Regularity of aperiodic minimal subshifts

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

A classification of aperiodic order via spectral metrics and Jarník sets

“…In particular, it was shown in [38] that power free and repulsive are equivalent and, as we will shortly see, we have that power free and 1-finite (Definition 2.13) are equivalent. The general result follows, by combining these observations with Theorem 3.1 of [24] where it is shown that 1-repulsive and 1-finite are equivalent.…”

“…The following proposition, which is proven in Section 5.1, relates the notions 1-repulsive and repulsive. In fact, in a sequel to this article [24], this result is shown to hold for general subshifts over finite alphabets. In particular, it was shown in [38] that power free and repulsive are equivalent and, as we will shortly see, we have that power free and 1-finite (Definition 2.13) are equivalent.…”

“…A very natural question is if the three combinatorical properties (α-repetitive, α-repulsive and α-finite) are equivalent outside of the setting of Sturmian sequences. This was one of the main motivations of the sequel to this article [24], where in Theorem 3.1 it is shown that in general α-repulsive and α-finite are equivalent. Moreover, examples are given which demonstrate that the notions of α-repetitive and α-repulsive are in fact different.…”

“…Further, we remark that a class of S-adic subshifts, referred to as (generalised) Grigorchuk subshifts, have been investigated in this context in a sequel to this article [24]. The construction of these subshifts were inspired by Lysenok's [48] presentation of the Grigorchuk group G (the first known finitely generated group to exhibit intermediate growth) and have been shown to exhibit a rich variety of behaviours, see for instance [24,30,31].…”