Title: Entropy Formula for Random $\mathbb{Z}^k$-actions,
Abstract: 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
Pesin's 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 Friedland's entropy for certain $C^{2}$ $\mathbb{Z}^k$-actions.