Preimage entropy for random dynamical systems

In [6], Cheng and Newhouse introduced and studied the new invariants - preimage entropies for deterministic dynamical systems. In this paper, the analogous notions, measure-theoretic preimage entropy and topological preimage entropy, are formulated for random dynamical systems. Analogues of many known results for entropies, such as the Shannon-McMillan-Breiman Theorem, the Kolmogorov-Sinai Theorem, the Abromov-Rokhlin formula and the power rule, are obtained for preimage entropies. In particular, a variational principle is given.