Asymptotic Average Shadowing Property and Chain Transitivity for Multiple Flow Systems

2016 ◽  
Vol 28 (2) ◽  
pp. 325-336 ◽  
Author(s):  
Mehdi Fatehi Nia
Keyword(s):  
Shadowing Property ◽  
Flow Systems ◽  
Chain Transitivity ◽  
Asymptotic Average Shadowing

Chain transitivity and variations of the shadowing property

2014 ◽  
Vol 35 (7) ◽  
pp. 2044-2052 ◽  
Author(s):  
WILLIAM R. BRIAN ◽  
JONATHAN MEDDAUGH ◽  
BRIAN E. RAINES
Keyword(s):  
Arbitrary Sequence ◽  
Ramsey Property ◽  
Shadowing Property ◽  
Chain Transitivity ◽  
A Chain

We show that, under the assumption of chain transitivity, the shadowing property is equivalent to the thick shadowing property. We also show that, if ${\mathcal{F}}$ is a family with the Ramsey property, then an arbitrary sequence of points in a chain transitive space can be ${\it\varepsilon}$-shadowed (for any ${\it\varepsilon}$) on a set in ${\mathcal{F}}$.

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].

Shadowing and average shadowing properties for iterated function systems

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Alireza Zamani Bahabadi
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Skew Product ◽  
Shadowing Property ◽  
Iterated Function ◽  
Chain Transitivity

AbstractIn this paper, we introduce the definitions of shadowing and average shadowing properties for iterated function systems and give some examples characterizing these definitions. We prove that an iterated function system has the shadowing property if and only if the step skew product corresponding to the iterated function system has the shadowing property. Also, we study some notions such as transitivity, chain transitivity, chain mixing and mixing for iterated function systems.

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