Blowup Points and Baby Mandelbrot Sets for a Family of Singularly Perturbed Rational Maps

We study the dynamics of the family of rational maps of the form $$\begin{aligned} f_{d,{\uplambda }}(z)={\uplambda }\left( z+\frac{1}{z^{d-1}}\right) , \quad d \ge 3, \quad {\uplambda }\in \mathbb C{\setminus } \{0\}. \end{aligned}$$ Among other things, we show that the parameter planes for these maps contain infinitely many copies of the Mandelbrot set as well as infinitely many “blowup points”, i.e., parameters for which the critical orbits map to $$\infty $$ , so the Julia set is the entire sphere. Our efforts are aided by the useful observation that for fixed $$d \ge 3$$ , this family is conformally conjugate on the entire Riemann sphere to the family of relaxed Newton maps for $$p_d(z) = z^d-1$$ . The conjugacy allows us to move from one family to the other in order to find simpler proofs of our results, as well as establishing a dictionary of results from one family to the other.