m-Generators of Fuzzy Dynamical Systems

Year 2012, Volume: 9 Issue: 2 , - , 01.04.2012

Mohammad Ebrahimi Uosef Mohamadi

https://izlik.org/JA93ZZ64GZ

Abstract

In this paper we prove that the entropy of a fuzzy measure preserving transformation
with respect to a sub-σ-algebra having finite atoms is affine and then we extend
the method of computing the entropy of a finite sub-σ-algebra to a sub-σ-algebra having
countable atoms, and we investigate the ergodic properties of fuzzy probability dynamical
systems. At the end by using this notion, a version of Kolmogorov-Sinai proposition [6, 9,
10] is given.

Keywords

Fuzzy probability space , entropy , fuzzy dynamical systems , m-equivalence , m-refinement , fuzzy m-generator

References

• [1] D. Dumitrescu, C. H˘aloiu and A. Dumitrescu, Generators of fuzzy dynamical systems, Fuzzy Sets and Systems 113 (2000), 447–452.
• [2] D. Dumitrescu, Entropy of a fuzzy dynamical system, Fuzzy Sets and Systems 70 (1995), 45–57.
• [3] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems 55 (1993), 169–177.
• [4] M. Ebrahimi, Generators of probability dynamical systems, Differential Geometry-Dynamical Systems 8 (2006), 90–97.
• [5] M. Ebrahimi and N. Mohamadi, The entropy function on an algebraic structure with infinite partition and m-preserving transformation generators, Applied Sciences 12 (2010), 48–63.
• [6] A. N. Kolmogorov, Entropy per unit time as a metric invariant of automorphism, Doklady of Russian Academy of Sciences 124 (1959), 754–755.
• [7] P. Serivastava, M. Khare and Y. K. Srivastava, A fuzzy measure algebra on a metric space,Fuzzy Sets and Systems 79 (1996), 395–400.
• [8] P. Serivastava, M. Khare and Y. K. Srivastava, m-Equivalence, entropy and F-dynamical systems, Fuzzy Sets and Systems 121 (2001), 275–283.
• [9] Ya. Sinai, On the notion of entropy of a dynamical system, Doklady ofRussian Academy of Sciences 124 (1959), 768–771.
• [10] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.