*Corresponding author
Received November 17, 2013
1 Available online at http://scik.org
Engineering Mathematics Letters, 2014, 2014:4 ISSN 2049-9337
TWO TYPES OF TRACING PROPERTIES IN NON-AUTONOMOUS
DYNAMICAL SYSTEM
XIAOQIAN SONG1,*, YING ZHANG2
1College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100,
China
2College of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China Copyright © 2014 Song and Zhang. 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: In this paper, we introduce the notions of asymptotic pseudo-orbit tracing property and pointwise pseudo-orbit tracing property in non-autonomous dynamical system, which are the generalizations of corresponding notions in autonomous dynamical system. We showed that if ( , )X F is topologically conjugate to( , )Y G , then Fhas asymptotic pseudo-orbit tracing property if and only if Ghas the same property. Also we discuss the relationship of pointwise pseudo-orbit tracing property between the product system and its subsystems.
Keywords: pointwise pseudo-orbit; asymptotic pseudo-orbit; topological conjugation; product system.
2000 AMS Subject Classification: 70F99
1. Introduction
Throughout this paper, by a non-autonomous dynamical system we mean a
pair( , )X F , whereXis a compact metric space with the metricd, and F fk k1is a
any point x1X x, 2 f x1 1 , x3 f2 x2 ,... , xn1fn xn ,... , n1, 2,... . If fk f, we call( , )X f an autonomous dynamical system.
Pseudo-orbit and its tracing skills are powerful tools in discussing autonomous
dynamic systems. Pseudo-orbit tracing property (POTP) has close relations with
chaotic properties of system. The concept of pseudo-orbit in autonomous dynamical
system has firstly appeared in the work of Anosov [1-2], and is closed related with the
property of stability [3] and chaos [4-5]. Li established the definition of pointwise
pseudo-orbit tracing property (PPOTP) in an autonomous dynamical system [6]. The
so-called asymptotic pseudo-orbit tracing property (APOTP) in autonomous
dynamical system was introduced by S. Y. Pilyugin. And many papers were devoted
to its study, see [7-9] for detail.
In this paper, we extend the notions of pointwise pseudo-orbit tracing property and
asymptotic pseudo-orbit tracing property to non-autonomous systems, and we show
that if the system( , )X F has the asymptotic pseudo-orbit tracing property, then any
system ( , )Y G conjugated with ( , )X F has the same property. Also we investigate the
relationship of pointwise pseudo-orbit tracing property between the product system
and its subsystem.
Our results extend the scope of the research on discrete dynamical system and
generalize the existing results to a very general case. The remainder of this paper is
organized as follows. Section 2 introduces some basic definitions and some notations,
also several new concepts are given in this part. Main results are established in
Section 3.
2. Preliminaries
Firstly we complete some notations and recall some concepts.
Let( , )X F be a non-autonomous dynamical system, for convenience, denote
2 1( )1 1( ),1 3 2 1( )1 2( ),1 ,
x f x F x x f f x F x xn1 fn fn1 f x1( )1 F xn( ),1 .
The following concept can be found in the classical mathematical theory. For the
sake of completeness, they are listed as follows.
Definition 2.1 The functionh X: Yis called a homeomorphism from a metric
space ( ,X dX) into a metric space ( ,Y dY) if, it is one-to-one and onto, and
bothhandh1are continuous.
Definition 2.2 Let( , )X F and( , )Y G be non-autonomous dynamical systems with metrics
X
d and dY respectively, where F fk k1 and G gk k1 . h X: Y be a
homeomorphism. If for anyk1,gk h x( )h f x xk( ), X, thenFandGare said to be conjugate orhconjugate.
Let( , )X F and( , )Y G be non-autonomous discrete systems and X Y be product
space with metric ''
1 1 2 2 1 2 1 2 (( , ), ( , )) max{ X( , ), Y( , )},
d x y x y d x x d y y where dX and dY are
metrics, F fk k 1
andG gk k1
are sequence of maps onX andY , respectively.
Let the sequence of maps F G {fi gi i} 1
be defined by ( , )x y X Y ,
(figi)( , )x y ( ( ),f x g yi i( )),i1, 2, . And it is easy to verify that
1 1 1 1
( ) ( , )i ( ) ( ) ( )( , ) ( )( , ).
i i i i i i
F G x y f g f g f g x y FG x y
Then we obtain the product system(X Y F G , )with metricd''.
Now, we introduce some new notions in non-autonomous discrete system.
Definition 2.3 Let( , )X F be a non-autonomous discrete systems. If 0, and for
any 0 i n , d f x( ( ),i i xi1), then the sequence{ ,x x1 2 ,xn}is called a -
pseudo-orbit of F (or -chain). If for any x y, X, 0, there exists a finite pseudo-orbit{ ,x x1 2 ,xn}of X , such that x1x x, ny , then { ,x x1 2 ,xn}is called a
-chain fromxtoy.
sequence{ ,x x1 2 ,xn}is-traced (>0) by the pointzinX if d F z x( n( ), n)for every
positive integern . We sayF has pointwise pseudo-orbit tracing property if for
any >0 there exists a real number 0 such that for any -pseudo-orbit
1 2
{ ,x x ,xn}ofF , {xN,xN1 ,}can be -traced for someN0, i.e. there exists a
pointzinXsuch thatd F z x( k( ), N k )fork0..
Definition 2.5 Let ( , )X F be a non-autonomous discrete systems. A sequence
1
{ }xi i in X is called an asymptotic pseudo-orbit of F if lim ( ( ),i i i1) 0
id f x x . A
sequence{ }xi i1
is said to be asymptoticcally pseudo-orbit tracing by the pointzinXif
lim ( n( ), n) 0
nd F z x . Fis said to have asymptotic pseudo-orbit tracing property if every
asymptotic pseudo orbit of Fcan be asymptotically pseudo-orbit tracing by some
point in X.
3. Main results
Theorem 3.1 Let X andY be compact metric spaces with metrics dX and dY
respectively, and F X: Xand G Y: Y be sequences of continuous maps. If Fis topologically conjugate toG, thenFhas the asymptotic pseudo-orbit tracing property
if and only if Ghas the same property.
Proof. It is enough to prove the necessity. SupposeFhas the asymptotic pseudo-orbit
tracing property, and let{ }yi i 0
be an asymptotic pseudo-orbit ofGandhbe a conjugate
function fromXtoY, by the compactness ofY, 1 :
h YX is uniformly continuous.
Then for any 0, there exists 0 such that
1 1
( , ) ( ( ), ( ))
Y X
d y y d h y h y fory y, Y.
As{ }yi i1is an asymptotic pseudo-orbit ofG, i.e.lim Y( ( ),i i i1) 0
id g y y , so there exists a
positive integerN1such that
1 ( ( ), )
Y i i i
Hence for eachiN1,
1 1
1 ( ( ( )), ( ))
X i i i
d h g y h y .
Let 1
( )
i i
x h y , the above inequality becomes
1
1
( ( )), )
X i i i
d h g h x x .
By the conjugation ofh, we have
1 ( ( )), )
X i i i
d f x x .
Hence
1 lim X( ( )),i i i ) 0
id f x x .
This shows that the sequence{ }xi i1
is an asymptotic pseudo-orbit ofF, sinceFhas the
asymptotic pseudo-orbit tracing property. Therefore there is a pointzinXsuch that
lim X( ( )), )i i 0
id F z x
.
To prove that{ }yi i1
can be asymptotically traced by some pointzinY, letzh z( ) , by the uniformly continuity ofh, for any 0, there exists 0such that
( , ) ( ( ), ( ))
X Y
d x x d h x h x forx x, X.
Then aslim X( ( )), )i i 0
id F z x
, there exists a positive integerN2such that
( ( )), )
X i i
d F z x for eachiN2.
Therefore for eachiN2, we have
( ( )), ( ))
Y i i
d h F z h x .
Ash FiGi h, the above inequality becomes
( ( )), ( ))
Y i i
d G h z h x ,
That isd G z yY( i( ), i), i.e.lim Y( i( ), i) 0
id G z y , this shows thatGhas the asymptotic
pseudo-orbit tracing property. It is similar to prove the sufficiency.
This completes the proof.
respectively, and F X: Xand G Y: Y be sequences of continuous maps. Then
the product mapF G has the pointwise pseudo-orbit tracing property if and only if
bothFandGhave the pointwise pseudo-orbit tracing properties.
Proof. Let sequences{ }xi i1
and{ }yi i1
be - pseudo-orbit ofFandGrespectively, i.e.
1 1
( ( ), ) , ( ( ), ) , 1, 2,
X i i i Y i i i
d f x x d g y y i .
Then
'' ''
1 1 1 1
(( ,i i)( ,i i), ( i , i )) (( ( ),i i i( )), (i i , i ))
d f g x y x y d f x g y x y
1 1
max{dX( ( ),f xi i xi),dY( ( ),g yi i yi)} .
This means the sequence{ , }x yi i i1
is the - pseudo-orbit ofF G . SinceF G has the
pointwise pseudo-orbit tracing property, for each 0, there exists a point ( , )z z in
X Y and a positive integerNsuch that
''(( ) ( , ), (k , )) ''(( k( ), k( ), ( , ))
N N N N
d F G z z x y d F z G z x y
max{ ( k( ), ), ( k( ), )} .
X N Y N
d F z x d G z y
Thus ( k( ), )
X N
d F z x and ( k( ), )
Y N
d G z y , this shows that FandGhave the pointwise
pseudo-orbit tracing property. It is similar to prove the sufficiency.
The proof of Theorem 3.2 is completed.
As an immediate consequence of Theorems 3.2, we have the following corollary.
Corollary 3.1 Let X1, ,Xn be compact metric spaces with metrics dXi on Xi
respectively, and F Xi: iXi be sequences of continuous maps(i1, ,n). Then the
product mapF1 Fnhas the pointwise pseudo-orbit tracing property if and only if
eachFihas the pointwise pseudo-orbit tracing property.
Conflict of Interests
REFERENCES
[1] Anosov V M. Geodesic. Flows on closed Riemannian manifolds with negative curvature [J]. Proc. Steklov Inst., 1967, 90.
[2] Slackov S V. Pseudo-orbit tracing property and structural stability of expanding maps of the interval [J]. Ergod. Th. Dynam. Sys., 1992, 12: 573-587.
[3] Komuro M. Lorenz attractors do not have the pseudo-orbit tracing property [J]. J. Math. Soc., 1985, 37: 489-514.
[4] Rongbao Gu. The asymptotic average shadowing property and transitivity [J]. Nonlinear Analysis: Theory, Methods & Applications, 2007:1680-1689.
[5] Manseob Lee. Stably asymptotic average shadowing property and dominated splitting [J]. Advances in Difference Equations, 2012: 2012:25.
[6] LI Ming jun, Pointwise Pseudo-Orbit Tracing Property and its Application [J]. J. Math. Res. Expo 2005, 25: 23-30.
[7] Sakai K. Various shadowing properties for positively expansive maps [J], Topology and its Appl., 2003, 131: 15-31.
[8] Gu R B. Recurrence and the asymptotic pseudo-orbit tracing property [J], Nonlinear Analysis, Nonlinear Analysis, 2007, 66: 1698–1706.
