An Intuitively Simple Property of Limit Figures of Quadratic Transformations

In this paper, we deal with intuitively simple properties of the limit figures of two-dimensional inhomogeneous quadratic transformations. The divergence-convergence boundary of homogeneous quadratic transformations was investigated in detail in Da-te (1978). In an inhomogeneous case, there exist, possibly, the region of initial points converging to a fixed point other than the origin due to the linear terms. Then, there appears a boundary with a finite area as a limit figure. Next, in certain cases, the convergence regions or the divergence regions consist of infinite number of separated regions. We show the examples of the properties, and investigate them.