N Expansive Property for Flows

ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.72031 Feb. 26, 2019 410 Journal of Applied Mathematics and Physics

N

-Expansive Property for Flows

Le Huy Tien, Le Duc Nhien

Department of Mathematics, Mechanics and Informatics, Vietnam National University at Hanoi, Hanoi, Vietnam

Abstract

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∈, 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.

Keywords

Expansive, Flow, N-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 dis-crete centralizers and unstable centralizers and proved that expansive homeo-morphisms 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 cor-responding positive counterparts. However, as far as we know, no one has con-sidered 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 re-cently is the n-expansive property (see [10] [11] [12] [13] [14]).

How to cite this paper: Tien, L.H. and Nhien, L.D. (2019) N-Expansive Property for Flows. Journal of Applied Mathematics and Physics, 7, 410-417.

https://doi.org/10.4236/jamp.2019.72031

Received: January 28, 2019 Accepted: February 23, 2019 Published: February 26, 2019

Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/jamp.2019.72031 411 Journal of Applied Mathematics and Physics In this paper, we introduce a notion of n-expansivity for flows which is gene-ralization 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× → X satisfy-ing φ

( )

x,0 =x _and φ φ

(

( )

x s t, ,

)

=φ

(

x s t, +

)

_for x X∈ and t s, ∈. For convenience, we will denote

( )

x s, s

( )

x and ( )a b,

( )

x

{

t

( )

x t:

( )

a b,

}

.

φ =φ φ = φ ∈

The set φ_

( )

x is called the orbit of φ through x X∈ and will be denoted by Orbφ

( )

x . We have the following several basis concepts (see [1] [15] [16]).

Definition 1.1. Let φ _{beaflowinametricspace}

(

X d,

)

. Wesaythat φ _is

n-expansive(n∈)ifthereexists c>0 suchthatforevery x X∈ theset

( )

x c, :

{

y X d;

(

φ_t

( ) ( )

x ,φ_t y

)

c t,

}

,

Γ = ∈ ≤ ∀ ∈_

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

0

T > _suchthat φ_{t T+}

( )

x =φ_t

( )

x ,∀ ∈t _{. Denotethat} π

( )

x _{istheperiod of x,}

whichisthesmallestnon-negativenumbersatisfyingthisequation.

Definition 1.3. Give δ,T ≥0 . We say that a sequence of pairs

(

x t_{i i i},

)

_∈⊂X×_{ isa}

(

δ,T

)

_{-pseudoorbitof} φ _if t T_i≥ _and

( )

(

ti i , i 1

)

,

d φ x x₊ ≤ ∀ ∈δ i _.

We define

1

0

1

, 0,

0, 0,

, 0,

i j j

i

j j i

t i

s i

t i

−

=

−

=

 _>

 

=_ =



− <



∑

∑

and x t0 =φt s−i

( )

xi whenever s t si≤ < i+1.

Definition 1.4. We say that φ _{is shadowing property if for each} >0

there is δ >0 such that for any

( )

δ,1 -pseudo orbit

(

x t_{i i i},

)

_∈, there exists

x X∈ and an orientation preserving homeomorphism h:_→_{ such that}

( )

0 0

h = _and d x t

(

₀ ,φ_{h t( )}

( )

x

)

≤.

Denote by Rep the set of orientation preserving homeomorphism h:→

such that h

( )

0 =0_.

Definition 1.5. Give two pointsp and q in X. We say p and q are

(

δ,T

)

-related if there are two

(

δ,T

)

_-chains

(

x t_{i i i},

)

m₌₀ and

(

y s_i, _{i i}

)

n₌₀ such that

0 n

p x= =y _and q y= ₀ =x_{m. Wesay thatpandqarerelated}

(

p q

)

ifthey are

(

δ,T

)

-related for every δ,T >0_{. The chain-recurrent class of a point}

DOI: 10.4236/jamp.2019.72031 412 Journal of Applied Mathematics and Physics Theorem 1.1. For every n∈_{, there is an n-expansive flow, define in a}

compactmetricspace, thatisnot

(

n−1

)

_{-expansive, hastheshadowing}

proper-tyandadmitsaninfinitenumberofchain-recurrentclasses.

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∈}, which we can suppose belong to different or-bits. Let E be an infinite set, such that there exists a bijection r:_→E_{. Let}

{

1, , 1

} { }

0,

( )

_k

)

,

k

Q n k π p

∈



=

_

_ − × _{× }

N

and note that there exists a bijection s Q: →_{ . 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: _{× →}+ _by

( )

( )

( )

(

)

(

)

( )

(

)

(

)

(

)

(

)

( ) ( )

(

)

(

)

(

)

0

0

0

0

,

0, ,

, , , ,

1 _{, , , , , ,}

1 _{, , , , , ,}

1 , , , , , , , ,

1 1 _{, , , , , , , , or .}

j k

j k

t k r m

d x y

x y

d x y x y M

d y p x q i k j y M

k

d x p x M y q i k j

k

x q i k j y q l k j i l k

d p p x q i k j y q l m r k m j r

k m

φ

φ

φ φ

= 

 _∈

 

+ = ∈

  

=  + ∈ =

 

= = ≠

  

+ + = = ≠ ≠



Now we prove that function d is a metric in X. Indeed, we see that

( )

, 0

d x y = 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

|M M 0

d × =d , and d0 is a metric in M. When

(

x y z, ,

)

∈M M E× × then

(

, ,

)

z q i k j= and

( )

, 1 0

(

, j

( )

k

)

0

(

,

)

1 0

(

, j

( )

k

)

(

,

)

(

, .

)

d x z d x p d x y d y p d x y d y z

k φ k φ

= + ≤ + + = +

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

( )

, 0

( )

, 2 0

(

, j

( )

k

)

0

(

, j

( )

k

)

0

(

,

)

0

(

, .

)

d x z d x z d x p d z p d x y d y z

k φ φ

= ≤ + + = +

DOI: 10.4236/jamp.2019.72031 413 Journal of Applied Mathematics and Physics k m≠ or j r≠ then

( )

(

( )

)

( )

(

)

(

( ) ( )

)

(

)

( )

0 0 0 1 , ,

2 1 _{, ,}

, , .

r m

j k j k r m

d x z d x p

m

d x p d p p

k m

d x y d y z

φ

φ φ φ

= +

< + + +

= +

If k m= _, j r= _and i l≠ _then

( )

, 1 0

(

, r

( )

m

)

1 1 0

(

, j

( )

k

)

(

,

)

(

, .

)

d x z d x p d x p d x y d y z

m φ k m φ

= + < + + = +

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,

(

,

)

(

,

)

1 1 0

(

, j

( )

k

)

0

(

, r

( )

m

)

d x y d y z d y p d y p

k m φ φ

+ = + + +

and

( )

0

(

( ) ( )

)

1 1 _{, if or ,}

,

1 _{if , and .}

j k r m

d p p k m j r

k m d x z

k m j r i l

k φ φ  + + ≠ ≠  =   _{= = ≠} 

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

(

1, ,1 1

)

,

(

2, ,2 2

)

,

(

3, ,3 3

)

x q i k j y q i k j= = z q i k j= . Case 1. If k1=k3 and j1= j3 we have

( )

1

1 ,

d x z k

= _{, and}

( )

( )

( )

( )

(

_{1 2}

)

(

₂

( )

₃

( )

)

1 2 3 1 2 3

1

0 1 2 0 2 3 1 3 2 1 3 2

1 2

, ,

2 , and ,

2 2 _{, , , or .}

j j j j

d x y d y z

k k k j j j

k

d k k d k k k k k j j j

k k φ φ φ φ

+  _{= = = =}  =   + + + = ≠ = ≠ 

It means that d x z

( )

, <d x y

( )

, +d y z

( )

, _{for both of 2 cases.}

Case 2. If k1≠k3 or j1 ≠ j3, we have

( )

0

(

₁

( )

1 ₃

( )

3

)

1 3

1 1

, j , j ,

d x z d k k

k k φ φ

= + + and

( ) ( )

( )

( )

(

)

( )

( )

(

)

( )

( )

(

)

(

( )

( )

)

2 3 1 2

1 2 2 3

0 2 3 1 2 1 2

1 3

0 1 2 2 3 2 3

1 3

0 1 2 0 2 3 1 2 3 1 2 3

1 2 3

, ,

2 1 _{, , and ,}

1 2 _{, , and ,}

1 2 1 _{, , , or .}

j j

j j

j j j j

d x y d y z

d k k k k j j

k k

d k k k k j j

k k

d k k d k k k k k j j j

k k k

φ φ

φ φ

φ φ φ φ

DOI: 10.4236/jamp.2019.72031 414 Journal of Applied Mathematics and Physics 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∈}∈X_{. We prove that this sequence has a convergent subsequence. If}

( )

x_{n n∈} has infinite elements in M, then the compactness of M and the fact

|M M 0

d × =d , so

( )

x_{n n}∈ has a convergent subsequence. We consider

( )

xn n∈ has finite elements in M; therefore, it has infinite elements in E. We can assume that

( )

x_{n n∈} ⊂E _then x_n=q i k j

(

_n, ,_{n n}

)

_{. If there is} N∈ such that

,

n

k <N n∀ ∈ then the set

{

x n_n; ∈

}

is finite, so at least one point of

( )

x_{n n∈} appears infinite times, forming a convergent subsequence. Now sup-pose

( )

kn n∈ is unbounded, therefore, lim_n→∞kn= ∞. We choose yn =φjn

( )

pkn , so

( )

y_{n n∈}⊂M and

(

n, n

)

1 ,

n

d x y n

k

= ∀ ∈. Since

( )

y_{n n∈} is a subset of

compact set M,

( )

y_{n n∈} has a subsequence

( )

y_{n ll}

∈ converging to y M∈ . Thus, we have

(

_,

) (

_,

) (

_,

)

1

(

_,

)

_{0 when .}

l l l l l

l

n n n n n

n

d x y d x y d y y d y y l

k

< + = + → → ∞

It implies that

( )

x_{n n∈} has a subsequence

( )

x_{n ll ∈}

 which converges to y. Thus,

(

X d,

)

is a compact metric space.

For all t∈_, we define a map ψt by

( )

t

₍

( )

_{, ,}

₍

₎

_mod

_{( )}

₎

if_if

₍

,_{, , .}

₎

t

k

x x M

x φ_{q i k j t p x q i k j}

ψ

π

∈ 

=  _{+ =}



We can see that j, t, j t+ _{cannot be in , but we can define a real number:}

( )

mod _k :

t π p =r_{, when}

( )

k , , 0

( )

k .

t m p= π +r m∈ ≤ <r π p

By definition of flow, it's easy to see that ψ _{is a flow of X. Indeed, we can}

prove that ψt s+ =ψ ψt s, ,∀t s∈. If x M∈ , we get

( )

( )

( )

( )

, , .

t s x t s x t s x t s x t s

ψ + =φ+ =φ φ =ψ ψ ∀ ∈

If x q i k j=

(

, ,

)

_{, we have}

( )

(

, ,

(

)

mod

( )

)

( )

.

t s x q i k j t s pk t s x

ψ+ = + + π =ψ ψ

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, _{, then} x y= . Sup-pose that

{

x1, , xn+1

}

are n+1 different points of X satisfying

( )

( )

(

t i , t j

)

, ,

( ) {

, 1, , 1

} {

1, , 1 .

}

d ψ x ψ x ≤a ∀ ∈t  ∀ i j ∈ _ n+ × _ n+

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

(

, ,

)

,

{

1, ,

}

m m m m

x =q i k j m∈ _ n _{. Because} i_m∈

{

1, ,_ n−1

}

_{and we have n}

DOI: 10.4236/jamp.2019.72031 415 Journal of Applied Mathematics and Physics 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∈_ we have

( )

(

)

(

)

(

(

( )

)

)

(

)

(

(

)

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

,

2

, , , , ,

, , , , , .

s j k s r k

s s

s s

d p d p

d q i k j q i k r k

d q i k j q i k r a

φ φ φ φ

ψ ψ

ψ ψ

= −

< <

This implies that φj

( )

pk =φr

( )

pk (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∈ we have:

( )

(

)

(

)

(

(

( )

)

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

(

(

)

)

(

)

,

1 1

, , , , ,

, , , , , .

s j k s r m

s s

s s

d p d p

d q i k j q i m r

k m

d q i k j q i m r a

φ φ φ φ

ψ ψ

ψ ψ

= − −

< <

So similarly, we have φj

( )

pk =φr

( )

pm ; hence, pm= pk, which is

contradic-tion 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∈ such that 1 a

k < , so we have

( ) (

)

(

_{j k} , , ,

)

1 , ,

{

1, , 1

}

d p q i k j a j i n

k

φ = < ∀ ∈ ∀ ∈  − . So Γ

(

p ak,

)

contain at

least n points

{

p q k_k, 1, ,0 , ,

(

)

_ q n

(

−1, ,0k

)

}

_{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 sha-dowing property, for each >0, we can consider δφ >0, so for any

( )

δφ,1

-pseudo-orbit in M we have the

2

_{-shadowing. Now consider}

₍

_,

₎

n n n

x t _∈ has

the

( )

δ,1 -pseudo-orbit by ψ _{in X. We assume that}

3 3

φ

δ

δ

< <_{. So we have}

( )

(

tn n , n 1

)

d ψ x x+ <δ . Let N is a smallest integer number such that _N1 <δ , and

we consider

(

x xn, n+1

)

in 3 cases.

Case 1. If

(

x xn, n+1

)

∈ ×E M, we have xn=q i k j

(

, ,

)

and

( )

(

tn n , n1

)

1 0

(

n1, j tn

( )

k

)

d x x d x p

k

ψ ₊ = + ₊ φ₊ , so 1_k<δ ; hence, k N≥ .

Case 2. If

(

x xn, n+1

)

∈M E× , we obtain xn+1=q i k j

(

, ,

)

and

( )

(

tn n , n 1

)

1 0

(

j

( ) ( )

k , j n

)

d x x d p x

k

ψ ₊ = + φ φ , so 1_k <δ ; hence, k N≥ _.

DOI: 10.4236/jamp.2019.72031 416 Journal of Applied Mathematics and Physics and jn+1= j tn+ ∀ ∈n, n . So one obtain sn = jn− j0, thus,

( ) ( )

(

t sn n , t 0

)

(

(

, , n n

) (

, , , 0

)

)

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

Therefore, the shadowing property is proved.

When x_i=q l k j

(

, ,

)

_{, then} k N> _{. Define a sequence}

(

y t_{n n n},

)

_∈ ⊂M _by

( )

ifif

(

,, , .

)

n n

n

j k n

x x M

y = _{φ p x} ∈_{=q l k j} 

Then

(

y t_{n n n},

)

_∈ is δφ-pseudo-orbit in M since

( )

(

)

(

( )

)

( )

( )

(

)

(

( )

)

(

)

1 1

1 1 1

, ,

, , ,

1 1 _.

n n

n n n

t n n t n n

t n t n t n n n n

d y y d y y

d y x d x x d x y

N N φ

φ ψ

ψ ψ ψ

δ δ

+ +

+ + +

=

≤ + +

< + + <

Hence, there exists y M∈ _{and a function} h Rep∈ _{such that}

( )

_{( )}

( )

(

t sn n , h t

)

₂, n n1.

d φ₋ y φ y < ∀ ≤ <s t s₊

So

( )

( )

( )

(

,

)

(

( )

,

( )

)

(

( )

, ( )

( )

)

1 _.

2

n n n n

t s n h t t s n t s n t s n h t

d x y d x y d y y

N

φ₋ φ < φ₋ φ₋ + φ₋ φ

< + < 

Therefore,

(

x t_{n n n},

)

_∈ is  -shadowing. Hence, ψ _{has the shadowing}

property.

Finally, we have ψ _{admits an infinite number of chain-recurrent classes.}

In-deed, if we have q i k l

(

, ,

)

∈E then

(

)

(

, , ,

)

1, \

{

(

, ,

)

}

.

d q i k j x x X q i k j

k

≥ ∀ ∈

So if 0 1

k

< < then the orbit of q i k j

(

, ,

)

cannot be connected by  -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

DOI: 10.4236/jamp.2019.72031 417 Journal of Applied Mathematics and Physics

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