VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

2009 ◽  
Vol 09 (02) ◽  
pp. 205-215 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

scholarly journals Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 309
Author(s):  
Xianfeng Ma ◽  
Zhongyue Wang ◽  
Hailin Tan
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Image Measure ◽  
Measure Preserving Transformations

A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.

scholarly journals Relative tail entropy for random bundle transformations

2017 ◽  
Vol 18 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Xianfeng Ma ◽  
Junqi Yang ◽  
Ercai Chen
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Random Dynamical Systems

We introduce the relative tail entropy to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail entropy is conserved by the principal extension.

scholarly journals On the random dynamics of Volterra quadratic operators

2015 ◽  
Vol 37 (1) ◽  
pp. 228-243 ◽  
Author(s):  
U. U. JAMILOV ◽  
M. SCHEUTZOW ◽  
M. WILKE-BERENGUER
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Special Class ◽  
Random Dynamical Systems ◽  
Random Point ◽  
Random Dynamical System ◽  
Quadratic Stochastic Operators ◽  
Point Attractor ◽  
Set Up

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex $S^{m-1}$. We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex $S^{m-1}$, implying the survival of only one species. We also show that the minimal random point attractor of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

2003 ◽  
Vol 67 (2) ◽  
Author(s):  
Ying-Cheng Lai ◽  
Zonghua Liu ◽  
Lora Billings ◽  
Ira B. Schwartz
Keyword(s):  
Dynamical Systems ◽  
Random Dynamical Systems ◽  
Transition To Chaos
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