Dynamical Systems seminar - Subshifts of very low complexity

Speaker: Ronnie Pavlov (University of Denver)

Abstract: The word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It's been known since work of Ferenczi that subshifts X with linear word complexity function (i.e. limsup p(n)/n finite) have highly constrained/structured behavior. I'll discuss recent work with Darren Creutz, where we show that if limsup p(n)/n < 4/3, then the subshift X must in fact have measurably discrete spectrum, i.e. it is isomorphic to a compact abelian group rotation. Our proof uses a substitutive/S-adic decomposition for such shifts, and I'll touch on connections to the so-called S-adic Pisot conjecture.