On semi-open codes and bi-continuing almost everywhere codes

TL;DR

This paper explores the properties of semi-open codes and their relation to synchronization, finite type shifts, and bi-continuity in symbolic dynamical systems, providing new characterizations and conditions for these concepts.

Contribution

It establishes the equivalence between synchronization and semi-open covers, characterizes semi-open codes on irreducible sofic shifts, and provides conditions for codes to be open or bi-continuing.

Findings

A system is synchronized iff it has a semi-open cover.

Any factor code on an irreducible sofic shift is semi-open.

Semi-open codes on synchronized systems are bi-continuing almost everywhere.

Abstract

We will show that a system is synchronized if and only if it has a cover whose cover map is semi-open. Also, any factor code on an irreducible sofic shift is semi-open and the image of a synchronized system by a semi-open code is synchronized. On the other side, right-closing semi-open extension of an irreducible shift of finite type is of finite type. Moreover, we give conditions on finite-to-one factor codes to be open and show that any semi-open code on a synchronized system is bi-continuing a.e.. We give some sufficient conditions for a right-continuing a.e. factor code being right-continuing everywhere.