The authors study, from a topological perspective, continuous flows associated to \(C^1\) structurally stable vector fields tangent to a two-dimensional compact subset \(M\) of \(\mathbb{R}^k\). They call these flows Gutierrez-Sotomayor (shortly denoted by GS) flows on manifolds \(M\) with simple singularities and they use Conley index theory to study them. The Conley indices of all simple singularities are computed and an Euler characteristic formula is obtained. By considering a stratification of \(M\) which decomposes it into a union of its regular and singular strata, certain Euler-type formulas which relate the topology of \(M\) and the dynamics on the strata are obtained. The existence of a Lyapunov function for GS flows without periodic orbits and singular cycles is established. Using long exact sequence analysis of index pairs the authors determine necessary and sufficient conditions for a GS flow to be defined on an isolating block. They organize this information combinatorially with the aid of Lyapunov graphs and using a Poincaré-Hopf equality to construct isolating blocks for all simple singularities.
The main results generalize results of the second author and R. D. Franzosa [Trans. Am. Math. Soc. 340, No. 2, 767–784 (1993; Zbl 0806.58042)], where Morse-Smale flows and more generally continuous flows on smooth surfaces are classified.