Complexity and recurrence in infinite words and related structures

TL;DR

This paper explores the complexity and recurrence properties of infinite words, providing new constructions and characterizations that answer longstanding open questions and have implications for algebraic structures.

Contribution

It constructs infinite recurrent words with near-linear complexity and unbounded derivatives, characterizes complexity functions of ergodic subshifts, and addresses algebraic recurrence questions.

Findings

Constructed recurrent words with complexity close to linear but unbounded derivatives.

Characterized complexity functions of strictly ergodic subshifts up to a linear factor.

Constructed simple algebras with prescribed filter dimensions.

Abstract

We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are arbitrarily close to linear, but whose discrete derivatives are not bounded from above by $p_w(n)/n$. Moreover, we construct words of polynomially bounded complexity whose discrete derivatives exceed $p_w(n)/n^\varepsilon$ infinitely often, for every given $\varepsilon>0$. These provide negative answers in a strong sense to an open question of Cassaigne from 1997, showing that his theorem on words of linear complexity is best possible. Next, we characterize, up to a linear multiplicative error, the complexity functions of strictly ergodic subshifts, showing that every non-decreasing, submultiplicative function arises in this setting. This gives the first `industrial' construction of strictly ergodic subshifts of prescribed subexponential complexity. We then investigate quantitative recurrence in uniformly recurrent words and, as an application, address a question of Bavula from 2006 related to holonomic inequalities on the spectrum of possible filter dimensions of simple associative algebras: we construct simple algebras of prescribed filter dimension in $[1,\infty)$ and essentially settling the problem entirely in the graded case. Throughout, we construct uniformly recurrent words of linear complexity and with arbitrary polynomial recurrence growth.