On some dynamical aspects of NIP theories

TL;DR

This paper explores the dynamical behavior of automorphisms in NIP theories, linking model-theoretic properties with dynamical systems concepts like entropy and invariant measures.

Contribution

It provides new characterizations of NIP theories and dividing lines using dynamics, invariant measures, and symbolic representations.

Findings

Characterizations of NIP theories via automorphism dynamics

Connections between invariant measures and model-theoretic dividing lines

Symbolic representations elucidate combinatorial configurations

Abstract

We study some dynamical aspects of the action of automorphisms in model theory in particular in the presence of invariant measures. We give some characterizations for NIP theories in terms of dynamics of automorphisms and invariant measures for example in terms of compact systems, entropy and measure algebras. Moreover, we study the concept of symbolic representation for models. Amongst the results, we give some characterizations for dividing lines and combinatorial configurations such as independence property, order property and strictly order property in terms of symbolic representations.