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arXiv · 1005.1335

Local entropy theory for a countable discrete amenable group action

Wen Huang·Xiangdong Ye·Guohua Zhang
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Abstract

In the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action. For such an action, we introduce entropy tuples in both topological and measure-theoretic settings and build the variational relation between these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover. Moreover, based the idea of topological entropy pairs, we introduce and study two special classes of such an action: uniformly positive entropy and completely positive entropy. Note that in the building of the local variational principle, following Romagnoli's ideas two kinds of measure-theoretic entropy are introduced for finite Borel covers. These two kinds of entropy turn out to be the same, where Danilenko's orbital approach becomes an inevitable tool.

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Wen Huang, Xiangdong Ye, Guohua Zhang. 2010-05-08. Local entropy theory for a countable discrete amenable group action. https://arxiv.org/abs/1005.1335

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