Research Article A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

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1 Discrete Dyamics i Nature ad Society Volume 2011, Article ID , 6 pages doi: /2011/ Research Article A Note o Ergodicity of Systems with the Asymptotic Average Shadowig Property Risog Li ad Xiaoliag Zhou School of Sciece, Guagdog Ocea Uiversity, Zhajiag , Chia Correspodece should be addressed to Risog Li, gdoulrs@163.com Received 23 July 2011; Revised 22 September 2011; Accepted 5 October 2011 Academic Editor: Victor S. Kozyaki Copyright q 2011 R. Li ad X. Zhou. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We prove that if a cotiuous, Lyapuov stable map f from a compact metric space X ito itself is topologically trasitive ad has the asymptotic average shadowig property, the X is cosistig of oe poit. As a applicatio, we prove that the idetity map i X : X X does ot have the asymptotic average shadowig property, where X is a compact metric space with at least two poits. 1. Itroductio It is well kow that the pseudoorbit tracig property is oe of the most importat otios i dyamical systems see 1 4. Blak 5 itroduced the otio of the average shadowig property to characterize Aosov diffeomorphisms see 6. Yag 7 proved that if a cotiuous map f : X X of a compact metric space X has the pseudoorbit tracig property ad is chai trasitive, the it is topologically ergodic. Gu ad Guo 8 discussed the relatio betwee the average shadowig property ad topological ergodicity for flows ad showed that a Lyapuov stable flow with the average shadowig property is topologically ergodic. I a recet work, the author 9 itroduced the otio of the asymptotic average shadowig property ad foud that the asymptotic average shadowig property is closely related with trasitivity. More recetly, Gu 10 showed that every Lyapuov stable map from a compact metric space oto itself is topologically ergodic, provided it has the asymptotic average shadowig property. However, i this paper, we will show that a cotiuous, Lyapuov stable map f from a compact metric space X ito itself is topologically trasitive ad has the asymptotic average shadowig property, the X is cosistig of oe poit. As a applicatio, it is show that the idetity map i X : X X does ot have the asymptotic average shadowig property,

2 2 Discrete Dyamics i Nature ad Society where X is a compact metric space with at least two poits. Moreover, we poit out that the proof of 10, Theorem 3.1 caot be true. The orgaizatio of this paper is as follows. I Sectio 2, we recall some cocepts ad useful lemmas. Mai results are established i Sectio Prelimiaries Firstly, we complete some otatios ad recall some cocepts. I this paper, by a dyamical system we mea a pair X, f, where f : X X is a cotiuous map ad X is a compact metric space with metric d. For x X, we write O x, f {f x } 0.Let A deote the cardiality of a set A ad Z {0, 1,...}. AsubsetS Z is called the positive upper desity if lim sup k 1 { 0 j k : j S } > 0. k For ay two oempty sets U, V X, we write N f U, V { Z : U f V / }. Obviously, we have N f U, V { Z : f U V / }. A map f : X X is topologically trasitive if N f U, V is oempty, for ay oempty ope sets U, V X. A map f : X X is topologically ergodic if N f U, V has positive upper desity, for ay oempty ope sets U, V X. A map f : X X is topologically weak mixig if f f is topologically trasitive. Let X, f be a dyamical system. For δ>0, a sequece {x i } of poits i X is called a δ pseudoorbit of f if d f x i,x i 1 <δfor all i 0. A sequece {x i } is said to be ε traced by some poit z X if d f i z,x i <εfor every i 0. A poit x X is said to be a stable poit of f if, for ay ε>0, there exists δ δ x > 0 satisfyig that d f x,f y <εfor ay y X with d x, y <δad ay 0. A map f : X X is called Lyapuov stable, if every poit of X is a stable poit of f. A map f : X X is said to have sesitive depedece o iitial coditios if every poit of X is ot a stable poit of f. A map f : X X is said to have the pseudoorbit tracig property, if, for ay ε>0, there exists a δ>0such that every δ pseudoorbit of f ca be ε traced by some poit i X. A sequece {x i } of poits i X is called a δ average pseudoorbit of f if there exists a iteger N N δ > 0 such that for ay iteger N ad every iteger k 0, d ( f x i k,x i k 1 <δ. 2.2 A map f : X X is said to have the average shadowig property, if for ay ε>0there exists a δ>0such that every δ average pseudoorbit {x i } of f is ε shadowed i average by some poit z i X, thatis, lim sup d (f i z,x i <ε. 2.3

3 Discrete Dyamics i Nature ad Society 3 A sequece {x i } of poits i X is called a asymptotic average pseudoorbit of f if lim d ( f x i,x i For a give dyamical system X, f, a sequece {x i } of poits i X is said to be asymptotically shadowed i average by some poit z X if lim d (f i z,x i A map f : X X is said to have the asymptotic average shadowig property, if every asymptotic average pseudoorbit of f is asymptotically shadowed i average by some poit i X. We kow from 9 that the pseudoorbit tracig property does ot imply the asymptotic average shadowig property ad the asymptotic average shadowig property does ot imply the pseudoorbit tracig property. 3. Mai Results The followig lemmas will be used i the proof of Theorem 3.5. Lemma 3.1 see 10. Let X be a compact metric space cotaiig at least two poits ad f : X X a cotiuous map. If f is topologically weakly mixig, the f has sesitive depedece o iitial coditios. Let X, f ad X,g be dyamical systems ad X X the product space with metric d x, x, y, y max{d x, y,d x,y }, where d ad d are, respectively, metrics for X ad X. Let the map f g be defied by f g x, x f x,g x x X, x X. Lemma 3.2. Let X, f be a dyamical system. The, f has the asymptotic average shadowig property if ad oly if f f has the asymptotic average shadowig property. Proof. Suppose that f has the asymptotic average shadowig property. Let { x i,y i } be a asymptotic average pseudoorbit of f f, thatis, lim d (( f f ( ( x i,y i, xi 1,y i This implies that lim d ( f x i,x i 1 0, lim d ( f ( y i,yi

4 4 Discrete Dyamics i Nature ad Society So {x i } ad {y i} are asymptotic average pseudoorbits of f. Hece, there are two poits z 1,z 2 X such that lim d (f i z 1,x i 0, lim d (f i z 2,y i By Lemma 2.3 i 9 ad 3.3, there is a set J 0 Z of zero desity such that ( lim d f j z 1,x j 0, 3.5 j where j/ J 0. Similarly, there is a set J 1 Z of zero desity such that ( lim d f j z 2,y j 0, 3.6 j where j/ J 1.LetJ J 0 J1. The, J is a subset of Z of zero desity ad lim d ( ( j z1 f f,z 2, ( x j,y j 0, j 3.7 where j/ J. So, by Lemma 2.3 i 9, we have lim d ( ( j z1 f f,z 2, ( x j,y j Cosequetly, f f has the asymptotic average shadowig property. Similarly, oe ca easily prove that if f f has the asymptotic average shadowig property, the so does f. Thus, the proof is eded. Lemma 3.3. A topologically trasitive map from a compact metric space ito itself is a surjective map. Proof. By the defiitio, the proof is easy ad is omitted here. We do ot kow whether a cotiuous self-map of a compact metric space with the asymptotic average shadowig property is topologically trasitive. However, we have the followig lemma which is from 11. Lemma 3.4. Let X, d be a compact metric space. If f : X X is a equicotiuous surjectio ad has the asymptotic average shadowig property, the f is topologically trasitive. I 10, Gu proved that every Lyapuov stable map from a compact metric space oto itself is topologically ergodic, provided it has the asymptotic average shadowig property. However, we obtai the followig theorem.

5 Discrete Dyamics i Nature ad Society 5 Theorem 3.5. If a Lyapuov stable map f from a compact metric space X ito itself is topologically trasitive ad has the asymptotic average shadowig property, the X is cosistig of oe poit. Proof. Let f : X X be a Lyapuov stable map of a compact metric space X, ad suppose that f is topologically trasitive ad has the asymptotic average shadowig property. Assumig that X cosists of at least two poits, we derive a cotradictio. First of all, we ote that every Lyapuov stable map is cotiuous by the defiitio ad that f is a surjective map by Lemma 3.3 sice f is topologically trasitive. Sice f f also has the asymptotic average shadowig property by Lemma 3.2, f f is topologically trasitive by Lemma 3.4.Thus,f is topologically weakly mixig by defiitio. Hece, by Lemma 3.1, f has sesitive depedece o iitial coditios, so that f is ot Lyapuov stable. This is a cotradictio. Remark 3.6. Theorem 3.5 shows that 10, Theorem 3.1 caot be true. Ideed, let X be a compact metric space cotaiig at least two poits, ad let f : X X be a cotiuous Lyapuov stable map. Suppose that f has the asymptotic average shadowig property. If 10, Theorem 3.1 is correct, the f is topologically ergodic, so that f is topologically trasitive by defiitio. Thus, by Theorem 3.5 proved above, X must be cosistig oe poit. This is a cotradictio, ad, thus, 10, Theorem 3.1 caot be true. To prove Theorem 3.1 i 10, the author first costruct a sequece {w i } as follows. For ay two oempty ope subsets U, V X ad ay ε>0, choose x U, y V with {z X : d x, z <ε} Uad {z X : d y, z <ε} V.Letw 0 x, w 1 y, w 2 x, w 3 y, w 4 x 1, w 5 x, w 6 y 1, w 7 y, w 2 k j x 2 k 1 1 j for j {0, 1,...,2 k 1 1}, adw 2 k j y 2 k 1 1 j 2 k 1 for j {2k 1, 2 k 1 1,...,2 k 1}, where f x t x t 1 for every t>0,x 0 x ad f y l y l 1 for every l>0,y 0 y. Ad the the author showed that the sequece {w i } is a asymptotic average pseudoorbit of f. Fially, the author defied J z {i : w i {z 2 i 1 1, z 2 i 1 2,...,z 1,z} ad d f i w,w i <δ} for z {x, y} with the positive costat δ satisfyig that for ay u, v X, d u, v <δimplies d f u,f v <εfor every iteger 0ad J m ( y { i Jy : w i y m } 3.9 for every m 0 ad showed that there is a iteger m 0 0 such that D J m0 y > 0 which leads to the fact that f is topologically ergodic, where D J m0 y is the upper desity of the set J m0. Remark 3.7. I fact, for a give iteger m 0aday m 1, let k l / l 2, where l / l 2 is the greatest iteger less tha or equal to l / l 2. The, by the defiitio of the sequece {w i }, D J m y k 1 /2 k, which implies D J m y 0. Cosequetly, the proof of Theorem 3.1 from 10 is ot correct. As a applicatio, we obtai the followig theorem which follows from Theorem 3.1 i 12. For completeess, we ow give a differet proof here. Theorem 3.8. Let X, d be a compact metric space with at least two poits. The the idetity map i X : X X does ot have the asymptotic average shadowig property. Proof. Clearly, the idetity map i X is Lyapuov stable. Assume that the idetity map i X : X X has the asymptotic average shadowig property. By Lemma 3.4, the idetity map i X is topologically trasitive. Sice X, d is a compact metric space with at least two poits, by

6 6 Discrete Dyamics i Nature ad Society Theorem 3.5, the idetity map i X does ot have the asymptotic average shadowig property. Thus, the proof is eded. Remark 3.9. Theorem 3.8 exteds Example 5.1 from 9, ad the proof of Theorem 3.8 is simpler tha that of Example 5.1 from 9. Ackowledgmets The authors thak the ukow referee of the paper for may valuable suggestios, i particular for givig a clear proof of Theorem 3.5 ad a good descriptio of Remark 3.6. This research was supported by the NSF of Guagdog Provice Grat o ad the Key Scietific ad Techological Research Project of Sciece ad Techology Departmet of Zhajiag City Grat o. 2010C Refereces 1 R. Bowe, Equilibrium States ad the Ergodic Theory of Axiom A Diffeomorphisms, Spriger, New York, NY, USA, P. Walters, O the Pseudo-Orbit Tracig Property ad its Relatioship to Stability, vol. 668 of Lecture Notes i Mathematics, Spriger, New York, NY, USA, R. S. Yag, Pseudo-orbit tracig ad chaos, Acta Mathematica Siica, vol. 39, o. 3, pp , 1996 Chiese. 4 R. S. Yag ad S. L. She, Pseudo-orbit-tracig ad completely positive etropy, Acta Mathematica Siica, vol. 42, o. 1, pp , 1999 Chiese. 5 M. L. Blak, Small perturbatios of chaotic dyamical systems, Russia Mathematical Surveys, vol. 44, o. 6, pp. 1 33, K. Sakai, Diffeomorphisms with the average-shadowig property o two dimesioal closed maifolds, The Rocky Moutai Joural of Mathematics, vol. 30, o. 3, pp. 1 9, R. S. Yag, Topologically ergodic maps, Acta Mathematica Siica, vol. 44, o. 6, pp , 2001 Chiese. 8 R. B. Gu ad W. J. Guo, The average-shadowig property ad topological ergodicity for flows, Chaos, vol. 25, o. 2, pp , R. B. Gu, The asymptotic average shadowig property ad trasitivity, Noliear Aalysis, vol. 67, o. 6, pp , R. B. Gu, O ergodicity of systems with the asymptotic average shadowig property, Computers & Mathematics with Applicatios, vol. 55, o. 6, pp , Y. X. Niu, Dyamical systems with the asymptotic average shadowig property, Applied Mathematics: A Joural of Chiese Uiversities: Series A, vol. 22, o. 4, pp , M. Kulczycki ad P. Oprocha, Explorig the asymptotic average shadowig property, Joural of Differece Equatios ad Applicatios, vol. 16, o. 10, pp , 2010.

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