1. Introduction and Preliminaries
The classical terms, expansive flows on a metric space are presented by Bowen and Walters  which generalized the similar notion by Anosov . Besides, Walters  investigated continuous transformations of metric spaces with discrete centralizers and unstable centralizers and proved that expansive homeomorphisms have unstable centralizers; other result was studied in . In discrete case, this concept originally introduced for bijective maps by Utz  has been generalized to positively expansiveness in which positive orbits are considered instead . Further generalizations are the pointwise expansiveness (with the above radius depending on the point ), the entropy-expansiveness , the continuum-wise expansiveness , the measure-expansiveness and their corresponding positive counterparts. However, as far as we know, no one has considered the generalization in which at most n companion orbits are allowed for a certain prefixed positive integer n. For simplicity we call these systems n-expansive (or positively n-expansive if positive orbits are considered instead). A generalization of the expansiveness property that has been given attention recently is the n-expansive property (see     ).
In this paper, we introduce a notion of n-expansivity for flows which is generalization of expansivity, and show that there is an n-expansive flow but not -expansive flow. Moreover, that flow is shadowable and has infinite number of chain-recurrent classes.
Let be a metric space. A flow on X is a map satisfying and for and . For convenience, we will denote
The set is called the orbit of through and will be denoted by . We have the following several basis concepts (see   ).
Definition 1.1. Let be a flow in a metric space . We say that is n-expansive ( ) if there exists such that for every the set
contains at most n different points of X.
We say that is finite expansive if there exists such that for every the set is finite.
Definition 1.2. Let . We say that x is a period point if there exists such that . Denote that is the period of x, which is the smallest non-negative number satisfying this equation.
Definition 1.3. Give . We say that a sequence of pairs is a -pseudo orbit of if and .
and whenever .
Definition 1.4. We say that is shadowing property if for each there is such that for any -pseudo orbit , there exists and an orientation preserving homeomorphism such that and .
Denote by Rep the set of orientation preserving homeomorphism such that .
Definition 1.5. Give two points p and q in X. We say p and q are -related if there are two -chains and such that and . We say that p and q are related if they are -related for every . The chain-recurrent class of a point is the set of all points such that .
Theorem 1.1. For every , there is an n-expansive flow, define in a compact metric space, that is not -expansive, has the shadowing property and admits an infinite number of chain-recurrent classes.
2. Proof of the Main Theorem
Consider a flow defined in a compact metric space , and has 1-expansive, and has the shadowing property. Further, suppose it has an infinite number of period points , which we can suppose belong to different orbits. Let E be an infinite set, such that there exists a bijection . Let
and note that there exists a bijection . Consider the bijection defined by
Let . Thus, any point has the form for some . Define a function by
Now we prove that function d is a metric in X. Indeed, we see that iff , and that for any pair . We shall prove that the triangle inequality for any triple . When we have that , and is a metric in M. When then and
Therefore, when , changing the role of x and z in the previous case, we obtain this result. When , we have and
When , we have and . If or then
If , and then
So if , change the role of x and z in previous case, and we get the result. If then and . Hence,
Thus, we always get the result for both of 2 cases. When , we let
Case 1. If and we have , and
It means that for both of 2 cases.
Case 2. If or , we have
It implies d is a metric in X.
Next, we prove that is a compact metric space. Let any sequences . We prove that this sequence has a convergent subsequence. If has infinite elements in M, then the compactness of M and the fact , so has a convergent subsequence. We consider has finite elements in M; therefore, it has infinite elements in E. We can assume that then . If there is such that then the set is finite, so at least one point of appears infinite times, forming a convergent subsequence. Now suppose is unbounded, therefore, . We choose ,
so and . Since is a subset of
compact set M, has a subsequence converging to . Thus, we have
It implies that has a subsequence which converges to y. Thus, is a compact metric space.
For all , we define a map by
We can see that j, t, cannot be in , but we can define a real number: , when
By definition of flow, it's easy to see that is a flow of X. Indeed, we can prove that . If , we get
If , we have
Therefore, is the flow with the previous properties.
In order to prove that is n-expansive, first we see that is 1-expansive; so there is such that if , then . Suppose that are different points of X satisfying
Hence, at most one of these points belong to M. Consequently, at least n of them belong to E. Without loss of generality, we get . Because and we have n number ; thus, by Pigeonhole principle, at least two of these points are of the form and . We prove that . Indeed, if , we have 2 points are and with (because all of points are different). For each we have
This implies that (by the Proposition of 1-expansive of ), which implies that and we obtain a contradiction. Therefore, .
Now we implies that: for every we have:
So similarly, we have ; hence, , which is contradiction with the fact that . Thus, we cannot choose points satisfy this proposition; it means is n-expansive in X.
Next, we prove that is not -expansive. For any , we choose such that , so we have
. So contain at
least n points and that is not -expansive, because there is not satisfies this define about -expansive.
Now we prove that has the shadowing property. Since has the shadowing property, for each , we can consider , so for any -pseudo-orbit in M we have the -shadowing. Now consider has
the -pseudo-orbit by in X. We assume that . So we have
. Let N is a smallest integer number such that , and we consider in 3 cases.
Case 1. If , we have and , so ; hence, .
Case 2. If , we obtain and , so ; hence, .
Case 3. If , we have and . So . Thus, if we want , we have either if , so (by similarly) or if , we have , such that . When is one of orbit , and . So one obtain , thus,
Therefore, the shadowing property is proved.
When , then . Define a sequence by
Then is -pseudo-orbit in M since
Hence, there exists and a function such that
Therefore, is -shadowing. Hence, has the shadowing property.
Finally, we have admits an infinite number of chain-recurrent classes. Indeed, if we have then
So if then the orbit of cannot be connected by -pseudo orbits with any other point of X. This proves that the chain-recurrent classes of contains only its orbit. Therefore different periodic orbits in E belong to different chain-recurrent classes and we conclude the proof.
The first author was supported in part by the VNU Project of Vietnam National University No. QG101-15.
How are the properties of the local stable (unstable) sets of n-expansive flows?