Regularity of aperiodic minimal subshifts

TL;DR

This paper explores the properties of aperiodic minimal subshifts, establishing equivalences and differences among various regularity features, and provides explicit complexity formulas and ergodic properties for specific subshifts derived from Grigorchuk's group.

Contribution

It proves the equivalence of $oldsymbol{ ext{α-repulsive}}$ and $oldsymbol{ ext{α-finite}}$ for general subshifts, and shows that $oldsymbol{ ext{α-repetitive}}$ is not equivalent to these for certain subshifts, with explicit complexity formulas.

Findings

Proved the equivalence of α-repulsive and α-finite for general subshifts.

Demonstrated that α-repulsive is not equivalent to α-repetitive for specific subshifts from Grigorchuk's group.

Derived explicit complexity functions and established unique ergodicity of these subshifts.

Abstract

At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely $\alpha$-repetitive, $\alpha$-repulsive and $\alpha$-finite ($\alpha \geq 1$), have been introduced and studied. We establish the equivalence of $\alpha$-repulsive and $\alpha$-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite $2$-group $G$. In particular, we show that these subshifts provide examples that demonstrate $\alpha$-repulsive (and hence $\alpha$-finite) is not equivalent to $\alpha$-repetitive, for $\alpha > 1$. We also give necessary and sufficient conditions for these subshifts to be $\alpha$-repetitive, and $\alpha$-repulsive (and hence $\alpha$-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.