Rotation, entropy, and equilibrium states

For a dynamical system (X, T) and function f : X --> R-d we consider the co rresponding generalised rotation set. This is the convex subset of R-d cons isting of all integrals of f with respect to T-invariant probability measur es. We study the entropy H(rho) of rotation vectors rho, and relate this to the directional entropy H (rho) of Geller & Misiurewicz. For (X, T) a mixi ng subshift of finite type, and f of summable variation, we prove that if t he rotation set is strictly convex then the functions H and H are in fact o ne and the same. For those rotation sets which are not strictly convex we p rove that H (rho) and H(rho) can differ only at non-exposed boundary points rho.