The space of all bi-infinite sequences is defined analogously: This is the space of all sequences of symbols such that the symbol p can be followed by the symbol q only if the (p,q)th entry of the matrix A is 1. If the sequence extends to infinity in only one direction, it is called a one-sided subshift of finite type, and if it is bilateral, it is called a two-sided subshift of finite type.įormally, one may define the sequence of edges as A subshift of finite type is then defined as a pair ( X, T) obtained in this way. Let T be the shift operator on such sequences it plays the role of the time-evolution operator of the dynamical system. Let Y be the set of all infinite admissible sequences of edges, where by admissible it is meant that the sequence is a walk of the graph. Using these elements we construct a directed graph G=( V, E) with V the set of vertices, the set of edges E defined with A: so x→ y is in E iff A x, y=1. Now let A be a adjacency matrix with entries in. A symbolic flow or subshift is a closed shift-invariant subset Y of ( X, T), and the associated language L Y is the set of finite subsequences of Y. We endow V with the discrete topology and X with the product topology. Let X denote the set V Z of all bi-infinite sequences of elements of V with T the shift operator. Let be a finite set of symbols (alphabet).
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