Combinatorial independence in measurable dynamics

TL;DR

This paper introduces a detailed local analysis of measure entropy using combinatorial independence, extending concepts from topological dynamics to measurable systems and exploring their connections with operator algebras.

Contribution

It develops measure IE- and IN-tuples, provides local characterizations of the Pinsker algebra, and generalizes to actions of discrete groups with an emphasis on amenability.

Findings

Introduced measure IE- and IN-tuples in measurable dynamics.

Provided local characterizations of the Pinsker von Neumann algebra.

Extended analysis to actions of general discrete groups, emphasizing amenability.

Abstract

We develop a fine-scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IE-tuples and measure IN-tuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations of the Pinsker von Neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence, l_1 geometry, and Voiculescu's completely positive approximation entropy. Among the novel features of our local study is the treatment of general discrete acting groups, with the structural assumption of amenability in the case of entropy.