Rotation, entropy, and equilibrium states

Abstract

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(e) of rotation vectors e, and relate this to the directional entropy lHl(L) 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 1H((e) and H(e) can differ only at non-exposed boundary points e.