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


A well-known formula for the topological entropy of a symbolic system is htop(X) = limn→∞ log N(Λn)/|Λn|, where Λn is the box of side n in ℤd and N(Λ) is the number of configurations of the system on the finite subset Λ of ℤd. We investigate the convergence of the above limit for sequences of regions other than Λn and show in particular that if Ξn is any sequence of finite 'convex' sets in ℤd whose inradii tend to infinity, then the sequence log N(Ξn)/|Ξn| converges to htop(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.

Publication Title

Illinois Journal of Mathematics