Directions for structurally stable flows on surfaces via rotation vectors

The concept of rotation number for circle maps has been extended to rotation vectors for maps and flows on the n-dimensional torus. In this paper a natural extension of rotation vector is presented in the setting of a continuous flow on a compact orientable surface M of genus g. A theorem is presented which classifies the local structure in this rotation set for structurally stable flows on M. In particular, it is shown that for g > 1 there exist at most 4g − 2 linearly independent directions in the rotation set, and that there exist continuous flows for which this upper bound on the number of directions is attained.