Decidability and Universality of Quasiminimal Subshifts

TL;DR

This paper introduces quasiminimal subshifts, explores their properties under different group actions, and constructs a universal example that refutes a previous conjecture, advancing understanding of subshift dynamics.

Contribution

It defines quasiminimal subshifts, analyzes their behavior under $ $- and $ $-actions, and constructs a universal system with a single proper subsystem, challenging existing conjectures.

Findings

Quasiminimal subshifts have finitely many subsystems.

Under $ $-actions, their theory aligns with minimal systems.

A universal system with one proper subsystem is constructed.

Abstract

We introduce the quasiminimal subshifts, subshifts having only finitely many subsystems. With $\mathbb{N}$-actions, their theory essentially reduces to the theory of minimal systems, but with $\mathbb{Z}$-actions, the class is much larger. We show many examples of such subshifts, and in particular construct a universal system with only a single proper subsystem, refuting a conjecture of [Delvenne, K\r{u}rka, Blondel, '05].