The asymptotic average-shadowing property and transitivity for flows

2009 ◽  
Vol 41 (5) ◽  
pp. 2234-2240 ◽  
Author(s):  
Rongbao Gu
Keyword(s):  
Shadowing Property ◽  
Asymptotic Average Shadowing

scholarly journals A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Risong Li ◽  
Xiaoliang Zhou
Keyword(s):  
Metric Space ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Stable Map ◽  
Asymptotic Average Shadowing

We prove that if a continuous, Lyapunov stable mapffrom a compact metric spaceXinto itself is topologically transitive and has the asymptotic average shadowing property, thenXis consisting of one point. As an application, we prove that the identity mapiX:X→Xdoes not have the asymptotic average shadowing property, whereXis a compact metric space with at least two points.

Various Shadowing in Linear Dynamical Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950042 ◽  
Author(s):  
Xinxing Wu ◽  
Xu Zhang ◽  
Xin Ma
Keyword(s):  
Dynamical Systems ◽  
Linear Transformation ◽  
Positive Answer ◽  
Transformation Formula ◽  
Nonsingular Matrix ◽  
Linear Dynamical Systems ◽  
Shadowing Property ◽  
Matrix Formula ◽  
Asymptotic Average Shadowing

This paper proves that the linear transformation [Formula: see text] on [Formula: see text] has the (asymptotic) average shadowing property if and only if [Formula: see text] is hyperbolic, where [Formula: see text] is a nonsingular matrix, giving a positive answer to a question in [Lee, 2012a]. Besides, it is proved that [Formula: see text] does not have the [Formula: see text]-shadowing property, thus does not have the ergodic shadowing property for every nonsingular matrix [Formula: see text].

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