Document Type : Research Paper
Author
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
Abstract
In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ \Theta $. The relative entropy of a sub-$ \sigma_\Theta $-algebra having finite atoms is defined and then the ergodic properties of relative semi-dynamical systems are investigated. Also, a relative version of Kolmogorov-Sinai theorem is given. Finally, it is proved that the relative entropy of a relative $ \Theta $-measure preserving transformations with respect to a relative sub-$\sigma_\Theta$-algebra having finite atoms is affine.
Keywords
- Relative entropy
- Relative semi-dynamical system
- $m_\Theta$-equivalence
- $m_\Theta$-generator
- $ (\Theta_1, \Theta_2) $-isomorphism
Main Subjects
[2] U. Mohammadi, Relative information functional of relative dynamical systems, J. Mahani. Mat. Res. Cent., 2 (2013), pp. 17-28.
[3] U. Mohammadi, Weighted information function of dynamical systems, J. Math. Comput. Sci., 10 (2014), pp. 72-77.
[4] U. Mohammadi, Weighted entropy function as an extension of the Kolmogorov-Sinai entropy, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 4 (2015), pp. 117-122.
[5] U. Mohammadi, Relative entropy functional of relative dynamical systems, Cankaya. Univ. J. Sci. Eng., 2 (2014), pp. 29-38.
[6] M.R. Molaei, Relative semi-dynamical systems, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 12 (2004), pp. 237-243.
[7] M.R. Molaei, Mathematical modeling of observer in physical systems, J. Dyn. Syst. Geom. Theor., 4 (2006), pp. 183-186.
[8] M.R. Molaei, Observational modeling of topological spaces, Chaos Solitons Fractals, 42 (2009), pp. 615-619.
[9] M.R. Molaei, M.H. Anvari, and T. Haqiri, On relative semi-dynamical systems, Intell. Autom. Soft Comput. Syst., 12 (2004), pp. 237-243.
[10] M.R. Molaei and B. Ghazanfari, Relative entropy of relative measure preserving maps with constant observers, J. Dyn. Syst. Geom. Theor., 5 (2007), pp. 179-191.
[11] Ya. Sinai, On the notion of entropy of a dynamical system, Dokl. Akad. Nauk. S.S.S.R, 125 (1959), pp. 768-771.
[12] P. Walters, An Introduction to Ergodic Theory, Springer Verlag, 1982.
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