Emergence of Mandelbrot-like and Julia-like Structures in Parameter Slices of Rational Maps

Abstract

We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit parametrizations, we explore these slices numerically by analyzing the behavior of the critical orbits and approximating the corresponding connectedness loci. The computations reveal rich parameter space structures closely analogous to those arising in cubic polynomial families, including Mandelbrot-like sets. In addition, we observe regions exhibiting Julia-like structures embedded in parameter space, arising from the interaction between bounded and escaping critical orbits. While the appearance of such structures is well established in polynomial dynamics, it remains comparatively less explored in the setting of rational maps. Our results provide numerical evidence that these parameter slices contain subsets closely related to the period-one and period-two slices of cubic polynomial families. More precisely, certain regions appear to exhibit geometric and dynamical features consistent with embedded copies of these classical parameter spaces. These observations highlight how classical structures from polynomial dynamics can emerge naturally within parameter slices of rational maps.