The Shannon-McMillan-Breiman theorem of random dynamical systems for amenable group actions

Abstract

The Shannon-McMillan-Breiman theorem is one of the most important results in information theory, which can describe the random ergodic process, and its proof uses the famous Birkhoff ergodic theorem, so it can be seen that it plays a crucial role in ergodic theory. In this paper, the Shannon-McMillan-Breiman theorem in the random dynamical systems is proved from the perspective of an amenable group action, which provides a boost for the development of entropy theory in the random dynamical systems for amenable group actions.