Rotation and entropy

For a given map f : X --> X and an observable phi : X --> R-d; rotation vec tors are the limits of ergodic averages of phi. We study which part of the topological entropy of f is associated to a given rotation vector and which part is associated with many rotation vectors. According to this distincti on, we introduce directional and lost entropies. We discuss their propertie s in the general case and analyze them more closely for subshifts of finite type and circle maps.