A Symmetry Group of a Thue-Morse Quasicrystal

We present a method of coding general self-similar structures. In particular, we construct a symmetry group of a one-dimensional Thue-Morse quasicrystal, i.e., of a nonperiodic ground state of a certain translation-invariant, exponentially decaying interaction. A symmetry group of a three-dimensional crystal consists of lattice translations, rotations, and reflexions. Starting from any point of a crystal, we can reach any other point, successively applying different elements of the symmetry group of the crystal. It was shown recently [1, 2] that certain one-dimensional quasicrystals can be build by successive applications, on one of its points, of elements of certain discrete affine semigroups. Here we describe a general method, based on ideas contained in [3, 4], of representing self-similar structures by one-sided sequences of two symbols. In particular, we construct a symmetry group of a Thue-Morse quasicrystal, i.e., of a nonperiodic ground state of a certain one-dimensional classical lattice-gas model. In one-dimensional classical lattice-gas models, every site of the lattice Z (the set of all integers) can be occupied by a particle or be empty. Configurations of such models are therefore elements of Ω = {0, 1}, where 1 denotes the presence and 0 the absence of a particle at any given lattice site. By X(i) we denote the projection of X to a lattice site i ∈ Z. The Thue-Morse Example. We use the Thue-Morse substitution rule to construct a configuration in Ω. We put 1 at the origin and perform alternatively to the right and to the left the following substitution: 1 → 10, 0 → 01. After the first right substitution we get 10, which is the configuration on [0, 1], then the substitution to the left gives us 0110 on [−2, 1] (the substitution to the left means that we replace 1 by 01 and 0 by 10), then we get 01101001 on [−2, 5] and so on. The effect of performing the substitution to the right or to the left is the same as taking a sequence of symbols already obtained, changing every 0 to 1 and every 1 to 0 and then placing the new sequence either right to or left to the previous sequence. In this manner we obtain a nonperiodic configuration XTM ∈ Ω. Notice that if all substitutions were performed to the right, we would get the well-known one-sided ThueMorse configuration: 1001011001101001... [5, 6, 7] Let T be a translation operator, i.e., T : Ω → Ω, T (X)(i) = X(i − 1), X ∈ Ω. Let GTM be a closure (in the product topology of the discrete topologies on {0, 1}) of the orbit of XTM by translations, i.e., GTM = {T (XTM), n ≥ 0}. We call elements of GTM two-sided Thue-Morse configurations. It can be shown that GTM supports exactly one translation-invariant probability measure μTM on Ω [8, 9]. Such measure is called uniquely ergodic and can be obtained as the limit of averaging over XTM and its translates: μTM = limn→∞(1/n) ∑n i=1 δ(T (XTM)), where δ(T (XTM)) is the probability measure assigning probability 1 to T (XTM). It means that all two-sided Thue-Morse configurations look locally identical any local pattern of particles appears in all of them with the same density (defined uniformly in space). It is also said that such configurations belong to the same isomorphism class. The Thue-Morse measure is the unique ground state, i.e., a measure supported by configurations with the minimal energy density, of certain exponentially decaying, translationinvariant, four-body interaction [10, 11, 12]. Multilayer Thue-Morse superlattice het-