Open maps between shift spaces

TL;DR

This paper explores properties of open maps between shift spaces, establishing conditions under which openness, closeness, and constant-to-one mappings imply each other, especially in the context of sofic shifts.

Contribution

It proves a new equivalence between openness, closeness, and constant-to-one properties for maps from shift spaces to irreducible sofic shifts, including bi-closing cases.

Findings

Open maps imply the other properties under certain conditions.

Closing open or constant-to-one extensions preserve sofic shift structure.

Results extend to non-sofic ranges with bi-closingness.

Abstract

Given a code from a shift space to an irreducible sofic shift, any two of the following three conditions -- open, constant-to-one, (right or left) closing -- imply the third. If the range is not sofic, then the same result holds when bi-closingness replaces closingness. Properties of open mappings between shift spaces are investigated in detail. In particular, we show that a closing open (or constant-to-one) extension preserves the structure of a sofic shift.