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Register a new userEntropy Formula for Random $mathbb{Z}^k$-actions
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Yujun Zhu
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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 Pesins 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 Friedlands entropy for certain $C^{2}$ $mathbb{Z}^k$-actions.
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We study directional mean dimension of $mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+infty]$-valued function on the torus) is not continuous. On the other hand, we prove that if a $mathbb{Z}^k$-action is continuum-wise expansive, then the values of its $(k-1)$-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamotos theorem on mean dimension and expansive multiparameter actions) of a classical result due to Ma~ne: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.
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