Properties of dynamical systems with the asymptotic average shadowing property

2011 ◽  
Vol 212 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Marcin Kulczycki ◽  
Piotr Oprocha
Keyword(s):  
Dynamical Systems ◽  
Shadowing Property ◽  
Asymptotic Average Shadowing

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].

Average Shadowing Property and Asymptotic Average Shadowing Property of Linear Dynamical Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950170
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Invertible Operator ◽  
Linear Dynamical Systems ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Asymptotic Average Shadowing ◽  
Necessary And Sufficient

This paper investigates the average shadowing property and the asymptotic average shadowing property of linear dynamical systems in Banach spaces. Firstly, necessary and sufficient conditions for an invertible operator [Formula: see text] on a Banach space to have the average shadowing property and the asymptotic average shadowing property are given, respectively. Then, it is concluded that both the average shadowing property and the asymptotic average shadowing property are preserved under iterations. Furthermore, if [Formula: see text] is hyperbolic, then [Formula: see text] has the (asymptotic) average shadowing property. However, the inverse implication fails in infinite-dimensional Banach spaces. Finally, it is proved that the (asymptotic) average shadowing property is equivalent to the hyperbolicity for dynamical systems in a finite-dimensional Banach space.

On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems

2020 ◽  
Vol 36 (12) ◽  
pp. 1384-1394
Author(s):  
Xiao Fang Luo ◽  
Xiao Xiao Nie ◽  
Jian Dong Yin
Keyword(s):  
Dynamical Systems ◽  
Shadowing Property

Dynamics of G-processes

2020 ◽  
Vol 20 (01) ◽  
pp. 2050037
Author(s):  
W. Jung ◽  
K. Lee ◽  
C.A. Morales
Keyword(s):  
Dynamical Systems ◽  
Group Structure ◽  
Global Solutions ◽  
Topological Stability ◽  
Natural Group ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Mathematical Sciences ◽  
The Relationship

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176 (Amer. Math. Soc., 2011)]) for which the role of evolution parameter is played by a general topological group [Formula: see text]. These processes are broad enough to include the [Formula: see text]-actions (characterized as autonomous [Formula: see text]-processes) and the two-parameter flows (where [Formula: see text]). We endow the space of [Formula: see text]-processes with a natural group structure. We introduce the notions of orbit, pseudo-orbit and shadowing property for [Formula: see text]-processes and analyze the relationship with the [Formula: see text]-processes group structure. We study the equicontinuous [Formula: see text]-processes and use it to construct nonautonomous [Formula: see text]-processes with the shadowing property. We study the global solutions of the [Formula: see text]-processes and the corresponding global shadowing property. We study the expansivity (global and pullback) of the [Formula: see text]-processes. We prove that there are nonautonomous expansive [Formula: see text]-processes and characterize the existence of expansive equicontinuous [Formula: see text]-processes. We define the topological stability for [Formula: see text]-processes and prove that every expansive [Formula: see text]-process with the shadowing property is topologically stable. Examples of nonautonomous topologically stable [Formula: see text]-processes are given.

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.

Shadowing and structural stability for operators

2020 ◽  
pp. 1-20
Author(s):  
NILSON C. BERNARDES ◽  
ALI MESSAOUDI
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Structural Stability ◽  
Invertible Operator ◽  
Hyperbolic Operator ◽  
Weighted Shifts ◽  
Shadowing Property ◽  
Finite Dimensional ◽  
Structurally Stable

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_{0}(\mathbb{Z})$ and $\ell _{p}(\mathbb{Z})$ ( $1\leq p<\infty$ ) that satisfy the shadowing property.

Sign in / Sign up

Export Citation Format

Share Document