadowing property which is equivalent to the topological stability, and such a shadowing property may be expected to have the property that f has the property iff f n has the property for some n > 0, as the standard shadowing property does. However, Corollary 1. 3 shows that such an attempt should fail in the Cantor spaces. We remark that when X = S 1 , it is known that f ∈ H(S 1 ) is topologically stable iff f is topologically conjugate to a Morse-Smale diffeomorphism [23], so it holds that f is topologically stable iff f n is so for some n > 0 iff f n is so for all n > 0.
As a complement to Theorem 1.2, we prove the following statement.
Proposition 1.1. Let (X, d) be a compact metric space and let f ∈ H(X). If dim X = 0 and X = P er(f ), then f is equicontinuous and satisfies the strict periodic shadowing property.
Note that in Proposition 1.1, not only dense but every point of X is assumed to be a periodic point for f . In fact, this proposition implies that if a homeomorphism f ∈ H(X) of a Cantor space (X, d) has the properties (2) and (4) above, then X must contain a compact f -invariant subset S ⊂ X such that f | S is topologically conjugate to an odometer (see Lemma 2.2 in Section 2). By Proposition 1.1 and Theorem 1.2, we obtain the following corollary. Corollary 1.4. Let (X, d) be a compact metric space. If dim X = 0, then the identity map id X : X → X is topologically stable.
This corollary shows that when the dimension of the space is zero, topologically stable homeomorphisms may exhibit a very non-hyperbolic behavior. Indeed, there is a conjecture that if a homeomorphism of a closed topological manifold is topologically stable, then its restriction to the non-wandering set is expansive (see [3,Remark 2.4.12]). This conjecture strictly fails when the space is totally disconnected.
The subsequent theorems are general results on the topological stability of the homeomorphisms of Cantor spaces. Here, let us remark that the conjugating map h ∈ C(X) which will appear in the proof of Theorem 1.2 in Section 4 is degenerate in the sense that its image is a finite set, so possibly far from the whole space. When X is a closed topological manifold, for sufficiently small > 0, h ∈ C(X) with d C 0 (h, id X ) < is surjective (see [3,Remark 2.4.6]). Note that if we put an additional assumption that the map h is surjective in the definition of the topological stability, then for example, the identity map id C of the Cantor ternary set C is not topologically stable. It is worth mentioning that such a phenomenon was already observed by Walters in [22]. In [22], it was proved that for any compact metric space (X, d) and f ∈ H(X), if f is expansive and has the shadowing property, then f is topologically stable, and also if > 0 is sufficiently small, the conjugating map h ∈ C(X) with d C 0 (h, id X ) < must be unique. Hence, for example, the full shift σ on two symbols is topologically stable, but there is a periodic homeomorphism g, i.e., g m = id for some m > 0, which is arbitrary close to σ, so for such g, the image of the unique map h near id with h • g = σ • h must be a finite set. Taking into account these observations, we make the following definition.
, f is said to be topologically stable in the strong sense (or s-topologically stable) if for any > 0, there is δ > 0 such that for every In this definition, the full shift on two symbols and the identity map of the Cantor ternary set are not s-topologically stable. Then, it would be natural to ask even whether there exists an s-topologically stable homeomorphism of a Cantor space or not. The following theorem states that such a homeomorphism should have quite simple dynamics as the Morse-Smale diffeomorphisms.
is topologically stable in the strong sense, then f has the shadowing property, and Ω(f ) is a finite set.
In the proof of Theorem 1.3, we use the fact that when (X, d) is a Cantor space, there exists g ∈ H(X) such that its conjugacy class is residual in (H(X), D), i.e., containing dense G δ set (see [2,11]). As a simple application of Theorem 1.1, we prove here that such g is not topologically stable, so the following theorem holds.
Proof. By Lemma 1.1 and Theorem 1.1, it is sufficient to prove that P er(g) = ∅. Assume the contrary, i.e., there are p ∈ X and m > 0 such that g m (p) = p. We take a homeomorphism F ∈ H(X) such that P er(F ) = ∅ (for instance, a homeomorphism which is topologically conjugate to an odometer). Then, since n and p n = φ n (p). Then, we have D(g m n , F m ) → 0 as n → ∞, and g m n (p n ) = p n for all n ≥ 1. By the compactness of X, there is an increasing sequence of integers 1 ≤ n 1 < n 2 < · · · such that lim k→∞ d(p n k , p) = 0 for some p ∈ X. From , p n k ) = 0, but this contradicts that P er(F ) = ∅.
This paper consists of six sections. Some preliminaries are given in Section 2. In Section 3, we prove Lemma 1.1, Theorem 1.1, and Corollary 1.1. We prove Theorem 1.2 and Proposition 1.1 in Section 4. In Section 5, the example of a homeomorphism of a Cantor space with the five properties listed above is given. Finally, we prove Theorem 1.3 in Section 6.
2. Preliminaries. In this section, we give some notations, definitions, and results used in this paper. As mentioned in Section 1, we prove the following lemma for the sake of completeness. Proof. It is obvious from the definition that for any > 0, D(f, g) < implies d C 0 (f, g) < for all f, g ∈ H(X). Conversely, let us prove that for any given f ∈ H(X) and > 0, there is δ > 0 such that : n ≥ 0} is minimal, and non-wandering if for every neighborhood U of x, we have f n (U ) ∩ U = ∅ for some n > 0. We denote by P er(f ), RR(f ), M (f ), and Ω(f ) the sets of periodic, regularly recurrent, minimal, and non-wandering points for f , respectively. As mentioned in Section 1, we also denote by CR(f ) the set of chain recurrent points for f . Note that

2.2.
Continuous shadowing property. Let (X, d) be a compact metric space and let X Z be the set of all bi-infinite sequences of points in X. We define a metric d on X Z bỹ which is compatible with the product topology. For f ∈ H(X) and δ > 0, let P (f, δ) denote the subset of X Z consisting of all δ-pseudo orbits of f . Then, for f ∈ H(X), we say that f has the continuous shadowing property if for any > 0, there are δ > 0 and a continuous map r : It is easy to see that, given two compact metric spaces (X i , d i ), i = 1, 2, if f 1 ∈ H(X 1 ) and f 2 ∈ H(X 2 ) are topologically conjugate, then f 1 has the continuous shadowing property iff f 2 has the same property.

TOPOLOGICAL STABILITY AND ZERO-DIMENSIONAL DYNAMICAL SYSTEMS 2749
2.3. Equicontinuity. Given f ∈ H(X), we say that f is equicontinuous at x ∈ X if for any > 0, there is δ > 0 such that d(x, y) ≤ δ implies sup i∈Z d(f i (x), f i (y)) ≤ for all y ∈ X. Then, f is said to be equicontinuous if for any > 0, there is δ > 0 such that d(x, y) ≤ δ implies sup i∈Z d(f i (x), f i (y)) ≤ for all x, y ∈ X. By the compactness of X, we easily see that f is equicontinuous iff f is equicontinuous at every x ∈ X. Every equicontinuous homeomorphism f ∈ H(X) is known to satisfy X = M (f ) and so X = CR(f ). As for the relation with the shadowing property, we have the following lemma.
Lemma 2.2. Let (X, d) be a compact metric space and let f ∈ H(X) be an equicontinuous homeomorphism. Then, the following holds.
(1) f has the shadowing property if and only if dim X = 0.
(2) If dim X = 0, then X = RR(f ), and for any x ∈ X, x ∈ P er(f ), is topologically conjugate to an odometer.
Proof. For (1), see [17,Theorem 4]. Then, we shall prove (2) as follows. In fact, Lemma 4.2 in Section 4 implies that if f ∈ H(X) is equicontinuous, and dim X = 0, then X = RR(f ). Note that, as On the other hand, due to [6, Corollary 2.5], we know that for any continuous selfmap g ∈ C(Y ) of a compact metric space Y , if g is minimal, and Y = RR(g), then Y is a periodic orbit of g, or g is topologically conjugate to an odometer. Thus, by , we obtain (2).
The set X m has the subspace topology induced by the product topology on k≥1 X(k), and the resulting dynamical system (X m , g) is called an odometer with the periodic structure m. It is immediate from the definition that g : X m → X m is an equicontinuous homeomorphism, and in fact, the odometers are characterized as the minimal equicontinuous systems on Cantor spaces (see [13] [1], and a proof is outlined there. However, we shall give a rather simple proof of the result below. Proof of Lemma 1.1. It is sufficient to prove the following claim: Given > 0 and an integer n > 0, suppose that two n-tuples ζ = (x 1 , x 2 , . . . , x n ), η = (y 1 , y 2 , . . . , y n ) ∈ X n are both proper and d n (ζ, η) < /3. Then, there exists φ ∈ H(X) such that D(φ, id X ) < and φ (n) (ζ) = η.
First, we consider the case where x i = y j for all 1 ≤ i, j ≤ n. Since dim X = 0, we can choose 2n pairwise disjoint clopen subsets X 1 , X 2 , . . . , X n , Y 1 , Y 2 , . . . , Y n of X such that x i ∈ X i , y i ∈ Y i , diam X i < /3, and diam Y i < /3 for every 1 ≤ i ≤ n. We put Z = X \ n i=1 (X i ∪ Y i ). Note that all non-empty clopen subsets of a Cantor space are Cantor spaces, so homeomorphic, and any Cantor space is homogeneous, i.e., for any two points, there is a homeomorphism of the space which maps one point to the other point. Hence, for each 1 ≤ i ≤ n, we can take a homeomorphism It is obvious that φ ∈ H(X), φ 2 = id X , and φ (n) (ζ) = η. For all x ∈ X i , 1 ≤ i ≤ n, we have Similarly, we have d(φ(x), x) < for all x ∈ Y i , 1 ≤ i ≤ n, so it holds that d C 0 (φ, id X ) = D(φ, id X ) < . This proves the above claim. Next, we deal with the general case without the assumption above. By d n (ζ, η) < /3, we can choose d n (ζ, η) < ρ < /3 and δ > 0 so that ρ + 2δ < /3. Since (X, d) is perfect, we can take an n-tuple θ = (z 1 , z 2 , . . . , z n ) ∈ X n satisfying the following properties.
Then, we give a proof of Theorem 1.1. As mentioned in Section 1, it was essentially proved by Walters in [22] that, under the assumption of Theorem 1.1, topologically stable homeomorphisms satisfy the shadowing property, so we may omit its proof. In the following proof, we prove that those homeomorphisms satisfy the strict periodic shadowing property by a modification of Walters' argument.
Suppose that (x i ) m i=0 is a δ/3-cycle of f , and let us prove that there is p ∈ X satisfying f m (p) = p and d(f i (p), x i ) ≤ for all 0 ≤ i ≤ m. Put x = x 0 = x m and ζ = (x 0 , x 1 , . . . , x m ) ∈ X m+1 . Then, since (X, d) is perfect, we can take an (m + 1)-tuple η = (z 0 , z 1 , . . . , z m ) ∈ X m+1 with the following properties.

TOPOLOGICAL STABILITY AND ZERO-DIMENSIONAL DYNAMICAL SYSTEMS 2751
Then, by (4) and (5), we have (3) and (6), there exists φ ∈ H(X) such that D(φ, id X ) < γ, and φ(y i ) = z i for every 1 ≤ i ≤ m. Put g = φ • f ∈ H(X) and note that we have (1) and (2), there is h ∈ C(X) such that Put p = h(z 0 ). Then, since z 0 = z m by (5), we have Moreover, for every 0 ≤ i ≤ m, we have and so (5) yields which finishes the proof.
To prove Corollary 1.1, we need the following simple lemma. Proof. It follows from Lemma 2.2 (1) that f satisfies the shadowing property. Hence, for any given > 0, we can take γ > 0 and δ > 0 with the following properties.
Define r : P (f, δ) → X by r(x) = x 0 for x = (x i ) i∈Z ∈ P (f, δ). Then, it is obvious that r is continuous, and moreover, for each x = (x i ) i∈Z ∈ P (f, δ), by (1) and (2), we have This shows that r satisfies the required property.
As the final proof of this section, we prove Corollary 1.1.
Proof of Corollary 1.1. Take an odometer g : X m → X m . Since X m is a Cantor space, so dim X m = 0, and g is equicontinuous, by Lemma 3.1, g has the continuous shadowing property. By Lemma 1.1 and Theorem 1.1, if g is topologically stable, we should have CR(g) = P er(g), but this is not the case because P er(g) = ∅. Hence, g is not topologically stable. Now, since X m and X are both Cantor spaces, there is a homeomorphism φ : X m → X. Put f = φ • g • φ −1 . Then, f has the same properties as g.

4.
Proof of Theorem 1.2 and Proposition 1.1. In this section, we prove Theorem 1.2 and Proposition 1.1. We begin with the following lemma which seems to be a 'folklore'. Here, for f ∈ C(X), we say that f has the pseudo periodic shadowing property if for any > 0, there is δ > 0 such that for every δ-cycle Note that p is not required to be a periodic point for f .
It is obvious from the definition that if f ∈ H(X) has the strict periodic shadowing property, then it has the pseudo periodic shadowing property. For any equicontinuous f ∈ H(X), we have X = M (f ) = CR(f ), so if it has the strict periodic shadowing property, then from Lemma 4.1 and Lemma 2.2 (1), it follows that dim X = 0, a part of the conclusion in Theorem 1.2. In this way, dim X = 0 is a relatively easy consequence of the equicontinuity and the strict periodic shadowing property of f . We remark here that dim X = 0 is a vital property in the proof of Theorem 1.2. However, for the above reason, even if we put it as an assumption of Theorem 1.2, the substance of Theorem 1.2 would not be lost. This is why we give only a simple reasoning of Lemma 4.1 below.
Proof of Lemma 4.1. We know that if f has the shadowing property, then so does f | CR(f ) . Its statement can be found in [3,Theorem 3.4.2] or [17, Lemma 1], and we could use the arguments given in there to deduce the same conclusion from the pseudo periodic shadowing property. For the converse, we could exploit the arguments given in the proof of [3, Theorem 3.1.6] (which states that CR(f | CR(f ) ) = CR(f )) to show that f has the pseudo periodic shadowing property.
Our proof of Theorem 1.2 and Proposition 1.1 relies on a sequence of partitions of X with respect to some equivalence relations defined by chains. We briefly describe it below. It should be remarked that the following is a generalization of the argument first given in [1,Exercise 8.22] and used in several papers so far. A more detailed description is given in [10].
Suppose that f ∈ C(X) satisfies X = CR(f ). For δ > 0, we define a relation ∼ δ on X as follows: Given x, y ∈ X, x ∼ δ y iff there are two δ-chains (x i ) k i=0 and (y i ) l i=0 of f such that x 0 = y l = x and x k = y 0 = y. It is obvious from the definition that ∼ δ is symmetric and transitive, and by X = CR(f ), we have x ∼ δ x for every x ∈ X. Hence, ∼ δ is an equivalence relation on X. Each equivalence class with respect to ∼ δ is called a δ-chain component. By X = CR(f ), we can show that x ∼ δ f (x) for every x ∈ X, and x ∼ δ y for all x, y ∈ X with d(x, y) < δ; therefore, every δ-chain component C is clopen and f -invariant, i.e., f (C) ⊂ C. Then, X is decomposed into finitely many δ-chain components, and such a decomposition is called a δ-chain decomposition.
Fix a δ-chain component C. Note that for any δ-cycle c = (x i ) n i=0 of f , if x i ∈ C for some 0 ≤ i ≤ n, then x i ∈ C for all 0 ≤ i ≤ n. In such a case, we write c ⊂ C. Set l(c) = n for every δ-cycle c = (x i ) n i=0 of f . Define N = {n ∈ N : ∃ δ-cycle c of f with c ⊂ C and l(c) = n} and put m = gcd N = max{j ∈ N : j|n for every n ∈ N }.
Then, we define a relation ∼ δ,m on C as follows: Given x, y ∈ C, x ∼ δ,m y iff there is a δ-chain (x i ) k i=0 of f with x 0 = x, x k = y and m|k. By the definition of m, we easily see that ∼ δ,m is an equivalence relation on C, and by X = CR(f ), we have x ∼ δ,m y for all x, y ∈ C with d(x, y) < δ. Hence, every equivalence class D with respect to ∼ δ,m is clopen in X as C is so. Take any p ∈ C and consider m points p, f (p), . . . , f m−1 (p). Then, it is easy to see that C = Now, let C i , 1 ≤ i ≤ K, be all the δ-chain components and for each 1 ≤ i ≤ K, a δ-cyclic decomposition of X. We denote by r(δ) the mesh of D(δ): Note that every equicontinuous f ∈ H(X) satisfies X = M (f ) = CR(f ), so we may consider the δ-cyclic decomposition of X for every δ > 0. In the above notation, we prove the following lemma. Proof. Let us assume lim sup δ→0 r(δ) > r > 0 and deduce a contradiction. Since f is equicontinuous and dim X = 0, by Lemma 2.2 (1), f satisfies the shadowing property. We take 0 < < r/2 and γ > 0 with the following properties.
(2) Every γ-pseudo orbit of f is /2-shadowed by some point of X. By the assumption, we can choose 0 < δ < γ so that r(δ) > r. Then, there is a component D i,j ∈ D(δ) such that diam D i,j > r. Take x, y ∈ D i,j with d(x, y) > r and any p ∈ D i,j . Then, by the property (D3) above, there is an integer L > 0 and two δ-chains (x i ) L i=0 and (y i ) L i=0 of f such that x 0 = y 0 = p, x L = x, and y L = y. By (2), (x i ) L i=0 and (y i ) L i=0 are /2-shadowed by some z and w, respectively. Then, we have This contradicts (1) and finishes the proof. Now, we prove Theorem 1.2.
Proof of Theorem 1.2. If f satisfies the strict periodic shadowing property, then it is obvious from the definition that f has the pseudo periodic shadowing property. Note that we have X = M (f ) = CR(f ) because f is equicontinuous. Hence, dim X = 0 is implied by Lemma 4.1 and Lemma 2.2 (1). Given any > 0, take 0 < δ < so small as in the definition of the strict periodic shadowing property. Then, by Lemma 4.2, there is γ > 0 such that Then, for each 1 ≤ i ≤ K, by the property (D2), is a δ-cycle of f , and so by the choice of δ, -shadowed by some Now, take β > 0 so small that d(D, D ) > β for any distinct elements D, D ∈ D(γ), and suppose that g ∈ H(X) satisfies D(f, g) < β. Then, for any x ∈ D i,j with 1 ≤ i ≤ K and 0 ≤ j ≤ m i − 1, since f (x) ∈ D i,j+1 by the property (D2) and d(f (x), g(x)) < β, by the choice of β, we have g(x) ∈ D i,j+1 . This implies that for any 1 ≤ i ≤ K and 0 ≤ j ≤ m i − 1. Then, for any x ∈ D i,j with 1 ≤ i ≤ K and 0 ≤ j ≤ m i − 1, we have and since g(x) ∈ D i,j+1 , h(g(x)) = f j+1 (p i ) (note that this holds true for j = m i −1 because of f mi (p i ) = p i ). Thus, h•g = f •h. Since g ∈ H(X) with D(f, g) < β is arbitrary, f is topologically stable.
Finally, we prove Proposition 1.1. A definition is needed before the proof. For any f ∈ H(X), we say that f is distal if inf i∈Z d(f i (x), f i (y)) > 0 whenever x, y ∈ X are distinct. If f is equicontinuous, then f is distal, but the converse does not hold in general. However, due to [5, Corollary 1.9], we know that the converse also holds when dim X = 0.
3. f has the property (4). Let us prove that f does not satisfy the strict periodic shadowing property. We shall assume the contrary and deduce a contradiction. Given > 0, take δ > 0 so small as in the definition of the property. Take a sufficiently large k ≥ 1 and x l ∈ J k,l ∩ X for each l ∈ {0, 1, . . . , 2 k − 1}. Then, (x 0 , x 1 , . . . , x 2 k −1 , x 0 ) is a δ-cycle of f , so there should be p ∈ X such that f 2 k (p) = p and d(x l , f l (p)) ≤ for each l ∈ {0, 1, . . . , 2 k − 1}. However, such a periodic point p has the least period 2 α for some α ≥ 0. This contradicts that f have only the periodic points whose least periods are in the form of 3 · 2 β for some β ≥ 0.
4. f has the property (5). Finally, we shall prove that F = f 3 satisfies the strict periodic shadowing property. Given > 0, take k ≥ 2 so large that the diameter of any element of {J k,l : l ∈ {0, 1, . . . , 2 k − 1}} is ≤ . Since F is equicontinuous, and dim X = 0, by Lemma 2.2 (1), F satisfies the shadowing property. By the equicontinuity of F , there is δ > 0 such that every δ-pseudo orbit (z i ) i≥0 of F is -shadowed by z 0 itself. If δ > 0 is taken to be sufficiently small, it also holds that, putting Suppose that (x i ) m i=0 is a δ-cycle of F and let us prove that there is p ∈ X such that F m (p) = p and d(x i , F i (p)) ≤ for all 0 ≤ i ≤ m. Note that Without loss of generality, we may assume that x 0 ∈ J k,0 or x 0 ∈ D j 0 for some 1 ≤ j ≤ k − 1. In the former case, by the choice of δ, we have for every 0 ≤ i ≤ m, and so 2 k |m because x m = x 0 ∈ J k,0 . Put p = p k 0 ∈ D k 0 and note that F 2 k (p) = f 3·2 k (p) = p, then F m (p) = p. Since D k 0 ⊂ J k,0 , we have F i (p) ∈ J k,3i (mod 2 k ) for every 0 ≤ i ≤ m. Therefore, by the choice of k, d(x i , F i (p)) ≤ for all 0 ≤ i ≤ m. In the latter case, note that f (D j l ) = D j l+1 (mod 3 · 2 j ) for every 0 ≤ l ≤ 3 · 2 j − 1. By the choice of δ, we have x i ∈ D j 3i (mod 3 · 2 j ) for every 0 ≤ i ≤ m, and so 2 j |m because x m = x 0 ∈ D j 0 . Put p = x 0 ∈ D j 0 and note that F 2 j (p) = f 3·2 j (p) = p, then F m (p) = p. Again by the choice of δ, it also holds that d(x i , F i (p)) = d(x i , F i (x 0 )) ≤ for all 0 ≤ i ≤ m, and this finishes the proof.
6. Proof of Theorem 1.3. In this section, we prove Theorem 1.3. For the proof, we need the fact that when (X, d) is a Cantor space, there exists g ∈ H(X) such that its conjugacy class {φ • g • φ −1 : φ ∈ H(X)} is residual in (H(X), D), i.e., containing dense G δ set, and topological entropy of such g (denoted by h top (g)) is zero (see [2,8,11]). We also use the fact that when (X, d) is a Cantor space, the following set T A(X) = {F ∈ H(X) : F is expansive and has the shadowing property} is dense in (H(X), D) (see [12,20]). For any F ∈ T A(X), we have CR(F ) = Ω(F ), and Ω(F ) admits the spectral decomposition, that is, where Ω i is clopen in Ω(F ) and F -invariant, and F | Ωi is transitive for each 1 ≤ i ≤ k. Then, each Ω i , 1 ≤ i ≤ k, is known to admit a decomposition where D i,j is clopen in Ω(F ), F (D i,j ) = D i,j+1 (mod m i ), and F mi | Di,j is topologically mixing for every 0 ≤ j ≤ m i − 1 (see [3,Theorem 3.1.11]).
Proof of Theorem 1.3. If f is s-topologically stable, then it is obvious from the definition that f is topologically stable, so by Lemma 1.1 and Theorem 1.1, f has the shadowing property. Fix > 0 and take δ > 0 so small as in the definition of the s-topological stability. Then, by the above fact, there is g ∈ H(X) such that D(f, g) < δ and h top (g) = 0. Given such g, since there is a surjective h ∈ C(X) with h • g = f • h, we have h top (f ) = 0. Since f has the shadowing property, and h top (f ) = 0, by [17,Corollary 6], f | Ω(f ) is equicontinuous and dim Ω(f ) = 0. By Lemma 2.2 (2), for each x ∈ Ω(f ), x ∈ P er(f ) or f | O f (x) is conjugate to an odometer. Then, by the other fact above, there is F ∈ T A(X) such that D(f, F ) < δ, and so H • F = f • H for some surjective H ∈ C(X). Let Ω(F ) = k i=1 Ω i be the spectral decomposition as above and put B i = H(Ω i ) for each 1 ≤ i ≤ k. For any fixed 1 ≤ i ≤ k, B i is f -invariant, and f | Bi is transitive, so there is x i ∈ B i such that B i = ω(x i , f ). By this, B i is a periodic orbit of f , or f | Bi is conjugate to an odometer. Let Ω i = mi−1 j=0 D i,j be the decomposition as above and put C i = H(D i,0 ) (⊂ B i ). Then, C i is f mi -invariant, and f mi | Ci is topologically mixing. It is clear that this is not possible when f | Bi is conjugate to an odometer, therefore B i is a periodic orbit of f . It remains to prove that Ω(f ) = k i=1 B i . Assume that x ∈ Ω(f ) \ k i=1 B i . Then, since H is surjective, there is y ∈ X such that H(y) = x. Then, there is 1 ≤ i ≤ k such that