Independence entropy of $\mathbb{Z}^{d}$ -shift spaces

Abstract

We introduce a concept of independence entropy for symbolic dynamical systems. This notion of entropy measures the extent to which one can freely insert symbols in positions without violating the constraint defined by the shift space. We show that for a certain class of one-dimensional shift spaces X, the independence entropy coincides with the limiting, as d tends to infinity, topological entropy of the dimensional shift defined by imposing the constraints of X in each of the d cardinal directions. This is of interest because for these shift spaces independence entropy is easy to compute. Thus, while in these cases, the topological entropy of the d-dimensional shift (d?2) is difficult to compute, the limiting topological entropy is easy to compute. In some cases, we also compute the rate of convergence of the sequence of d-dimensional entropies. This work generalizes earlier work on constrained systems with unconstrained positions.