Logical entropy of dynamical systems

Abstract

Abstract The main purpose of the paper is to extend the results of Ellerman (Int. J. Semant. Comput. 7:121–145, 2013) to the case of dynamical systems. We define the logical entropy and conditional logical entropy of finite measurable partitions and derive the basic properties of these measures. Subsequently, the suggested concept of logical entropy of finite measurable partitions is used to define the logical entropy of a dynamical system. It is proved that two metrically isomorphic dynamical systems have the same logical entropy. Finally, we provide a logical version of the Kolmogorov–Sinai theorem on generators. So it is shown that by replacing the Shannon entropy function by the logical entropy function we obtain the results analogous to the case of classical Kolmogorov–Sinai entropy theory of dynamical systems.

Keywords

• Measurable partition
• Logical entropy
• Conditional logical entropy
• Measure-preserving transformation
• Dynamical system