Independence entropy of Z-shift spaces

Topological entropy is the most fundamental numerical invariant associated to a d-dimensional shift space. When d = 1 and the shift space is a shift of finite type (SFT) or sofic shift, the topological entropy is easy to compute as the log of the largest eigenvalue of a nonnegative integer matrix. However, when d = 2 there is no known explicit expression for the topological entropy of SFTs, even for the simplest nontrivial examples. For instance, the one-dimensional golden mean shift is the set of all bi-infinite sequences of ‘0’s and ‘1’s such that 11 never appears. The topological entropy of this shift is well known to be the log of the golden mean, which is approximately ln(1.618). For the two-dimensional golden mean shift, defined to be the set of all configurations of ‘0’s and ‘1’s on Z such that 11 and 1 never appear, there is no known expression for the topological entropy. One analogously defines the d-dimensional golden mean shift for any d ≥ 1. It is well known that the topological entropy is a non-increasing function in d and thus has a limit. It is not so well known that the value of this limit is known in this case; namely it is ln(2) 2 ≈ ln(1.414) (see [11, 17]). One way of interpreting this result is as follows. For any d, call a site in Z even (resp., odd) if the sum of its coordinates is even (resp., odd). Now, if we assign value 0 to each even site, then we are free to assign arbitrary binary values to each of the odd sites; the ‘0’s in the even sites prevent two adjacent sites from both taking value 1. Equivalently, for any configuration of ‘0’s and ‘1’s on all of Z such that all even sites have value 0, then if we freely change the values at all odd sites in any way, the resulting configuration belongs to the golden mean shift. The number of possible restrictions of configurations of this form to a d-dimensional rectangular cube of even side length L is 2 d/2. The asymptotic growth rate of the