On Topological Classification of Gradient-like Flows on an $n$-sphere in the Sense of Topological Conjugacy

In this paper, we study gradient-like flows without heteroclinic intersections on an $n$-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. This result implies that for these flows with the same (up to a change of coordinates) partitions into trajectories, the partitions for elements, composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomenon strongly contrasts with the situation for flows with periodic orbits and connections, where one class of equivalence contains continuum classes of conjugacy. In addition, we realize every connected bicolor tree by a gradient-like flow without heteroclinic intersections on the $n$-sphere. In addition, we present a linear-time algorithm on the number of vertices for distinguishing these trees.