Observational Modeling of Logical Entropy of Dynamical Systems with local approach

Abstract

In this paper, we introduce a new kind of the logical entropy through a local relative approach. The notions of local relative logical entropy and local relative conditional logical entropy from an observer's viewpoint on local relative probability measure space are introduced and some of their ergodic properties are studied. Some properties of the local relative logical entropy of independent partitions are investigated and the concavity property for the local relative logical entropy has been proved. We show that, the basic properties of Shannon entropy of partitions on probability measure spaces, are established for the case of the local relative logical entropy. So the suggested measures can be used besides of the Shannon entropy of partitions. Using the concept of the local relative logical entropy of partitions, we define the local relative logical entropy of a dynamical system and present some of its properties. Finally, it is shown that the local relative logical entropy of dynamical systems is invariant under isomorphism. So the notion of local relative logical entropy of dynamical systems can be a new tool for distinction of non-isomorphic relative dynamical systems.