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 n-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bi-colour tree corresponding to a skeleton formed by co-dimension one separatrices. Moreover, we show that such a tree is a complete invariant for these
flows with respect to the topological equivalence also. The obtained result means that for considered flows with the same (up to a change of coordinates) partitions on trajectories, the partitions for elements,
composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomena strongly contrasts with the situation for flows with periodic orbits and connections, where one class of the equivalence contains continuum classes of the conjugacy. Additionally, we realize every connected bi-colour tree by a gradient-like flow without heteroclinic intersections on n-sphere. Besides, we present a linear-time algorithm on the number of vertices for distinguishing these trees.