Such subshifts are called subshifts of finite type. Given an alphabet \(\Sigma\) and a 01-matrix \(A\) indexed by \(\Sigma\times\Sigma\), one can obtain a subshift by taking all sequences \((x_n)\) of elements of \(\Sigma\) such that \(A_=1\) for every \(n\). Arguably the main open problem of this type is to determine whether the problem is decidable for a class of subshifts that are created as follows. In the light of this, attention has turned to investigating more specific classes of subshifts. However, that turns out to be hopeless, as the problem of determining whether two subshifts are isomorphic has been shown to be undecidable. Subshifts are important partly because many dynamical systems can be represented in this form (very roughly, given a map \(T\) defined on some manifold \(M\), one divides \(M\) up into a finite number of small regions and associates with each point of the manifold the sequence of regions it visits under iterates of \(T\)), and partly because they are interesting in their own right and lead to many attractive questions.Ī very basic question is to classify subshifts. Symbolic dynamics is the study of topological dynamical systems \((X,S)\) where \(X\) is a shift-invariant space of singly or doubly infinite sequences of words in a finite alphabet, with the product topology, and \(S\) is the shift map. Decidability of the isomorphism and the factorization between minimal substitution subshifts, Discrete Analysis 2022:7, 65 pp.
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