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

TL;DR

This paper derives formulas for measure-theoretic and topological entropy of random $bZ^k$-actions, extending classical entropy concepts to random and more general dynamical systems using Pesin's theory and Lyapunov spectra.

Contribution

It introduces a measure-theoretic entropy formula for $C^{2}$ random $bZ^k$-actions based on Lyapunov spectra and provides bounds and formulas for topological entropy in broader contexts.

Findings

Derived measure-theoretic entropy formula using Lyapunov spectra.

Established bounds for topological entropy of general maps.

Provided a formula for Friedland's entropy in specific cases.

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.