Peixoto's theorem

In the theory of dynamical systems, Peixoto's theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properties:


 * The set of non-wandering points consists only of periodic orbits and fixed points.
 * The set of fixed points is finite and consists only of hyperbolic equilibrium points.
 * Finiteness of attracting or repelling periodic orbits.
 * Absence of saddle-to-saddle connections.

Moreover, they form an open set in the space of all flows endowed with C1 topology.