Pattern Complexity of Aperiodic Substitutive Subshifts

TL;DR

This paper investigates the pattern complexity of aperiodic substitutive subshifts, establishing quadratic bounds and demonstrating the optimality of recent complexity bounds in symbolic dynamics.

Contribution

It proves a quadratic lower bound on pattern complexity for certain substitutive subshifts and confirms the optimality of recent bounds for aperiodic subshifts.

Findings

Quadratic lower bound on pattern complexity for specific substitutive subshifts

Pattern complexity of these subshifts is in b1(n^2)

Recent bounds by Kari and Moutot are shown to be optimal

Abstract

This paper aims to better understand the link better understand the links between aperiodicity in subshifts and pattern complexity. Our main contribution deals with substitutive subshifts, an equivalent to substitutive tilings in the context of symbolic dynamics. For a class of substitutive subshifts, we prove a quadratic lower bound on their pattern complexity. Together with an already known upper bound, this shows that this class of substitutive subshifts has a pattern complexity in $\Theta(n^2)$. We also prove that the recent bound of Kari and Moutot, showing that any aperiodic subshift has pattern complexity at least $mn+1$, is optimal for fixed $m$ and $n$.