Dependency of the positive and negative long-time behaviors of flows on surfaces

Abstract

Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors of flows on surfaces are captured by the Poincaré-Bendixson theorem using the ω-limit sets. This paper demonstrates that the positive and negative long-time behaviors are not independent. In fact, we show the dependence between the ω-limit sets and the α-limit sets of points of flows on surfaces, which partially generalizes the Poincaré-Bendixson theorem. Applying the dependency result to solve what kinds of the ω-limit sets appear in the area-preserving (or, more generally, non-wandering) flows on compact surfaces, we show that the ω-limit set of any non-closed orbit of such a flow on a compact surface is either a subset of singular points or a locally dense Q-set. Moreover, we show the wildness of surgeries to add totally disconnected singular points and the tameness of those to add finitely many singular points for flows on surfaces.