On the Interplay of Direct Topological Factorizations and Cellular Automata Dynamics on Beta-Shifts

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts Sβ and its relation to direct topological factorizations. We show that any reversible CA F:Sβ→Sβ has an almost equicontinuous direction whenever Sβ is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations X×Y of two nontrivial subshifts X and Y. We also give a simple criterion to determine whether Snγ is conjugate to Sn×Sγ for a given integer n≥1 and a given real γ>1 when Sγ is a subshift of finite type. When Sγ is strictly sofic, we show that such a conjugacy is not possible at least when γ is a quadratic Pisot number of degree 2. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.</p

Direct prime subshifts and canonical covers

We present a new sufficient criterion to prove that a non-sofic half-synchronized subshift is direct prime. The criterion is based on conjugacy invariant properties of Fischer graphs of half-synchronized shifts. We use this criterion to show as a new result that all n-Dyck shifts are direct prime, and we also give new proofs of direct primeness of non-sofic beta-shifts and non-sofic S-gap shifts. We also construct a class of non-sofic synchronized direct prime subshifts which additionally admit reversible cellular automata with all directions sensitive

2011 IMSAloquium, Student Investigation Showcase

Inquiry Without Boundaries reflects our students’ infinite possibilities to explore their unique passions, develop new interests, and collaborate with experts around the globe.https://digitalcommons.imsa.edu/archives_sir/1003/thumbnail.jp