One-dimensional chain recurrent sets of flows in the 2-sphere
The classical Poincaré-Bendixson theory is concerned with the study of the structure of the limit sets of flows in the 2-sphere \(S^2\) and the behavior of the orbits near them. A limit set of a flow in \(S^2\) which contains at least one non-singular point is a 1-dimensional invariant continuum and the restricted flow on it is chain recurrent. The motivation of this work was to examine what properties of limit sets can be extended to the class of 1-dimensional invariant chain recurrent continua for flows in \(S^2\). An assertion similar to the Poincaré-Bendixson theorem is true in this wider class. More precisely, let \(\phi\) be a flow in \(S^2\) and let \(X\subset S^2\) be a 1-dimensional, chain recurrent, invariant continuum. If \(X\) contains no singular point, then it is a periodic orbit. It is known that any 1-dimensional invariant chain recurrent continuum with at least one non-singular point of a flow in \(S^2\) separates \(S^2\). However, such a set may not be locally an arc at its non-singular points, as simple examples show. It is proved in this work that every 1-dimensional chain component of a flow in \(S^2\) with finitely many singularities is homeomorphic to a finite graph. The assumption that there are finitely many singularities is essential, as it is shown by an example.
scientific article; zbMATH DE number 1028275