Computer generated images for quadratic rational maps with a periodic critical point

Abstract: We describe an algorithm for distinguishing hyperbolic components in the parameter space of quadratic rational maps with a periodic critical point. We then illustrate computer images of the hyperbolic components of the parameter spaces $V_1-V_4$, which were produced using our algorithm. We also resolve the singularities of the projective closure of $V_5$ by blowups, giving an alternative proof that as an algebraic curve, the geometric genus of $V_5$ is 1. This explains why we are unable to produce an image for $V_5$.

Keywords: rational map, complex dynamics, plane curve singularities, geometric genus, hyperbolic maps, Mandelbrot set

Classification (MSC2000): 37F45; 37F10,14H50

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