Regularity of aperiodic minimal subshifts. (English) Zbl 1407.37023
Summary: 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.
MSC:
| 37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |
| 20E08 | Groups acting on trees |
| 37B10 | Symbolic dynamics |
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