Arithmetical complexity of the language of generic limit sets of cellular automata

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 Törmä. 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 ${\Pi }_2^0$ language if they are inclusion-minimal, a ${\Sigma }_1^0$ language if the cellular automaton has equicontinuous points, and that these bounds are tight. We also prove that many chain mixing ${\Pi }_2^0$ subshifts and all chain mixing ${\Delta }_2^0$ subshifts are realizable as generic limit sets. As a corollary, we characterize the minimal subshifts that occur as generic limit sets.