Directions for structurally stable flows on surfaces via rotation vectors (Q1902989)

Continuous flows on a compact orientable surface \(M\) of genus \(g\) are considered. The definition of rotation set for such flows is introduced. This set is a subset of the tangent bundle of the universal covering space \(\widetilde {M}\). It generalizes the concept of rotation number for circle maps. A classification of the local structure in this rotation set for structurally stable flows is given. It is shown that for \(g >1\) there exist at most \(4g - 2\) linearly independent directions in the rotation set, and there exist continuous flows for which this upper bound on the number of directions is attained.