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 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.

AN OPEN SET OF UNBOUNDED COCYCLES WITH SIMPLE LYAPUNOV SPECTRUM AND NO EXPONENTIAL SEPARATION

2007 ◽  
Vol 07 (03) ◽  
pp. 335-355 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
DOAN THAI SON
Keyword(s):  
Ergodic Theorem ◽  
Integrability Condition ◽  
Lyapunov Spectrum ◽  
Separation Property ◽  
Open Set ◽  
Exponential Separation ◽  
Multiplicative Ergodic Theorem

We give an example of an open set of unbounded cocycles satisfying the integrability condition of the multiplicative ergodic theorem such that all the cocycles in this open set have simple Lyapunov spectrum but have no exponential separation. Thus, unlike the bounded case, the exponential separation property is nongeneric in the space of unbounded cocycles.

scholarly journals Information Gain in Event Space Reflects Chance and Necessity Components of an Event

Information ◽  
2019 ◽  
Vol 10 (11) ◽  
pp. 358 ◽  
Author(s):  
Georg F. Weber
Keyword(s):  
Phase Space ◽  
Ergodic Theorem ◽  
Information Gain ◽  
Characteristic Exponent ◽  
Vector Spaces ◽  
Event Space ◽  
Lyapunov Characteristic Exponent ◽  
Multiplicative Ergodic Theorem ◽  
Non Linear Systems ◽  
Eigenvectors And Eigenvalues

Information flow for occurrences in phase space can be assessed through the application of the Lyapunov characteristic exponent (multiplicative ergodic theorem), which is positive for non-linear systems that act as information sources and is negative for events that constitute information sinks. Attempts to unify the reversible descriptions of dynamics with the irreversible descriptions of thermodynamics have replaced phase space models with event space models. The introduction of operators for time and entropy in lieu of traditional trajectories has consequently limited—to eigenvectors and eigenvalues—the extent of knowable details about systems governed by such depictions. In this setting, a modified Lyapunov characteristic exponent for vector spaces can be used as a descriptor for the evolution of information, which is reflective of the associated extent of undetermined features. This novel application of the multiplicative ergodic theorem leads directly to the formulation of a dimension that is a measure for the information gain attributable to the occurrence. Thus, it provides a readout for the magnitudes of chance and necessity that contribute to an event. Related algorithms express a unification of information content, degree of randomness, and complexity (fractal dimension) in event space.

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