Regularity of aperiodic minimal subshifts
Fabian Dreher, Marc Keßeböhmer, A. Mosbach, Tony Samuel, M. Steffens · Bulletin of Mathematical Sciences · 2017
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 ( $$\alpha \ge 1$$ ), 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.