Entropy along convex shapes, random tilings and shifts of finite type

A well-known formula for the topological entropy of a symbolic system is h(top)(X) = lim(n-->infinity) logN(Lambda(n))/\Lambda(n)\, where Lambda(n) is the box of side n in Z(d) and N(Lambda) is the number of. configurations of the system on the finite subset Lambda of Z(d). We investigate the convergence of the above limit for sequences of regions other than Lambda(n) and show in particular that if Xi(n) is any sequence of finite 'convex' sets in Z(d) whose inradii tend to infinity, then the sequence logN(Xi(n))/\Xi(n)\ converges to h(top) (X). We apply this to give a concrete proof of a 'strong Variational Principle', that is, the result that for certain higher dimensional systems the topological entropy of the system is the supremum of the measure-theoretic entropies taken over the set of all invariant measures with the Bernoulli property.