Local entropy theory for a countable discrete amenable group action

By Wen Huang, Xiangdong Ye, Guohua Zhang. 2010

106 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/920a5c2304c58d7ca6595ead86e080ad3c6a8996

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Citations in the corpus

Homoclinic groups, IE groups, and expansive algebraic actions (2011, background): “[7–10,24,26,31–35]), and has been found to be related to combinatorial independence [39,40], which appeared first in Rosenthal’s [69] work on Banach spaces containing 1.”
Local entropy theory (2009, methodology): “Together with results from Huang et al [54] and Huang and Ye [57], one can show that a minimal topological K -system is strongly mixing.”