Regularity of aperiodic minimal subshifts

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.Comment: 15 page

Local Complexity of Delone Sets and Crystallinity

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package

Avoiding Abelian powers in binary words with bounded Abelian complexity

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian $k$-powers for some integer $k$, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian $k$-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian $k$-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb\"uhler.Comment: 16 pages, submitte

A note on palindromicity

Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence. The second provides a constructive method to build palindromic minimal sequences based upon regular, generic model sets with centro-symmetric window. These give rise to diagonal tight-binding models in one dimension with purely singular continuous spectrum.Comment: 12 page

On the noncommutative geometry of tilings

This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry