<i>N</i>-Expansive Property for Flows

In this paper, we discuss the dynamics of n -expansive homeomorphisms with the shadowing property defined on compact metric spaces in continuous case. For every n ∈N , we exhibit an n-expansive homeomorphism but not (n-1) -expansive. Furthermore, that flow has the shadowing property and admits an infinite number of chain-recurrent classes.

• Expansive
• Flow
• <i>N</i>-Expansive
• Shadowing

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 ( n − 1 ) -expansive flow. Moreover, that flow is shadowable and has infinite number of chain-recurrent classes.

Let ( X , d ) be a metric space. A flow on X is a map ϕ : X × ℝ → X satisfying ϕ ( x , 0 ) = x and ϕ ( ϕ ( x , s ) , t ) = ϕ ( x , s + t ) for x ∈ X and t , s ∈ ℝ . For convenience, we will denote

ϕ ( x , s ) = ϕ s ( x )       and       ϕ ( a , b ) ( x ) = { ϕ t ( x ) : t ∈ ( a , b ) } .

The set ϕ ℝ ( x ) is called the orbit of ϕ through x ∈ X and will be denoted by Orb ϕ ( x ) . We have the following several basis concepts (see [1] [15] [16]).

Definition 1.1. Let ϕ be a flow in a metric space ( X , d ) . We say that ϕ is n-expansive ( n ∈ ℕ ) if there exists c > 0 such that for every x ∈ X the set

Γ ( x , c ) : = { y ∈ X ; d ( ϕ t ( x ) , ϕ t ( y ) ) ≤ c , ∀ t ∈ ℝ } ,

contains at most n different points of X.

We say that ϕ is finite expansive if there exists c > 0 such that for every x ∈ X the set Γ ( x , c ) is finite.

Definition 1.2. Let x ∈ X . We say that x is a period point if there exists T > 0 such that ϕ t + T ( x ) = ϕ t ( x ) , ∀ t ∈ ℝ . Denote that π ( x ) is the period of x, which is the smallest non-negative number satisfying this equation.

Definition 1.3. Give δ , T ≥ 0 . We say that a sequence of pairs ( x i , t i ) i ∈ ℤ ⊂ X × ℝ is a ( δ , T ) -pseudo orbit of ϕ if t i ≥ T and d ( ϕ t i ( x i ) , x i + 1 ) ≤ δ , ∀ i ∈ ℤ .

We define

s i = ( ∑ j = 0 i − 1 t j , i > 0 , 0 , i = 0 , − ∑ j = i − 1 t j , i < 0 ,

and x 0 ⋆ t = ϕ t − s i ( x i ) whenever s i ≤ t < s i + 1 .

Definition 1.4. We say that ϕ is shadowing property if for each ϵ > 0 there is δ > 0 such that for any ( δ ,1 ) -pseudo orbit ( x i , t i ) i ∈ ℤ , there exists x ∈ X and an orientation preserving homeomorphism h : ℝ → ℝ such that h ( 0 ) = 0 and d ( x 0 ⋆ t , ϕ h ( t ) ( x ) ) ≤ ϵ .

Denote by Rep the set of orientation preserving homeomorphism h : ℝ → ℝ such that h ( 0 ) = 0 .

Definition 1.5. Give two points p and q in X. We say p and q are ( δ , T ) -related if there are two ( δ , T ) -chains ( x i , t i ) i = 0 m and ( y i , s i ) i = 0 n such that p = x 0 = y n and q = y 0 = x m . We say that p and q are related ( p ∼ q ) if they are ( δ , T ) -related for every δ , T > 0 . The chain-recurrent class of a point p ∈ X is the set of all points q ∈ X such that p ∼ q .

Theorem 1.1. For every n ∈ ℕ , there is an n-expansive flow, define in a compact metric space, that is not ( n − 1 ) -expansive, has the shadowing property and admits an infinite number of chain-recurrent classes.

2. Proof of the Main Theorem

Consider a flow ϕ defined in a compact metric space ( M , d 0 ) , and ϕ has 1-expansive, and has the shadowing property. Further, suppose it has an infinite number of period points { p k } k ∈ ℕ , which we can suppose belong to different orbits. Let E be an infinite set, such that there exists a bijection r : ℝ → E . Let

Q = ∪ k ∈ N { 1, ⋯ , n − 1 } × { k } × [ 0, π ( p k ) ) ,

and note that there exists a bijection s : Q → ℝ . Consider the bijection q : Q → E defined by

q ( i , k , j ) = r ∘ s ( i , k , j ) .

Let X = M ∪ E . Thus, any point x ∈ E has the form x = q ( i , k , j ) for some ( i , k , j ) ∈ Q . Define a function d : X × X → ℝ + by

d ( x , y ) = ( 0 , x = y , d 0 ( x , y ) , x , y ∈ M , 1 k + d 0 ( y , ϕ j ( p k ) ) , x = q ( i , k , j ) ,   y ∈ M , 1 k + d 0 ( x , ϕ j ( p k ) ) , x ∈ M ,     y = q ( i , k , j ) , 1 k , x = q ( i , k , j ) ,   y = q ( l , k , j ) ,       i ≠ l , 1 k + 1 m + d 0 ( ϕ t ( p k ) , ϕ r ( p m ) ) , x = q ( i , k , j ) ,   y = q ( l , m , r ) ,   k ≠ m   or   j ≠ r .

Now we prove that function d is a metric in X. Indeed, we see that d ( x , y ) = 0 iff x = y , and that d ( x , y ) = d ( y , x ) for any pair ( x , y ) ∈ X × X . We shall prove that the triangle inequality d ( x , z ) ≤ d ( x , y ) + d ( y , z ) for any triple ( x , y , z ) ∈ X × X × X . When ( x , y , z ) ∈ M × M × M we have that d | M × M = d 0 , and d 0 is a metric in M. When ( x , y , z ) ∈ M × M × E then z = q ( i , k , j ) and

d ( x , z ) = 1 k + d 0 ( x , ϕ j ( p k ) ) ≤ d 0 ( x , y ) + 1 k + d 0 ( y , ϕ j ( p k ) ) = d ( x , y ) + d ( y , z ) .

Therefore, when ( x , y , z ) ∈ E × M × M , changing the role of x and z in the previous case, we obtain this result. When ( x , y , z ) ∈ M × E × M , we have y = q ( i , k , j ) and

d ( x , z ) = d 0 ( x , z ) ≤ 2 k + d 0 ( x , ϕ j ( p k ) ) + d 0 ( z , ϕ j ( p k ) ) = d 0 ( x , y ) + d 0 ( y , z ) .

When ( x , y , z ) ∈ M × E × E , we have y = q ( i , k , j ) and z = ( l , m , r ) . If k ≠ m or j ≠ r then

d ( x , z ) = 1 m + d 0 ( x , ϕ r ( p m ) ) < 2 k + 1 m + d 0 ( x , ϕ j ( p k ) ) + d 0 ( ϕ j ( p k ) , ϕ r ( p m ) ) = d ( x , y ) + d ( y , z ) .

If k = m , j = r and i ≠ l then

d ( x , z ) = 1 m + d 0 ( x , ϕ r ( p m ) ) < 1 k + 1 m + d 0 ( x , ϕ j ( p k ) ) = d ( x , y ) + d ( y , z ) .

So if ( x , y , z ) ∈ E × E × M , change the role of x and z in previous case, and we get the result. If ( x , y , z ) ∈ E × M × E then x = q ( i , k , j ) and z = q ( l , m , r ) . Hence,

d ( x , y ) + d ( y , z ) = 1 k + 1 m + d 0 ( y , ϕ j ( p k ) ) + d 0 ( y , ϕ r ( p m ) )

and

d ( x , z ) = ( 1 k + 1 m + d 0 ( ϕ j ( p k ) , ϕ r ( p m ) ) if   k ≠ m   or   j ≠ r , 1 k if   k = m ,       j = r   and   i ≠ l .

Thus, we always get the result d ( x , z ) < d ( x , y ) + d ( y , z ) for both of 2 cases. When ( x , y , z ) ∈ E × E × E , we let

x = q ( i 1 , k 1 , j 1 ) , y = q ( i 2 , k 2 , j 2 ) , z = q ( i 3 , k 3 , j 3 ) .

Case 1. If k 1 = k 3 and j 1 = j 3 we have d ( x , z ) = 1 k 1 , and

d ( x , y ) + d ( y , z ) = ( 2 k 1 , k 1 = k 2 = k 3   and   j 1 = j 2 = j 3 , 2 k 1 + 2 k 2 + d 0 ( ϕ j 1 ( k 1 ) , ϕ j 2 ( k 2 ) ) + d 0 ( ϕ j 2 ( k 2 ) , ϕ j 3 ( k 3 ) ) , k 1 = k 3 ≠ k 2   or   j 1 = j 3 ≠ j 2 .

It means that d ( x , z ) < d ( x , y ) + d ( y , z ) for both of 2 cases.

Case 2. If k 1 ≠ k 3 or j 1 ≠ j 3 , we have

d ( x , z ) = 1 k 1 + 1 k 3 + d 0 ( ϕ j 1 ( k 1 ) , ϕ j 3 ( k 3 ) ) ,

and

d ( x , y ) + d ( y , z ) = ( 2 k 1 + 1 k 3 + d 0 ( ϕ j 2 ( k 2 ) , ϕ j 3 ( k 3 ) ) , k 1 = k 2   and   j 1 = j 2 , 1 k 1 + 2 k 3 + d 0 ( ϕ j 1 ( k 1 ) , ϕ j 2 ( k 2 ) ) , k 2 = k 3   and   j 2 = j 3 , 1 k 1 + 2 k 2 + 1 k 3 + d 0 ( ϕ j 1 ( k 1 ) , ϕ j 2 ( k 2 ) ) + d 0 ( ϕ j 2 ( k 2 ) , ϕ j 3 ( k 3 ) ) , k 1 ≠ k 2 ≠ k 3   or   j 1 ≠ j 2 ≠ j 3 .

Hence, d ( x , z ) < d ( x , y ) + d ( y , z ) .

It implies d is a metric in X.

Next, we prove that ( X , d ) is a compact metric space. Let any sequences ( x n ) n ∈ ℕ ∈ X . We prove that this sequence has a convergent subsequence. If ( x n ) n ∈ ℕ has infinite elements in M, then the compactness of M and the fact d | M × M = d 0 , so ( x n ) n ∈ ℕ has a convergent subsequence. We consider ( x n ) n ∈ ℕ has finite elements in M; therefore, it has infinite elements in E. We can assume that ( x n ) n ∈ ℕ ⊂ E then x n = q ( i n , k n , j n ) . If there is N ∈ ℕ such that k n < N , ∀ n ∈ ℕ then the set { x n ; n ∈ ℕ } is finite, so at least one point of ( x n ) n ∈ ℕ appears infinite times, forming a convergent subsequence. Now suppose ( k n ) n ∈ ℕ is unbounded, therefore, lim n → ∞ k n = ∞ . We choose y n = ϕ j n ( p k n ) ,

so ( y n ) n ∈ ℕ ⊂ M and d ( x n , y n ) = 1 k n , ∀ n ∈ ℕ . Since ( y n ) n ∈ ℕ is a subset of

compact set M, ( y n ) n ∈ ℕ has a subsequence ( y n l ) l ∈ ℕ converging to y ∈ M . Thus, we have

d ( x n l , y ) < d ( x n l , y n l ) + d ( y n l , y ) = 1 k n l + d ( y n l , y ) → 0     when   l → ∞ .

It implies that ( x n ) n ∈ ℕ has a subsequence ( x n l ) l ∈ ℕ which converges to y. Thus, ( X , d ) is a compact metric space.

For all t ∈ ℝ , we define a map ψ t by

ψ t ( x ) = ( ϕ t ( x ) if   x ∈ M , q ( i , k , ( j + t ) mod π ( p k ) ) if   x = q ( i , k , j ) .

We can see that j, t, j + t cannot be in ℕ , but we can define a real number: t mod π ( p k ) : = r , when

t = m π ( p k ) + r ,       m ∈ ℤ ,       0 ≤ r < π ( p k ) .

By definition of flow, it's easy to see that ψ is a flow of X. Indeed, we can prove that ψ t + s = ψ t ∘ ψ s , ∀ t , s ∈ ℝ . If x ∈ M , we get

ψ t + s ( x ) = ϕ t + s ( x ) = ϕ t ∘ ϕ s ( x ) = ψ t ∘ ψ s ( x ) ,       ∀ t , s ∈ ℝ .

If x = q ( i , k , j ) , we have

ψ t + s ( x ) = q ( i , k , ( j + t + s ) mod π ( p k ) ) = ψ t ∘ ψ s ( x ) .

Therefore, ψ is the flow with the previous properties.

In order to prove that ψ is n-expansive, first we see that ϕ is 1-expansive; so there is a > 0 such that if d ( ϕ t ( x ) , ϕ t ( y ) ) ≤ a , ∀ t ∈ ℕ , then x = y . Suppose that { x 1 , ⋯ , x n + 1 } are n + 1 different points of X satisfying

d ( ψ t ( x i ) , ψ t ( x j ) ) ≤ a ,       ∀ t ∈ ℝ ,       ∀ ( i , j ) ∈ { 1, ⋯ , n + 1 } × { 1, ⋯ , n + 1 } .

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 x m = q ( i m , k m , j m ) , m ∈ { 1 , ⋯ , n } . Because i m ∈ { 1, ⋯ , n − 1 } and we have n number i m ; thus, by Pigeonhole principle, at least two of these points are of the form q ( i , k , j ) and q ( i , m , r ) . We prove that k ≠ m . Indeed, if k = m , we have 2 points are q ( i , k , j ) and q ( i , k , r ) with j ≠ r (because all of n + 1 points are different). For each s ∈ ℝ we have

d ( ϕ s ( ϕ j ( p k ) ) ) , d ( ϕ s ( ϕ r ( p k ) ) ) = d ( ψ s ( q ( i , k , j ) , ψ s ( q ( i , k , r ) ) ) ) − 2 k < d ( ψ s ( q ( i , k , j ) ) , ψ s ( q ( i , k , r ) ) ) < a .

This implies that ϕ j ( p k ) = ϕ r ( p k ) (by the Proposition of 1-expansive of ϕ ), which implies that j = r and we obtain a contradiction. Therefore, k ≠ m .

Now we implies that: for every s ∈ ℝ we have:

d ( ϕ s ( ϕ j ( p k ) ) ) , d ( ϕ s ( ϕ r ( p m ) ) ) = d ( ψ s ( q ( i , k , j ) ) , ψ s ( q ( i , m , r ) ) ) − 1 k − 1 m < d ( ψ s ( q ( i , k , j ) ) , ψ s ( q ( i , m , r ) ) ) < a .

So similarly, we have ϕ j ( p k ) = ϕ r ( p m ) ; hence, p m = p k , which is contradiction with the fact that k ≠ m . Thus, we cannot choose n + 1 points satisfy this proposition; it means ψ is n-expansive in X.

Next, we prove that ψ is not ( n − 1 ) -expansive. For any a > 0 , we choose k ∈ ℕ such that 1 k < a , so we have

d ( ϕ j ( p k ) , q ( i , k , j ) ) = 1 k < a   , ∀ j ∈ ℝ , ∀ i ∈ { 1 , ⋯ , n − 1 } . So Γ ( p k , a ) contain at

least n points { p k , q ( 1, k ,0 ) , ⋯ , q ( n − 1, k ,0 ) } and that ψ is not ( n − 1 ) -expansive, because there is not a > 0 satisfies this define about ( n − 1 ) -expansive.

Now we prove that ψ has the shadowing property. Since ϕ has the shadowing property, for each ϵ > 0 , we can consider δ ϕ > 0 , so for any ( δ ϕ ,1 ) -pseudo-orbit in M we have the ϵ 2 -shadowing. Now consider ( x n , t n ) n ∈ ℤ has

the ( δ ,1 ) -pseudo-orbit by ψ in X. We assume that δ < δ ϕ 3 < ϵ 3 . So we have

d ( ψ t n ( x n ) , x n + 1 ) < δ . Let N is a smallest integer number such that 1 N < δ , and we consider ( x n , x n + 1 ) in 3 cases.

Case 1. If ( x n , x n + 1 ) ∈ E × M , we have x n = q ( i , k , j ) and d ( ψ t n ( x n ) , x n + 1 ) = 1 k + d 0 ( x n + 1 , ϕ j + t n ( p k ) ) , so 1 k < δ ; hence, k ≥ N .

Case 2. If ( x n , x n + 1 ) ∈ M × E , we obtain x n + 1 = q ( i , k , j ) and d ( ψ t n ( x n ) , x n + 1 ) = 1 k + d 0 ( ϕ j ( p k ) , ϕ j ( x n ) ) , so 1 k < δ ; hence, k ≥ N .

Case 3. If ( x n , x n + 1 ) ∈ E × E , we have x n = q ( i , k , j ) and x n + 1 = q ( l , m , r ) . So ψ t n ( x n ) = q ( i , k , j + t n ) . Thus, if we want d ( ψ t n ( x n ) , x n + 1 ) < δ , we have either if k ≥ N , so m ≥ N (by similarly) or if k < N , we have x n + 1 = ψ t n ( x n ) , such that x n + 1 = q ( i , k , j + t n ) . When ( x n , t n ) n ∈ ℤ is one of orbit { q ( l , k , j n ) } n ∈ ℤ , and j n + 1 = j n + t n , ∀ n ∈ ℤ . So one obtain s n = j n − j 0 , thus,

d ( ψ t − s n ( x n ) , ψ t ( x 0 ) ) = d ( q ( l , k , t − s n + j n ) , q ( l , k , t + j 0 ) ) = 0 ,       s n ≤ t < s n + 1 .

Therefore, the shadowing property is proved.

When x i = q ( l , k , j ) , then k > N . Define a sequence ( y n , t n ) n ∈ ℤ ⊂ M by

y n = ( x n if   x n ∈ M , ϕ j ( p k ) if   x n = q ( l , k , j ) .

Then ( y n , t n ) n ∈ ℤ is δ ϕ -pseudo-orbit in M since

d ( ϕ t n ( y n ) , y n + 1 ) = d ( ψ t n ( y n ) , y n + 1 ) ≤ d ( ψ t n ( y n ) , ψ t n ( x n ) ) + d ( ψ t n ( x n ) , x n + 1 ) + d ( x n + 1 , y n + 1 ) < 1 N + δ + 1 N < δ ϕ .

Hence, there exists y ∈ M and a function h ∈ R e p such that

d ( ϕ t − s n ( y n ) , ϕ h ( t ) ( y ) ) < ϵ 2 ,       ∀ s n ≤ t < s n + 1 .

So

d ( ϕ t − s n ( x n ) , ϕ h ( t ) ( y ) ) < d ( ϕ t − s n ( x n ) , ϕ t − s n ( y n ) ) + d ( ϕ t − s n ( y n ) , ϕ h ( t ) ( y ) ) < 1 N + ϵ 2 < ϵ .

Therefore, ( x n , t n ) n ∈ ℤ is ϵ -shadowing. Hence, ψ has the shadowing property.

Finally, we have ψ admits an infinite number of chain-recurrent classes. Indeed, if we have q ( i , k , l ) ∈ E then

d ( q ( i , k , j ) , x ) ≥ 1 k ,       ∀ x ∈ X \ { q ( i , k , j ) } .

So if 0 < ϵ < 1 k then the orbit of q ( i , k , j ) cannot be connected by ϵ -pseudo orbits with any other point of X. This proves that the chain-recurrent classes of q ( i , k , j ) 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?

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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