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arXiv · 1701.00563

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

Yujun Zhu
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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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Yujun Zhu. 2017-01-03. Entropy Formula for Random $\mathbb{Z}^k$-actions. https://arxiv.org/abs/1701.00563

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