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 α -repetitive, α -repulsive and α -finite ( α ≥ 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.