Model theoretic properties of dynamics on the Cantor set

TL;DR

This paper explores the model theoretic properties of dynamical systems on the Cantor set, particularly focusing on the generic homeomorphism and its characterization as the prime model in continuous logic.

Contribution

It demonstrates that the generic homeomorphism of the Cantor set is the prime model of its theory and clarifies the distinction between different notions of genericity.

Findings

The generic homeomorphism is the prime model of its theory.

The notion of 'generic' differs between topological dynamics and Fraisse theory.

Provides a model-theoretic perspective on Cantor set dynamics.

Abstract

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of "generic" used by Akin, Glasner, and Weiss is distinct from the notion of "generic" encountered in Fraisse theory.