It has grown rapidly over the past three decades on the one hand, due to the experimental discovery of physical solid substances, called quasicrystals, exhibiting such features and on the other hand, due to intrinsic mathematical interest in describing the very border between crystallinity and aperiodicity. The theory of aperiodic order is a relatively young field of mathematics, which has attracted considerable attention in recent years, see for instance. Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.Īperiodic subshifts over finite alphabets play a vital role in various branches of mathematics, physics, and computer science. We also give necessary and sufficient conditions for these subshifts to be \(\alpha \)-repetitive, and \(\alpha \)-repulsive (and hence \(\alpha \)-finite).
In particular, we show that these subshifts provide examples that demonstrate \(\alpha \)-repulsive (and hence \(\alpha \)-finite) is not equivalent to \(\alpha \)-repetitive, for \(\alpha > 1\). Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk’s infinite 2-group G. We establish the equivalence of \(\alpha \)-repulsive and \(\alpha \)-finite for general subshifts over finite alphabets. Since then, generalisations and extensions of these features, namely \(\alpha \)-repetitive, \(\alpha \)-repulsive and \(\alpha \)-finite ( \(\alpha \ge 1\)), have been introduced and studied. 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.
0 kommentar(er)
