Ranks in Ellis semigroups and model theory

TL;DR

This paper generalizes a rank concept for Ellis semigroups to broader topological spaces and links it to model-theoretic stability, showing the rank's ordinal nature characterizes NIP theories.

Contribution

It extends the notion of rank in Ellis semigroups beyond metric spaces and connects it to key model-theoretic dividing lines like NIP.

Findings

Rank is ordinal-valued iff the theory is NIP.

Generalization of the rank applies to any compact Hausdorff space.

Links between Ellis semigroup dynamics and model theory are established.

Abstract

We slightly generalize a notion of rank introduced by Glasner and Megrelishvili, which captures the oscillations of elements of Ellis semigroups, so that it can be applied to any compact Hausdorff space instead of being limited to the metric case. Then, we relate this rank to classical dividing lines in the model-theoretic stability hierarchy. For example, that the rank is ordinal-valued if and only if the background theory is NIP.