Krieger's finite generator theorem for actions of countable groups I

• 成果类型: Article
• 署名作者: Seward, Brandon
• 署名单位: New York University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0826-9
• 发表日期: 2019
• 页码: 265-310
• 关键词: entropy; shifts

摘要

For an ergodic p.m.p. action G?(X,) of a countable group G, we define the Rokhlin entropy hGRok(X,) to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov-Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if then there exists a generating partition consisting of k sets.