The average-shadowing property and topological ergodicity

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1 Joural of Computatioal ad Applied Mathematics 206 (2007) The average-shadowig property ad topological ergodicity Rogbao Gu School of Fiace, Najig Uiversity of Fiace ad Ecoomics, Najig , PR Chia Received 6 August 2006; received i revised form 26 August 2006 Abstract I this ote we prove that a Lyapuov stable map havig the average-shadowig property from a compact metric space oto itself is topologically ergodic, but it is ot topologically weakly mixig Elsevier B.V. All rights reserved. MSC: 37A25; 34D20 Keywords: The average-shadowig property; Lyapuov stable; Topologically ergodic 1. Itroductio By a discrete dyamical system, we mea a pair (X, f ), where X is a metric space with metric d ad f is a cotiuous map from X ito itself. It is kow that a umerous class of real problems are modelled by a discrete dyamical system x +1 = f(x ), = 0, 1, 2,.... The basic goal of the theory of discrete dyamical systems is to uderstad the ature of all orbits x,f(x),f 2 (x),..., f (x) as becomes large ad, geerally, this is a impossible task. I cocrete situatios, we are ofte uable to compute the iitial coditio x exactly. We just compute a value x 0 close to x. It may also be the case that we caot compute f(x 0 ) exactly, but just a value x 1 close to f(x 0 ). The we compute a value x 2 close to f(x 1 ) ad so o. I this way, we obtai a sequece x 0,x 1,x 2,...that ca be thought of as the predicted behavior of the system (X, f ) at x. It is atural to ask whether or ot this predicted behavior is close to the actual behavior of the system. This leads to research o shadowig properties. The pseudo-orbit tracig property is oe of the most importat otios i dyamical systems (see [2]), which is closely related to stability ad chaos of systems, see, for istace, [5,7,8]. I[9], Yag discussed the relatioship betwee the pseudo-orbit tracig property ad topological ergodicity, ad showed that a chai trasitive system (X, f ) havig the pseudo-orbit tracig property is topologically ergodic. I a recet work, Blak [1] itroduced the otio of Project supported by the Natioal Natural Sciece Foudatio of Chia ( ) ad the Natural Sciece Foudatio of the Committee of Educatio of Jiagsu Provice (05KJB110033). address: rbgu@sia.com /$ - see frot matter 2006 Elsevier B.V. All rights reserved. doi: /j.cam

2 R. Gu / Joural of Computatioal ad Applied Mathematics 206 (2007) the average-shadowig property i studyig chaotic dyamical systems, which is a good tool to characterize Aosov diffeomorphisms (see [4]). Now a atural questio arises: which system havig the average-shadowig property is topologically ergodic? I this ote, we try to discuss this questio ad show that a Lyapuov stable system (X, f ) havig the average-shadowig property is topologically ergodic. I additio, we also show that such a system is ot topologically weakly mixig. 2. Some basic termiology Let (X, f ) be a dyamical system with metric d. Ifx X the the trajectory of x is the sequece O(x, f ) = {f (x)} 0 <. For δ > 0, a sequece {x i } 0 i< of poits i X is called a δ-average-pseudo-orbit of f if there is a positive iteger N = N(δ)>0 such that for every iteger N ad every oegative iteger k, 1 1 d(f(x i+k ), x i+k+1 )<δ. A map f is said to have the average-shadowig property, if for ay ε > 0 there is a δ > 0 such that every δ-averagepseudo-orbit {x i } 0 i< is ε-shadowed i average by some poit z X, that is, 1 1 d(f i (z), x i )<ε. A poit x X is said to be stable poit of f if for ay ε > 0 there is a δ > 0 such that d(f (x), f (y)) < ε for every y X with d(x,y)<δ ad every positive iteger. The f is called Lyapuov stable if every poit of X is a stable poit of f. The f is called sesitive depedece o iitial coditios if every poit of X is ot stable poit of f. If U ad V are two oempty subsets of X, the we let N(U,V)={ : f (U) V =, 0 < }. A map f is called topologically trasitive if for ay two oempty ope subsets U ad V of X, N(U,V) =. The f is called topologically weakly mixig if f f is topologically trasitive. The f is called topologically mixig if for ay two oempty ope subsets U ad V of X there is a positive iteger N such that N(U,V) {N,N + 1,...}. A map f is called topologically ergodic if for ay two oempty ope subsets U ad V of X, N(U,V) has positive upper desity, that is, D(N(U, V )) = Card[N(U,V) {0, 1,..., 1}] where Card(E) deotes the umber of members i the fiite set E. It is well kow that mixig 3. Results ad proofs weakly mixig ergodic trasitive. The mai result of the paper is the followig theorem. Theorem 3.1. Let X be a compact metric space ad f : X X be a Lyapuov stable map from X oto itself. If f has the average-shadowig property, the f is topologically ergodic. Proof. Suppose that U ad V are two oempty ope subsets of X. We choose x U, y V ad ε > 0 such that B(x,ε) U ad B(y,ε) V, where B(a,ε) ={b X : d(a,b)<ε}. > 0,

3 798 R. Gu / Joural of Computatioal ad Applied Mathematics 206 (2007) Sice f is Lyapuov stable, every poit x X is a stable poit of f, hece for ay ε > 0 there is a δ x > 0 such that d(f (x), f (y)) < ε for every positive iteger ad every poit y B(x,δ x ). By the compactess of X, there is a δ > 0 such that for ay u, v X, d(u, v) < δ implies d(f (u), f (v)) < ε for every positive iteger. Sice f has the average-shadowig property, let δ 1 = δ 1 (δ/2) be a positive umber as i the defiitio of the average-shadowig property. Let D = diam(x) = sup{d(x,y) : x,y X} be the diameter of X. We choose a positive iteger N 0 such that 3D/N 0 < δ 1. Defie a periodic sequece {w i } 0 i< such that w i = x N0 +[i mod 2N 0 ] if [i mod 2N 0 ] {1, 2,...,N 0 }, w i = y 2N0 +[i mod 2N 0 ] if [i mod 2N 0 ] {N 0 + 1,N 0 + 2,...,2N 0 }, where x 0 = x, y 0 = y, ad x i f 1 (x i+1 ), y i f 1 (y i+1 ), for i = 1, 2,...,N 0 1. That is, the terms of the sequece from i = 1to2N 0 are x N0 +1,x N0 +2,...,x 1,x 0, y N0 +1,y N0 +2,...,y 1,y 0. It is easy to see that for N 0 ad 0 k<, 1 1 d(f(w k+i ), w k+i+1 )< [/N 0] 3D 3D N 0 < δ 1. Thus {w i } 0 i< is a periodic δ 1 -average-pseudo-orbit of f. Hece, it ca be δ/2-shadowed i average by some w X, that is, 1 1 For z {x,y}, let d(f i (w), w i )< δ 2. J z ={i : w i {z N0 +1,z N0 +2,...,z 1,z} ad d(f i (w), w i )<δ}. We have the followig claim: Claim. J z has positive upper desity, that is, D(J z )>0. Proof. Without loss of geerality, we assume z = x. Suppose o the cotrary that D(J x ) = 0, the we have Let Card(J x {0, 1,..., 1}) lim = 0. J x ={i : w i {x N0 +1,x N0 +2,...,x 1,x} ad d(f i (w), w i ) δ}. The Card(J x {0, 1,..., 1}) lim = 1 2. Hece, for ay ρ (0, 1/2) there is a positive iteger N such that Card(J x {0, 1,..., 1}) > 1 2 ρ,

4 R. Gu / Joural of Computatioal ad Applied Mathematics 206 (2007) for every N. Thus, 1 1 Sice ρ is arbitrary, we have 1 1 d(f i (w), w i ) d(f i (w), w i ) 1 2 δ. 1 Card(J x δ ( ) 1 δ 2 ρ. i J x {0,1,..., 1} d(f i (w), w i ) {0, 1,..., 1}) This is a cotradictio. Therefore D(J x )>0. The proof of claim is completed. Now, let J m (y)={i J y : w i =y m }, for 0 m N 0 1. The, by claim, there is a iteger m 0 with 0 m 0 N 0 1 such that D(J m0 (y)) > 0. Choose i 0 N 0 ad 0 k 0 N 0 1 such that f i 0(w) B(x k0, δ). For ay j J m0 (y) with j i 0 + k 0,wehave f j (w) B(y m0, δ). Sice f is Lyapuov stable, we have f i 0+k 0 (w) B(x,ε) ad f j+m 0 (w) B(y,ε). Let j = (j + m 0 ) (i 0 + k 0 ), the f j (B(x, ε)) B(y,ε) =. So, f j (U) V =. Hece, D(K(U, V )) D(J m0 (y)) > 0. This shows that f is topologically ergodic. The proof of Theorem 3.1 is completed. The followig theorem is due to Yag [9]. For completeess, we give a proof of this theorem. Theorem 3.2. Let X be a compact metric space cotaiig at least two poits ad f : X X be cotiuous map. If f is topologically weakly mixig the f is sesitive depedet o iitial coditios. Proof. Suppose o the cotrary that f is ot sesitive depedet o iitial coditios. There is at least oe stable poit z of f. Suppose that a ad b are two distict poits of X. We choose two ope subsets U ad V of X such that a U U, b V V ad U V =. Let d = d(u,v). Clearly, d>0. For ay ε (0,d), there is δ > 0 such that if d(z,z )<δ the we have d(f (z), f (z )) < ε/2 for all positive iteger. Sice f is topologically weakly mixig, there is a positive iteger N such that (f f) N (B(z, δ) B(z,δ)) (U V) =. It follows that f N (B(z, δ)) U = ad f N (B(z, δ)) V =. Hece, there are x B(z,δ) such that f N (x) U ad y B(z,δ) such that f N (y) V. Sice z is stable poit for f,wehave d(f N (x), f N (z)) < ε 2 ad d(f N (y), f N (z)) < ε 2. Thus, d(f N (x), f N (y)) d(f N (x), f N (z)) + d(f N (z), f N (y)) < ε 2 + ε 2 = ε. Note that f N (x) U ad f N (y) V,wehaved(U,V)<ε. O the other had, d(u,v) d(u,v)= d>ε. This is a cotradictio. Therefore, f is sesitive depedet o iitial coditios. The proof of Theorem 3.2 is completed.

5 800 R. Gu / Joural of Computatioal ad Applied Mathematics 206 (2007) Remark 3.3. Lardjae [3] studied the relatioship betwee topological mixig ad sesitive depedece o iitial coditios ad proved that if f is topologically mixig the it is sesitive depedet o iitial coditios. This coclusio of Lardjae is a corollary of Theorem 3.2, because topological mixig implies topologically weak mixig. Theorem 3.4. Let X be a compact metric space ad f : X X be a Lyapuov stable map from X oto itself havig the average-shadowig property, the f is topologically ergodic, but f is ot topologically weakly mixig. Further, f is ot topologically mixig. Proof. It is obtaied directly from Theorems 3.1 ad Coclusio We have kow from Xiog [6] that topological trasitivity is strictly weaker tha topological ergodicity. I this paper we show that topological ergodicity is strictly weaker tha topological mixig. Moreover, we also show that topological ergodicity is differet to topologically weak mixig, although both properties lie betwee topological trasitivity ad topological mixig. Refereces [1] M.L. Blak, Small perturbatios of chaotic dyamical systems, Russia Math. Surveys 44 (1989) [2] R. Bowe, Equilibrium States ad the Ergodic Theory of Axiom A Diffeomorphisms, Spriger, New York, 1975, pp [3] S. Lardjae, O some stochastic property i Devaey s chaos, Chaos Solitos Fractals 28 (2006) [4] K. Sakai, Diffeomorphisms with the average-shadowig property o two dimesioal closed maifold, Rocky Moutai J. Math. 3 (2000) 1 9. [5] P. Walters, O the pseudo-orbit tracig property ad its relatioship to stability, Lecture Notes i Mathematics, vol. 668, Spriger, Berli, 1978, pp [6] J.C. Xiog, Chaos o topological trasitive systems, Sci. Chia Ser. A: Math. 35 (2005) [7] R.S. Yag, The pseudo-orbit tracig property ad chaos, Acta Math. Siica 39 (1996) (i Chiese). [8] R.S. Yag, Pseudo-orbit tracig property ad completely positive etropy, Acta Math. Siica 42 (1999) (i Chiese). [9] R.S. Yag, Topologically ergodic map, Acta Math. Siica 44 (2001) (i Chiese).