Almost automorphic subshifts with finiteness conditions for the boundary of the separating cover

TL;DR

This paper investigates the structure of almost automorphic subshifts with finiteness conditions on their boundary, analyzing orbit behavior, complexity, and factor subshifts, with explicit constructions for Toeplitz subshifts.

Contribution

It introduces finiteness conditions on the boundary of separating covers in almost automorphic subshifts and explores their implications, including conditions for factor subshifts with finite boundaries.

Findings

Finiteness conditions influence orbit and complexity properties.

Explicit constructions illustrate the necessity of assumptions.

Conditions for the existence of factor subshifts with finite boundaries.

Abstract

In this article we study orbits of proximal pairs in almost automorphic subshifts. The corresponding orbits in the maximal equicontinuous factor are precisely those orbits that intersect the boundary of the subshift's separating cover. We impose certain finiteness conditions on this boundary and explore the resulting consequences for the subshift, for instance in terms of complexity or the relations between proximal and asymptotic pairs. The last part of our article deals with Toeplitz subshifts without a finite boundary. There we treat the question of necessary conditions and sufficient conditions for the existence of a factor subshift with a finite boundary. Throughout the whole article, we provide explicit constructions for Toeplitz subshifts to illustrate our findings and the necessity of our assumptions.