Path sets in one-sided symbolic dynamics

TL;DR

This paper studies path sets in one-sided symbolic dynamics, revealing their structure, properties, and their role as a generalization of sofic shifts, with applications in fractal geometry.

Contribution

It introduces path sets as a broader class of symbolic spaces, extending the theory of sofic shifts and analyzing their fundamental properties.

Findings

Path sets are a strict generalization of one-sided sofic shifts.

Basic properties and structure of path sets are established.

Path sets naturally appear in geometric fractal constructions.

Abstract

Path sets are spaces of one-sided infinite symbol sequences associated to pointed graphs (G_v_0), which are edge-labeled directed graphs G with a distinguished vertex v_0. Such sets arise naturally as address labels in geometric fractal constructions and in other contexts. The resulting set of symbol sequences need not be closed under the one-sided shift. this paper establishes basic properties of the structure and symbolic dynamics of path sets, and shows they are a strict generalization of one-sided sofic shifts.