Limit sets of stable Cellular Automata

TL;DR

This paper characterizes the limit sets of stable cellular automata using symbolic dynamics, focusing on sofic shifts and factor maps, and provides conditions for when these shifts can be realized as automaton limit sets.

Contribution

It establishes new criteria involving steady factor maps and minimal right-resolving covers for sofic shifts to be limit sets of stable cellular automata.

Findings

Characterization of right-closing almost-everywhere steady maps between sofic shifts.

Conditions for sofic shifts to be limit sets of stable cellular automata.

Characterization of AFT shifts via properties of steady maps.

Abstract

We study limit sets of stable cellular automata standing from a symbolic dynamics point of view where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. In terms of cellular automata, this translates into: A sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a fullshift and there exists a right- closing almost-everywhere factor map from the sofic shift onto its minimal right- resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a fullshift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the Almost of Finite Type shifts (AFT) in terms of a property of steady maps that have them as range.