On the complexity of subshifts and infinite words

TL;DR

This paper characterizes the growth patterns of complexity functions in subshifts and infinite words, establishing conditions for their asymptotic equivalence and providing explicit constructions.

Contribution

It provides a complete characterization of complexity functions of subshifts, including explicit, algorithmic constructions for recurrent and minimal subshifts.

Findings

Every aperiodic complexity function is non-decreasing, submultiplicative, and grows at least linearly.

Every such function is asymptotically equivalent to a recurrent subshift's complexity function.

Every non-decreasing submultiplicative function is asymptotically similar to a minimal subshift's complexity function.

Abstract

We characterize the complexity functions of subshifts up to asymptotic equivalence. The complexity function of every aperiodic function 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 a linear error term, to the complexity function of a minimal subshift.