Local entropy theory, combinatorics, and local theory of Banach spaces

TL;DR

This paper explores the relationship between local entropy theory for group actions on spaces and measures, introduces a new combinatorial lemma, and applies it to Banach space theory.

Contribution

It establishes connections between entropy theories for actions on spaces and measures, and introduces a novel combinatorial lemma with applications to Banach spaces.

Findings

Comparison of local entropy for actions on spaces and measures

Introduction of a new combinatorial lemma

Application of the lemma to Banach space theory

Abstract

Each continuous action of a countably infinite discrete group $\Gamma$ on a compact metrizable space X induces a continuous action of $\Gamma$ on the space M(X) of Borel probability measures on X. We compare the local entropy theory for these two actions, and describe the relation between their IE-tuples. Several other types of tuples are also studied. Our main tool is a new combinatorial lemma. We also give an application of the combinatorial lemma to the local theory of Banach spaces.