Rotation, entropy, and equilibrium states

For a dynamical system $(X,T)$ and function f:X→ Rᵈ we consider the corresponding generalised rotation set. This is the convex subset of Rᵈ consisting of all integrals of f with respect to T-invariant probability measures. We study the entropy H( ) of rotation vectors , and relate this to the directional entropy H( ) of Geller & Misiurewicz. For $(X,T)$ a mixing subshift of finite type, and f of summable variation, we prove that if the rotation set is strictly convex then the functions H and H are in fact one and the same. For those rotation sets which are not strictly convex we prove that H( ) and H( ) can differ only at non-exposed boundary points .