On the topological dynamics of automorphism groups; a model-theoretic perspective

TL;DR

This paper provides a model-theoretic framework for understanding the topological dynamics of automorphism groups, extending classical results to broader structures beyond the countable case.

Contribution

It introduces a novel model-theoretic approach to analyze automorphism groups' dynamics, generalizing existing theories to non-countable structures.

Findings

Describes the universal ambit as a space of types in an expanded language

Recovers key results of Kechris-Pestov-Todorčević and others in a broader context

Extends topological dynamics results to automorphism groups of non-countable structures

Abstract

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todor\v{c}evi\'{c} theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover various results of Kechris-Pestov-Todor\v{c}evi\'{c}, Moore, Ngyuen Van Th\'{e}, in the context of automorphism groups of not necessarily countable structures, as well as Zucker.