Examples and Nonexamples of Distal Metric Structures

TL;DR

This paper explores examples of distal metric structures, including metric valued fields and a topological dynamics structure called Dual Linear Continua, analyzing their properties and indiscernible sequences.

Contribution

It provides new examples of distal metric structures and characterizes models of Dual Linear Continua, including their indiscernible sequences and explicit distal cell decompositions.

Findings

Real closed metric valued fields are distal.

Algebraically closed metric valued fields are stable and have the strong Erdős-Hajnal property.

Models of Dual Linear Continua are distal with explicit cell decompositions.

Abstract

This article provides examples of distal metric structures. One source of examples are metric valued fields. By analyzing indiscernible sequences, we show that real closed metric valued fields are distal, and conclude that algebraically closed metric valued fields, while stable, have the strong Erd\H{o}s-Hajnal property, which we define appropriately for metric structures. We find another example in topological dynamics: we study a metric structure whose automorphism group is the well-understood Polish group $\mathrm{Hom}^+([0,1])$ of increasing homeomorphisms of $[0,1]$. This was known to be NIP and highly unstable, and further properties were established in arXiv:1510.00238. We characterize models of its theory of this structure, which we call Dual Linear Continua, up to isomorphism. We analyze their indiscernible sequences and prove that they are distal, as well as constructing explicit distal cell decompositions.