Asymptotic behavior of the shape of planar polygons by linear flows

In this work we analyze the asymptotic behavior of the shape of a closed polygonal curve on the plane (which we will call n-gon) whose vertexes evolve according to a linear complex flow. We prove that the ω-limit set of an n-gon is a point, a closed curve or a k-torus in the space of n-gons.