scholarly journals On random linear dynamical systems in a Banach space. I. Multiplicative Ergodic Theorem and Krein–Rutman type Theorems

2017 ◽  
Vol 312 ◽  
pp. 374-424 ◽  
Author(s):  
Zeng Lian ◽  
Yi Wang
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Ergodic Theorem ◽  
Linear Dynamical Systems ◽  
Multiplicative Ergodic Theorem

ON THE ABSOLUTE REGULARITY OF LINEAR RANDOM DYNAMICAL SYSTEMS

2003 ◽  
Vol 03 (04) ◽  
pp. 453-461 ◽  
Author(s):  
LUU HOANG DUC
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theorem ◽  
Random Dynamical Systems ◽  
Absolute Regularity ◽  
Integrability Conditions ◽  
Multiplicative Ergodic Theorem ◽  
Lyapunov Regularity ◽  
The Absolute

We introduce a concept of absolute regularity of linear random dynamical systems (RDS) that is stronger than Lyapunov regularity. We prove that a linear RDS that satisfies the integrability conditions of the multiplicative ergodic theorem of Oseledets is not merely Lyapunov regular but absolutely regular.

scholarly journals Non-recurrence sets for weakly mixing linear dynamical systems

2012 ◽  
Vol 34 (1) ◽  
pp. 132-152 ◽  
Author(s):  
SOPHIE GRIVAUX
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Gaussian Measure ◽  
Bounded Linear Operator ◽  
Probability Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Weakly Mixing

AbstractWe study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space$X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets$\{n_{k}\}$such that$n_{k+1}/n_{k}\to +\infty $, or such that$n_{k}$divides$n_{k+1}$for each$k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each$r\ge 1$, of$r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

scholarly journals Notes on the multiplicative ergodic theorem

2017 ◽  
Vol 39 (5) ◽  
pp. 1153-1189 ◽  
Author(s):  
SIMION FILIP
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theorem ◽  
Basic Result ◽  
Multiplicative Ergodic Theorem ◽  
Summer Schools

The Oseledets multiplicative ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.

Pseudo-orbit tracing property for random diffeomorphisms

1996 ◽  
Vol 126 (5) ◽  
pp. 1027-1033
Author(s):  
Liu Peidong ◽  
Qian Minping ◽  
Tang Fuchang
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theorem ◽  
Graph Transformation ◽  
Multiplicative Ergodic Theorem ◽  
Deterministic Dynamical Systems ◽  
Transformation Methods ◽  
Hyperbolic Sets

In this paper we consider the pseudo-orbit tracing property for dynamical systems generated by iterations of random diffeomorphisms. We first define a type of hyperbolicity by means of a ‘random’ multiplicative ergodic theorem, and then prove our shadowing result by employing the graph transformation methods. That result applies to, for example, the case of small random diffeomorphisms type perturbations of hyperbolic sets of deterministic dynamical systems.

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.

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