Entropy formula for random ℤ^{𝕜}-actions

Zhu, Yujun

doi:10.1090/tran/6798

Entropy formula for random $\mathbb {Z}^k$-actions
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Trans. Amer. Math. Soc. 369 (2017), 4517-4544 Request permission

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.

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