1. Introduction and Preliminaries
The classical terms, expansive flows on a metric space are presented by Bowen and Walters [1] which generalized the similar notion by Anosov [2]. Besides, Walters [3] 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 [4]. In discrete case, this concept originally introduced for bijective maps by Utz [5] has been generalized to positively expansiveness in which positive orbits are considered instead [6]. Further generalizations are the pointwise expansiveness (with the above radius depending on the point [7]), the entropy-expansiveness [8], the continuum-wise expansiveness [9], 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 [10] [11] [12] [13] [14]).
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 [1] [15] [16]).
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
.
We define
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,
and
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
and
Hence,
.
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
So
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.
Acknowledgements
The first author was supported in part by the VNU Project of Vietnam National University No. QG101-15.
Open Questions
How are the properties of the local stable (unstable) sets of n-expansive flows?