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

TL;DR

This paper extends the Shannon-McMillan-Breiman theorem to random dynamical systems influenced by amenable group actions, enhancing the understanding of entropy in such complex systems.

Contribution

It proves the Shannon-McMillan-Breiman theorem within the context of random dynamical systems for amenable group actions, advancing entropy theory in this area.

Findings

The theorem is established for random dynamical systems with amenable group actions.

Provides new tools for analyzing entropy in complex dynamical systems.

Strengthens the connection between ergodic theory and information theory.

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.