2023/02/20 by Darren Creutz, Creutz, Darren, Ronnie Pavlov +1 · 2 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Cellular Automata and Applications #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2302.10336
We prove results about subshifts with linear (word) complexity, meaning that \limsup (p(n))/(n) < ∞, where for every n, p(n) is the number of n-letter words appearing in sequences in the subshift. Denoting this limsup by C, we show that when C < (4)/(3), the subshift has discrete spectrum, i.e. is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with C = (3)/(2) which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether C = (5)/(3) was the minimum possible among such subshifts; our results show that the infimum in fact lies in [(4)/(3), (3)/(2)]. All results are consequences of a general S-adic/substitutive structure proved when C < (4)/(3).