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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 237820, 10 pages
http://dx.doi.org/10.1155/2012/237820
Research Article

Asymptotic Properties of -Expansive Homeomorphisms on a Metric -Space

Department of Mathematics, Faculty of Science, The M. S. University of Baroda, Vadodara 390002, India

Received 21 August 2012; Accepted 27 September 2012

Academic Editor: Ngai-Ching Wong

Copyright © 2012 Ruchi Das and Tarun Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define and study the notions of positively and negatively -asymptotic points for a homeomorphism on a metric -space. We obtain necessary and sufficient conditions for two points to be positively/negatively -asymptotic. Also, we show 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.

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 -expansive homeomorphism is defined for a self-homeomorphism on a metric -space. It is observed that the notion of expansiveness and the notion of -expansiveness under a nontrivial action of are independent of each other. Conditions under which an expansive homeomorphism on a metric -space is -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 -spaces. Further, in [14], the notion of generator in -spaces termed as -generator is defined and a characterization of -expansive homeomorphisms is obtained using -generator. Some interesting consequences have been obtained regarding existence of -expansive homeomorphisms. In [15, 16] we have studied some more properties of -expansive homeomorphisms. For some other dynamical properties on -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 -asymptotic points for a homeomorphism on a metric -space. It is observed that this notion under the trivial action of on coincides with positively/negatively asymptotic points. However under a nontrivial action of on , while positively/negatively asymptotic points are positively/negatively -asymptotic, examples are provided to justify that the converse is not true. Studying -asymptotic points in relation to -generators for a homeomorphism on a compact metric -space, we obtain necessary and sufficient condition for two points to be positively/negatively -asymptotic. In Section 4, we show 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, we obtain 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 extending William’s result [6].

2. Preliminaries

Throughout denotes the collection of all self-homeomorphisms of a topological space , denotes the set of integers, and denotes the set of positive integers. By a -space [19, 20] we mean a triple , where is a Hausdorff space, is a topological group, and is a continuous action of on . Henceforth, will be denoted by . For , the set is called the -orbit of in . Note that -orbits and of points in are either disjoint or equal. A subset of is called -invariant if . Let and be the natural quotient map taking to , then endowed with the quotient topology is called the orbit space of (with respect to ). The map which is called the orbit map, is continuous and open and if is compact then is also a closed map. An action of on is called trivial if , for every and . If are -spaces, then a continuous map is called equivariant if for each in and each in . We call pseudoequivariant if for each in . 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 -space, we mean a normed linear space on which a topological group acts.

Recall that if is a metric space with metric and is a self homeomorphism of then is called expansive, if there exists a such that whenever then there exists an integer satisfying is then called an expansive constant for . Distinct points are called positively (resp., negatively) asymptotic under if for each , there exists such that (resp., ) implies . Given a compact Hausdorff space and a self-homeomorphism of , a finite open cover of is called a generator for [22] if for each bisequence of members of , contains at most one point. If is a metric -space with metric then a self-homeomorphism of is called -expansive with -expansive constant if whenever with then there exists an integer satisfying , for all and . Given a compact Hausdorff -space and a self-homeomorphism of , a finite cover of consisting of -invariant open sets is called a -generator for if for each bisequence of members of , contains at most one -orbit. Under the trivial action of on , a -generator is equivalent to a generator but in [14] examples are provided to justify that under a nontrivial action both are independent.

3. -Generators and -Asymptotic Points

Definition 3.1. Let be a metric -space and be a homeomorphism. Then are called positively -asymptotic (resp., negatively -asymptotic) points with respect to if for given there exists an integer such that whenever (resp., ), , for some .

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

Example 3.3. Let under usual metric and define defined by then . Let discrete group act on by and , . Then the points and are seen to be positively -asymptotic but are not positively asymptotic with respect to .

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

Lemma 3.4. Let be a compact metric -space, , and be a -generator for . Then for each nonnegative integer , there exists such that for with , for some implies the existence of in such that . Conversely, for each , there exists a positive integer such that with and in implies for some .

Proof. Since is compact and being a -generator is an open cover of , has a Lebesgue number, say . Fix a nonnegative integer, say, . Since is a compact metric space therefore , are uniformly continuous. Thus for above , there exists an such that implies for all . Now if for some then using the fact that is a Lebesgue number for , for each , we find an such that and hence Conversely, suppose is given. If the required result is not true, then for each positive integer , there exist with distinct -orbits and such that for all . Since is compact, sequences and will converge. Suppose they converge to and , respectively, then (*) implies . Since is a finite open cover, infinitely many of are same, say and therefore for infinitely many , . But this gives . Similarly, for each integer , infinitely many of coincide and hence one gets in such that . Thus This contradicts the fact that be a -generator for .

Theorem 3.5. Let be a compact metric -space, be equivariant and be a -generator for . Then with distinct -orbits are positively -asymptotic with respect to if and only if there exists an such that for each , there exists an with .

Proof. Suppose with distinct -orbits are positively -asymptotic points. Then for a given , there exists such that wherein . Take to be a Lebesgue number of . Then for each , there exists in such that for some and hence using equivariancy of , we obtain .
Conversely, suppose that there exists an integer such that for each , there exists an such that . Let . Then by Lemma 3.4, obtain a positive integer such that if with and in then for some . Let . Then implies Therefore, Also implies and from equivariancy of , we obtain that and hence for some . Now equivariancy of gives . Thus given , there exists such that whenever , for some , we have which proves that are positively -asymptotic points with respect to .

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

Theorem 3.6. Let be a compact metric -space, be equivariant and be a -generator for . Then with distinct -orbits are negatively -asymptotic with respect to if and only if there exists an integer such that for each , there exists an with .

4. Linearization of -Expansive Homeomorphisms

We show 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.

Let be a normed linear -Space with norm and act on in such a way that defined by is linear for every .

Let and for , let Let be defined by , for every and for every . For , define and for a scalar , define by . Define by . With this norm is a normed linear space.

Using the above notations we have the following results.

Theorem 4.1. Let be a normed linear -Space, be a bounded subset of and be an equivariant homeomorphism. Then defined by , for each and each integer , satisfies .

Proof. Let and then being bounded and being equivariant, we have Hence .

Theorem 4.2. Let be a normed linear -Space, be a bounded subset of and be an equivariant homeomorphism. The map is a linear homeomorphism of onto itself under which is invariant. Moreover, is bounded and is a homeomorphism of onto . Also, is -expansive on if and only if is -expansive on .

Proof. Let . Then for every . Therefore . Also, implies . Hence is linear. If in then for some which implies and hence . Thus is one-one. If then , where and , which proves that is onto. If then therefore is continuous. Similarly is continuous. Next, we show that . Let then which implies . Clearly is bounded. It is easy to observe that is a homeomorphism of onto . Suppose is -expansive on with -expansive constant . Let with . Let . Since is equivariant, is also equivariant and hence . Further -expansivity of on gives existence of an integer such that for all . Now being linear and being equivariant, we get Therefore is -expansive on with -expansive constant .
Conversely, suppose is -expansive on with -expansive constant . We show that is -expansive on with -expansive constant . Suppose not. Then there exist with such that for some and for all . Let then being equivariant homeomorphism, . Now being linear and being equivariant, we have for all a contradiction to the fact that is -expansive with -expansive constant . Thus is a -expansive constant for .

Theorem 4.3. Let be a normed linear -Space, be a bounded subset of and be an equivariant homeomorphism. Points are positively (negatively) -asymptotic under if and only if and are positively (negatively) -asymptotic under .

Proof. Suppose are positively -asymptotic under . Let . Then there exists such that for all and for some , we have Since we get Thus are positively -asymptotic under .
Conversely, suppose are positively -asymptotic under . Let be given then there exist and such that for all ,
Choose such that
Then for , we have
Hence for and for above , being equivariant we get, implying are positively -asymptotic under .
The proof for the case of negatively asymptotic points is similar.

Acknowledgment

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

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