Asymptotic Properties of 𝐺-Expansive Homeomorphisms on a Metric 𝐺-Space

Abstract

We define and study the notions of positively and negatively 𝐺-asymptotic points for a homeomorphism on a metric 𝐺-space. We obtain necessary and sufficient conditions for two points to be positively/negatively 𝐺-asymptotic. Also, we show that the problem of studying 𝐺-expansive homeomorphisms on a bounded subset of a normed linear 𝐺-space is equivalent to the problem of studying linear 𝐺-expansive homeomorphisms on a bounded subset of another normed linear 𝐺-space.

1. Introduction

Expansiveness, introduced by Utz [1] in 1950 for homeomorphisms on metric spaces, is one of the important dynamical properties studied for dynamical systems. Expansive homeomorphisms have lots of applications in topological dynamics, ergodic theory, continuum theory, symbolic dynamics, and so forth. The notion of asymptotic points for a homeomorphism on a metric space was defined by Utz in [1]. On metric spaces, the existence of asymptotic points under expansive homeomorphisms is studied by Utz [1], Bryant [2, 3], Wine [4], Williams [5, 6], and others. In [7], authors have used this notion to classify all homeomorphisms of the circle without periodic points. Using the concept of generators, Bryant and Walters in [8] have obtained necessary and sufficient conditions for two points to be positively/negatively asymptotic under a homeomorphism on a compact metric space.

In [6], Williams has shown that the problem of studying expansive homeomorphisms on a bounded subset of a normed linear space is equivalent to the problem of studying linear expansive homeomorphisms on a bounded subset of another normed linear space. Using the above equivalence, Williams has obtained a necessary and sufficient condition for two points to be positively/negatively asymptotic under a homeomorphism on a bounded subset of a normed linear space. For study of expansive automorphisms on Banach spaces, one can refer to [9, 10].

With the intention of studying various dynamical properties of maps under the continuous action of a topological group, in [11], the notion of expansiveness termed as 𝐺-expansive homeomorphism is defined for a self-homeomorphism on a metric 𝐺-space. It is observed that the notion of expansiveness and the notion of 𝐺-expansiveness under a nontrivial action of 𝐺 are independent of each other. Conditions under which an expansive homeomorphism on a metric 𝐺-space is 𝐺-expansive and viceversa are also obtained. Recently Choi and Kim in [12] have used this concept to generalize topological decomposition theorem proved in [13] due to Aoki and Hiraide for compact metric 𝐺-spaces. Further, in [14], the notion of generator in 𝐺-spaces termed as 𝐺-generator is defined and a characterization of 𝐺-expansive homeomorphisms is obtained using 𝐺-generator. Some interesting consequences have been obtained regarding existence of 𝐺-expansive homeomorphisms. In [15, 16] we have studied some more properties of 𝐺-expansive homeomorphisms. For some other dynamical properties on 𝐺-spaces, one can refer to [17, 18]. In Section 2, we give the preliminaries required for remaining sections. In Section 3, we define the notion of positively/negatively 𝐺-asymptotic points for a homeomorphism on a metric 𝐺-space. It is observed that this notion under the trivial action of 𝐺 on 𝑋 coincides with positively/negatively asymptotic points. However under a nontrivial action of 𝐺 on 𝑋, while positively/negatively asymptotic points are positively/negatively 𝐺-asymptotic, examples are provided to justify that the converse is not true. Studying 𝐺-asymptotic points in relation to 𝐺-generators for a homeomorphism on a compact metric 𝐺-space, we obtain necessary and sufficient condition for two points to be positively/negatively 𝐺-asymptotic. In Section 4, we show that the problem of studying 𝐺-expansive homeomorphisms on a bounded subset of a normed linear 𝐺-space is equivalent to the problem of studying linear 𝐺-expansive homeomorphisms on a bounded subset of another normed linear 𝐺-space. Using the above equivalence, we obtain a necessary and sufficient condition for two points to be positively/negatively 𝐺-asymptotic under a homeomorphism on a bounded subset of a normed linear 𝐺-space extending William’s result [6].

2. Preliminaries

Throughout 𝐻(𝑋) denotes the collection of all self-homeomorphisms of a topological space 𝑋, ℤ denotes the set of integers, and ℕ denotes the set of positive integers. By a 𝐺-space [19, 20] we mean a triple (𝑋,𝐺,𝜃), where 𝑋 is a Hausdorff space, 𝐺 is a topological group, and 𝜃∶𝐺×𝑋→𝑋 is a continuous action of 𝐺 on 𝑋. Henceforth, 𝜃(𝑔,𝑥) will be denoted by 𝑔𝑥. For 𝑥∈𝑋, the set 𝐺(𝑥)={𝑔𝑥∣𝑔∈𝐺} is called the 𝐺-orbit of 𝑥 in 𝑋. Note that 𝐺-orbits 𝐺(𝑥) and 𝐺(𝑦) of points 𝑥,𝑦 in 𝑋 are either disjoint or equal. A subset 𝑆 of 𝑋 is called 𝐺-invariant if 𝜃(𝐺×𝑆)⊆𝑆. Let 𝑋/𝐺={𝐺(𝑥)|𝑥∈𝑋} and 𝑝_𝑋∶𝑋→𝑋/𝐺 be the natural quotient map taking 𝑥 to 𝐺(𝑥),𝑥∈𝑋, then 𝑋/𝐺 endowed with the quotient topology is called the orbit space of 𝑋 (with respect to 𝐺). The map 𝑝_𝑋 which is called the orbit map, is continuous and open and if 𝐺 is compact then 𝑝_𝑋 is also a closed map. An action of 𝐺 on 𝑋 is called trivial if 𝑔𝑥=𝑥, for every 𝑔∈𝐺 and 𝑥∈𝑋. If 𝑋,𝑌 are 𝐺-spaces, then a continuous map ℎ∶𝑋→𝑌 is called equivariant if ℎ(𝑔𝑥)=𝑔ℎ(𝑥) for each 𝑔 in 𝐺 and each 𝑥 in 𝑋. We call ℎ pseudoequivariant if ℎ(𝐺(𝑥))=𝐺(ℎ(𝑥)) for each 𝑥 in 𝑋. An equivariant map is clearly pseudoequivariant but converse need not be true [11]. We have studied properties of such maps in detail in [21]. By a normed linear 𝐺-space, we mean a normed linear space on which a topological group 𝐺 acts.

Recall that if 𝑋 is a metric space with metric 𝑑 and ℎ is a self homeomorphism of 𝑋 then ℎ is called expansive, if there exists a 𝛿>0 such that whenever 𝑥,𝑦∈𝑋,𝑥≠𝑦 then there exists an integer 𝑛 satisfying 𝑑(ℎ^𝑛(𝑥),ℎ^𝑛(𝑦))>𝛿;𝛿 is then called an expansive constant for ℎ. Distinct points 𝑥,𝑦∈𝑋 are called positively (resp., negatively) asymptotic under ℎ if for each 𝜖>0, there exists 𝑁∈ℕ such that 𝑛≥𝑁 (resp., 𝑛≤𝑁) implies 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝜖. Given a compact Hausdorff space 𝑋 and a self-homeomorphism ℎ of 𝑋, a finite open cover 𝕌 of 𝑋 is called a generator for (𝑋,ℎ) [22] if for each bisequence (𝑈_𝑖)_𝑖∈ℤ of members of 𝕌, ∩^∞_𝑖=−∞ℎ^−𝑖(𝑐𝑙𝑈_𝑖) contains at most one point. If 𝑋 is a metric 𝐺-space with metric 𝑑 then a self-homeomorphism ℎ of 𝑋 is called 𝐺-expansive with 𝐺-expansive constant 𝛿>0 if whenever 𝑥,𝑦∈𝑋 with 𝐺(𝑥)≠𝐺(𝑦) then there exists an integer 𝑛 satisfying 𝑑(ℎ^𝑛(𝑢),ℎ^𝑛(𝑣))>𝛿, for all 𝑢∈𝐺(𝑥) and 𝑣∈𝐺(𝑦). Given a compact Hausdorff 𝐺-space 𝑋 and a self-homeomorphism ℎ of 𝑋, a finite cover 𝕌 of 𝑋 consisting of 𝐺-invariant open sets is called a 𝐺-generator for (𝑋,ℎ) if for each bisequence (𝑈_𝑖)_𝑖∈ℤ of members of 𝕌, ∩^∞_𝑖=−∞ℎ^−𝑖(𝑐𝑙𝑈_𝑖) contains at most one 𝐺-orbit. Under the trivial action of 𝐺 on 𝑋, a 𝐺-generator is equivalent to a generator but in [14] examples are provided to justify that under a nontrivial action both are independent.

3. 𝐺-Generators and 𝐺-Asymptotic Points

Definition 3.1. Let (𝑋,𝑑) be a metric 𝐺-space and ℎ∶𝑋→𝑋 be a homeomorphism. Then 𝑥,𝑦∈𝑋 are called positively 𝐺-asymptotic (resp., negatively 𝐺-asymptotic) points with respect to ℎ if for given 𝜀>0 there exists an integer 𝑁 such that whenever 𝑛≥𝑁 (resp., 𝑛≤𝑁), 𝑑(ℎ^𝑛(𝑔𝑥),ℎ^𝑛(𝑘𝑦))<𝜀, for some 𝑔,𝑘∈𝐺.

Remark 3.2. Under the trivial action of a 𝐺 on 𝑋 the notion of positively (resp., negatively) 𝐺-asymptotic points coincides with the notion of positively (resp., negatively) asymptotic points. On the other hand, under a nontrivial action of 𝐺 on 𝑋, clearly positively (resp., negatively) asymptotic points with respect to a homeomorphism on 𝑋 are positively (resp., negatively) 𝐺-asymptotic points: in fact take 𝑔=𝑘= the identity element of 𝐺. However, the fact that the converse need not be true can be seen from the following example.

Example 3.3. Let 𝑋={±(1/𝑚),±(1−1/𝑚)∣𝑚∈ℕ} under usual metric and define ℎ∶𝑋→𝑋 defined by ℎ(𝑥)=𝑥,if𝑥∈{−1,0,1},thepointof𝑋whichisimmediatenexttoright(left)of𝑥,if𝑥>0(𝑥<0), (3.1) then ℎ∈𝐻(𝑋). Let discrete group 𝐺={−1,1} act on 𝑋 by −1⋅𝑥=−𝑥 and 1⋅𝑥=𝑥, 𝑥∈𝑋. Then the points 𝑥=−1/8 and 𝑦=1/4 are seen to be positively 𝐺-asymptotic but are not positively asymptotic with respect to ℎ.

We obtain a necessary and sufficient condition for two points to be positively/negatively 𝐺-asymptotic with respect to a homeomorphism on a compact metric 𝐺-space having a 𝐺-generator. We first prove the following lemma for 𝐺- generators.

Lemma 3.4. Let 𝑋 be a compact metric 𝐺-space, ℎ∈𝐻(𝑋), and ℑ be a 𝐺-generator for (𝑋,ℎ). Then for each nonnegative integer 𝑛, there exists 𝜀>0 such that for 𝑥,𝑦∈𝑋 with 𝐺(𝑥)≠𝐺(𝑦), 𝑑(𝑔𝑥,𝑘𝑦)<𝜀 for some 𝑔,𝑘∈𝐺 implies the existence of 𝐴_−𝑛,….,𝐴₀,…,𝐴_𝑛 in ℑ such that 𝑔𝑥,𝑘𝑦∈∩^𝑛_𝑖=−𝑛ℎ^−𝑖(𝐴_𝑖). Conversely, for each 𝜀>0, there exists a positive integer 𝑛 such that 𝑥,𝑦∈∩^𝑛_𝑖=−𝑛ℎ^−𝑖(𝐴_𝑖) with 𝐺(𝑥)≠𝐺(𝑦) and 𝐴_−𝑛,….,𝐴₀,…,𝐴_𝑛 in ℑ implies 𝑑(𝑔𝑥,𝑘𝑦)<𝜀 for some 𝑔,𝑘∈𝐺.

Proof. Since 𝑋 is compact and ℑ being a 𝐺-generator is an open cover of 𝑋, ℑ has a Lebesgue number, say 𝜂. Fix a nonnegative integer, say, 𝑛. Since 𝑋 is a compact metric space therefore ℎ^𝑖,|𝑖|≤𝑛, are uniformly continuous. Thus for above 𝜂, there exists an 𝜀>0 such that 𝑑(𝑥,𝑦)<𝜀 implies 𝑑(ℎ^𝑖(𝑥),ℎ^𝑖(𝑦))<𝜂 for all 𝑖,|𝑖|≤𝑛. Now if for some 𝑔,𝑘∈𝐺,𝑑(𝑔𝑥,𝑘𝑦)<𝜀 then using the fact that 𝜂 is a Lebesgue number for ℑ, for each 𝑖,|𝑖|≤𝑛, we find an 𝐴_𝑖∈ℑ such that ℎ^𝑖(𝑔𝑥),ℎ^𝑖(𝑘𝑦)∈𝐴_𝑖 and hence 𝑔𝑥,𝑘𝑦∈^𝑛_𝑖=−𝑛ℎ^−𝑖𝐴_𝑖. (3.2) Conversely, suppose 𝜀>0 is given. If the required result is not true, then for each positive integer 𝑗, there exist 𝑥_𝑗,𝑦_𝑗∈𝑋 with distinct 𝐺-orbits and {𝐴_𝑗,𝑖}_{−𝑗≤𝑖≤𝑗}⊂ℑ such that 𝑥_𝑗,𝑦_𝑗∈^𝑗_𝑖=−𝑗ℎ^−𝑖𝐴_𝑗,𝑖,𝑑𝑔𝑥_𝑗,𝑘𝑦_𝑗≥𝜀 (∗) for all 𝑔,𝑘∈𝐺. Since 𝑋 is compact, sequences {𝑥_𝑗} and {𝑦_𝑗} will converge. Suppose they converge to 𝑥 and 𝑦, respectively, then (*) implies 𝐺(𝑥)≠𝐺(𝑦). Since ℑ is a finite open cover, infinitely many of 𝐴_𝑗,0 are same, say 𝐴₀ and therefore for infinitely many 𝑗^′𝑠, 𝑥_𝑗,𝑦_𝑗∈𝐴₀. But this gives 𝑥,𝑦∈𝐶𝑙𝐴₀. Similarly, for each integer 𝑛, infinitely many of 𝐴_𝑗,𝑛 coincide and hence one gets 𝐴_𝑛 in ℑ such that 𝑥,𝑦∈ℎ^−𝑛(𝐶𝑙𝐴_𝑛). Thus 𝑥,𝑦∈^∞_𝑛=−∞ℎ^−𝑛𝐶𝑙𝐴_𝑛. (3.3) This contradicts the fact that ℑ be a 𝐺-generator for (𝑋,ℎ).

Theorem 3.5. Let 𝑋 be a compact metric 𝐺-space, ℎ∈𝐻(𝑋) be equivariant and ℑ be a 𝐺-generator for (𝑋,ℎ). Then 𝑥,𝑦∈𝑋 with distinct 𝐺-orbits are positively 𝐺-asymptotic with respect to ℎ if and only if there exists an 𝑁∈ℕ such that for each 𝑖≥𝑁, there exists an 𝐴_𝑖∈ℑ with 𝑥,𝑦∈∩^∞_𝑖=𝑁ℎ^−𝑖(𝐴_𝑖).

Proof. Suppose 𝑥,𝑦∈𝑋 with distinct 𝐺-orbits are positively 𝐺-asymptotic points. Then for a given 𝜀>0, there exists 𝑁∈ℕ such that 𝑑ℎ^𝑖𝑔𝑥,ℎ^𝑖𝑘𝑦<𝜀forsome𝑔,𝑘∈𝐺, (3.4) wherein 𝑖≥𝑁. Take 𝜀 to be a Lebesgue number of ℑ. Then for each 𝑖≥𝑁, there exists 𝐴_𝑖 in ℑ such that ℎ^𝑖(𝑔𝑥),ℎ^𝑖(𝑘𝑦)∈𝐴_𝑖 for some 𝑔,𝑘∈𝐺 and hence using equivariancy of ℎ, we obtain 𝑥,𝑦∈∩^∞_𝑖=𝑁ℎ^−𝑖(𝐴_𝑖).
Conversely, suppose that there exists an integer 𝑁 such that for each 𝑖≥𝑁, there exists an 𝐴_𝑖∈ℑ such that 𝑥,𝑦∈∩^∞_𝑖=𝑁ℎ^−𝑖(𝐴_𝑖). Let 𝜀>0. Then by Lemma 3.4, obtain a positive integer 𝑛 such that if 𝑥,𝑦∈∩^∞_𝑖=𝑁ℎ^−𝑖(𝐴_𝑖) with 𝐺(𝑥)≠𝐺(𝑦) and 𝐴_−𝑛,….,𝐴₀,…,𝐴_𝑛 in ℑ then 𝑑(𝑔𝑥,𝑘𝑦)<𝜀 for some 𝑔,𝑘∈𝐺. Let 𝑝≥𝑁+𝑛. Then 𝑥,𝑦∈∩^∞_𝑖=𝑁ℎ^−𝑖(𝐴_𝑖) implies 𝑥,𝑦∈^𝑝+𝑛_{𝑖=𝑝−𝑛}ℎ^−𝑖𝐴_𝑖. (3.5) Therefore, ℎ^𝑝(𝑥),ℎ^𝑝𝑦∈^𝑝+𝑛_{𝑖=𝑝−𝑛}ℎ^{−(𝑖−𝑝)}𝐴_𝑖=^𝑛_𝑗=−𝑛ℎ^−𝑗𝐴_𝑗+𝑝. (3.6) Also 𝐺(𝑥)≠𝐺(𝑦) implies ℎ^𝑝(𝐺(𝑥))∩ℎ^𝑝((𝐺(𝑦))=∅ and from equivariancy of ℎ, we obtain that 𝐺(ℎ^𝑝(𝑥))≠𝐺(ℎ^𝑝(𝑦)) and hence for some 𝑔,𝑘∈𝐺,𝑑(𝑔ℎ^𝑝(𝑥),𝑘ℎ^𝑝(𝑦))<𝜖. Now equivariancy of ℎ gives 𝑑(ℎ^𝑝(𝑔𝑥),ℎ^𝑝(𝑘𝑦))<𝜖. Thus given 𝜖>0, there exists 𝑁∈ℕ such that whenever 𝑛≥𝑁, for some 𝑔,𝑘∈𝐺, we have 𝑑(ℎ^𝑛(𝑔𝑥),ℎ^𝑛(𝑘𝑦))<𝜖 which proves that 𝑥,𝑦 are positively 𝐺-asymptotic points with respect to ℎ.

The following result concerning negatively 𝐺-asymptotic points can be proved similarly.

Theorem 3.6. Let 𝑋 be a compact metric 𝐺-space, ℎ∈𝐻(𝑋) be equivariant and ℑ be a 𝐺-generator for (𝑋,ℎ). Then 𝑥,𝑦∈𝑋 with distinct 𝐺-orbits are negatively 𝐺-asymptotic with respect to ℎ if and only if there exists an integer 𝑁 such that for each 𝑖≤𝑁, there exists an 𝐴_𝑖∈ℑ with 𝑥,𝑦∈∩^𝑁_𝑖=−∞ℎ^−𝑖(𝐴_𝑖).

4. Linearization of 𝐺-Expansive Homeomorphisms

We show that the problem of studying 𝐺-expansive homeomorphisms on a bounded subset of a normed linear 𝐺-space is equivalent to the problem of studying linear 𝐺-expansive homeomorphisms on a bounded subset of another normed linear 𝐺-space.

Let 𝐻 be a normed linear 𝐺-Space with norm || and 𝐺 act on 𝐻 in such a way that 𝑇_𝑘∶𝐻→𝐻 defined by 𝑇_𝑘(𝑥)=𝑘𝑥,𝑥∈𝐻 is linear for every 𝑘∈𝐺.

Let 𝑆(𝐻)={𝑧∶ℤ→𝐻} (4.1) and for 𝑧∈𝑆(𝐻), let 𝑧(𝑖)=𝑧_𝑖𝑁,𝑖∈ℤ,_𝐺(𝐻)=𝑧∈𝑆(𝐻)∣^∞_𝑖=−∞2^−|𝑖|𝑘𝑧_𝑖²<∞,𝑘∈𝐺. (4.2) Let ℎ∶𝑁_𝐺(𝐻)→𝑁_𝐺(𝐻) be defined by (ℎ(𝑧))_𝑖=(𝑧_𝑖+1), for every 𝑧∈𝑁_𝐺(𝐻) and for every 𝑖∈ℤ. For 𝑧,𝑤∈𝑁_𝐺(𝐻), define (𝑧+𝑤)_𝑖=𝑧_𝑖+𝑤_𝑖 and for a scalar 𝑐, define 𝑐𝑧 by (𝑐𝑧)_𝑖=𝑐𝑧_𝑖. Define ‖𝑧‖ by (∑^∞_𝑖=−∞2^−|𝑖||𝑧_𝑖|²)^−1/2. With this norm 𝑁_𝐺(𝐻) is a normed linear space.

Using the above notations we have the following results.

Theorem 4.1. Let 𝐻 be a normed linear 𝐺-Space, 𝑋 be a bounded subset of 𝐻 and 𝑓∶𝑋→𝑋 be an equivariant homeomorphism. Then 𝑔∶𝑋→𝑆(𝐻) defined by (𝑔(𝑥))_𝑖=𝑓^𝑖(𝑥), for each 𝑥∈𝑋 and each integer 𝑖, satisfies 𝑔(𝑋)⊆𝑁_𝐺(𝐻).

Proof. Let 𝑥∈𝑋 and 𝑘∈𝐺 then 𝑋 being bounded and 𝑓 being equivariant, we have ^∞_𝑖=−∞2^−|𝑖|𝑘𝑔(𝑥)_𝑖²=^∞_𝑖=−∞2^−|𝑖|𝑘𝑓^𝑖(𝑥)²=^∞_𝑖=−∞2^−|𝑖|𝑓^𝑖(𝑘𝑥)²<∞. (4.3) Hence 𝑔(𝑋)⊆𝑁_𝐺(𝐻).

Theorem 4.2. Let 𝐻 be a normed linear 𝐺-Space, 𝑋 be a bounded subset of 𝐻 and 𝑓∶𝑋→𝑋 be an equivariant homeomorphism. The map ℎ is a linear homeomorphism of 𝑁_𝐺(𝐻) onto itself under which 𝑔(𝑋) is invariant. Moreover, 𝑔(𝑋) is bounded and 𝑔 is a homeomorphism of 𝑋 onto 𝑔(𝑋). Also, ℎ is 𝐺-expansive on 𝑔(𝑋) if and only if 𝑓 is 𝐺-expansive on 𝑋.

Proof. Let 𝑧,𝑤∈𝑁_𝐺(𝐻). Then (ℎ(𝑧+𝑤))_𝑖=(𝑧+𝑤)_𝑖+1=𝑧_𝑖+1+𝑤_𝑖+1=(ℎ(𝑧))_𝑖+(ℎ(𝑤))_𝑖=(ℎ(𝑧)+ℎ(𝑤))_𝑖, (4.4) for every 𝑖∈𝑍. Therefore ℎ(𝑧+𝑤)=ℎ(𝑧)+ℎ(𝑤). Also, (ℎ(𝑐𝑧))_𝑖=(𝑐𝑧)_𝑖+1=𝑐𝑧_𝑖+1=𝑐(ℎ(𝑧))_𝑖 implies ℎ(𝑐𝑧)=𝑐(ℎ(𝑧)). Hence ℎ is linear. If 𝑧≠𝑤 in 𝑁_𝐺(𝐻) then for some 𝑖∈ℤ,𝑧_𝑖≠𝑤_𝑖 which implies (ℎ(𝑧))_𝑖−1≠(ℎ(𝑤))_𝑖−1 and hence ℎ(𝑧)≠ℎ(𝑤). Thus ℎ is one-one. If 𝑤∈𝑁_𝐺(𝐻) then 𝑤^′∈𝑁_𝐺(𝐻), where (𝑤^′)_𝑖=𝑤_𝑖−1 and ℎ(𝑤^′)=𝑤, which proves that ℎ is onto. If 𝑧_𝑛→0 then ℎ(𝑧_𝑛)→0 therefore ℎ is continuous. Similarly ℎ⁻¹ is continuous. Next, we show that ℎ(𝑔(𝑋))⊆𝑔(𝑋). Let 𝑥∈𝑋 then ℎ𝑔(𝑥)_𝑖=𝑔(𝑥)_𝑖+1=𝑓^𝑖+1(𝑥)=𝑓^𝑖𝑓(𝑥)=𝑔𝑓(𝑥)_𝑖 (4.5) which implies ℎ(𝑔(𝑋))⊆𝑔(𝑋). Clearly 𝑔(𝑋) is bounded. It is easy to observe that 𝑔 is a homeomorphism of 𝑋 onto 𝑔(𝑋). Suppose 𝑓 is 𝐺-expansive on 𝑋 with 𝐺-expansive constant 𝛿. Let 𝑧,𝑤∈𝑔(𝑋) with 𝐺(𝑧)≠𝐺(𝑤). Let 𝑧=𝑔(𝑧₀),𝑤=𝑔(𝑤₀),𝑧₀,𝑤₀∈𝑋. Since 𝑓 is equivariant, 𝑔 is also equivariant and hence 𝐺(𝑧₀)≠𝐺(𝑤₀). Further 𝐺-expansivity of 𝑓 on 𝑋 gives existence of an integer 𝑛 such that 𝑓^𝑛𝑘𝑧₀−𝑓^𝑛𝑝𝑤₀>𝛿, (4.6) for all 𝑘,𝑝∈𝐺. Now ℎ being linear and 𝑔 being equivariant, we get ℎ^𝑛(𝑘𝑧)−ℎ^𝑛𝑝𝑤=ℎ^𝑛=√𝑘𝑧−𝑝𝑤_{∞𝑖=−∞}2^−|𝑖|ℎ^𝑛𝑘𝑧−𝑝𝑤_𝑖²=√^∞_𝑖=−∞2^−|𝑖|ℎ^𝑛(𝑘𝑧)_𝑖−ℎ^𝑛𝑝𝑤_𝑖²=√^∞_𝑖=−∞2^−|𝑖|(𝑘𝑧)_𝑛+𝑖−𝑝𝑤_𝑛+𝑖²≥(𝑘𝑧)_𝑛−𝑝𝑤_𝑛=𝑘𝑔𝑧_0𝑛−𝑝𝑔𝑤_0𝑛=𝑔𝑘𝑧_0𝑛−𝑔𝑝𝑤_0𝑛=𝑓^𝑛𝑘𝑧₀−𝑓^𝑛𝑝𝑤₀>𝛿. (4.7) Therefore ℎ is 𝐺-expansive on 𝑔(𝑋) with 𝐺-expansive constant 𝛿.
Conversely, suppose ℎ is 𝐺-expansive on 𝑔(𝑋) with 𝐺-expansive constant 𝛿. We show that 𝑓 is 𝐺-expansive on 𝑋 with 𝐺-expansive constant 𝛿/√3. Suppose not. Then there exist 𝑧₀,𝑤₀∈𝑋 with 𝐺(𝑧₀)≠𝐺(𝑤₀) such that 𝑓^𝑛𝑘𝑧₀−𝑓^𝑛𝑝𝑤₀≤𝛿√3, (4.8) for some 𝑘,𝑝∈𝐺 and for all 𝑛∈ℤ. Let 𝑧=𝑔(𝑧₀),𝑤=𝑔(𝑤₀) then 𝑔 being equivariant homeomorphism, 𝐺(𝑧)≠𝐺(𝑤). Now ℎ being linear and 𝑔 being equivariant, we have for all 𝑛∈ℤℎ^𝑛(𝑘𝑧)−ℎ^𝑛𝑝𝑤=ℎ^𝑛𝑔𝑘𝑧₀−𝑔𝑝𝑤₀=√^∞_𝑖=−∞2^−|𝑖|𝑔𝑘𝑧_0𝑛+𝑖−𝑔𝑝𝑤_0𝑛+𝑖²=√^∞_𝑖=−∞2^−|𝑖|𝑓^𝑛+𝑖𝑘𝑧₀−𝑓^𝑛+𝑖𝑝𝑤₀²≤√^∞_𝑖=−∞2^−|𝑖|𝛿²3=𝛿, (4.9) a contradiction to the fact that ℎ is 𝐺-expansive with 𝐺-expansive constant 𝛿. Thus 𝛿/√3 is a 𝐺-expansive constant for 𝑓.

Theorem 4.3. Let 𝐻 be a normed linear 𝐺-Space, 𝑋 be a bounded subset of 𝐻 and 𝑓∶𝑋→𝑋 be an equivariant homeomorphism. Points 𝑥,𝑦∈𝑋 are positively (negatively) 𝐺-asymptotic under 𝑓 if and only if 𝑔(𝑥) and 𝑔(𝑦) are positively (negatively) 𝐺-asymptotic under ℎ.

Proof. Suppose 𝑔(𝑥),𝑔(𝑦) are positively 𝐺-asymptotic under ℎ. Let 𝜖>0. Then there exists 𝑁∈ℕ such that for all 𝑛≥𝑁 and for some 𝑘,𝑝∈𝐺, we have ℎ^𝑛𝑘𝑔(𝑥)−ℎ^𝑛𝑝𝑔𝑦=ℎ^𝑛𝑔(𝑘𝑥)−ℎ^𝑛𝑔𝑝𝑦<𝜖. (4.10) Since 𝑓^𝑛(𝑘𝑥)−𝑓^𝑛𝑝𝑦≤ℎ^𝑛𝑔(𝑘𝑥)−ℎ^𝑛𝑔𝑝𝑦, (4.11) we get 𝑓^𝑛(𝑘𝑥)−𝑓^𝑛𝑝𝑦<𝜖. (4.12) Thus 𝑥,𝑦 are positively 𝐺-asymptotic under 𝑓.
Conversely, suppose 𝑥,𝑦 are positively 𝐺-asymptotic under 𝑓. Let 𝜖>0 be given then there exist 𝑁₁∈ℕ and 𝑘,𝑝∈𝐺 such that for all 𝑛≥𝑁₁, 𝑓^𝑛(𝑘𝑥)−𝑓^𝑛<𝜖𝑝𝑦2. (4.13)
Choose 𝑁₂∈ℕ,𝑁₂<𝑁₁ such that _𝑖<𝑁22^−|𝑖|(diam𝑋)²<𝜖²4. (4.14)
Then for 𝑛>(𝑁₁−𝑁₂), we have ℎ^𝑛𝑔(𝑘𝑥)−ℎ^𝑛𝑔𝑝𝑦²=_𝑖≤𝑁22^−|𝑖|𝑓^𝑛+𝑖(𝑘𝑥)−𝑓^𝑛+𝑖𝑝𝑦²+_𝑖≥𝑁22^−|𝑖|𝑓^𝑛+𝑖(𝑘𝑥)−𝑓^𝑛+𝑖𝑝𝑦²<𝜖²4+𝜖²4_𝑖≥𝑁22^−|𝑖|<𝜖²4+𝜖²4^∞_𝑖=−∞2^−|𝑖|=𝜖²4+3𝜖²4=𝜖². (4.15)
Hence for 𝑛>(𝑁₁−𝑁₂) and for above 𝑘,𝑝∈𝐺, 𝑔 being equivariant we get, ℎ^𝑛𝑘𝑔(𝑥)−ℎ^𝑛𝑝𝑔𝑦<𝜖, (4.16) implying 𝑔(𝑥),𝑔(𝑦) are positively 𝐺-asymptotic under ℎ.
The proof for the case of negatively asymptotic points is similar.

Acknowledgment

The authors express sincere thanks to the referees for their suggestions.