Decidability of the isomorphism and the factorization between minimal substitution subshifts

Classification is a central problem for dynamical systems, in particular for families that arise in a wide range of topics, like substitution subshifts. It is important to be able to distinguish whether two such subshifts are isomorphic, but the existing invariants are not sufficient for this purpose. We first show that given two minimal substitution subshifts, there exists a computable constant $R$ such that any factor map between these subshifts (if any) is the composition of a factor map with a radius smaller than $R$ and some power of the shift map. Then we prove that it is decidable to check whether a given sliding block code is a factor map between two prescribed minimal substitution subshifts. As a consequence of these two results, we provide an algorithm that, given two minimal substitution subshifts, decides whether one is a factor of the other and, as a straightforward corollary, whether they are isomorphic.Comment: 54 page

Decidability of the isomorphism and the factorization between minimal substitution subshifts

68 pagesClassification is a central problem for dynamical systems, in particular for families that arise in a wide range of topics, like substitution subshifts. It is important to be able to distinguish whether two such subshifts are isomorphic, but the existing invariants are not sufficient for this purpose. We first show that given two minimal substitution subshifts, there exists a computable constant R such that any factor map between these sub-shifts (if any) is the composition of a factor map with a radius smaller than R and some power of the shift map. Then we prove that it is decid-able to check whether a given sliding block code is a factor map between two prescribed minimal substitution subshifts. As a consequence of these two results, we provide an algorithm that, given two minimal substitution subshifts, decides whether one is a factor of the other and, as a straightforward corollary, whether they are isomorphic

On the subword equivalence problem for morphic words

AbstractTwo infinite words x and y are said to be subword equivalent if they have the same set of finite subwords (factors). The subword equivalence problem is the question whether two infinite words are subword equivalent. We show that, under mild hypotheses, the decidability of the subword equivalence problem implies the decidability of the ω-sequence equivalence problem, a problem which has been shown to be decidable by Čulik and Harju for morphic words (i.e. words generated by iterating a morphism). Yet, we do use the decidability of the ω-sequence equivalence problem to prove our result.We prove that the subword equivalence problem is decidable for two morphic words, provided the morphisms are primitive and have bounded delays. We also prove that the sub-word equivalence problem is decidable for any pair of morphic words in the case of a binary alphabet.Our results hold in fact for a stronger version, namely for the subword inclusion problem