Fixed points on null and tame flows for groups of automorphisms

TL;DR

This paper establishes fixed point theorems for null and tame flows of automorphism groups of Fraïssé structures, extending the Kechris-Pestov-Todorčević correspondence to broader classes of actions.

Contribution

It generalizes fixed point results for automorphism groups of Fraïssé structures using a new approach based on a generalized correspondence.

Findings

Null flows have fixed points when the age is a free joint embedding class.

Tame flows have fixed points when the age is a free amalgamation class.

The results extend the scope of fixed point theorems in topological dynamics.

Abstract

Using a generalization of the Kechris-Pestov-Todor\v{c}evi\'{c} correspondence due to Nguyen Van Th\'{e} we obtain fixed point theorems for null and tame actions of groups of the form $\mathrm{Aut}(\mathcal F)$, where $\mathcal{F}$ is a Fra\"{i}ss\'{e} structure. In particular we show that if $\mathrm{Age}(\mathcal F)$ is a free joint embedding class, then every null flow $\mathrm{Aut}(\mathcal F)\curvearrowright X$ has a fixed point, while if $\mathrm{Age}(\mathcal F)$ is a free amalgamation class, then every tame flow $\mathrm{Aut}(\mathcal F)\curvearrowright X$ has a fixed point.