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

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