UVictoria Dynamics and Probability Seminar: Ronnie Pavlov

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 linear word complexity highly constrains the dynamical behavior of a subshift. In recent work with Darren Creutz, we show that if X is a transitive subshift with limsup p(n)/n < 3/2, then X is measure-theoretically isomorphic to a compact abelian group rotation. On the other hand, limsup p(n)/n = 3/2 can occur even for X measurably weak mixing. Our proofs rely on a substitutive/S-adic decomposition for such subshifts. I'll give some background/history on linear complexity, discuss our results, and will describe several ways in which 3/2 turns out to be a key threshold (for limsup p(n)/n) for several different types of dynamical behavior.