Symbolic recurrence plot for uniform binary substitutions

Diagonal lines in symbolic recurrence plots are closely related to the identification and characterization of specific biprolongable words within a sequence. In this paper we focus on the recurrence plot of a fixed point of a uniform binary substitution. We show that, if the substitution is primitive and aperiodic, the set of all diagonal line lengths of the recurrence plot has zero density. However, if a line of a specific length exists in the recurrence plot, the density of (the set of starting points of) all diagonal lines with that length is strictly positive. On the other hand, we demonstrate that the recurrence plot of a non-primitive substitution contains lines of any given length. Nonetheless, for any given length, the density of lines with that length is zero

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

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