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Exploring the asymptotic average shadowing property. (English) Zbl 1205.54034
Let $$f: X\to X$$ be a continuous selfmap of a compact metric space $$X$$. A sequence $$\{x_i\}^\infty_{i=0}\subseteq X$$ is called an asymptotic average pseudo-orbit of $$f$$ if
$\lim_{n\to\infty} {1\over n+1} \sum^n_{i=0} d(f(x_i), x_{i+1})= 0,$
and is said to be asymptotic shadowed in average by the point $$z\in X$$ if
$\lim_{n\to\infty} {1\over n+1} \sum^n_{i=0} d(x_i, f^i(z))= 0.$
A map $$f$$ is said to have the asymptotic average shadowing property (abbreviated AASP) if every asymptotic average pseudo-orbit of $$f$$ is asymptotically shawdowed in average by some point in $$X$$. The authors prove the following statement: Assume that there is a point $$x\in X$$ that has the closure of its orbit contained in some open $$f$$-invariant subset $$U\subseteq X$$. Assume that $$y\in X$$ is a point whose orbit is metrically separated from $$U$$. Then the map $$f$$ does not have the property AASP.

##### MSC:
 54H20 Topological dynamics (MSC2010) 34D05 Asymptotic properties of solutions to ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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##### References:
 [1] Akin E., Graduate Studies in Mathematics 1 (1993) [2] Aoki N., Recent Advances. North-Holland Mathematical Library 52 (1994) [3] DOI: 10.1017/S0143385797069885 · Zbl 0921.54029 · doi:10.1017/S0143385797069885 [4] DOI: 10.1307/mmj/1029003477 · Zbl 0655.58023 · doi:10.1307/mmj/1029003477 [5] DOI: 10.1017/S014338570000451X · Zbl 0659.58031 · doi:10.1017/S014338570000451X [6] Blank M.L., Translations of Mathematical Monographs 161 (1997) [7] Blokh A.M., Dynam. Report. Expositions Dynam. Systems (N.S.) 4 (1995) [8] Bowen R., Trans. Amer. Math. Soc. 154 pp 377– (1971) [9] Bowen R., Lecture Notes in Mathematics 470 (1975) [10] DOI: 10.1090/S0002-9947-97-01873-4 · Zbl 0870.58017 · doi:10.1090/S0002-9947-97-01873-4 [11] Conley R., Regional Conference Series in Mathematics 38 (1978) [12] Denker M., Lecture Notes in Mathematics 527 (1976) [13] Downarowicz T., Algebraic and Topological Dynamics, Contemporary Mathematics 385 (2005) · Zbl 1096.37002 [14] DOI: 10.1080/01630569708816748 · Zbl 0881.58049 · doi:10.1080/01630569708816748 [15] DOI: 10.1016/j.na.2006.07.040 · Zbl 1121.37011 · doi:10.1016/j.na.2006.07.040 [16] DOI: 10.3934/dcds.2005.13.533 · Zbl 1078.37015 · doi:10.3934/dcds.2005.13.533 [17] Pilyugin S.Yu., Lecture Notes in Mathematics 1706 (1999) [18] DOI: 10.1007/s10884-007-9073-2 · Zbl 1128.37018 · doi:10.1007/s10884-007-9073-2 [19] Plamenevskaya O., Vestnik St. Petersburg Univ. Math. 30 pp 27– (1997) [20] Rosenholtz I., Fund. Math. 91 pp 1– (1976) [21] Sakai K., Topology Appl. 131 pp 15– (2003) · Zbl 1024.37016 · doi:10.1016/S0166-8641(02)00260-2 [22] DOI: 10.1090/S0002-9947-1974-0352411-X · doi:10.1090/S0002-9947-1974-0352411-X [23] Walters P., An Introduction to Ergodic Theory (1982) · Zbl 0475.28009 · doi:10.1007/978-1-4612-5775-2
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