Notes on the multiplicative ergodic theorem

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.

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

Advances in Mathematics ◽

10.1016/j.aim.2017.03.024 ◽

2017 ◽

Vol 312 ◽

pp. 374-424 ◽

Cited By ~ 4

Author(s):

Zeng Lian ◽

Yi Wang

Keyword(s):

Banach Space ◽

Dynamical Systems ◽

Ergodic Theorem ◽

Linear Dynamical Systems ◽

Multiplicative Ergodic Theorem

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ON THE ABSOLUTE REGULARITY OF LINEAR RANDOM DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s021949370300084x ◽

2003 ◽

Vol 03 (04) ◽

pp. 453-461 ◽

Cited By ~ 1

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.

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Pseudo-orbit tracing property for random diffeomorphisms

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023234 ◽

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.

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Induced transformations of recurrent a.m.s. dynamical systems

Stochastics and Dynamics ◽

10.1142/s0219493715500100 ◽

2015 ◽

Vol 15 (02) ◽

pp. 1550010

Author(s):

Sheng Huang ◽

Mikael Skoglund

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Random Process ◽

Ergodic Theorem ◽

Finite Measure ◽

Stationary Random Process ◽

Finite State ◽

Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

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