Entropy formula for random ℤ^{𝕜}-actions

Abstract

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