Dynamics and entropy of dynamical systems arising from two-dimensional random substitutions.

A symbolic random substitution assigns to each letter a finite set of possible images with a probability distribution producing complex dynamical systems with positive entropy, mixed spectral types and nuanced mixing properties. In this study, we consider a two-dimensional random substitution where each letter is mapped to a finite set of a nonempty array of letters called rectangular words, such that any letter can be iterated infinitely many times via canonical concatenation to produce larger and larger rectangular words. We show that such systems are topologically transitive, have either empty or dense sets of periodic points, and have dense sets of linearly repetitive elements. For the case of a primitive block random substitution subshift, we show that its inflation block entropy coincides with its topological entropy.