Construction of $\mu$-Limit Sets

TL;DR

This paper presents a method to construct cellular automata that realize specific subshifts as their $$-limit sets under the uniform Bernoulli measure, expanding understanding of cellular automaton behaviors.

Contribution

It introduces a construction technique for cellular automata to realize given subshifts as $$-limit sets, for a broad class of subshifts.

Findings

Constructed cellular automata for specific subshifts as $$-limit sets

Demonstrated realization under the uniform Bernoulli measure

Extended the class of subshifts achievable as $$-limit sets

Abstract

The $\mu$-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to in- finity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as $\mu$-limit set where $\mu$ is the uniform Bernoulli measure.