A Construction of Two-Dimensional Random Substitution Systems

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Conference Proceeding

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A two-dimensional substitution is a function that maps every letter in an alphabet to a predetermined rectangular word. It is said to be rectangular-preserving if any letter can be iterated infinitely many times via the canonical concatenation to produce larger and larger rectangular words. This type of substitution is a natural generalization of the one-dimensional deterministic substitution. In this study, we construct a random generalization of the two-dimensional rectangular-preserving substitutions. In particular, we extend the notion of rectangular-preserving to two-dimensional finite-set-valued substitution, a function where every letter is assigned a finite set of nonempty rectangular words, in order to define what we call as two-dimensional rectangular-preserving random substitutions. We give a simple necessary and sufficient condition for a two-dimensional finite-set-valued substitution to be rectangular-preserving. We also define a family of one-dimensional random substitutions such that the product of any two random substitutions in this family give rise to a two-dimensional rectangular-preserving random substitution. Finally, we discuss the associated two-dimensional subshifts to rectangular-preserving random substitutions and present some dynamical properties of the corresponding systems.