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In this paper we introduce the concept of mutual entropy map for continuous maps on metric spaces. It is a non-negative extended real number which depends on two measures which are preserved by a system. Then we will extract the Kolmogorov entropy of ergodic systems from the mutual entropy as a special case when the two measures are equal.

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In this paper, we review some properties of the entropy of random dynamical systems. We define
a local entropy map for random dynamical systems and study some of its properties. We extract the
entropy of random dynamical systems from the introduced map.

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In this paper, we use some bi-sequences of positive numbers to define weighted dynamical metrics. Then we show that, replacing the Bowen dynamical metric by the weighted metric, the definition of pressure for asymptotically sub-additive potentials, including measure-theoretic and topological, is not affected. This generalizes some known results for pressure, defined using mean metrics and continuous potentials.