The $ω$-limit set of a flow with arbitrarily many singular points on a surface and surgeries to add singular points

Abstract

The ω-limit set is one of the fundamental objects in dynamical systems. Using the ω-limit sets, the Poincaré-Bendixson theorem captures the limit behaviors of orbits of flows on surfaces. It was generalized in several ways and was applied to various phenomena. In this paper, we consider what kinds of the ω-limit sets do appear in the non-wandering flows on compact surfaces, and show that the ω-limit set of any non-closed orbit of a non-wandering flow with arbitrarily many singular points on a compact surface is either a subset of singular points or a locally dense Q-set. To show this, we demonstrate a similar result holds for locally dense orbits without non-wanderingness. Moreover, the ω-limit set of any non-closed orbits of a Hamiltonian flow with arbitrarily many singular points on a compact surface consists of singular points.