On a Class of Generalized $K$-Entropies and Bernoulli Shifts

Abstract

The paper presents the construction of an example important for the discussion of some generalizations of the $K$-entropy. These generalizations have been introduced in a previous paper. A formula for the generalized entropies of a process is calculated for the case where the process is given by a Bernoulli shift and a partition consisting only of cylinder sets. Furthermore, a special optimization problem on the set of all probability vectors of a given entropy is solved using a new method. The results of these computations are combined to the cited example.