Arithmetical Complexity of the Language of Generic Limit Sets of Cellular Automata

TL;DR

This paper investigates the computational complexity of the languages of generic limit sets in cellular automata, establishing bounds and realizability results for various classes of these sets.

Contribution

It provides tight complexity bounds for generic limit sets and characterizes which minimal subshifts can occur as such sets in cellular automata.

Findings

Generic limit sets have a 2_0 language if minimal

They have a _1 language if automaton has equicontinuous points

Many chain mixing 2_0 and all _2 chain mixing subshifts are realizable as generic limit sets

Abstract

The generic limit set of a dynamical system is the smallest set that attracts most of the space in a topological sense: it is the smallest closed set with a comeager basin of attraction. Introduced by Milnor, it has been studied in the context of one-dimensional cellular automata by Djenaoui and Guillon, Delacourt, and T\"orm\"a. In this article we present complexity bounds on realizations of generic limit sets of cellular automata with prescribed properties. We show that generic limit sets have a $\Pi^0_2$ language if they are inclusion-minimal, a $\Sigma^0_1$ language if the cellular automaton has equicontinuous points, and that these bounds are tight. We also prove that many chain mixing $\Pi^0_2$ subshifts and all chain mixing $\Delta^0_2$ subshifts are realizable as generic limit sets. As a corollary, we characterize the minimal subshifts that occur as generic limit sets.