Local entropy theory for a countable discrete amenable group action

2010/05/08 by Wen Huang, Xiangdong Ye, Huang, Wen +3 · 4 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Computability, Logic, AI Algorithms #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1005.1335

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.

Citations

• On Groups of Measure Preserving Transformations. I

Cited by

• Metric mean dimension and the variational principle for actions of amenable groups
• Lowering topological entropy over subsets for amenable group actions
• Mean dimension and rate-distortion function revisited
• Local intricacy and average sample complexity for amenable group actions

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