Regularity of aperiodic minimal subshifts

Abstract

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

Keywords

• Aperiodic order
• Complexity
• Subshifts
• Grigorchuk group
• Unique ergodicity