On the Dynamic Geometry of Kasner Polygons with Complex Parameter

In this paper we explore the dynamics of the sequence of Kasner polygons \((A^1_nA^2_n\dots A^k_n)_{n\ge 0}\) situated in the plane, defined for a complex parameter \(\alpha \) , and find the parameter values ensuring that the iterations are convergent, periodic or divergent. The results generalize and extend previous research on Kasner triangles and quadrilaterals with a fixed real parameter, where it was found that iterations were convergent if and only if \(0< \alpha < 1\) , that is polygons in the sequence are nested.