Regularity of aperiodic minimal subshifts - F. Dreher; M. Kesseböhmer; A. Mosbach; T. Samuel; M. Steffens (Bulletin of Mathematical Sciences, 2017)
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 $$α $$ α -repetitive, $$α $$ α -repulsive and $$α $$ α -finite ($$α \ge 1$$ α≥1 ), have been introduced and studied. We establish the equivalence of $$α $$ α -repulsive and $$α $$ 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 $$α $$ α -repulsive (and hence $$α $$ α -finite) is not equivalent to $$α $$ α -repetitive, for $$α > 1$$ α>1 . We also give necessary and sufficient conditions for these subshifts to be $$α $$ α -repetitive, and $$α $$ α -repulsive (and hence $$α $$ α -finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.