On the complexity of subshifts and infinite words

Abstract

We characterize the complexity functions of subshifts up to asymptotic equivalence. The complexity function of every aperiodic subshift is non-decreasing, submultiplicative and grows at least linearly. We prove that, conversely, every function satisfying these conditions is asymptotically equivalent to the complexity function of a recurrent subshift, equivalently, a recurrent infinite word. Our construction is explicit, algorithmic in nature and is philosophically based on constructing certain ‘Cantor sets of integers’, whose ‘gaps’ correspond to blocks of zeros. We also prove that every non-decreasing submultiplicative function is asymptotically equivalent, up to a linear error term, to the complexity function of a minimal subshift.