Degenerated ground-states in a class of 1D Ising-like models: a characterization by symbolic dynamics

In this note we study a class of one-dimensional Ising chain having a highly degenerated set of ground-state configurations. The model consists of spin chain having infinite-range pair interactions with a given structure. We show that the set of ground-state configurations of such a model can be fully characterized by means of symbolic dynamics. Particularly we found that the set ground- state configurations defines what in symbolic dynamics is called sofic shift space. Finally we prove that this system has a non-vanishing residual entropy (the topological entropy of the shift space), which can be exactly calculated.