Sliding block codes between shift spaces over infinite alphabets

TL;DR

This paper generalizes the concept of sliding block codes for shift spaces over infinite alphabets and establishes conditions under which these codes coincide with continuous shift-commuting maps, extending the Curtis-Hedlund-Lyndon theorem.

Contribution

It introduces a broader definition of sliding block codes for Ott-Tomforde-Willis shift spaces and proves a Curtis-Hedlund-Lyndon type theorem for them.

Findings

Proposed a more general definition of sliding block codes.

Proved necessary and sufficient conditions for equivalence with continuous shift-commuting maps.

Extended classical theorems to infinite alphabet shift spaces.

Abstract

Recently Ott, Tomforde and Willis introduced a notion of one-sided shifts over infinite alphabets and proposed a definition for sliding block codes between such shift spaces. In this work we propose a more general definition for sliding block codes between Ott-Tomforde-Willis shift spaces and then we prove Curtis-Hedlund-Lyndon type theorems for them, finding sufficient and necessary conditions under which the class of the sliding block codes coincides with the class of continuous shift-commuting maps.