Entropy Formula for Random ℤ^k-actions

2017/01/03 by Zhu, Yujun
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.00563

Abstract

In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random ℤ^k-actions which are generated by random compositions of the generators of ℤ^k-actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of C² random ℤ^k-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random ℤ^k(or ℤ₊^k )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and C¹ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedland's entropy for certain C² ℤ^k-actions.

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