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 -repetitive, alpha -repulsive and alpha -finite (alpha ge 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.
Highlights
Aperiodic subshifts over finite alphabets play a vital role in various branches of mathematics, physics, and computer science
Through the work of Durand [13], and Lagarias and Pleasants [31] it has become apparent that key features of aperiodic minimal subshifts to be studied are linearly repetitive, repulsive and power free
We show that l-Grigorchuk subshifts are aperiodic and minimal (Proposition 4.4 and Corollary 4.16)
Summary
Aperiodic subshifts over finite alphabets play a vital role in various branches of mathematics, physics, and computer science. For α > 1, we establish that α-repetitive is not necessarily equivalent to α-repulsive, and nor α-finite (Theorems 4.5 and 4.10) This latter result is provided by a class of subshifts stemming from Grigorchuk’s infinite 2-group. When l is the constant one sequence, the resulting l-Grigorchuk subshift is intimately related to Lysenok group presentation of Grigorchuk’s infinite 2-group G By studying this subshift, very recently [20,21] the spectral type of the Laplacian on the Schreier graphs describing the action of Grigorchuk’s infinite 2-group G on the boundary of the infinite binary rooted tree were determined and it has been shown that it is different in the isotropic and anisotropic cases. 4.2 and 4.3 we provide necessary and sufficient conditions on a sequence l which ensures that the associated l-Grigorchuk subshift is α-repulsive (and α-finite), and α-repetitive respectively; after which, in Sect.
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