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.