Ground States, Shifts of Finite Type and Quasicrystals for Lattice Systems

This paper deals with ground states for finite-range interaction potentials in lattice models on S d , where S is a finite set. First, we analyse the projective structure of the set of finite, translation-invariant, measures. Second, we show some drastic differences between d = 1 and d = 2 which shed light on crystalline and quasicrystalline ground states. The latter are modelled as uniquely ergodic shifts of finite type (SFT for short) which are showed to have necessarily zero entropy. For d = 1, a uniquely ergodic SFT can only be a single periodic orbit, whereas, for d = 2, we prove the existence of a uniquely ergodic SFT without periodic points. To that end we use Wang tilings and pull back some properties of the Penrose tilings. This example reflects in particular the link between substitution dynamical systems and subshifts of finite type which exists for d ≥ 2.