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

Each continuous action of a countably infinite discrete group $\mathrm{\Gamma }$ on a compact metrizable space X induces a continuous action of $\mathrm{\Gamma }$ 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.