Regularity of aperiodic minimal subshifts

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.