On graph induced symbolic systems

In this paper, we investigate shift spaces arising from a multidimensional graph G. In particular, we investigate non-emptiness and exis-tence of periodic points for a multidimensional shift space. We derive sufficient conditions under which these questions can be answered af-firmatively. We investigate the structure of the shift space using the generating matrices. We prove that any d-dimensional shift of finite type is finite if and only if it is conjugate to a shift generated through permutation matrices. We prove that if any triangular pattern of the form (formula presented) can be extended to a 1 × 1 square then the two dimensional shift space possesses periodic points. We introduce the notion of an E-pair for a two dimensional shift space. Using the notion of an E-pair, we derive sufficient conditions for non-emptiness of the two dimensional shift space under discussion.