Pasting Together Julia Sets: A Worked Out Example of Mating

Abstract

The operation of "mating'' two suitable complex polynomial maps {\small $f_1$} and {\small $f_2$} constructs a new dynamical system by carefully pasting together the boundaries of their filled Julia sets so as to obtain a copy of the Riemann sphere, together with a rational map {\small $f_1 \mat f_2$} from this sphere to itself. This construction is particularly hard to visualize when the filled Julia sets {\small $K(f_i)$} are dendrites, with no interior. This note will work out an explicit example of this type, with effectively computable maps from {\small $K(f_1)$} and {\small $K(f_2)$} onto the Riemann sphere.