Local variational principle concerning entropy of a sofic group action

TL;DR

This paper develops a local variational principle for entropy in actions of sofic groups, connecting local and classical entropy concepts and extending prior global principles to more localized settings.

Contribution

It introduces a localized version of the variational principle for entropy in sofic group actions, bridging measure-theoretic and topological entropy.

Findings

Local variational principle established for finite open covers

Local entropy coincides with classical entropy for amenable groups

Extends global variational principles to localized contexts

Abstract

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropy coincide with their classical counterparts when the acting group is an infinite amenable group.