Entropy Formula for Random $mathbb{Z}^k$-actions

In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $mathbb{Z}^k$-actions which are generated by random compositions of the generators of $mathbb{Z}^k$-actions. Applying Pesins theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $mathbb{Z}^k$(or $mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and $C^{1}$ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{Z}^k$-actions.