Observers and Relative ‎E‎ntropy ‎Functional‎

Subject Areas : International Journal of Industrial Mathematics

عادل گروهی ¹ , یوسف محمدی ² , محمد ابراهیمی ³

1 - Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran.
2 - Department of Mathematics, University of Jiroft, Jiroft, Iran.
3 - Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran.

Received: 2021-02-06 Accepted : 2022-10-09 Published : 2023-01-01

Keywords: Relative entropy ‎functional‎, Kolmogorov entropy, Invariant, Relative dynamical system,

Abstract :

‎In this paper‎, ‎we will use the mathematical modeling of one-dimensional observers to present the notion of the \emph{relative entropy functional} for relative dynamical systems‎. ‎Also‎, ‎the invariance of the entropy of a system under topological conjugacy is generalized to the relative entropy functional‎. ‎Moreover‎, ‎from observer viewpoint‎, ‎a new version of the Jacobs Theorem is obtained‎. ‎It has been proved that relative entropy functional is equivalent to the Kolmogorov entropy for dynamical systems‎, ‎from the viewpoint of observer $ \chi_X $‎, ‎where $ \chi_X $ is the characteristic function on compact metric space $X$‎.

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