Finite expansive homeomorphisms

Topology and its Applications 253 (2019) 95–112

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Topology and its Applications www.elsevier.com/locate/topol

Finite expansive homeomorphisms Ali Barzanouni Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Iran

a r t i c l e

i n f o

Article history: Received 27 December 2017 Received in revised form 14 November 2018 Accepted 22 November 2018 Available online 5 December 2018 MSC: 54H20 37B20 Keywords: Metric finite expansive Orbit finite expansive and topologically finite expansive

a b s t r a c t We introduce the notions of metric finite expansive homeomorphism, orbit finite expansive homeomorphism, and topologically finite expansive homeomorphism on compact metric spaces, topological spaces, and uniform spaces, respectively. These notions coincide on compact metric spaces, but they are not the same on general metric spaces. We give some examples to show that the metric finite expansivity, orbit finite expansivity, and topological finite expansivity are weaker than the metric n-expansivity, orbit expansivity and topological expansivity, respectively. We state suitable conditions to imply that the metric finite expansivity is equal to the metric expansivity. We show that any regular recurrent point of a metric finite expansive homeomorphism f : X → X is a periodic point and if X is an uncountable compact metric space, then Ω(f ) is an infinite set. It is known that if there is an orbit expansive homeomorphism on X, then X is a T1 -space, in spite of it, we give an orbit finite expansive homeomorphism f : X → X such that X is not a T1 -space. Then we show that if f : X → X is an orbit finite expansive homeomorphism with the orbit finite expansive covering {Ui }n i=1 , then X − Ui is an infinite set for all 1 ≤ i ≤ n. Finally we show that if f : X → X is a homeomorphism on a compact Hausdorff space X, then f has a weak finite generator if and only if f is a topologically finite expansive homeomorphism. © 2018 Elsevier B.V. All rights reserved.

1. Introduction A discrete invertible expansive system is a dynamical system such that every point of the underlying space has a distinctive behavior. A homeomorphism f from a compact metric space X onto X is expansive if there exists c > 0 (called the expansivity constant of f ), such that if x, y ∈ X and d(f n (x), f n (y)) ≤ c for every n ∈ Z, then x = y. Thus, if x = y, then d(f n (x), f n (y)) > c for some n. The study of expansive homeomorphisms has started in 1950 when Utz defined this systems with the name of the unstable homeomorphism [19]. In Utz’s paper, the examples on compact spaces were sub-dynamics of shift maps, thus we can say that the theory of expansive homeomorphisms started with the symbolic dynamics but it quickly developed by itself. The definition of expansiveness has shown that small variations on the definition are also interesting.

E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2018.11.018 0166-8641/© 2018 Elsevier B.V. All rights reserved.

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The first of such variations appeared in 1952, when Schwartzman [18] considered what now is called the positive expansiveness. His definition requires that different points are separated in the positive time. In 1970, Reddy [16] introduced the pointwise expansiveness, a variation that does not require the existence of a uniform expansive constant but each point has a positive one. He proved that even on a compact space, pointwise expansiveness does not imply expansiveness. In 1993, Kato [9] defined the continuumwise expansiveness by requiring that if a continuum has a small diameter for all the time, then it is a singleton. This definition seems to be based on the techniques developed for expansive homeomorphisms; it was designed in order to be able of extending important results. In 2012, Morales [14] defined another variation called an n-expansiveness. Now a set of points whose orbits are close for all the time, has the cardinality smaller than n. Also he introduced the countable expansive homeomorphism which means the set of points whose orbits are close for all the time being countable. Artigue in [4] gave an example of the countable expansive homeomorphism f : (X, d) → (X, d) and x ∈ X such that Γδ (x, f ) is an infinite set for arbitrarily small value of δ. Hence not only f is not an n-expansive homeomorphism, for all n ∈ N, also the dynamical ball Γδ (x, f ) is not finite. It motivates us to study another variation of the expansivity between the n-expansive homeomorphism and the countable expansive homeomorphism that we call it the finite expansive homeomorphism. Indeed, a homeomorphism is a metric finite expansive whenever dynamic balls have a finite number of elements. However it seems that the concept of the metric finite expansive homeomorphism has not been defined and studied before this research. There are some important homeomorphisms which are metric finite expansive, but not n-expansive for all n ∈ N. However many theorems concerning the expansive homeomorphism will be generalized to the case of the metric finite expansive homeomorphism. Another goal of the present paper is to investigate the topological definitions of the finite expansivity which can extend the known result of the metric finite expansivity to the topological setting. We introduce two of such definitions. In order to explain the first one, let us consider a metric finite expansive homeomorphism f as above. Since X is compact, there exists a finite open cover U = {U1 , U2 , . . . , Un }, such that for every infinite set A ⊆ X, there is m ∈ Z with f m (A)  Ui for all i = 1, 2, . . . , n. We use this property of the finite expansivity to extend the definition for nonmetric topological spaces. If the above covering U = {U1 , . . . , Ul } exists, then we will say that f is orbit finite expansive. Let us explain the second definition. Consider again a metric finite expansive homeomorphism f with a finite expansive constant c. If Ac = d−1 ([0, c]), then for every x ∈ X, the set {y|(f n (y), f n (x)) ∈ Ac , ∀n ∈ Z} is finite. This motivates us to define an A-finite expansive homeomorphism f : X → X when (X, U) is a uniform space and A ∈ U. If there is A ∈ U such that for every x ∈ X, {y|(f n (y), f n (x)) ∈ A, ∀n ∈ Z} is a finite set, then we say that f is a topologically finite expansive homeomorphism. The paper is organized as follows. In Section 2, we introduce the notion of metric finite expansive homeomorphism. This section consists of three subsections. In Subsection 2.1, we study the existence of finite expansive homeomorphisms on zero-dimensional spaces. In [12], it is shown that there is a dense subset R of the space of homeomorphisms on the Cantor set C, such that every f ∈ R is an expansive homeomorphism, hence the set of finite expansive homeomorphisms is dense in the space of homeomorphism on C. In Example 2.2, we give a zero-dimensional space such that it does not admit any finite expansive homeomorphism. In Theorem 2.3, we prove that a countable compactum X admits a finite expansive homeomorphism if and only if it admits an expansive homeomorphism. It is known that there is no continuum-wise expansive homeomorphism on one-dimensional compact manifolds; hence compact manifolds with dimension one, do not support a finite expansive homeomorphism. In Subsection 2.2, we study the existence of finite expansive homeomorphism on surface spaces. A continuum C ⊆ S is stable with diameter  > 0, if diam(f n (C)) ≤  for all n ≥ 0. It is known that every stable continuum of any expansive homeomorphisms on surfaces is locally connected. But Artigue in [4] showed that the genus two surface admits a continuum-wise expansive home-

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omorphism with a fixed point whose stable continuum is not locally connected. He also proved that for an n-expansive surface homeomorphism, every stable and unstable continuum is locally connected, we extend this result to finite expansive homeomorphisms. Indeed, in Proposition 2.5, we show that every stable and unstable continuum of a finite expansive surface homeomorphism are locally connected. In Corollary 2.6, we prove that for the surface homeomorphism without biasymptotic sector, the notion of finite expansive surface homeomorphism coincides with the notion of expansive surface homeomorphism, we recall that a disc bounded by the union of a stable arc and an unstable arc is called a biasymptotic sector. In Subsection 2.3, we recall the notions of positively expansive map and positively continuum-wise expansive. Although, there is no injective continuous positively expansive map on an infinite compact metric space, but there are many infinite compact metric spaces X which admit the positively expansive local homeomorphism on X, for example all shifts of finite type and all the maps z → z n , n = 0, ±1, of the unit circle are the positively expansive local homeomorphism. In Proposition 2.7, we show that the notions of positively finite expansive local homeomorphism on S 1 and positively expansive local homeomorphism on S 1 coincide. In Section 3, we give some properties of the metric finite expansive homeomorphisms. Examples 3.1 and 3.2 show that a finite expansive homeomorphism is not an n-expansive homeomorphism, in general. This section consists of two subsections. The purpose of Subsection 3.1 is to describe some of the results of an investigation of the lifting, projection, and iteration of finite expansive homeomorphisms. In Theorem 3.5, we prove that for homeomorphisms f, g on compact metric spaces X, Y , respectively, if there is a covering map ϕ of the compact metric space X onto Y with ϕ ◦ f = g ◦ ϕ, then f is a metric finite expansive homeomorphism if and only if g is too. Also, in Theorem 3.6, we prove that f : X → X is a metric finite expansive homeomorphism on the compact metric space X if and only if f m is a metric finite expansive homeomorphism for all integer m = 0. Note that compactness is essential in Theorems 3.6 and 3.5, for example, if f is the homeomorphism constructed in [5], then f m is a metric finite expansive homeomorphism if and only if m = ±1. In Subsection 3.2, we first recall the notions of periodic point, recurrent point, regular recurrent point, and nonwandering point. We study the relation between them when the system is a finite expansive homeomorphism. In Proposition 3.7, we show that every regular recurrent point is periodic if the system is metric finite expansive, hence we can imply that for metric finite expansive homeomorphisms with the shadowing property, the set of periodic points is dense in chain recurrent points; see Theorem 3.8. We also in Theorem 3.10, show that for every metric finite expansive homeomorphism on an uncountable compact metric space, the set of nonwandering points is an infinite set. In Section 4, we introduce the notion of orbit finite expansive homeomorphism. This notion is weaker than of the orbit expansive homeomorphism that introduced in [1]; see Example 4.3. In [1, Proposition 2.5], it is shown that if X admits an orbit expansive homeomorphism, then X is T1 . But in Example 4.3, we show that a space which is not a T1 -space may admit an orbit finite expansive homeomorphism. In Proposition 4.2, we show that if f : X → X is an orbit finite expansive homeomorphism with the orbit finite expansive covering U = {Ui }ni=1 , then X − Ui is infinite for all 1 ≤ i ≤ n. Hence if X is infinite with the cofinite topology, then no homeomorphism of X is orbit finite expansive. We also show that an orbit finite expansivity is preserved under iteration; see Theorem 4.5. In Theorem 4.6, we show that every homeomorphism on a compact metric space is orbit finite expansive if and only if it is metric finite expansive. In Section 5, we introduce the notion of topologically finite expansivity that is weaker than the metric finite expansivity on metric spaces; see Example 5.2. This notion is preserved under iteration. We also define the notions weak finite generator and finite generator. We prove that they coincide with each other on the paracompact Hausdorff spaces; see Theorem 5.8. Moreover, in Theorem 5.7, we show that for every compact Hausdorff space X, the system (X, f ) has a weak finite generator if and only if it is a topologically finite expansive homeomorphism.

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2. Variations of expansiveness Let f : (X, d) → (X, d) be a homeomorphism. Fix x ∈ X and c > 0. We define the dynamical c-ball at x, Γc (x, f ) = {y|d(f n (x), f n (y)) ≤ c ∀n ∈ Z}

(2.1)

Let β be the Borel σ-algebra on X. Denote by M(X) the set of all Borel probability measures on X endowed with weak ∗ topology. We say that μ ∈ M(X) is atomic if there is a point x ∈ X such that μ({x}) > 0. Let M∗ (X) = {μ ∈ M(X) : μ is non-atomic}, and let M∗f (X) = {μ ∈ M∗ (X) : μ is f -invariant}. We say that • f is an expansive homeomorphism, if there is c > 0 such that Γc (x, f ) = {x} for all x ∈ X. • For n ∈ N, the homeomorphism f is n-expansive, if there is c > 0 such that Γc (x, f ) has at most n elements for all x ∈ X. • f is a countable expansive homeomorphism, if there is c > 0 such that Γc (x, f ) is a countable set for all x ∈ X. • f is a measure-expansive homeomorphism, if there is c > 0 such that μ(Γc (x, f )) = 0 for all x ∈ X and all μ ∈ M∗f (X). • f is a continuum-wise expansive homeomorphism, if there is c > 0 such that if A ⊆ X is a continuum (compact connected) and diam(f n (A)) ≤ c for all n ∈ Z, then A is a singleton. • f is a pointwise expansive homeomorphism, if for every x ∈ X, there is c(x) > 0 such that Γc(x) (x, f ) = {x}. In this paper we introduce the notion of finite expansivity. Definition 2.1. Let (X, d) be a metric space. A homeomorphism f : X → X is called metric finite expansive with the expansive constant c > 0 whenever Γc (x, f ) is a finite set for every x ∈ X. In [3], it is shown that the countable expansivity is equivalent to the measure expansivity, hence we can say the following: expansivity ⇒ n-expansivity ⇒ finite expansivity ⇒ countable expansivity ⇒ continuum-wise expansivity

(2.2)

The converse does not hold in general. • In [2, Theorem 5.3.1], it is shown that the 2-expansivity does not imply the expansivity. This implies that n-expansiveness  expansiveness • In Examples 3.1 and 3.2, we show that finite expansiveness  n-expansiveness • Artigue in [4] gave an example to show that the genus two surface admits a continuum-wise expansive homeomorphism with a fixed point whose local stable set is not locally connected. One can check that this example is countable expansive but there are some points with |Γδ (x, f )| = ∞ for arbitrarily small value of δ. This implies that countable expansiveness  finite expansiveness. • On totally disconnected spaces, as the Cantor set, every homeomorphism f is continuum-wise expansive but it may be happen that f is not countable expansive (consider for example, the identity on the Cantor set). This implies that continuum-wise expansiveness  countable expansiveness. • Finite expansiveness implies pointwise expansiveness, but in general pointwise expansiveness does not imply finite expansiveness. For instance Artigue in [2, Example 3.4.10] presented an example of a pointwise expansive homeomorphism that is not continuum-wise expansive; hence it is not finite expansive.

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Note that if f is pointwise expansive on a compact metric space X and the function c : X → R is continuous, then f is finite expansive, since X is compact. One natural question is what kind of spaces can admit finite expansive homeomorphisms. In subsection 2.1, we study this question for zero-dimensional spaces. It is known that every one-dimensional compact manifold is homeomorphic either to the circle S 1 or the interval [0, 1]. Also we know that these spaces do not admit a continuum-wise expansive homeomorphism, hence they do not support any finite expansive homeomorphisms. In subsection 2.2, we study above question for surface spaces. In subsection 2.3, we show that the notions of positively finite local homeomorphism on S 1 and positively expansive local homeomorphism on S 1 coincide. 2.1. Zero-dimensional space and metric finite expansivity In the following, we give a zero-dimensional space having no finite expansive homeomorphism. First, we note that if f is a finite expansive homeomorphism on a compact metric space (X, d), then F ix(f ) is finite because if F ix(f ) is an infinite set, then there is an accumulation point x in F ix(f ), and so Γc (x, f ) is infinite for all c > 0. Example 2.2 (See [12]). Let C ⊆ [0, 1] be the Cantor set and let S = {0} ∪ { n1 : n ∈ N}. Let Xn = (C ⊕ S n )/{0, 0n } be the quotient space obtained by identifying {0, 0n } to a point xn , where 0 ∈ C and  0n = (0, 0, . . . , 0) ∈ S n , for every n ∈ N, and let Xo = {x0 } be a one-point space. Let X = {Xn : n ∈ N ∪ {0}}. We give X a topology as follows. Let B(x) = {U : U is a neighborhood of x in Xn } for every  x ∈ X and n ∈ N and B(x0 ) = { {Xi : j ≤ i} ∪ X0 : j ∈ N}. The space X with the topology generated by {B(x) : x ∈ X} is compact, metrizable, and zero-dimensional. The point xn is the only point that has arbitrarily small neighborhoods containing a set homeomorphic to the Cantor set, a set homeomorphic to S n , and no set homeomorphic to S n+1 . Therefore if f : X → X is a homeomorphism, then f (xn ) = xn for all n ∈ N. This implies that f has an infinite number of fixed points. Hence f is not a finite expansive homeomorphism. Kimora in [12] proved that the set of all expansive homeomorphisms with shadowing property of the Cantor set C is dense in H(C); in particular there is a dense subset R ⊆ H(C) such that every f ∈ R is a finite expansive homeomorphism. A point p of X is an accumulation point of the set X if p ∈ X − {p}, otherwise p is an isolated point. The collection of accumulation points of X is said to be the derived set of X, denoted by X d . The derived set of  X of order α is defined by the conditions; X (0) = X, X (1) = X d , X (α+1) = (X (α) )d , and X (λ) = α<λ X (α) if λ is a limit ordinal number. We denote the derived degree of X by d(X) and write d(X) = α if X (α) = ∅ but X (α+1) = ∅. It is well known that a compactum X is a countable set if and only if d(X) exists and it is a countable ordinal number. Also, it is clear that if d(X) = α, then X (α) is a finite set. We refer the readers to the book by Kuratowski [13, p.261] for more details. Here, a natural question to ask is that what kinds of countable compacta admit finite expansive homeomorphisms. In [10, Theorem 2.2] it is proved that if X is a countable compactum with d(X) = α, then X admits an expansive homeomorphism if and only if α is not a limit ordinal number. The proof of [10, Theorem 2.2] can be applied to show the following theorem. Theorem 2.3. Let X be a countable compactum with d(X) = α. Then the following are equivalent: 1. α is not a limit ordinal number; 2. X admits an expansive homeomorphism;

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3. X admits a finite expansive homeomorphism. 2.2. Surfaces and finite expansive homeomorphism A continuum C ⊂ X is stable with diameter  > 0 if diam(f k (C)) ≤  for all k ≥ 0. If f : S → S is a continuum-wise expansive homeomorphism on a compact surface S, then there is δ > 0 such that for all x ∈ S there is a stable continuum C containing x with the diameter δ and also limn→∞ diam(f n (C)) = 0. Recently, Artigue in [4], showed that the genus two surface admits a continuum-wise expansive homeomorphism with a fixed point whose stable continuum is not locally connected. Definition 2.4. [4] A surface homeomorphism f : S → S is a regular continuum-wise expansive homeomorphism, if it is continuum-wise expansive and stable and unstable continua are locally connected. Artigue in [2, Theorem 5.1.2.] showed that every n-expansive surface homeomorphism is a regular continuum-wise expansive homeomorphism. The proof of this theorem can be applied to show the following proposition. Proposition 2.5. Every finite expansive surface homeomorphism is a regular continuum-wise expansive homeomorphism. A disc bounded by the union of a stable arc and an unstable arc is called a biasymptotic sector. Theorem 5.1.8. in [2], shows that if f is a regular continuum-wise expansive homeomorphism without biasymptotic sector, then there is c > 0 such that Wcs (x) ∩ Wcu (x) = {x} for all x ∈ X, hence by Proposition 2.5 we can obtain the following corollary. Corollary 2.6. If f : S → S is a homeomorphism of a compact surface without biasymptotic sector, then the following statements are equivalent: • The homeomorphism f is expansive ; • The homeomorphism f is finite expansive. 2.3. Existence of positive finite expansivity on a space It is known [7] that if a compact metric space admits a continuous and one-to-one positively expansive map, then the space has only a finite number of points. Recall that f : X → X is positively expansive if there is c > 0 such that if x = y, then d(f n (x), f n (y)) > c for some n ≥ 0. Therefore if X is an infinite compact metric space, then, for every injective continuous map f : X → X and for all  > 0, there is C = {x, y : x = y} ⊆ X such that diam(f n (C)) <  for all n ≥ 0. It is clear that C ⊆ X is not a continuum set. But there are many infinite compact spaces like X, which admit a homeomorphism f : X → X such that for every nontrivial continuum C ⊆ X, there is n ≥ 0 such that diamf n (C) > c. In this case f is called a positively continuum-wise expansive homeomorphism. Although there is no injective continuous positively expansive map on an infinite compact metric space but there are many infinite compact metric spaces like X which admit a positively expansive local homeomorphism on X, for example all shifts of finite type and all the maps z → z n , n = 0, ±1, of the unit circle are positively expansive local homeomorphisms. In the following, we show that the notions of positively finite expansive local homeomorphisms on S 1 and positively expansive local homeomorphisms on S 1 coincide. We say that f is a positively finite expansive map, if there is c > 0, such that for every infinite set A ⊆ X, there is n ≥ 0 such that diamf n (A) ≥ c. Before we proceed our goal, consider x, y ∈ S 1 in such a way that x = eis , y = eit , s < t, and t − s < 2π. Let arc(x, y) be the image of [s, t] ⊂ R under the map e : R → S 1 , e(x) = eix . But e is a covering, so that

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f : S 1 → S 1 lifts to a map f˜ : R → R such that e ◦ f˜ = f ◦ e. This lift is not unique, but if f˜i are any two lifts, we have f˜2 (x) = f˜1 (x) + 2kπ for some k ∈ Z. We say that f is increasing on arc(x, y) if ˜f is increasing on [s, t]. We can check that the following are equivalent: (1) The homeomorphism f is local. (2) The map ˜f is a local homeomorphism. (3) The map ˜f is a homeomorphism. (4) The map ˜f is strictly monotonic. (5) The map ˜f is locally strictly monotonic. (6) The map ˜f is strictly monotonic on each interval [a, b] of length b − a < 2π (this is the “correct” interpretation of f being strictly monotonic on each arc in S 1 ). ˜ (7) There exists δ˜ > 0 such that ˜f is strictly monotonic on each interval [a, b] of length b − a < δ. Proposition 2.7. Let f : S 1 → S 1 be a local homeomorphism of the circle S 1 . Then f is a positively finite expansive local homeomorphism if and only if f is a positively expansive local homeomorphism. Proof. Since f is a local homeomorphism, for every x ∈ S 1 , there is an arc Ix such that f is increasing (or decreasing) on Ix . Let δ > 0 be the Lebesgue number for the cover {Ix : x ∈ S 1 }. We can say that for x, y ∈ S 1 , if diam arc(x, y) < δ, then f is increasing (or decreasing) on arc(x, y). We claim that if f : S 1 → S 1 is a positively finite expansive local homeomorphism with positively finite expansive constant δ, then it is a positively expansive local homeomorphism, with the positively expansive constant δ. If it is not true, then there is x = y such that d(f n (x), f n (y)) < δ for all n ≥ 0. Take z ∈ arc(x, y), then d(f (z), f (x)) ≤ d(f (y), f (x)) < δ, because diamf (arc(x, y)) = diam arc(f (x), f (y)) and f (z) ∈ f (arc(x, y)). By induction, we have d(f n (z), f n (x)) < c, this implies that arc(x, y) ⊆ Γ+ c (x, f ) that is a contradiction + with the finiteness of Γc (x, f ). 2 3. Metric finite expansive homeomorphism Let M (X) denote the set of all metrics on X. The metric d ∈ M (X) is called finite discrete on A ⊆ X, if there is c > 0 such that Bd [x, c] ∩ A is a finite set for all x ∈ A, where Bd [x, c] = {y : d(y, x) ≤ c}. Similarly, d is called n-discrete on A ⊆ X, if there is c > 0 such that Bd [x, c] ∩ A has at most n-elements for all x ∈ A. If Lf : M (X) → M (X) is defined by Lf (d) = supi∈Z d(f i , f i ), then it is easy to see the following: • The homeomorphism f is metric finite expansive if and only if Lf (d) is finite discrete. • The homeomorphism f is an n-expansive if and only if Lf (d) is n-discrete. In the following, we show that there is a metric ρ ∈ M (X) such that f : (X, ρ) → (X, ρ) is a metric finite expansive homeomorphism but it is not an n-expansive homeomorphism for all n ∈ N. Example 3.1. Let f : X → X be a bijection map with an infinite orbit. Choose an infinite sequence {xi }∞ / O(xj , f ) for all i = j. Take B1 = {x1 }, B2 = {x2 , x3 }, B3 = {x4 , x5 , x6 }, . . .. It i=1 ⊆ X such that xi ∈ i is clear that f (Bj ) ∩ Bk = ∅ for j = k; thus if Bji := f i (Bj ), then Bji ∩ Bnk = ∅. Define ρ : X × X → [0, ∞) by  ρ(x, y) =

1 4+|i|+|j| ,

δ(x, y),

{x, y} ⊆ Bji and x = y, otherwise,

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where δ(x, x) = 0 and for x = y, δ(x, y) = 1. It is easy to see that if ρ(x, y) < 1 and x ∈ Bi , then y ∈ Bi , also ρ(x, y) < 1 for {x, y} ⊆ Bi . This implies that ρ is finite discrete but it is not n-discrete for all n ∈ N. For every x, y ∈ X if x, y ∈ Bji , then f n (x), f n (y) ∈ Bji+n . Thus ρ(f n (x), f n (y)) ≤ ρ(x, y) ≤

1 . 4 + |i| + |j|

(3.1)

It implies that Lf (ρ) = ρ. Since ρ is finite discrete, f is metric finite expansive. Moreover, ρ is not n-discrete and hence it is not n- expansive. In [6], Carvalho and Cordeiro gave an example of a nonexpansive being a 2-expansive homeomorphism. Briefly, they started with a homeomorphism f : (X, d) → (X, d) which is expansive and has the shadowing property and a dense set of periodic orbits {On }n∈N . Then they added some copies of these periodic orbits and obtained a new compact metric space, (Y = X ∪ {On }, d ). Essentially, the new metric satisfies d (On , On ) = n1 . Also, the new map g : Y → Y is equal to f on X and it is periodic in each On . This implies that for any  > 0, if n1 < , then the dynamical ball over On has two orbits. Thus it is not expansive but it is 2-expansive. In the following, we use the technique of this example to show that there is a finite expansive homeomorphism on a compact space that is not an n-expansive homeomorphism for all n ∈ N. Example 3.2. Let g : M → M be an expansive homeomorphism on a compact metric space (M, d0 ) with an infinite number of periodic points. Let {pk }k∈N ⊆ P er(g) be an infinite sequence such that Og (pi ) ∩Og (pj ) = ∅. Take 

Q=

{1, . . . , k} × {k} × {0, 1, . . . , π(pk ) − 1}

k∈N

such that π(pk ) is the period of pk . Let E be an infinite countable set such that q : Q → E is a bijection. Thus, any point x ∈ E has the form x = q(i, k, j) for some (i, k, j) ∈ Q. Let X = M ∪ E, and define a function d : X × X → R+ by ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ d 0 (x, y), ⎪ ⎪ ⎨ 1 + d (g j (p ), y), 0 k d(x, y) = k1 j ⎪ + d (g (p 0 k ), x), ⎪ k ⎪ ⎪ 1 ⎪ , ⎪ ⎪ ⎩ k1 1 j r k + m + d0 (g (pk ), g (pm ))),

x = y, x, y ∈ M , x = q(i, j, k), y ∈ M , y = q(i, j, k), x ∈ M , x = q(i, j, k), y = q(l, k, j), l = i, x = q(i, j, k), y = q(i, m, r) and k = m or j = r.

Then similarly as in Proposition 3.1 in [6], one can show that (X, d) is a compact metric space. Define a map f : X → X by  f (x) =

g(x), q(i, k, j + 1 mod pk ),

x ∈ M, x = q(i, k, j).

A similar procedure as in Proposition 3.2 in [6] follows to prove that f is a homeomorphism. For every δ > 0 choose k ∈ N such that k1 < δ and note that for every i ∈ {1, . . . , k}, the point q(i, k, 0) belongs to Γf (pk , k1 ). This implies that Γf (pk , δ) contains at least k + 1 different points; hence f is not n-expansive for all n ∈ N. In the following, we show that f is 2c -finite expansive if g is a c-expansive homeomorphism. If this is not true, then there is x ∈ X such that Γf (x, 2c ) is infinite. This implies that there is (i, k, j) ∈ Q such that Γc (q(i, k, j) ∩ E is infinite. Choose q(l, m, r) ∈ Γc (q(i, k, j) ∩ E such that m = k, hence for every s ∈ Z, we have

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d0 (g s (g r (pm )), g s (g j (pk ))) = d(f s (q(l, m, r)), f s (q(i, k, j))) −

103

1 1 − < c. m k

Since c > 0 is the constant of expansivity for g, we can say that g r (pm ) = g j (pk ), that contradicts Og (pm ) ∩ Og (pk ) = ∅. 3.1. Projecting, lifting, and iteration of finite expansive homeomorphisms Definition 3.3. Let X and Y be topological spaces, a surjective map ϕ : X → Y is called a covering map, if there is an open cover {Uα } of Y such that for every α, the space ϕ−1 (Uα ) is a disjoint union of open sets in X, each of which is mapped by ϕ homeomorphically onto Uα . The map R → S 1 given by t → eit is a covering, indeed a cover {Uα } can be taken to consist of any two open arcs whose union is S 1 . It is clear that every covering map is a local homeomorphism but the converse is not true, for example, restrict the exponential map R → S 1 to some intervals like (0, 1.5), so that the fibers have different cardinalities. It is known that any local homeomorphism from a compact space to a connected Hausdorff space is a covering map. One can see that if X is a compact space and ϕ : X → Y is a covering map, then the map g : Y → N defined by g(x) = Cardϕ−1 (x) is continuous. Thus if Y is compact, then the covering map ϕ : X → Y is at most a k-to-one map for some k ∈ N. Lemma 3.4. Let (X, d) be a compact metric space. If ϕ : X → Y is a covering map, then there is β > 0 such that d(x, y) > β, whenever ϕ(x) = ϕ(y) and x = y. Proof. Suppose that there is a sequence of pairs {xn , yn } ⊆ X such that ϕ(xn ) = ϕ(yn ) and d(xn , yn ) < n1 . Let {xn }, {yn } converge to the common limit point z. Thus, there is no neighborhood of z such that ϕ is one-to-one on it. This is a contradiction because ϕ is a covering map. 2 Theorem 3.5. Suppose that f : (X, d1 ) → (X, d1 ) and g : (Y, d2 ) → (Y, d2 ) are homeomorphisms on the compact metric spaces (X, d1 ) and (Y, d2 ), respectively. If ϕ is a covering map of the compact X onto Y with ϕ ◦ f = g ◦ ϕ, then f is a finite expansive homeomorphism if and only if g is. Proof. Assume g has a finite expansive constant c > 0. If f is not finite expansive, then corresponding to each  > 0, there is x ∈ X such that Γ (x , f ) is an infinite set. Since ϕ is a covering map, there is β > 0 satisfying in Lemma 3.4. Let 0 <  < β2 . For every x, y ∈ Γ (x , f ) since d1 (x, y) < β, we have ϕ(x) = ϕ(y). Pick a finite open covering {Ui } of X such that diam(ϕ(Ui )) < 2c for all i = 1, . . . , n. Suppose that 0 > 0 is a Lebesgue number of the covering {Ui }; then for every n ∈ Z and every y ∈ Γ (x , f ) (0 <  < min{0 , β2 }), we have f n (y ), f n (x ) ∈ Ui . Thus d2 (ϕ(f n (y ), ϕ(f n (x )))) < 2c . Since ϕ ◦ f n = g n ◦ ϕ, we obtain d(g n (ϕ(x )), g n (ϕ(y ))) < 2c ∀n ∈ Z. This implies that Γ 2c (ϕ(x ), g) is infinite that is a contradiction with the finite expansivity of g. Conversely let the finite expansive constant of f be c. By Lemma 3.4, there is η > 0 such that d1 (x, y) > η whenever ϕ(x) = ϕ(y). Since Y is compact and ϕ is a covering map, there is an integer k such that ϕ is at most k-to-1. Corresponding to each sufficiently small open neighborhood Vy of y, there is a collection of open sets {Uy,i }ki=1 such that ϕ−1 (Vy ) = ∪ki=1 Uy,i and ϕ|Uy,i is a homeomorphism of Uy,i onto Vy . Let the diameter of the sets Uy,i be so small that Uy,i , f (Uy,i ), and f −1 (Uy,i ) have diameters less than min{c, η2 } for all y and i. Choose a finite covering {Vj }rj=1 from the collection {Vy }y∈Y , and let the corresponding U ’s be called {Uji }, j = 1, . . . , r and i = 1, . . . , k. Let β > 0 be a Lebesgue number for each of the coverings {Vj }, {gVj } and {g −1 Vj }. If g is not a finite expansive homeomorphism, then corresponding to each  > 0, there is an infinite set A ⊆ Y such that diam(g n (A )) <  for all n ∈ Z. Let  < β. Then to each n, there is

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j = j(n) such that g n (A ) ⊆ Vj(n) . Hence for a given n, corresponding to each i (i = 1, 2, . . . , k) there is a unique infinite set in Uj(n),i whose image under ϕ is g n (A ). Pick i, and choose an infinite set B ⊆ Uj(n),i with ϕ(B ) = A . Since f has the finite expansive constant c, there is m = 0, the least in absolute value, such that f m (B ) fails to belong to the same element of the covering {Uj,i }. We assume that m is positive. There is {s, t} such that f m−1 (B ) ⊆ Us,t . It follows that f m (B ) ⊆ f (Us,t ) and diam(f m (B )) < η2 . Choose p, q ∈ B such that f m (p) and f m (q) fail to belong to the same element of the covering {Uj,i } but {f m−1 (p), f m−1 (q)} ⊆ Us,t . It follows that {f m (p), f m (q)} ⊆ f (Us,t ) and d1 (f m (p), f m (q)) < η2 . Let ϕ(p) = a and ϕ(q) = b. Then {g m (a), g m (b)} ⊆ g m (A ) and d2 (g m (a), g m (b)) < β. It follows that there is an index u such that {g m (a), g m (b)} ⊆ Vu , 1 ≤ u ≤ r. Choose 1 ≤ i ≤ k such that f m (p) ∈ Uu,i , and denote ϕ−1 ϕf m (q) = ϕ−1 g m (b) in Uu,v by t. Hence d1 (f m (p), t) < η2 . Since f m (p) and f m (q) fail to belong to the same element of the covering {Uu,i }, it follows that t = f m (q). Hence d1 (t, f m (q)) ≤ d1 (t, f m (p)) + d1 (f m (p), f m (q)) < η. That is a contradiction, because by Lemma 3.4, the equality ϕ(t) d1 (t, f m (q)) > η. 2

=

ϕ(f m (q)) implies that

Note that the compactness condition is essential in Theorem 3.5. For example, suppose that f : R → R is defined by f (x) = x +1, d1 (x, y) = |x −y|, and d2 (x, y) = |ex −ey |. It is easy to see that f : (R, d2 ) → (R, d2 ) is a finite expansive homeomorphism, but f : (R, d1 ) → (R, d1 ) is not a finite expansive homeomorphism. Theorem 3.6. Suppose that f : (X, d) → (X, d) is a homeomorphism on the compact metric space (X, d). Then f is metric finite expansive if and only if f m is metric finite expansive for all m ∈ Z − {0}. Proof. It is clear that Γc (x, f ) ⊆ Γc (x, f m ) for all x ∈ X and m ∈ Z − {0}. Thus, if f m is metric finite expansive for some m ∈ Z − {0}, then f is metric finite expansive. Conversely let m ∈ Z −{0} be given. Since f is metric finite expansive, there is c > 0 such that B Lf (d) [x, c] is a finite set for all x ∈ X, where B Lf (d) [x, c] = {y : Lf (d)(x, y) ≤ c}. Since f is uniform continuous, there is δ > 0 such that if d(x, y) < δ, then d(f i (x), f i (y)) < c for −m ≤ i ≤ m. Then B Lf m (d) [x, δ] ⊆ B Lf (d) [x, c] for all x ∈ X. Since Lf (d) is finite discrete, Lf m (d) is finite discrete, therefore f m is metric finite expansive. 2 We emphasize that the compactness condition is essential in Theorem 3.6, for example, if f is the homeomorphism constructed in [5], then f m is metric finite expansive if and only if m = ±1. 3.2. Periodic points, regular recurrent points, and chain recurrent points of finite expansive homeomorphism We recall the following concepts. • A point x ∈ X is nonwandering, x ∈ Ω(f ), if for every neighborhood U of x, there is a positive integer  k, such that f k (U ) U = ∅. • A point x ∈ X is a regular recurrent point, x ∈ RR(f ), if for every neighborhood U of x, there is k ∈ N such that f kn (x) ∈ U for all n ∈ Z. • A point x ∈ X is a recurrent point, x ∈ R(f ), if for every neighborhood U of x, there is k ∈ Nsuch that f k (x) ∈ U . • A point x ∈ X is a chain recurrent, x ∈ CR(f ), if for every δ > 0 there is a finite sequence {xi }ki=0 satisfying x0 = x = xn and d(f (xi ), xi+1 ) < δ for i = 0, . . . , k − 1.

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It is easy to see that P er(f ) ⊆ RR(f ) ⊆ Ω(f ) ⊆ CR(f ).

(3.2)

We know that the set of all fixed points for a finite expansive homeomorphism is finite. Hence by Theorem 3.5, if f is metric finite expansive, then P er(f ) is countable. Lemma 3.7. If f : X → X is a metric finite expansive homeomorphism on a compact metric space (X, d), then P er(f ) = RR(f ). Proof. It is known that if x ∈ RR(f ), then Of (x) is a minimal set. Since f is a metric finite expansive homeomorphism with the finite expansive constant c > 0, it is continuum-wise expansive and we can apply [9, Theorem 5.2] to conclude that Of (x) is totally disconnected. It implies that for c > 0, there is  > 0 such that diam([z] ) ≤ c for all z ∈ O(f ), where [z] denotes the set of all y ∈ X that there is {zn }kn=0 such that z0 = z, zk = y, and d(zn , zn+1 ) <  for all n = 1, 2, . . . , k − 1. Since x ∈ RR(f ), there is k ∈ N such that kZ ⊆ N (x, B 2 (x))for  > 0. Hence if B = {f kn (x) : n ∈ Z}, then B ⊆ [x] . It is easy to see that f n (B) ⊆ [f n (x)] for all n ∈ Z. Thus, diamf n (B) < c, where diamf n (B) = sup{d(x, y) : x, y ∈ f n (B)}. Since f is the metric finite expansive homeomorphism, we have B is finite, hence x ∈ P er(f ). 2 A sequence {xn }n∈Z is called a δ-pseudo orbit of f whenever d(f (xn ), xn+1 ) < δ for all n ∈ Z. A system f : X → X has the shadowing property if for every  > 0 there is δ > 0 such that if {xn }n∈Z is a δ-pseudo orbit, then there is x ∈ X such that d(f n (x), xn ) <  for all n ∈ Z. In the following, we show that the converse of the inclusions given in (3.2) hold if f is a metric finite expansive homeomorphism with the shadowing property. Theorem 3.8. If f : X → X is a metric finite expansive homeomorphism and it has shadowing property, then P er(f ) = CR(f ). Proof. It is known that if f has shadowing property, then RR(f ) = CR(f ). Hence by Lemma 3.7, we have P er(f ) = CR(f ). 2 Following [15], we say that z is a point with converging semi-orbits under a bijective map f : X → X, if both αf (z) and ωf (z) reduce to singleton, where ωf (z) = {y : f nk (z) → y for some subsequence {nk } ⊆ N, with nk → ∞} and αf (z) = ωf −1 (z). Denote by A(f ) the set of all points with converging semi-orbits under f . Theorem 1.26 in [15] stated that if f : X → X is an expansive homeomorphism, then A(f ) is countable. Similar to the proof of Theorem 1.26 in [15], one can prove the following. Lemma 3.9. The set of all points with converging semi-orbits under a finite expansive homeomorphism of a compact metric space is countable. If f is a homeomorphism of [0, 1] onto itself, every point has converging semi-orbits under f , this implies that there exists no finite expansive homeomorphism on an arc. By the heteroclinic point of a bijective map f : X → X on a metric space X, we mean any point for which both the alpha and the omega-limit sets reduce to periodic orbits. Denote by Het(f ) the set of all heteroclinic points of f . By Lemma 1.30 in [15], we have

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Het(f ) ⊆



A(f n ).

(3.3)

n∈N

Theorem 3.10. Let f : X → X be a finite expansive homeomorphism on an uncountable compact metric space. Then Ω(f ) is infinite. Proof. If the assertion is not true, then Ω(f ) is finite. This implies that every point for such homeomorphism is heteroclinic that is a contradiction, because by the Relation (3.3) and Lemma 3.9, if f is a finite expansive homeomorphism, then the set of heteroclinic points of f is countable. 2 Corollary 3.11. If X is an uncountable compact metric space and f : X → X is a homeomorphism with finite non-wandering set, then f is not a finite expansive homeomorphism. 4. Orbit finite expansive homeomorphism Let (X, d) be a compact metric space. Since X is compact, there are x1 , x2 , . . . , xn ∈ X such that n X = i=1 B(xi , 2c ) for c > 0. It is easy to see that if f : X → X is a c-finite expansive homeomorphism, then for every infinite set A ⊆ X, there is n ∈ Z such that f n (A)  B(xi , 2c ) for all i = 1, 2, . . . , n. We use this property of the finite expansivity to extend the definition for nonmetric topological spaces. Let (X, τ ) be a topological space, and assume that f : X → X is a homeomorphism. For a set A ⊆ X and a cover C of X, we write A ≺ C if there exists C ∈ C such that A ⊆ C. If A is a family of subsets of X, then A ≺ C means that A ≺ C for all A ∈ A. Definition 4.1. We say that f : (X, τ ) → (X, τ ) is an orbit finite expansive homeomorphism, when there is a finite open cover U ⊆ τ of X such that for every A ⊆ X if f n (A) ≺ U for all n ∈ Z, then A is a finite set. In this case we call U an orbit finite expansive covering. Proposition 4.2. Let f : X → X be an orbit finite expansive homeomorphism with an orbit finite expansive covering U = {Ui }ni=1 . Then X − Ui is an infinite set for all 1 ≤ i ≤ n. Proof. If it is not true, then there is 1 ≤ k ≤ n such that X − Uk is a finite set. If all of the points of X are periodic, then Uk ∩ P er(f ) is infinite. Since X − Uk is finite, there is an infinite set A ⊆ Uk such that if x ∈ A, then O(x, f ) ⊆ Uk . It implies that f n (A) ⊆ Uk , for every n ∈ Z, that is a contradiction with the orbit finite expansivity of f . Let x ∈ / P er(f ); then there is N ∈ N such that f i (x) ∈ / X − Uk for all |i| ≥ N . 2nN ∞ n Take A = {f (x)}n=1 , one can check that f (A) ≺ U for every n ∈ Z. That is a contradiction because U is an orbit finite expansive covering of f . 2 It is clear that if U is an orbit finite expansive covering of f and V ≺ U, then V is an orbit finite expansive covering of f , too. For every two covers U and V, define their join as U ∨ V = {U ∩ V |U ∈ U, V ∈ V}. Since U ∨ V ≺ U, if U is an orbit finite expansive covering, then U ∨ V is an orbit finite expansive covering of f for every open cover V of X. Also in this case, since f −n (U) is an open cover of X, one can say that −n ∨∞ (U) is an orbit finite expansive covering of f . n=1 f We recall from [1] that f is an orbit finite expansive homeomorphism, when there is a finite open cover U of X such that, for every x, y ∈ X, if {f n (x), f n (y)} ≺ U for all n ∈ Z, then x = y. In [1, Proposition 2.5], it is shown that if X admits an orbit expansive homeomorphism, then X is T1 . In the following, we show that a space which is not a T1 -space may admit orbit finite expansive homeomorphisms and also every orbit finite expansive homeomorphism is not an orbit expansive homeomorphism, in general. Example 4.3. Let f : (X, τ ) → (X, τ ) be an orbit finite expansive homeomorphism with orbit finite expansive covering U. Let a, b ∈ / X, and let τ0 = {U ∪ {a, b} : U ∈ τ } ∪ {∅}. Then (Y = X ∪ {a, b}), τ0 ) is a topological

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space that is not T1 -space. Let g : (Y, τ0 ) → (Y, τ0 ) be defined by g|X = f and g(x) = x if x ∈ {a, b}. One can check that g is an orbit finite expansive homeomorphism with the orbit finite expansive covering U0 = {U ∪ {a, b} : U ∈ U}. Also g : (Y, τ0 ) → (Y, τ0 ) is not an orbit expansive homeomorphism, because Y is not a T1 -space. Remark 4.4. Similar to the proof of Proposition 2.5 in [1], we can show that if f : X → X is an orbit finite expansive homeomorphism and P er(f ) = ∅, then X is a T1 -space. The following results extend the well known properties of the metric finite expansivity. Theorem 4.5. Let f be a homeomorphism, and let r = 0 be an integer. Then f is an orbit finite expansive homeomorphism if and only if f r is. Proof. It is clear that every orbit finite expansive covering of f r is an orbit finite expansive covering of f . Conversely, we prove that if f is an orbit finite expansive homeomorphism, then f 2 is an orbit finite expansive homeomorphism. Let U = {U1 , U2 , . . . , Un } be an orbit finite expansive covering of f , and let V = {Ui ∩ f (Uj )|i, j ∈ {1, 2, . . . , n}}. We claim that if f 2k (A) ≺ V for all k ∈ Z, then A is finite. The fact f 2k (A) ≺ V implies that f 2k+2 (A) ⊆ Ui ∩ f (Uj ) for some j, i ∈ {1, 2, . . . , n}. Thus we have f 2k+1 (A) ⊆ Ui , that is, f m (A) ≺ U for all m ∈ Z. Since U = {U1 , U2 , . . . , Un } is an orbit finite expansive covering of f , so A is finite. 2 If U = {U1 , U2 , . . . , Un } is an orbit finite expansive covering of f : X → X, then for Y ⊆ X with f (Y ) = Y , the open cover {U1 ∩ Y, U2 ∩ Y, . . . , Un ∩ Y } is an orbit finite expansive covering of f : Y → Y . Also U × U := {Ui × Uj : i, j ∈ {1, 2, . . . , n}} is an orbit finite expansive covering of f × f . Conversely, if V = {V1 , V2 , . . . , Vn } is an orbit finite expansive covering of f × f and πi (x1 , x2 ) = xi for i = 1, 2, then {πi (Uj )|(i, j) ∈ {1, 2} × {1, 2, . . . , n}} is an orbit expansive covering of f . Thus if f : X → X is an orbit finite expansive, Y ⊆ X, and f (Y ) = Y , then f : Y → Y and f × f : X × X → X × X are orbit finite expansive homeomorphisms. If f : (X, τ1 ) → (X, τ1 ) and g : (Y, τ2 ) → (Y, τ2 ) are homeomorphisms such that hof = goh for a homeomorphism h : X → Y , then we say that (X, f ) is conjugate with (Y, g) under the homeomorphism h : X → Y . If U is a finite open cover of X such that h(U) is an orbit finite expansive covering of g, then U is an orbit finite expansive covering of f . This shows that if a homeomorphism f : (X, τ1 ) → (X, τ1 ) is conjugate with a homeomorphism g : (Y, τ2 ) → (Y, τ2 ), then f is orbit finite expansive if and only if g is. Theorem 4.6. Let f be a homeomorphism on a compact metric space (X, d). Then f is an orbit finite expansive homeomorphism if and only if it is a finite expansive homeomorphism. Proof. If f is a c-finite expansive homeomorphism, then, for c > 0, there are x1 , x2 , . . . , xn ∈ X such that V = {B(xi , 2c ) : i = 1, . . . , n} is a finite open cover for X. If, for A ⊆ X, f n (A) ≺ V for all n ∈ Z, then A ⊆ Γc (a, f ) for some a ∈ A. Since f is finite expansive, A is finite. Conversely, let V be a finite expansive covering. If δ > 0 is a Lebesgue number for open cover V, then we show that for every 0 < c < 2δ and every x ∈ X, if A ⊆ Γc (x, f ), then A is finite. For all n ∈ Z, we have diamf n (A) < δ, thus there is U ∈ U such that f n (A) ⊆ U , that is, f n (A) ≺ U for all n ∈ Z. It implies that A is finite. 2 5. Topologically finite expansive homeomorphism In a metric space (X, d), the point x is -close to y if d(x, y) < , that is, x, y ∈ d−1 ([0, ]). So, we can say that x is close to y whenever (x, y) is in a given neighborhood of diagonal of X. This approach

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is useful for extending the dynamical properties from the compact setting to the noncompact setting. We make the standing assumption that all topological spaces are first countable, locally compact, paracompact, and Hausdorff and that all maps are homeomorphisms although some of the results apply more generally (every locally compact, Hausdorff space is Tychonoff, and so has a uniform structure). In the following, we give some information about the uniform spaces. Recall that a relation is a set of ordered pairs and if U is a relation, then U −1 = {(y, x) : (x, y) ∈ U }. If U = U −1 , then U is called symmetric. If U and V are relations, then the composition U oV is the set of all pairs (x, z) such that (x, y) ∈ V and (y, z) ∈ U for some y. The set of all pairs (x, x) for x in X is called the identity relation or the diagonal, which is denoted by X . For every subset A of X, the set U [A] is defined to be {y : (x, y) ∈ U f or some x ∈ A}, and if x is a point of X, then U [x] is U [{x}]. A uniformity for a set X is a nonempty family U of subsets of X × X such that 1. 2. 3. 4. 5.

If U ∈ U, then U −1 ∈ U. Each member of U contains the diagonal of X. If U ∈ U, then V oV ⊆ U for some V ∈ U. If U and V are elements of U, then U ∩ V ∈ U. If U ∈ U and U ⊆ V ⊆ X × X, then V ∈ U.

The pair (X, U) is called a uniform space. There may be many different uniformities for a set X, the largest of them is the family of all subsets of X × X which contain diagonal of X and the smallest one is the family whose only member is X × X If (X, U) is a uniform space, the topology τ of the uniformity U or the uniform topology is the set of all subsets A ⊆ X such that there is U ∈ U with U [x] ⊆ A for every x ∈ A. The interior of a subset A of X with respect to the uniform topology is the set of all x such that U [x] ⊆ A for some U ∈ U; see [11, Theorem 4, P. 178]. Hence U [x] is a neighborhood of x for every U ∈ U. Every member of U is a neighborhood of ΔX , but the converse is not true in general. For example, the usual uniformity for R is the family U of all subsets U of R × R such that {(x, y)||x − y| < r} ⊆ U, f or some r > 0. 1 Every member of U is a neighborhood of ΔR , but {(x, y)||x − y| < 1+|y| } is a neighborhood of ΔR which is not a member of U. If (X, U) is a compact uniform space, then every neighborhood of diagonal ΔX in X × X is a member of U; see [11, Theorem 29, P. 197].  Note that the closure of the set {x} is {U [x] : U ∈ U}, because every uniform space is always completely  regular, therefore the space is Hausdorff if and only if {U |U ∈ U} is the diagonal ΔX . By [11, Theorem 7, P.179], the closure of a subset M of X × X with respect to the uniform topology, is ∩{U oM oU : U ∈ U}. It implies that if U ∈ U and V is a member of U such that V oV oV ⊆ U , then V ⊆ U . Thus the family of closed members of a uniformity U is a basis of U. In 2007, the authors of [17] studied two topological generalizations of the positively expansivity. For a given dynamical system f : X → X, they used the product map F = f ×f on X ×X and the neighborhoods of the diagonal ΔX = {(x, x) : x ∈ X} to extend the notion of positive expansivity. In 2013, the authors of [8] introduced the topological expansivity. Indeed f : (X, U) → (X, U) is called a topologically expansive homeomorphism, if there is a closed member D ∈ U such that ΓD {x} = {x}, for all x ∈ X, where ΓD {x} = {y : (f n (y), f n (x)) ∈ D, ∀n ∈ Z}. If f : (X, d) → (X, d) is a finite expansive homeomorphism with constant finite expansive c and Ac = d−1 ([0, c]), then {y|(f n (y), f n (x)) ∈ Ac , ∀n ∈ Z} is finite set,

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for every x ∈ X. Therefore it motivates to define an A-finite expansivity of a homeomorphism f : X → X when (X, U) is a uniform space and A ∈ U. Definition 5.1. Let X be an uniform space. f : X → X is called a topologically finite expansive homeomorphism, if there is a closed neighborhood A of ΔX = {(x, x) : x ∈ X} such that for every x ∈ X, ΓA (x, f ) = {y|(f n (x), f n (y)) ∈ A ∀n ∈ Z} is a finite set. If f : X → X is an A-finite expansive homeomorphism, X ⊆ A, and Y ⊆ X is a subspace of X with f (Y ) = Y , then it is easy to see that f : Y → Y is a B-finite expansive homeomorphism where B = A ∩ X × X. Also for the topological spaces X, Y and the regular closed sets X ⊆ A and Y ⊆ B, if f : X → X is A-finite expansive and g : Y → Y is B-finitely expansive, then f × g : X × Y → X × Y is a C-finite expansive homeomorphism, where C = h−1 (A × B) in which h : (X × Y )2 → X 2 × Y 2 is defined by h((x1 , y1 ), (x2 , y2 )) = (x1 , x2 , y1 , y2 ). If c > 0 is a finite expansive constant, then it is easy to see that A = d−1 [0, c] is a finite expansive neighborhood. Thus the finite expansivity in the case of metric spaces, implies the finite expansivity in the case of topological space, but in the following, we show that the converse is not true. Example 5.2. Let (S 2 , d) be a metric space with d being the Euclidean metric inherited from R3 . Using the stereographic projection P : S 2 − {(0, 0, 1)} → R2 , we define the metric ρ on R2 to be given by ρ(P (x1 , y1 , z1 ), P (x2 , y2 , z2 )) = d((x1 , y1 , z1 ), (x2 , y2 , z2 )). Note that P −1 : R2 → S 2 − {(0, 0, 1)} is given by

(X, Y ) →

2Y X2 + Y 2 − 1 2X , 2 , 2 2 2 2 X +Y +1 X +Y +1 X +Y2+1

 . 

Now consider the linear function f : (R2 , ρ) → (R2 , ρ) determined by a matrix M = diag(eλ , eλ ) for some λ, λ = 0. In the following, we show that {(eμ , 0)| cosh μ < 1 +

2 } ⊆ Γ ((1, 0), f ). 2

Consider the points A0 = (eμ , 0) and B0 = (1, 0). Also, for simplicity, let An = f n (eμ , 0) = (enλ+μ , 0) and Bn = f n (1, 0) = (enλ , 0). We have P −1 (An ) =



 1 , 0, tanh(nλ + μ) , cosh(nλ + μ)

P −1 (Bn ) =



 1 , 0, tanh(nλ) . cosh(nλ)

It is easy to see that ρ(An , Bn ) = 2

2[cosh(μ) − 1] ≤ 2[cosh(μ) − 1]. cosh(nλ + μ) cosh(nλ) 2

It implies that if cosh μ < 1 + 2 , then ρ(f n (eμ , 0), f n (1, 0)) < . Thus for all μ ∈ R with cosh μ < 1 + 2 , we have (eμ , 0) ∈ Γ ((1, 0), f ), that is, f : (R2 , ρ) → (R2 , ρ) cannot be -finite expansive for any  > 0. The induced topology of ||.|| and ρ are the same on R2 , and it is easy to see that f : (R2 , ||.||) → (R2 , ||.||) is expansive. It implies that if U is the usual uniformity for R2 , that is, U is the family of all subsets U of R2 × R2 such that {(x, y) : ||x − y|| < r} ⊆ U for some r > 0, then f : (R2 , U) → (R2 , U) is topologically finite expansive, but f : (R2 , ρ) → (R2 , ρ) cannot be -finite expansive for any  > 0.

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Lemma 5.3. Let f : X → X be a homeomorphism on a compact Hausdorff space X. Then f is topologically finite expansive if and only if there is a closed neighborhood U of X such that x is an isolated point in ΓU (x, f ) for all x ∈ X. Proof. If f is a topologically finite expansive homeomorphism, then there is a closed neighborhood U of X such that ΓU (x, f ) is a finite set. Thus x is an isolated point in ΓU (x, f ). Conversely, letx is an isolated point in ΓU (x, f ) for all x ∈ X. Choose a closed neighborhood V of X such that V oV ⊆ U . We claim that ΓV (x, f ) is a finite set for all x ∈ X. By contradiction, let ΓV (y, f ) be infinite for some y ∈ X. Since X is compact, ΓV (y, f ) has an accumulation point x and x ∈ ΓV (y, f ), because ΓV (y, f ) is a closed set. Also we have ΓV (y, f ) ⊆ ΓV oV (x, f ). Thus x is not isolated in ΓU (x, f ), that is a contradiction. 2 Let (X, U1 ), (Y, U2 ) be compact Hausdorff uniform spaces, and let (X, f ) be conjugate with (Y, g) under a homeomorphism h : X → Y . Then for each closed neighborhood D of Y , there is a closed neighborhood U of X such that h × h(U ) ⊆ D. It implies that if (Y, g) is finite expansive, then (X, f ) is finite expansive. So on the compact Hausdorff uniform spaces, topologically finite expansive property is preserved under conjugacy. Thus we conclude the following. Theorem 5.4. Let (X, U1 ), (Y, U2 ) be compact Hausdorff uniform spaces, and let (X, f ) be conjugate with (Y, g) under a homeomorphism h : X → Y . Then for any closed neighborhood D of Y , there is a closed neighborhood U of X such that x is an isolated point in ΓU (x, f ) if and only if h(x) is an isolated point in ΓD (x, g). In Example 5.2, the finite expansive homeomorphism f : (R2 , d) → (R2 , d) is conjugate with the nonfinite expansive homeomorphism f : (R2 , ρ) → (R2 , ρ). It implies that compactness condition is essential in Theorem 5.4. Theorem 5.5. Let (X, U) be a uniform space. A homeomorphism f : X → X is topologically finite expansive if and only if f n is a topologically finite expansive homeomorphism for all n ∈ Z − {0}. Proof. Note that A is a finite expansive neighborhood for f if and only if it is a finite expansive neighborhood for f −1 . Thus we will assume that n is a positive integer. It is clear that if A is a finite expansive neighborhood for f n , then it is a finite expansive neighborhood for f . Conversely if A is a finite expansive neighborhood −i n for f , then ∩n−1 i=0 (f × f ) (A) is a finite expansive neighborhood for f . 2 Definition 5.6. Let X be a paracompact Hausdorff space, and let f be a homeomorphism on X. Then  • A locally finite open covering U of X is called finite generator for X if i∈Z f −i (Ui ) is finite for every bisequence {Ui }i∈Z of members of U.  • An open covering U of X is called weak finite generator for X if i∈Z f −i (Ui ) is finite for every bisequence {Ui }i∈Z of members of U. It is clear that if U is a finite generator, then U is a finite weak generator. Conversely the following theorem holds. Theorem 5.7. Let X be a paracompact Hausdorff space, and let f be a homeomorphism on X. Then if A is a finite weak generator, then A is a finite generator. Proof. Suppose that A is a finite weak generator for (X, f ). Since A is an open cover of X, for each x ∈ X, there is Ax ∈ A such that x ∈ Ax . Now the regularity of X guarantees the existence of an open set Gx such

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that x ∈ Gx ⊆ Gx ⊆ Ax . Obviously G = {Gx |x ∈ X} is an open cover of X. Since X is paracompact and Hausdorff, there exists a locally finite open refinement, say, V = {Vβ |β ∈ B}. We claim that V is a finite generator. Choose any bisequence {Vi }i∈Z of members of V. For each index i, there exist a Gi in G and an Ai in A such that Vi ⊆ Gi ⊆ Gi ⊆ Ai . Clearly {Ai }i∈Z is a bisequence of members of A, and as A is a weak    generator, so i∈Z f −i (Ai ) is finite. Thus the inclusion i∈Z f −i (Vi ) ⊆ f −i (Ai ) implies that i∈Z f −i (Vi ) is finite. 2 Theorem 5.8. Let X be a compact Hausdorff space, and let f be a homeomorphism on X. The following conditions are equivalent: 1. The homeomorphism f is topologically finite expansive. 2. There is an open set U ⊆ X × X of the diagonal such that if B ⊆ X × X is infinite, then there is n ∈ Z such that f n × f n (B)  U . 3. The homeomorphism f is orbit finite expansive.  4. There is a finite open covering U of X such that i∈Z f i (Ui ) is finite for every bisequence {Ui }i∈Z of members of U.  5. There is a finite open cover U of X, if i∈Z f i (Ui ) is finite for every bisequence {Ui }i∈Z of member of U. Proof. (1) ⇒ (2). Let A be the finite expansive constant, take U an open neighborhood of X such that U ⊆ A. By contrary, assume that there is an infinite set B ⊆ X × X such that f n × f n (B) ⊆ U . It implies that ΓA (x, f ) is infinite for every x ∈ B. This is a contradiction because A is a finite expansive constant for f . (2) ⇒ (1) Choose a closed neighborhood A of the diagonal such that AoA ⊆ U . If there is a point x ∈ X, such that ΓA (x, f ) is infinite, then for the infinite set B = ΓA (x, f ), there exist n ∈ Z and y, z ∈ ΓA (x, f ) / U . But the fact that y, z ∈ ΓA (x, f ) implies that (f n (y), f n (z)) ∈ AoA ⊆ U such that (f n (y), f n (z)) ∈ that is a contradiction. Thus ΓA (x, f ) is finite for every x ∈ X. (2) ⇒ (3) For every x ∈ X, choose a neighborhood Ux such that Ux × Ux ⊆ U . Since X is Hausdorff, X is a closed set in the compact space X; thus it is a compact set. Consider the finite cover {Uxi × Uxi }li=1 of X . Then taking Ui = Uxi for i = 1, 2, . . . , l, we obtain an orbit finite covering {U1 , U2 , . . . , Ul }. (3) ⇒ (2) Let U = {U1 , U2 , . . . , Un } be an orbit finite expansive covering of f . Take U = ∪ni=1 Ui × Ui . If there is an infinite set B ⊆ X × X such that f n × f n (B) ⊆ U for every n ∈ Z, then for infinite set πi (B), (i ∈ {1, 2}), we have f n (πi (B)) ≺ U, that is a contradiction.  (3) ⇔ (4) One can see that U is a finite expansive covering if and only if card( j∈Z f j (Ukj )) < ∞ for all (kj )j∈Z ∈ {1, 2, . . . , n}Z , where card stands for the cardinality of the set. (4) ⇔ (5) See Theorem 5.7. 2 References [1] M. Achigar, A. Artigue, I. Monteverde, Expansive homeomorphisms on non-Hausdorff spaces, Topol. Appl. 207 (2016) 109–122. [2] A. Artigue, Expansive Dynamical System, Doctoral Thesis, 2015. [3] A. Artigue, D. Carrasco-Olivera, A note on measure-expansive diffeomorphisms, J. Math. Anal. Appl. 428 (1) (2015) 713–716. [4] A. Artigue, Anomalous Cw-expansive surface homeomorphisms, Discrete Contin. Dyn. Syst. 36 (2016) 3511–3518. [5] B.F. Bryant, D.B. Coleman, Some expansive homeomorphisms of the reals, Am. Math. Mon. 73 (1966) 370–373. [6] B. Carvalho, W. Cordeiro, N -expansive homeomorphisms with the shadowing property, J. Differ. Equ. 261 (6) (2016) 3734–3755. [7] E.M. Coven, M. Keane, Every Compact Metric Space that Supports a Positively Expansive Homeomorphism is Finite, IMS Lecture Notes Monogr. Ser., Dynamics Stochastics, vol. 48, 2006, pp. 304–305. [8] T. Das, K. Lee, D. Richeson, J. Wiseman, Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topol. Appl. 160 (1) (2013) 149–158.

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