Regularity of aperiodic minimal 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 $α$-repetitive, $α$-repulsive and $α$-finite ($α\geq 1$), have been introduced and studied. We establish the equivalence of $α$-repulsive and $α$-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 $α$-repulsive (and hence $α$-finite) is not equivalent to $α$-repetitive, for $α> 1$. We also give necessary and sufficient conditions for these subshifts to be $α$-repetitive, and $α$-repulsive (and hence $α$-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.